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-{"size":22413,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn\n-\/\nimport category_theory.fully_faithful\nimport category_theory.whiskering\nimport category_theory.essential_image\nimport tactic.slice\n\n\/-!\n# Equivalence of categories\n\nAn equivalence of categories `C` and `D` is a pair of functors `F : C \u2964 D` and `G : D \u2964 C` such\nthat `\u03b7 : \ud835\udfed C \u2245 F \u22d9 G` and `\u03b5 : G \u22d9 F \u2245 \ud835\udfed D`. In many situations, equivalences are a better\nnotion of \"sameness\" of categories than the stricter isomorphims of categories.\n\nRecall that one way to express that two functors `F : C \u2964 D` and `G : D \u2964 C` are adjoint is using\ntwo natural transformations `\u03b7 : \ud835\udfed C \u27f6 F \u22d9 G` and `\u03b5 : G \u22d9 F \u27f6 \ud835\udfed D`, called the unit and the\ncounit, such that the compositions `F \u27f6 FGF \u27f6 F` and `G \u27f6 GFG \u27f6 G` are the identity. Unfortunately,\nit is not the case that the natural isomorphisms `\u03b7` and `\u03b5` in the definition of an equivalence\nautomatically give an adjunction. However, it is true that\n* if one of the two compositions is the identity, then so is the other, and\n* given an equivalence of categories, it is always possible to refine `\u03b7` in such a way that the\n  identities are satisfied.\n\nFor this reason, in mathlib we define an equivalence to be a \"half-adjoint equivalence\", which is\na tuple `(F, G, \u03b7, \u03b5)` as in the first paragraph such that the composite `F \u27f6 FGF \u27f6 F` is the\nidentity. By the remark above, this already implies that the tuple is an \"adjoint equivalence\",\ni.e., that the composite `G \u27f6 GFG \u27f6 G` is also the identity.\n\nWe also define essentially surjective functors and show that a functor is an equivalence if and only\nif it is full, faithful and essentially surjective.\n\n## Main definitions\n\n* `equivalence`: bundled (half-)adjoint equivalences of categories\n* `is_equivalence`: type class on a functor `F` containing the data of the inverse `G` as well as\n  the natural isomorphisms `\u03b7` and `\u03b5`.\n* `ess_surj`: type class on a functor `F` containing the data of the preimages and the isomorphisms\n  `F.obj (preimage d) \u2245 d`.\n\n## Main results\n\n* `equivalence.mk`: upgrade an equivalence to a (half-)adjoint equivalence\n* `equivalence_of_fully_faithfully_ess_surj`: a fully faithful essentially surjective functor is an\n  equivalence.\n\n## Notations\n\nWe write `C \u224c D` (`\\backcong`, not to be confused with `\u2245`\/`\\cong`) for a bundled equivalence.\n\n-\/\n\nnamespace category_theory\nopen category_theory.functor nat_iso category\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\n\/-- We define an equivalence as a (half)-adjoint equivalence, a pair of functors with\n  a unit and counit which are natural isomorphisms and the triangle law `F\u03b7 \u226b \u03b5F = 1`, or in other\n  words the composite `F \u27f6 FGF \u27f6 F` is the identity.\n\n  In `unit_inverse_comp`, we show that this is actually an adjoint equivalence, i.e., that the\n  composite `G \u27f6 GFG \u27f6 G` is also the identity.\n\n  The triangle equation is written as a family of equalities between morphisms, it is more\n  complicated if we write it as an equality of natural transformations, because then we would have\n  to insert natural transformations like `F \u27f6 F1`.\n\nSee https:\/\/stacks.math.columbia.edu\/tag\/001J\n-\/\nstructure equivalence (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2082) [category.{v\u2082} D] :=\nmk' ::\n(functor : C \u2964 D)\n(inverse : D \u2964 C)\n(unit_iso   : \ud835\udfed C \u2245 functor \u22d9 inverse)\n(counit_iso : inverse \u22d9 functor \u2245 \ud835\udfed D)\n(functor_unit_iso_comp' : \u2200(X : C), functor.map ((unit_iso.hom : \ud835\udfed C \u27f6 functor \u22d9 inverse).app X) \u226b\n  counit_iso.hom.app (functor.obj X) = \ud835\udfd9 (functor.obj X) . obviously)\n\nrestate_axiom equivalence.functor_unit_iso_comp'\n\ninfixr ` \u224c `:10  := equivalence\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n\nnamespace equivalence\n\n\/-- The unit of an equivalence of categories. -\/\nabbreviation unit (e : C \u224c D) : \ud835\udfed C \u27f6 e.functor \u22d9 e.inverse := e.unit_iso.hom\n\/-- The counit of an equivalence of categories. -\/\nabbreviation counit (e : C \u224c D) : e.inverse \u22d9 e.functor \u27f6 \ud835\udfed D := e.counit_iso.hom\n\/-- The inverse of the unit of an equivalence of categories. -\/\nabbreviation unit_inv (e : C \u224c D) : e.functor \u22d9 e.inverse \u27f6 \ud835\udfed C := e.unit_iso.inv\n\/-- The inverse of the counit of an equivalence of categories. -\/\nabbreviation counit_inv (e : C \u224c D) : \ud835\udfed D \u27f6 e.inverse \u22d9 e.functor := e.counit_iso.inv\n\n\/- While these abbreviations are convenient, they also cause some trouble,\npreventing structure projections from unfolding. -\/\n@[simp] lemma equivalence_mk'_unit (functor inverse unit_iso counit_iso f) :\n  (\u27e8functor, inverse, unit_iso, counit_iso, f\u27e9 : C \u224c D).unit = unit_iso.hom := rfl\n@[simp] lemma equivalence_mk'_counit (functor inverse unit_iso counit_iso f) :\n  (\u27e8functor, inverse, unit_iso, counit_iso, f\u27e9 : C \u224c D).counit = counit_iso.hom := rfl\n@[simp] lemma equivalence_mk'_unit_inv (functor inverse unit_iso counit_iso f) :\n  (\u27e8functor, inverse, unit_iso, counit_iso, f\u27e9 : C \u224c D).unit_inv = unit_iso.inv := rfl\n@[simp] lemma equivalence_mk'_counit_inv (functor inverse unit_iso counit_iso f) :\n  (\u27e8functor, inverse, unit_iso, counit_iso, f\u27e9 : C \u224c D).counit_inv = counit_iso.inv := rfl\n\n@[simp] lemma functor_unit_comp (e : C \u224c D) (X : C) :\n  e.functor.map (e.unit.app X) \u226b e.counit.app (e.functor.obj X) = \ud835\udfd9 (e.functor.obj X) :=\ne.functor_unit_iso_comp X\n\n@[simp] lemma counit_inv_functor_comp (e : C \u224c D) (X : C) :\n  e.counit_inv.app (e.functor.obj X) \u226b e.functor.map (e.unit_inv.app X) = \ud835\udfd9 (e.functor.obj X) :=\nbegin\n  erw [iso.inv_eq_inv\n    (e.functor.map_iso (e.unit_iso.app X) \u226a\u226b e.counit_iso.app (e.functor.obj X)) (iso.refl _)],\n  exact e.functor_unit_comp X\nend\n\nlemma counit_inv_app_functor (e : C \u224c D) (X : C) :\n  e.counit_inv.app (e.functor.obj X) = e.functor.map (e.unit.app X) :=\nby { symmetry, erw [\u2190iso.comp_hom_eq_id (e.counit_iso.app _), functor_unit_comp], refl }\n\nlemma counit_app_functor (e : C \u224c D) (X : C) :\n  e.counit.app (e.functor.obj X) = e.functor.map (e.unit_inv.app X) :=\nby { erw [\u2190iso.hom_comp_eq_id (e.functor.map_iso (e.unit_iso.app X)), functor_unit_comp], refl }\n\n\/-- The other triangle equality. The proof follows the following proof in Globular:\n  http:\/\/globular.science\/1905.001 -\/\n@[simp] lemma unit_inverse_comp (e : C \u224c D) (Y : D) :\n  e.unit.app (e.inverse.obj Y) \u226b e.inverse.map (e.counit.app Y) = \ud835\udfd9 (e.inverse.obj Y) :=\nbegin\n  rw [\u2190id_comp (e.inverse.map _), \u2190map_id e.inverse, \u2190counit_inv_functor_comp, map_comp,\n      \u2190iso.hom_inv_id_assoc (e.unit_iso.app _) (e.inverse.map (e.functor.map _)),\n      app_hom, app_inv],\n  slice_lhs 2 3 { erw [e.unit.naturality] },\n  slice_lhs 1 2 { erw [e.unit.naturality] },\n  slice_lhs 4 4\n  { rw [\u2190iso.hom_inv_id_assoc (e.inverse.map_iso (e.counit_iso.app _)) (e.unit_inv.app _)] },\n  slice_lhs 3 4 { erw [\u2190map_comp e.inverse, e.counit.naturality],\n    erw [(e.counit_iso.app _).hom_inv_id, map_id] }, erw [id_comp],\n  slice_lhs 2 3 { erw [\u2190map_comp e.inverse, e.counit_iso.inv.naturality, map_comp] },\n  slice_lhs 3 4 { erw [e.unit_inv.naturality] },\n  slice_lhs 4 5 { erw [\u2190map_comp (e.functor \u22d9 e.inverse), (e.unit_iso.app _).hom_inv_id, map_id] },\n  erw [id_comp],\n  slice_lhs 3 4 { erw [\u2190e.unit_inv.naturality] },\n  slice_lhs 2 3 { erw [\u2190map_comp e.inverse, \u2190e.counit_iso.inv.naturality,\n    (e.counit_iso.app _).hom_inv_id, map_id] }, erw [id_comp, (e.unit_iso.app _).hom_inv_id], refl\nend\n\n@[simp] lemma inverse_counit_inv_comp (e : C \u224c D) (Y : D) :\n  e.inverse.map (e.counit_inv.app Y) \u226b e.unit_inv.app (e.inverse.obj Y) = \ud835\udfd9 (e.inverse.obj Y) :=\nbegin\n  erw [iso.inv_eq_inv\n    (e.unit_iso.app (e.inverse.obj Y) \u226a\u226b e.inverse.map_iso (e.counit_iso.app Y)) (iso.refl _)],\n  exact e.unit_inverse_comp Y\nend\n\nlemma unit_app_inverse (e : C \u224c D) (Y : D) :\n  e.unit.app (e.inverse.obj Y) = e.inverse.map (e.counit_inv.app Y) :=\nby { erw [\u2190iso.comp_hom_eq_id (e.inverse.map_iso (e.counit_iso.app Y)), unit_inverse_comp], refl }\n\nlemma unit_inv_app_inverse (e : C \u224c D) (Y : D) :\n  e.unit_inv.app (e.inverse.obj Y) = e.inverse.map (e.counit.app Y) :=\nby { symmetry, erw [\u2190iso.hom_comp_eq_id (e.unit_iso.app _), unit_inverse_comp], refl }\n\n@[simp] lemma fun_inv_map (e : C \u224c D) (X Y : D) (f : X \u27f6 Y) :\n  e.functor.map (e.inverse.map f) = e.counit.app X \u226b f \u226b e.counit_inv.app Y :=\n(nat_iso.naturality_2 (e.counit_iso) f).symm\n\n@[simp] lemma inv_fun_map (e : C \u224c D) (X Y : C) (f : X \u27f6 Y) :\n  e.inverse.map (e.functor.map f) = e.unit_inv.app X \u226b f \u226b e.unit.app Y :=\n(nat_iso.naturality_1 (e.unit_iso) f).symm\n\nsection\n-- In this section we convert an arbitrary equivalence to a half-adjoint equivalence.\nvariables {F : C \u2964 D} {G : D \u2964 C} (\u03b7 : \ud835\udfed C \u2245 F \u22d9 G) (\u03b5 : G \u22d9 F \u2245 \ud835\udfed D)\n\n\/-- If `\u03b7 : \ud835\udfed C \u2245 F \u22d9 G` is part of a (not necessarily half-adjoint) equivalence, we can upgrade it\nto a refined natural isomorphism `adjointify_\u03b7 \u03b7 : \ud835\udfed C \u2245 F \u22d9 G` which exhibits the properties\nrequired for a half-adjoint equivalence. See `equivalence.mk`. -\/\ndef adjointify_\u03b7 : \ud835\udfed C \u2245 F \u22d9 G :=\ncalc\n  \ud835\udfed C \u2245 F \u22d9 G               : \u03b7\n  ... \u2245 F \u22d9 (\ud835\udfed D \u22d9 G)      : iso_whisker_left F (left_unitor G).symm\n  ... \u2245 F \u22d9 ((G \u22d9 F) \u22d9 G) : iso_whisker_left F (iso_whisker_right \u03b5.symm G)\n  ... \u2245 F \u22d9 (G \u22d9 (F \u22d9 G)) : iso_whisker_left F (associator G F G)\n  ... \u2245 (F \u22d9 G) \u22d9 (F \u22d9 G) : (associator F G (F \u22d9 G)).symm\n  ... \u2245 \ud835\udfed C \u22d9 (F \u22d9 G)      : iso_whisker_right \u03b7.symm (F \u22d9 G)\n  ... \u2245 F \u22d9 G               : left_unitor (F \u22d9 G)\n\nlemma adjointify_\u03b7_\u03b5 (X : C) :\n  F.map ((adjointify_\u03b7 \u03b7 \u03b5).hom.app X) \u226b \u03b5.hom.app (F.obj X) = \ud835\udfd9 (F.obj X) :=\nbegin\n  dsimp [adjointify_\u03b7], simp,\n  have := \u03b5.hom.naturality (F.map (\u03b7.inv.app X)), dsimp at this, rw [this], clear this,\n  rw [\u2190assoc _ _ (F.map _)],\n  have := \u03b5.hom.naturality (\u03b5.inv.app $ F.obj X), dsimp at this, rw [this], clear this,\n  have := (\u03b5.app $ F.obj X).hom_inv_id, dsimp at this, rw [this], clear this,\n  rw [id_comp], have := (F.map_iso $ \u03b7.app X).hom_inv_id, dsimp at this, rw [this]\nend\n\nend\n\n\/-- Every equivalence of categories consisting of functors `F` and `G` such that `F \u22d9 G` and\n    `G \u22d9 F` are naturally isomorphic to identity functors can be transformed into a half-adjoint\n    equivalence without changing `F` or `G`. -\/\nprotected definition mk (F : C \u2964 D) (G : D \u2964 C)\n  (\u03b7 : \ud835\udfed C \u2245 F \u22d9 G) (\u03b5 : G \u22d9 F \u2245 \ud835\udfed D) : C \u224c D :=\n\u27e8F, G, adjointify_\u03b7 \u03b7 \u03b5, \u03b5, adjointify_\u03b7_\u03b5 \u03b7 \u03b5\u27e9\n\n\/-- Equivalence of categories is reflexive. -\/\n@[refl, simps] def refl : C \u224c C :=\n\u27e8\ud835\udfed C, \ud835\udfed C, iso.refl _, iso.refl _, \u03bb X, category.id_comp _\u27e9\n\ninstance : inhabited (C \u224c C) :=\n\u27e8refl\u27e9\n\n\/-- Equivalence of categories is symmetric. -\/\n@[symm, simps] def symm (e : C \u224c D) : D \u224c C :=\n\u27e8e.inverse, e.functor, e.counit_iso.symm, e.unit_iso.symm, e.inverse_counit_inv_comp\u27e9\n\nvariables {E : Type u\u2083} [category.{v\u2083} E]\n\n\/-- Equivalence of categories is transitive. -\/\n@[trans, simps] def trans (e : C \u224c D) (f : D \u224c E) : C \u224c E :=\n{ functor := e.functor \u22d9 f.functor,\n  inverse := f.inverse \u22d9 e.inverse,\n  unit_iso :=\n  begin\n    refine iso.trans e.unit_iso _,\n    exact iso_whisker_left e.functor (iso_whisker_right f.unit_iso e.inverse) ,\n  end,\n  counit_iso :=\n  begin\n    refine iso.trans _ f.counit_iso,\n    exact iso_whisker_left f.inverse (iso_whisker_right e.counit_iso f.functor)\n  end,\n  -- We wouldn't have needed to give this proof if we'd used `equivalence.mk`,\n  -- but we choose to avoid using that here, for the sake of good structure projection `simp`\n  -- lemmas.\n  functor_unit_iso_comp' := \u03bb X,\n  begin\n    dsimp,\n    rw [\u2190 f.functor.map_comp_assoc, e.functor.map_comp, \u2190counit_inv_app_functor, fun_inv_map,\n        iso.inv_hom_id_app_assoc, assoc, iso.inv_hom_id_app, counit_app_functor,\n        \u2190 functor.map_comp],\n    erw [comp_id, iso.hom_inv_id_app, functor.map_id],\n  end }\n\n\/-- Composing a functor with both functors of an equivalence yields a naturally isomorphic\nfunctor. -\/\ndef fun_inv_id_assoc (e : C \u224c D) (F : C \u2964 E) : e.functor \u22d9 e.inverse \u22d9 F \u2245 F :=\n(functor.associator _ _ _).symm \u226a\u226b iso_whisker_right e.unit_iso.symm F \u226a\u226b F.left_unitor\n\n@[simp] lemma fun_inv_id_assoc_hom_app (e : C \u224c D) (F : C \u2964 E) (X : C) :\n  (fun_inv_id_assoc e F).hom.app X = F.map (e.unit_inv.app X) :=\nby { dsimp [fun_inv_id_assoc], tidy }\n\n@[simp] lemma fun_inv_id_assoc_inv_app (e : C \u224c D) (F : C \u2964 E) (X : C) :\n  (fun_inv_id_assoc e F).inv.app X = F.map (e.unit.app X) :=\nby { dsimp [fun_inv_id_assoc], tidy }\n\n\/-- Composing a functor with both functors of an equivalence yields a naturally isomorphic\nfunctor. -\/\ndef inv_fun_id_assoc (e : C \u224c D) (F : D \u2964 E) : e.inverse \u22d9 e.functor \u22d9 F \u2245 F :=\n(functor.associator _ _ _).symm \u226a\u226b iso_whisker_right e.counit_iso F \u226a\u226b F.left_unitor\n\n@[simp] lemma inv_fun_id_assoc_hom_app (e : C \u224c D) (F : D \u2964 E) (X : D) :\n  (inv_fun_id_assoc e F).hom.app X = F.map (e.counit.app X) :=\nby { dsimp [inv_fun_id_assoc], tidy }\n\n@[simp] lemma inv_fun_id_assoc_inv_app (e : C \u224c D) (F : D \u2964 E) (X : D) :\n  (inv_fun_id_assoc e F).inv.app X = F.map (e.counit_inv.app X) :=\nby { dsimp [inv_fun_id_assoc], tidy }\n\n\/-- If `C` is equivalent to `D`, then `C \u2964 E` is equivalent to `D \u2964 E`. -\/\n@[simps functor inverse unit_iso counit_iso]\ndef congr_left (e : C \u224c D) : (C \u2964 E) \u224c (D \u2964 E) :=\nequivalence.mk\n  ((whiskering_left _ _ _).obj e.inverse)\n  ((whiskering_left _ _ _).obj e.functor)\n  (nat_iso.of_components (\u03bb F, (e.fun_inv_id_assoc F).symm) (by tidy))\n  (nat_iso.of_components (\u03bb F, e.inv_fun_id_assoc F) (by tidy))\n\n\/-- If `C` is equivalent to `D`, then `E \u2964 C` is equivalent to `E \u2964 D`. -\/\n@[simps functor inverse unit_iso counit_iso]\ndef congr_right (e : C \u224c D) : (E \u2964 C) \u224c (E \u2964 D) :=\nequivalence.mk\n  ((whiskering_right _ _ _).obj e.functor)\n  ((whiskering_right _ _ _).obj e.inverse)\n  (nat_iso.of_components\n    (\u03bb F, F.right_unitor.symm \u226a\u226b iso_whisker_left F e.unit_iso \u226a\u226b functor.associator _ _ _)\n    (by tidy))\n  (nat_iso.of_components\n    (\u03bb F, functor.associator _ _ _ \u226a\u226b iso_whisker_left F e.counit_iso \u226a\u226b F.right_unitor)\n    (by tidy))\n\nsection cancellation_lemmas\nvariables (e : C \u224c D)\n\n\/- We need special forms of `cancel_nat_iso_hom_right(_assoc)` and\n`cancel_nat_iso_inv_right(_assoc)` for units and counits, because neither `simp` or `rw` will apply\nthose lemmas in this setting without providing `e.unit_iso` (or similar) as an explicit argument.\nWe also provide the lemmas for length four compositions, since they're occasionally useful.\n(e.g. in proving that equivalences take monos to monos) -\/\n\n@[simp] lemma cancel_unit_right {X Y : C}\n  (f f' : X \u27f6 Y) :\n  f \u226b e.unit.app Y = f' \u226b e.unit.app Y \u2194 f = f' :=\nby simp only [cancel_mono]\n\n@[simp] lemma cancel_unit_inv_right {X Y : C}\n  (f f' : X \u27f6 e.inverse.obj (e.functor.obj Y))   :\n  f \u226b e.unit_inv.app Y = f' \u226b e.unit_inv.app Y \u2194 f = f' :=\nby simp only [cancel_mono]\n\n@[simp] lemma cancel_counit_right {X Y : D}\n  (f f' : X \u27f6 e.functor.obj (e.inverse.obj Y))   :\n  f \u226b e.counit.app Y = f' \u226b e.counit.app Y \u2194 f = f' :=\nby simp only [cancel_mono]\n\n@[simp] lemma cancel_counit_inv_right {X Y : D}\n  (f f' : X \u27f6 Y) :\n  f \u226b e.counit_inv.app Y = f' \u226b e.counit_inv.app Y \u2194 f = f' :=\nby simp only [cancel_mono]\n\n@[simp] lemma cancel_unit_right_assoc {W X X' Y : C}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y) :\n  f \u226b g \u226b e.unit.app Y = f' \u226b g' \u226b e.unit.app Y \u2194 f \u226b g = f' \u226b g' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\n@[simp] lemma cancel_counit_inv_right_assoc {W X X' Y : D}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y) :\n  f \u226b g \u226b e.counit_inv.app Y = f' \u226b g' \u226b e.counit_inv.app Y \u2194 f \u226b g = f' \u226b g' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\n@[simp] lemma cancel_unit_right_assoc' {W X X' Y Y' Z : C}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (h : Y \u27f6 Z) (f' : W \u27f6 X') (g' : X' \u27f6 Y') (h' : Y' \u27f6 Z) :\n  f \u226b g \u226b h \u226b e.unit.app Z = f' \u226b g' \u226b h' \u226b e.unit.app Z \u2194 f \u226b g \u226b h = f' \u226b g' \u226b h' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\n@[simp] lemma cancel_counit_inv_right_assoc' {W X X' Y Y' Z : D}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (h : Y \u27f6 Z) (f' : W \u27f6 X') (g' : X' \u27f6 Y') (h' : Y' \u27f6 Z) :\n  f \u226b g \u226b h \u226b e.counit_inv.app Z = f' \u226b g' \u226b h' \u226b e.counit_inv.app Z \u2194 f \u226b g \u226b h = f' \u226b g' \u226b h' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\nend cancellation_lemmas\n\nsection\n\n-- There's of course a monoid structure on `C \u224c C`,\n-- but let's not encourage using it.\n-- The power structure is nevertheless useful.\n\n\/-- Powers of an auto-equivalence. -\/\ndef pow (e : C \u224c C) : \u2124 \u2192 (C \u224c C)\n| (int.of_nat 0) := equivalence.refl\n| (int.of_nat 1) := e\n| (int.of_nat (n+2)) := e.trans (pow (int.of_nat (n+1)))\n| (int.neg_succ_of_nat 0) := e.symm\n| (int.neg_succ_of_nat (n+1)) := e.symm.trans (pow (int.neg_succ_of_nat n))\n\ninstance : has_pow (C \u224c C) \u2124 := \u27e8pow\u27e9\n\n@[simp] lemma pow_zero (e : C \u224c C) : e^(0 : \u2124) = equivalence.refl := rfl\n@[simp] lemma pow_one (e : C \u224c C) : e^(1 : \u2124) = e := rfl\n@[simp] lemma pow_neg_one (e : C \u224c C) : e^(-1 : \u2124) = e.symm := rfl\n\n-- TODO as necessary, add the natural isomorphisms `(e^a).trans e^b \u2245 e^(a+b)`.\n-- At this point, we haven't even defined the category of equivalences.\n\nend\n\nend equivalence\n\n\n\/-- A functor that is part of a (half) adjoint equivalence -\/\nclass is_equivalence (F : C \u2964 D) :=\nmk' ::\n(inverse    : D \u2964 C)\n(unit_iso   : \ud835\udfed C \u2245 F \u22d9 inverse)\n(counit_iso : inverse \u22d9 F \u2245 \ud835\udfed D)\n(functor_unit_iso_comp' : \u2200 (X : C), F.map ((unit_iso.hom : \ud835\udfed C \u27f6 F \u22d9 inverse).app X) \u226b\n  counit_iso.hom.app (F.obj X) = \ud835\udfd9 (F.obj X) . obviously)\n\nrestate_axiom is_equivalence.functor_unit_iso_comp'\n\nnamespace is_equivalence\n\ninstance of_equivalence (F : C \u224c D) : is_equivalence F.functor :=\n{ ..F }\n\ninstance of_equivalence_inverse (F : C \u224c D) : is_equivalence F.inverse :=\nis_equivalence.of_equivalence F.symm\n\nopen equivalence\n\/-- To see that a functor is an equivalence, it suffices to provide an inverse functor `G` such that\n    `F \u22d9 G` and `G \u22d9 F` are naturally isomorphic to identity functors. -\/\nprotected definition mk {F : C \u2964 D} (G : D \u2964 C)\n  (\u03b7 : \ud835\udfed C \u2245 F \u22d9 G) (\u03b5 : G \u22d9 F \u2245 \ud835\udfed D) : is_equivalence F :=\n\u27e8G, adjointify_\u03b7 \u03b7 \u03b5, \u03b5, adjointify_\u03b7_\u03b5 \u03b7 \u03b5\u27e9\n\nend is_equivalence\n\n\nnamespace functor\n\n\/-- Interpret a functor that is an equivalence as an equivalence. -\/\ndef as_equivalence (F : C \u2964 D) [is_equivalence F] : C \u224c D :=\n\u27e8F, is_equivalence.inverse F, is_equivalence.unit_iso, is_equivalence.counit_iso,\n  is_equivalence.functor_unit_iso_comp\u27e9\n\ninstance is_equivalence_refl : is_equivalence (\ud835\udfed C) :=\nis_equivalence.of_equivalence equivalence.refl\n\n\/-- The inverse functor of a functor that is an equivalence. -\/\ndef inv (F : C \u2964 D) [is_equivalence F] : D \u2964 C :=\nis_equivalence.inverse F\n\ninstance is_equivalence_inv (F : C \u2964 D) [is_equivalence F] : is_equivalence F.inv :=\nis_equivalence.of_equivalence F.as_equivalence.symm\n\n@[simp] lemma as_equivalence_functor (F : C \u2964 D) [is_equivalence F] :\n  F.as_equivalence.functor = F := rfl\n\n@[simp] lemma as_equivalence_inverse (F : C \u2964 D) [is_equivalence F] :\n  F.as_equivalence.inverse = inv F := rfl\n\n@[simp] lemma inv_inv (F : C \u2964 D) [is_equivalence F] :\n  inv (inv F) = F := rfl\n\nvariables {E : Type u\u2083} [category.{v\u2083} E]\n\ninstance is_equivalence_trans (F : C \u2964 D) (G : D \u2964 E) [is_equivalence F] [is_equivalence G] :\n  is_equivalence (F \u22d9 G) :=\nis_equivalence.of_equivalence (equivalence.trans (as_equivalence F) (as_equivalence G))\n\nend functor\n\nnamespace equivalence\n\n@[simp]\nlemma functor_inv (E : C \u224c D) : E.functor.inv = E.inverse := rfl\n\n@[simp]\nlemma inverse_inv (E : C \u224c D) : E.inverse.inv = E.functor := rfl\n\n@[simp]\nlemma functor_as_equivalence (E : C \u224c D) : E.functor.as_equivalence = E :=\nby { cases E, congr, }\n\n@[simp]\nlemma inverse_as_equivalence (E : C \u224c D) : E.inverse.as_equivalence = E.symm :=\nby { cases E, congr, }\n\nend equivalence\n\nnamespace is_equivalence\n\n@[simp] lemma fun_inv_map (F : C \u2964 D) [is_equivalence F] (X Y : D) (f : X \u27f6 Y) :\n  F.map (F.inv.map f) = F.as_equivalence.counit.app X \u226b f \u226b F.as_equivalence.counit_inv.app Y :=\nbegin\n  erw [nat_iso.naturality_2],\n  refl\nend\n@[simp] lemma inv_fun_map (F : C \u2964 D) [is_equivalence F] (X Y : C) (f : X \u27f6 Y) :\n  F.inv.map (F.map f) = F.as_equivalence.unit_inv.app X \u226b f \u226b F.as_equivalence.unit.app Y :=\nbegin\n  erw [nat_iso.naturality_1],\n  refl\nend\n\nend is_equivalence\n\nnamespace equivalence\n\n\/--\nAn equivalence is essentially surjective.\n\nSee https:\/\/stacks.math.columbia.edu\/tag\/02C3.\n-\/\nlemma ess_surj_of_equivalence (F : C \u2964 D) [is_equivalence F] : ess_surj F :=\n\u27e8\u03bb Y, \u27e8F.inv.obj Y, \u27e8F.as_equivalence.counit_iso.app Y\u27e9\u27e9\u27e9\n\n\/--\nAn equivalence is faithful.\n\nSee https:\/\/stacks.math.columbia.edu\/tag\/02C3.\n-\/\n@[priority 100] -- see Note [lower instance priority]\ninstance faithful_of_equivalence (F : C \u2964 D) [is_equivalence F] : faithful F :=\n{ map_injective' := \u03bb X Y f g w,\n  begin\n    have p := congr_arg (@category_theory.functor.map _ _ _ _ F.inv _ _) w,\n    simpa only [cancel_epi, cancel_mono, is_equivalence.inv_fun_map] using p\n  end }.\n\n\/--\nAn equivalence is full.\n\nSee https:\/\/stacks.math.columbia.edu\/tag\/02C3.\n-\/\n@[priority 100] -- see Note [lower instance priority]\ninstance full_of_equivalence (F : C \u2964 D) [is_equivalence F] : full F :=\n{ preimage := \u03bb X Y f, F.as_equivalence.unit.app X \u226b F.inv.map f \u226b F.as_equivalence.unit_inv.app Y,\n  witness' := \u03bb X Y f, F.inv.map_injective $\n  by simpa only [is_equivalence.inv_fun_map, assoc, iso.inv_hom_id_app_assoc, iso.inv_hom_id_app]\n    using comp_id _ }\n\n@[simps] private noncomputable def equivalence_inverse (F : C \u2964 D) [full F] [faithful F]\n  [ess_surj F] : D \u2964 C :=\n{ obj  := \u03bb X, F.obj_preimage X,\n  map := \u03bb X Y f, F.preimage ((F.obj_obj_preimage_iso X).hom \u226b f \u226b (F.obj_obj_preimage_iso Y).inv),\n  map_id' := \u03bb X, begin apply F.map_injective, tidy end,\n  map_comp' := \u03bb X Y Z f g, by apply F.map_injective; simp }\n\n\/--\nA functor which is full, faithful, and essentially surjective is an equivalence.\n\nSee https:\/\/stacks.math.columbia.edu\/tag\/02C3.\n-\/\nnoncomputable def equivalence_of_fully_faithfully_ess_surj\n  (F : C \u2964 D) [full F] [faithful F] [ess_surj F] : is_equivalence F :=\nis_equivalence.mk (equivalence_inverse F)\n  (nat_iso.of_components\n    (\u03bb X, (preimage_iso $ F.obj_obj_preimage_iso $ F.obj X).symm)\n    (\u03bb X Y f, by { apply F.map_injective, obviously }))\n  (nat_iso.of_components F.obj_obj_preimage_iso (by tidy))\n\n@[simp] lemma functor_map_inj_iff (e : C \u224c D) {X Y : C} (f g : X \u27f6 Y) :\n  e.functor.map f = e.functor.map g \u2194 f = g :=\n\u27e8\u03bb h, e.functor.map_injective h, \u03bb h, h \u25b8 rfl\u27e9\n\n@[simp] lemma inverse_map_inj_iff (e : C \u224c D) {X Y : D} (f g : X \u27f6 Y) :\n  e.inverse.map f = e.inverse.map g \u2194 f = g :=\nfunctor_map_inj_iff e.symm f g\n\nend equivalence\n\nend category_theory\n","avg_line_length":41.2003676471,"max_line_length":100,"alphanum_fraction":0.6695667693}
-{"size":4392,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"def StateT (m : Type \u2192 Type) (\u03c3 : Type) (\u03b1 : Type) := (Unit \u00d7 \u03c3) \u2192 m (\u03b1 \u00d7 \u03c3)\nnamespace StateT\nvariables {m : Type \u2192 Type} [Monad m] {\u03c3 : Type} {\u03b1 \u03b2 : Type}\n@[inline] protected def pure (a : \u03b1) : StateT m \u03c3 \u03b1 := \u03bb \u27e8_, s\u27e9, pure (a, s)\n@[inline] protected def bind (x : StateT m \u03c3 \u03b1) (f : \u03b1 \u2192 StateT m \u03c3 \u03b2) : StateT m \u03c3 \u03b2 := \u03bb p, do (a, s') \u2190 x p, f a ((), s')\n@[inline] def read : StateT m \u03c3 \u03c3 := \u03bb \u27e8_, s\u27e9, pure (s, s)\n@[inline] def write (s' : \u03c3) : StateT m \u03c3 Unit := \u03bb \u27e8_, s\u27e9, pure ((), s')\n@[inline] def updt (f : \u03c3 \u2192 \u03c3) : StateT m \u03c3 Unit := \u03bb \u27e8_, s\u27e9, pure ((), f s)\ninstance : Monad (StateT m \u03c3) :=\n{pure := @StateT.pure _ _ _, bind := @StateT.bind _ _ _}\nend StateT\n\ndef ExceptT (m : Type \u2192 Type) (\u03b5 : Type) (\u03b1 : Type) := { e : Except \u03b5 Unit \/\/ e = Except.ok () } \u2192 m (Except \u03b5 \u03b1)\nnamespace ExceptT\nvariables {m : Type \u2192 Type} [Monad m] {\u03b5 : Type} {\u03b1 \u03b2 : Type}\n@[inline] protected def pure (a : \u03b1) : ExceptT m \u03b5 \u03b1 :=\n\u03bb e, match e with\n     | \u27e8Except.ok _, h\u27e9    := pure (Except.ok a)\n     | \u27e8Except.error _, h\u27e9 := Except.noConfusion h\n@[inline] protected def bind (x : ExceptT m \u03b5 \u03b1) (f : \u03b1 \u2192 ExceptT m \u03b5 \u03b2) : ExceptT m \u03b5 \u03b2 :=\n\u03bb e, do v \u2190 x e,\n        match v with\n        | Except.error e := pure (Except.error e)\n        | Except.ok a    := f a \u27e8Except.ok (), rfl\u27e9\n@[inline] def error (e : \u03b5) : ExceptT m \u03b5 \u03b1 :=\n\u03bb e', match e' with\n     | \u27e8Except.ok _, h\u27e9    := pure (Except.error e)\n     | \u27e8Except.error _, h\u27e9 := Except.noConfusion h\n@[inline] def lift (x : m \u03b1) : ExceptT m \u03b5 \u03b1 := \u03bb e, do {a \u2190 x, pure (Except.ok a)}\ninstance : Monad (ExceptT m \u03b5) :=\n{pure := @ExceptT.pure _ _ _, bind := @ExceptT.bind _ _ _}\nend ExceptT\n\nabbreviation Node := Nat\n\nstructure nodeData :=\n(find : Node) (rank : Nat := 0)\n\nabbreviation ufData := Array nodeData\n\nabbreviation M (\u03b1 : Type) := ExceptT (StateT id ufData) String \u03b1\n@[inline] def read : M ufData := ExceptT.lift StateT.read\n@[inline] def write (s : ufData) : M Unit := ExceptT.lift (StateT.write s)\n@[inline] def updt (f : ufData \u2192 ufData) : M Unit := ExceptT.lift (StateT.updt f)\n@[inline] def error {\u03b1 : Type} (e : String) : M \u03b1 := ExceptT.error e\ndef run {\u03b1 : Type} (x : M \u03b1) (s : ufData := Array.nil) : Except String \u03b1 \u00d7 ufData :=\nx \u27e8Except.ok (), rfl\u27e9 ((), s)\n\ndef capacity : M Nat :=\ndo d \u2190 read, pure d.size\n\ndef findEntryAux : Nat \u2192 Node \u2192 M nodeData\n| 0     n := error \"out of fuel\"\n| (i+1) n :=\n  do s \u2190 read,\n     if h : n < s.size then\n       do { let e := s.read \u27e8n, h\u27e9,\n            if e.find = n then pure e\n            else do e\u2081 \u2190 findEntryAux i e.find,\n                    updt (\u03bb s, s.write' n e\u2081),\n                    pure e\u2081 }\n     else error \"invalid Node\"\n\ndef findEntry (n : Node) : M nodeData :=\ndo c \u2190 capacity,\n   findEntryAux c n\n\ndef find (n : Node) : M Node :=\ndo e \u2190 findEntry n, pure e.find\n\ndef mk : M Node :=\ndo n \u2190 capacity,\n   updt $ \u03bb s, s.push {find := n, rank := 1},\n   pure n\n\ndef union (n\u2081 n\u2082 : Node) : M Unit :=\ndo r\u2081 \u2190 findEntry n\u2081,\n   r\u2082 \u2190 findEntry n\u2082,\n   if r\u2081.find = r\u2082.find then pure ()\n   else updt $ \u03bb s,\n     if r\u2081.rank < r\u2082.rank then s.write' r\u2081.find { find := r\u2082.find }\n     else if r\u2081.rank = r\u2082.rank then\n        let s\u2081 := s.write' r\u2081.find { find := r\u2082.find } in\n        s\u2081.write' r\u2082.find { rank := r\u2082.rank + 1, .. r\u2082}\n     else s.write' r\u2082.find { find := r\u2081.find }\n\n\ndef mkNodes : Nat \u2192 M Unit\n| 0     := pure ()\n| (n+1) := mk *> mkNodes n\n\ndef checkEq (n\u2081 n\u2082 : Node) : M Unit :=\ndo r\u2081 \u2190 find n\u2081, r\u2082 \u2190 find n\u2082,\n   unless (r\u2081 = r\u2082) $ error \"nodes are not equal\"\n\ndef mergePackAux : Nat \u2192 Nat \u2192 Nat \u2192 M Unit\n| 0     _ _ := pure ()\n| (i+1) n d :=\n  do c \u2190 capacity,\n  if (n+d) < c\n  then union n (n+d) *> mergePackAux i (n+1) d\n  else pure ()\n\ndef mergePack (d : Nat) : M Unit :=\ndo c \u2190 capacity, mergePackAux c 0 d\n\ndef numEqsAux : Nat \u2192 Node \u2192 Nat \u2192 M Nat\n| 0     _ r := pure r\n| (i+1) n r :=\n  do c \u2190 capacity,\n     if n < c\n     then do { n\u2081 \u2190 find n, numEqsAux i (n+1) (if n = n\u2081 then r else r+1) }\n     else pure r\n\ndef numEqs : M Nat :=\ndo c \u2190 capacity,\n   numEqsAux c 0 0\n\ndef test (n : Nat) : M Nat :=\nif n < 2 then error \"input must be greater than 1\"\nelse do\n  mkNodes n,\n  mergePack 50000,\n  mergePack 10000,\n  mergePack 5000,\n  mergePack 1000,\n  numEqs\n\ndef main (xs : List String) : IO UInt32 :=\nlet n := xs.head.toNat in\nmatch run (test n) with\n| (Except.ok v, s)    := IO.println (\"ok \" ++ toString v) *> pure 0\n| (Except.error e, s) := IO.println (\"Error : \" ++ e) *> pure 1\n","avg_line_length":32.776119403,"max_line_length":124,"alphanum_fraction":0.5626138434}
-{"size":742,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import Lake\nopen System Lake DSL\n\ndef cDir : FilePath := \"c\"\ndef ffiSrc := cDir \/ \"ffi.cpp\"\ndef buildDir := defaultBuildDir\n\ndef ffiOTarget (pkgDir : FilePath) : FileTarget :=\n  let oFile := pkgDir \/ buildDir \/ cDir \/ \"ffi.o\"\n  let srcTarget := inputFileTarget <| pkgDir \/ ffiSrc\n  fileTargetWithDep oFile srcTarget fun srcFile => do\n    compileO oFile srcFile #[\"-I\", (\u2190 getLeanIncludeDir).toString] \"c++\"\n\ndef cLibTarget (pkgDir : FilePath) : FileTarget :=\n  let libFile := pkgDir \/ buildDir \/ cDir \/ \"libffi.a\"\n  staticLibTarget libFile #[ffiOTarget pkgDir]\n\npackage ffi (pkgDir) (args) {\n  -- customize layout\n  srcDir := \"lib\"\n  libRoots := #[`Ffi]\n  -- specify the lib as an additional target\n  moreLibTargets := #[cLibTarget pkgDir]\n}\n","avg_line_length":29.68,"max_line_length":72,"alphanum_fraction":0.6900269542}
-{"size":1169,"ext":"lean","lang":"Lean","max_stars_count":134.0,"content":"import ddl.basic\n\nnamespace ddl.host\n\n  \/- The type syntax of the host language -\/\n  inductive type (\u2113 : Type) : Type\n    | bool {} : type\n    | nat {} : type\n    | union_nil {} : type\n    | union_cons : \u2113 \u2192 type \u2192 type \u2192 type\n    | struct_nil {} : type\n    | struct_cons : \u2113 \u2192 type \u2192 type \u2192 type\n    | array {} : type \u2192 type\n\n  namespace type\n\n    variables {\u2113 : Type}\n\n    def lookup (l : \u2113) [decidable_eq \u2113] : type \u2113 \u2192 option (type \u2113)\n      | (struct_cons l' t tr) := if l = l' then some t else lookup tr\n      | _                     := none\n\n  end type\n\n\n  \/- Binary operators -\/\n  inductive binop : Type\n    | add\n    | mul\n\n\n  \/- The expression syntax of the host language -\/\n  inductive expr (\u2113 : Type) : Type\n    | bool {} : bool \u2192 expr\n    | nat {} : \u2115 \u2192 expr\n    | app_binop {} : binop \u2192 expr \u2192 expr \u2192 expr\n    | proj : expr \u2192 \u2113 \u2192 expr\n\n  instance has_coe_to_bool {\u2113} : has_coe bool (expr \u2113) := \u27e8expr.bool\u27e9\n  instance has_coe_to_nat {\u2113} : has_coe \u2115 (expr \u2113)  := \u27e8expr.nat\u27e9\n\n  namespace expr\n\n    variables {\u2113 : Type}\n\n    instance : has_add (expr \u2113) := \u27e8app_binop binop.add\u27e9\n    instance : has_mul (expr \u2113) := \u27e8app_binop binop.mul\u27e9\n\n  end expr\n\nend ddl.host\n","avg_line_length":22.4807692308,"max_line_length":69,"alphanum_fraction":0.5714285714}
-{"size":16206,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2021 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca\n-\/\n\nimport ring_theory.trace\nimport ring_theory.norm\nimport number_theory.number_field\n\n\/-!\n# Discriminant of a family of vectors\n\nGiven an `A`-algebra `B` and `b`, an `\u03b9`-indexed family of elements of `B`, we define the\n*discriminant* of `b` as the determinant of the matrix whose `(i j)`-th element is the trace of\n`b i * b j`.\n\n## Main definition\n\n* `algebra.discr A b` : the discriminant of `b : \u03b9 \u2192 B`.\n\n## Main results\n\n* `algebra.discr_zero_of_not_linear_independent` : if `b` is not linear independent, then\n  `algebra.discr A b = 0`.\n* `algebra.discr_of_matrix_vec_mul` and `discr_of_matrix_mul_vec` : formulas relating\n  `algebra.discr A \u03b9 b` with `algebra.discr A ((P.map (algebra_map A B)).vec_mul b)` and\n  `algebra.discr A ((P.map (algebra_map A B)).mul_vec b)`.\n* `algebra.discr_not_zero_of_basis` : over a field, if `b` is a basis, then\n  `algebra.discr K b \u2260 0`.\n* `algebra.discr_eq_det_embeddings_matrix_reindex_pow_two` : if `L\/K` is a field extension and\n  `b : \u03b9 \u2192 L`, then `discr K b` is the square of the determinant of the matrix whose `(i, j)`\n  coefficient is `\u03c3\u2c7c (b i)`, where `\u03c3\u2c7c : L \u2192\u2090[K] E` is the embedding in an algebraically closed\n  field `E` corresponding to `j : \u03b9` via a bijection `e : \u03b9 \u2243 (L \u2192\u2090[K] E)`.\n* `algebra.discr_of_power_basis_eq_prod` : the discriminant of a power basis.\n* `discr_is_integral` : if `K` and `L` are fields and `is_scalar_tower R K L`, is `b : \u03b9 \u2192 L`\n  satisfies ` \u2200 i, is_integral R (b i)`, then `is_integral R (discr K b)`.\n* `discr_mul_is_integral_mem_adjoin` : let `K` be the fraction field of an integrally closed domain\n  `R` and let `L` be a finite separable extension of `K`. Let `B : power_basis K L` be such that\n  `is_integral R B.gen`. Then for all, `z : L` we have\n  `(discr K B.basis) \u2022 z \u2208 adjoin R ({B.gen} : set L)`.\n\n## Implementation details\n\nOur definition works for any `A`-algebra `B`, but note that if `B` is not free as an `A`-module,\nthen `trace A B = 0` by definition, so `discr A b = 0` for any `b`.\n-\/\n\nuniverses u v w z\n\nopen_locale matrix big_operators\n\nopen matrix finite_dimensional fintype polynomial finset intermediate_field\n\nnamespace algebra\n\nvariables (A : Type u) {B : Type v} (C : Type z) {\u03b9 : Type w}\nvariables [comm_ring A] [comm_ring B] [algebra A B] [comm_ring C] [algebra A C]\n\nsection discr\n\n\/-- Given an `A`-algebra `B` and `b`, an `\u03b9`-indexed family of elements of `B`, we define\n`discr A \u03b9 b` as the determinant of `trace_matrix A \u03b9 b`. -\/\nnoncomputable\ndef discr (A : Type u) {B : Type v} [comm_ring A] [comm_ring B] [algebra A B] [fintype \u03b9]\n  (b : \u03b9 \u2192 B) := by { classical, exact (trace_matrix A b).det }\n\nlemma discr_def [decidable_eq \u03b9] [fintype \u03b9] (b : \u03b9 \u2192 B) :\n  discr A b = (trace_matrix A b).det := by convert rfl\n\nvariables {\u03b9' : Type*} [fintype \u03b9'] [fintype \u03b9]\n\nsection basic\n\n@[simp] lemma discr_reindex (b : basis \u03b9 A B) (f : \u03b9 \u2243 \u03b9') :\n  discr A (b \u2218 \u21d1(f.symm)) = discr A b :=\nbegin\n  classical,\n  rw [\u2190 basis.coe_reindex, discr_def, trace_matrix_reindex, det_reindex_self, \u2190 discr_def]\nend\n\n\/-- If `b` is not linear independent, then `algebra.discr A b = 0`. -\/\nlemma discr_zero_of_not_linear_independent [is_domain A] {b : \u03b9 \u2192 B}\n  (hli : \u00aclinear_independent A b) : discr A b = 0 :=\nbegin\n  classical,\n  obtain \u27e8g, hg, i, hi\u27e9 := fintype.not_linear_independent_iff.1 hli,\n  have : (trace_matrix A b).mul_vec g = 0,\n  { ext i,\n    have : \u2200 j, (trace A B) (b i * b j) * g j = (trace A B) (((g j) \u2022 (b j)) * b i),\n    { intro j, simp [mul_comm], },\n    simp only [mul_vec, dot_product, trace_matrix, pi.zero_apply, trace_form_apply,\n      \u03bb j, this j, \u2190 linear_map.map_sum, \u2190 sum_mul, hg, zero_mul, linear_map.map_zero] },\n  by_contra h,\n  rw discr_def at h,\n  simpa [matrix.eq_zero_of_mul_vec_eq_zero h this] using hi,\nend\n\nvariable {A}\n\n\/-- Relation between `algebra.discr A \u03b9 b` and\n`algebra.discr A ((P.map (algebra_map A B)).vec_mul b)`. -\/\nlemma discr_of_matrix_vec_mul [decidable_eq \u03b9] (b : \u03b9 \u2192 B) (P : matrix \u03b9 \u03b9 A) :\n  discr A ((P.map (algebra_map A B)).vec_mul b) = P.det ^ 2 * discr A b :=\nby rw [discr_def, trace_matrix_of_matrix_vec_mul, det_mul, det_mul, det_transpose, mul_comm,\n    \u2190 mul_assoc, discr_def, pow_two]\n\n\/-- Relation between `algebra.discr A \u03b9 b` and\n`algebra.discr A ((P.map (algebra_map A B)).mul_vec b)`. -\/\nlemma discr_of_matrix_mul_vec [decidable_eq \u03b9] (b : \u03b9 \u2192 B) (P : matrix \u03b9 \u03b9 A) :\n  discr A ((P.map (algebra_map A B)).mul_vec b) = P.det ^ 2 * discr A b :=\nby rw [discr_def, trace_matrix_of_matrix_mul_vec, det_mul, det_mul, det_transpose,\n  mul_comm, \u2190 mul_assoc, discr_def, pow_two]\n\nend basic\n\nsection field\n\nvariables (K : Type u) {L : Type v} (E : Type z) [field K] [field L] [field E]\nvariables [algebra K L] [algebra K E]\nvariables [module.finite K L]  [is_alg_closed E]\n\n\/-- Over a field, if `b` is a basis, then `algebra.discr K b \u2260 0`. -\/\nlemma discr_not_zero_of_basis [is_separable K L] (b : basis \u03b9 K L) : discr K b \u2260 0 :=\nbegin\n  by_cases h : nonempty \u03b9,\n  { classical,\n    have := span_eq_top_of_linear_independent_of_card_eq_finrank b.linear_independent\n      (finrank_eq_card_basis b).symm,\n    rw [discr_def, trace_matrix_def],\n    simp_rw [\u2190 basis.mk_apply b.linear_independent this],\n    rw [\u2190 trace_matrix_def, trace_matrix_of_basis, \u2190 bilin_form.nondegenerate_iff_det_ne_zero],\n    exact trace_form_nondegenerate _ _  },\n  letI := not_nonempty_iff.1 h,\n  simp [discr],\nend\n\n\/-- Over a field, if `b` is a basis, then `algebra.discr K b` is a unit. -\/\nlemma discr_is_unit_of_basis [is_separable K L] (b : basis \u03b9 K L) : is_unit (discr K b) :=\nis_unit.mk0 _ (discr_not_zero_of_basis _ _)\n\nvariables (b : \u03b9 \u2192 L) (pb : power_basis K L)\n\n\/-- If `L\/K` is a field extension and `b : \u03b9 \u2192 L`, then `discr K b` is the square of the\ndeterminant of the matrix whose `(i, j)` coefficient is `\u03c3\u2c7c (b i)`, where `\u03c3\u2c7c : L \u2192\u2090[K] E` is the\nembedding in an algebraically closed field `E` corresponding to `j : \u03b9` via a bijection\n`e : \u03b9 \u2243 (L \u2192\u2090[K] E)`. -\/\nlemma discr_eq_det_embeddings_matrix_reindex_pow_two [decidable_eq \u03b9] [is_separable K L]\n  (e : \u03b9 \u2243 (L \u2192\u2090[K] E)) : algebra_map K E (discr K b) =\n  (embeddings_matrix_reindex K E b e).det ^ 2 :=\nby rw [discr_def, ring_hom.map_det, ring_hom.map_matrix_apply,\n    trace_matrix_eq_embeddings_matrix_reindex_mul_trans, det_mul, det_transpose, pow_two]\n\n\/-- The discriminant of a power basis. -\/\nlemma discr_power_basis_eq_prod (e : fin pb.dim \u2243 (L \u2192\u2090[K] E)) [is_separable K L] :\n  algebra_map K E (discr K pb.basis) =\n  \u220f i : fin pb.dim, \u220f j in finset.univ.filter (\u03bb j, i < j), (e j pb.gen- (e i pb.gen)) ^ 2 :=\nbegin\n  rw [discr_eq_det_embeddings_matrix_reindex_pow_two K E pb.basis e,\n    embeddings_matrix_reindex_eq_vandermonde, det_transpose, det_vandermonde, \u2190 prod_pow],\n  congr, ext i,\n  rw [\u2190 prod_pow]\nend\n\n\/-- A variation of `of_power_basis_eq_prod`. -\/\nlemma discr_power_basis_eq_prod' [is_separable K L] (e : fin pb.dim \u2243 (L \u2192\u2090[K] E)) :\n  algebra_map K E (discr K pb.basis) =\n  \u220f i : fin pb.dim, \u220f j in finset.univ.filter (\u03bb j, i < j),\n  -((e j pb.gen- (e i pb.gen)) * (e i pb.gen- (e j pb.gen))) :=\nbegin\n  rw [discr_power_basis_eq_prod _ _ _ e],\n  congr, ext i, congr, ext j,\n  ring\nend\n\nlocal notation `n` := finrank K L\n\n\/-- A variation of `of_power_basis_eq_prod`. -\/\nlemma discr_power_basis_eq_prod'' [is_separable K L] (e : fin pb.dim \u2243 (L \u2192\u2090[K] E)) :\n  algebra_map K E (discr K pb.basis) =\n  (-1) ^ (n * (n - 1) \/ 2) * \u220f i : fin pb.dim, \u220f j in finset.univ.filter (\u03bb j, i < j),\n  ((e j pb.gen- (e i pb.gen)) * (e i pb.gen- (e j pb.gen))) :=\nbegin\n  rw [discr_power_basis_eq_prod' _ _ _ e],\n  simp_rw [\u03bb i j, neg_eq_neg_one_mul ((e j pb.gen- (e i pb.gen)) * (e i pb.gen- (e j pb.gen))),\n    prod_mul_distrib],\n  congr,\n  simp only [prod_pow_eq_pow_sum, prod_const],\n  congr,\n  simp_rw [fin.card_filter_lt],\n  apply (@nat.cast_inj \u211a _ _ _ _ _).1,\n  rw [nat.cast_sum],\n  have : \u2200 (x : fin pb.dim), (\u2191x + 1) \u2264 pb.dim := by simp [nat.succ_le_iff, fin.is_lt],\n  simp_rw [nat.sub_sub],\n  simp only [nat.cast_sub, this, finset.card_fin, nsmul_eq_mul, sum_const, sum_sub_distrib,\n    nat.cast_add, nat.cast_one, sum_add_distrib, mul_one],\n  rw [\u2190 nat.cast_sum, \u2190 @finset.sum_range \u2115 _ pb.dim (\u03bb i, i), sum_range_id ],\n  have hn : n = pb.dim,\n  { rw [\u2190 alg_hom.card K L E, \u2190 fintype.card_fin pb.dim],\n    exact card_congr (equiv.symm e) },\n  have h\u2082 : 2 \u2223 (pb.dim * (pb.dim - 1)) := even_iff_two_dvd.1 (nat.even_mul_self_pred _),\n  have hne : ((2 : \u2115) : \u211a) \u2260 0 := by simp,\n  have hle : 1 \u2264 pb.dim,\n  { rw [\u2190 hn, nat.one_le_iff_ne_zero, \u2190 zero_lt_iff, finite_dimensional.finrank_pos_iff],\n    apply_instance },\n  rw [hn, nat.cast_div h\u2082 hne, nat.cast_mul, nat.cast_sub hle],\n  field_simp,\n  ring,\nend\n\n\/-- Formula for the discriminant of a power basis using the norm of the field extension. -\/\nlemma discr_power_basis_eq_norm [is_separable K L] : discr K pb.basis =\n  (-1) ^ (n * (n - 1) \/ 2) * (norm K (aeval pb.gen (minpoly K pb.gen).derivative)) :=\nbegin\n  let E := algebraic_closure L,\n  letI := \u03bb (a b : E), classical.prop_decidable (eq a b),\n\n  have e : fin pb.dim \u2243 (L \u2192\u2090[K] E),\n  { refine equiv_of_card_eq _,\n    rw [fintype.card_fin, alg_hom.card],\n    exact (power_basis.finrank pb).symm },\n  have hnodup : (map (algebra_map K E) (minpoly K pb.gen)).roots.nodup :=\n    nodup_roots (separable.map (is_separable.separable K pb.gen)),\n  have hroots : \u2200 \u03c3 : L \u2192\u2090[K] E, \u03c3 pb.gen \u2208 (map (algebra_map K E) (minpoly K pb.gen)).roots,\n  { intro \u03c3,\n    rw [mem_roots, is_root.def, eval_map, \u2190 aeval_def, aeval_alg_hom_apply],\n    repeat { simp [minpoly.ne_zero (is_separable.is_integral K pb.gen)] } },\n\n  apply (algebra_map K E).injective,\n  rw [ring_hom.map_mul, ring_hom.map_pow, ring_hom.map_neg, ring_hom.map_one,\n    discr_power_basis_eq_prod'' _ _ _ e],\n  congr,\n  rw [norm_eq_prod_embeddings, fin.prod_filter_lt_mul_neg_eq_prod_off_diag],\n  conv_rhs { congr, skip, funext,\n    rw [\u2190 aeval_alg_hom_apply, aeval_root_derivative_of_splits (minpoly.monic\n      (is_separable.is_integral K pb.gen)) (is_alg_closed.splits_codomain _) (hroots \u03c3),\n      \u2190 finset.prod_mk _ (hnodup.erase _)] },\n  rw [prod_sigma', prod_sigma'],\n  refine prod_bij (\u03bb i hi, \u27e8e i.2, e i.1 pb.gen\u27e9) (\u03bb i hi, _) (\u03bb i hi, by simp at hi)\n    (\u03bb i j hi hj hij, _) (\u03bb \u03c3 h\u03c3, _),\n  { simp only [true_and, finset.mem_mk, mem_univ, mem_sigma],\n    rw [multiset.mem_erase_of_ne (\u03bb h, _)],\n    { exact hroots _ },\n    { simp only [true_and, mem_filter, mem_univ, ne.def, mem_sigma] at hi,\n      refine hi (equiv.injective e (equiv.injective (power_basis.lift_equiv pb) _)),\n      rw [\u2190 power_basis.lift_equiv_apply_coe, \u2190 power_basis.lift_equiv_apply_coe] at h,\n      exact subtype.eq h } },\n  { simp only [equiv.apply_eq_iff_eq, heq_iff_eq] at hij,\n    have h := hij.2,\n    rw [\u2190 power_basis.lift_equiv_apply_coe, \u2190 power_basis.lift_equiv_apply_coe] at h,\n    refine sigma.eq (equiv.injective e (equiv.injective _ (subtype.eq h))) (by simp [hij.1]) },\n  { simp only [true_and, finset.mem_mk, mem_univ, mem_sigma] at h\u03c3,\n    simp only [sigma.exists, true_and, exists_prop, mem_filter, mem_univ, ne.def, mem_sigma],\n    refine \u27e8e.symm (power_basis.lift pb \u03c3.2 _), e.symm \u03c3.1, \u27e8\u03bb h, _, sigma.eq _ _\u27e9\u27e9,\n    { rw [aeval_def, eval\u2082_eq_eval_map, \u2190 is_root.def, \u2190 mem_roots],\n      { exact multiset.erase_subset _ _ h\u03c3 },\n      { simp [minpoly.ne_zero (is_separable.is_integral K pb.gen)] } },\n    { replace h := alg_hom.congr_fun (equiv.injective _ h) pb.gen,\n      rw [power_basis.lift_gen] at h,\n      rw [\u2190 h] at h\u03c3,\n      exact hnodup.not_mem_erase h\u03c3 },\n    all_goals { simp } }\nend\n\nsection integral\n\nvariables {R : Type z} [comm_ring R] [algebra R K] [algebra R L] [is_scalar_tower R K L]\n\nlocal notation `is_integral` := _root_.is_integral\n\n\/-- If `K` and `L` are fields and `is_scalar_tower R K L`, and `b : \u03b9 \u2192 L` satisfies\n` \u2200 i, is_integral R (b i)`, then `is_integral R (discr K b)`. -\/\nlemma discr_is_integral {b : \u03b9 \u2192 L} (h : \u2200 i, is_integral R (b i)) :\n  is_integral R (discr K b) :=\nbegin\n  classical,\n  rw [discr_def],\n  exact is_integral.det (\u03bb i j, is_integral_trace (is_integral_mul (h i) (h j)))\nend\n\n\/-- If `b` and `b'` are `\u211a`-bases of a number field `K` such that\n`\u2200 i j, is_integral \u2124 (b.to_matrix b' i j)` and `\u2200 i j, is_integral \u2124 (b'.to_matrix b i j)` then\n`discr \u211a b = discr \u211a b'`. -\/\nlemma discr_eq_discr_of_to_matrix_coeff_is_integral [number_field K] {b : basis \u03b9 \u211a K}\n  {b' : basis \u03b9' \u211a K} (h : \u2200 i j, is_integral \u2124 (b.to_matrix b' i j))\n  (h' : \u2200 i j, is_integral \u2124 (b'.to_matrix b i j)) :\n  discr \u211a b = discr \u211a b' :=\nbegin\n  replace h' : \u2200 i j, is_integral \u2124 (b'.to_matrix ((b.reindex (b.index_equiv b'))) i j),\n  { intros i j,\n    convert h' i ((b.index_equiv b').symm j),\n    simpa },\n  classical,\n  rw [\u2190 (b.reindex (b.index_equiv b')).to_matrix_map_vec_mul b', discr_of_matrix_vec_mul,\n    \u2190 one_mul (discr \u211a b), basis.coe_reindex, discr_reindex],\n  congr,\n  have hint : is_integral \u2124 (((b.reindex (b.index_equiv b')).to_matrix b').det) :=\n    is_integral.det (\u03bb i j, h _ _),\n  obtain \u27e8r, hr\u27e9 := is_integrally_closed.is_integral_iff.1 hint,\n  have hunit : is_unit r,\n  { have : is_integral \u2124 ((b'.to_matrix (b.reindex (b.index_equiv b'))).det) :=\n      is_integral.det (\u03bb i j, h' _ _),\n    obtain \u27e8r', hr'\u27e9 := is_integrally_closed.is_integral_iff.1 this,\n    refine is_unit_iff_exists_inv.2 \u27e8r', _\u27e9,\n    suffices : algebra_map \u2124 \u211a (r * r') = 1,\n    { rw [\u2190 ring_hom.map_one (algebra_map \u2124 \u211a)] at this,\n      exact (is_fraction_ring.injective \u2124 \u211a) this },\n    rw [ring_hom.map_mul, hr, hr', \u2190 det_mul, basis.to_matrix_mul_to_matrix_flip, det_one] },\n  rw [\u2190 ring_hom.map_one (algebra_map \u2124 \u211a), \u2190 hr],\n  cases int.is_unit_iff.1 hunit with hp hm,\n  { simp [hp] },\n  { simp [hm] }\nend\n\n\/-- Let `K` be the fraction field of an integrally closed domain `R` and let `L` be a finite\nseparable extension of `K`. Let `B : power_basis K L` be such that `is_integral R B.gen`.\nThen for all, `z : L` that are integral over `R`, we have\n`(discr K B.basis) \u2022 z \u2208 adjoin R ({B.gen} : set L)`. -\/\nlemma discr_mul_is_integral_mem_adjoin [is_domain R] [is_separable K L] [is_integrally_closed R]\n  [is_fraction_ring R K] {B : power_basis K L} (hint : is_integral R B.gen) {z : L}\n  (hz : is_integral R z) : (discr K B.basis) \u2022 z \u2208 adjoin R ({B.gen} : set L) :=\nbegin\n  have hinv : is_unit (trace_matrix K B.basis).det :=\n    by simpa [\u2190 discr_def] using discr_is_unit_of_basis _ B.basis,\n\n  have H : (trace_matrix K B.basis).det \u2022 (trace_matrix K B.basis).mul_vec (B.basis.equiv_fun z) =\n    (trace_matrix K B.basis).det \u2022 (\u03bb i, trace K L (z * B.basis i)),\n  { congr, exact trace_matrix_of_basis_mul_vec _ _ },\n  have cramer := mul_vec_cramer (trace_matrix K B.basis) (\u03bb i, trace K L (z * B.basis i)),\n\n  suffices : \u2200 i, ((trace_matrix K B.basis).det \u2022 (B.basis.equiv_fun z)) i \u2208 (\u22a5 : subalgebra R K),\n  { rw [\u2190 B.basis.sum_repr z, finset.smul_sum],\n    refine subalgebra.sum_mem _ (\u03bb i hi, _),\n    replace this := this i,\n    rw [\u2190 discr_def, pi.smul_apply, mem_bot] at this,\n    obtain \u27e8r, hr\u27e9 := this,\n    rw [basis.equiv_fun_apply] at hr,\n    rw [\u2190 smul_assoc, \u2190 hr, algebra_map_smul],\n    refine subalgebra.smul_mem _ _ _,\n    rw [B.basis_eq_pow i],\n    refine subalgebra.pow_mem _ (subset_adjoin (set.mem_singleton _)) _},\n  intro i,\n  rw [\u2190 H, \u2190 mul_vec_smul] at cramer,\n  replace cramer := congr_arg (mul_vec (trace_matrix K B.basis)\u207b\u00b9) cramer,\n  rw [mul_vec_mul_vec, nonsing_inv_mul _ hinv, mul_vec_mul_vec, nonsing_inv_mul _ hinv,\n    one_mul_vec, one_mul_vec] at cramer,\n  rw [\u2190 congr_fun cramer i, cramer_apply, det_apply],\n  refine subalgebra.sum_mem _ (\u03bb \u03c3 _, subalgebra.zsmul_mem _ (subalgebra.prod_mem _ (\u03bb j _, _)) _),\n  by_cases hji : j = i,\n  { simp only [update_column_apply, hji, eq_self_iff_true, power_basis.coe_basis],\n    exact mem_bot.2 (is_integrally_closed.is_integral_iff.1 $ is_integral_trace $\n      is_integral_mul hz $ is_integral.pow hint _) },\n  { simp only [update_column_apply, hji, power_basis.coe_basis],\n    exact mem_bot.2 (is_integrally_closed.is_integral_iff.1 $ is_integral_trace\n      $ is_integral_mul (is_integral.pow hint _) (is_integral.pow hint _)) }\nend\n\nend integral\n\nend field\n\nend discr\n\nend algebra\n","avg_line_length":44.4,"max_line_length":99,"alphanum_fraction":0.6743181538}
-{"size":37789,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-\/\nimport analysis.specific_limits\nimport order.iterate\nimport order.semiconj_Sup\nimport algebra.iterate_hom\n\n\/-!\n# Translation number of a monotone real map that commutes with `x \u21a6 x + 1`\n\nLet `f : \u211d \u2192 \u211d` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `\u03c4`, `\u03c1`, `rot`, etc\n\nIn this file we define a structure `circle_deg1_lift` for bundled maps with these properties, define\ntranslation number of `f : circle_deg1_lift`, prove some estimates relating `f^n(x)-x` to `\u03c4(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `\u03c4(f)=m\/n`.\n\nMaps of this type naturally appear as lifts of orientation preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation preserving homeomorphism of the circle $S^1=\u211d\/\u2124$, and\nconsider a real number `a` such that\n`\u27e6a\u27e7 = f 0`, where `\u27e6\u27e7` means the natural projection `\u211d \u2192 \u211d\/\u2124`. Then there exists a unique\ncontinuous function `F : \u211d \u2192 \u211d` such that `F 0 = a` and `\u27e6F x\u27e7 = f \u27e6x\u27e7` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `\u27e6\u03c4 F\u27e7 : \u211d \/ \u2124` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `circle_deg1_lift`: a monotone map `f : \u211d \u2192 \u211d` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `circle_deg1_lift` is equipped with `lattice` and `monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `circle_deg1_lift.translation_number`: translation number of `f : circle_deg1_lift`.\n\n## Main statements\n\nWe prove the following properties of `circle_deg1_lift.translation_number`.\n\n* `circle_deg1_lift.translation_number_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : \u2115`, then `f` and `g` have equal\n  translation numbers.\n\n* `circle_deg1_lift.translation_number_eq_of_semiconj_by`: if two `circle_deg1_lift` maps `f`, `g`\n  are semiconjugate by a `circle_deg1_lift` map, then `\u03c4 f = \u03c4 g`.\n\n* `circle_deg1_lift.translation_number_units_inv`: if `f` is an invertible `circle_deg1_lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f\u207b\u00b9` is the negative of the translation number of `f`.\n\n* `circle_deg1_lift.translation_number_mul_of_commute`: if `f` and `g` commute, then\n  `\u03c4 (f * g) = \u03c4 f + \u03c4 g`.\n\n* `circle_deg1_lift.translation_number_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m \/ n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `circle_deg1_lift.semiconj_of_bijective_of_translation_number_eq`: if `f` and `g` are two\n  bijective `circle_deg1_lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `circle_deg1_lift.semiconj_of_group_action_of_forall_translation_number_eq`: let `f\u2081` and `f\u2082` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f\u2081` and `f\u2082` are two\n  homomorphisms from `G \u2192* circle_deg1_lift`). If the translation numbers of `f\u2081 g` and `f\u2082 g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some `F :\n  circle_deg1_lift`. This is a version of Proposition 5.4 from [\u00c9tienne Ghys, Groupes\n  d'homeomorphismes du cercle et cohomologie bornee][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `\u03c4` for the translation number of `f : circle_deg1_lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : circle_deg1_lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 \/ (2 ^ n)`, then prove that `((f ^ n) x - x) \/ n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : \u211d \u2192 \u211d` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincar\u00e9 maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [\u00c9tienne Ghys, Groupes d'homeomorphismes du cercle et cohomologie bornee][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use `units\n  circle_deg1_lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `semiconj_by` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `conditionally_complete_lattice` structure, use it in the proof of\n  `circle_deg1_lift.semiconj_of_group_action_of_forall_translation_number_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `circle_deg1_lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-\/\n\nopen filter set function (hiding commute)\nopen_locale topological_space classical\n\n\/-!\n### Definition and monoid structure\n-\/\n\n\/-- A lift of a monotone degree one map `S\u00b9 \u2192 S\u00b9`. -\/\nstructure circle_deg1_lift : Type :=\n(to_fun : \u211d \u2192 \u211d)\n(monotone' : monotone to_fun)\n(map_add_one' : \u2200 x, to_fun (x + 1) = to_fun x + 1)\n\nnamespace circle_deg1_lift\n\ninstance : has_coe_to_fun circle_deg1_lift := \u27e8\u03bb _, \u211d \u2192 \u211d, circle_deg1_lift.to_fun\u27e9\n\n@[simp] lemma coe_mk (f h\u2081 h\u2082) : \u21d1(mk f h\u2081 h\u2082) = f := rfl\n\nvariables (f g : circle_deg1_lift)\n\nprotected lemma monotone  : monotone f := f.monotone'\n\n@[mono] lemma mono {x y} (h : x \u2264 y) : f x \u2264 f y := f.monotone h\n\nlemma strict_mono_iff_injective : strict_mono f \u2194 injective f :=\nf.monotone.strict_mono_iff_injective\n\n@[simp] lemma map_add_one : \u2200 x, f (x + 1) = f x + 1 := f.map_add_one'\n\n@[simp] lemma map_one_add (x : \u211d) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm]\n\ntheorem coe_inj : \u2200 \u2983f g : circle_deg1_lift \u2984, (f : \u211d \u2192 \u211d) = g \u2192 f = g :=\nassume \u27e8f, fm, fd\u27e9 \u27e8g, gm, gd\u27e9 h, by congr; exact h\n\n@[ext] theorem ext \u2983f g : circle_deg1_lift \u2984 (h : \u2200 x, f x = g x) : f = g :=\ncoe_inj $ funext h\n\ntheorem ext_iff {f g : circle_deg1_lift} : f = g \u2194 \u2200 x, f x = g x :=\n\u27e8\u03bb h x, h \u25b8 rfl, \u03bb h, ext h\u27e9\n\ninstance : monoid circle_deg1_lift :=\n{ mul := \u03bb f g,\n  { to_fun := f \u2218 g,\n    monotone' := f.monotone.comp g.monotone,\n    map_add_one' := \u03bb x, by simp [map_add_one] },\n  one := \u27e8id, monotone_id, \u03bb _, rfl\u27e9,\n  mul_one := \u03bb f, coe_inj $ function.comp.right_id f,\n  one_mul := \u03bb f, coe_inj $ function.comp.left_id f,\n  mul_assoc := \u03bb f\u2081 f\u2082 f\u2083, coe_inj rfl }\n\ninstance : inhabited circle_deg1_lift := \u27e81\u27e9\n\n@[simp] lemma coe_mul : \u21d1(f * g) = f \u2218 g := rfl\n\nlemma mul_apply (x) : (f * g) x = f (g x) := rfl\n\n@[simp] lemma coe_one : \u21d1(1 : circle_deg1_lift) = id := rfl\n\ninstance units_has_coe_to_fun : has_coe_to_fun (units circle_deg1_lift) :=\n\u27e8\u03bb _, \u211d \u2192 \u211d, \u03bb f, \u21d1(f : circle_deg1_lift)\u27e9\n\n@[simp, norm_cast] lemma units_coe (f : units circle_deg1_lift) : \u21d1(f : circle_deg1_lift) = f := rfl\n\n@[simp] lemma units_inv_apply_apply (f : units circle_deg1_lift) (x : \u211d) :\n  (f\u207b\u00b9 : units circle_deg1_lift) (f x) = x :=\nby simp only [\u2190 units_coe, \u2190 mul_apply, f.inv_mul, coe_one, id]\n\n@[simp] lemma units_apply_inv_apply (f : units circle_deg1_lift) (x : \u211d) :\n  f ((f\u207b\u00b9 : units circle_deg1_lift) x) = x :=\nby simp only [\u2190 units_coe, \u2190 mul_apply, f.mul_inv, coe_one, id]\n\n\/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -\/\ndef to_order_iso : units circle_deg1_lift \u2192* \u211d \u2243o \u211d :=\n{ to_fun := \u03bb f,\n    { to_fun := f,\n      inv_fun := \u21d1(f\u207b\u00b9),\n      left_inv := units_inv_apply_apply f,\n      right_inv := units_apply_inv_apply f,\n      map_rel_iff' := \u03bb x y, \u27e8\u03bb h, by simpa using mono \u2191(f\u207b\u00b9) h, mono f\u27e9 },\n  map_one' := rfl,\n  map_mul' := \u03bb f g, rfl }\n\n@[simp] lemma coe_to_order_iso (f : units circle_deg1_lift) : \u21d1(to_order_iso f) = f := rfl\n\n@[simp] lemma coe_to_order_iso_symm (f : units circle_deg1_lift) :\n  \u21d1(to_order_iso f).symm = (f\u207b\u00b9 : units circle_deg1_lift) := rfl\n\n@[simp] lemma coe_to_order_iso_inv (f : units circle_deg1_lift) :\n  \u21d1(to_order_iso f)\u207b\u00b9 = (f\u207b\u00b9 : units circle_deg1_lift) := rfl\n\nlemma is_unit_iff_bijective {f : circle_deg1_lift} : is_unit f \u2194 bijective f :=\n\u27e8\u03bb \u27e8u, h\u27e9, h \u25b8 (to_order_iso u).bijective, \u03bb h, is_unit_unit\n  { val := f,\n    inv := { to_fun := (equiv.of_bijective f h).symm,\n             monotone' := \u03bb x y hxy, (f.strict_mono_iff_injective.2 h.1).le_iff_le.1\n               (by simp only [equiv.of_bijective_apply_symm_apply f h, hxy]),\n             map_add_one' := \u03bb x, h.1 $\n               by simp only [equiv.of_bijective_apply_symm_apply f, f.map_add_one] },\n    val_inv := ext $ equiv.of_bijective_apply_symm_apply f h,\n    inv_val := ext $ equiv.of_bijective_symm_apply_apply f h }\u27e9\n\nlemma coe_pow : \u2200 n : \u2115, \u21d1(f^n) = (f^[n])\n| 0 := rfl\n| (n+1) := by {ext x, simp [coe_pow n, pow_succ'] }\n\nlemma semiconj_by_iff_semiconj {f g\u2081 g\u2082 : circle_deg1_lift} :\n  semiconj_by f g\u2081 g\u2082 \u2194 semiconj f g\u2081 g\u2082 :=\next_iff\n\nlemma commute_iff_commute {f g : circle_deg1_lift} :\n  commute f g \u2194 function.commute f g :=\next_iff\n\n\/-!\n### Translate by a constant\n-\/\n\n\/-- The map `y \u21a6 x + y` as a `circle_deg1_lift`. More precisely, we define a homomorphism from\n`multiplicative \u211d` to `units circle_deg1_lift`, so the translation by `x` is\n`translation (multiplicative.of_add x)`. -\/\ndef translate : multiplicative \u211d \u2192* units circle_deg1_lift :=\nby refine (units.map _).comp to_units.to_monoid_hom; exact\n{ to_fun := \u03bb x, \u27e8\u03bb y, x.to_add + y, \u03bb y\u2081 y\u2082 h, add_le_add_left h _, \u03bb y, (add_assoc _ _ _).symm\u27e9,\n  map_one' := ext $ zero_add,\n  map_mul' := \u03bb x y, ext $ add_assoc _ _ }\n\n@[simp] lemma translate_apply (x y : \u211d) : translate (multiplicative.of_add x) y = x + y := rfl\n\n@[simp]\nlemma translate_inv_apply (x y : \u211d) : (translate $ multiplicative.of_add x)\u207b\u00b9 y = -x + y := rfl\n\n@[simp] lemma translate_gpow (x : \u211d) (n : \u2124) :\n  (translate (multiplicative.of_add x))^n = translate (multiplicative.of_add $ \u2191n * x) :=\nby simp only [\u2190 gsmul_eq_mul, of_add_gsmul, monoid_hom.map_gpow]\n\n@[simp] lemma translate_pow (x : \u211d) (n : \u2115) :\n  (translate (multiplicative.of_add x))^n = translate (multiplicative.of_add $ \u2191n * x) :=\ntranslate_gpow x n\n\n@[simp] lemma translate_iterate (x : \u211d) (n : \u2115) :\n  (translate (multiplicative.of_add x))^[n] = translate (multiplicative.of_add $ \u2191n * x) :=\nby rw [\u2190 units_coe, \u2190 coe_pow, \u2190 units.coe_pow, translate_pow, units_coe]\n\n\/-!\n### Commutativity with integer translations\n\nIn this section we prove that `f` commutes with translations by an integer number.\nFirst we formulate these statements (for a natural or an integer number,\naddition on the left or on the right, addition or subtraction) using `function.commute`,\nthen reformulate as `simp` lemmas `map_int_add` etc.\n-\/\n\nlemma commute_nat_add (n : \u2115) : function.commute f ((+) n) :=\nby simpa only [nsmul_one, add_left_iterate] using function.commute.iterate_right f.map_one_add n\n\nlemma commute_add_nat (n : \u2115) : function.commute f (\u03bb x, x + n) :=\nby simp only [add_comm _ (n:\u211d), f.commute_nat_add n]\n\nlemma commute_sub_nat (n : \u2115) : function.commute f (\u03bb x, x - n) :=\nby simpa only [sub_eq_add_neg] using\n  (f.commute_add_nat n).inverses_right (equiv.add_right _).right_inv (equiv.add_right _).left_inv\n\nlemma commute_add_int : \u2200 n : \u2124, function.commute f (\u03bb x, x + n)\n| (n:\u2115) := f.commute_add_nat n\n| -[1+n] := by simpa only [sub_eq_add_neg] using f.commute_sub_nat (n + 1)\n\nlemma commute_int_add (n : \u2124) : function.commute f ((+) n) :=\nby simpa only [add_comm _ (n:\u211d)] using f.commute_add_int n\n\nlemma commute_sub_int (n : \u2124) : function.commute f (\u03bb x, x - n) :=\nby simpa only [sub_eq_add_neg] using\n  (f.commute_add_int n).inverses_right (equiv.add_right _).right_inv (equiv.add_right _).left_inv\n\n@[simp] lemma map_int_add (m : \u2124) (x : \u211d) : f (m + x) = m + f x :=\nf.commute_int_add m x\n\n@[simp] lemma map_add_int (x : \u211d) (m : \u2124) : f (x + m) = f x + m :=\nf.commute_add_int m x\n\n@[simp] lemma map_sub_int (x : \u211d) (n : \u2124) : f (x - n) = f x - n :=\nf.commute_sub_int n x\n\n@[simp] lemma map_add_nat (x : \u211d) (n : \u2115) : f (x + n) = f x + n :=\nf.map_add_int x n\n\n@[simp] lemma map_nat_add (n : \u2115) (x : \u211d) : f (n + x) = n + f x :=\nf.map_int_add n x\n\n@[simp] lemma map_sub_nat (x : \u211d) (n : \u2115) : f (x - n) = f x - n :=\nf.map_sub_int x n\n\nlemma map_int_of_map_zero (n : \u2124) : f n = f 0 + n :=\nby rw [\u2190 f.map_add_int, zero_add]\n\n@[simp] lemma map_fract_sub_fract_eq (x : \u211d) :\n  f (fract x) - fract x = f x - x :=\nby conv_rhs { rw [\u2190 fract_add_floor x, f.map_add_int, add_sub_comm, sub_self, add_zero] }\n\n\/-!\n### Pointwise order on circle maps\n-\/\n\n\/-- Monotone circle maps form a lattice with respect to the pointwise order -\/\nnoncomputable instance : lattice circle_deg1_lift :=\n{ sup := \u03bb f g,\n  { to_fun := \u03bb x, max (f x) (g x),\n    monotone' := \u03bb x y h, max_le_max (f.mono h) (g.mono h), -- TODO: generalize to `monotone.max`\n    map_add_one' := \u03bb x, by simp [max_add_add_right] },\n  le := \u03bb f g, \u2200 x, f x \u2264 g x,\n  le_refl := \u03bb f x, le_refl (f x),\n  le_trans := \u03bb f\u2081 f\u2082 f\u2083 h\u2081\u2082 h\u2082\u2083 x, le_trans (h\u2081\u2082 x) (h\u2082\u2083 x),\n  le_antisymm := \u03bb f\u2081 f\u2082 h\u2081\u2082 h\u2082\u2081, ext $ \u03bb x, le_antisymm (h\u2081\u2082 x) (h\u2082\u2081 x),\n  le_sup_left := \u03bb f g x, le_max_left (f x) (g x),\n  le_sup_right := \u03bb f g x, le_max_right (f x) (g x),\n  sup_le := \u03bb f\u2081 f\u2082 f\u2083 h\u2081 h\u2082 x, max_le (h\u2081 x) (h\u2082 x),\n  inf := \u03bb f g,\n  { to_fun := \u03bb x, min (f x) (g x),\n    monotone' := \u03bb x y h, min_le_min (f.mono h) (g.mono h),\n    map_add_one' := \u03bb x, by simp [min_add_add_right] },\n  inf_le_left := \u03bb f g x, min_le_left (f x) (g x),\n  inf_le_right := \u03bb f g x, min_le_right (f x) (g x),\n  le_inf := \u03bb f\u2081 f\u2082 f\u2083 h\u2082 h\u2083 x, le_min (h\u2082 x) (h\u2083 x) }\n\n@[simp] lemma sup_apply (x : \u211d) : (f \u2294 g) x = max (f x) (g x) := rfl\n\n@[simp] lemma inf_apply (x : \u211d) : (f \u2293 g) x = min (f x) (g x) := rfl\n\nlemma iterate_monotone (n : \u2115) : monotone (\u03bb f : circle_deg1_lift, f^[n]) :=\n\u03bb f g h, f.monotone.iterate_le_of_le h _\n\nlemma iterate_mono {f g : circle_deg1_lift} (h : f \u2264 g) (n : \u2115) : f^[n] \u2264 (g^[n]) :=\niterate_monotone n h\n\nlemma pow_mono {f g : circle_deg1_lift} (h : f \u2264 g) (n : \u2115) : f^n \u2264 g^n :=\n\u03bb x, by simp only [coe_pow, iterate_mono h n x]\n\nlemma pow_monotone (n : \u2115) : monotone (\u03bb f : circle_deg1_lift, f^n) :=\n\u03bb f g h, pow_mono h n\n\n\/-!\n### Estimates on `(f * g) 0`\n\nWe prove the estimates `f 0 + \u230ag 0\u230b \u2264 f (g 0) \u2264 f 0 + \u2308g 0\u2309` and some corollaries with added\/removed\nfloors and ceils.\n\nWe also prove that for two semiconjugate maps `g\u2081`, `g\u2082`, the distance between `g\u2081 0` and `g\u2082 0`\nis less than two.\n-\/\n\nlemma map_le_of_map_zero (x : \u211d) : f x \u2264 f 0 + \u2308x\u2309 :=\ncalc f x \u2264 f \u2308x\u2309     : f.monotone $ le_ceil _\n     ... = f 0 + \u2308x\u2309 : f.map_int_of_map_zero _\n\nlemma map_map_zero_le : f (g 0) \u2264 f 0 + \u2308g 0\u2309 := f.map_le_of_map_zero (g 0)\n\nlemma floor_map_map_zero_le : \u230af (g 0)\u230b \u2264 \u230af 0\u230b + \u2308g 0\u2309 :=\ncalc \u230af (g 0)\u230b \u2264 \u230af 0 + \u2308g 0\u2309\u230b : floor_mono $ f.map_map_zero_le g\n           ... = \u230af 0\u230b + \u2308g 0\u2309 : floor_add_int _ _\n\nlemma ceil_map_map_zero_le : \u2308f (g 0)\u2309 \u2264 \u2308f 0\u2309 + \u2308g 0\u2309 :=\ncalc \u2308f (g 0)\u2309 \u2264 \u2308f 0 + \u2308g 0\u2309\u2309 : ceil_mono $ f.map_map_zero_le g\n           ... = \u2308f 0\u2309 + \u2308g 0\u2309 : ceil_add_int _ _\n\nlemma map_map_zero_lt : f (g 0) < f 0 + g 0 + 1 :=\ncalc f (g 0) \u2264 f 0 + \u2308g 0\u2309     : f.map_map_zero_le g\n         ... < f 0 + (g 0 + 1) : add_lt_add_left (ceil_lt_add_one _) _\n         ... = f 0 + g 0 + 1   : (add_assoc _ _ _).symm\n\nlemma le_map_of_map_zero (x : \u211d) : f 0 + \u230ax\u230b \u2264 f x :=\ncalc f 0 + \u230ax\u230b = f \u230ax\u230b : (f.map_int_of_map_zero _).symm\n           ... \u2264 f x   : f.monotone $ floor_le _\n\nlemma le_map_map_zero : f 0 + \u230ag 0\u230b \u2264 f (g 0) := f.le_map_of_map_zero (g 0)\n\nlemma le_floor_map_map_zero : \u230af 0\u230b + \u230ag 0\u230b \u2264 \u230af (g 0)\u230b :=\ncalc \u230af 0\u230b + \u230ag 0\u230b = \u230af 0 + \u230ag 0\u230b\u230b : (floor_add_int _ _).symm\n               ... \u2264 \u230af (g 0)\u230b     : floor_mono $ f.le_map_map_zero g\n\nlemma le_ceil_map_map_zero : \u2308f 0\u2309 + \u230ag 0\u230b \u2264 \u2308(f * g) 0\u2309 :=\ncalc \u2308f 0\u2309 + \u230ag 0\u230b = \u2308f 0 + \u230ag 0\u230b\u2309 : (ceil_add_int _ _).symm\n               ... \u2264 \u2308f (g 0)\u2309     : ceil_mono $ f.le_map_map_zero g\n\nlemma lt_map_map_zero : f 0 + g 0 - 1 < f (g 0) :=\ncalc f 0 + g 0 - 1 = f 0 + (g 0 - 1) : add_sub_assoc _ _ _\n               ... < f 0 + \u230ag 0\u230b     : add_lt_add_left (sub_one_lt_floor _) _\n               ... \u2264 f (g 0)         : f.le_map_map_zero g\n\nlemma dist_map_map_zero_lt : dist (f 0 + g 0) (f (g 0)) < 1 :=\nbegin\n  rw [dist_comm, real.dist_eq, abs_lt, lt_sub_iff_add_lt', sub_lt_iff_lt_add', \u2190 sub_eq_add_neg],\n  exact \u27e8f.lt_map_map_zero g, f.map_map_zero_lt g\u27e9\nend\n\nlemma dist_map_zero_lt_of_semiconj {f g\u2081 g\u2082 : circle_deg1_lift} (h : function.semiconj f g\u2081 g\u2082) :\n  dist (g\u2081 0) (g\u2082 0) < 2 :=\ncalc dist (g\u2081 0) (g\u2082 0) \u2264 dist (g\u2081 0) (f (g\u2081 0) - f 0) + dist _ (g\u2082 0) : dist_triangle _ _ _\n... = dist (f 0 + g\u2081 0) (f (g\u2081 0)) + dist (g\u2082 0 + f 0) (g\u2082 (f 0)) :\n  by simp only [h.eq, real.dist_eq, sub_sub, add_comm (f 0), sub_sub_assoc_swap, abs_sub (g\u2082 (f 0))]\n... < 2 : add_lt_add (f.dist_map_map_zero_lt g\u2081) (g\u2082.dist_map_map_zero_lt f)\n\nlemma dist_map_zero_lt_of_semiconj_by {f g\u2081 g\u2082 : circle_deg1_lift} (h : semiconj_by f g\u2081 g\u2082) :\n  dist (g\u2081 0) (g\u2082 0) < 2 :=\ndist_map_zero_lt_of_semiconj $ semiconj_by_iff_semiconj.1 h\n\n\/-!\n### Limits at infinities and continuity\n-\/\n\nprotected lemma tendsto_at_bot : tendsto f at_bot at_bot :=\ntendsto_at_bot_mono f.map_le_of_map_zero $ tendsto_at_bot_add_const_left _ _ $\n  tendsto_at_bot_mono (\u03bb x, (ceil_lt_add_one x).le) $ tendsto_at_bot_add_const_right _ _ tendsto_id\n\nprotected lemma tendsto_at_top : tendsto f at_top at_top :=\ntendsto_at_top_mono f.le_map_of_map_zero $ tendsto_at_top_add_const_left _ _ $\n  tendsto_at_top_mono (\u03bb x, (sub_one_lt_floor x).le) $\n    by simpa [sub_eq_add_neg] using tendsto_at_top_add_const_right _ _ tendsto_id\n\nlemma continuous_iff_surjective : continuous f \u2194 function.surjective f :=\n\u27e8\u03bb h, h.surjective f.tendsto_at_top f.tendsto_at_bot, f.monotone.continuous_of_surjective\u27e9\n\n\/-!\n### Estimates on `(f^n) x`\n\nIf we know that `f x` is `\u2264`\/`<`\/`\u2265`\/`>`\/`=` to `x + m`, then we have a similar estimate on\n`f^[n] x` and `x + n * m`.\n\nFor `\u2264`, `\u2265`, and `=` we formulate both `of` (implication) and `iff` versions because implications\nwork for `n = 0`. For `<` and `>` we formulate only `iff` versions.\n-\/\n\nlemma iterate_le_of_map_le_add_int {x : \u211d} {m : \u2124} (h : f x \u2264 x + m) (n : \u2115) :\n  f^[n] x \u2264 x + n * m :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).iterate_le_of_map_le f.monotone (monotone_id.add_const m) h n\n\nlemma le_iterate_of_add_int_le_map {x : \u211d} {m : \u2124} (h : x + m \u2264 f x) (n : \u2115) :\n  x + n * m \u2264 (f^[n]) x :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).symm.iterate_le_of_map_le (monotone_id.add_const m) f.monotone h n\n\nlemma iterate_eq_of_map_eq_add_int {x : \u211d} {m : \u2124} (h : f x = x + m) (n : \u2115) :\n  f^[n] x = x + n * m :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).iterate_eq_of_map_eq n h\n\nlemma iterate_pos_le_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n  f^[n] x \u2264 x + n * m \u2194 f x \u2264 x + m :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).iterate_pos_le_iff_map_le f.monotone (strict_mono_id.add_const m) hn\n\nlemma iterate_pos_lt_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n  f^[n] x < x + n * m \u2194 f x < x + m :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).iterate_pos_lt_iff_map_lt f.monotone (strict_mono_id.add_const m) hn\n\nlemma iterate_pos_eq_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n  f^[n] x = x + n * m \u2194 f x = x + m :=\nby simpa only [nsmul_eq_mul, add_right_iterate]\nusing (f.commute_add_int m).iterate_pos_eq_iff_map_eq f.monotone (strict_mono_id.add_const m) hn\n\nlemma le_iterate_pos_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n  x + n * m \u2264 (f^[n]) x \u2194 x + m \u2264 f x :=\nby simpa only [not_lt] using not_congr (f.iterate_pos_lt_iff hn)\n\nlemma lt_iterate_pos_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n  x + n * m < (f^[n]) x \u2194 x + m < f x :=\nby simpa only [not_le] using not_congr (f.iterate_pos_le_iff hn)\n\nlemma mul_floor_map_zero_le_floor_iterate_zero (n : \u2115) : \u2191n * \u230af 0\u230b \u2264 \u230a(f^[n] 0)\u230b :=\nbegin\n  rw [le_floor, int.cast_mul, int.cast_coe_nat, \u2190 zero_add ((n : \u211d) * _)],\n  apply le_iterate_of_add_int_le_map,\n  simp [floor_le]\nend\n\n\/-!\n### Definition of translation number\n-\/\nnoncomputable theory\n\n\/-- An auxiliary sequence used to define the translation number. -\/\ndef transnum_aux_seq (n : \u2115) : \u211d := (f^(2^n)) 0 \/ 2^n\n\n\/-- The translation number of a `circle_deg1_lift`, $\u03c4(f)=\\lim_{n\u2192\u221e}\\frac{f^n(x)-x}{n}$. We use\nan auxiliary sequence `\\frac{f^{2^n}(0)}{2^n}` to define `\u03c4(f)` because some proofs are simpler\nthis way. -\/\ndef translation_number : \u211d :=\nlim at_top f.transnum_aux_seq\n\n-- TODO: choose two different symbols for `circle_deg1_lift.translation_number` and the future\n-- `circle_mono_homeo.rotation_number`, then make them `localized notation`s\nlocal notation `\u03c4` := translation_number\n\nlemma transnum_aux_seq_def : f.transnum_aux_seq = \u03bb n : \u2115, (f^(2^n)) 0 \/ 2^n := rfl\n\nlemma translation_number_eq_of_tendsto_aux {\u03c4' : \u211d}\n  (h : tendsto f.transnum_aux_seq at_top (\ud835\udcdd \u03c4')) :\n  \u03c4 f = \u03c4' :=\nh.lim_eq\n\nlemma translation_number_eq_of_tendsto\u2080 {\u03c4' : \u211d}\n  (h : tendsto (\u03bb n:\u2115, f^[n] 0 \/ n) at_top (\ud835\udcdd \u03c4')) :\n  \u03c4 f = \u03c4' :=\nf.translation_number_eq_of_tendsto_aux $\nby simpa [(\u2218), transnum_aux_seq_def, coe_pow]\nusing h.comp (nat.tendsto_pow_at_top_at_top_of_one_lt one_lt_two)\n\nlemma translation_number_eq_of_tendsto\u2080' {\u03c4' : \u211d}\n  (h : tendsto (\u03bb n:\u2115, f^[n + 1] 0 \/ (n + 1)) at_top (\ud835\udcdd \u03c4')) :\n  \u03c4 f = \u03c4' :=\nf.translation_number_eq_of_tendsto\u2080 $ (tendsto_add_at_top_iff_nat 1).1 h\n\nlemma transnum_aux_seq_zero : f.transnum_aux_seq 0 = f 0 := by simp [transnum_aux_seq]\n\nlemma transnum_aux_seq_dist_lt (n : \u2115) :\n  dist (f.transnum_aux_seq n) (f.transnum_aux_seq (n+1)) < (1 \/ 2) \/ (2^n) :=\nbegin\n  have : 0 < (2^(n+1):\u211d) := pow_pos zero_lt_two _,\n  rw [div_div_eq_div_mul, \u2190 pow_succ, \u2190 abs_of_pos this],\n  replace := abs_pos.2 (ne_of_gt this),\n  convert (div_lt_div_right this).2 ((f^(2^n)).dist_map_map_zero_lt (f^(2^n))),\n  simp_rw [transnum_aux_seq, real.dist_eq],\n  rw [\u2190 abs_div, sub_div, pow_succ', pow_succ, \u2190 two_mul,\n    mul_div_mul_left _ _ (@two_ne_zero \u211d _ _),\n    pow_mul, pow_two, mul_apply]\nend\n\nlemma tendsto_translation_number_aux : tendsto f.transnum_aux_seq at_top (\ud835\udcdd $ \u03c4 f) :=\n(cauchy_seq_of_le_geometric_two 1 (\u03bb n, le_of_lt $ f.transnum_aux_seq_dist_lt n)).tendsto_lim\n\nlemma dist_map_zero_translation_number_le : dist (f 0) (\u03c4 f) \u2264 1 :=\nf.transnum_aux_seq_zero \u25b8 dist_le_of_le_geometric_two_of_tendsto\u2080 1\n  (\u03bb n, le_of_lt $ f.transnum_aux_seq_dist_lt n) f.tendsto_translation_number_aux\n\nlemma tendsto_translation_number_of_dist_bounded_aux (x : \u2115 \u2192 \u211d) (C : \u211d)\n  (H : \u2200 n : \u2115, dist ((f^n) 0) (x n) \u2264 C) :\n  tendsto (\u03bb n : \u2115, x (2^n) \/ (2^n)) at_top (\ud835\udcdd $ \u03c4 f) :=\nbegin\n  refine f.tendsto_translation_number_aux.congr_dist (squeeze_zero (\u03bb _, dist_nonneg) _ _),\n  { exact \u03bb n, C \/ 2^n },\n  { intro n,\n    have : 0 < (2^n:\u211d) := pow_pos zero_lt_two _,\n    convert (div_le_div_right this).2 (H (2^n)),\n    rw [transnum_aux_seq, real.dist_eq, \u2190 sub_div, abs_div, abs_of_pos this, real.dist_eq] },\n  { exact mul_zero C \u25b8 tendsto_const_nhds.mul (tendsto_inv_at_top_zero.comp $\n      tendsto_pow_at_top_at_top_of_one_lt one_lt_two) }\nend\n\nlemma translation_number_eq_of_dist_bounded {f g : circle_deg1_lift} (C : \u211d)\n  (H : \u2200 n : \u2115, dist ((f^n) 0) ((g^n) 0) \u2264 C) :\n  \u03c4 f = \u03c4 g :=\neq.symm $ g.translation_number_eq_of_tendsto_aux $\n  f.tendsto_translation_number_of_dist_bounded_aux _ C H\n\n@[simp] lemma translation_number_one : \u03c4 1 = 0 :=\ntranslation_number_eq_of_tendsto\u2080 _ $ by simp [tendsto_const_nhds]\n\nlemma translation_number_eq_of_semiconj_by {f g\u2081 g\u2082 : circle_deg1_lift} (H : semiconj_by f g\u2081 g\u2082) :\n  \u03c4 g\u2081 = \u03c4 g\u2082 :=\ntranslation_number_eq_of_dist_bounded 2 $ \u03bb n, le_of_lt $\n  dist_map_zero_lt_of_semiconj_by $ H.pow_right n\n\nlemma translation_number_eq_of_semiconj {f g\u2081 g\u2082 : circle_deg1_lift}\n  (H : function.semiconj f g\u2081 g\u2082) :\n  \u03c4 g\u2081 = \u03c4 g\u2082 :=\ntranslation_number_eq_of_semiconj_by $ semiconj_by_iff_semiconj.2 H\n\nlemma translation_number_mul_of_commute {f g : circle_deg1_lift} (h : commute f g) :\n  \u03c4 (f * g) = \u03c4 f + \u03c4 g :=\nbegin\n  have : tendsto (\u03bb n : \u2115, ((\u03bb k, (f^k) 0 + (g^k) 0) (2^n)) \/ (2^n)) at_top (\ud835\udcdd $ \u03c4 f + \u03c4 g) :=\n    ((f.tendsto_translation_number_aux.add g.tendsto_translation_number_aux).congr $\n      \u03bb n, (add_div ((f^(2^n)) 0) ((g^(2^n)) 0) ((2:\u211d)^n)).symm),\n  refine tendsto_nhds_unique\n    ((f * g).tendsto_translation_number_of_dist_bounded_aux _ 1 (\u03bb n, _))\n    this,\n  rw [h.mul_pow, dist_comm],\n  exact le_of_lt ((f^n).dist_map_map_zero_lt (g^n))\nend\n\n@[simp] lemma translation_number_units_inv (f : units circle_deg1_lift) :\n  \u03c4 \u2191(f\u207b\u00b9) = -\u03c4 f :=\neq_neg_iff_add_eq_zero.2 $\n  by simp [\u2190 translation_number_mul_of_commute (commute.refl _).units_inv_left]\n\n@[simp] lemma translation_number_pow :\n  \u2200 n : \u2115, \u03c4 (f^n) = n * \u03c4 f\n| 0 := by simp\n| (n+1) := by rw [pow_succ', translation_number_mul_of_commute (commute.pow_self f n),\n  translation_number_pow n, nat.cast_add_one, add_mul, one_mul]\n\n@[simp] lemma translation_number_gpow (f : units circle_deg1_lift) :\n  \u2200 n : \u2124, \u03c4 (f ^ n : units _) = n * \u03c4 f\n| (n : \u2115) := by simp [translation_number_pow f n]\n| -[1+n] := by { simp,  ring }\n\n@[simp] lemma translation_number_conj_eq (f : units circle_deg1_lift) (g : circle_deg1_lift) :\n  \u03c4 (\u2191f * g * \u2191(f\u207b\u00b9)) = \u03c4 g :=\n(translation_number_eq_of_semiconj_by (f.mk_semiconj_by g)).symm\n\n@[simp] lemma translation_number_conj_eq' (f : units circle_deg1_lift) (g : circle_deg1_lift) :\n  \u03c4 (\u2191(f\u207b\u00b9) * g * f) = \u03c4 g :=\ntranslation_number_conj_eq f\u207b\u00b9 g\n\nlemma dist_pow_map_zero_mul_translation_number_le (n:\u2115) :\n  dist ((f^n) 0) (n * f.translation_number) \u2264 1 :=\nf.translation_number_pow n \u25b8 (f^n).dist_map_zero_translation_number_le\n\nlemma tendsto_translation_number\u2080' :\n  tendsto (\u03bb n:\u2115, (f^(n+1)) 0 \/ (n+1)) at_top (\ud835\udcdd $ \u03c4 f) :=\nbegin\n  refine (tendsto_iff_dist_tendsto_zero.2 $ squeeze_zero (\u03bb _, dist_nonneg) (\u03bb n, _)\n    ((tendsto_const_div_at_top_nhds_0_nat 1).comp (tendsto_add_at_top_nat 1))),\n  dsimp,\n  have : (0:\u211d) < n + 1 := n.cast_add_one_pos,\n  rw [real.dist_eq, div_sub' _ _ _ (ne_of_gt this), abs_div, \u2190 real.dist_eq, abs_of_pos this,\n    div_le_div_right this, \u2190 nat.cast_add_one],\n  apply dist_pow_map_zero_mul_translation_number_le\nend\n\nlemma tendsto_translation_number\u2080 :\n  tendsto (\u03bb n:\u2115, ((f^n) 0) \/ n) at_top (\ud835\udcdd $ \u03c4 f) :=\n(tendsto_add_at_top_iff_nat 1).1 f.tendsto_translation_number\u2080'\n\n\/-- For any `x : \u211d` the sequence $\\frac{f^n(x)-x}{n}$ tends to the translation number of `f`.\nIn particular, this limit does not depend on `x`. -\/\nlemma tendsto_translation_number (x : \u211d) :\n  tendsto (\u03bb n:\u2115, ((f^n) x - x) \/ n) at_top (\ud835\udcdd $ \u03c4 f) :=\nbegin\n  rw [\u2190 translation_number_conj_eq' (translate $ multiplicative.of_add x)],\n  convert tendsto_translation_number\u2080 _,\n  ext n,\n  simp [sub_eq_neg_add, units.conj_pow']\nend\n\nlemma tendsto_translation_number' (x : \u211d) :\n  tendsto (\u03bb n:\u2115, ((f^(n+1)) x - x) \/ (n+1)) at_top (\ud835\udcdd $ \u03c4 f) :=\n(tendsto_add_at_top_iff_nat 1).2 (f.tendsto_translation_number x)\n\nlemma translation_number_mono : monotone \u03c4 :=\n\u03bb f g h, le_of_tendsto_of_tendsto' f.tendsto_translation_number\u2080\n  g.tendsto_translation_number\u2080 $ \u03bb n, div_le_div_of_le_of_nonneg (pow_mono h n 0) n.cast_nonneg\n\nlemma translation_number_translate (x : \u211d) :\n  \u03c4 (translate $ multiplicative.of_add x) = x :=\ntranslation_number_eq_of_tendsto\u2080' _ $\n  by simp [nat.cast_add_one_ne_zero, mul_div_cancel_left, tendsto_const_nhds]\n\nlemma translation_number_le_of_le_add {z : \u211d} (hz : \u2200 x, f x \u2264 x + z) : \u03c4 f \u2264 z :=\ntranslation_number_translate z \u25b8 translation_number_mono\n  (\u03bb x, trans_rel_left _ (hz x) (add_comm _ _))\n\nlemma le_translation_number_of_add_le {z : \u211d} (hz : \u2200 x, x + z \u2264 f x) : z \u2264 \u03c4 f :=\ntranslation_number_translate z \u25b8 translation_number_mono\n  (\u03bb x, trans_rel_right _ (add_comm _ _) (hz x))\n\nlemma translation_number_le_of_le_add_int {x : \u211d} {m : \u2124} (h : f x \u2264 x + m) : \u03c4 f \u2264 m :=\nle_of_tendsto' (f.tendsto_translation_number' x) $ \u03bb n,\n(div_le_iff' (n.cast_add_one_pos : (0 : \u211d) < _)).mpr $ sub_le_iff_le_add'.2 $\n(coe_pow f (n + 1)).symm \u25b8 f.iterate_le_of_map_le_add_int h (n + 1)\n\nlemma translation_number_le_of_le_add_nat {x : \u211d} {m : \u2115} (h : f x \u2264 x + m) : \u03c4 f \u2264 m :=\n@translation_number_le_of_le_add_int f x m h\n\nlemma le_translation_number_of_add_int_le {x : \u211d} {m : \u2124} (h : x + m \u2264 f x) : \u2191m \u2264 \u03c4 f :=\nge_of_tendsto' (f.tendsto_translation_number' x) $ \u03bb n,\n(le_div_iff (n.cast_add_one_pos : (0 : \u211d) < _)).mpr $ le_sub_iff_add_le'.2 $\nby simp only [coe_pow, mul_comm (m:\u211d), \u2190 nat.cast_add_one, f.le_iterate_of_add_int_le_map h]\n\nlemma le_translation_number_of_add_nat_le {x : \u211d} {m : \u2115} (h : x + m \u2264 f x) : \u2191m \u2264 \u03c4 f :=\n@le_translation_number_of_add_int_le f x m h\n\n\/-- If `f x - x` is an integer number `m` for some point `x`, then `\u03c4 f = m`.\nOn the circle this means that a map with a fixed point has rotation number zero. -\/\nlemma translation_number_of_eq_add_int {x : \u211d} {m : \u2124} (h : f x = x + m) : \u03c4 f = m :=\nle_antisymm (translation_number_le_of_le_add_int f $ le_of_eq h)\n  (le_translation_number_of_add_int_le f $ le_of_eq h.symm)\n\nlemma floor_sub_le_translation_number (x : \u211d) : \u2191\u230af x - x\u230b \u2264 \u03c4 f :=\nle_translation_number_of_add_int_le f $ le_sub_iff_add_le'.1 (floor_le $ f x - x)\n\nlemma translation_number_le_ceil_sub (x : \u211d) : \u03c4 f \u2264 \u2308f x - x\u2309 :=\ntranslation_number_le_of_le_add_int f $ sub_le_iff_le_add'.1 (le_ceil $ f x - x)\n\nlemma map_lt_of_translation_number_lt_int {n : \u2124} (h : \u03c4 f < n) (x : \u211d) : f x < x + n :=\nnot_le.1 $ mt f.le_translation_number_of_add_int_le $ not_le.2 h\n\nlemma map_lt_of_translation_number_lt_nat {n : \u2115} (h : \u03c4 f < n) (x : \u211d) : f x < x + n :=\n@map_lt_of_translation_number_lt_int f n h x\n\nlemma map_lt_add_floor_translation_number_add_one (x : \u211d) : f x < x + \u230a\u03c4 f\u230b + 1 :=\nbegin\n  rw [add_assoc],\n  norm_cast,\n  refine map_lt_of_translation_number_lt_int _ _ _,\n  push_cast,\n  exact lt_floor_add_one _\nend\n\nlemma map_lt_add_translation_number_add_one (x : \u211d) : f x < x + \u03c4 f + 1 :=\ncalc f x < x + \u230a\u03c4 f\u230b + 1 : f.map_lt_add_floor_translation_number_add_one x\n... \u2264 x + \u03c4 f + 1 : by { mono*, exact floor_le (\u03c4 f) }\n\nlemma lt_map_of_int_lt_translation_number {n : \u2124} (h : \u2191n < \u03c4 f) (x : \u211d) : x + n < f x :=\nnot_le.1 $ mt f.translation_number_le_of_le_add_int $ not_le.2 h\n\nlemma lt_map_of_nat_lt_translation_number {n : \u2115} (h : \u2191n < \u03c4 f) (x : \u211d) : x + n < f x :=\n@lt_map_of_int_lt_translation_number f n h x\n\n\/-- If `f^n x - x`, `n > 0`, is an integer number `m` for some point `x`, then\n`\u03c4 f = m \/ n`. On the circle this means that a map with a periodic orbit has\na rational rotation number. -\/\nlemma translation_number_of_map_pow_eq_add_int {x : \u211d} {n : \u2115} {m : \u2124}\n  (h : (f^n) x = x + m) (hn : 0 < n) :\n  \u03c4 f = m \/ n :=\nbegin\n  have := (f^n).translation_number_of_eq_add_int h,\n  rwa [translation_number_pow, mul_comm, \u2190 eq_div_iff] at this,\n  exact nat.cast_ne_zero.2 (ne_of_gt hn)\nend\n\n\/-- If a predicate depends only on `f x - x` and holds for all `0 \u2264 x \u2264 1`,\nthen it holds for all `x`. -\/\nlemma forall_map_sub_of_Icc (P : \u211d \u2192 Prop)\n  (h : \u2200 x \u2208 Icc (0:\u211d) 1, P (f x - x)) (x : \u211d) : P (f x - x) :=\nf.map_fract_sub_fract_eq x \u25b8 h _ \u27e8fract_nonneg _, le_of_lt (fract_lt_one _)\u27e9\n\nlemma translation_number_lt_of_forall_lt_add (hf : continuous f) {z : \u211d}\n  (hz : \u2200 x, f x < x + z) : \u03c4 f < z :=\nbegin\n  obtain \u27e8x, xmem, hx\u27e9 : \u2203 x \u2208 Icc (0:\u211d) 1, \u2200 y \u2208 Icc (0:\u211d) 1, f y - y \u2264 f x - x,\n    from compact_Icc.exists_forall_ge (nonempty_Icc.2 zero_le_one)\n      (hf.sub continuous_id).continuous_on,\n  refine lt_of_le_of_lt _ (sub_lt_iff_lt_add'.2 $ hz x),\n  apply translation_number_le_of_le_add,\n  simp only [\u2190 sub_le_iff_le_add'],\n  exact f.forall_map_sub_of_Icc (\u03bb a, a \u2264 f x - x) hx\nend\n\nlemma lt_translation_number_of_forall_add_lt (hf : continuous f) {z : \u211d}\n  (hz : \u2200 x, x + z < f x) : z < \u03c4 f :=\nbegin\n  obtain \u27e8x, xmem, hx\u27e9 : \u2203 x \u2208 Icc (0:\u211d) 1, \u2200 y \u2208 Icc (0:\u211d) 1, f x - x \u2264 f y - y,\n    from compact_Icc.exists_forall_le (nonempty_Icc.2 zero_le_one)\n      (hf.sub continuous_id).continuous_on,\n  refine lt_of_lt_of_le (lt_sub_iff_add_lt'.2 $ hz x) _,\n  apply le_translation_number_of_add_le,\n  simp only [\u2190 le_sub_iff_add_le'],\n  exact f.forall_map_sub_of_Icc _ hx\nend\n\n\/-- If `f` is a continuous monotone map `\u211d \u2192 \u211d`, `f (x + 1) = f x + 1`, then there exists `x`\nsuch that `f x = x + \u03c4 f`. -\/\nlemma exists_eq_add_translation_number (hf : continuous f) :\n  \u2203 x, f x = x + \u03c4 f :=\nbegin\n  obtain \u27e8a, ha\u27e9 : \u2203 x, f x \u2264 x + f.translation_number,\n  { by_contradiction H,\n    push_neg at H,\n    exact lt_irrefl _ (f.lt_translation_number_of_forall_add_lt hf H) },\n  obtain \u27e8b, hb\u27e9 : \u2203 x, x + \u03c4 f \u2264 f x,\n  { by_contradiction H,\n    push_neg at H,\n    exact lt_irrefl _ (f.translation_number_lt_of_forall_lt_add hf H) },\n  exact intermediate_value_univ\u2082 hf (continuous_id.add continuous_const) ha hb\nend\n\nlemma translation_number_eq_int_iff (hf : continuous f) {m : \u2124} :\n  \u03c4 f = m \u2194 \u2203 x, f x = x + m :=\nbegin\n  refine \u27e8\u03bb h, h \u25b8 f.exists_eq_add_translation_number hf, _\u27e9,\n  rintros \u27e8x, hx\u27e9,\n  exact f.translation_number_of_eq_add_int hx\nend\n\nlemma continuous_pow (hf : continuous f) (n : \u2115) :\n  continuous \u21d1(f^n : circle_deg1_lift) :=\nby { rw coe_pow, exact hf.iterate n }\n\nlemma translation_number_eq_rat_iff (hf : continuous f) {m : \u2124}\n  {n : \u2115} (hn : 0 < n) :\n  \u03c4 f = m \/ n \u2194 \u2203 x, (f^n) x = x + m :=\nbegin\n  rw [eq_div_iff, mul_comm, \u2190 translation_number_pow]; [skip, exact ne_of_gt (nat.cast_pos.2 hn)],\n  exact (f^n).translation_number_eq_int_iff (f.continuous_pow hf n)\nend\n\n\/-- Consider two actions `f\u2081 f\u2082 : G \u2192* circle_deg1_lift` of a group on the real line by lifts of\norientation preserving circle homeomorphisms. Suppose that for each `g : G` the homeomorphisms\n`f\u2081 g` and `f\u2082 g` have equal rotation numbers. Then there exists `F : circle_deg1_lift`  such that\n`F * f\u2081 g = f\u2082 g * F` for all `g : G`.\n\nThis is a version of Proposition 5.4 from [\u00c9tienne Ghys, Groupes d'homeomorphismes du cercle et\ncohomologie bornee][ghys87:groupes]. -\/\nlemma semiconj_of_group_action_of_forall_translation_number_eq\n  {G : Type*} [group G] (f\u2081 f\u2082 : G \u2192* circle_deg1_lift)\n  (h : \u2200 g, \u03c4 (f\u2081 g) = \u03c4 (f\u2082 g)) :\n  \u2203 F : circle_deg1_lift, \u2200 g, semiconj F (f\u2081 g) (f\u2082 g) :=\nbegin\n  -- Equality of translation number guarantees that for each `x`\n  -- the set `{f\u2082 g\u207b\u00b9 (f\u2081 g x) | g : G}` is bounded above.\n  have : \u2200 x, bdd_above (range $ \u03bb g, f\u2082 g\u207b\u00b9 (f\u2081 g x)),\n  { refine \u03bb x, \u27e8x + 2, _\u27e9,\n    rintro _ \u27e8g, rfl\u27e9,\n    have : \u03c4 (f\u2082 g\u207b\u00b9) = -\u03c4 (f\u2082 g),\n      by rw [\u2190 monoid_hom.coe_to_hom_units, monoid_hom.map_inv,\n        translation_number_units_inv, monoid_hom.coe_to_hom_units],\n    calc f\u2082 g\u207b\u00b9 (f\u2081 g x) \u2264 f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2081 g) + 1) :\n      mono _ (map_lt_add_translation_number_add_one _ _).le\n    ... = f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2082 g)) + 1 :\n      by rw [h, map_add_one]\n    ... \u2264 x + \u03c4 (f\u2082 g) + \u03c4 (f\u2082 g\u207b\u00b9) + 1 + 1 :\n      by { mono, exact (map_lt_add_translation_number_add_one _ _).le }\n    ... = x + 2 : by simp [this, bit0, add_assoc] },\n  -- We have a theorem about actions by `order_iso`, so we introduce auxiliary maps\n  -- to `\u211d \u2243o \u211d`.\n  set F\u2081 := to_order_iso.comp f\u2081.to_hom_units,\n  set F\u2082 := to_order_iso.comp f\u2082.to_hom_units,\n  have hF\u2081 : \u2200 g, \u21d1(F\u2081 g) = f\u2081 g := \u03bb _, rfl,\n  have hF\u2082 : \u2200 g, \u21d1(F\u2082 g) = f\u2082 g := \u03bb _, rfl,\n  simp only [\u2190 hF\u2081, \u2190 hF\u2082],\n  -- Now we apply `cSup_div_semiconj` and go back to `f\u2081` and `f\u2082`.\n  refine \u27e8\u27e8_, \u03bb x y hxy, _, \u03bb x, _\u27e9, cSup_div_semiconj F\u2082 F\u2081 (\u03bb x, _)\u27e9;\n    simp only [hF\u2081, hF\u2082, \u2190 monoid_hom.map_inv, coe_mk],\n  { refine csupr_le_csupr (this y) (\u03bb g, _),\n    exact mono _ (mono _ hxy) },\n  { simp only [map_add_one],\n    exact (map_csupr_of_continuous_at_of_monotone (continuous_at_id.add continuous_at_const)\n      (monotone_id.add_const (1 : \u211d)) (this x)).symm },\n  { exact this x }\nend\n\n\/-- If two lifts of circle homeomorphisms have the same translation number, then they are\nsemiconjugate by a `circle_deg1_lift`. This version uses arguments `f\u2081 f\u2082 : units circle_deg1_lift`\nto assume that `f\u2081` and `f\u2082` are homeomorphisms. -\/\nlemma units_semiconj_of_translation_number_eq {f\u2081 f\u2082 : units circle_deg1_lift}\n  (h : \u03c4 f\u2081 = \u03c4 f\u2082) :\n  \u2203 F : circle_deg1_lift, semiconj F f\u2081 f\u2082 :=\nbegin\n  have : \u2200 n : multiplicative \u2124, \u03c4 ((units.coe_hom _).comp (gpowers_hom _ f\u2081) n) =\n    \u03c4 ((units.coe_hom _).comp (gpowers_hom _ f\u2082) n),\n  { intro n, simp [h] },\n  exact (semiconj_of_group_action_of_forall_translation_number_eq _ _ this).imp\n    (\u03bb F hF, hF (multiplicative.of_add 1))\nend\n\n\/-- If two lifts of circle homeomorphisms have the same translation number, then they are\nsemiconjugate by a `circle_deg1_lift`. This version uses assumptions `is_unit f\u2081` and `is_unit f\u2082`\nto assume that `f\u2081` and `f\u2082` are homeomorphisms. -\/\nlemma semiconj_of_is_unit_of_translation_number_eq {f\u2081 f\u2082 : circle_deg1_lift}\n  (h\u2081 : is_unit f\u2081) (h\u2082 : is_unit f\u2082) (h : \u03c4 f\u2081 = \u03c4 f\u2082) :\n  \u2203 F : circle_deg1_lift, semiconj F f\u2081 f\u2082 :=\nby { rcases \u27e8h\u2081, h\u2082\u27e9 with \u27e8\u27e8f\u2081, rfl\u27e9, \u27e8f\u2082, rfl\u27e9\u27e9, exact units_semiconj_of_translation_number_eq h }\n\n\/-- If two lifts of circle homeomorphisms have the same translation number, then they are\nsemiconjugate by a `circle_deg1_lift`. This version uses assumptions `bijective f\u2081` and\n`bijective f\u2082` to assume that `f\u2081` and `f\u2082` are homeomorphisms. -\/\nlemma semiconj_of_bijective_of_translation_number_eq {f\u2081 f\u2082 : circle_deg1_lift}\n  (h\u2081 : bijective f\u2081) (h\u2082 : bijective f\u2082) (h : \u03c4 f\u2081 = \u03c4 f\u2082) :\n  \u2203 F : circle_deg1_lift, semiconj F f\u2081 f\u2082 :=\nsemiconj_of_is_unit_of_translation_number_eq\n  (is_unit_iff_bijective.2 h\u2081) (is_unit_iff_bijective.2 h\u2082) h\n\nend circle_deg1_lift\n","avg_line_length":42.8446712018,"max_line_length":100,"alphanum_fraction":0.6749583212}
-{"size":9806,"ext":"lean","lang":"Lean","max_stars_count":1538.0,"content":"\/-\nCopyright (c) 2020 Marc Huisinga. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Marc Huisinga, Wojciech Nawrocki\n-\/\nimport Init.Control\nimport Init.System.IO\nimport Std.Data.RBTree\nimport Lean.Data.Json\n\n\/-! Implementation of JSON-RPC 2.0 (https:\/\/www.jsonrpc.org\/specification)\nfor use in the LSP server. -\/\n\nnamespace Lean.JsonRpc\n\nopen Json\nopen Std (RBNode)\n\ninductive RequestID where\n  | str (s : String)\n  | num (n : JsonNumber)\n  | null\n  deriving Inhabited, BEq, Ord\n\ninstance : ToString RequestID where\n  toString\n  | RequestID.str s => s!\"\\\"s\\\"\"\n  | RequestID.num n => toString n\n  | RequestID.null => \"null\"\n\n\/-- Error codes defined by JSON-RPC and LSP. -\/\ninductive ErrorCode where\n  | parseError\n  | invalidRequest\n  | methodNotFound\n  | invalidParams\n  | internalError\n  | serverNotInitialized\n  | unknownErrorCode\n  -- LSP-specific codes below.\n  | contentModified\n  | requestCancelled\n  -- Lean-specific codes below.\n  | rpcNeedsReconnect\n  deriving Inhabited, BEq\n\ninstance : FromJson ErrorCode := \u27e8fun\n  | num (-32700 : Int) => ErrorCode.parseError\n  | num (-32600 : Int) => ErrorCode.invalidRequest\n  | num (-32601 : Int) => ErrorCode.methodNotFound\n  | num (-32602 : Int) => ErrorCode.invalidParams\n  | num (-32603 : Int) => ErrorCode.internalError\n  | num (-32002 : Int) => ErrorCode.serverNotInitialized\n  | num (-32001 : Int) => ErrorCode.unknownErrorCode\n  | num (-32801 : Int) => ErrorCode.contentModified\n  | num (-32800 : Int) => ErrorCode.requestCancelled\n  | num (-32900 : Int) => ErrorCode.rpcNeedsReconnect\n  | _  => throw \"expected error code\"\u27e9\n\ninstance : ToJson ErrorCode := \u27e8fun\n  | ErrorCode.parseError           => (-32700 : Int)\n  | ErrorCode.invalidRequest       => (-32600 : Int)\n  | ErrorCode.methodNotFound       => (-32601 : Int)\n  | ErrorCode.invalidParams        => (-32602 : Int)\n  | ErrorCode.internalError        => (-32603 : Int)\n  | ErrorCode.serverNotInitialized => (-32002 : Int)\n  | ErrorCode.unknownErrorCode     => (-32001 : Int)\n  | ErrorCode.contentModified      => (-32801 : Int)\n  | ErrorCode.requestCancelled     => (-32800 : Int)\n  | ErrorCode.rpcNeedsReconnect    => (-32900 : Int)\u27e9\n\n\/- Uses separate constructors for notifications and errors because client and server\nbehavior is expected to be wildly different for both. -\/\ninductive Message where\n  | request (id : RequestID) (method : String) (params? : Option Structured)\n  | notification (method : String) (params? : Option Structured)\n  | response (id : RequestID) (result : Json)\n  | responseError (id : RequestID) (code : ErrorCode) (message : String) (data? : Option Json)\n\ndef Batch := Array Message\n\n-- Compound type with simplified APIs for passing around\n-- jsonrpc data\nstructure Request (\u03b1 : Type u) where\n  id     : RequestID\n  method : String\n  param  : \u03b1\n  deriving Inhabited, BEq\n\ninstance [ToJson \u03b1] : Coe (Request \u03b1) Message :=\n  \u27e8fun r => Message.request r.id r.method (toStructured? r.param).toOption\u27e9\n\nstructure Notification (\u03b1 : Type u) where\n  method : String\n  param  : \u03b1\n  deriving Inhabited, BEq\n\ninstance [ToJson \u03b1] : Coe (Notification \u03b1) Message :=\n  \u27e8fun r => Message.notification r.method (toStructured? r.param).toOption\u27e9\n\nstructure Response (\u03b1 : Type u) where\n  id     : RequestID\n  result : \u03b1\n  deriving Inhabited, BEq\n\ninstance [ToJson \u03b1] : Coe (Response \u03b1) Message :=\n  \u27e8fun r => Message.response r.id (toJson r.result)\u27e9\n\nstructure ResponseError (\u03b1 : Type u) where\n  id      : RequestID\n  code    : ErrorCode\n  message : String\n  data?   : Option \u03b1 := none\n  deriving Inhabited, BEq\n\ninstance [ToJson \u03b1] : Coe (ResponseError \u03b1) Message :=\n  \u27e8fun r => Message.responseError r.id r.code r.message (r.data?.map toJson)\u27e9\n\ninstance : Coe String RequestID := \u27e8RequestID.str\u27e9\ninstance : Coe JsonNumber RequestID := \u27e8RequestID.num\u27e9\n\nprivate def RequestID.lt : RequestID \u2192 RequestID \u2192 Bool\n  | RequestID.str a, RequestID.str b            => a < b\n  | RequestID.num a, RequestID.num b            => a < b\n  | RequestID.null,  RequestID.num _            => true\n  | RequestID.null,  RequestID.str _            => true\n  | RequestID.num _, RequestID.str _            => true\n  | _, _ \/- str < *, num < null, null < null -\/ => false\n\nprivate def RequestID.ltProp : LT RequestID :=\n  \u27e8fun a b => RequestID.lt a b = true\u27e9\n\ninstance : LT RequestID :=\n  RequestID.ltProp\n\ninstance (a b : RequestID) : Decidable (a < b) :=\n  inferInstanceAs (Decidable (RequestID.lt a b = true))\n\ninstance : FromJson RequestID := \u27e8fun j =>\n  match j with\n  | str s => RequestID.str s\n  | num n => RequestID.num n\n  | _     => throw \"a request id needs to be a number or a string\"\u27e9\n\ninstance : ToJson RequestID := \u27e8fun rid =>\n  match rid with\n  | RequestID.str s => s\n  | RequestID.num n => num n\n  | RequestID.null  => null\u27e9\n\ninstance : ToJson Message := \u27e8fun m =>\n  mkObj $ \u27e8\"jsonrpc\", \"2.0\"\u27e9 :: match m with\n  | Message.request id method params? =>\n    [ \u27e8\"id\", toJson id\u27e9,\n      \u27e8\"method\", method\u27e9\n    ] ++ opt \"params\" params?\n  | Message.notification method params? =>\n    \u27e8\"method\", method\u27e9 ::\n    opt \"params\" params?\n  | Message.response id result =>\n    [ \u27e8\"id\", toJson id\u27e9,\n      \u27e8\"result\", result\u27e9]\n  | Message.responseError id code message data? =>\n    [ \u27e8\"id\", toJson id\u27e9,\n      \u27e8\"error\", mkObj $ [\n          \u27e8\"code\", toJson code\u27e9,\n          \u27e8\"message\", message\u27e9\n        ] ++ opt \"data\" data?\u27e9\n    ]\u27e9\n\ninstance : FromJson Message where\n  fromJson? j := do\n    let \"2.0\" \u2190 j.getObjVal? \"jsonrpc\" | throw \"only version 2.0 of JSON RPC is supported\"\n    (do let id \u2190 j.getObjValAs? RequestID \"id\"\n        let method \u2190 j.getObjValAs? String \"method\"\n        let params? := j.getObjValAs? Structured \"params\"\n        pure (Message.request id method params?.toOption)) <|>\n    (do let method \u2190 j.getObjValAs? String \"method\"\n        let params? := j.getObjValAs? Structured \"params\"\n        pure (Message.notification method params?.toOption)) <|>\n    (do let id \u2190 j.getObjValAs? RequestID \"id\"\n        let result \u2190 j.getObjVal? \"result\"\n        pure (Message.response id result)) <|>\n    (do let id \u2190 j.getObjValAs? RequestID \"id\"\n        let err \u2190 j.getObjVal? \"error\"\n        let code \u2190 err.getObjValAs? ErrorCode \"code\"\n        let message \u2190 err.getObjValAs? String \"message\"\n        let data? := err.getObjVal? \"data\"\n        pure (Message.responseError id code message data?.toOption))\n\n-- TODO(WN): temporary until we have deriving FromJson\ninstance [FromJson \u03b1] : FromJson (Notification \u03b1) where\n  fromJson? j := do\n    let msg : Message \u2190 fromJson? j\n    if let Message.notification method params? := msg then\n      let params \u2190 params?\n      let param : \u03b1 \u2190 fromJson? (toJson params)\n      pure $ \u27e8method, param\u27e9\n    else throw \"not a notfication\"\n\nend Lean.JsonRpc\n\nnamespace IO.FS.Stream\n\nopen Lean\nopen Lean.JsonRpc\n\nsection\n  def readMessage (h : FS.Stream) (nBytes : Nat) : IO Message := do\n    let j \u2190 h.readJson nBytes\n    match fromJson? j with\n    | Except.ok m => pure m\n    | Except.error inner => throw $ userError s!\"JSON '{j.compress}' did not have the format of a JSON-RPC message.\\n{inner}\"\n\n  def readRequestAs (h : FS.Stream) (nBytes : Nat) (expectedMethod : String) (\u03b1) [FromJson \u03b1] : IO (Request \u03b1) := do\n    let m \u2190 h.readMessage nBytes\n    match m with\n    | Message.request id method params? =>\n      if method = expectedMethod then\n        let j := toJson params?\n        match fromJson? j with\n        | Except.ok v => pure \u27e8id, expectedMethod, v\u27e9\n        | Except.error inner => throw $ userError s!\"Unexpected param '{j.compress}' for method '{expectedMethod}'\\n{inner}\"\n      else\n        throw $ userError s!\"Expected method '{expectedMethod}', got method '{method}'\"\n    | _ => throw $ userError s!\"Expected JSON-RPC request, got: '{(toJson m).compress}'\"\n\n  def readNotificationAs (h : FS.Stream) (nBytes : Nat) (expectedMethod : String) (\u03b1) [FromJson \u03b1] : IO (Notification \u03b1) := do\n    let m \u2190 h.readMessage nBytes\n    match m with\n    | Message.notification method params? =>\n      if method = expectedMethod then\n        let j := toJson params?\n        match fromJson? j with\n        | Except.ok v => pure \u27e8expectedMethod, v\u27e9\n        | Except.error inner => throw $ userError s!\"Unexpected param '{j.compress}' for method '{expectedMethod}'\\n{inner}\"\n      else\n        throw $ userError s!\"Expected method '{expectedMethod}', got method '{method}'\"\n    | _ => throw $ userError s!\"Expected JSON-RPC notification, got: '{(toJson m).compress}'\"\n\n  partial def readResponseAs (h : FS.Stream) (nBytes : Nat) (expectedID : RequestID) (\u03b1) [FromJson \u03b1] : IO (Response \u03b1) := do\n    let m \u2190 h.readMessage nBytes\n    match m with\n    | Message.response id result =>\n      if id == expectedID then\n        match fromJson? result with\n        | Except.ok v => pure \u27e8expectedID, v\u27e9\n        | Except.error inner => throw $ userError s!\"Unexpected result '{result.compress}'\\n{inner}\"\n      else\n        throw $ userError s!\"Expected id {expectedID}, got id {id}\"\n    | Message.notification .. => readResponseAs h nBytes expectedID \u03b1\n    | _ => throw $ userError s!\"Expected JSON-RPC response, got: '{(toJson m).compress}'\"\nend\n\nsection\n  variable [ToJson \u03b1]\n\n  def writeMessage (h : FS.Stream) (m : Message) : IO Unit :=\n    h.writeJson (toJson m)\n\n  def writeRequest (h : FS.Stream) (r : Request \u03b1) : IO Unit :=\n    h.writeMessage r\n\n  def writeNotification (h : FS.Stream) (n : Notification \u03b1) : IO Unit :=\n    h.writeMessage n\n\n  def writeResponse (h : FS.Stream) (r : Response \u03b1) : IO Unit :=\n    h.writeMessage r\n\n  def writeResponseError (h : FS.Stream) (e : ResponseError Unit) : IO Unit :=\n    h.writeMessage (Message.responseError e.id e.code e.message none)\n\n  def writeResponseErrorWithData (h : FS.Stream) (e : ResponseError \u03b1) : IO Unit :=\n    h.writeMessage e\nend\n\nend IO.FS.Stream\n","avg_line_length":35.146953405,"max_line_length":126,"alphanum_fraction":0.6521517438}
-{"size":45302,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes H\u00f6lzl\n-\/\nimport algebra.abs\nimport algebra.order.sub\nimport order.order_dual\n\n\/-!\n# Ordered groups\n\nThis file develops the basics of ordered groups.\n\n## Implementation details\n\nUnfortunately, the number of `'` appended to lemmas in this file\nmay differ between the multiplicative and the additive version of a lemma.\nThe reason is that we did not want to change existing names in the library.\n-\/\n\nset_option old_structure_cmd true\nopen function\n\nuniverse u\nvariable {\u03b1 : Type u}\n\n@[to_additive]\ninstance group.covariant_class_le.to_contravariant_class_le\n  [group \u03b1] [has_le \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] : contravariant_class \u03b1 \u03b1 (*) (\u2264) :=\ngroup.covconv\n\n@[to_additive]\ninstance group.swap.covariant_class_le.to_contravariant_class_le [group \u03b1] [has_le \u03b1]\n  [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] : contravariant_class \u03b1 \u03b1 (swap (*)) (\u2264) :=\n{ elim := \u03bb a b c bc, calc  b = b * a * a\u207b\u00b9 : eq_mul_inv_of_mul_eq rfl\n                          ... \u2264 c * a * a\u207b\u00b9 : mul_le_mul_right' bc a\u207b\u00b9\n                          ... = c           : mul_inv_eq_of_eq_mul rfl }\n\n@[to_additive]\ninstance group.covariant_class_lt.to_contravariant_class_lt\n  [group \u03b1] [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] : contravariant_class \u03b1 \u03b1 (*) (<) :=\n{ elim := \u03bb a b c bc, calc  b = a\u207b\u00b9 * (a * b) : eq_inv_mul_of_mul_eq rfl\n                          ... < a\u207b\u00b9 * (a * c) : mul_lt_mul_left' bc a\u207b\u00b9\n                          ... = c             : inv_mul_cancel_left a c }\n\n@[to_additive]\ninstance group.swap.covariant_class_lt.to_contravariant_class_lt [group \u03b1] [has_lt \u03b1]\n  [covariant_class \u03b1 \u03b1 (swap (*)) (<)] : contravariant_class \u03b1 \u03b1 (swap (*)) (<) :=\n{ elim := \u03bb a b c bc, calc  b = b * a * a\u207b\u00b9 : eq_mul_inv_of_mul_eq rfl\n                          ... < c * a * a\u207b\u00b9 : mul_lt_mul_right' bc a\u207b\u00b9\n                          ... = c           : mul_inv_eq_of_eq_mul rfl }\n\n\/-- An ordered additive commutative group is an additive commutative group\nwith a partial order in which addition is strictly monotone. -\/\n@[protect_proj, ancestor add_comm_group partial_order]\nclass ordered_add_comm_group (\u03b1 : Type u) extends add_comm_group \u03b1, partial_order \u03b1 :=\n(add_le_add_left : \u2200 a b : \u03b1, a \u2264 b \u2192 \u2200 c : \u03b1, c + a \u2264 c + b)\n\n\/-- An ordered commutative group is an commutative group\nwith a partial order in which multiplication is strictly monotone. -\/\n@[protect_proj, ancestor comm_group partial_order]\nclass ordered_comm_group (\u03b1 : Type u) extends comm_group \u03b1, partial_order \u03b1 :=\n(mul_le_mul_left : \u2200 a b : \u03b1, a \u2264 b \u2192 \u2200 c : \u03b1, c * a \u2264 c * b)\nattribute [to_additive] ordered_comm_group\n\n@[to_additive]\ninstance ordered_comm_group.to_covariant_class_left_le (\u03b1 : Type u) [ordered_comm_group \u03b1] :\n  covariant_class \u03b1 \u03b1 (*) (\u2264) :=\n{ elim := \u03bb a b c bc, ordered_comm_group.mul_le_mul_left b c bc a }\n\n\/--The units of an ordered commutative monoid form an ordered commutative group. -\/\n@[to_additive]\ninstance units.ordered_comm_group [ordered_comm_monoid \u03b1] : ordered_comm_group \u03b1\u02e3 :=\n{ mul_le_mul_left := \u03bb a b h c, (mul_le_mul_left' (h : (a : \u03b1) \u2264 b) _ :  (c : \u03b1) * a \u2264 c * b),\n  .. units.partial_order,\n  .. units.comm_group }\n\n@[priority 100, to_additive]    -- see Note [lower instance priority]\ninstance ordered_comm_group.to_ordered_cancel_comm_monoid (\u03b1 : Type u)\n  [s : ordered_comm_group \u03b1] :\n  ordered_cancel_comm_monoid \u03b1 :=\n{ mul_left_cancel       := \u03bb a b c, (mul_right_inj a).mp,\n  le_of_mul_le_mul_left := \u03bb a b c, (mul_le_mul_iff_left a).mp,\n  ..s }\n\n@[priority 100, to_additive]\ninstance ordered_comm_group.has_exists_mul_of_le (\u03b1 : Type u)\n  [ordered_comm_group \u03b1] :\n  has_exists_mul_of_le \u03b1 :=\n\u27e8\u03bb a b hab, \u27e8b * a\u207b\u00b9, (mul_inv_cancel_comm_assoc a b).symm\u27e9\u27e9\n\n@[to_additive] instance [h : has_inv \u03b1] : has_inv (order_dual \u03b1) := h\n@[to_additive] instance [h : has_div \u03b1] : has_div (order_dual \u03b1) := h\n@[to_additive] instance [h : div_inv_monoid \u03b1] : div_inv_monoid (order_dual \u03b1) := h\n@[to_additive] instance [h : group \u03b1] : group (order_dual \u03b1) := h\n@[to_additive] instance [h : comm_group \u03b1] : comm_group (order_dual \u03b1) := h\n\n@[to_additive] instance [ordered_comm_group \u03b1] : ordered_comm_group (order_dual \u03b1) :=\n{ .. order_dual.ordered_comm_monoid, .. order_dual.group }\n\nsection group\nvariables [group \u03b1]\n\nsection typeclasses_left_le\nvariables [has_le \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] {a b c d : \u03b1}\n\n\/--  Uses `left` co(ntra)variant. -\/\n@[simp, to_additive left.neg_nonpos_iff]\nlemma left.inv_le_one_iff :\n  a\u207b\u00b9 \u2264 1 \u2194 1 \u2264 a :=\nby { rw [\u2190 mul_le_mul_iff_left a], simp }\n\n\/--  Uses `left` co(ntra)variant. -\/\n@[simp, to_additive left.nonneg_neg_iff]\nlemma left.one_le_inv_iff :\n  1 \u2264 a\u207b\u00b9 \u2194 a \u2264 1 :=\nby { rw [\u2190 mul_le_mul_iff_left a], simp }\n\n@[simp, to_additive]\nlemma le_inv_mul_iff_mul_le : b \u2264 a\u207b\u00b9 * c \u2194 a * b \u2264 c :=\nby { rw \u2190 mul_le_mul_iff_left a, simp }\n\n@[simp, to_additive]\nlemma inv_mul_le_iff_le_mul : b\u207b\u00b9 * a \u2264 c \u2194 a \u2264 b * c :=\nby rw [\u2190 mul_le_mul_iff_left b, mul_inv_cancel_left]\n\n@[to_additive neg_le_iff_add_nonneg']\nlemma inv_le_iff_one_le_mul' : a\u207b\u00b9 \u2264 b \u2194 1 \u2264 a * b :=\n(mul_le_mul_iff_left a).symm.trans $ by rw mul_inv_self\n\n@[to_additive]\nlemma le_inv_iff_mul_le_one_left : a \u2264 b\u207b\u00b9 \u2194 b * a \u2264 1 :=\n(mul_le_mul_iff_left b).symm.trans $ by rw mul_inv_self\n\n@[to_additive]\nlemma le_inv_mul_iff_le : 1 \u2264 b\u207b\u00b9 * a \u2194 b \u2264 a :=\nby rw [\u2190 mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]\n\n@[to_additive]\nlemma inv_mul_le_one_iff : a\u207b\u00b9 * b \u2264 1 \u2194 b \u2264 a :=\ntrans (inv_mul_le_iff_le_mul) $ by rw mul_one\n\nend typeclasses_left_le\n\nsection typeclasses_left_lt\nvariables [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] {a b c : \u03b1}\n\n\/--  Uses `left` co(ntra)variant. -\/\n@[simp, to_additive left.neg_pos_iff]\nlemma left.one_lt_inv_iff :\n  1 < a\u207b\u00b9 \u2194 a < 1 :=\nby rw [\u2190 mul_lt_mul_iff_left a, mul_inv_self, mul_one]\n\n\/--  Uses `left` co(ntra)variant. -\/\n@[simp, to_additive left.neg_neg_iff]\nlemma left.inv_lt_one_iff :\n  a\u207b\u00b9 < 1 \u2194 1 < a :=\nby rw [\u2190 mul_lt_mul_iff_left a, mul_inv_self, mul_one]\n\n@[simp, to_additive]\nlemma lt_inv_mul_iff_mul_lt : b < a\u207b\u00b9 * c \u2194 a * b < c :=\nby { rw [\u2190 mul_lt_mul_iff_left a], simp }\n\n@[simp, to_additive]\nlemma inv_mul_lt_iff_lt_mul : b\u207b\u00b9 * a < c \u2194 a < b * c :=\nby rw [\u2190 mul_lt_mul_iff_left b, mul_inv_cancel_left]\n\n@[to_additive]\nlemma inv_lt_iff_one_lt_mul' : a\u207b\u00b9 < b \u2194 1 < a * b :=\n(mul_lt_mul_iff_left a).symm.trans $ by rw mul_inv_self\n\n@[to_additive]\nlemma lt_inv_iff_mul_lt_one' : a < b\u207b\u00b9 \u2194 b * a < 1 :=\n(mul_lt_mul_iff_left b).symm.trans $ by rw mul_inv_self\n\n@[to_additive]\nlemma lt_inv_mul_iff_lt : 1 < b\u207b\u00b9 * a \u2194 b < a :=\nby rw [\u2190 mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]\n\n@[to_additive]\nlemma inv_mul_lt_one_iff : a\u207b\u00b9 * b < 1 \u2194 b < a :=\ntrans (inv_mul_lt_iff_lt_mul) $ by rw mul_one\n\nend typeclasses_left_lt\n\nsection typeclasses_right_le\nvariables [has_le \u03b1] [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] {a b c : \u03b1}\n\n\/--  Uses `right` co(ntra)variant. -\/\n@[simp, to_additive right.neg_nonpos_iff]\nlemma right.inv_le_one_iff :\n  a\u207b\u00b9 \u2264 1 \u2194 1 \u2264 a :=\nby { rw [\u2190 mul_le_mul_iff_right a], simp }\n\n\/--  Uses `right` co(ntra)variant. -\/\n@[simp, to_additive right.nonneg_neg_iff]\nlemma right.one_le_inv_iff :\n  1 \u2264 a\u207b\u00b9 \u2194 a \u2264 1 :=\nby { rw [\u2190 mul_le_mul_iff_right a], simp }\n\n@[to_additive neg_le_iff_add_nonneg]\nlemma inv_le_iff_one_le_mul : a\u207b\u00b9 \u2264 b \u2194 1 \u2264 b * a :=\n(mul_le_mul_iff_right a).symm.trans $ by rw inv_mul_self\n\n@[to_additive]\nlemma le_inv_iff_mul_le_one_right : a \u2264 b\u207b\u00b9 \u2194 a * b \u2264 1 :=\n(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_self\n\n@[simp, to_additive]\nlemma mul_inv_le_iff_le_mul : a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 c * b :=\n(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right\n\n@[simp, to_additive]\nlemma le_mul_inv_iff_mul_le : c \u2264 a * b\u207b\u00b9 \u2194 c * b \u2264 a :=\n(mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right\n\n@[simp, to_additive]\nlemma mul_inv_le_one_iff_le : a * b\u207b\u00b9 \u2264 1 \u2194 a \u2264 b :=\nmul_inv_le_iff_le_mul.trans $ by rw one_mul\n\n@[to_additive]\nlemma le_mul_inv_iff_le : 1 \u2264 a * b\u207b\u00b9 \u2194 b \u2264 a :=\nby rw [\u2190 mul_le_mul_iff_right b, one_mul, inv_mul_cancel_right]\n\n@[to_additive]\nlemma mul_inv_le_one_iff : b * a\u207b\u00b9 \u2264 1 \u2194 b \u2264 a :=\ntrans (mul_inv_le_iff_le_mul) $ by rw one_mul\n\nend typeclasses_right_le\n\nsection typeclasses_right_lt\nvariables [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (swap (*)) (<)] {a b c : \u03b1}\n\n\/--  Uses `right` co(ntra)variant. -\/\n@[simp, to_additive right.neg_neg_iff]\nlemma right.inv_lt_one_iff :\n  a\u207b\u00b9 < 1 \u2194 1 < a :=\nby rw [\u2190 mul_lt_mul_iff_right a, inv_mul_self, one_mul]\n\n\/--  Uses `right` co(ntra)variant. -\/\n@[simp, to_additive right.neg_pos_iff]\nlemma right.one_lt_inv_iff :\n  1 < a\u207b\u00b9 \u2194 a < 1 :=\nby rw [\u2190 mul_lt_mul_iff_right a, inv_mul_self, one_mul]\n\n@[to_additive]\nlemma inv_lt_iff_one_lt_mul : a\u207b\u00b9 < b \u2194 1 < b * a :=\n(mul_lt_mul_iff_right a).symm.trans $ by rw inv_mul_self\n\n@[to_additive]\nlemma lt_inv_iff_mul_lt_one : a < b\u207b\u00b9 \u2194 a * b < 1 :=\n(mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_self\n\n@[simp, to_additive]\nlemma mul_inv_lt_iff_lt_mul : a * b\u207b\u00b9 < c \u2194 a < c * b :=\nby rw [\u2190 mul_lt_mul_iff_right b, inv_mul_cancel_right]\n\n@[simp, to_additive]\nlemma lt_mul_inv_iff_mul_lt : c < a * b\u207b\u00b9 \u2194 c * b < a :=\n(mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right\n\n@[simp, to_additive]\nlemma inv_mul_lt_one_iff_lt : a * b\u207b\u00b9 < 1 \u2194 a < b :=\nby rw [\u2190 mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]\n\n@[to_additive]\nlemma lt_mul_inv_iff_lt : 1 < a * b\u207b\u00b9 \u2194 b < a :=\nby rw [\u2190 mul_lt_mul_iff_right b, one_mul, inv_mul_cancel_right]\n\n@[to_additive]\nlemma mul_inv_lt_one_iff : b * a\u207b\u00b9 < 1 \u2194 b < a :=\ntrans (mul_inv_lt_iff_lt_mul) $ by rw one_mul\n\nend typeclasses_right_lt\n\nsection typeclasses_left_right_le\nvariables [has_le \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)]\n  {a b c d : \u03b1}\n\n@[simp, to_additive]\nlemma inv_le_inv_iff : a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a :=\nby { rw [\u2190 mul_le_mul_iff_left a, \u2190 mul_le_mul_iff_right b], simp }\n\nalias neg_le_neg_iff \u2194 le_of_neg_le_neg _\n\nsection\n\nvariable (\u03b1)\n\n\/-- `x \u21a6 x\u207b\u00b9` as an order-reversing equivalence. -\/\n@[to_additive \"`x \u21a6 -x` as an order-reversing equivalence.\", simps]\ndef order_iso.inv : \u03b1 \u2243o order_dual \u03b1 :=\n{ to_equiv := (equiv.inv \u03b1).trans order_dual.to_dual,\n  map_rel_iff' := \u03bb a b, @inv_le_inv_iff \u03b1 _ _ _ _ _ _ }\n\nend\n\n@[to_additive neg_le]\nlemma inv_le' : a\u207b\u00b9 \u2264 b \u2194 b\u207b\u00b9 \u2264 a :=\n(order_iso.inv \u03b1).symm_apply_le\n\nalias inv_le' \u2194 inv_le_of_inv_le' _\nattribute [to_additive neg_le_of_neg_le] inv_le_of_inv_le'\n\n@[to_additive le_neg]\nlemma le_inv' : a \u2264 b\u207b\u00b9 \u2194 b \u2264 a\u207b\u00b9 :=\n(order_iso.inv \u03b1).le_symm_apply\n\n@[to_additive]\nlemma mul_inv_le_inv_mul_iff : a * b\u207b\u00b9 \u2264 d\u207b\u00b9 * c \u2194 d * a \u2264 c * b :=\nby rw [\u2190 mul_le_mul_iff_left d, \u2190 mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,\n    inv_mul_cancel_right]\n\n@[simp, to_additive] lemma div_le_self_iff (a : \u03b1) {b : \u03b1} : a \/ b \u2264 a \u2194 1 \u2264 b :=\nby simp [div_eq_mul_inv]\n\n@[simp, to_additive] lemma le_div_self_iff (a : \u03b1) {b : \u03b1} : a \u2264 a \/ b \u2194 b \u2264 1 :=\nby simp [div_eq_mul_inv]\n\nalias sub_le_self_iff \u2194 _ sub_le_self\n\nend typeclasses_left_right_le\n\nsection typeclasses_left_right_lt\nvariables [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] [covariant_class \u03b1 \u03b1 (swap (*)) (<)]\n  {a b c d : \u03b1}\n\n@[simp, to_additive]\nlemma inv_lt_inv_iff : a\u207b\u00b9 < b\u207b\u00b9 \u2194 b < a :=\nby { rw [\u2190 mul_lt_mul_iff_left a, \u2190 mul_lt_mul_iff_right b], simp }\n\n@[to_additive neg_lt]\nlemma inv_lt' : a\u207b\u00b9 < b \u2194 b\u207b\u00b9 < a :=\nby rw [\u2190 inv_lt_inv_iff, inv_inv]\n\n@[to_additive lt_neg]\nlemma lt_inv' : a < b\u207b\u00b9 \u2194 b < a\u207b\u00b9 :=\nby rw [\u2190 inv_lt_inv_iff, inv_inv]\n\nalias lt_inv' \u2194 lt_inv_of_lt_inv _\nattribute [to_additive] lt_inv_of_lt_inv\n\nalias inv_lt' \u2194 inv_lt_of_inv_lt' _\nattribute [to_additive neg_lt_of_neg_lt] inv_lt_of_inv_lt'\n\n@[to_additive]\nlemma mul_inv_lt_inv_mul_iff : a * b\u207b\u00b9 < d\u207b\u00b9 * c \u2194 d * a < c * b :=\nby rw [\u2190 mul_lt_mul_iff_left d, \u2190 mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,\n    inv_mul_cancel_right]\n\n@[simp, to_additive] lemma div_lt_self_iff (a : \u03b1) {b : \u03b1} : a \/ b < a \u2194 1 < b :=\nby simp [div_eq_mul_inv]\n\nalias sub_lt_self_iff \u2194 _ sub_lt_self\n\nend typeclasses_left_right_lt\n\nsection pre_order\nvariable [preorder \u03b1]\n\nsection left_le\nvariables [covariant_class \u03b1 \u03b1 (*) (\u2264)] {a : \u03b1}\n\n@[to_additive]\nlemma left.inv_le_self (h : 1 \u2264 a) : a\u207b\u00b9 \u2264 a :=\nle_trans (left.inv_le_one_iff.mpr h) h\n\nalias left.neg_le_self \u2190 neg_le_self\n\n@[to_additive]\nlemma left.self_le_inv (h : a \u2264 1) : a \u2264 a\u207b\u00b9 :=\nle_trans h (left.one_le_inv_iff.mpr h)\n\nend left_le\n\nsection left_lt\nvariables [covariant_class \u03b1 \u03b1 (*) (<)] {a : \u03b1}\n\n@[to_additive]\nlemma left.inv_lt_self (h : 1 < a) : a\u207b\u00b9 < a :=\n(left.inv_lt_one_iff.mpr h).trans h\n\nalias left.neg_lt_self \u2190 neg_lt_self\n\n@[to_additive]\nlemma left.self_lt_inv (h : a < 1) : a < a\u207b\u00b9 :=\nlt_trans h (left.one_lt_inv_iff.mpr h)\n\nend left_lt\n\nsection right_le\nvariables [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] {a : \u03b1}\n\n@[to_additive]\nlemma right.inv_le_self (h : 1 \u2264 a) : a\u207b\u00b9 \u2264 a :=\nle_trans (right.inv_le_one_iff.mpr h) h\n\n@[to_additive]\nlemma right.self_le_inv (h : a \u2264 1) : a \u2264 a\u207b\u00b9 :=\nle_trans h (right.one_le_inv_iff.mpr h)\n\nend right_le\n\nsection right_lt\nvariables [covariant_class \u03b1 \u03b1 (swap (*)) (<)] {a : \u03b1}\n\n@[to_additive]\nlemma right.inv_lt_self (h : 1 < a) : a\u207b\u00b9 < a :=\n(right.inv_lt_one_iff.mpr h).trans h\n\n@[to_additive]\nlemma right.self_lt_inv (h : a < 1) : a < a\u207b\u00b9 :=\nlt_trans h (right.one_lt_inv_iff.mpr h)\n\nend right_lt\n\nend pre_order\n\nend group\n\nsection comm_group\nvariables [comm_group \u03b1]\n\nsection has_le\nvariables [has_le \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] {a b c d : \u03b1}\n\n@[to_additive]\nlemma inv_mul_le_iff_le_mul' : c\u207b\u00b9 * a \u2264 b \u2194 a \u2264 b * c :=\nby rw [inv_mul_le_iff_le_mul, mul_comm]\n\n@[simp, to_additive]\nlemma mul_inv_le_iff_le_mul' : a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 b * c :=\nby rw [\u2190 inv_mul_le_iff_le_mul, mul_comm]\n\n@[to_additive add_neg_le_add_neg_iff]\nlemma mul_inv_le_mul_inv_iff' : a * b\u207b\u00b9 \u2264 c * d\u207b\u00b9 \u2194 a * d \u2264 c * b :=\nby rw [mul_comm c, mul_inv_le_inv_mul_iff, mul_comm]\n\nend has_le\n\nsection has_lt\nvariables [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] {a b c d : \u03b1}\n\n@[to_additive]\nlemma inv_mul_lt_iff_lt_mul' : c\u207b\u00b9 * a < b \u2194 a < b * c :=\nby rw [inv_mul_lt_iff_lt_mul, mul_comm]\n\n@[simp, to_additive]\nlemma mul_inv_lt_iff_le_mul' : a * b\u207b\u00b9 < c \u2194 a < b * c :=\nby rw [\u2190 inv_mul_lt_iff_lt_mul, mul_comm]\n\n@[to_additive add_neg_lt_add_neg_iff]\nlemma mul_inv_lt_mul_inv_iff' : a * b\u207b\u00b9 < c * d\u207b\u00b9 \u2194 a * d < c * b :=\nby rw [mul_comm c, mul_inv_lt_inv_mul_iff, mul_comm]\n\nend has_lt\n\nend comm_group\n\nalias le_inv' \u2194 le_inv_of_le_inv _\nattribute [to_additive] le_inv_of_le_inv\n\nalias left.inv_le_one_iff \u2194 one_le_of_inv_le_one _\nattribute [to_additive] one_le_of_inv_le_one\n\nalias left.one_le_inv_iff \u2194 le_one_of_one_le_inv _\nattribute [to_additive nonpos_of_neg_nonneg] le_one_of_one_le_inv\n\nalias inv_lt_inv_iff \u2194 lt_of_inv_lt_inv _\nattribute [to_additive] lt_of_inv_lt_inv\n\nalias left.inv_lt_one_iff \u2194 one_lt_of_inv_lt_one _\nattribute [to_additive] one_lt_of_inv_lt_one\n\nalias left.inv_lt_one_iff \u2190 inv_lt_one_iff_one_lt\nattribute [to_additive] inv_lt_one_iff_one_lt\n\nalias left.inv_lt_one_iff \u2190 inv_lt_one'\nattribute [to_additive neg_lt_zero] inv_lt_one'\n\nalias left.one_lt_inv_iff \u2194  inv_of_one_lt_inv _\nattribute [to_additive neg_of_neg_pos] inv_of_one_lt_inv\n\nalias left.one_lt_inv_iff \u2194 _ one_lt_inv_of_inv\nattribute [to_additive neg_pos_of_neg] one_lt_inv_of_inv\n\nalias le_inv_mul_iff_mul_le \u2194 mul_le_of_le_inv_mul _\nattribute [to_additive] mul_le_of_le_inv_mul\n\nalias le_inv_mul_iff_mul_le \u2194 _ le_inv_mul_of_mul_le\nattribute [to_additive] le_inv_mul_of_mul_le\n\nalias inv_mul_le_iff_le_mul \u2194 _ inv_mul_le_of_le_mul\nattribute [to_additive] inv_mul_le_iff_le_mul\n\nalias lt_inv_mul_iff_mul_lt \u2194 mul_lt_of_lt_inv_mul _\nattribute [to_additive] mul_lt_of_lt_inv_mul\n\nalias lt_inv_mul_iff_mul_lt \u2194 _ lt_inv_mul_of_mul_lt\nattribute [to_additive] lt_inv_mul_of_mul_lt\n\nalias inv_mul_lt_iff_lt_mul \u2194 lt_mul_of_inv_mul_lt inv_mul_lt_of_lt_mul\nattribute [to_additive] lt_mul_of_inv_mul_lt\nattribute [to_additive] inv_mul_lt_of_lt_mul\n\nalias lt_mul_of_inv_mul_lt \u2190 lt_mul_of_inv_mul_lt_left\nattribute [to_additive] lt_mul_of_inv_mul_lt_left\n\nalias left.inv_le_one_iff \u2190 inv_le_one'\nattribute [to_additive neg_nonpos] inv_le_one'\n\nalias left.one_le_inv_iff \u2190 one_le_inv'\nattribute [to_additive neg_nonneg] one_le_inv'\n\nalias left.one_lt_inv_iff \u2190 one_lt_inv'\nattribute [to_additive neg_pos] one_lt_inv'\n\nalias mul_lt_mul_left' \u2190 ordered_comm_group.mul_lt_mul_left'\nattribute [to_additive ordered_add_comm_group.add_lt_add_left] ordered_comm_group.mul_lt_mul_left'\n\nalias le_of_mul_le_mul_left' \u2190 ordered_comm_group.le_of_mul_le_mul_left\nattribute [to_additive ordered_add_comm_group.le_of_add_le_add_left]\n  ordered_comm_group.le_of_mul_le_mul_left\n\nalias lt_of_mul_lt_mul_left' \u2190 ordered_comm_group.lt_of_mul_lt_mul_left\nattribute [to_additive ordered_add_comm_group.lt_of_add_lt_add_left]\n  ordered_comm_group.lt_of_mul_lt_mul_left\n\n\/-- Pullback an `ordered_comm_group` under an injective map.\nSee note [reducible non-instances]. -\/\n@[reducible, to_additive function.injective.ordered_add_comm_group\n\"Pullback an `ordered_add_comm_group` under an injective map.\"]\ndef function.injective.ordered_comm_group [ordered_comm_group \u03b1] {\u03b2 : Type*}\n  [has_one \u03b2] [has_mul \u03b2] [has_inv \u03b2] [has_div \u03b2]\n  (f : \u03b2 \u2192 \u03b1) (hf : function.injective f) (one : f 1 = 1)\n  (mul : \u2200 x y, f (x * y) = f x * f y)\n  (inv : \u2200 x, f (x\u207b\u00b9) = (f x)\u207b\u00b9)\n  (div : \u2200 x y, f (x \/ y) = f x \/ f y) :\n  ordered_comm_group \u03b2 :=\n{ ..partial_order.lift f hf,\n  ..hf.ordered_comm_monoid f one mul,\n  ..hf.comm_group f one mul inv div }\n\n\/-  Most of the lemmas that are primed in this section appear in ordered_field. -\/\n\/-  I (DT) did not try to minimise the assumptions. -\/\nsection group\nvariables [group \u03b1] [has_le \u03b1]\n\nsection right\nvariables [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] {a b c d : \u03b1}\n\n@[simp, to_additive]\nlemma div_le_div_iff_right (c : \u03b1) : a \/ c \u2264 b \/ c \u2194 a \u2264 b :=\nby simpa only [div_eq_mul_inv] using mul_le_mul_iff_right _\n\n@[to_additive sub_le_sub_right]\nlemma div_le_div_right' (h : a \u2264 b) (c : \u03b1) : a \/ c \u2264 b \/ c :=\n(div_le_div_iff_right c).2 h\n\n@[simp, to_additive sub_nonneg]\nlemma one_le_div' : 1 \u2264 a \/ b \u2194 b \u2264 a :=\nby rw [\u2190 mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias sub_nonneg \u2194 le_of_sub_nonneg sub_nonneg_of_le\n\n@[simp, to_additive sub_nonpos]\nlemma div_le_one' : a \/ b \u2264 1 \u2194 a \u2264 b :=\nby rw [\u2190 mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias sub_nonpos \u2194 le_of_sub_nonpos sub_nonpos_of_le\n\n@[to_additive]\nlemma le_div_iff_mul_le : a \u2264 c \/ b \u2194 a * b \u2264 c :=\nby rw [\u2190 mul_le_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias le_sub_iff_add_le \u2194 add_le_of_le_sub_right le_sub_right_of_add_le\n\n@[to_additive]\nlemma div_le_iff_le_mul : a \/ c \u2264 b \u2194 a \u2264 b * c :=\nby rw [\u2190 mul_le_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]\n\n-- TODO: Should we get rid of `sub_le_iff_le_add` in favor of\n-- (a renamed version of) `tsub_le_iff_right`?\n@[priority 100] -- see Note [lower instance priority]\ninstance add_group.to_has_ordered_sub {\u03b1 : Type*} [add_group \u03b1] [has_le \u03b1]\n  [covariant_class \u03b1 \u03b1 (swap (+)) (\u2264)] : has_ordered_sub \u03b1 :=\n\u27e8\u03bb a b c, sub_le_iff_le_add\u27e9\n\/-- `equiv.mul_right` as an `order_iso`. See also `order_embedding.mul_right`. -\/\n@[to_additive \"`equiv.add_right` as an `order_iso`. See also `order_embedding.add_right`.\",\n  simps to_equiv apply {simp_rhs := tt}]\ndef order_iso.mul_right (a : \u03b1) : \u03b1 \u2243o \u03b1 :=\n{ map_rel_iff' := \u03bb _ _, mul_le_mul_iff_right a, to_equiv := equiv.mul_right a }\n\n@[simp, to_additive] lemma order_iso.mul_right_symm (a : \u03b1) :\n  (order_iso.mul_right a).symm = order_iso.mul_right a\u207b\u00b9 :=\nby { ext x, refl }\n\nend right\n\nsection left\nvariables [covariant_class \u03b1 \u03b1 (*) (\u2264)]\n\n\/-- `equiv.mul_left` as an `order_iso`. See also `order_embedding.mul_left`. -\/\n@[to_additive \"`equiv.add_left` as an `order_iso`. See also `order_embedding.add_left`.\",\n  simps to_equiv apply  {simp_rhs := tt}]\ndef order_iso.mul_left (a : \u03b1) : \u03b1 \u2243o \u03b1 :=\n{ map_rel_iff' := \u03bb _ _, mul_le_mul_iff_left a, to_equiv := equiv.mul_left a }\n\n@[simp, to_additive] lemma order_iso.mul_left_symm (a : \u03b1) :\n  (order_iso.mul_left a).symm = order_iso.mul_left a\u207b\u00b9 :=\nby { ext x, refl }\n\nvariables [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] {a b c : \u03b1}\n\n@[simp, to_additive]\nlemma div_le_div_iff_left (a : \u03b1) : a \/ b \u2264 a \/ c \u2194 c \u2264 b :=\nby rw [div_eq_mul_inv, div_eq_mul_inv, \u2190 mul_le_mul_iff_left a\u207b\u00b9, inv_mul_cancel_left,\n    inv_mul_cancel_left, inv_le_inv_iff]\n\n@[to_additive sub_le_sub_left]\nlemma div_le_div_left' (h : a \u2264 b) (c : \u03b1) : c \/ b \u2264 c \/ a :=\n(div_le_div_iff_left c).2 h\n\nend left\n\nend group\n\nsection comm_group\nvariables [comm_group \u03b1]\n\nsection has_le\nvariables [has_le \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] {a b c d : \u03b1}\n\n@[to_additive sub_le_sub_iff]\nlemma div_le_div_iff' : a \/ b \u2264 c \/ d \u2194 a * d \u2264 c * b :=\nby simpa only [div_eq_mul_inv] using mul_inv_le_mul_inv_iff'\n\n@[to_additive]\nlemma le_div_iff_mul_le' : b \u2264 c \/ a \u2194 a * b \u2264 c :=\nby rw [le_div_iff_mul_le, mul_comm]\n\nalias le_sub_iff_add_le' \u2194 add_le_of_le_sub_left le_sub_left_of_add_le\n\n@[to_additive]\nlemma div_le_iff_le_mul' : a \/ b \u2264 c \u2194 a \u2264 b * c :=\nby rw [div_le_iff_le_mul, mul_comm]\n\nalias sub_le_iff_le_add' \u2194 le_add_of_sub_left_le sub_left_le_of_le_add\n\n@[simp, to_additive]\nlemma inv_le_div_iff_le_mul : b\u207b\u00b9 \u2264 a \/ c \u2194 c \u2264 a * b :=\nle_div_iff_mul_le.trans inv_mul_le_iff_le_mul'\n\n@[to_additive]\nlemma inv_le_div_iff_le_mul' : a\u207b\u00b9 \u2264 b \/ c \u2194 c \u2264 a * b :=\nby rw [inv_le_div_iff_le_mul, mul_comm]\n\n@[to_additive sub_le]\nlemma div_le'' : a \/ b \u2264 c \u2194 a \/ c \u2264 b :=\ndiv_le_iff_le_mul'.trans div_le_iff_le_mul.symm\n\n@[to_additive le_sub]\nlemma le_div'' : a \u2264 b \/ c \u2194 c \u2264 b \/ a :=\nle_div_iff_mul_le'.trans le_div_iff_mul_le.symm\n\nend has_le\n\nsection preorder\nvariables [preorder \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)] {a b c d : \u03b1}\n\n@[to_additive sub_le_sub]\nlemma div_le_div'' (hab : a \u2264 b) (hcd : c \u2264 d) :\n  a \/ d \u2264 b \/ c :=\nbegin\n  rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_le_inv_mul_iff, mul_comm],\n  exact mul_le_mul' hab hcd\nend\n\nend preorder\n\nend comm_group\n\n\/-  Most of the lemmas that are primed in this section appear in ordered_field. -\/\n\/-  I (DT) did not try to minimise the assumptions. -\/\nsection group\nvariables [group \u03b1] [has_lt \u03b1]\n\nsection right\nvariables [covariant_class \u03b1 \u03b1 (swap (*)) (<)] {a b c d : \u03b1}\n\n@[simp, to_additive]\nlemma div_lt_div_iff_right (c : \u03b1) : a \/ c < b \/ c \u2194 a < b :=\nby simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _\n\n@[to_additive sub_lt_sub_right]\nlemma div_lt_div_right' (h : a < b) (c : \u03b1) : a \/ c < b \/ c :=\n(div_lt_div_iff_right c).2 h\n\n@[simp, to_additive sub_pos]\nlemma one_lt_div' : 1 < a \/ b \u2194 b < a :=\nby rw [\u2190 mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias sub_pos \u2194 lt_of_sub_pos sub_pos_of_lt\n\n@[simp, to_additive sub_neg]\nlemma div_lt_one' : a \/ b < 1 \u2194 a < b :=\nby rw [\u2190 mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias sub_neg \u2194 lt_of_sub_neg sub_neg_of_lt\n\nalias sub_neg \u2190 sub_lt_zero\n\n@[to_additive]\nlemma lt_div_iff_mul_lt : a < c \/ b \u2194 a * b < c :=\nby rw [\u2190 mul_lt_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias lt_sub_iff_add_lt \u2194 add_lt_of_lt_sub_right lt_sub_right_of_add_lt\n\n@[to_additive]\nlemma div_lt_iff_lt_mul : a \/ c < b \u2194 a < b * c :=\nby rw [\u2190 mul_lt_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]\n\nalias sub_lt_iff_lt_add \u2194 lt_add_of_sub_right_lt sub_right_lt_of_lt_add\n\nend right\n\nsection left\nvariables [covariant_class \u03b1 \u03b1 (*) (<)] [covariant_class \u03b1 \u03b1 (swap (*)) (<)] {a b c : \u03b1}\n\n@[simp, to_additive]\nlemma div_lt_div_iff_left (a : \u03b1) : a \/ b < a \/ c \u2194 c < b :=\nby rw [div_eq_mul_inv, div_eq_mul_inv, \u2190 mul_lt_mul_iff_left a\u207b\u00b9, inv_mul_cancel_left,\n    inv_mul_cancel_left, inv_lt_inv_iff]\n\n@[simp, to_additive]\nlemma inv_lt_div_iff_lt_mul : a\u207b\u00b9 < b \/ c \u2194 c < a * b :=\nby rw [div_eq_mul_inv, lt_mul_inv_iff_mul_lt, inv_mul_lt_iff_lt_mul]\n\n@[to_additive sub_lt_sub_left]\nlemma div_lt_div_left' (h : a < b) (c : \u03b1) : c \/ b < c \/ a :=\n(div_lt_div_iff_left c).2 h\n\nend left\n\nend group\n\nsection comm_group\nvariables [comm_group \u03b1]\n\nsection has_lt\nvariables [has_lt \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] {a b c d : \u03b1}\n\n@[to_additive sub_lt_sub_iff]\nlemma div_lt_div_iff' : a \/ b < c \/ d \u2194 a * d < c * b :=\nby simpa only [div_eq_mul_inv] using mul_inv_lt_mul_inv_iff'\n\n@[to_additive]\nlemma lt_div_iff_mul_lt' : b < c \/ a \u2194 a * b < c :=\nby rw [lt_div_iff_mul_lt, mul_comm]\n\nalias lt_sub_iff_add_lt' \u2194 add_lt_of_lt_sub_left lt_sub_left_of_add_lt\n\n@[to_additive]\nlemma div_lt_iff_lt_mul' : a \/ b < c \u2194 a < b * c :=\nby rw [div_lt_iff_lt_mul, mul_comm]\n\nalias sub_lt_iff_lt_add' \u2194 lt_add_of_sub_left_lt sub_left_lt_of_lt_add\n\n@[to_additive]\nlemma inv_lt_div_iff_lt_mul' : b\u207b\u00b9 < a \/ c \u2194 c < a * b :=\nlt_div_iff_mul_lt.trans inv_mul_lt_iff_lt_mul'\n\n@[to_additive sub_lt]\nlemma div_lt'' : a \/ b < c \u2194 a \/ c < b :=\ndiv_lt_iff_lt_mul'.trans div_lt_iff_lt_mul.symm\n\n@[to_additive lt_sub]\nlemma lt_div'' : a < b \/ c \u2194 c < b \/ a :=\nlt_div_iff_mul_lt'.trans lt_div_iff_mul_lt.symm\n\nend has_lt\n\nsection preorder\nvariables [preorder \u03b1] [covariant_class \u03b1 \u03b1 (*) (<)] {a b c d : \u03b1}\n\n@[to_additive sub_lt_sub]\nlemma div_lt_div'' (hab : a < b) (hcd : c < d) :\n  a \/ d < b \/ c :=\nbegin\n  rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_lt_inv_mul_iff, mul_comm],\n  exact mul_lt_mul_of_lt_of_lt hab hcd\nend\n\nend preorder\n\nend comm_group\n\nsection linear_order\nvariables [group \u03b1] [linear_order \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)]\n\nsection variable_names\nvariables {a b c : \u03b1}\n\n@[to_additive]\nlemma le_of_forall_one_lt_lt_mul (h : \u2200 \u03b5 : \u03b1, 1 < \u03b5 \u2192 a < b * \u03b5) : a \u2264 b :=\nle_of_not_lt (\u03bb h\u2081, lt_irrefl a (by simpa using (h _ (lt_inv_mul_iff_lt.mpr h\u2081))))\n\n@[to_additive]\nlemma le_iff_forall_one_lt_lt_mul : a \u2264 b \u2194 \u2200 \u03b5, 1 < \u03b5 \u2192 a < b * \u03b5 :=\n\u27e8\u03bb h \u03b5, lt_mul_of_le_of_one_lt h, le_of_forall_one_lt_lt_mul\u27e9\n\n\/-  I (DT) introduced this lemma to prove (the additive version `sub_le_sub_flip` of)\n`div_le_div_flip` below.  Now I wonder what is the point of either of these lemmas... -\/\n@[to_additive]\nlemma div_le_inv_mul_iff [covariant_class \u03b1 \u03b1 (swap (*)) (\u2264)] :\n  a \/ b \u2264 a\u207b\u00b9 * b \u2194 a \u2264 b :=\nbegin\n  rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff],\n  exact \u27e8\u03bb h, not_lt.mp (\u03bb k, not_lt.mpr h (mul_lt_mul''' k k)), \u03bb h, mul_le_mul' h h\u27e9,\nend\n\n\/-  What is the point of this lemma?  See comment about `div_le_inv_mul_iff` above. -\/\n@[simp, to_additive]\nlemma div_le_div_flip {\u03b1 : Type*} [comm_group \u03b1] [linear_order \u03b1] [covariant_class \u03b1 \u03b1 (*) (\u2264)]\n  {a b : \u03b1}:\n  a \/ b \u2264 b \/ a \u2194 a \u2264 b :=\nbegin\n  rw [div_eq_mul_inv b, mul_comm],\n  exact div_le_inv_mul_iff,\nend\n\n@[simp, to_additive] lemma max_one_div_max_inv_one_eq_self (a : \u03b1) :\n  max a 1 \/ max a\u207b\u00b9 1 = a :=\nby { rcases le_total a 1 with h|h; simp [h] }\n\nalias max_zero_sub_max_neg_zero_eq_self \u2190 max_zero_sub_eq_self\n\nend variable_names\n\nsection densely_ordered\nvariables [densely_ordered \u03b1] {a b c : \u03b1}\n\n@[to_additive]\nlemma le_of_forall_one_lt_le_mul (h : \u2200 \u03b5 : \u03b1, 1 < \u03b5 \u2192 a \u2264 b * \u03b5) : a \u2264 b :=\nle_of_forall_le_of_dense $ \u03bb c hc,\ncalc a \u2264 b * (b\u207b\u00b9 * c) : h _ (lt_inv_mul_iff_lt.mpr hc)\n   ... = c             : mul_inv_cancel_left b c\n\n@[to_additive]\nlemma le_of_forall_lt_one_mul_le (h : \u2200 \u03b5 < 1, a * \u03b5 \u2264 b) : a \u2264 b :=\n@le_of_forall_one_lt_le_mul (order_dual \u03b1) _ _ _ _ _ _ h\n\n@[to_additive]\nlemma le_of_forall_one_lt_div_le (h : \u2200 \u03b5 : \u03b1, 1 < \u03b5 \u2192 a \/ \u03b5 \u2264 b) : a \u2264 b :=\nle_of_forall_lt_one_mul_le $ \u03bb \u03b5 \u03b51,\n  by simpa only [div_eq_mul_inv, inv_inv]  using h \u03b5\u207b\u00b9 (left.one_lt_inv_iff.2 \u03b51)\n\n@[to_additive]\nlemma le_iff_forall_one_lt_le_mul : a \u2264 b \u2194 \u2200 \u03b5, 1 < \u03b5 \u2192 a \u2264 b * \u03b5 :=\n\u27e8\u03bb h \u03b5 \u03b5_pos, le_mul_of_le_of_one_le h \u03b5_pos.le, le_of_forall_one_lt_le_mul\u27e9\n\n@[to_additive]\nlemma le_iff_forall_lt_one_mul_le : a \u2264 b \u2194 \u2200 \u03b5 < 1, a * \u03b5 \u2264 b :=\n@le_iff_forall_one_lt_le_mul (order_dual \u03b1) _ _ _ _ _ _\n\nend densely_ordered\n\nend linear_order\n\n\/-!\n### Linearly ordered commutative groups\n-\/\n\n\/-- A linearly ordered additive commutative group is an\nadditive commutative group with a linear order in which\naddition is monotone. -\/\n@[protect_proj, ancestor ordered_add_comm_group linear_order]\nclass linear_ordered_add_comm_group (\u03b1 : Type u) extends ordered_add_comm_group \u03b1, linear_order \u03b1\n\n\/-- A linearly ordered commutative monoid with an additively absorbing `\u22a4` element.\n  Instances should include number systems with an infinite element adjoined.` -\/\n@[protect_proj, ancestor linear_ordered_add_comm_monoid_with_top sub_neg_monoid nontrivial]\nclass linear_ordered_add_comm_group_with_top (\u03b1 : Type*)\n  extends linear_ordered_add_comm_monoid_with_top \u03b1, sub_neg_monoid \u03b1, nontrivial \u03b1 :=\n(neg_top : - (\u22a4 : \u03b1) = \u22a4)\n(add_neg_cancel : \u2200 a:\u03b1, a \u2260 \u22a4 \u2192 a + (- a) = 0)\n\n\/-- A linearly ordered commutative group is a\ncommutative group with a linear order in which\nmultiplication is monotone. -\/\n@[protect_proj, ancestor ordered_comm_group linear_order, to_additive]\nclass linear_ordered_comm_group (\u03b1 : Type u) extends ordered_comm_group \u03b1, linear_order \u03b1\n\n@[to_additive] instance [linear_ordered_comm_group \u03b1] :\n  linear_ordered_comm_group (order_dual \u03b1) :=\n{ .. order_dual.ordered_comm_group, .. order_dual.linear_order \u03b1 }\n\nsection linear_ordered_comm_group\nvariables [linear_ordered_comm_group \u03b1] {a b c : \u03b1}\n\n@[priority 100, to_additive] -- see Note [lower instance priority]\ninstance linear_ordered_comm_group.to_linear_ordered_cancel_comm_monoid :\n  linear_ordered_cancel_comm_monoid \u03b1 :=\n{ le_of_mul_le_mul_left := \u03bb x y z, le_of_mul_le_mul_left',\n  mul_left_cancel := \u03bb x y z, mul_left_cancel,\n  ..\u2039linear_ordered_comm_group \u03b1\u203a }\n\n\/-- Pullback a `linear_ordered_comm_group` under an injective map.\nSee note [reducible non-instances]. -\/\n@[reducible, to_additive function.injective.linear_ordered_add_comm_group\n\"Pullback a `linear_ordered_add_comm_group` under an injective map.\"]\ndef function.injective.linear_ordered_comm_group {\u03b2 : Type*}\n  [has_one \u03b2] [has_mul \u03b2] [has_inv \u03b2] [has_div \u03b2]\n  (f : \u03b2 \u2192 \u03b1) (hf : function.injective f) (one : f 1 = 1)\n  (mul : \u2200 x y, f (x * y) = f x * f y)\n  (inv : \u2200 x, f (x\u207b\u00b9) = (f x)\u207b\u00b9)\n  (div : \u2200 x y, f (x \/ y) = f x \/ f y)  :\n  linear_ordered_comm_group \u03b2 :=\n{ ..linear_order.lift f hf,\n  ..hf.ordered_comm_group f one mul inv div }\n\n@[to_additive linear_ordered_add_comm_group.add_lt_add_left]\nlemma linear_ordered_comm_group.mul_lt_mul_left'\n  (a b : \u03b1) (h : a < b) (c : \u03b1) : c * a < c * b :=\nmul_lt_mul_left' h c\n\n@[to_additive min_neg_neg]\nlemma min_inv_inv' (a b : \u03b1) : min (a\u207b\u00b9) (b\u207b\u00b9) = (max a b)\u207b\u00b9 :=\neq.symm $ @monotone.map_max \u03b1 (order_dual \u03b1) _ _ has_inv.inv a b $ \u03bb a b, inv_le_inv_iff.mpr\n\n@[to_additive max_neg_neg]\nlemma max_inv_inv' (a b : \u03b1) : max (a\u207b\u00b9) (b\u207b\u00b9) = (min a b)\u207b\u00b9 :=\neq.symm $ @monotone.map_min \u03b1 (order_dual \u03b1) _ _ has_inv.inv a b $ \u03bb a b, inv_le_inv_iff.mpr\n\n@[to_additive min_sub_sub_right]\nlemma min_div_div_right' (a b c : \u03b1) : min (a \/ c) (b \/ c) = min a b \/ c :=\nby simpa only [div_eq_mul_inv] using min_mul_mul_right a b (c\u207b\u00b9)\n\n@[to_additive max_sub_sub_right]\nlemma max_div_div_right' (a b c : \u03b1) : max (a \/ c) (b \/ c) = max a b \/ c :=\nby simpa only [div_eq_mul_inv] using max_mul_mul_right a b (c\u207b\u00b9)\n\n@[to_additive min_sub_sub_left]\nlemma min_div_div_left' (a b c : \u03b1) : min (a \/ b) (a \/ c) = a \/ max b c :=\nby simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']\n\n@[to_additive max_sub_sub_left]\nlemma max_div_div_left' (a b c : \u03b1) : max (a \/ b) (a \/ c) = a \/ min b c :=\nby simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']\n\n@[to_additive eq_zero_of_neg_eq]\nlemma eq_one_of_inv_eq' (h : a\u207b\u00b9 = a) : a = 1 :=\nmatch lt_trichotomy a 1 with\n| or.inl h\u2081 :=\n  have 1 < a, from h \u25b8 one_lt_inv_of_inv h\u2081,\n  absurd h\u2081 this.asymm\n| or.inr (or.inl h\u2081) := h\u2081\n| or.inr (or.inr h\u2081) :=\n  have a < 1, from h \u25b8 inv_lt_one'.mpr h\u2081,\n  absurd h\u2081 this.asymm\nend\n\n@[to_additive exists_zero_lt]\nlemma exists_one_lt' [nontrivial \u03b1] : \u2203 (a:\u03b1), 1 < a :=\nbegin\n  obtain \u27e8y, hy\u27e9 := decidable.exists_ne (1 : \u03b1),\n  cases hy.lt_or_lt,\n  { exact \u27e8y\u207b\u00b9, one_lt_inv'.mpr h\u27e9 },\n  { exact \u27e8y, h\u27e9 }\nend\n\n@[priority 100, to_additive] -- see Note [lower instance priority]\ninstance linear_ordered_comm_group.to_no_max_order [nontrivial \u03b1] :\n  no_max_order \u03b1 :=\n\u27e8 begin\n    obtain \u27e8y, hy\u27e9 : \u2203 (a:\u03b1), 1 < a := exists_one_lt',\n    exact \u03bb a, \u27e8a * y, lt_mul_of_one_lt_right' a hy\u27e9\n  end \u27e9\n\n@[priority 100, to_additive] -- see Note [lower instance priority]\ninstance linear_ordered_comm_group.to_no_min_order [nontrivial \u03b1] : no_min_order \u03b1 :=\n\u27e8 begin\n    obtain \u27e8y, hy\u27e9 : \u2203 (a:\u03b1), 1 < a := exists_one_lt',\n    exact \u03bb a, \u27e8a \/ y, (div_lt_self_iff a).mpr hy\u27e9\n  end \u27e9\n\nend linear_ordered_comm_group\n\nsection covariant_add_le\n\nsection has_neg\n\n\/-- `abs a` is the absolute value of `a`. -\/\n@[to_additive, priority 100] -- see Note [lower instance priority]\ninstance has_inv_lattice_has_abs [has_inv \u03b1] [lattice \u03b1] : has_abs (\u03b1)  := \u27e8\u03bba, a \u2294 (a\u207b\u00b9)\u27e9\n\n@[to_additive]\nlemma abs_eq_sup_inv [has_inv \u03b1] [lattice \u03b1] (a : \u03b1) : |a| = a \u2294 a\u207b\u00b9 :=\nrfl\n\nvariables [has_neg \u03b1] [linear_order \u03b1] {a b: \u03b1}\n\nlemma abs_eq_max_neg : abs a = max a (-a) :=\nrfl\n\nlemma abs_choice (x : \u03b1) : |x| = x \u2228 |x| = -x := max_choice _ _\n\nlemma abs_le' : |a| \u2264 b \u2194 a \u2264 b \u2227 -a \u2264 b := max_le_iff\n\nlemma le_abs : a \u2264 |b| \u2194 a \u2264 b \u2228 a \u2264 -b := le_max_iff\n\nlemma le_abs_self (a : \u03b1) : a \u2264 |a| := le_max_left _ _\n\nlemma neg_le_abs_self (a : \u03b1) : -a \u2264 |a| := le_max_right _ _\n\nlemma lt_abs : a < |b| \u2194 a < b \u2228 a < -b := lt_max_iff\n\ntheorem abs_le_abs (h\u2080 : a \u2264 b) (h\u2081 : -a \u2264 b) : |a| \u2264 |b| :=\n(abs_le'.2 \u27e8h\u2080, h\u2081\u27e9).trans (le_abs_self b)\n\nlemma abs_by_cases (P : \u03b1 \u2192 Prop) {a : \u03b1} (h1 : P a) (h2 : P (-a)) : P (|a|) :=\nsup_ind _ _ h1 h2\n\nend has_neg\n\nsection add_group\nvariables [add_group \u03b1] [linear_order \u03b1]\n\n@[simp] lemma abs_neg (a : \u03b1) : | -a| = |a| :=\nbegin\n  rw [abs_eq_max_neg, max_comm, neg_neg, abs_eq_max_neg]\nend\n\nlemma eq_or_eq_neg_of_abs_eq {a b : \u03b1} (h : |a| = b) : a = b \u2228 a = -b :=\nby simpa only [\u2190 h, eq_comm, eq_neg_iff_eq_neg] using abs_choice a\n\nlemma abs_eq_abs {a b : \u03b1} : |a| = |b| \u2194 a = b \u2228 a = -b :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { obtain rfl | rfl := eq_or_eq_neg_of_abs_eq h;\n    simpa only [neg_eq_iff_neg_eq, neg_inj, or.comm, @eq_comm _ (-b)] using abs_choice b },\n  { cases h; simp only [h, abs_neg] },\nend\n\nlemma abs_sub_comm (a b : \u03b1) : |a - b| = |b - a| :=\ncalc  |a - b| = | - (b - a)| : congr_arg _ (neg_sub b a).symm\n              ... = |b - a|     : abs_neg (b - a)\n\nvariables [covariant_class \u03b1 \u03b1 (+) (\u2264)] {a b c : \u03b1}\n\nlemma abs_of_nonneg (h : 0 \u2264 a) : |a| = a :=\nmax_eq_left $ (neg_nonpos.2 h).trans h\n\nlemma abs_of_pos (h : 0 < a) : |a| = a :=\nabs_of_nonneg h.le\n\nlemma abs_of_nonpos (h : a \u2264 0) : |a| = -a :=\nmax_eq_right $ h.trans (neg_nonneg.2 h)\n\nlemma abs_of_neg (h : a < 0) : |a| = -a :=\nabs_of_nonpos h.le\n\n@[simp] lemma abs_zero : |0| = (0:\u03b1) :=\nabs_of_nonneg le_rfl\n\n@[simp] lemma abs_pos : 0 < |a| \u2194 a \u2260 0 :=\nbegin\n  rcases lt_trichotomy a 0 with (ha|rfl|ha),\n  { simp [abs_of_neg ha, neg_pos, ha.ne, ha] },\n  { simp },\n  { simp [abs_of_pos ha, ha, ha.ne.symm] }\nend\n\nlemma abs_pos_of_pos (h : 0 < a) : 0 < |a| := abs_pos.2 h.ne.symm\n\nlemma abs_pos_of_neg (h : a < 0) : 0 < |a| := abs_pos.2 h.ne\n\nlemma neg_abs_le_self (a : \u03b1) : -|a| \u2264 a :=\nbegin\n  cases le_total 0 a with h h,\n  { calc -|a| = - a   : congr_arg (has_neg.neg) (abs_of_nonneg h)\n            ... \u2264 0     : neg_nonpos.mpr h\n            ... \u2264 a     : h },\n  { calc -|a| = - - a : congr_arg (has_neg.neg) (abs_of_nonpos h)\n            ... \u2264 a     : (neg_neg a).le }\nend\n\nlemma abs_nonneg (a : \u03b1) : 0 \u2264 |a| :=\n(le_total 0 a).elim (\u03bb h, h.trans (le_abs_self a)) (\u03bb h, (neg_nonneg.2 h).trans $ neg_le_abs_self a)\n\n@[simp] lemma abs_abs (a : \u03b1) : | |a| | = |a| :=\nabs_of_nonneg $ abs_nonneg a\n\n@[simp] lemma abs_eq_zero : |a| = 0 \u2194 a = 0 :=\ndecidable.not_iff_not.1 $ ne_comm.trans $ (abs_nonneg a).lt_iff_ne.symm.trans abs_pos\n\n@[simp] lemma abs_nonpos_iff {a : \u03b1} : |a| \u2264 0 \u2194 a = 0 :=\n(abs_nonneg a).le_iff_eq.trans abs_eq_zero\n\nvariable [covariant_class \u03b1 \u03b1 (swap (+)) (\u2264)]\n\nlemma abs_lt : |a| < b \u2194 - b < a \u2227 a < b :=\nmax_lt_iff.trans $ and.comm.trans $ by rw [neg_lt]\n\nlemma neg_lt_of_abs_lt (h : |a| < b) : -b < a := (abs_lt.mp h).1\n\nlemma lt_of_abs_lt (h : |a| < b) : a < b := (abs_lt.mp h).2\n\nlemma max_sub_min_eq_abs' (a b : \u03b1) : max a b - min a b = |a - b| :=\nbegin\n  cases le_total a b with ab ba,\n  { rw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub], rwa sub_nonpos },\n  { rw [max_eq_left ba, min_eq_right ba, abs_of_nonneg], rwa sub_nonneg }\nend\n\nlemma max_sub_min_eq_abs (a b : \u03b1) : max a b - min a b = |b - a| :=\nby { rw abs_sub_comm, exact max_sub_min_eq_abs' _ _ }\n\nend add_group\n\nend covariant_add_le\n\nsection linear_ordered_add_comm_group\n\nvariables [linear_ordered_add_comm_group \u03b1] {a b c d : \u03b1}\n\nlemma abs_le : |a| \u2264 b \u2194 - b \u2264 a \u2227 a \u2264 b := by rw [abs_le', and.comm, neg_le]\n\nlemma le_abs' : a \u2264 |b| \u2194 b \u2264 -a \u2228 a \u2264 b := by rw [le_abs, or.comm, le_neg]\n\nlemma neg_le_of_abs_le (h : |a| \u2264 b) : -b \u2264 a := (abs_le.mp h).1\n\nlemma le_of_abs_le (h : |a| \u2264 b) : a \u2264 b := (abs_le.mp h).2\n\n\/--\nThe **triangle inequality** in `linear_ordered_add_comm_group`s.\n-\/\nlemma abs_add (a b : \u03b1) : |a + b| \u2264 |a| + |b| :=\nabs_le.2 \u27e8(neg_add (|a|) (|b|)).symm \u25b8\n  add_le_add (neg_le.2 $ neg_le_abs_self _) (neg_le.2 $ neg_le_abs_self _),\n  add_le_add (le_abs_self _) (le_abs_self _)\u27e9\n\ntheorem abs_sub (a b : \u03b1) :\n  |a - b| \u2264 |a| + |b| :=\nby { rw [sub_eq_add_neg, \u2190abs_neg b], exact abs_add a _ }\n\nlemma abs_sub_le_iff : |a - b| \u2264 c \u2194 a - b \u2264 c \u2227 b - a \u2264 c :=\nby rw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add']\n\nlemma abs_sub_lt_iff : |a - b| < c \u2194 a - b < c \u2227 b - a < c :=\nby rw [abs_lt, neg_lt_sub_iff_lt_add', sub_lt_iff_lt_add', and_comm, sub_lt_iff_lt_add']\n\nlemma sub_le_of_abs_sub_le_left (h : |a - b| \u2264 c) : b - c \u2264 a :=\nsub_le.1 $ (abs_sub_le_iff.1 h).2\n\nlemma sub_le_of_abs_sub_le_right (h : |a - b| \u2264 c) : a - c \u2264 b :=\nsub_le_of_abs_sub_le_left (abs_sub_comm a b \u25b8 h)\n\nlemma sub_lt_of_abs_sub_lt_left (h : |a - b| < c) : b - c < a :=\nsub_lt.1 $ (abs_sub_lt_iff.1 h).2\n\nlemma sub_lt_of_abs_sub_lt_right (h : |a - b| < c) : a - c < b :=\nsub_lt_of_abs_sub_lt_left (abs_sub_comm a b \u25b8 h)\n\nlemma abs_sub_abs_le_abs_sub (a b : \u03b1) : |a| - |b| \u2264 |a - b| :=\nsub_le_iff_le_add.2 $\ncalc |a| = |a - b + b|     : by rw [sub_add_cancel]\n       ... \u2264 |a - b| + |b| : abs_add _ _\n\nlemma abs_abs_sub_abs_le_abs_sub (a b : \u03b1) : | |a| - |b| | \u2264 |a - b| :=\nabs_sub_le_iff.2 \u27e8abs_sub_abs_le_abs_sub _ _, by rw abs_sub_comm; apply abs_sub_abs_le_abs_sub\u27e9\n\nlemma abs_eq (hb : 0 \u2264 b) : |a| = b \u2194 a = b \u2228 a = -b :=\nbegin\n  refine \u27e8eq_or_eq_neg_of_abs_eq, _\u27e9,\n  rintro (rfl|rfl); simp only [abs_neg, abs_of_nonneg hb]\nend\n\nlemma abs_le_max_abs_abs (hab : a \u2264 b)  (hbc : b \u2264 c) : |b| \u2264 max (|a|) (|c|) :=\nabs_le'.2\n  \u27e8by simp [hbc.trans (le_abs_self c)],\n   by simp [(neg_le_neg_iff.mpr hab).trans (neg_le_abs_self a)]\u27e9\n\nlemma eq_of_abs_sub_eq_zero {a b : \u03b1} (h : |a - b| = 0) : a = b :=\nsub_eq_zero.1 $ abs_eq_zero.1 h\n\nlemma abs_sub_le (a b c : \u03b1) : |a - c| \u2264 |a - b| + |b - c| :=\ncalc\n    |a - c| = |a - b + (b - c)|     : by rw [sub_add_sub_cancel]\n            ... \u2264 |a - b| + |b - c| : abs_add _ _\n\nlemma abs_add_three (a b c : \u03b1) : |a + b + c| \u2264 |a| + |b| + |c| :=\n(abs_add _ _).trans (add_le_add_right (abs_add _ _) _)\n\nlemma dist_bdd_within_interval {a b lb ub : \u03b1} (hal : lb \u2264 a) (hau : a \u2264 ub)\n      (hbl : lb \u2264 b) (hbu : b \u2264 ub) : |a - b| \u2264 ub - lb :=\nabs_sub_le_iff.2 \u27e8sub_le_sub hau hbl, sub_le_sub hbu hal\u27e9\n\nlemma eq_of_abs_sub_nonpos (h : |a - b| \u2264 0) : a = b :=\neq_of_abs_sub_eq_zero (le_antisymm h (abs_nonneg (a - b)))\n\nlemma max_sub_max_le_max (a b c d : \u03b1) : max a b - max c d \u2264 max (a - c) (b - d) :=\nbegin\n  simp only [sub_le_iff_le_add, max_le_iff], split,\n  calc a = a - c + c : (sub_add_cancel a c).symm\n  ... \u2264 max (a - c) (b - d) + max c d : add_le_add (le_max_left _ _) (le_max_left _ _),\n  calc b = b - d + d : (sub_add_cancel b d).symm\n  ... \u2264 max (a - c) (b - d) + max c d : add_le_add (le_max_right _ _) (le_max_right _ _)\nend\n\nlemma abs_max_sub_max_le_max (a b c d : \u03b1) : |max a b - max c d| \u2264 max (|a - c|) (|b - d|) :=\nbegin\n  refine abs_sub_le_iff.2 \u27e8_, _\u27e9,\n  { exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) },\n  { rw [abs_sub_comm a c, abs_sub_comm b d],\n    exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) }\nend\n\nlemma abs_min_sub_min_le_max (a b c d : \u03b1) : |min a b - min c d| \u2264 max (|a - c|) (|b - d|) :=\nby simpa only [max_neg_neg, neg_sub_neg, abs_sub_comm]\n  using abs_max_sub_max_le_max (-a) (-b) (-c) (-d)\n\nlemma abs_max_sub_max_le_abs (a b c : \u03b1) : |max a c - max b c| \u2264 |a - b| :=\nby simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg _)]\n  using abs_max_sub_max_le_max a c b c\n\ninstance with_top.linear_ordered_add_comm_group_with_top :\n  linear_ordered_add_comm_group_with_top (with_top \u03b1) :=\n{ neg            := option.map (\u03bb a : \u03b1, -a),\n  neg_top        := @option.map_none _ _ (\u03bb a : \u03b1, -a),\n  add_neg_cancel := begin\n    rintro (a | a) ha,\n    { exact (ha rfl).elim },\n    { exact with_top.coe_add.symm.trans (with_top.coe_eq_coe.2 (add_neg_self a)) }\n  end,\n  .. with_top.linear_ordered_add_comm_monoid_with_top,\n  .. option.nontrivial }\n\n@[simp, norm_cast]\nlemma with_top.coe_neg (a : \u03b1) : ((-a : \u03b1) : with_top \u03b1) = -a := rfl\n\nend linear_ordered_add_comm_group\n\nnamespace add_comm_group\n\n\/-- A collection of elements in an `add_comm_group` designated as \"non-negative\".\nThis is useful for constructing an `ordered_add_commm_group`\nby choosing a positive cone in an exisiting `add_comm_group`. -\/\n@[nolint has_inhabited_instance]\nstructure positive_cone (\u03b1 : Type*) [add_comm_group \u03b1] :=\n(nonneg          : \u03b1 \u2192 Prop)\n(pos             : \u03b1 \u2192 Prop := \u03bb a, nonneg a \u2227 \u00ac nonneg (-a))\n(pos_iff         : \u2200 a, pos a \u2194 nonneg a \u2227 \u00ac nonneg (-a) . order_laws_tac)\n(zero_nonneg     : nonneg 0)\n(add_nonneg      : \u2200 {a b}, nonneg a \u2192 nonneg b \u2192 nonneg (a + b))\n(nonneg_antisymm : \u2200 {a}, nonneg a \u2192 nonneg (-a) \u2192 a = 0)\n\n\/-- A positive cone in an `add_comm_group` induces a linear order if\nfor every `a`, either `a` or `-a` is non-negative. -\/\n@[nolint has_inhabited_instance]\nstructure total_positive_cone (\u03b1 : Type*) [add_comm_group \u03b1] extends positive_cone \u03b1 :=\n(nonneg_decidable : decidable_pred nonneg)\n(nonneg_total : \u2200 a : \u03b1, nonneg a \u2228 nonneg (-a))\n\n\/-- Forget that a `total_positive_cone` is total. -\/\nadd_decl_doc total_positive_cone.to_positive_cone\n\nend add_comm_group\n\nnamespace ordered_add_comm_group\n\nopen add_comm_group\n\n\/-- Construct an `ordered_add_comm_group` by\ndesignating a positive cone in an existing `add_comm_group`. -\/\ndef mk_of_positive_cone {\u03b1 : Type*} [add_comm_group \u03b1] (C : positive_cone \u03b1) :\n  ordered_add_comm_group \u03b1 :=\n{ le               := \u03bb a b, C.nonneg (b - a),\n  lt               := \u03bb a b, C.pos (b - a),\n  lt_iff_le_not_le := \u03bb a b, by simp; rw [C.pos_iff]; simp,\n  le_refl          := \u03bb a, by simp [C.zero_nonneg],\n  le_trans         := \u03bb a b c nab nbc, by simp [-sub_eq_add_neg];\n    rw \u2190 sub_add_sub_cancel; exact C.add_nonneg nbc nab,\n  le_antisymm      := \u03bb a b nab nba, eq_of_sub_eq_zero $\n    C.nonneg_antisymm nba (by rw neg_sub; exact nab),\n  add_le_add_left  := \u03bb a b nab c, by simpa [(\u2264), preorder.le] using nab,\n  ..\u2039add_comm_group \u03b1\u203a }\n\nend ordered_add_comm_group\n\nnamespace linear_ordered_add_comm_group\n\nopen add_comm_group\n\n\/-- Construct a `linear_ordered_add_comm_group` by\ndesignating a positive cone in an existing `add_comm_group`\nsuch that for every `a`, either `a` or `-a` is non-negative. -\/\ndef mk_of_positive_cone {\u03b1 : Type*} [add_comm_group \u03b1] (C : total_positive_cone \u03b1) :\n  linear_ordered_add_comm_group \u03b1 :=\n{ le_total := \u03bb a b, by { convert C.nonneg_total (b - a), change C.nonneg _ = _, congr, simp, },\n  decidable_le := \u03bb a b, C.nonneg_decidable _,\n  ..ordered_add_comm_group.mk_of_positive_cone C.to_positive_cone }\n\nend linear_ordered_add_comm_group\n\nnamespace prod\n\nvariables {G H : Type*}\n\n@[to_additive]\ninstance [ordered_comm_group G] [ordered_comm_group H] :\n  ordered_comm_group (G \u00d7 H) :=\n{ .. prod.comm_group, .. prod.partial_order G H, .. prod.ordered_cancel_comm_monoid }\n\nend prod\n\nsection type_tags\n\ninstance [ordered_add_comm_group \u03b1] : ordered_comm_group (multiplicative \u03b1) :=\n{ ..multiplicative.comm_group,\n  ..multiplicative.ordered_comm_monoid }\n\ninstance [ordered_comm_group \u03b1] : ordered_add_comm_group (additive \u03b1) :=\n{ ..additive.add_comm_group,\n  ..additive.ordered_add_comm_monoid }\n\ninstance [linear_ordered_add_comm_group \u03b1] : linear_ordered_comm_group (multiplicative \u03b1) :=\n{ ..multiplicative.linear_order,\n  ..multiplicative.ordered_comm_group }\n\ninstance [linear_ordered_comm_group \u03b1] : linear_ordered_add_comm_group (additive \u03b1) :=\n{ ..additive.linear_order,\n  ..additive.ordered_add_comm_group }\n\nend type_tags\n\nsection norm_num_lemmas\n\/- The following lemmas are stated so that the `norm_num` tactic can use them with the\nexpected signatures.  -\/\nvariables [ordered_comm_group \u03b1] {a b : \u03b1}\n\n@[to_additive neg_le_neg]\nlemma inv_le_inv' : a \u2264 b \u2192 b\u207b\u00b9 \u2264 a\u207b\u00b9 :=\ninv_le_inv_iff.mpr\n\n@[to_additive neg_lt_neg]\nlemma inv_lt_inv' : a < b \u2192 b\u207b\u00b9 < a\u207b\u00b9 :=\ninv_lt_inv_iff.mpr\n\n\/-  The additive version is also a `linarith` lemma. -\/\n@[to_additive]\ntheorem inv_lt_one_of_one_lt : 1 < a \u2192 a\u207b\u00b9 < 1 :=\ninv_lt_one_iff_one_lt.mpr\n\n\/-  The additive version is also a `linarith` lemma. -\/\n@[to_additive]\nlemma inv_le_one_of_one_le : 1 \u2264 a \u2192 a\u207b\u00b9 \u2264 1 :=\ninv_le_one'.mpr\n\n@[to_additive neg_nonneg_of_nonpos]\nlemma one_le_inv_of_le_one :  a \u2264 1 \u2192 1 \u2264 a\u207b\u00b9 :=\none_le_inv'.mpr\n\nend norm_num_lemmas\n","avg_line_length":33.1639824305,"max_line_length":100,"alphanum_fraction":0.6895059821}
-{"size":10854,"ext":"lean","lang":"Lean","max_stars_count":1131.0,"content":"\/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nComputational realization of filters (experimental).\n-\/\nimport order.filter.cofinite\nopen set filter\n\n\/-- A `cfilter \u03b1 \u03c3` is a realization of a filter (base) on `\u03b1`,\n  represented by a type `\u03c3` together with operations for the top element and\n  the binary inf operation. -\/\nstructure cfilter (\u03b1 \u03c3 : Type*) [partial_order \u03b1] :=\n(f : \u03c3 \u2192 \u03b1)\n(pt : \u03c3)\n(inf : \u03c3 \u2192 \u03c3 \u2192 \u03c3)\n(inf_le_left : \u2200 a b : \u03c3, f (inf a b) \u2264 f a)\n(inf_le_right : \u2200 a b : \u03c3, f (inf a b) \u2264 f b)\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03c3 : Type*} {\u03c4 : Type*}\n\nnamespace cfilter\nsection\nvariables [partial_order \u03b1] (F : cfilter \u03b1 \u03c3)\n\ninstance : has_coe_to_fun (cfilter \u03b1 \u03c3) := \u27e8_, cfilter.f\u27e9\n\n@[simp] theorem coe_mk (f pt inf h\u2081 h\u2082 a) : (@cfilter.mk \u03b1 \u03c3 _ f pt inf h\u2081 h\u2082) a = f a := rfl\n\n\/-- Map a cfilter to an equivalent representation type. -\/\ndef of_equiv (E : \u03c3 \u2243 \u03c4) : cfilter \u03b1 \u03c3 \u2192 cfilter \u03b1 \u03c4\n| \u27e8f, p, g, h\u2081, h\u2082\u27e9 :=\n  { f            := \u03bb a, f (E.symm a),\n    pt           := E p,\n    inf          := \u03bb a b, E (g (E.symm a) (E.symm b)),\n    inf_le_left  := \u03bb a b, by simpa using h\u2081 (E.symm a) (E.symm b),\n    inf_le_right := \u03bb a b, by simpa using h\u2082 (E.symm a) (E.symm b) }\n\n@[simp] theorem of_equiv_val (E : \u03c3 \u2243 \u03c4) (F : cfilter \u03b1 \u03c3) (a : \u03c4) :\n  F.of_equiv E a = F (E.symm a) := by cases F; refl\n\nend\n\n\/-- The filter represented by a `cfilter` is the collection of supersets of\n  elements of the filter base. -\/\ndef to_filter (F : cfilter (set \u03b1) \u03c3) : filter \u03b1 :=\n{ sets             := {a | \u2203 b, F b \u2286 a},\n  univ_sets        := \u27e8F.pt, subset_univ _\u27e9,\n  sets_of_superset := \u03bb x y \u27e8b, h\u27e9 s, \u27e8b, subset.trans h s\u27e9,\n  inter_sets       := \u03bb x y \u27e8a, h\u2081\u27e9 \u27e8b, h\u2082\u27e9, \u27e8F.inf a b,\n    subset_inter (subset.trans (F.inf_le_left _ _) h\u2081) (subset.trans (F.inf_le_right _ _) h\u2082)\u27e9 }\n\n@[simp] theorem mem_to_filter_sets (F : cfilter (set \u03b1) \u03c3) {a : set \u03b1} :\n  a \u2208 F.to_filter \u2194 \u2203 b, F b \u2286 a := iff.rfl\n\nend cfilter\n\n\/-- A realizer for filter `f` is a cfilter which generates `f`. -\/\nstructure filter.realizer (f : filter \u03b1) :=\n(\u03c3 : Type*)\n(F : cfilter (set \u03b1) \u03c3)\n(eq : F.to_filter = f)\n\nprotected def cfilter.to_realizer (F : cfilter (set \u03b1) \u03c3) : F.to_filter.realizer := \u27e8\u03c3, F, rfl\u27e9\n\nnamespace filter.realizer\n\ntheorem mem_sets {f : filter \u03b1} (F : f.realizer) {a : set \u03b1} : a \u2208 f \u2194 \u2203 b, F.F b \u2286 a :=\nby cases F; subst f; simp\n\n-- Used because it has better definitional equalities than the eq.rec proof\ndef of_eq {f g : filter \u03b1} (e : f = g) (F : f.realizer) : g.realizer :=\n\u27e8F.\u03c3, F.F, F.eq.trans e\u27e9\n\n\/-- A filter realizes itself. -\/\ndef of_filter (f : filter \u03b1) : f.realizer := \u27e8f.sets,\n{ f            := subtype.val,\n  pt           := \u27e8univ, univ_mem\u27e9,\n  inf          := \u03bb \u27e8x, h\u2081\u27e9 \u27e8y, h\u2082\u27e9, \u27e8_, inter_mem h\u2081 h\u2082\u27e9,\n  inf_le_left  := \u03bb \u27e8x, h\u2081\u27e9 \u27e8y, h\u2082\u27e9, inter_subset_left x y,\n  inf_le_right := \u03bb \u27e8x, h\u2081\u27e9 \u27e8y, h\u2082\u27e9, inter_subset_right x y },\nfilter_eq $ set.ext $ \u03bb x, set_coe.exists.trans exists_mem_subset_iff\u27e9\n\n\/-- Transfer a filter realizer to another realizer on a different base type. -\/\ndef of_equiv {f : filter \u03b1} (F : f.realizer) (E : F.\u03c3 \u2243 \u03c4) : f.realizer :=\n\u27e8\u03c4, F.F.of_equiv E, by refine eq.trans _ F.eq; exact filter_eq (set.ext $ \u03bb x,\n\u27e8\u03bb \u27e8s, h\u27e9, \u27e8E.symm s, by simpa using h\u27e9, \u03bb \u27e8t, h\u27e9, \u27e8E t, by simp [h]\u27e9\u27e9)\u27e9\n\n@[simp] theorem of_equiv_\u03c3 {f : filter \u03b1} (F : f.realizer) (E : F.\u03c3 \u2243 \u03c4) :\n  (F.of_equiv E).\u03c3 = \u03c4 := rfl\n@[simp] theorem of_equiv_F {f : filter \u03b1} (F : f.realizer) (E : F.\u03c3 \u2243 \u03c4) (s : \u03c4) :\n  (F.of_equiv E).F s = F.F (E.symm s) := by delta of_equiv; simp\n\n\/-- `unit` is a realizer for the principal filter -\/\nprotected def principal (s : set \u03b1) : (principal s).realizer := \u27e8unit,\n{ f            := \u03bb _, s,\n  pt           := (),\n  inf          := \u03bb _ _, (),\n  inf_le_left  := \u03bb _ _, le_refl _,\n  inf_le_right := \u03bb _ _, le_refl _ },\nfilter_eq $ set.ext $ \u03bb x,\n\u27e8\u03bb \u27e8_, s\u27e9, s, \u03bb h, \u27e8(), h\u27e9\u27e9\u27e9\n\n@[simp] theorem principal_\u03c3 (s : set \u03b1) : (realizer.principal s).\u03c3 = unit := rfl\n@[simp] theorem principal_F (s : set \u03b1) (u : unit) : (realizer.principal s).F u = s := rfl\n\n\/-- `unit` is a realizer for the top filter -\/\nprotected def top : (\u22a4 : filter \u03b1).realizer :=\n(realizer.principal _).of_eq principal_univ\n\n@[simp] theorem top_\u03c3 : (@realizer.top \u03b1).\u03c3 = unit := rfl\n@[simp] theorem top_F (u : unit) : (@realizer.top \u03b1).F u = univ := rfl\n\n\/-- `unit` is a realizer for the bottom filter -\/\nprotected def bot : (\u22a5 : filter \u03b1).realizer :=\n(realizer.principal _).of_eq principal_empty\n\n@[simp] theorem bot_\u03c3 : (@realizer.bot \u03b1).\u03c3 = unit := rfl\n@[simp] theorem bot_F (u : unit) : (@realizer.bot \u03b1).F u = \u2205 := rfl\n\n\/-- Construct a realizer for `map m f` given a realizer for `f` -\/\nprotected def map (m : \u03b1 \u2192 \u03b2) {f : filter \u03b1} (F : f.realizer) : (map m f).realizer := \u27e8F.\u03c3,\n{ f            := \u03bb s, image m (F.F s),\n  pt           := F.F.pt,\n  inf          := F.F.inf,\n  inf_le_left  := \u03bb a b, image_subset _ (F.F.inf_le_left _ _),\n  inf_le_right := \u03bb a b, image_subset _ (F.F.inf_le_right _ _) },\nfilter_eq $ set.ext $ \u03bb x, by simp [cfilter.to_filter]; rw F.mem_sets; refl \u27e9\n\n@[simp] theorem map_\u03c3 (m : \u03b1 \u2192 \u03b2) {f : filter \u03b1} (F : f.realizer) : (F.map m).\u03c3 = F.\u03c3 := rfl\n@[simp] theorem map_F (m : \u03b1 \u2192 \u03b2) {f : filter \u03b1} (F : f.realizer) (s) :\n  (F.map m).F s = image m (F.F s) := rfl\n\n\/-- Construct a realizer for `comap m f` given a realizer for `f` -\/\nprotected def comap (m : \u03b1 \u2192 \u03b2) {f : filter \u03b2} (F : f.realizer) : (comap m f).realizer := \u27e8F.\u03c3,\n{ f            := \u03bb s, preimage m (F.F s),\n  pt           := F.F.pt,\n  inf          := F.F.inf,\n  inf_le_left  := \u03bb a b, preimage_mono (F.F.inf_le_left _ _),\n  inf_le_right := \u03bb a b, preimage_mono (F.F.inf_le_right _ _) },\nfilter_eq $ set.ext $ \u03bb x, by cases F; subst f; simp [cfilter.to_filter, mem_comap]; exact\n\u27e8\u03bb \u27e8s, h\u27e9, \u27e8_, \u27e8s, subset.refl _\u27e9, h\u27e9,\n \u03bb \u27e8y, \u27e8s, h\u27e9, h\u2082\u27e9, \u27e8s, subset.trans (preimage_mono h) h\u2082\u27e9\u27e9\u27e9\n\n\/-- Construct a realizer for the sup of two filters -\/\nprotected def sup {f g : filter \u03b1} (F : f.realizer) (G : g.realizer) :\n  (f \u2294 g).realizer := \u27e8F.\u03c3 \u00d7 G.\u03c3,\n{ f            := \u03bb \u27e8s, t\u27e9, F.F s \u222a G.F t,\n  pt           := (F.F.pt, G.F.pt),\n  inf          := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, (F.F.inf a b, G.F.inf a' b'),\n  inf_le_left  := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, union_subset_union (F.F.inf_le_left _ _) (G.F.inf_le_left _ _),\n  inf_le_right := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, union_subset_union (F.F.inf_le_right _ _)\n                    (G.F.inf_le_right _ _) },\nfilter_eq $ set.ext $ \u03bb x, by cases F; cases G; substs f g; simp [cfilter.to_filter]; exact\n\u27e8\u03bb \u27e8s, t, h\u27e9, \u27e8\u27e8s, subset.trans (subset_union_left _ _) h\u27e9,\n               \u27e8t, subset.trans (subset_union_right _ _) h\u27e9\u27e9,\n \u03bb \u27e8\u27e8s, h\u2081\u27e9, \u27e8t, h\u2082\u27e9\u27e9, \u27e8s, t, union_subset h\u2081 h\u2082\u27e9\u27e9\u27e9\n\n\/-- Construct a realizer for the inf of two filters -\/\nprotected def inf {f g : filter \u03b1} (F : f.realizer) (G : g.realizer) :\n  (f \u2293 g).realizer := \u27e8F.\u03c3 \u00d7 G.\u03c3,\n{ f            := \u03bb \u27e8s, t\u27e9, F.F s \u2229 G.F t,\n  pt           := (F.F.pt, G.F.pt),\n  inf          := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, (F.F.inf a b, G.F.inf a' b'),\n  inf_le_left  := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _),\n  inf_le_right := \u03bb \u27e8a, a'\u27e9 \u27e8b, b'\u27e9, inter_subset_inter (F.F.inf_le_right _ _)\n                    (G.F.inf_le_right _ _) },\n begin\n   ext x,\n   cases F; cases G; substs f g; simp [cfilter.to_filter],\n   split,\n   { rintro \u27e8s : F_\u03c3, t : G_\u03c3, h\u27e9,\n     apply mem_inf_of_inter _ _ h,\n     use s,\n     use t, },\n   { rintros \u27e8s, \u27e8a, ha\u27e9, t, \u27e8b, hb\u27e9, rfl\u27e9,\n     exact \u27e8a, b, inter_subset_inter ha hb\u27e9 }\n end\u27e9\n\n\/-- Construct a realizer for the cofinite filter -\/\nprotected def cofinite [decidable_eq \u03b1] : (@cofinite \u03b1).realizer := \u27e8finset \u03b1,\n{ f            := \u03bb s, {a | a \u2209 s},\n  pt           := \u2205,\n  inf          := (\u222a),\n  inf_le_left  := \u03bb s t a, mt (finset.mem_union_left _),\n  inf_le_right := \u03bb s t a, mt (finset.mem_union_right _) },\nfilter_eq $ set.ext $ \u03bb x,\n\u27e8\u03bb \u27e8s, h\u27e9, s.finite_to_set.subset (compl_subset_comm.1 h),\n \u03bb \u27e8fs\u27e9, by exactI \u27e8x\u1d9c.to_finset, \u03bb a (h : a \u2209 x\u1d9c.to_finset),\n  classical.by_contradiction $ \u03bb h', h (mem_to_finset.2 h')\u27e9\u27e9\u27e9\n\n\/-- Construct a realizer for filter bind -\/\nprotected def bind {f : filter \u03b1} {m : \u03b1 \u2192 filter \u03b2} (F : f.realizer) (G : \u2200 i, (m i).realizer) :\n  (f.bind m).realizer :=\n\u27e8\u03a3 s : F.\u03c3, \u03a0 i \u2208 F.F s, (G i).\u03c3,\n{ f            := \u03bb \u27e8s, f\u27e9, \u22c3 i \u2208 F.F s, (G i).F (f i H),\n  pt           := \u27e8F.F.pt, \u03bb i H, (G i).F.pt\u27e9,\n  inf          := \u03bb \u27e8a, f\u27e9 \u27e8b, f'\u27e9, \u27e8F.F.inf a b, \u03bb i h,\n    (G i).F.inf (f i (F.F.inf_le_left _ _ h)) (f' i (F.F.inf_le_right _ _ h))\u27e9,\n  inf_le_left  := \u03bb \u27e8a, f\u27e9 \u27e8b, f'\u27e9 x,\n    show (x \u2208 \u22c3 (i : \u03b1) (H : i \u2208 F.F (F.F.inf a b)), _) \u2192\n          x \u2208 \u22c3 i (H : i \u2208 F.F a), ((G i).F) (f i H), by simp; exact\n    \u03bb i h\u2081 h\u2082, \u27e8i, F.F.inf_le_left _ _ h\u2081, (G i).F.inf_le_left _ _ h\u2082\u27e9,\n  inf_le_right := \u03bb \u27e8a, f\u27e9 \u27e8b, f'\u27e9 x,\n    show (x \u2208 \u22c3 (i : \u03b1) (H : i \u2208 F.F (F.F.inf a b)), _) \u2192\n          x \u2208 \u22c3 i (H : i \u2208 F.F b), ((G i).F) (f' i H), by simp; exact\n    \u03bb i h\u2081 h\u2082, \u27e8i, F.F.inf_le_right _ _ h\u2081, (G i).F.inf_le_right _ _ h\u2082\u27e9 },\nfilter_eq $ set.ext $ \u03bb x,\nby cases F with _ F _; subst f; simp [cfilter.to_filter, mem_bind]; exact\n\u27e8\u03bb \u27e8s, f, h\u27e9, \u27e8F s, \u27e8s, subset.refl _\u27e9, \u03bb i H, (G i).mem_sets.2\n   \u27e8f i H, \u03bb a h', h \u27e8_, \u27e8i, rfl\u27e9, _, \u27e8H, rfl\u27e9, h'\u27e9\u27e9\u27e9,\n \u03bb \u27e8y, \u27e8s, h\u27e9, f\u27e9,\n  let \u27e8f', h'\u27e9 := classical.axiom_of_choice (\u03bb i:F s, (G i).mem_sets.1 (f i (h i.2))) in\n  \u27e8s, \u03bb i h, f' \u27e8i, h\u27e9, \u03bb a \u27e8_, \u27e8i, rfl\u27e9, _, \u27e8H, rfl\u27e9, m\u27e9, h' \u27e8_, H\u27e9 m\u27e9\u27e9\u27e9\n\n\/-- Construct a realizer for indexed supremum -\/\nprotected def Sup {f : \u03b1 \u2192 filter \u03b2} (F : \u2200 i, (f i).realizer) : (\u2a06 i, f i).realizer :=\nlet F' : (\u2a06 i, f i).realizer :=\n  ((realizer.bind realizer.top F).of_eq $\n    filter_eq $ set.ext $ by simp [filter.bind, eq_univ_iff_forall, supr_sets_eq]) in\nF'.of_equiv $ show (\u03a3 u:unit, \u03a0 (i : \u03b1), true \u2192 (F i).\u03c3) \u2243 \u03a0 i, (F i).\u03c3, from\n\u27e8\u03bb\u27e8_,f\u27e9 i, f i \u27e8\u27e9, \u03bb f, \u27e8(), \u03bb i _, f i\u27e9,\n \u03bb \u27e8\u27e8\u27e9, f\u27e9, by dsimp; congr; simp, \u03bb f, rfl\u27e9\n\n\/-- Construct a realizer for the product of filters -\/\nprotected def prod {f g : filter \u03b1} (F : f.realizer) (G : g.realizer) : (f.prod g).realizer :=\n(F.comap _).inf (G.comap _)\n\ntheorem le_iff {f g : filter \u03b1} (F : f.realizer) (G : g.realizer) :\n  f \u2264 g \u2194 \u2200 b : G.\u03c3, \u2203 a : F.\u03c3, F.F a \u2264 G.F b :=\n\u27e8\u03bb H t, F.mem_sets.1 (H (G.mem_sets.2 \u27e8t, subset.refl _\u27e9)),\n \u03bb H x h, F.mem_sets.2 $\n   let \u27e8s, h\u2081\u27e9 := G.mem_sets.1 h, \u27e8t, h\u2082\u27e9 := H s in \u27e8t, subset.trans h\u2082 h\u2081\u27e9\u27e9\n\ntheorem tendsto_iff (f : \u03b1 \u2192 \u03b2) {l\u2081 : filter \u03b1} {l\u2082 : filter \u03b2} (L\u2081 : l\u2081.realizer)\n  (L\u2082 : l\u2082.realizer) :\n  tendsto f l\u2081 l\u2082 \u2194 \u2200 b, \u2203 a, \u2200 x \u2208 L\u2081.F a, f x \u2208 L\u2082.F b :=\n(le_iff (L\u2081.map f) L\u2082).trans $ forall_congr $ \u03bb b, exists_congr $ \u03bb a, image_subset_iff\n\ntheorem ne_bot_iff {f : filter \u03b1} (F : f.realizer) :\n  f \u2260 \u22a5 \u2194 \u2200 a : F.\u03c3, (F.F a).nonempty :=\nbegin\n  classical,\n  rw [not_iff_comm, \u2190 le_bot_iff, F.le_iff realizer.bot, not_forall],\n  simp only [set.not_nonempty_iff_eq_empty],\n  exact \u27e8\u03bb \u27e8x, e\u27e9 _, \u27e8x, le_of_eq e\u27e9,\n    \u03bb h, let \u27e8x, h\u27e9 := h () in \u27e8x, le_bot_iff.1 h\u27e9\u27e9\nend\n\nend filter.realizer\n","avg_line_length":43.0714285714,"max_line_length":100,"alphanum_fraction":0.5665192556}
-{"size":5583,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Wojciech Nawrocki. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Wojciech Nawrocki\n-\/\nimport Lean.Data.Json\nimport Lean.Data.Lsp.Basic\n\nnamespace Lean\nnamespace Lsp\n\nopen Json\n\nstructure CompletionOptions where\n  triggerCharacters? : Option (Array String) := none\n  allCommitCharacters?: Option (Array String) := none\n  resolveProvider?: Bool := false\n  deriving FromJson, ToJson\n\nstructure CompletionItem where\n  label : String\n  detail? : Option String\n  documentation? : Option MarkupContent\n  \/-\n  kind? : CompletionItemKind\n  tags? : CompletionItemTag[]\n  deprecated? : boolean\n  preselect? : boolean\n  sortText? : string\n  filterText? : string\n  insertText? : string\n  insertTextFormat? : InsertTextFormat\n  insertTextMode? : InsertTextMode\n  textEdit? : TextEdit | InsertReplaceEdit\n  additionalTextEdits? : TextEdit[]\n  commitCharacters? : string[]\n  command? : Command\n  data? : any -\/\n  deriving FromJson, ToJson, Inhabited\n\nstructure CompletionList where\n  isIncomplete : Bool\n  items : Array CompletionItem\n  deriving FromJson, ToJson\n\nstructure CompletionParams extends TextDocumentPositionParams where\n  -- context? : CompletionContext\n  deriving FromJson, ToJson\n\nstructure Hover where\n  \/- NOTE we should also accept MarkedString\/MarkedString[] here\n  but they are deprecated, so maybe can get away without. -\/\n  contents : MarkupContent\n  range? : Option Range := none\n  deriving ToJson, FromJson\n\nstructure HoverParams extends TextDocumentPositionParams\n  deriving FromJson, ToJson\n\nstructure DeclarationParams extends TextDocumentPositionParams\n  deriving FromJson, ToJson\n\nstructure DefinitionParams extends TextDocumentPositionParams\n  deriving FromJson, ToJson\n\nstructure TypeDefinitionParams extends TextDocumentPositionParams\n  deriving FromJson, ToJson\n\nstructure DocumentHighlightParams extends TextDocumentPositionParams\n  deriving FromJson, ToJson\n\ninductive DocumentHighlightKind where\n  | text\n  | read\n  | write\n\ninstance : ToJson DocumentHighlightKind where\n toJson\n\t| DocumentHighlightKind.text => 1\n\t| DocumentHighlightKind.read => 2\n\t| DocumentHighlightKind.write => 3\n\nstructure DocumentHighlight where\n  range : Range\n  kind? : Option DocumentHighlightKind := none\n  deriving ToJson\n\nabbrev DocumentHighlightResult := Array DocumentHighlight\n\nstructure DocumentSymbolParams where\n  textDocument : TextDocumentIdentifier\n  deriving FromJson, ToJson\n\ninductive SymbolKind where\n\t| file\n\t| module\n\t| \u00abnamespace\u00bb\n\t| package\n\t| \u00abclass\u00bb\n\t| method\n\t| property\n\t| field\n\t| constructor\n\t| enum\n\t| interface\n\t| function\n\t| \u00abvariable\u00bb\n\t| \u00abconstant\u00bb\n\t| string\n\t| number\n\t| boolean\n\t| array\n\t| object\n\t| key\n\t| null\n\t| enumMember\n\t| struct\n\t| event\n\t| operator\n\t| typeParameter\n\ninstance : ToJson SymbolKind where\n toJson\n\t| SymbolKind.file => 1\n\t| SymbolKind.module => 2\n\t| SymbolKind.namespace => 3\n\t| SymbolKind.package => 4\n\t| SymbolKind.class => 5\n\t| SymbolKind.method => 6\n\t| SymbolKind.property => 7\n\t| SymbolKind.field => 8\n\t| SymbolKind.constructor => 9\n\t| SymbolKind.enum => 10\n\t| SymbolKind.interface => 11\n\t| SymbolKind.function => 12\n\t| SymbolKind.variable => 13\n\t| SymbolKind.constant => 14\n\t| SymbolKind.string => 15\n\t| SymbolKind.number => 16\n\t| SymbolKind.boolean => 17\n\t| SymbolKind.array => 18\n\t| SymbolKind.object => 19\n\t| SymbolKind.key => 20\n\t| SymbolKind.null => 21\n\t| SymbolKind.enumMember => 22\n\t| SymbolKind.struct => 23\n\t| SymbolKind.event => 24\n\t| SymbolKind.operator => 25\n\t| SymbolKind.typeParameter => 26\n\nstructure DocumentSymbolAux (Self : Type) where\n  name : String\n  detail? : Option String := none\n  kind : SymbolKind\n  -- tags? : Array SymbolTag\n  range : Range\n  selectionRange : Range\n  children? : Option (Array Self) := none\n  deriving ToJson\n\ninductive DocumentSymbol where\n  | mk (sym : DocumentSymbolAux DocumentSymbol)\n\npartial instance : ToJson DocumentSymbol where\n  toJson :=\n    let rec go\n      | DocumentSymbol.mk sym =>\n        have ToJson DocumentSymbol := \u27e8go\u27e9\n        toJson sym\n    go\n\nstructure DocumentSymbolResult where\n  syms : Array DocumentSymbol\n\ninstance : ToJson DocumentSymbolResult where\n  toJson dsr := toJson dsr.syms\n\ninductive SemanticTokenType where\n  | keyword\n  | \u00abvariable\u00bb\n  | property\n  \/-\n  | \u00abnamespace\u00bb\n  | type\n  | \u00abclass\u00bb\n  | enum\n  | interface\n  | struct\n  | typeParameter\n  | parameter\n  | enumMember\n  | event\n  | function\n  | method\n  | \u00abmacro\u00bb\n  | modifier\n  | comment\n  | string\n  | number\n  | regexp\n  | operator\n  -\/\n\ndef SemanticTokenType.names : Array String :=\n  #[\"keyword\", \"variable\", \"property\"]\n\n-- must be the correct index in `names`\ndef SemanticTokenType.toNat : SemanticTokenType \u2192 Nat\n  | keyword    => 0\n  | \u00abvariable\u00bb => 1\n  | property   => 2\n\n\/-\ninductive SemanticTokenModifier where\n  | declaration\n  | definition\n  | readonly\n  | static\n  | deprecated\n  | abstract\n  | async\n  | modification\n  | documentation\n  | defaultLibrary\n-\/\n\nstructure SemanticTokensLegend where\n  tokenTypes : Array String\n  tokenModifiers : Array String\n  deriving FromJson, ToJson\n\nstructure SemanticTokensOptions where\n  legend : SemanticTokensLegend\n  range : Bool\n  full : Bool \/- | {\n    delta?: boolean;\n  } -\/\n  deriving FromJson, ToJson\n\nstructure SemanticTokensParams where\n  textDocument : TextDocumentIdentifier\n  deriving FromJson, ToJson\n\nstructure SemanticTokensRangeParams where\n  textDocument : TextDocumentIdentifier\n  range : Range\n  deriving FromJson, ToJson\n\nstructure SemanticTokens where\n  -- resultId?: string;\n  data : Array Nat\n  deriving FromJson, ToJson\n\nend Lsp\nend Lean\n","avg_line_length":21.7237354086,"max_line_length":68,"alphanum_fraction":0.7345513165}
-{"size":49612,"ext":"lean","lang":"Lean","max_stars_count":2.0,"content":"\/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-\/\nimport algebra.big_operators.finsupp\nimport linear_algebra.finsupp\nimport linear_algebra.prod\nimport linear_algebra.pi\nimport order.zorn\nimport data.finset.order\nimport data.equiv.fin\n\n\/-!\n\n# Linear independence\n\nThis file defines linear independence in a module or vector space.\n\nIt is inspired by Isabelle\/HOL's linear algebra, and hence indirectly by HOL Light.\n\nWe define `linear_independent R v` as `ker (finsupp.total \u03b9 M R v) = \u22a5`. Here `finsupp.total` is the\nlinear map sending a function `f : \u03b9 \u2192\u2080 R` with finite support to the linear combination of vectors\nfrom `v` with these coefficients. Then we prove that several other statements are equivalent to this\none, including injectivity of `finsupp.total \u03b9 M R v` and some versions with explicitly written\nlinear combinations.\n\n## Main definitions\n\nAll definitions are given for families of vectors, i.e. `v : \u03b9 \u2192 M` where `M` is the module or\nvector space and `\u03b9 : Type*` is an arbitrary indexing type.\n\n* `linear_independent R v` states that the elements of the family `v` are linearly independent.\n\n* `linear_independent.repr hv x` returns the linear combination representing `x : span R (range v)`\non the linearly independent vectors `v`, given `hv : linear_independent R v`\n(using classical choice). `linear_independent.repr hv` is provided as a linear map.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `fintype.linear_independent_iff`: if `\u03b9` is a finite type, then any function `f : \u03b9 \u2192 R` has\n  finite support, so we can reformulate the statement using `\u2211 i : \u03b9, f i \u2022 v i` instead of a sum\n  over an auxiliary `s : finset \u03b9`;\n* `linear_independent_empty_type`: a family indexed by an empty type is linearly independent;\n* `linear_independent_unique_iff`: if `\u03b9` is a singleton, then `linear_independent K v` is\n  equivalent to `v (default \u03b9) \u2260 0`;\n* linear_independent_option`, `linear_independent_sum`, `linear_independent_fin_cons`,\n  `linear_independent_fin_succ`: type-specific tests for linear independence of families of vector\n  fields;\n* `linear_independent_insert`, `linear_independent_union`, `linear_independent_pair`,\n  `linear_independent_singleton`: linear independence tests for set operations.\n\nIn many cases we additionally provide dot-style operations (e.g., `linear_independent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\nWe also prove that any family of vectors includes a linear independent subfamily spanning the same\nsubmodule.\n\n## Implementation notes\n\nWe use families instead of sets because it allows us to say that two identical vectors are linearly\ndependent.\n\nIf you want to use sets, use the family `(\u03bb x, x : s \u2192 M)` given a set `s : set M`. The lemmas\n`linear_independent.to_subtype_range` and `linear_independent.of_subtype_range` connect those two\nworlds.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-\/\n\nnoncomputable theory\n\nopen function set submodule\nopen_locale classical big_operators\n\nuniverse u\n\nvariables {\u03b9 : Type*} {\u03b9' : Type*} {R : Type*} {K : Type*}\nvariables {M : Type*} {M' M'' : Type*} {V : Type u} {V' : Type*}\n\n\nsection module\n\nvariables {v : \u03b9 \u2192 M}\nvariables [semiring R] [add_comm_monoid M] [add_comm_monoid M'] [add_comm_monoid M'']\nvariables [module R M] [module R M'] [module R M'']\nvariables {a b : R} {x y : M}\n\nvariables (R) (v)\n\n\/-- `linear_independent R v` states the family of vectors `v` is linearly independent over `R`. -\/\ndef linear_independent : Prop := (finsupp.total \u03b9 M R v).ker = \u22a5\nvariables {R} {v}\n\ntheorem linear_independent_iff : linear_independent R v \u2194\n  \u2200l, finsupp.total \u03b9 M R v l = 0 \u2192 l = 0 :=\nby simp [linear_independent, linear_map.ker_eq_bot']\n\ntheorem linear_independent_iff' : linear_independent R v \u2194\n  \u2200 s : finset \u03b9, \u2200 g : \u03b9 \u2192 R, \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 i \u2208 s, g i = 0 :=\nlinear_independent_iff.trans\n\u27e8\u03bb hf s g hg i his, have h : _ := hf (\u2211 i in s, finsupp.single i (g i)) $\n      by simpa only [linear_map.map_sum, finsupp.total_single] using hg, calc\n    g i = (finsupp.lapply i : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] R) (finsupp.single i (g i)) :\n      by rw [finsupp.lapply_apply, finsupp.single_eq_same]\n    ... = \u2211 j in s, (finsupp.lapply i : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] R) (finsupp.single j (g j)) :\n      eq.symm $ finset.sum_eq_single i\n        (\u03bb j hjs hji, by rw [finsupp.lapply_apply, finsupp.single_eq_of_ne hji])\n        (\u03bb hnis, hnis.elim his)\n    ... = (\u2211 j in s, finsupp.single j (g j)) i : (finsupp.lapply i : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] R).map_sum.symm\n    ... = 0 : finsupp.ext_iff.1 h i,\n\u03bb hf l hl, finsupp.ext $ \u03bb i, classical.by_contradiction $ \u03bb hni, hni $ hf _ _ hl _ $\n  finsupp.mem_support_iff.2 hni\u27e9\n\ntheorem linear_independent_iff'' :\n  linear_independent R v \u2194 \u2200 (s : finset \u03b9) (g : \u03b9 \u2192 R) (hg : \u2200 i \u2209 s, g i = 0),\n    \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 i, g i = 0 :=\nlinear_independent_iff'.trans \u27e8\u03bb H s g hg hv i, if his : i \u2208 s then H s g hv i his else hg i his,\n\u03bb H s g hg i hi, by { convert H s (\u03bb j, if j \u2208 s then g j else 0) (\u03bb j hj, if_neg hj)\n    (by simp_rw [ite_smul, zero_smul, finset.sum_extend_by_zero, hg]) i,\n  exact (if_pos hi).symm }\u27e9\n\ntheorem linear_dependent_iff : \u00ac linear_independent R v \u2194\n  \u2203 s : finset \u03b9, \u2203 g : \u03b9 \u2192 R, (\u2211 i in s, g i \u2022 v i) = 0 \u2227 (\u2203 i \u2208 s, g i \u2260 0) :=\nbegin\n  rw linear_independent_iff',\n  simp only [exists_prop, not_forall],\nend\n\ntheorem fintype.linear_independent_iff [fintype \u03b9] :\n  linear_independent R v \u2194 \u2200 g : \u03b9 \u2192 R, \u2211 i, g i \u2022 v i = 0 \u2192 \u2200 i, g i = 0 :=\nbegin\n  refine \u27e8\u03bb H g, by simpa using linear_independent_iff'.1 H finset.univ g,\n    \u03bb H, linear_independent_iff''.2 $ \u03bb s g hg hs i, H _ _ _\u27e9,\n  rw \u2190 hs,\n  refine (finset.sum_subset (finset.subset_univ _) (\u03bb i _ hi, _)).symm,\n  rw [hg i hi, zero_smul]\nend\n\n\/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : \u03b9 \u2192 R` to `\u2211 i, c i \u2022 v i` has the trivial kernel. -\/\ntheorem fintype.linear_independent_iff' [fintype \u03b9] :\n  linear_independent R v \u2194\n    (linear_map.lsum R (\u03bb i : \u03b9, R) \u2115 (\u03bb i, linear_map.id.smul_right (v i))).ker = \u22a5 :=\nby simp [fintype.linear_independent_iff, linear_map.ker_eq_bot', funext_iff]\n\nlemma linear_independent_empty_type (h : \u00ac nonempty \u03b9) : linear_independent R v :=\nbegin\n rw [linear_independent_iff],\n intros,\n ext i,\n exact false.elim (h \u27e8i\u27e9)\nend\n\nlemma linear_independent.ne_zero [nontrivial R]\n  (i : \u03b9) (hv : linear_independent R v) : v i \u2260 0 :=\n\u03bb h, @zero_ne_one R _ _ $ eq.symm begin\n  suffices : (finsupp.single i 1 : \u03b9 \u2192\u2080 R) i = 0, {simpa},\n  rw linear_independent_iff.1 hv (finsupp.single i 1),\n  { simp },\n  { simp [h] }\nend\n\n\/-- A subfamily of a linearly independent family (i.e., a composition with an injective map) is a\nlinearly independent family. -\/\nlemma linear_independent.comp\n  (h : linear_independent R v) (f : \u03b9' \u2192 \u03b9) (hf : injective f) : linear_independent R (v \u2218 f) :=\nbegin\n  rw [linear_independent_iff, finsupp.total_comp],\n  intros l hl,\n  have h_map_domain : \u2200 x, (finsupp.map_domain f l) (f x) = 0,\n    by rw linear_independent_iff.1 h (finsupp.map_domain f l) hl; simp,\n  ext x,\n  convert h_map_domain x,\n  rw [finsupp.map_domain_apply hf]\nend\n\n\/-- If `v` is a linearly independent family of vectors and the kernel of a linear map `f` is\ndisjoint with the sumodule spaned by the vectors of `v`, then `f \u2218 v` is a linearly independent\nfamily of vectors. See also `linear_independent.map'` for a special case assuming `ker f = \u22a5`. -\/\nlemma linear_independent.map (hv : linear_independent R v) {f : M \u2192\u2097[R] M'}\n  (hf_inj : disjoint (span R (range v)) f.ker) : linear_independent R (f \u2218 v) :=\nbegin\n  rw [disjoint, \u2190 set.image_univ, finsupp.span_image_eq_map_total, map_inf_eq_map_inf_comap,\n    map_le_iff_le_comap, comap_bot, finsupp.supported_univ, top_inf_eq] at hf_inj,\n  unfold linear_independent at hv \u22a2,\n  rw [hv, le_bot_iff] at hf_inj,\n  haveI : inhabited M := \u27e80\u27e9,\n  rw [finsupp.total_comp, @finsupp.lmap_domain_total _ _ R _ _ _ _ _ _ _ _ _ _ f,\n    linear_map.ker_comp, hf_inj],\n  exact \u03bb _, rfl,\nend\n\n\/-- An injective linear map sends linearly independent families of vectors to linearly independent\nfamilies of vectors. See also `linear_independent.map` for a more general statement. -\/\nlemma linear_independent.map' (hv : linear_independent R v) (f : M \u2192\u2097[R] M')\n  (hf_inj : f.ker = \u22a5) : linear_independent R (f \u2218 v) :=\nhv.map $ by simp [hf_inj]\n\n\/-- If the image of a family of vectors under a linear map is linearly independent, then so is\nthe original family. -\/\nlemma linear_independent.of_comp (f : M \u2192\u2097[R] M') (hfv : linear_independent R (f \u2218 v)) :\n  linear_independent R v :=\nlinear_independent_iff'.2 $ \u03bb s g hg i his,\nhave \u2211 (i : \u03b9) in s, g i \u2022 f (v i) = 0,\n  by simp_rw [\u2190 f.map_smul, \u2190 f.map_sum, hg, f.map_zero],\nlinear_independent_iff'.1 hfv s g this i his\n\n\/-- If `f` is an injective linear map, then the family `f \u2218 v` is linearly independent\nif and only if the family `v` is linearly independent. -\/\nprotected lemma linear_map.linear_independent_iff (f : M \u2192\u2097[R] M') (hf_inj : f.ker = \u22a5) :\n  linear_independent R (f \u2218 v) \u2194 linear_independent R v :=\n\u27e8\u03bb h, h.of_comp f, \u03bb h, h.map $ by simp only [hf_inj, disjoint_bot_right]\u27e9\n\n@[nontriviality]\nlemma linear_independent_of_subsingleton [subsingleton R] : linear_independent R v :=\nlinear_independent_iff.2 (\u03bb l hl, subsingleton.elim _ _)\n\ntheorem linear_independent_equiv (e : \u03b9 \u2243 \u03b9') {f : \u03b9' \u2192 M} :\n  linear_independent R (f \u2218 e) \u2194 linear_independent R f :=\n\u27e8\u03bb h, function.comp.right_id f \u25b8 e.self_comp_symm \u25b8 h.comp _ e.symm.injective,\n\u03bb h, h.comp _ e.injective\u27e9\n\ntheorem linear_independent_equiv' (e : \u03b9 \u2243 \u03b9') {f : \u03b9' \u2192 M} {g : \u03b9 \u2192 M} (h : f \u2218 e = g) :\n  linear_independent R g \u2194 linear_independent R f :=\nh \u25b8 linear_independent_equiv e\n\ntheorem linear_independent_subtype_range {\u03b9} {f : \u03b9 \u2192 M} (hf : injective f) :\n  linear_independent R (coe : range f \u2192 M) \u2194 linear_independent R f :=\niff.symm $ linear_independent_equiv' (equiv.of_injective f hf) rfl\n\nalias linear_independent_subtype_range \u2194 linear_independent.of_subtype_range _\n\ntheorem linear_independent_image {\u03b9} {s : set \u03b9} {f : \u03b9 \u2192 M} (hf : set.inj_on f s) :\n  linear_independent R (\u03bb x : s, f x) \u2194 linear_independent R (\u03bb x : f '' s, (x : M)) :=\nlinear_independent_equiv' (equiv.set.image_of_inj_on _ _ hf) rfl\n\nlemma linear_independent_span (hs : linear_independent R v) :\n  @linear_independent \u03b9 R (span R (range v))\n      (\u03bb i : \u03b9, \u27e8v i, subset_span (mem_range_self i)\u27e9) _ _ _ :=\nlinear_independent.of_comp (span R (range v)).subtype hs\n\n\/-- See `linear_independent.fin_cons` for a family of elements in a vector space. -\/\nlemma linear_independent.fin_cons' {m : \u2115} (x : M) (v : fin m \u2192 M)\n  (hli : linear_independent R v)\n  (x_ortho : (\u2200 (c : R) (y : submodule.span R (set.range v)), c \u2022 x + y = (0 : M) \u2192 c = 0)) :\n  linear_independent R (fin.cons x v : fin m.succ \u2192 M) :=\nbegin\n  rw fintype.linear_independent_iff at hli \u22a2,\n  rintros g total_eq j,\n  have zero_not_mem : (0 : fin m.succ) \u2209 finset.univ.image (fin.succ : fin m \u2192 fin m.succ),\n  { rw finset.mem_image,\n    rintro \u27e8x, hx, succ_eq\u27e9,\n    exact fin.succ_ne_zero _ succ_eq },\n  simp only [submodule.coe_mk, fin.univ_succ, finset.sum_insert zero_not_mem,\n  fin.cons_zero, fin.cons_succ,\n  forall_true_iff, imp_self, fin.succ_inj, finset.sum_image] at total_eq,\n  have : g 0 = 0,\n  { refine x_ortho (g 0) \u27e8\u2211 (i : fin m), g i.succ \u2022 v i, _\u27e9 total_eq,\n    exact sum_mem _ (\u03bb i _, smul_mem _ _ (subset_span \u27e8i, rfl\u27e9)) },\n  refine fin.cases this (\u03bb j, _) j,\n  apply hli (\u03bb i, g i.succ),\n  simpa only [this, zero_smul, zero_add] using total_eq\nend\n\n\/-- A set of linearly independent vectors in a module `M` over a semiring `K` is also linearly\nindependent over a subring `R` of `K`.\nThe implementation uses minimal assumptions about the relationship between `R`, `K` and `M`.\nThe version where `K` is an `R`-algebra is `linear_independent.restrict_scalars_algebras`.\n -\/\nlemma linear_independent.restrict_scalars [semiring K] [smul_with_zero R K] [module K M]\n  [is_scalar_tower R K M]\n  (hinj : function.injective (\u03bb r : R, r \u2022 (1 : K))) (li : linear_independent K v) :\n  linear_independent R v :=\nbegin\n  refine linear_independent_iff'.mpr (\u03bb s g hg i hi, hinj (eq.trans _ (zero_smul _ _).symm)),\n  refine (linear_independent_iff'.mp li : _) _ _ _ i hi,\n  simp_rw [smul_assoc, one_smul],\n  exact hg,\nend\n\nsection subtype\n\/-! The following lemmas use the subtype defined by a set in `M` as the index set `\u03b9`. -\/\n\ntheorem linear_independent_comp_subtype {s : set \u03b9} :\n  linear_independent R (v \u2218 coe : s \u2192 M) \u2194\n  \u2200 l \u2208 (finsupp.supported R R s), (finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0 :=\nbegin\n  simp only [linear_independent_iff, (\u2218), finsupp.mem_supported, finsupp.total_apply,\n    set.subset_def, finset.mem_coe],\n  split,\n  { intros h l hl\u2081 hl\u2082,\n    have := h (l.subtype_domain s) ((finsupp.sum_subtype_domain_index hl\u2081).trans hl\u2082),\n    exact (finsupp.subtype_domain_eq_zero_iff hl\u2081).1 this },\n  { intros h l hl,\n    refine finsupp.emb_domain_eq_zero.1 (h (l.emb_domain $ function.embedding.subtype s) _ _),\n    { suffices : \u2200 i hi, \u00acl \u27e8i, hi\u27e9 = 0 \u2192 i \u2208 s, by simpa,\n      intros, assumption },\n    { rwa [finsupp.emb_domain_eq_map_domain, finsupp.sum_map_domain_index],\n      exacts [\u03bb _, zero_smul _ _, \u03bb _ _ _, add_smul _ _ _] } }\nend\n\nlemma linear_dependent_comp_subtype' {s : set \u03b9} :\n  \u00ac linear_independent R (v \u2218 coe : s \u2192 M) \u2194\n  \u2203 f : \u03b9 \u2192\u2080 R, f \u2208 finsupp.supported R R s \u2227 finsupp.total \u03b9 M R v f = 0 \u2227 f \u2260 0 :=\nby simp [linear_independent_comp_subtype]\n\n\/-- A version of `linear_dependent_comp_subtype'` with `finsupp.total` unfolded. -\/\nlemma linear_dependent_comp_subtype {s : set \u03b9} :\n  \u00ac linear_independent R (v \u2218 coe : s \u2192 M) \u2194\n  \u2203 f : \u03b9 \u2192\u2080 R, f \u2208 finsupp.supported R R s \u2227 \u2211 i in f.support, f i \u2022 v i = 0 \u2227 f \u2260 0 :=\nlinear_dependent_comp_subtype'\n\ntheorem linear_independent_subtype {s : set M} :\n  linear_independent R (\u03bb x, x : s \u2192 M) \u2194\n  \u2200 l \u2208 (finsupp.supported R R s), (finsupp.total M M R id) l = 0 \u2192 l = 0 :=\nby apply @linear_independent_comp_subtype _ _ _ id\n\ntheorem linear_independent_comp_subtype_disjoint {s : set \u03b9} :\n  linear_independent R (v \u2218 coe : s \u2192 M) \u2194\n  disjoint (finsupp.supported R R s) (finsupp.total \u03b9 M R v).ker :=\nby rw [linear_independent_comp_subtype, linear_map.disjoint_ker]\n\ntheorem linear_independent_subtype_disjoint {s : set M} :\n  linear_independent R (\u03bb x, x : s \u2192 M) \u2194\n  disjoint (finsupp.supported R R s) (finsupp.total M M R id).ker :=\nby apply @linear_independent_comp_subtype_disjoint _ _ _ id\n\ntheorem linear_independent_iff_total_on {s : set M} :\n  linear_independent R (\u03bb x, x : s \u2192 M) \u2194 (finsupp.total_on M M R id s).ker = \u22a5 :=\nby rw [finsupp.total_on, linear_map.ker, linear_map.comap_cod_restrict, map_bot, comap_bot,\n  linear_map.ker_comp, linear_independent_subtype_disjoint, disjoint, \u2190 map_comap_subtype,\n  map_le_iff_le_comap, comap_bot, ker_subtype, le_bot_iff]\n\nlemma linear_independent.restrict_of_comp_subtype {s : set \u03b9}\n  (hs : linear_independent R (v \u2218 coe : s \u2192 M)) :\n  linear_independent R (s.restrict v) :=\nhs\n\nvariables (R M)\nlemma linear_independent_empty : linear_independent R (\u03bb x, x : (\u2205 : set M) \u2192 M) :=\nby simp [linear_independent_subtype_disjoint]\nvariables {R M}\n\nlemma linear_independent.mono {t s : set M} (h : t \u2286 s) :\n  linear_independent R (\u03bb x, x : s \u2192 M) \u2192 linear_independent R (\u03bb x, x : t \u2192 M) :=\nbegin\n simp only [linear_independent_subtype_disjoint],\n exact (disjoint.mono_left (finsupp.supported_mono h))\nend\n\nlemma linear_independent_of_finite (s : set M)\n  (H : \u2200 t \u2286 s, finite t \u2192 linear_independent R (\u03bb x, x : t \u2192 M)) :\n  linear_independent R (\u03bb x, x : s \u2192 M) :=\nlinear_independent_subtype.2 $\n  \u03bb l hl, linear_independent_subtype.1 (H _ hl (finset.finite_to_set _)) l (subset.refl _)\n\nlemma linear_independent_Union_of_directed {\u03b7 : Type*}\n  {s : \u03b7 \u2192 set M} (hs : directed (\u2286) s)\n  (h : \u2200 i, linear_independent R (\u03bb x, x : s i \u2192 M)) :\n  linear_independent R (\u03bb x, x : (\u22c3 i, s i) \u2192 M) :=\nbegin\n  by_cases h\u03b7 : nonempty \u03b7,\n  { resetI,\n    refine linear_independent_of_finite (\u22c3 i, s i) (\u03bb t ht ft, _),\n    rcases finite_subset_Union ft ht with \u27e8I, fi, hI\u27e9,\n    rcases hs.finset_le fi.to_finset with \u27e8i, hi\u27e9,\n    exact (h i).mono (subset.trans hI $ bUnion_subset $\n      \u03bb j hj, hi j (fi.mem_to_finset.2 hj)) },\n  { refine (linear_independent_empty _ _).mono _,\n    rintro _ \u27e8_, \u27e8i, _\u27e9, _\u27e9, exact h\u03b7 \u27e8i\u27e9 }\nend\n\nlemma linear_independent_sUnion_of_directed {s : set (set M)}\n  (hs : directed_on (\u2286) s)\n  (h : \u2200 a \u2208 s, linear_independent R (\u03bb x, x : (a : set M) \u2192 M)) :\n  linear_independent R (\u03bb x, x : (\u22c3\u2080 s) \u2192 M) :=\nby rw sUnion_eq_Union; exact\nlinear_independent_Union_of_directed hs.directed_coe (by simpa using h)\n\nlemma linear_independent_bUnion_of_directed {\u03b7} {s : set \u03b7} {t : \u03b7 \u2192 set M}\n  (hs : directed_on (t \u207b\u00b9'o (\u2286)) s) (h : \u2200a\u2208s, linear_independent R (\u03bb x, x : t a \u2192 M)) :\n  linear_independent R (\u03bb x, x : (\u22c3a\u2208s, t a) \u2192 M) :=\nby rw bUnion_eq_Union; exact\nlinear_independent_Union_of_directed (directed_comp.2 $ hs.directed_coe) (by simpa using h)\n\nend subtype\n\nend module\n\n\/-! ### Properties which require `ring R` -\/\n\nsection module\nvariables {v : \u03b9 \u2192 M}\nvariables [ring R] [add_comm_group M] [add_comm_group M'] [add_comm_group M'']\nvariables [module R M] [module R M'] [module R M'']\nvariables {a b : R} {x y : M}\n\ntheorem linear_independent_iff_injective_total : linear_independent R v \u2194\n  function.injective (finsupp.total \u03b9 M R v) :=\nlinear_independent_iff.trans (finsupp.total \u03b9 M R v).to_add_monoid_hom.injective_iff.symm\n\nalias linear_independent_iff_injective_total \u2194 linear_independent.injective_total _\n\nlemma linear_independent.injective [nontrivial R] (hv : linear_independent R v) :\n  injective v :=\nbegin\n  intros i j hij,\n  let l : \u03b9 \u2192\u2080 R := finsupp.single i (1 : R) - finsupp.single j 1,\n  have h_total : finsupp.total \u03b9 M R v l = 0,\n  { simp_rw [linear_map.map_sub, finsupp.total_apply],\n    simp [hij] },\n  have h_single_eq : finsupp.single i (1 : R) = finsupp.single j 1,\n  { rw linear_independent_iff at hv,\n    simp [eq_add_of_sub_eq' (hv l h_total)] },\n  simpa [finsupp.single_eq_single_iff] using h_single_eq\nend\n\ntheorem linear_independent.to_subtype_range {\u03b9} {f : \u03b9 \u2192 M} (hf : linear_independent R f) :\n  linear_independent R (coe : range f \u2192 M) :=\nbegin\n  nontriviality R,\n  exact (linear_independent_subtype_range hf.injective).2 hf\nend\n\ntheorem linear_independent.to_subtype_range' {\u03b9} {f : \u03b9 \u2192 M} (hf : linear_independent R f)\n  {t} (ht : range f = t) :\n  linear_independent R (coe : t \u2192 M) :=\nht \u25b8 hf.to_subtype_range\n\ntheorem linear_independent.image_of_comp {\u03b9 \u03b9'} (s : set \u03b9) (f : \u03b9 \u2192 \u03b9') (g : \u03b9' \u2192 M)\n  (hs : linear_independent R (\u03bb x : s, g (f x))) :\n  linear_independent R (\u03bb x : f '' s, g x) :=\nbegin\n  nontriviality R,\n  have : inj_on f s, from inj_on_iff_injective.2 hs.injective.of_comp,\n  exact (linear_independent_equiv' (equiv.set.image_of_inj_on f s this) rfl).1 hs\nend\n\ntheorem linear_independent.image {\u03b9} {s : set \u03b9} {f : \u03b9 \u2192 M}\n  (hs : linear_independent R (\u03bb x : s, f x)) : linear_independent R (\u03bb x : f '' s, (x : M)) :=\nby convert linear_independent.image_of_comp s f id hs\n\nlemma linear_independent.group_smul\n  {G : Type*} [hG : group G] [distrib_mul_action G R] [distrib_mul_action G M]\n  [is_scalar_tower G R M] [smul_comm_class G R M] {v : \u03b9 \u2192 M} (hv : linear_independent R v)\n  (w : \u03b9 \u2192 G) : linear_independent R (w \u2022 v) :=\nbegin\n  rw linear_independent_iff'' at hv \u22a2,\n  intros s g hgs hsum i,\n  refine (smul_eq_zero_iff_eq (w i)).1 _,\n  refine hv s (\u03bb i, w i \u2022 g i) (\u03bb i hi, _) _ i,\n  { dsimp only,\n    exact (hgs i hi).symm \u25b8 smul_zero _ },\n  { rw [\u2190 hsum, finset.sum_congr rfl _],\n    intros, erw [pi.smul_apply, smul_assoc, smul_comm] },\nend\n\n-- This lemma cannot be proved with `linear_independent.group_smul` since the action of\n-- `units R` on `R` is not commutative.\nlemma linear_independent.units_smul {v : \u03b9 \u2192 M} (hv : linear_independent R v)\n  (w : \u03b9 \u2192 units R) : linear_independent R (w \u2022 v) :=\nbegin\n  rw linear_independent_iff'' at hv \u22a2,\n  intros s g hgs hsum i,\n  rw \u2190 (w i).mul_left_eq_zero,\n  refine hv s (\u03bb i, g i \u2022 w i) (\u03bb i hi, _) _ i,\n  { dsimp only,\n    exact (hgs i hi).symm \u25b8 zero_smul _ _ },\n  { rw [\u2190 hsum, finset.sum_congr rfl _],\n    intros,\n    erw [pi.smul_apply, smul_assoc],\n    refl }\nend\n\nsection subtype\n\/-! The following lemmas use the subtype defined by a set in `M` as the index set `\u03b9`. -\/\n\nlemma linear_independent.disjoint_span_image (hv : linear_independent R v) {s t : set \u03b9}\n  (hs : disjoint s t) :\n  disjoint (submodule.span R $ v '' s) (submodule.span R $ v '' t) :=\nbegin\n  simp only [disjoint_def, finsupp.mem_span_image_iff_total],\n  rintros _ \u27e8l\u2081, hl\u2081, rfl\u27e9 \u27e8l\u2082, hl\u2082, H\u27e9,\n  rw [hv.injective_total.eq_iff] at H, subst l\u2082,\n  have : l\u2081 = 0 := finsupp.disjoint_supported_supported hs (submodule.mem_inf.2 \u27e8hl\u2081, hl\u2082\u27e9),\n  simp [this]\nend\n\nlemma linear_independent_sum {v : \u03b9 \u2295 \u03b9' \u2192 M} :\n  linear_independent R v \u2194 linear_independent R (v \u2218 sum.inl) \u2227\n    linear_independent R (v \u2218 sum.inr) \u2227\n      disjoint (submodule.span R (range (v \u2218 sum.inl))) (submodule.span R (range (v \u2218 sum.inr))) :=\nbegin\n  rw [range_comp v, range_comp v],\n  refine \u27e8\u03bb h, \u27e8h.comp _ sum.inl_injective, h.comp _ sum.inr_injective,\n    h.disjoint_span_image is_compl_range_inl_range_inr.1\u27e9, _\u27e9,\n  rintro \u27e8hl, hr, hlr\u27e9,\n  rw [linear_independent_iff'] at *,\n  intros s g hg i hi,\n  have : \u2211 i in s.preimage sum.inl (sum.inl_injective.inj_on _), (\u03bb x, g x \u2022 v x) (sum.inl i) +\n    \u2211 i in s.preimage sum.inr (sum.inr_injective.inj_on _), (\u03bb x, g x \u2022 v x) (sum.inr i) = 0,\n  { rw [finset.sum_preimage', finset.sum_preimage', \u2190 finset.sum_union, \u2190 finset.filter_or],\n    { simpa only [\u2190 mem_union, range_inl_union_range_inr, mem_univ, finset.filter_true] },\n    { exact finset.disjoint_filter.2 (\u03bb x hx, disjoint_left.1 is_compl_range_inl_range_inr.1) } },\n  { rw \u2190 eq_neg_iff_add_eq_zero at this,\n    rw [disjoint_def'] at hlr,\n    have A := hlr _ (sum_mem _ $ \u03bb i hi, _) _ (neg_mem _ $ sum_mem _ $ \u03bb i hi, _) this,\n    { cases i with i i,\n      { exact hl _ _ A i (finset.mem_preimage.2 hi) },\n      { rw [this, neg_eq_zero] at A,\n        exact hr _ _ A i (finset.mem_preimage.2 hi) } },\n    { exact smul_mem _ _ (subset_span \u27e8sum.inl i, mem_range_self _, rfl\u27e9) },\n    { exact smul_mem _ _ (subset_span \u27e8sum.inr i, mem_range_self _, rfl\u27e9) } }\nend\n\nlemma linear_independent.sum_type {v' : \u03b9' \u2192 M} (hv : linear_independent R v)\n  (hv' : linear_independent R v')\n  (h : disjoint (submodule.span R (range v)) (submodule.span R (range v'))) :\n  linear_independent R (sum.elim v v') :=\nlinear_independent_sum.2 \u27e8hv, hv', h\u27e9\n\nlemma linear_independent.union {s t : set M}\n  (hs : linear_independent R (\u03bb x, x : s \u2192 M)) (ht : linear_independent R (\u03bb x, x : t \u2192 M))\n  (hst : disjoint (span R s) (span R t)) :\n  linear_independent R (\u03bb x, x : (s \u222a t) \u2192 M) :=\n(hs.sum_type ht $ by simpa).to_subtype_range' $ by simp\n\nlemma linear_independent_Union_finite_subtype {\u03b9 : Type*} {f : \u03b9 \u2192 set M}\n  (hl : \u2200i, linear_independent R (\u03bb x, x : f i \u2192 M))\n  (hd : \u2200i, \u2200t:set \u03b9, finite t \u2192 i \u2209 t \u2192 disjoint (span R (f i)) (\u2a06i\u2208t, span R (f i))) :\n  linear_independent R (\u03bb x, x : (\u22c3i, f i) \u2192 M) :=\nbegin\n  rw [Union_eq_Union_finset f],\n  apply linear_independent_Union_of_directed,\n  apply directed_of_sup,\n  exact (assume t\u2081 t\u2082 ht, Union_subset_Union $ assume i, Union_subset_Union_const $ assume h, ht h),\n  assume t, rw [set.Union, \u2190 finset.sup_eq_supr],\n  refine t.induction_on _ _,\n  { rw finset.sup_empty,\n    apply linear_independent_empty_type (not_nonempty_iff_imp_false.2 _),\n    exact \u03bb x, set.not_mem_empty x (subtype.mem x) },\n  { rintros i s his ih,\n    rw [finset.sup_insert],\n    refine (hl _).union ih _,\n    rw [finset.sup_eq_supr],\n    refine (hd i _ _ his).mono_right _,\n    { simp only [(span_Union _).symm],\n      refine span_mono (@supr_le_supr2 (set M) _ _ _ _ _ _),\n      rintros i, exact \u27e8i, le_refl _\u27e9 },\n    { exact s.finite_to_set } }\nend\n\nlemma linear_independent_Union_finite {\u03b7 : Type*} {\u03b9s : \u03b7 \u2192 Type*}\n  {f : \u03a0 j : \u03b7, \u03b9s j \u2192 M}\n  (hindep : \u2200j, linear_independent R (f j))\n  (hd : \u2200i, \u2200t:set \u03b7, finite t \u2192 i \u2209 t \u2192\n      disjoint (span R (range (f i))) (\u2a06i\u2208t, span R (range (f i)))) :\n  linear_independent R (\u03bb ji : \u03a3 j, \u03b9s j, f ji.1 ji.2) :=\nbegin\n  nontriviality R,\n  apply linear_independent.of_subtype_range,\n  { rintros \u27e8x\u2081, x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9 hxy,\n    by_cases h_cases : x\u2081 = y\u2081,\n    subst h_cases,\n    { apply sigma.eq,\n      rw linear_independent.injective (hindep _) hxy,\n      refl },\n    { have h0 : f x\u2081 x\u2082 = 0,\n      { apply disjoint_def.1 (hd x\u2081 {y\u2081} (finite_singleton y\u2081)\n          (\u03bb h, h_cases (eq_of_mem_singleton h))) (f x\u2081 x\u2082) (subset_span (mem_range_self _)),\n        rw supr_singleton,\n        simp only at hxy,\n        rw hxy,\n        exact (subset_span (mem_range_self y\u2082)) },\n      exact false.elim ((hindep x\u2081).ne_zero _ h0) } },\n  rw range_sigma_eq_Union_range,\n  apply linear_independent_Union_finite_subtype (\u03bb j, (hindep j).to_subtype_range) hd,\nend\n\nend subtype\n\nsection repr\nvariables (hv : linear_independent R v)\n\n\/-- Canonical isomorphism between linear combinations and the span of linearly independent vectors.\n-\/\ndef linear_independent.total_equiv (hv : linear_independent R v) :\n  (\u03b9 \u2192\u2080 R) \u2243\u2097[R] span R (range v) :=\nbegin\napply linear_equiv.of_bijective\n  (linear_map.cod_restrict (span R (range v)) (finsupp.total \u03b9 M R v) _),\n{ rw linear_map.ker_cod_restrict,\n  apply hv },\n{ rw [linear_map.range_eq_map, linear_map.map_cod_restrict, \u2190 linear_map.range_le_iff_comap,\n    range_subtype, map_top],\n  rw finsupp.range_total,\n  apply le_refl (span R (range v)) },\n{ intro l,\n  rw \u2190 finsupp.range_total,\n  rw linear_map.mem_range,\n  apply mem_range_self l }\nend\n\n\/-- Linear combination representing a vector in the span of linearly independent vectors.\n\nGiven a family of linearly independent vectors, we can represent any vector in their span as\na linear combination of these vectors. These are provided by this linear map.\nIt is simply one direction of `linear_independent.total_equiv`. -\/\ndef linear_independent.repr (hv : linear_independent R v) :\n  span R (range v) \u2192\u2097[R] \u03b9 \u2192\u2080 R := hv.total_equiv.symm\n\nlemma linear_independent.total_repr (x) : finsupp.total \u03b9 M R v (hv.repr x) = x :=\nsubtype.ext_iff.1 (linear_equiv.apply_symm_apply hv.total_equiv x)\n\nlemma linear_independent.total_comp_repr :\n  (finsupp.total \u03b9 M R v).comp hv.repr = submodule.subtype _ :=\nlinear_map.ext $ hv.total_repr\n\nlemma linear_independent.repr_ker : hv.repr.ker = \u22a5 :=\nby rw [linear_independent.repr, linear_equiv.ker]\n\nlemma linear_independent.repr_range : hv.repr.range = \u22a4 :=\nby rw [linear_independent.repr, linear_equiv.range]\n\nlemma linear_independent.repr_eq\n  {l : \u03b9 \u2192\u2080 R} {x} (eq : finsupp.total \u03b9 M R v l = \u2191x) :\n  hv.repr x = l :=\nbegin\n  have : \u2191((linear_independent.total_equiv hv : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] span R (range v)) l)\n      = finsupp.total \u03b9 M R v l := rfl,\n  have : (linear_independent.total_equiv hv : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] span R (range v)) l = x,\n  { rw eq at this,\n    exact subtype.ext_iff.2 this },\n  rw \u2190linear_equiv.symm_apply_apply hv.total_equiv l,\n  rw \u2190this,\n  refl,\nend\n\nlemma linear_independent.repr_eq_single (i) (x) (hx : \u2191x = v i) :\n  hv.repr x = finsupp.single i 1 :=\nbegin\n  apply hv.repr_eq,\n  simp [finsupp.total_single, hx]\nend\n\n-- TODO: why is this so slow?\nlemma linear_independent_iff_not_smul_mem_span :\n  linear_independent R v \u2194 (\u2200 (i : \u03b9) (a : R), a \u2022 (v i) \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0) :=\n\u27e8 \u03bb hv i a ha, begin\n  rw [finsupp.span_image_eq_map_total, mem_map] at ha,\n  rcases ha with \u27e8l, hl, e\u27e9,\n  rw sub_eq_zero.1 (linear_independent_iff.1 hv (l - finsupp.single i a) (by simp [e])) at hl,\n  by_contra hn,\n  exact (not_mem_of_mem_diff (hl $ by simp [hn])) (mem_singleton _),\nend, \u03bb H, linear_independent_iff.2 $ \u03bb l hl, begin\n  ext i, simp only [finsupp.zero_apply],\n  by_contra hn,\n  refine hn (H i _ _),\n  refine (finsupp.mem_span_image_iff_total _).2 \u27e8finsupp.single i (l i) - l, _, _\u27e9,\n  { rw finsupp.mem_supported',\n    intros j hj,\n    have hij : j = i :=\n      not_not.1\n          (\u03bb hij : j \u2260 i, hj ((mem_diff _).2 \u27e8mem_univ _, \u03bb h, hij (eq_of_mem_singleton h)\u27e9)),\n    simp [hij] },\n  { simp [hl] }\nend\u27e9\n\nvariable (R)\n\nlemma exists_maximal_independent' (s : \u03b9 \u2192 M) :\n  \u2203 I : set \u03b9, linear_independent R (\u03bb x : I, s x) \u2227\n    \u2200 J : set \u03b9, I \u2286 J \u2192 linear_independent R (\u03bb x : J, s x) \u2192 I = J :=\nbegin\n  let indep : set \u03b9 \u2192 Prop := \u03bb I, linear_independent R (s \u2218 coe : I \u2192 M),\n  let X := { I : set \u03b9 \/\/ indep I },\n  let r : X \u2192 X \u2192 Prop := \u03bb I J, I.1 \u2286 J.1,\n  have key : \u2200 c : set X, zorn.chain r c \u2192 indep (\u22c3 (I : X) (H : I \u2208 c), I),\n  { intros c hc,\n    dsimp [indep],\n    rw [linear_independent_comp_subtype],\n    intros f hsupport hsum,\n    rcases eq_empty_or_nonempty c with rfl | \u27e8a, hac\u27e9,\n    { simpa using hsupport },\n    haveI : is_refl X r := \u27e8\u03bb _, set.subset.refl _\u27e9,\n    obtain \u27e8I, I_mem, hI\u27e9 : \u2203 I \u2208 c, (f.support : set \u03b9) \u2286 I :=\n      finset.exists_mem_subset_of_subset_bUnion_of_directed_on hac hc.directed_on hsupport,\n    exact linear_independent_comp_subtype.mp I.2 f hI hsum },\n  have trans : transitive r := \u03bb I J K, set.subset.trans,\n  obtain \u27e8\u27e8I, hli : indep I\u27e9, hmax : \u2200 a, r \u27e8I, hli\u27e9 a \u2192 r a \u27e8I, hli\u27e9\u27e9 :=\n    @zorn.exists_maximal_of_chains_bounded _ r\n    (\u03bb c hc, \u27e8\u27e8\u22c3 I \u2208 c, (I : set \u03b9), key c hc\u27e9, \u03bb I, set.subset_bUnion_of_mem\u27e9) trans,\n  exact \u27e8I, hli, \u03bb J hsub hli, set.subset.antisymm hsub (hmax \u27e8J, hli\u27e9 hsub)\u27e9,\nend\n\nlemma exists_maximal_independent (s : \u03b9 \u2192 M) : \u2203 I : set \u03b9, linear_independent R (\u03bb x : I, s x) \u2227\n  \u2200 i \u2209 I, \u2203 a : R, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I) :=\nbegin\n  classical,\n  rcases exists_maximal_independent' R s with \u27e8I, hIlinind, hImaximal\u27e9,\n  use [I, hIlinind],\n  intros i hi,\n  specialize hImaximal (I \u222a {i}) (by simp),\n  set J := I \u222a {i} with hJ,\n  have memJ : \u2200 {x}, x \u2208 J \u2194 x = i \u2228 x \u2208 I, by simp [hJ],\n  have hiJ : i \u2208 J := by simp,\n  have h := mt hImaximal _, swap,\n  { intro h2,\n    rw h2 at hi,\n    exact absurd hiJ hi },\n  obtain \u27e8f, supp_f, sum_f, f_ne\u27e9 := linear_dependent_comp_subtype.mp h,\n  have hfi : f i \u2260 0,\n  { contrapose hIlinind,\n    refine linear_dependent_comp_subtype.mpr \u27e8f, _, sum_f, f_ne\u27e9,\n    simp only [finsupp.mem_supported, hJ] at \u22a2 supp_f,\n    rintro x hx,\n    refine (memJ.mp (supp_f hx)).resolve_left _,\n    rintro rfl,\n    exact hIlinind (finsupp.mem_support_iff.mp hx) },\n  use [f i, hfi],\n  have hfi' : i \u2208 f.support := finsupp.mem_support_iff.mpr hfi,\n  rw [\u2190 finset.insert_erase hfi', finset.sum_insert (finset.not_mem_erase _ _),\n      add_eq_zero_iff_eq_neg] at sum_f,\n  rw sum_f,\n  refine neg_mem _ (sum_mem _ (\u03bb c hc, smul_mem _ _ (subset_span \u27e8c, _, rfl\u27e9))),\n  exact (memJ.mp (supp_f (finset.erase_subset _ _ hc))).resolve_left (finset.ne_of_mem_erase hc),\nend\nend repr\n\nlemma surjective_of_linear_independent_of_span [nontrivial R]\n  (hv : linear_independent R v) (f : \u03b9' \u21aa \u03b9)\n  (hss : range v \u2286 span R (range (v \u2218 f))) :\n  surjective f :=\nbegin\n  intros i,\n  let repr : (span R (range (v \u2218 f)) : Type*) \u2192 \u03b9' \u2192\u2080 R := (hv.comp f f.injective).repr,\n  let l := (repr \u27e8v i, hss (mem_range_self i)\u27e9).map_domain f,\n  have h_total_l : finsupp.total \u03b9 M R v l = v i,\n  { dsimp only [l],\n    rw finsupp.total_map_domain,\n    rw (hv.comp f f.injective).total_repr,\n    { refl },\n    { exact f.injective } },\n  have h_total_eq : (finsupp.total \u03b9 M R v) l = (finsupp.total \u03b9 M R v) (finsupp.single i 1),\n    by rw [h_total_l, finsupp.total_single, one_smul],\n  have l_eq : l = _ := linear_map.ker_eq_bot.1 hv h_total_eq,\n  dsimp only [l] at l_eq,\n  rw \u2190finsupp.emb_domain_eq_map_domain at l_eq,\n  rcases finsupp.single_of_emb_domain_single (repr \u27e8v i, _\u27e9) f i (1 : R) zero_ne_one.symm l_eq\n    with \u27e8i', hi'\u27e9,\n  use i',\n  exact hi'.2\nend\n\nlemma eq_of_linear_independent_of_span_subtype [nontrivial R] {s t : set M}\n  (hs : linear_independent R (\u03bb x, x : s \u2192 M)) (h : t \u2286 s) (hst : s \u2286 span R t) : s = t :=\nbegin\n  let f : t \u21aa s := \u27e8\u03bb x, \u27e8x.1, h x.2\u27e9, \u03bb a b hab, subtype.coe_injective (subtype.mk.inj hab)\u27e9,\n  have h_surj : surjective f,\n  { apply surjective_of_linear_independent_of_span hs f _,\n    convert hst; simp [f, comp], },\n  show s = t,\n  { apply subset.antisymm _ h,\n    intros x hx,\n    rcases h_surj \u27e8x, hx\u27e9 with \u27e8y, hy\u27e9,\n    convert y.mem,\n    rw \u2190 subtype.mk.inj hy,\n    refl }\nend\n\nopen linear_map\n\nlemma linear_independent.image_subtype {s : set M} {f : M \u2192\u2097 M'}\n  (hs : linear_independent R (\u03bb x, x : s \u2192 M))\n  (hf_inj : disjoint (span R s) f.ker) : linear_independent R (\u03bb x, x : f '' s \u2192 M') :=\nbegin\n  rw [\u2190 @subtype.range_coe _ s] at hf_inj,\n  refine (hs.map hf_inj).to_subtype_range' _,\n  simp [set.range_comp f]\nend\n\nlemma linear_independent.inl_union_inr {s : set M} {t : set M'}\n  (hs : linear_independent R (\u03bb x, x : s \u2192 M))\n  (ht : linear_independent R (\u03bb x, x : t \u2192 M')) :\n  linear_independent R (\u03bb x, x : inl R M M' '' s \u222a inr R M M' '' t \u2192 M \u00d7 M') :=\nbegin\n  refine (hs.image_subtype _).union (ht.image_subtype _) _; [simp, simp, skip],\n  simp only [span_image],\n  simp [disjoint_iff, prod_inf_prod]\nend\n\nlemma linear_independent_inl_union_inr' {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'}\n  (hv : linear_independent R v) (hv' : linear_independent R v') :\n  linear_independent R (sum.elim (inl R M M' \u2218 v) (inr R M M' \u2218 v')) :=\n(hv.map' (inl R M M') ker_inl).sum_type (hv'.map' (inr R M M') ker_inr) $\nbegin\n  refine is_compl_range_inl_inr.disjoint.mono _ _;\n    simp only [span_le, range_coe, range_comp_subset_range],\nend\n\n\/-- Dedekind's linear independence of characters -\/\n-- See, for example, Keith Conrad's note\n--  <https:\/\/kconrad.math.uconn.edu\/blurbs\/galoistheory\/linearchar.pdf>\ntheorem linear_independent_monoid_hom (G : Type*) [monoid G] (L : Type*) [comm_ring L]\n  [no_zero_divisors L] :\n  @linear_independent _ L (G \u2192 L) (\u03bb f, f : (G \u2192* L) \u2192 (G \u2192 L)) _ _ _ :=\nby letI := classical.dec_eq (G \u2192* L);\n   letI : mul_action L L := distrib_mul_action.to_mul_action;\n-- We prove linear independence by showing that only the trivial linear combination vanishes.\nexact linear_independent_iff'.2\n-- To do this, we use `finset` induction,\n(\u03bb s, finset.induction_on s (\u03bb g hg i, false.elim) $ \u03bb a s has ih g hg,\n-- Here\n-- * `a` is a new character we will insert into the `finset` of characters `s`,\n-- * `ih` is the fact that only the trivial linear combination of characters in `s` is zero\n-- * `hg` is the fact that `g` are the coefficients of a linear combination summing to zero\n-- and it remains to prove that `g` vanishes on `insert a s`.\n\n-- We now make the key calculation:\n-- For any character `i` in the original `finset`, we have `g i \u2022 i = g i \u2022 a` as functions on the\n-- monoid `G`.\nhave h1 : \u2200 i \u2208 s, (g i \u2022 i : G \u2192 L) = g i \u2022 a, from \u03bb i his, funext $ \u03bb x : G,\n  -- We prove these expressions are equal by showing\n  -- the differences of their values on each monoid element `x` is zero\n  eq_of_sub_eq_zero $ ih (\u03bb j, g j * j x - g j * a x)\n    (funext $ \u03bb y : G, calc\n    -- After that, it's just a chase scene.\n          (\u2211 i in s, ((g i * i x - g i * a x) \u2022 i : G \u2192 L)) y\n        = \u2211 i in s, (g i * i x - g i * a x) * i y : finset.sum_apply _ _ _\n    ... = \u2211 i in s, (g i * i x * i y - g i * a x * i y) : finset.sum_congr rfl\n      (\u03bb _ _, sub_mul _ _ _)\n    ... = \u2211 i in s, g i * i x * i y - \u2211 i in s, g i * a x * i y : finset.sum_sub_distrib\n    ... = (g a * a x * a y + \u2211 i in s, g i * i x * i y)\n          - (g a * a x * a y + \u2211 i in s, g i * a x * i y) : by rw add_sub_add_left_eq_sub\n    ... = \u2211 i in insert a s, g i * i x * i y - \u2211 i in insert a s, g i * a x * i y :\n      by rw [finset.sum_insert has, finset.sum_insert has]\n    ... = \u2211 i in insert a s, g i * i (x * y) - \u2211 i in insert a s, a x * (g i * i y) :\n      congr (congr_arg has_sub.sub (finset.sum_congr rfl $ \u03bb i _, by rw [i.map_mul, mul_assoc]))\n        (finset.sum_congr rfl $ \u03bb _ _, by rw [mul_assoc, mul_left_comm])\n    ... = (\u2211 i in insert a s, (g i \u2022 i : G \u2192 L)) (x * y)\n          - a x * (\u2211 i in insert a s, (g i \u2022 i : G \u2192 L)) y :\n      by rw [finset.sum_apply, finset.sum_apply, finset.mul_sum]; refl\n    ... = 0 - a x * 0 : by rw hg; refl\n    ... = 0 : by rw [mul_zero, sub_zero])\n    i\n    his,\n-- On the other hand, since `a` is not already in `s`, for any character `i \u2208 s`\n-- there is some element of the monoid on which it differs from `a`.\nhave h2 : \u2200 i : G \u2192* L, i \u2208 s \u2192 \u2203 y, i y \u2260 a y, from \u03bb i his,\n  classical.by_contradiction $ \u03bb h,\n  have hia : i = a, from monoid_hom.ext $ \u03bb y, classical.by_contradiction $ \u03bb hy, h \u27e8y, hy\u27e9,\n  has $ hia \u25b8 his,\n-- From these two facts we deduce that `g` actually vanishes on `s`,\nhave h3 : \u2200 i \u2208 s, g i = 0, from \u03bb i his, let \u27e8y, hy\u27e9 := h2 i his in\n  have h : g i \u2022 i y = g i \u2022 a y, from congr_fun (h1 i his) y,\n  or.resolve_right (mul_eq_zero.1 $ by rw [mul_sub, sub_eq_zero]; exact h) (sub_ne_zero_of_ne hy),\n-- And so, using the fact that the linear combination over `s` and over `insert a s` both vanish,\n-- we deduce that `g a = 0`.\nhave h4 : g a = 0, from calc\n  g a = g a * 1 : (mul_one _).symm\n  ... = (g a \u2022 a : G \u2192 L) 1 : by rw \u2190 a.map_one; refl\n  ... = (\u2211 i in insert a s, (g i \u2022 i : G \u2192 L)) 1 : begin\n      rw finset.sum_eq_single a,\n      { intros i his hia, rw finset.mem_insert at his,\n        rw [h3 i (his.resolve_left hia), zero_smul] },\n      { intros haas, exfalso, apply haas, exact finset.mem_insert_self a s }\n    end\n  ... = 0 : by rw hg; refl,\n-- Now we're done; the last two facts together imply that `g` vanishes on every element\n-- of `insert a s`.\n(finset.forall_mem_insert _ _ _).2 \u27e8h4, h3\u27e9)\n\nlemma le_of_span_le_span [nontrivial R] {s t u: set M}\n  (hl : linear_independent R (coe : u \u2192 M )) (hsu : s \u2286 u) (htu : t \u2286 u)\n  (hst : span R s \u2264 span R t) : s \u2286 t :=\nbegin\n  have := eq_of_linear_independent_of_span_subtype\n    (hl.mono (set.union_subset hsu htu))\n    (set.subset_union_right _ _)\n    (set.union_subset (set.subset.trans subset_span hst) subset_span),\n  rw \u2190 this, apply set.subset_union_left\nend\n\nlemma span_le_span_iff [nontrivial R] {s t u: set M}\n  (hl : linear_independent R (coe : u \u2192 M)) (hsu : s \u2286 u) (htu : t \u2286 u) :\n  span R s \u2264 span R t \u2194 s \u2286 t :=\n\u27e8le_of_span_le_span hl hsu htu, span_mono\u27e9\n\nend module\n\nsection nontrivial\n\nvariables [ring R] [nontrivial R] [add_comm_group M] [add_comm_group M']\nvariables [module R M] [no_zero_smul_divisors R M] [module R M']\nvariables {v : \u03b9 \u2192 M} {s t : set M} {x y z : M}\n\nlemma linear_independent_unique_iff\n  (v : \u03b9 \u2192 M) [unique \u03b9] :\n  linear_independent R v \u2194 v (default \u03b9) \u2260 0 :=\nbegin\n  simp only [linear_independent_iff, finsupp.total_unique, smul_eq_zero],\n  refine \u27e8\u03bb h hv, _, \u03bb hv l hl, finsupp.unique_ext $ hl.resolve_right hv\u27e9,\n  have := h (finsupp.single (default \u03b9) 1) (or.inr hv),\n  exact one_ne_zero (finsupp.single_eq_zero.1 this)\nend\n\nalias linear_independent_unique_iff \u2194 _ linear_independent_unique\n\nlemma linear_independent_singleton {x : M} (hx : x \u2260 0) :\n  linear_independent R (\u03bb x, x : ({x} : set M) \u2192 M) :=\nlinear_independent_unique coe hx\n\nend nontrivial\n\n\/-!\n### Properties which require `division_ring K`\n\nThese can be considered generalizations of properties of linear independence in vector spaces.\n-\/\n\nsection module\n\nvariables [division_ring K] [add_comm_group V] [add_comm_group V']\nvariables [module K V] [module K V']\nvariables {v : \u03b9 \u2192 V} {s t : set V} {x y z : V}\n\nopen submodule\n\n\/- TODO: some of the following proofs can generalized with a zero_ne_one predicate type class\n   (instead of a data containing type class) -\/\n\nlemma mem_span_insert_exchange : x \u2208 span K (insert y s) \u2192 x \u2209 span K s \u2192 y \u2208 span K (insert x s) :=\nbegin\n  simp [mem_span_insert],\n  rintro a z hz rfl h,\n  refine \u27e8a\u207b\u00b9, -a\u207b\u00b9 \u2022 z, smul_mem _ _ hz, _\u27e9,\n  have a0 : a \u2260 0, {rintro rfl, simp * at *},\n  simp [a0, smul_add, smul_smul]\nend\n\nlemma linear_independent_iff_not_mem_span :\n  linear_independent K v \u2194 (\u2200i, v i \u2209 span K (v '' (univ \\ {i}))) :=\nbegin\n  apply linear_independent_iff_not_smul_mem_span.trans,\n  split,\n  { intros h i h_in_span,\n    apply one_ne_zero (h i 1 (by simp [h_in_span])) },\n  { intros h i a ha,\n    by_contradiction ha',\n    exact false.elim (h _ ((smul_mem_iff _ ha').1 ha)) }\nend\n\nlemma linear_independent.insert (hs : linear_independent K (\u03bb b, b : s \u2192 V)) (hx : x \u2209 span K s) :\n  linear_independent K (\u03bb b, b : insert x s \u2192 V) :=\nbegin\n  rw \u2190 union_singleton,\n  have x0 : x \u2260 0 := mt (by rintro rfl; apply zero_mem _) hx,\n  apply hs.union (linear_independent_singleton x0),\n  rwa [disjoint_span_singleton' x0]\nend\n\nlemma linear_independent_option' :\n  linear_independent K (\u03bb o, option.cases_on' o x v : option \u03b9 \u2192 V) \u2194\n    linear_independent K v \u2227 (x \u2209 submodule.span K (range v)) :=\nbegin\n  rw [\u2190 linear_independent_equiv (equiv.option_equiv_sum_punit \u03b9).symm, linear_independent_sum,\n    @range_unique _ punit, @linear_independent_unique_iff punit, disjoint_span_singleton],\n  dsimp [(\u2218)],\n  refine \u27e8\u03bb h, \u27e8h.1, \u03bb hx, h.2.1 $ h.2.2 hx\u27e9, \u03bb h, \u27e8h.1, _, \u03bb hx, (h.2 hx).elim\u27e9\u27e9,\n  rintro rfl,\n  exact h.2 (zero_mem _)\nend\n\nlemma linear_independent.option (hv : linear_independent K v)\n  (hx : x \u2209 submodule.span K (range v)) :\n  linear_independent K (\u03bb o, option.cases_on' o x v : option \u03b9 \u2192 V) :=\nlinear_independent_option'.2 \u27e8hv, hx\u27e9\n\nlemma linear_independent_option {v : option \u03b9 \u2192 V} :\n  linear_independent K v \u2194\n    linear_independent K (v \u2218 coe : \u03b9 \u2192 V) \u2227 v none \u2209 submodule.span K (range (v \u2218 coe : \u03b9 \u2192 V)) :=\nby simp only [\u2190 linear_independent_option', option.cases_on'_none_coe]\n\ntheorem linear_independent_insert' {\u03b9} {s : set \u03b9} {a : \u03b9} {f : \u03b9 \u2192 V} (has : a \u2209 s) :\n  linear_independent K (\u03bb x : insert a s, f x) \u2194\n  linear_independent K (\u03bb x : s, f x) \u2227 f a \u2209 submodule.span K (f '' s) :=\nby { rw [\u2190 linear_independent_equiv ((equiv.option_equiv_sum_punit _).trans\n  (equiv.set.insert has).symm), linear_independent_option], simp [(\u2218), range_comp f] }\n\ntheorem linear_independent_insert (hxs : x \u2209 s) :\n  linear_independent K (\u03bb b : insert x s, (b : V)) \u2194\n  linear_independent K (\u03bb b : s, (b : V)) \u2227 x \u2209 submodule.span K s :=\n(@linear_independent_insert' _ _ _ _ _ _ _ _ id hxs).trans $ by simp\n\nlemma linear_independent_pair {x y : V} (hx : x \u2260 0) (hy : \u2200 a : K, a \u2022 x \u2260 y) :\n  linear_independent K (coe : ({x, y} : set V) \u2192 V) :=\npair_comm y x \u25b8 (linear_independent_singleton hx).insert $ mt mem_span_singleton.1\n  (not_exists.2 hy)\n\nlemma linear_independent_fin_cons {n} {v : fin n \u2192 V} :\n  linear_independent K (fin.cons x v : fin (n + 1) \u2192 V) \u2194\n    linear_independent K v \u2227 x \u2209 submodule.span K (range v) :=\nbegin\n  rw [\u2190 linear_independent_equiv (fin_succ_equiv n).symm, linear_independent_option],\n  convert iff.rfl,\n  { ext,\n    -- TODO: why doesn't simp use `fin_succ_equiv_symm_coe` here?\n    rw [comp_app, comp_app, fin_succ_equiv_symm_coe, fin.cons_succ] },\n  { rw [comp_app, fin_succ_equiv_symm_none, fin.cons_zero] },\n  { ext,\n    rw [comp_app, comp_app, fin_succ_equiv_symm_coe, fin.cons_succ] }\nend\n\nlemma linear_independent_fin_snoc {n} {v : fin n \u2192 V} :\n  linear_independent K (fin.snoc v x : fin (n + 1) \u2192 V) \u2194\n    linear_independent K v \u2227 x \u2209 submodule.span K (range v) :=\nby rw [fin.snoc_eq_cons_rotate, linear_independent_equiv, linear_independent_fin_cons]\n\n\/-- See `linear_independent.fin_cons'` for an uglier version that works if you\nonly have a module over a semiring. -\/\nlemma linear_independent.fin_cons {n} {v : fin n \u2192 V} (hv : linear_independent K v)\n  (hx : x \u2209 submodule.span K (range v)) :\n  linear_independent K (fin.cons x v : fin (n + 1) \u2192 V) :=\nlinear_independent_fin_cons.2 \u27e8hv, hx\u27e9\n\nlemma linear_independent_fin_succ {n} {v : fin (n + 1) \u2192 V} :\n  linear_independent K v \u2194\n    linear_independent K (fin.tail v) \u2227 v 0 \u2209 submodule.span K (range $ fin.tail v) :=\nby rw [\u2190 linear_independent_fin_cons, fin.cons_self_tail]\n\nlemma linear_independent_fin_succ' {n} {v : fin (n + 1) \u2192 V} :\n  linear_independent K v \u2194\n    linear_independent K (fin.init v) \u2227 v (fin.last _) \u2209 submodule.span K (range $ fin.init v) :=\nby rw [\u2190 linear_independent_fin_snoc, fin.snoc_init_self]\n\nlemma linear_independent_fin2 {f : fin 2 \u2192 V} :\n  linear_independent K f \u2194 f 1 \u2260 0 \u2227 \u2200 a : K, a \u2022 f 1 \u2260 f 0 :=\nby rw [linear_independent_fin_succ, linear_independent_unique_iff, range_unique,\n  mem_span_singleton, not_exists,\n  show fin.tail f (default (fin 1)) = f 1, by rw \u2190 fin.succ_zero_eq_one; refl]\n\nlemma exists_linear_independent (hs : linear_independent K (\u03bb x, x : s \u2192 V)) (hst : s \u2286 t) :\n  \u2203b\u2286t, s \u2286 b \u2227 t \u2286 span K b \u2227 linear_independent K (\u03bb x, x : b \u2192 V) :=\nbegin\n  rcases zorn.zorn_subset_nonempty {b | b \u2286 t \u2227 linear_independent K (\u03bb x, x : b \u2192 V)} _ _\n    \u27e8hst, hs\u27e9 with \u27e8b, \u27e8bt, bi\u27e9, sb, h\u27e9,\n  { refine \u27e8b, bt, sb, \u03bb x xt, _, bi\u27e9,\n    by_contra hn,\n    apply hn,\n    rw \u2190 h _ \u27e8insert_subset.2 \u27e8xt, bt\u27e9, bi.insert hn\u27e9 (subset_insert _ _),\n    exact subset_span (mem_insert _ _) },\n  { refine \u03bb c hc cc c0, \u27e8\u22c3\u2080 c, \u27e8_, _\u27e9, \u03bb x, _\u27e9,\n    { exact sUnion_subset (\u03bb x xc, (hc xc).1) },\n    { exact linear_independent_sUnion_of_directed cc.directed_on (\u03bb x xc, (hc xc).2) },\n    { exact subset_sUnion_of_mem } }\nend\n\n\/-- `linear_independent.extend` adds vectors to a linear independent set `s \u2286 t` until it spans\nall elements of `t`. -\/\nnoncomputable def linear_independent.extend (hs : linear_independent K (\u03bb x, x : s \u2192 V))\n  (hst : s \u2286 t) : set V :=\nclassical.some (exists_linear_independent hs hst)\n\nlemma linear_independent.extend_subset (hs : linear_independent K (\u03bb x, x : s \u2192 V))\n  (hst : s \u2286 t) : hs.extend hst \u2286 t :=\nlet \u27e8hbt, hsb, htb, hli\u27e9 := classical.some_spec (exists_linear_independent hs hst) in hbt\n\nlemma linear_independent.subset_extend (hs : linear_independent K (\u03bb x, x : s \u2192 V))\n  (hst : s \u2286 t) : s \u2286 hs.extend hst :=\nlet \u27e8hbt, hsb, htb, hli\u27e9 := classical.some_spec (exists_linear_independent hs hst) in hsb\n\nlemma linear_independent.subset_span_extend (hs : linear_independent K (\u03bb x, x : s \u2192 V))\n  (hst : s \u2286 t) : t \u2286 span K (hs.extend hst) :=\nlet \u27e8hbt, hsb, htb, hli\u27e9 := classical.some_spec (exists_linear_independent hs hst) in htb\n\nlemma linear_independent.linear_independent_extend (hs : linear_independent K (\u03bb x, x : s \u2192 V))\n  (hst : s \u2286 t) : linear_independent K (coe : hs.extend hst \u2192 V) :=\nlet \u27e8hbt, hsb, htb, hli\u27e9 := classical.some_spec (exists_linear_independent hs hst) in hli\n\nvariables {K V}\n\n-- TODO(Mario): rewrite?\nlemma exists_of_linear_independent_of_finite_span {t : finset V}\n  (hs : linear_independent K (\u03bb x, x : s \u2192 V)) (hst : s \u2286 (span K \u2191t : submodule K V)) :\n  \u2203t':finset V, \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 t'.card = t.card :=\nhave \u2200t, \u2200(s' : finset V), \u2191s' \u2286 s \u2192 s \u2229 \u2191t = \u2205 \u2192 s \u2286 (span K \u2191(s' \u222a t) : submodule K V) \u2192\n  \u2203t':finset V, \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 t'.card = (s' \u222a t).card :=\nassume t, finset.induction_on t\n  (assume s' hs' _ hss',\n    have s = \u2191s',\n      from eq_of_linear_independent_of_span_subtype hs hs' $\n          by simpa using hss',\n    \u27e8s', by simp [this]\u27e9)\n  (assume b\u2081 t hb\u2081t ih s' hs' hst hss',\n    have hb\u2081s : b\u2081 \u2209 s,\n      from assume h,\n      have b\u2081 \u2208 s \u2229 \u2191(insert b\u2081 t), from \u27e8h, finset.mem_insert_self _ _\u27e9,\n      by rwa [hst] at this,\n    have hb\u2081s' : b\u2081 \u2209 s', from assume h, hb\u2081s $ hs' h,\n    have hst : s \u2229 \u2191t = \u2205,\n      from eq_empty_of_subset_empty $ subset.trans\n        (by simp [inter_subset_inter, subset.refl]) (le_of_eq hst),\n    classical.by_cases\n      (assume : s \u2286 (span K \u2191(s' \u222a t) : submodule K V),\n        let \u27e8u, hust, hsu, eq\u27e9 := ih _ hs' hst this in\n        have hb\u2081u : b\u2081 \u2209 u, from assume h, (hust h).elim hb\u2081s hb\u2081t,\n        \u27e8insert b\u2081 u, by simp [insert_subset_insert hust],\n          subset.trans hsu (by simp), by simp [eq, hb\u2081t, hb\u2081s', hb\u2081u]\u27e9)\n      (assume : \u00ac s \u2286 (span K \u2191(s' \u222a t) : submodule K V),\n        let \u27e8b\u2082, hb\u2082s, hb\u2082t\u27e9 := not_subset.mp this in\n        have hb\u2082t' : b\u2082 \u2209 s' \u222a t, from assume h, hb\u2082t $ subset_span h,\n        have s \u2286 (span K \u2191(insert b\u2082 s' \u222a t) : submodule K V), from\n          assume b\u2083 hb\u2083,\n          have \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : set V),\n            by simp [insert_eq, -singleton_union, -union_singleton, union_subset_union, subset.refl,\n              subset_union_right],\n          have hb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : set V)),\n            from span_mono this (hss' hb\u2083),\n          have s \u2286 (span K (insert b\u2081 \u2191(s' \u222a t)) : submodule K V),\n            by simpa [insert_eq, -singleton_union, -union_singleton] using hss',\n          have hb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t)),\n            from mem_span_insert_exchange (this hb\u2082s) hb\u2082t,\n          by rw [span_insert_eq_span hb\u2081] at hb\u2083; simpa using hb\u2083,\n        let \u27e8u, hust, hsu, eq\u27e9 := ih _ (by simp [insert_subset, hb\u2082s, hs']) hst this in\n        \u27e8u, subset.trans hust $ union_subset_union (subset.refl _) (by simp [subset_insert]),\n          hsu, by simp [eq, hb\u2082t', hb\u2081t, hb\u2081s']\u27e9)),\nbegin\n  have eq : t.filter (\u03bbx, x \u2208 s) \u222a t.filter (\u03bbx, x \u2209 s) = t,\n  { ext1 x,\n    by_cases x \u2208 s; simp * },\n  apply exists.elim (this (t.filter (\u03bbx, x \u2209 s)) (t.filter (\u03bbx, x \u2208 s))\n    (by simp [set.subset_def]) (by simp [set.ext_iff] {contextual := tt}) (by rwa [eq])),\n  intros u h,\n  exact \u27e8u, subset.trans h.1 (by simp [subset_def, and_imp, or_imp_distrib] {contextual:=tt}),\n    h.2.1, by simp only [h.2.2, eq]\u27e9\nend\n\nlemma exists_finite_card_le_of_finite_of_linear_independent_of_span\n  (ht : finite t) (hs : linear_independent K (\u03bb x, x : s \u2192 V)) (hst : s \u2286 span K t) :\n  \u2203h : finite s, h.to_finset.card \u2264 ht.to_finset.card :=\nhave s \u2286 (span K \u2191(ht.to_finset) : submodule K V), by simp; assumption,\nlet \u27e8u, hust, hsu, eq\u27e9 := exists_of_linear_independent_of_finite_span hs this in\nhave finite s, from u.finite_to_set.subset hsu,\n\u27e8this, by rw [\u2190eq]; exact (finset.card_le_of_subset $ finset.coe_subset.mp $ by simp [hsu])\u27e9\n\nend module\n","avg_line_length":42.9913344887,"max_line_length":100,"alphanum_fraction":0.6652624365}
-{"size":457,"ext":"lean","lang":"Lean","max_stars_count":12.0,"content":"\/-\nCopyright (c) 2021 OpenAI. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f\n-\/\nimport mathzoo.imports.miniF2F\n\nopen_locale nat rat real big_operators topological_space\n\ntheorem mathd_algebra_451\n  (\u03c3 : equiv \u211d \u211d)\n  (h\u2080 : \u03c3.2 (-15) = 0)\n  (h\u2081 : \u03c3.2 0 = 3)\n  (h\u2082 : \u03c3.2 3 = 9)\n  (h\u2083 : \u03c3.2 9 = 20) :\n  \u03c3.1 (\u03c3.1 9) = 0 :=\nbegin\n  simp [\u2190 h\u2081, \u2190 h\u2082],\nend","avg_line_length":24.0526315789,"max_line_length":67,"alphanum_fraction":0.6630196937}
-{"size":852,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-\/\nimport data.nat.sqrt\nimport data.int.basic\n\nnamespace int\n\n\/-- `sqrt n` is the square root of an integer `n`. If `n` is not a\n  perfect square, and is positive, it returns the largest `k:\u2124` such\n  that `k*k \u2264 n`. If it is negative, it returns 0. For example,\n  `sqrt 2 = 1` and `sqrt 1 = 1` and `sqrt (-1) = 0` -\/\ndef sqrt (n : \u2124) : \u2124 :=\nnat.sqrt $ int.to_nat n\n\ntheorem sqrt_eq (n : \u2124) : sqrt (n*n) = n.nat_abs :=\nby rw [sqrt, \u2190 nat_abs_mul_self, to_nat_coe_nat, nat.sqrt_eq]\n\ntheorem exists_mul_self (x : \u2124) :\n  (\u2203 n, n * n = x) \u2194 sqrt x * sqrt x = x :=\n\u27e8\u03bb \u27e8n, hn\u27e9, by rw [\u2190 hn, sqrt_eq, \u2190 int.coe_nat_mul, nat_abs_mul_self],\n\u03bb h, \u27e8sqrt x, h\u27e9\u27e9\n\ntheorem sqrt_nonneg (n : \u2124) : 0 \u2264 sqrt n := trivial\n\nend int\n","avg_line_length":29.3793103448,"max_line_length":71,"alphanum_fraction":0.646713615}
-{"size":1610,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon, Alex Keizer\n-\/\n\nimport Qpf.MathlibPort.Fin2\nimport Qpf.Util.HEq\nimport Lean.Elab.Tactic.Conv\n\nuniverse u v w\n\nabbrev DVec {n : Nat} (\u03b1s : Fin2 n \u2192 Type u)  : Type _\n  := (i : Fin2 n) \u2192 \u03b1s i\n\nabbrev Vec (\u03b1 : Type _) (n : Nat)\n  := @DVec n fun _ => \u03b1\n\nnamespace Vec\n  def append1 {\u03b1 : Type u} {n} (tl : Vec \u03b1 n) (hd : \u03b1) : Vec \u03b1 (n+1)\n    | (Fin2.fs i) => tl i\n    | Fin2.fz     => hd\n\n  -- infixl:67 \" ::: \" => append1\n\n  \/-- Drop the last element from a `Vec` -\/\n  def drop (v : Vec \u03b1 (n+1)) : Vec \u03b1 n\n    := fun i => v (Fin2.fs i)\n\n  def constVec {\u03b1 : Type _} (a : \u03b1) (n : Nat) : Vec \u03b1 n\n    := fun _ => a\nend Vec\n\nunif_hint (n : Nat) where |- Fin2 n \u2192 Type u =?= Vec (Type u) n\nunif_hint {\u03b1 : Type _} (n : Nat) where |- DVec (Vec.constVec \u03b1 n) =?= Vec \u03b1 n\n\nnamespace DVec\n  \/-- Return the last element from a `DVec` -\/\n  def last (v : @DVec (n+1) \u03b1s ) : \u03b1s 0\n    := v 0\n\n  \/-- Drop the last element from a `DVec` -\/\n  def drop (v : DVec \u03b1s) : DVec (Vec.drop \u03b1s)\n    := fun i => v (Fin2.fs i)\n\n  @[reducible]\n  def nil : @DVec 0 \u03b1s\n    := fun emp => by contradiction\n\n  @[reducible]\n  def append1 {\u03b1 : Type u} {\u03b1s : Vec (Type u) n} (tl : DVec \u03b1s) (hd : \u03b1) : DVec (Vec.append1 \u03b1s \u03b1)\n    | (Fin2.fs i) => tl i\n    | Fin2.fz     => hd\n  \n\n  -- infixl:67 \" ::: \" => append1\nend DVec\n\nnamespace Vec\n  variable {\u03b1 : Type _} {n : Nat}\n\n  abbrev nil  : Vec \u03b1 0           := DVec.nil\n  abbrev last : Vec \u03b1 n.succ \u2192 \u03b1  := DVec.last\nend Vec\n","avg_line_length":24.7692307692,"max_line_length":98,"alphanum_fraction":0.5664596273}
-{"size":3579,"ext":"lean","lang":"Lean","max_stars_count":71.0,"content":"\/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-\/\nimport Lean.Parser.Term\nimport Mathlib.Init.SetNotation\n\n\/-!\n\n# Sets\n\nThis file sets up the theory of sets whose elements have a given type.\n\n## Main definitions\n\nGiven a type `X` and a predicate `p : X \u2192 Prop`:\n\n* `Set X` : the type of sets whose elements have type `X`\n* `{a : X | p a} : Set X` : the set of all elements of `X` satisfying `p`\n* `{a | p a} : Set X` : a more concise notation for `{a : X | p a}`\n* `{a \u2208 S | p a} : Set X` : given `S : Set X`, the subset of `S` consisting of\n   its elements satisfying `p`.\n\n## Implementation issues\n\nAs in Lean 3, `Set X := X \u2192 Prop`\n\nI didn't call this file Data.Set.Basic because it contains core Lean 3\nstuff which happens before mathlib3's data.set.basic .\nThis file is a port of the core Lean 3 file `lib\/lean\/library\/init\/data\/set.lean`.\n\n-\/\n\ndef Set (\u03b1 : Type u) := \u03b1 \u2192 Prop\n\ndef setOf {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) : Set \u03b1 :=\np\n\nnamespace Set\n\nprotected def mem (a : \u03b1) (s : Set \u03b1) :=\ns a\n\ninstance : Mem \u03b1 (Set \u03b1) :=\n\u27e8Set.mem\u27e9\n\nprotected def subset (s\u2081 s\u2082 : Set \u03b1) :=\n\u2200 {a}, a \u2208 s\u2081 \u2192 a \u2208 s\u2082\n\ninstance : Subset (Set \u03b1) :=\n\u27e8Set.subset\u27e9\n\ninstance : EmptyCollection (Set \u03b1) :=\n\u27e8\u03bb a => false\u27e9\n\nopen Mathlib.ExtendedBinder in\nsyntax \"{ \" extBinder \" | \" term \" }\" : term\n\nmacro_rules\n  | `({ $x:ident | $p }) => `(setOf fun $x:ident => $p)\n  | `({ $x:ident : $t | $p }) => `(setOf fun $x:ident : $t => $p)\n  | `({ $x:ident $b:binderPred | $p }) =>\n    `(setOf fun $x:ident => satisfiesBinderPred% $x $b \u2227 $p)\n\n@[appUnexpander setOf]\ndef setOf.unexpander : Lean.PrettyPrinter.Unexpander\n  | `(setOf fun $x:ident => $p) => `({ $x:ident | $p })\n  | `(setOf fun $x:ident : $ty:term => $p) => `({ $x:ident : $ty:term | $p })\n  | _ => throw ()\n\nopen Mathlib.ExtendedBinder in\nmacro (priority := low) \"{ \" t:term \" | \" bs:extBinders \" }\" : term =>\n  `({ x | \u2203\u1d49 $bs:extBinders, $t = x })\n\ndef univ : Set \u03b1 := {a | True }\n\nprotected def insert (a : \u03b1) (s : Set \u03b1) : Set \u03b1 :=\n{b | b = a \u2228 b \u2208 s}\n\nprotected def singleton (a : \u03b1) : Set \u03b1 :=\n{b | b = a}\n\nprotected def union (s\u2081 s\u2082 : Set \u03b1) : Set \u03b1 :=\n{a | a \u2208 s\u2081 \u2228 a \u2208 s\u2082}\n\ninstance : Union (Set \u03b1) :=\n\u27e8Set.union\u27e9\n\nprotected def inter (s\u2081 s\u2082 : Set \u03b1) : Set \u03b1 :=\n{a | a \u2208 s\u2081 \u2227 a \u2208 s\u2082}\n\ninstance : Inter (Set \u03b1) :=\n\u27e8Set.inter\u27e9\n\ndef compl (s : Set \u03b1) : Set \u03b1 :=\n{a | a \u2209 s}\n\nprotected def diff (s t : Set \u03b1) : Set \u03b1 :=\n{a \u2208 s | a \u2209 t}\n\ninstance : Sdiff (Set \u03b1) :=\n\u27e8Set.diff\u27e9\n\ndef powerset (s : Set \u03b1) : Set (Set \u03b1) :=\n{t | t \u2286 s}\n\nprefix:100 \"\ud835\udcab\" => powerset\n\n@[reducible]\ndef sUnion (s : Set (Set \u03b1)) : Set \u03b1 := {t | \u2203 a \u2208 s, t \u2208 a}\n\nprefix:110 \"\u22c3\u2080\" => sUnion\n\ndef image (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) : Set \u03b2 :=\n  { f a | a \u2208 s }\n\ninstance : Functor Set :=\n{ map := @Set.image }\n\ninstance : LawfulFunctor Set where\n  id_map s := funext $ \u03bb b => propext \u27e8\u03bb \u27e8_, sb, rfl\u27e9 => sb, \u03bb sb => \u27e8_, sb, rfl\u27e9\u27e9\n  comp_map g h s := funext $ \u03bb c => propext\n    \u27e8\u03bb \u27e8a, \u27e8h\u2081, h\u2082\u27e9\u27e9 => \u27e8g a, \u27e8\u27e8a, \u27e8h\u2081, rfl\u27e9\u27e9, h\u2082\u27e9\u27e9,\n     \u03bb \u27e8b, \u27e8\u27e8a, \u27e8h\u2081, h\u2082\u27e9\u27e9, h\u2083\u27e9\u27e9 => \u27e8a, \u27e8h\u2081, show h (g a) = c from h\u2082 \u25b8 h\u2083\u27e9\u27e9\u27e9\n  map_const := rfl\n\nsyntax (priority := high) \"{\" term,+ \"}\" : term\n\nmacro_rules\n  | `({$x}) => `(Set.singleton $x)\n  | `({$x, $xs:term,*}) => `(Set.insert $x {$xs,*})\n\n@[appUnexpander Set.singleton]\ndef singletonUnexpander : Lean.PrettyPrinter.Unexpander\n| `(Set.singleton $a) => `({ $a })\n| _ => throw ()\n\n@[appUnexpander Set.insert]\ndef insertUnexpander : Lean.PrettyPrinter.Unexpander\n| `(Set.insert $a { $ts,* }) => `({$a, $ts,*})\n| _ => throw ()\n\nend Set\n","avg_line_length":24.6827586207,"max_line_length":82,"alphanum_fraction":0.5736239173}
-{"size":24132,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Heather Macbeth.\n-\/\nimport analysis.calculus.times_cont_diff\nimport topology.local_homeomorph\nimport topology.metric_space.contracting\n\n\/-!\n# Inverse function theorem\n\nIn this file we prove the inverse function theorem. It says that if a map `f : E \u2192 F`\nhas an invertible strict derivative `f'` at `a`, then it is locally invertible,\nand the inverse function has derivative `f' \u207b\u00b9`.\n\nWe define `has_strict_deriv_at.to_local_homeomorph` that repacks a function `f`\nwith a `hf : has_strict_fderiv_at f f' a`, `f' : E \u2243L[\ud835\udd5c] F`, into a `local_homeomorph`.\nThe `to_fun` of this `local_homeomorph` is `defeq` to `f`, so one can apply theorems\nabout `local_homeomorph` to `hf.to_local_homeomorph f`, and get statements about `f`.\n\nThen we define `has_strict_fderiv_at.local_inverse` to be the `inv_fun` of this `local_homeomorph`,\nand prove two versions of the inverse function theorem:\n\n* `has_strict_fderiv_at.to_local_inverse`: if `f` has an invertible derivative `f'` at `a` in the\n  strict sense (`hf`), then `hf.local_inverse f f' a` has derivative `f'.symm` at `f a` in the\n  strict sense;\n\n* `has_strict_fderiv_at.to_local_left_inverse`: if `f` has an invertible derivative `f'` at `a` in\n  the strict sense and `g` is locally left inverse to `f` near `a`, then `g` has derivative\n  `f'.symm` at `f a` in the strict sense.\n\nIn the one-dimensional case we reformulate these theorems in terms of `has_strict_deriv_at` and\n`f'\u207b\u00b9`.\n\nWe also reformulate the theorems in terms of `times_cont_diff`, to give that `C^k` (respectively,\nsmooth) inputs give `C^k` (smooth) inverses.  These versions require that continuous\ndifferentiability implies strict differentiability; this is false over a general field, true over\n`\u211d` (the setting in which it is implemented here), and true but (TODO) not yet implemented over `\u2102`.\n\nSome related theorems, providing the derivative and higher regularity assuming that we already know\nthe inverse function, are formulated in `fderiv.lean`, `deriv.lean`, and `times_cont_diff.lean`.\n\n## Notations\n\nIn the section about `approximates_linear_on` we introduce some `local notation` to make formulas\nshorter:\n\n* by `N` we denote `\u2225f'\u207b\u00b9\u2225`;\n* by `g` we denote the auxiliary contracting map `x \u21a6 x + f'.symm (y - f x)` used to prove that\n  `{x | f x = y}` is nonempty.\n\n## Tags\n\nderivative, strictly differentiable, continuously differentiable, smooth, inverse function\n-\/\n\nopen function set filter metric\nopen_locale topological_space classical nnreal\n\nnoncomputable theory\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\nvariables {E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\nvariables {F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\nvariables {G : Type*} [normed_group G] [normed_space \ud835\udd5c G]\nvariables {G' : Type*} [normed_group G'] [normed_space \ud835\udd5c G']\n\nopen asymptotics filter metric set\nopen continuous_linear_map (id)\n\n\/-!\n### Non-linear maps approximating close to affine maps\n\nIn this section we study a map `f` such that `\u2225f x - f y - f' (x - y)\u2225 \u2264 c * \u2225x - y\u2225` on an open set\n`s`, where `f' : E \u2243L[\ud835\udd5c] F` is a continuous linear equivalence and `c < \u2225f'\u207b\u00b9\u2225`. Maps of this type\nbehave like `f a + f' (x - a)` near each `a \u2208 s`.\n\nIf `E` is a complete space, we prove that the image `f '' s` is open, and `f` is a homeomorphism\nbetween `s` and `f '' s`. More precisely, we define `approximates_linear_on.to_local_homeomorph` to\nbe a `local_homeomorph` with `to_fun = f`, `source = s`, and `target = f '' s`.\n\nMaps of this type naturally appear in the proof of the inverse function theorem (see next section),\nand `approximates_linear_on.to_local_homeomorph` will imply that the locally inverse function\nexists.\n\nWe define this auxiliary notion to split the proof of the inverse function theorem into small\nlemmas. This approach makes it possible\n\n- to prove a lower estimate on the size of the domain of the inverse function;\n\n- to reuse parts of the proofs in the case if a function is not strictly differentiable. E.g., for a\n  function `f : E \u00d7 F \u2192 G` with estimates on `f x y\u2081 - f x y\u2082` but not on `f x\u2081 y - f x\u2082 y`.\n-\/\n\n\/-- We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`,\nif `\u2225f x - f y - f' (x - y)\u2225 \u2264 c * \u2225x - y\u2225` whenever `x, y \u2208 s`.\n\nThis predicate is defined to facilitate the splitting of the inverse function theorem into small\nlemmas. Some of these lemmas can be useful, e.g., to prove that the inverse function is defined\non a specific set. -\/\ndef approximates_linear_on (f : E \u2192 F) (f' : E \u2192L[\ud835\udd5c] F) (s : set E) (c : \u211d\u22650) : Prop :=\n\u2200 (x \u2208 s) (y \u2208 s), \u2225f x - f y - f' (x - y)\u2225 \u2264 c * \u2225x - y\u2225\n\nnamespace approximates_linear_on\n\nvariables [cs : complete_space E] {f : E \u2192 F}\n\n\/-! First we prove some properties of a function that `approximates_linear_on` a (not necessarily\ninvertible) continuous linear map. -\/\n\nsection\n\nvariables {f' : E \u2192L[\ud835\udd5c] F} {s t : set E} {c c' : \u211d\u22650}\n\ntheorem mono_num (hc : c \u2264 c') (hf : approximates_linear_on f f' s c) :\n  approximates_linear_on f f' s c' :=\n\u03bb x hx y hy, le_trans (hf x hx y hy) (mul_le_mul_of_nonneg_right hc $ norm_nonneg _)\n\ntheorem mono_set (hst : s \u2286 t) (hf : approximates_linear_on f f' t c) :\n  approximates_linear_on f f' s c :=\n\u03bb x hx y hy, hf x (hst hx) y (hst hy)\n\nlemma lipschitz_sub (hf : approximates_linear_on f f' s c) :\n  lipschitz_with c (\u03bb x : s, f x - f' x) :=\nbegin\n  refine lipschitz_with.of_dist_le_mul (\u03bb x y, _),\n  rw [dist_eq_norm, subtype.dist_eq, dist_eq_norm],\n  convert hf x x.2 y y.2 using 2,\n  rw [f'.map_sub], abel\nend\n\nprotected lemma lipschitz (hf : approximates_linear_on f f' s c) :\n  lipschitz_with (nnnorm f' + c) (s.restrict f) :=\nby simpa only [restrict_apply, add_sub_cancel'_right]\n  using (f'.lipschitz.restrict s).add hf.lipschitz_sub\n\nprotected lemma continuous (hf : approximates_linear_on f f' s c) :\n  continuous (s.restrict f) :=\nhf.lipschitz.continuous\n\nprotected lemma continuous_on (hf : approximates_linear_on f f' s c) :\n  continuous_on f s :=\ncontinuous_on_iff_continuous_restrict.2 hf.continuous\n\nend\n\n\/-!\nFrom now on we assume that `f` approximates an invertible continuous linear map `f : E \u2243L[\ud835\udd5c] F`.\n\nWe also assume that either `E = {0}`, or `c < \u2225f'\u207b\u00b9\u2225\u207b\u00b9`. We use `N` as an abbreviation for `\u2225f'\u207b\u00b9\u2225`.\n-\/\n\nvariables {f' : E \u2243L[\ud835\udd5c] F} {s : set E} {c : \u211d\u22650}\n\nlocal notation `N` := nnnorm (f'.symm : F \u2192L[\ud835\udd5c] E)\n\nprotected lemma antilipschitz (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) :\n  antilipschitz_with (N\u207b\u00b9 - c)\u207b\u00b9 (s.restrict f) :=\nbegin\n  cases hc with hE hc,\n  { haveI : subsingleton s := \u27e8\u03bb x y, subtype.eq $ @subsingleton.elim _ hE _ _\u27e9,\n    exact antilipschitz_with.of_subsingleton },\n  convert (f'.antilipschitz.restrict s).add_lipschitz_with hf.lipschitz_sub hc,\n  simp [restrict]\nend\n\nprotected lemma injective (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) :\n  injective (s.restrict f) :=\n(hf.antilipschitz hc).injective\n\nprotected lemma inj_on (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) :\n  inj_on f s :=\ninj_on_iff_injective.2 $ hf.injective hc\n\n\/-- A map approximating a linear equivalence on a set defines a local equivalence on this set.\nShould not be used outside of this file, because it is superseded by `to_local_homeomorph` below.\n\nThis is a first step towards the inverse function. -\/\ndef to_local_equiv (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) : local_equiv E F :=\n(hf.inj_on hc).to_local_equiv _ _\n\n\/-- The inverse function is continuous on `f '' s`. Use properties of `local_homeomorph` instead. -\/\nlemma inverse_continuous_on (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) :\n  continuous_on (hf.to_local_equiv hc).symm (f '' s) :=\nbegin\n  apply continuous_on_iff_continuous_restrict.2,\n  refine ((hf.antilipschitz hc).to_right_inv_on' _ (hf.to_local_equiv hc).right_inv').continuous,\n  exact (\u03bb x hx, (hf.to_local_equiv hc).map_target hx)\nend\n\n\/-!\nNow we prove that `f '' s` is an open set. This follows from the fact that the restriction of `f`\non `s` is an open map. More precisely, we show that the image of a closed ball $$\\bar B(a, \u03b5) \u2286 s$$\nunder `f` includes the closed ball $$\\bar B\\left(f(a), \\frac{\u03b5}{\u2225{f'}\u207b\u00b9\u2225\u207b\u00b9-c}\\right)$$.\n\nIn order to do this, we introduce an auxiliary map $$g_y(x) = x + {f'}\u207b\u00b9 (y - f x)$$. Provided that\n$$\u2225y - f a\u2225 \u2264 \\frac{\u03b5}{\u2225{f'}\u207b\u00b9\u2225\u207b\u00b9-c}$$, we prove that $$g_y$$ contracts in $$\\bar B(a, \u03b5)$$ and `f`\nsends the fixed point of $$g_y$$ to `y`.\n-\/\n\nsection\n\nvariables (f f')\n\n\/-- Iterations of this map converge to `f\u207b\u00b9 y`. The formula is very similar to the one\nused in Newton's method, but we use the same `f'.symm` for all `y` instead of evaluating\nthe derivative at each point along the orbit. -\/\ndef inverse_approx_map (y : F) (x : E) : E := x + f'.symm (y - f x)\n\nend\n\nsection inverse_approx_map\n\nvariables (y : F) {x x' : E} {\u03b5 : \u211d}\n\nlocal notation `g` := inverse_approx_map f f' y\n\nlemma inverse_approx_map_sub (x x' : E) : g x - g x' = (x - x') - f'.symm (f x - f x') :=\nby { simp only [inverse_approx_map, f'.symm.map_sub], abel }\n\nlemma inverse_approx_map_dist_self (x : E) :\n  dist (g x) x = dist (f'.symm $ f x) (f'.symm y) :=\nby simp only [inverse_approx_map, dist_eq_norm, f'.symm.map_sub, add_sub_cancel', norm_sub_rev]\n\nlemma inverse_approx_map_dist_self_le (x : E) :\n  dist (g x) x \u2264 N * dist (f x) y :=\nby { rw inverse_approx_map_dist_self, exact f'.symm.lipschitz.dist_le_mul (f x) y }\n\nlemma inverse_approx_map_fixed_iff {x : E} :\n  g x = x \u2194 f x = y :=\nby rw [\u2190 dist_eq_zero, inverse_approx_map_dist_self, dist_eq_zero, f'.symm.injective.eq_iff]\n\nlemma inverse_approx_map_contracts_on (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  {x x'} (hx : x \u2208 s) (hx' : x' \u2208 s) :\n  dist (g x) (g x') \u2264 N * c * dist x x' :=\nbegin\n  rw [dist_eq_norm, dist_eq_norm, inverse_approx_map_sub, norm_sub_rev],\n  suffices : \u2225f'.symm (f x - f x' - f' (x - x'))\u2225 \u2264 N * (c * \u2225x - x'\u2225),\n    by simpa only [f'.symm.map_sub, f'.symm_apply_apply, mul_assoc] using this,\n  exact (f'.symm : F \u2192L[\ud835\udd5c] E).le_op_norm_of_le (hf x hx x' hx')\nend\n\nvariable {y}\n\nlemma inverse_approx_map_maps_to (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) {b : E} (hb : b \u2208 s) (h\u03b5 : closed_ball b \u03b5 \u2286 s)\n  (hy : y \u2208 closed_ball (f b) ((N\u207b\u00b9 - c) * \u03b5)) :\n  maps_to g (closed_ball b \u03b5) (closed_ball b \u03b5) :=\nbegin\n  cases hc with hE hc,\n  { exactI \u03bb x hx, subsingleton.elim x (g x) \u25b8 hx },\n  assume x hx,\n  simp only [mem_closed_ball] at hx hy \u22a2,\n  rw [dist_comm] at hy,\n  calc dist (inverse_approx_map f f' y x) b \u2264\n    dist (inverse_approx_map f f' y x) (inverse_approx_map f f' y b) +\n      dist (inverse_approx_map f f' y b) b : dist_triangle _ _ _\n  ... \u2264 N * c * dist x b + N * dist (f b) y :\n    add_le_add (hf.inverse_approx_map_contracts_on y (h\u03b5 hx) hb)\n      (inverse_approx_map_dist_self_le _ _)\n  ... \u2264 N * c * \u03b5 + N * ((N\u207b\u00b9 - c) * \u03b5) :\n    add_le_add (mul_le_mul_of_nonneg_left hx (mul_nonneg (nnreal.coe_nonneg _) c.coe_nonneg))\n      (mul_le_mul_of_nonneg_left hy (nnreal.coe_nonneg _))\n  ... = N * (c + (N\u207b\u00b9 - c)) * \u03b5 : by simp only [mul_add, add_mul, mul_assoc]\n  ... = \u03b5 : by { rw [add_sub_cancel'_right, mul_inv_cancel, one_mul],\n    exact ne_of_gt (inv_pos.1 $ lt_of_le_of_lt c.coe_nonneg hc) }\nend\n\nend inverse_approx_map\n\ninclude cs\n\nvariable {\u03b5 : \u211d}\n\ntheorem surj_on_closed_ball (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) {b : E} (\u03b50 : 0 \u2264 \u03b5) (h\u03b5 : closed_ball b \u03b5 \u2286 s) :\n  surj_on f (closed_ball b \u03b5) (closed_ball (f b) ((N\u207b\u00b9 - c) * \u03b5)) :=\nbegin\n  cases hc with hE hc,\n  { resetI,\n    haveI hF : subsingleton F := f'.symm.to_linear_equiv.to_equiv.subsingleton,\n    intros y hy,\n    exact \u27e8b, mem_closed_ball_self \u03b50, subsingleton.elim _ _\u27e9 },\n  intros y hy,\n  have : contracting_with (N * c) ((hf.inverse_approx_map_maps_to (or.inr hc)\n    (h\u03b5 $ mem_closed_ball_self \u03b50) h\u03b5 hy).restrict _ _ _),\n  { split,\n    { rwa [mul_comm, \u2190 nnreal.lt_inv_iff_mul_lt],\n      exact ne_of_gt (inv_pos.1 $ lt_of_le_of_lt c.coe_nonneg hc) },\n    { exact lipschitz_with.of_dist_le_mul (\u03bb x x', hf.inverse_approx_map_contracts_on\n        y (h\u03b5 x.mem) (h\u03b5 x'.mem)) } },\n  refine \u27e8this.efixed_point' _ _ _ b (mem_closed_ball_self \u03b50) (edist_lt_top _ _), _, _\u27e9,\n  { exact is_closed_ball.is_complete },\n  { apply contracting_with.efixed_point_mem' },\n  { exact (inverse_approx_map_fixed_iff y).1 (this.efixed_point_is_fixed_pt' _ _ _ _) }\nend\n\nsection\n\nvariables (f s)\n\n\/-- Given a function `f` that approximates a linear equivalence on an open set `s`,\nreturns a local homeomorph with `to_fun = f` and `source = s`. -\/\ndef to_local_homeomorph (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) (hs : is_open s) : local_homeomorph E F :=\n{ to_local_equiv := hf.to_local_equiv hc,\n  open_source := hs,\n  open_target :=\n    begin\n      cases hc with hE hc,\n      { resetI,\n        haveI hF : subsingleton F := f'.to_linear_equiv.to_equiv.symm.subsingleton,\n        apply is_open_discrete },\n      change is_open (f '' s),\n      simp only [is_open_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs \u22a2,\n      intros x hx,\n      rcases hs x hx with \u27e8\u03b5, \u03b50, h\u03b5\u27e9,\n      refine \u27e8(N\u207b\u00b9 - c) * \u03b5, mul_pos (sub_pos.2 hc) \u03b50, _\u27e9,\n      exact (hf.surj_on_closed_ball (or.inr hc) (le_of_lt \u03b50) h\u03b5).mono h\u03b5 (subset.refl _)\n    end,\n  continuous_to_fun := hf.continuous_on,\n  continuous_inv_fun := hf.inverse_continuous_on hc }\n\nend\n\n@[simp] lemma to_local_homeomorph_coe (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) (hs : is_open s) :\n  (hf.to_local_homeomorph f s hc hs : E \u2192 F) = f := rfl\n\n@[simp] lemma to_local_homeomorph_source (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) (hs : is_open s) :\n  (hf.to_local_homeomorph f s hc hs).source = s := rfl\n\n@[simp] lemma to_local_homeomorph_target (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) (hs : is_open s) :\n  (hf.to_local_homeomorph f s hc hs).target = f '' s := rfl\n\nlemma closed_ball_subset_target (hf : approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s c)\n  (hc : subsingleton E \u2228 c < N\u207b\u00b9) (hs : is_open s) {b : E} (\u03b50 : 0 \u2264 \u03b5) (h\u03b5 : closed_ball b \u03b5 \u2286 s) :\n  closed_ball (f b) ((N\u207b\u00b9 - c) * \u03b5) \u2286 (hf.to_local_homeomorph f s hc hs).target :=\n(hf.surj_on_closed_ball hc \u03b50 h\u03b5).mono h\u03b5 (subset.refl _)\n\nend approximates_linear_on\n\n\/-!\n### Inverse function theorem\n\nNow we prove the inverse function theorem. Let `f : E \u2192 F` be a map defined on a complete vector\nspace `E`. Assume that `f` has an invertible derivative `f' : E \u2243L[\ud835\udd5c] F` at `a : E` in the strict\nsense. Then `f` approximates `f'` in the sense of `approximates_linear_on` on an open neighborhood\nof `a`, and we can apply `approximates_linear_on.to_local_homeomorph` to construct the inverse\nfunction. -\/\n\nnamespace has_strict_fderiv_at\n\n\/-- If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`\nwith constant `c` on some neighborhood of `a`. -\/\nlemma approximates_deriv_on_nhds {f : E \u2192 F} {f' : E \u2192L[\ud835\udd5c] F} {a : E}\n  (hf : has_strict_fderiv_at f f' a) {c : \u211d\u22650} (hc : subsingleton E \u2228 0 < c) :\n  \u2203 s \u2208 \ud835\udcdd a, approximates_linear_on f f' s c :=\nbegin\n  cases hc with hE hc,\n  { refine \u27e8univ, mem_nhds_sets is_open_univ trivial, \u03bb x hx y hy, _\u27e9,\n    simp [@subsingleton.elim E hE x y] },\n  have := hf.def hc,\n  rw [nhds_prod_eq, filter.eventually, mem_prod_same_iff] at this,\n  rcases this with \u27e8s, has, hs\u27e9,\n  exact \u27e8s, has, \u03bb x hx y hy, hs (mk_mem_prod hx hy)\u27e9\nend\n\nvariables [cs : complete_space E] {f : E \u2192 F} {f' : E \u2243L[\ud835\udd5c] F} {a : E}\n\nlemma approximates_deriv_on_open_nhds (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  \u2203 (s : set E) (hs : a \u2208 s \u2227 is_open s),\n    approximates_linear_on f (f' : E \u2192L[\ud835\udd5c] F) s ((nnnorm (f'.symm : F \u2192L[\ud835\udd5c] E))\u207b\u00b9 \/ 2) :=\nbegin\n  refine ((nhds_basis_opens a).exists_iff _).1 _,\n  exact (\u03bb s t, approximates_linear_on.mono_set),\n  exact (hf.approximates_deriv_on_nhds $ f'.subsingleton_or_nnnorm_symm_pos.imp id $\n    \u03bb hf', nnreal.half_pos $ nnreal.inv_pos.2 $ hf')\nend\n\ninclude cs\n\nvariable (f)\n\n\/-- Given a function with an invertible strict derivative at `a`, returns a `local_homeomorph`\nwith `to_fun = f` and `a \u2208 source`. This is a part of the inverse function theorem.\nThe other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function\nof this `local_homeomorph` has derivative `f'.symm`. -\/\ndef to_local_homeomorph (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) : local_homeomorph E F :=\napproximates_linear_on.to_local_homeomorph f\n  (classical.some hf.approximates_deriv_on_open_nhds)\n  (classical.some_spec hf.approximates_deriv_on_open_nhds).snd\n  (f'.subsingleton_or_nnnorm_symm_pos.imp id $ \u03bb hf', nnreal.half_lt_self $ ne_of_gt $\n    nnreal.inv_pos.2 $ hf')\n  (classical.some_spec hf.approximates_deriv_on_open_nhds).fst.2\n\nvariable {f}\n\n@[simp] lemma to_local_homeomorph_coe (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  (hf.to_local_homeomorph f : E \u2192 F) = f := rfl\n\nlemma mem_to_local_homeomorph_source (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  a \u2208 (hf.to_local_homeomorph f).source :=\n(classical.some_spec hf.approximates_deriv_on_open_nhds).fst.1\n\nlemma image_mem_to_local_homeomorph_target (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  f a \u2208 (hf.to_local_homeomorph f).target :=\n(hf.to_local_homeomorph f).map_source hf.mem_to_local_homeomorph_source\n\nvariables (f f' a)\n\n\/-- Given a function `f` with an invertible derivative, returns a function that is locally inverse\nto `f`. -\/\ndef local_inverse (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) : F \u2192 E :=\n(hf.to_local_homeomorph f).symm\n\nvariables {f f' a}\n\nlemma local_inverse_def (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  hf.local_inverse f _ _ = (hf.to_local_homeomorph f).symm :=\nrfl\n\nlemma eventually_left_inverse (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  \u2200\u1da0 x in \ud835\udcdd a, hf.local_inverse f f' a (f x) = x :=\n(hf.to_local_homeomorph f).eventually_left_inverse hf.mem_to_local_homeomorph_source\n\n@[simp] lemma local_inverse_apply_image (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  hf.local_inverse f f' a (f a) = a :=\nhf.eventually_left_inverse.self_of_nhds\n\nlemma eventually_right_inverse (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  \u2200\u1da0 y in \ud835\udcdd (f a), f (hf.local_inverse f f' a y) = y :=\n(hf.to_local_homeomorph f).eventually_right_inverse' hf.mem_to_local_homeomorph_source\n\nlemma local_inverse_continuous_at (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  continuous_at (hf.local_inverse f f' a) (f a) :=\n(hf.to_local_homeomorph f).continuous_at_symm hf.image_mem_to_local_homeomorph_target\n\nlemma local_inverse_tendsto (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  tendsto (hf.local_inverse f f' a) (\ud835\udcdd $ f a) (\ud835\udcdd a) :=\n(hf.to_local_homeomorph f).tendsto_symm hf.mem_to_local_homeomorph_source\n\nlemma local_inverse_unique (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) {g : F \u2192 E}\n  (hg : \u2200\u1da0 x in \ud835\udcdd a, g (f x) = x) :\n  \u2200\u1da0 y in \ud835\udcdd (f a), g y = local_inverse f f' a hf y :=\neventually_eq_of_left_inv_of_right_inv hg hf.eventually_right_inverse $\n  (hf.to_local_homeomorph f).tendsto_symm hf.mem_to_local_homeomorph_source\n\n\/-- If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,\nthen the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. -\/\ntheorem to_local_inverse (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) :\n  has_strict_fderiv_at (hf.local_inverse f f' a) (f'.symm : F \u2192L[\ud835\udd5c] E) (f a) :=\n(hf.to_local_homeomorph f).has_strict_fderiv_at_symm hf.image_mem_to_local_homeomorph_target $\n  by simpa [\u2190 local_inverse_def] using hf\n\n\/-- If `f : E \u2192 F` has an invertible derivative `f'` at `a` in the sense of strict differentiability\nand `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.\n\nFor a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[complete_space E]`\nsee `of_local_left_inverse`.  -\/\ntheorem to_local_left_inverse (hf : has_strict_fderiv_at f (f' : E \u2192L[\ud835\udd5c] F) a) {g : F \u2192 E}\n  (hg : \u2200\u1da0 x in \ud835\udcdd a, g (f x) = x) :\n  has_strict_fderiv_at g (f'.symm : F \u2192L[\ud835\udd5c] E) (f a) :=\nhf.to_local_inverse.congr_of_eventually_eq $ (hf.local_inverse_unique hg).mono $ \u03bb _, eq.symm\n\nend has_strict_fderiv_at\n\n\/-!\n### Inverse function theorem, 1D case\n\nIn this case we prove a version of the inverse function theorem for maps `f : \ud835\udd5c \u2192 \ud835\udd5c`.\nWe use `continuous_linear_equiv.units_equiv_aut` to translate `has_strict_deriv_at f f' a` and\n`f' \u2260 0` into `has_strict_fderiv_at f (_ : \ud835\udd5c \u2243L[\ud835\udd5c] \ud835\udd5c) a`.\n-\/\n\nnamespace has_strict_deriv_at\n\nvariables [cs : complete_space \ud835\udd5c] {f : \ud835\udd5c \u2192 \ud835\udd5c} {f' a : \ud835\udd5c} (hf : has_strict_deriv_at f f' a)\n  (hf' : f' \u2260 0)\n\ninclude cs\n\nvariables (f f' a)\n\n\/-- A function that is inverse to `f` near `a`. -\/\n@[reducible] def local_inverse : \ud835\udd5c \u2192 \ud835\udd5c :=\n(hf.has_strict_fderiv_at_equiv hf').local_inverse _ _ _\n\nvariables {f f' a}\n\ntheorem to_local_inverse : has_strict_deriv_at (hf.local_inverse f f' a hf') f'\u207b\u00b9 (f a) :=\n(hf.has_strict_fderiv_at_equiv hf').to_local_inverse\n\ntheorem to_local_left_inverse {g : \ud835\udd5c \u2192 \ud835\udd5c} (hg : \u2200\u1da0 x in \ud835\udcdd a, g (f x) = x) :\n  has_strict_deriv_at g f'\u207b\u00b9 (f a) :=\n(hf.has_strict_fderiv_at_equiv hf').to_local_left_inverse hg\n\nend has_strict_deriv_at\n\n\/-!\n### Inverse function theorem, smooth case\n\n-\/\n\nnamespace times_cont_diff_at\nvariables {E' : Type*} [normed_group E'] [normed_space \u211d E']\nvariables {F' : Type*} [normed_group F'] [normed_space \u211d F']\nvariables [complete_space E'] (f : E' \u2192 F') {f' : E' \u2243L[\u211d] F'} {a : E'}\n\n\/-- Given a `times_cont_diff` function over `\u211d` with an invertible derivative at `a`, returns a\n`local_homeomorph` with `to_fun = f` and `a \u2208 source`. -\/\ndef to_local_homeomorph\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  local_homeomorph E' F' :=\n(hf.has_strict_fderiv_at' hf' hn).to_local_homeomorph f\n\nvariable {f}\n\n@[simp] lemma to_local_homeomorph_coe\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  (hf.to_local_homeomorph f hf' hn : E' \u2192 F') = f := rfl\n\nlemma mem_to_local_homeomorph_source\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  a \u2208 (hf.to_local_homeomorph f hf' hn).source :=\n(hf.has_strict_fderiv_at' hf' hn).mem_to_local_homeomorph_source\n\nlemma image_mem_to_local_homeomorph_target\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  f a \u2208 (hf.to_local_homeomorph f hf' hn).target :=\n(hf.has_strict_fderiv_at' hf' hn).image_mem_to_local_homeomorph_target\n\n\/-- Given a `times_cont_diff` function over `\u211d` with an invertible derivative at `a`, returns a\nfunction that is locally inverse to `f`. -\/\ndef local_inverse\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  F' \u2192 E' :=\n(hf.has_strict_fderiv_at' hf' hn).local_inverse f f' a\n\nlemma local_inverse_apply_image\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  hf.local_inverse hf' hn (f a) = a :=\n(hf.has_strict_fderiv_at' hf' hn).local_inverse_apply_image\n\n\/-- Given a `times_cont_diff` function over `\u211d` with an invertible derivative at `a`, the inverse\nfunction (produced by `times_cont_diff.to_local_homeomorph`) is also `times_cont_diff`. -\/\nlemma to_local_inverse\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f a) (hf' : has_fderiv_at f (f' : E' \u2192L[\u211d] F') a)\n  (hn : 1 \u2264 n) :\n  times_cont_diff_at \u211d n (hf.local_inverse hf' hn) (f a) :=\nbegin\n  have := hf.local_inverse_apply_image hf' hn,\n  apply (hf.to_local_homeomorph f hf' hn).times_cont_diff_at_symm\n    (image_mem_to_local_homeomorph_target hf hf' hn),\n  { convert hf' },\n  { convert hf }\nend\n\nend times_cont_diff_at\n","avg_line_length":42.0418118467,"max_line_length":100,"alphanum_fraction":0.6883391348}
-{"size":205,"ext":"lean","lang":"Lean","max_stars_count":2232.0,"content":"axiom I : Type\ndefinition F (X : Type) : Type := (X \u2192 Prop) \u2192 Prop\naxiom unfoldd : I \u2192 F I\naxiom foldd   : F I \u2192 I\naxiom iso1 : \u2200x, foldd (unfoldd x) = x\n\ntheorem iso2 : \u2200x, foldd (unfoldd x) = x\n:= sorry\n","avg_line_length":22.7777777778,"max_line_length":51,"alphanum_fraction":0.5951219512}
-{"size":3175,"ext":"hlean","lang":"Lean","max_stars_count":130.0,"content":"\/-\nCopyright (c) 2015 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n\nDeclaration of the primitive hits in Lean\n-\/\n\nprelude\n\nimport .trunc .pathover\n\nopen is_trunc eq\n\n\/-\n  We take two higher inductive types (hits) as primitive notions in Lean. We define all other hits\n  in terms of these two hits. The hits which are primitive are\n    - n-truncation\n    - quotients (not truncated)\n  For each of the hits we add the following constants:\n    - the type formation\n    - the term and path constructors\n    - the dependent recursor\n  We add the computation rule for point constructors judgmentally to the kernel of Lean.\n  The computation rules for the path constructors are added (propositionally) as axioms\n\n  In this file we only define the dependent recursor. For the nondependent recursor and all other\n  uses of these hits, see the folder ..\/hit\/\n-\/\n\nconstant trunc.{u} (n : \u2115\u208b\u2082) (A : Type.{u}) : Type.{u}\n\nnamespace trunc\n  constant tr {n : \u2115\u208b\u2082} {A : Type} (a : A) : trunc n A\n  constant is_trunc_trunc (n : \u2115\u208b\u2082) (A : Type) : is_trunc n (trunc n A)\n\n  attribute is_trunc_trunc [instance]\n\n  protected constant rec {n : \u2115\u208b\u2082} {A : Type} {P : trunc n A \u2192 Type}\n    [Pt : \u03a0aa, is_trunc n (P aa)] (H : \u03a0a, P (tr a)) : \u03a0aa, P aa\n\n  protected definition rec_on [reducible] {n : \u2115\u208b\u2082} {A : Type}\n    {P : trunc n A \u2192 Type} (aa : trunc n A) [Pt : \u03a0aa, is_trunc n (P aa)] (H : \u03a0a, P (tr a))\n    : P aa :=\n  trunc.rec H aa\nend trunc\n\nconstant quotient.{u v} {A : Type.{u}} (R : A \u2192 A \u2192 Type.{v}) : Type.{max u v}\n\nnamespace quotient\n\n  constant class_of {A : Type} (R : A \u2192 A \u2192 Type) (a : A) : quotient R\n\n  constant eq_of_rel {A : Type} (R : A \u2192 A \u2192 Type) \u2983a a' : A\u2984 (H : R a a')\n    : class_of R a = class_of R a'\n\n  protected constant rec {A : Type} {R : A \u2192 A \u2192 Type} {P : quotient R \u2192 Type}\n    (Pc : \u03a0(a : A), P (class_of R a)) (Pp : \u03a0\u2983a a' : A\u2984 (H : R a a'), Pc a =[eq_of_rel R H] Pc a')\n    (x : quotient R) : P x\n\n  protected definition rec_on [reducible] {A : Type} {R : A \u2192 A \u2192 Type} {P : quotient R \u2192 Type}\n    (x : quotient R) (Pc : \u03a0(a : A), P (class_of R a))\n    (Pp : \u03a0\u2983a a' : A\u2984 (H : R a a'), Pc a =[eq_of_rel R H] Pc a') : P x :=\n  quotient.rec Pc Pp x\n\nend quotient\n\ninit_hits -- Initialize builtin computational rules for trunc and quotient\n\nnamespace trunc\n  definition rec_tr [reducible] {n : \u2115\u208b\u2082} {A : Type} {P : trunc n A \u2192 Type}\n    [Pt : \u03a0aa, is_trunc n (P aa)] (H : \u03a0a, P (tr a)) (a : A) : trunc.rec H (tr a) = H a :=\n  idp\nend trunc\n\nnamespace quotient\n  definition rec_class_of {A : Type} {R : A \u2192 A \u2192 Type} {P : quotient R \u2192 Type}\n    (Pc : \u03a0(a : A), P (class_of R a)) (Pp : \u03a0\u2983a a' : A\u2984 (H : R a a'), Pc a =[eq_of_rel R H] Pc a')\n    (a : A) : quotient.rec Pc Pp (class_of R a) = Pc a :=\n  idp\n\n  constant rec_eq_of_rel {A : Type} {R : A \u2192 A \u2192 Type} {P : quotient R \u2192 Type}\n    (Pc : \u03a0(a : A), P (class_of R a)) (Pp : \u03a0\u2983a a' : A\u2984 (H : R a a'), Pc a =[eq_of_rel R H] Pc a')\n    {a a' : A} (H : R a a') : apd (quotient.rec Pc Pp) (eq_of_rel R H) = Pp H\nend quotient\n\nattribute quotient.class_of trunc.tr [constructor]\nattribute quotient.rec_on trunc.rec_on [unfold 4]\n","avg_line_length":35.6741573034,"max_line_length":98,"alphanum_fraction":0.6201574803}
-{"size":142,"ext":"lean","lang":"Lean","max_stars_count":19.0,"content":"theorem test (p q : Prop) (hp : p) (hq : q) : p \u2227 q \u2227 p :=\n  by apply And.intro\n     exact hp\n     apply And.intro\n     exact hq\n     exact hp","avg_line_length":23.6666666667,"max_line_length":58,"alphanum_fraction":0.5281690141}
-{"size":11350,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-\/\nimport Lean.Compiler.IR.Format\nimport Lean.Compiler.IR.Basic\nimport Lean.Compiler.IR.CompilerM\n\nnamespace Lean.IR.UnreachableBranches\n\n\/-- Value used in the abstract interpreter -\/\ninductive Value where\n  | bot -- undefined\n  | top -- any value\n  | ctor (i : CtorInfo) (vs : Array Value)\n  | choice (vs : List Value)\n  deriving Inhabited, Repr\n\nprotected partial def Value.toFormat : Value \u2192 Format\n  | Value.bot => \"\u22a5\"\n  | Value.top => \"\u22a4\"\n  | Value.ctor info vs =>\n    if vs.isEmpty then\n      format info.name\n    else\n      Format.paren <| format info.name ++ Std.Format.join (vs.toList.map fun v => \" \" ++ Value.toFormat v)\n  | Value.choice vs =>\n    Format.paren <| Std.Format.joinSep (vs.map Value.toFormat) \" | \"\n\ninstance : ToFormat Value where\n  format := Value.toFormat\n\ninstance : ToString Value where\n  toString v := toString (format v)\n\nnamespace Value\n\nprotected partial def beq : Value \u2192 Value \u2192 Bool\n  | bot, bot => true\n  | top, top => true\n  | ctor i\u2081 vs\u2081, ctor i\u2082 vs\u2082 => i\u2081 == i\u2082 && Array.isEqv vs\u2081 vs\u2082 Value.beq\n  | choice vs\u2081, choice vs\u2082 =>\n    vs\u2081.all (fun v\u2081 => vs\u2082.any fun v\u2082 => Value.beq v\u2081 v\u2082)\n    &&\n    vs\u2082.all (fun v\u2082 => vs\u2081.any fun v\u2081 => Value.beq v\u2081 v\u2082)\n  | _, _ => false\n\ninstance : BEq Value := \u27e8Value.beq\u27e9\n\npartial def addChoice (merge : Value \u2192 Value \u2192 Value) : List Value \u2192 Value \u2192 List Value\n  | [], v => [v]\n  | v\u2081@(ctor i\u2081 _) :: cs, v\u2082@(ctor i\u2082 _) =>\n    if i\u2081 == i\u2082 then merge v\u2081 v\u2082 :: cs\n    else v\u2081 :: addChoice merge cs v\u2082\n  | _, _ => panic! \"invalid addChoice\"\n\npartial def merge (v\u2081 v\u2082 : Value) : Value :=\n  match v\u2081, v\u2082 with\n  | bot, v => v\n  | v, bot => v\n  | top, _ => top\n  | _, top => top\n  | v\u2081@(ctor i\u2081 vs\u2081), v\u2082@(ctor i\u2082 vs\u2082) =>\n    if i\u2081 == i\u2082 then ctor i\u2081 <| vs\u2081.size.fold (init := #[]) fun i r => r.push (merge vs\u2081[i] vs\u2082[i])\n    else choice [v\u2081, v\u2082]\n  | choice vs\u2081, choice vs\u2082 => choice <| vs\u2081.foldl (addChoice merge) vs\u2082\n  | choice vs, v => choice <| addChoice merge vs v\n  | v, choice vs => choice <| addChoice merge vs v\n\nprotected partial def format : Value \u2192 Format\n  | top => \"top\"\n  | bot => \"bot\"\n  | choice vs => format \"@\" ++ @List.format _ \u27e8Value.format\u27e9 vs\n  | ctor i vs => format \"#\" ++ if vs.isEmpty then format i.name else Format.paren (format i.name ++ @formatArray _ \u27e8Value.format\u27e9 vs)\n\ninstance : ToFormat Value := \u27e8Value.format\u27e9\ninstance : ToString Value := \u27e8Format.pretty \u2218 Value.format\u27e9\n\n\/--\n  In `truncate`, we approximate a value as `top` if depth > `truncateMaxDepth`.\n  TODO: add option to control this parameter.\n-\/\ndef truncateMaxDepth := 8\n\n\/--\n  Make sure constructors of recursive inductive datatypes can only occur once in each path.\n  Values at depth > truncateMaxDepth are also approximated at `top`.\n  We use this function this function to implement a simple widening operation for our abstract\n  interpreter.\n  Recall the widening functions is used to ensure termination in abstract interpreters.\n-\/\npartial def truncate (env : Environment) (v : Value) (s : NameSet) : Value :=\n  go v s truncateMaxDepth\nwhere\n  go (v : Value) (s : NameSet) (depth : Nat) : Value :=\n    match depth with\n    | 0 => top\n    | depth+1 =>\n      match v, s with\n      | ctor i vs, found =>\n        let I := i.name.getPrefix\n        if found.contains I then\n          top\n        else\n          let cont (found' : NameSet) : Value :=\n            ctor i (vs.map fun v => go v found' depth)\n          match env.find? I with\n          | some (ConstantInfo.inductInfo d) =>\n            if d.isRec then cont (found.insert I)\n            else cont found\n          | _ => cont found\n      | choice vs, found =>\n        let newVs := vs.map fun v => go v found depth\n        if newVs.elem top then top\n        else choice newVs\n      | v, _ => v\n\n\/-- Widening operator that guarantees termination in our abstract interpreter. -\/\ndef widening (env : Environment) (v\u2081 v\u2082 : Value) : Value :=\n  truncate env (merge v\u2081 v\u2082) {}\n\nend Value\n\nabbrev FunctionSummaries := SMap FunId Value\n\nbuiltin_initialize functionSummariesExt : SimplePersistentEnvExtension (FunId \u00d7 Value) FunctionSummaries \u2190\n  registerSimplePersistentEnvExtension {\n    name       := `unreachBranchesFunSummary,\n    addImportedFn := fun as =>\n      let cache : FunctionSummaries := mkStateFromImportedEntries (fun s (p : FunId \u00d7 Value) => s.insert p.1 p.2) {} as\n      cache.switch,\n    addEntryFn := fun s \u27e8e, n\u27e9 => s.insert e n\n  }\n\ndef addFunctionSummary (env : Environment) (fid : FunId) (v : Value) : Environment :=\n  functionSummariesExt.addEntry env (fid, v)\n\ndef getFunctionSummary? (env : Environment) (fid : FunId) : Option Value :=\n  (functionSummariesExt.getState env).find? fid\n\nabbrev Assignment := Std.HashMap VarId Value\n\nstructure InterpContext where\n  currFnIdx : Nat := 0\n  decls     : Array Decl\n  env       : Environment\n  lctx      : LocalContext := {}\n\nstructure InterpState where\n  assignments : Array Assignment\n  funVals     : Std.PArray Value -- we take snapshots during fixpoint computations\n\nabbrev M := ReaderT InterpContext (StateM InterpState)\n\nopen Value\n\ndef findVarValue (x : VarId) : M Value := do\n  let ctx \u2190 read\n  let s \u2190 get\n  let assignment := s.assignments[ctx.currFnIdx]\n  return assignment.findD x bot\n\ndef findArgValue (arg : Arg) : M Value :=\n  match arg with\n  | Arg.var x => findVarValue x\n  | _         => pure top\n\ndef updateVarAssignment (x : VarId) (v : Value) : M Unit := do\n  let v' \u2190 findVarValue x\n  let ctx \u2190 read\n  modify fun s => { s with assignments := s.assignments.modify ctx.currFnIdx fun a => a.insert x (merge v v') }\n\ndef resetVarAssignment (x : VarId) : M Unit := do\n  let ctx \u2190 read\n  modify fun s => { s with assignments := s.assignments.modify ctx.currFnIdx fun a => a.insert x Value.bot }\n\ndef resetParamAssignment (y : Param) : M Unit :=\n  resetVarAssignment y.x\n\npartial def projValue : Value \u2192 Nat \u2192 Value\n  | ctor _ vs, i => vs.getD i bot\n  | choice vs, i => vs.foldl (fun r v => merge r (projValue v i)) bot\n  | v, _         => v\n\ndef interpExpr : Expr \u2192 M Value\n  | Expr.ctor i ys => return ctor i (\u2190 ys.mapM fun y => findArgValue y)\n  | Expr.proj i x  => return projValue (\u2190 findVarValue x) i\n  | Expr.fap fid _  => do\n    let ctx \u2190 read\n    match getFunctionSummary? ctx.env fid with\n    | some v => pure v\n    | none   => do\n      let s \u2190 get\n      match ctx.decls.findIdx? (fun decl => decl.name == fid) with\n      | some idx => pure s.funVals[idx]\n      | none     => pure top\n  | _ => pure top\n\npartial def containsCtor : Value \u2192 CtorInfo \u2192 Bool\n  | top,       _ => true\n  | ctor i _,  j => i == j\n  | choice vs, j => vs.any fun v => containsCtor v j\n  | _,         _ => false\n\ndef updateCurrFnSummary (v : Value) : M Unit := do\n  let ctx \u2190 read\n  let currFnIdx := ctx.currFnIdx\n  modify fun s => { s with funVals := s.funVals.modify currFnIdx (fun v' => widening ctx.env v v') }\n\n\/-- Return true if the assignment of at least one parameter has been updated. -\/\ndef updateJPParamsAssignment (ys : Array Param) (xs : Array Arg) : M Bool := do\n  let ctx \u2190 read\n  let currFnIdx := ctx.currFnIdx\n  ys.size.foldM (init := false) fun i r => do\n    let y := ys[i]\n    let x := xs[i]\n    let yVal \u2190 findVarValue y.x\n    let xVal \u2190 findArgValue x\n    let newVal := merge yVal xVal\n    if newVal == yVal then\n      pure r\n    else\n      modify fun s => { s with assignments := s.assignments.modify currFnIdx fun a => a.insert y.x newVal }\n      pure true\n\nprivate partial def resetNestedJPParams : FnBody \u2192 M Unit\n  | FnBody.jdecl _ ys _ k => do\n    ys.forM resetParamAssignment\n    \/- Remark we don't need to reset the parameters of joint-points\n      nested in `b` since they will be reset if this JP is used. -\/\n    resetNestedJPParams k\n  | FnBody.case _ _ _ alts =>\n    alts.forM fun alt => match alt with\n      | Alt.ctor _ b  => resetNestedJPParams b\n      | Alt.default b => resetNestedJPParams b\n  | e => do unless e.isTerminal do resetNestedJPParams e.body\n\npartial def interpFnBody : FnBody \u2192 M Unit\n  | FnBody.vdecl x _ e b => do\n    let v \u2190 interpExpr e\n    updateVarAssignment x v\n    interpFnBody b\n  | FnBody.jdecl j ys v b =>\n    withReader (fun ctx => { ctx with lctx := ctx.lctx.addJP j ys v }) do\n      interpFnBody b\n  | FnBody.case _ x _ alts => do\n    let v \u2190 findVarValue x\n    alts.forM fun alt => do\n      match alt with\n      | Alt.ctor i b  => if containsCtor v i then interpFnBody b\n      | Alt.default b => interpFnBody b\n  | FnBody.ret x => do\n    let v \u2190 findArgValue x\n    updateCurrFnSummary v\n  | FnBody.jmp j xs => do\n    let ctx \u2190 read\n    let ys := (ctx.lctx.getJPParams j).get!\n    let b  := (ctx.lctx.getJPBody j).get!\n    let updated \u2190 updateJPParamsAssignment ys xs\n    if updated then\n      -- We must reset the value of nested join-point parameters since they depend on `ys` values\n      resetNestedJPParams b\n      interpFnBody b\n  | e => do\n    unless e.isTerminal do\n      interpFnBody e.body\n\ndef inferStep : M Bool := do\n  let ctx \u2190 read\n  modify fun s => { s with assignments := ctx.decls.map fun _ => {} }\n  ctx.decls.size.foldM (init := false) fun idx modified => do\n    match ctx.decls[idx] with\n    | Decl.fdecl (xs := ys) (body := b) .. => do\n      let s \u2190 get\n      let currVals := s.funVals[idx]\n      withReader (fun ctx => { ctx with currFnIdx := idx }) do\n        ys.forM fun y => updateVarAssignment y.x top\n        interpFnBody b\n        let s \u2190 get\n        let newVals := s.funVals[idx]\n        pure (modified || currVals != newVals)\n    | Decl.extern _ _ _ _ => pure modified\n\npartial def inferMain : M Unit := do\n  let modified \u2190 inferStep\n  if modified then inferMain else pure ()\n\npartial def elimDeadAux (assignment : Assignment) : FnBody \u2192 FnBody\n  | FnBody.vdecl x t e b  => FnBody.vdecl x t e (elimDeadAux assignment b)\n  | FnBody.jdecl j ys v b => FnBody.jdecl j ys (elimDeadAux assignment v) (elimDeadAux assignment b)\n  | FnBody.case tid x xType alts =>\n    let v := assignment.findD x bot\n    let alts := alts.map fun alt =>\n      match alt with\n      | Alt.ctor i b  => Alt.ctor i <| if containsCtor v i then elimDeadAux assignment b else FnBody.unreachable\n      | Alt.default b => Alt.default (elimDeadAux assignment b)\n    FnBody.case tid x xType alts\n  | e =>\n    if e.isTerminal then e\n    else\n      let (instr, b) := e.split\n      let b := elimDeadAux assignment b\n      instr.setBody b\n\npartial def elimDead (assignment : Assignment) (d : Decl) : Decl :=\n  match d with\n  | Decl.fdecl (body := b) .. => d.updateBody! <| elimDeadAux assignment b\n  | other => other\n\nend UnreachableBranches\n\nopen UnreachableBranches\n\ndef elimDeadBranches (decls : Array Decl) : CompilerM (Array Decl) := do\n  let s \u2190 get\n  let env := s.env\n  let assignments : Array Assignment := decls.map fun _ => {}\n  let funVals := Std.mkPArray decls.size Value.bot\n  let ctx : InterpContext := { decls := decls, env := env }\n  let s : InterpState := { assignments := assignments, funVals := funVals }\n  let (_, s) := (inferMain ctx).run s\n  let funVals := s.funVals\n  let assignments := s.assignments\n  modify fun s =>\n    let env := decls.size.fold (init := s.env) fun i env =>\n      addFunctionSummary env decls[i].name funVals[i]\n    { s with env := env }\n  return decls.mapIdx fun i decl => elimDead assignments[i] decl\n\nend Lean.IR\n","avg_line_length":34.186746988,"max_line_length":133,"alphanum_fraction":0.6436123348}
-{"size":50012,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-\/\nimport algebra.iterate_hom\nimport data.equiv.ring_aut\nimport algebra.module.basic\nimport linear_algebra.basic\nimport tactic.abel\n\n\/-!\n# Algebras over commutative semirings\n\nIn this file we define `algebra`s over commutative (semi)rings, algebra homomorphisms `alg_hom`,\nand algebra equivalences `alg_equiv`.\nWe also define the usual operations on `alg_hom`s (`id`, `comp`).\n\n`subalgebra`s are defined in `algebra.algebra.subalgebra`.\n\nIf `S` is an `R`-algebra and `A` is an `S`-algebra then `algebra.comap.algebra R S A` can be used\nto provide `A` with a structure of an `R`-algebra. Other than that, `algebra.comap` is now\ndeprecated and replaced with `is_scalar_tower`.\n\nFor the category of `R`-algebras, denoted `Algebra R`, see the file\n`algebra\/category\/Algebra\/basic.lean`.\n\n## Notations\n\n* `A \u2192\u2090[R] B` : `R`-algebra homomorphism from `A` to `B`.\n* `A \u2243\u2090[R] B` : `R`-algebra equivalence from `A` to `B`.\n-\/\n\nuniverses u v w u\u2081 v\u2081\n\nopen_locale big_operators\n\nsection prio\n-- We set this priority to 0 later in this file\nset_option extends_priority 200 \/- control priority of\n`instance [algebra R A] : has_scalar R A` -\/\n\n\/--\nGiven a commutative (semi)ring `R`, an `R`-algebra is a (possibly noncommutative)\n(semi)ring `A` endowed with a morphism of rings `R \u2192+* A` which lands in the\ncenter of `A`.\n\nFor convenience, this typeclass extends `has_scalar R A` where the scalar action must\nagree with left multiplication by the image of the structure morphism.\n\nGiven an `algebra R A` instance, the structure morphism `R \u2192+* A` is denoted `algebra_map R A`.\n-\/\n@[nolint has_inhabited_instance]\nclass algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A]\n  extends has_scalar R A, R \u2192+* A :=\n(commutes' : \u2200 r x, to_fun r * x = x * to_fun r)\n(smul_def' : \u2200 r x, r \u2022 x = to_fun r * x)\nend prio\n\n\/-- Embedding `R \u2192+* A` given by `algebra` structure. -\/\ndef algebra_map (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : R \u2192+* A :=\nalgebra.to_ring_hom\n\n\/-- Creating an algebra from a morphism to the center of a semiring. -\/\ndef ring_hom.to_algebra' {R S} [comm_semiring R] [semiring S] (i : R \u2192+* S)\n  (h : \u2200 c x, i c * x = x * i c) :\n  algebra R S :=\n{ smul := \u03bb c x, i c * x,\n  commutes' := h,\n  smul_def' := \u03bb c x, rfl,\n  to_ring_hom := i}\n\n\/-- Creating an algebra from a morphism to a commutative semiring. -\/\ndef ring_hom.to_algebra {R S} [comm_semiring R] [comm_semiring S] (i : R \u2192+* S) :\n  algebra R S :=\ni.to_algebra' $ \u03bb _, mul_comm _\n\nlemma ring_hom.algebra_map_to_algebra {R S} [comm_semiring R] [comm_semiring S]\n  (i : R \u2192+* S) :\n  @algebra_map R S _ _ i.to_algebra = i :=\nrfl\n\nnamespace algebra\n\nvariables {R : Type u} {S : Type v} {A : Type w} {B : Type*}\n\n\/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 1) * x = x * (r \u2022 1) = r \u2022 x` for all `r : R` and `x : A`, then `A` is an `algebra`\nover `R`.\n\nSee note [reducible non-instances]. -\/\n@[reducible]\ndef of_module' [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x : A), (r \u2022 1) * x = r \u2022 x)\n  (h\u2082 : \u2200 (r : R) (x : A), x * (r \u2022 1) = r \u2022 x) : algebra R A :=\n{ to_fun := \u03bb r, r \u2022 1,\n  map_one' := one_smul _ _,\n  map_mul' := \u03bb r\u2081 r\u2082, by rw [h\u2081, mul_smul],\n  map_zero' := zero_smul _ _,\n  map_add' := \u03bb r\u2081 r\u2082, add_smul r\u2081 r\u2082 1,\n  commutes' := \u03bb r x, by simp only [h\u2081, h\u2082],\n  smul_def' := \u03bb r x, by simp only [h\u2081] }\n\n\/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 x) * y = x * (r \u2022 y) = r \u2022 (x * y)` for all `r : R` and `x y : A`, then `A`\nis an `algebra` over `R`.\n\nSee note [reducible non-instances]. -\/\n@[reducible]\ndef of_module [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x y : A), (r \u2022 x) * y = r \u2022 (x * y))\n  (h\u2082 : \u2200 (r : R) (x y : A), x * (r \u2022 y) = r \u2022 (x * y)) : algebra R A :=\nof_module' (\u03bb r x, by rw [h\u2081, one_mul]) (\u03bb r x, by rw [h\u2082, mul_one])\n\nsection semiring\n\nvariables [comm_semiring R] [comm_semiring S]\nvariables [semiring A] [algebra R A] [semiring B] [algebra R B]\n\n\/-- We keep this lemma private because it picks up the `algebra.to_has_scalar` instance\nwhich we set to priority 0 shortly. See `smul_def` below for the public version. -\/\nprivate lemma smul_def'' (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\n\/--\nTo prove two algebra structures on a fixed `[comm_semiring R] [semiring A]` agree,\nit suffices to check the `algebra_map`s agree.\n-\/\n-- We'll later use this to show `algebra \u2124 M` is a subsingleton.\n@[ext]\nlemma algebra_ext {R : Type*} [comm_semiring R] {A : Type*} [semiring A] (P Q : algebra R A)\n  (w : \u2200 (r : R), by { haveI := P, exact algebra_map R A r } =\n    by { haveI := Q, exact algebra_map R A r }) :\n  P = Q :=\nbegin\n  unfreezingI { rcases P with \u27e8\u27e8P\u27e9\u27e9, rcases Q with \u27e8\u27e8Q\u27e9\u27e9 },\n  congr,\n  { funext r a,\n    replace w := congr_arg (\u03bb s, s * a) (w r),\n    simp only [\u2190smul_def''] at w,\n    apply w, },\n  { ext r,\n    exact w r, },\n  { apply proof_irrel_heq, },\n  { apply proof_irrel_heq, },\nend\n\n@[priority 200] -- see Note [lower instance priority]\ninstance to_module : module R A :=\n{ one_smul := by simp [smul_def''],\n  mul_smul := by simp [smul_def'', mul_assoc],\n  smul_add := by simp [smul_def'', mul_add],\n  smul_zero := by simp [smul_def''],\n  add_smul := by simp [smul_def'', add_mul],\n  zero_smul := by simp [smul_def''] }\n\n-- From now on, we don't want to use the following instance anymore.\n-- Unfortunately, leaving it in place causes deterministic timeouts later in mathlib.\nattribute [instance, priority 0] algebra.to_has_scalar\n\nlemma smul_def (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\nlemma algebra_map_eq_smul_one (r : R) : algebra_map R A r = r \u2022 1 :=\ncalc algebra_map R A r = algebra_map R A r * 1 : (mul_one _).symm\n                   ... = r \u2022 1                 : (algebra.smul_def r 1).symm\n\nlemma algebra_map_eq_smul_one' : \u21d1(algebra_map R A) = \u03bb r, r \u2022 (1 : A) :=\nfunext algebra_map_eq_smul_one\n\n\/-- `mul_comm` for `algebra`s when one element is from the base ring. -\/\ntheorem commutes (r : R) (x : A) : algebra_map R A r * x = x * algebra_map R A r :=\nalgebra.commutes' r x\n\n\/-- `mul_left_comm` for `algebra`s when one element is from the base ring. -\/\ntheorem left_comm (x : A) (r : R) (y : A) :\n  x * (algebra_map R A r * y) = algebra_map R A r * (x * y) :=\nby rw [\u2190 mul_assoc, \u2190 commutes, mul_assoc]\n\n\/-- `mul_right_comm` for `algebra`s when one element is from the base ring. -\/\ntheorem right_comm (x : A) (r : R) (y : A) :\n  (x * algebra_map R A r) * y = (x * y) * algebra_map R A r :=\nby rw [mul_assoc, commutes, \u2190mul_assoc]\n\ninstance _root_.is_scalar_tower.right : is_scalar_tower R A A :=\n\u27e8\u03bb x y z, by rw [smul_eq_mul, smul_eq_mul, smul_def, smul_def, mul_assoc]\u27e9\n\n\/-- This is just a special case of the global `mul_smul_comm` lemma that requires less typeclass\nsearch (and was here first). -\/\n@[simp] protected lemma mul_smul_comm (s : R) (x y : A) :\n  x * (s \u2022 y) = s \u2022 (x * y) :=\n-- TODO: set up `is_scalar_tower.smul_comm_class` earlier so that we can actually prove this using\n-- `mul_smul_comm s x y`.\nby rw [smul_def, smul_def, left_comm]\n\n\/-- This is just a special case of the global `smul_mul_assoc` lemma that requires less typeclass\nsearch (and was here first). -\/\n@[simp] protected lemma smul_mul_assoc (r : R) (x y : A) :\n  (r \u2022 x) * y = r \u2022 (x * y) :=\nsmul_mul_assoc r x y\n\nsection\nvariables {r : R} {a : A}\n\n@[simp] lemma bit0_smul_one : bit0 r \u2022 (1 : A) = bit0 (r \u2022 (1 : A)) :=\nby simp [bit0, add_smul]\nlemma bit0_smul_one' : bit0 r \u2022 (1 : A) = r \u2022 2 :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit0 : bit0 r \u2022 bit0 a = r \u2022 (bit0 (bit0 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit1 : bit0 r \u2022 bit1 a = r \u2022 (bit0 (bit1 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit1_smul_one : bit1 r \u2022 (1 : A) = bit1 (r \u2022 (1 : A)) :=\nby simp [bit1, add_smul]\nlemma bit1_smul_one' : bit1 r \u2022 (1 : A) = r \u2022 2 + 1 :=\nby simp [bit1, bit0, add_smul, smul_add]\n@[simp] lemma bit1_smul_bit0 : bit1 r \u2022 bit0 a = r \u2022 (bit0 (bit0 a)) + bit0 a :=\nby simp [bit1, add_smul, smul_add]\n@[simp] lemma bit1_smul_bit1 : bit1 r \u2022 bit1 a = r \u2022 (bit0 (bit1 a)) + bit1 a :=\nby { simp only [bit0, bit1, add_smul, smul_add, one_smul], abel }\n\nend\n\nvariables (R A)\n\n\/--\nThe canonical ring homomorphism `algebra_map R A : R \u2192* A` for any `R`-algebra `A`,\npackaged as an `R`-linear map.\n-\/\nprotected def linear_map : R \u2192\u2097[R] A :=\n{ map_smul' := \u03bb x y, by simp [algebra.smul_def],\n  ..algebra_map R A }\n\n@[simp]\nlemma linear_map_apply (r : R) : algebra.linear_map R A r = algebra_map R A r := rfl\n\nlemma coe_linear_map : \u21d1(algebra.linear_map R A) = algebra_map R A := rfl\n\ninstance id : algebra R R := (ring_hom.id R).to_algebra\n\nvariables {R A}\n\nnamespace id\n\n@[simp] lemma map_eq_id : algebra_map R R = ring_hom.id _ := rfl\n\nlemma map_eq_self (x : R) : algebra_map R R x = x := rfl\n\n@[simp] lemma smul_eq_mul (x y : R) : x \u2022 y = x * y := rfl\n\nend id\n\nsection prod\nvariables (R A B)\n\ninstance : algebra R (A \u00d7 B) :=\n{ commutes' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [commutes r a, commutes r b] },\n  smul_def' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [smul_def r a, smul_def r b] },\n  .. prod.module,\n  .. ring_hom.prod (algebra_map R A) (algebra_map R B) }\n\nvariables {R A B}\n\n@[simp] lemma algebra_map_prod_apply (r : R) :\n  algebra_map R (A \u00d7 B) r = (algebra_map R A r, algebra_map R B r) := rfl\n\nend prod\n\n\/-- Algebra over a subsemiring. This builds upon `subsemiring.module`. -\/\ninstance of_subsemiring (S : subsemiring R) : algebra S A :=\n{ smul := (\u2022),\n  commutes' := \u03bb r x, algebra.commutes r x,\n  smul_def' := \u03bb r x, algebra.smul_def r x,\n  .. (algebra_map R A).comp S.subtype }\n\n\/-- Algebra over a subring. This builds upon `subring.module`. -\/\ninstance of_subring {R A : Type*} [comm_ring R] [ring A] [algebra R A]\n  (S : subring R) : algebra S A :=\n{ smul := (\u2022),\n  .. algebra.of_subsemiring S.to_subsemiring,\n  .. (algebra_map R A).comp S.subtype }\n\nlemma algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192+* R) = subring.subtype S := rfl\n\nlemma coe_algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192 R) = subtype.val := rfl\n\nlemma algebra_map_of_subring_apply {R : Type*} [comm_ring R] (S : subring R) (x : S) :\n  algebra_map S R x = x := rfl\n\n\/-- Explicit characterization of the submonoid map in the case of an algebra.\n`S` is made explicit to help with type inference -\/\ndef algebra_map_submonoid (S : Type*) [semiring S] [algebra R S]\n  (M : submonoid R) : (submonoid S) :=\nsubmonoid.map (algebra_map R S : R \u2192* S) M\n\nlemma mem_algebra_map_submonoid_of_mem [algebra R S] {M : submonoid R} (x : M) :\n  (algebra_map R S x) \u2208 algebra_map_submonoid S M :=\nset.mem_image_of_mem (algebra_map R S) x.2\n\nend semiring\n\nsection ring\nvariables [comm_ring R]\n\nvariables (R)\n\n\/-- A `semiring` that is an `algebra` over a commutative ring carries a natural `ring` structure.\nSee note [reducible non-instances]. -\/\n@[reducible]\ndef semiring_to_ring [semiring A] [algebra R A] : ring A := {\n  ..module.add_comm_monoid_to_add_comm_group R,\n  ..(infer_instance : semiring A) }\n\nvariables {R}\n\nlemma mul_sub_algebra_map_commutes [ring A] [algebra R A] (x : A) (r : R) :\n  x * (x - algebra_map R A r) = (x - algebra_map R A r) * x :=\nby rw [mul_sub, \u2190commutes, sub_mul]\n\nlemma mul_sub_algebra_map_pow_commutes [ring A] [algebra R A] (x : A) (r : R) (n : \u2115) :\n  x * (x - algebra_map R A r) ^ n = (x - algebra_map R A r) ^ n * x :=\nbegin\n  induction n with n ih,\n  { simp },\n  { rw [pow_succ, \u2190mul_assoc, mul_sub_algebra_map_commutes,\n      mul_assoc, ih, \u2190mul_assoc], }\nend\n\nend ring\n\nend algebra\n\nnamespace no_zero_smul_divisors\n\nvariables {R A : Type*}\n\nopen algebra\n\nsection ring\n\nvariables [comm_ring R]\n\n\/-- If `algebra_map R A` is injective and `A` has no zero divisors,\n`R`-multiples in `A` are zero only if one of the factors is zero.\n\nCannot be an instance because there is no `injective (algebra_map R A)` typeclass.\n-\/\nlemma of_algebra_map_injective\n  [semiring A] [algebra R A] [no_zero_divisors A]\n  (h : function.injective (algebra_map R A)) : no_zero_smul_divisors R A :=\n\u27e8\u03bb c x hcx, (mul_eq_zero.mp ((smul_def c x).symm.trans hcx)).imp_left\n  ((algebra_map R A).injective_iff.mp h _)\u27e9\n\nvariables (R A)\nlemma algebra_map_injective [ring A] [nontrivial A]\n  [algebra R A] [no_zero_smul_divisors R A] :\n  function.injective (algebra_map R A) :=\nsuffices function.injective (\u03bb (c : R), c \u2022 (1 : A)),\nby { convert this, ext, rw [algebra.smul_def, mul_one] },\nsmul_left_injective R one_ne_zero\n\nvariables {R A}\nlemma iff_algebra_map_injective [ring A] [is_domain A] [algebra R A] :\n  no_zero_smul_divisors R A \u2194 function.injective (algebra_map R A) :=\n\u27e8@@no_zero_smul_divisors.algebra_map_injective R A _ _ _ _,\n no_zero_smul_divisors.of_algebra_map_injective\u27e9\n\nend ring\n\nsection field\n\nvariables [field R] [semiring A] [algebra R A]\n\n@[priority 100] -- see note [lower instance priority]\ninstance algebra.no_zero_smul_divisors [nontrivial A] [no_zero_divisors A] :\n  no_zero_smul_divisors R A :=\nno_zero_smul_divisors.of_algebra_map_injective (algebra_map R A).injective\n\nend field\n\nend no_zero_smul_divisors\n\nnamespace opposite\n\nvariables {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]\n\ninstance : algebra R A\u1d52\u1d56 :=\n{ to_ring_hom := (algebra_map R A).to_opposite $ \u03bb x y, algebra.commutes _ _,\n  smul_def' := \u03bb c x, unop_injective $\n    by { dsimp, simp only [op_mul, algebra.smul_def, algebra.commutes, op_unop] },\n  commutes' := \u03bb r, opposite.rec $ \u03bb x, by dsimp; simp only [\u2190 op_mul, algebra.commutes],\n  ..opposite.has_scalar A R }\n\n@[simp] lemma algebra_map_apply (c : R) : algebra_map R A\u1d52\u1d56 c = op (algebra_map R A c) := rfl\n\nend opposite\n\nnamespace module\nvariables (R : Type u) (M : Type v) [comm_semiring R] [add_comm_monoid M] [module R M]\n\ninstance : algebra R (module.End R M) :=\nalgebra.of_module smul_mul_assoc (\u03bb r f g, (smul_comm r f g).symm)\n\nlemma algebra_map_End_eq_smul_id (a : R) :\n  (algebra_map R (End R M)) a = a \u2022 linear_map.id := rfl\n\n@[simp] lemma algebra_map_End_apply (a : R) (m : M) :\n  (algebra_map R (End R M)) a m = a \u2022 m := rfl\n\n@[simp] lemma ker_algebra_map_End (K : Type u) (V : Type v)\n  [field K] [add_comm_group V] [module K V] (a : K) (ha : a \u2260 0) :\n  ((algebra_map K (End K V)) a).ker = \u22a5 :=\nlinear_map.ker_smul _ _ ha\n\nend module\n\nset_option old_structure_cmd true\n\/-- Defining the homomorphism in the category R-Alg. -\/\n@[nolint has_inhabited_instance]\nstructure alg_hom (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] extends ring_hom A B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nrun_cmd tactic.add_doc_string `alg_hom.to_ring_hom \"Reinterpret an `alg_hom` as a `ring_hom`\"\n\ninfixr ` \u2192\u2090 `:25 := alg_hom _\nnotation A ` \u2192\u2090[`:25 R `] ` B := alg_hom R A B\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {C : Type u\u2081} {D : Type v\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A] [semiring B] [semiring C] [semiring D]\nvariables [algebra R A] [algebra R B] [algebra R C] [algebra R D]\n\ninstance : has_coe_to_fun (A \u2192\u2090[R] B) (\u03bb _, A \u2192 B) := \u27e8alg_hom.to_fun\u27e9\n\ninitialize_simps_projections alg_hom (to_fun \u2192 apply)\n\n@[simp] lemma to_fun_eq_coe (f : A \u2192\u2090[R] B) : f.to_fun = f := rfl\n\ninstance coe_ring_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+* B) := \u27e8alg_hom.to_ring_hom\u27e9\n\ninstance coe_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192* B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\ninstance coe_add_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+ B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\n@[simp, norm_cast] lemma coe_mk {f : A \u2192 B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  \u21d1(\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := rfl\n\n-- make the coercion the simp-normal form\n@[simp] lemma to_ring_hom_eq_coe (f : A \u2192\u2090[R] B) : f.to_ring_hom = f := rfl\n\n@[simp, norm_cast] lemma coe_to_ring_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+* B) = f := rfl\n\n-- as `simp` can already prove this lemma, it is not tagged with the `simp` attribute.\n@[norm_cast] lemma coe_to_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192* B) = f := rfl\n\n-- as `simp` can already prove this lemma, it is not tagged with the `simp` attribute.\n@[norm_cast] lemma coe_to_add_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+ B) = f := rfl\n\nvariables (\u03c6 : A \u2192\u2090[R] B)\n\ntheorem coe_fn_injective : @function.injective (A \u2192\u2090[R] B) (A \u2192 B) coe_fn :=\nby { intros \u03c6\u2081 \u03c6\u2082 H, cases \u03c6\u2081, cases \u03c6\u2082, congr, exact H }\n\ntheorem coe_fn_inj {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : (\u03c6\u2081 : A \u2192 B) = \u03c6\u2082 \u2194 \u03c6\u2081 = \u03c6\u2082 := coe_fn_injective.eq_iff\n\ntheorem coe_ring_hom_injective : function.injective (coe : (A \u2192\u2090[R] B) \u2192 (A \u2192+* B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 H, coe_fn_injective $ show ((\u03c6\u2081 : (A \u2192+* B)) : A \u2192 B) = ((\u03c6\u2082 : (A \u2192+* B)) : A \u2192 B),\n  from congr_arg _ H\n\ntheorem coe_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192* B)) :=\nring_hom.coe_monoid_hom_injective.comp coe_ring_hom_injective\n\ntheorem coe_add_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192+ B)) :=\nring_hom.coe_add_monoid_hom_injective.comp coe_ring_hom_injective\n\nprotected lemma congr_fun {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u03c6\u2081 = \u03c6\u2082) (x : A) : \u03c6\u2081 x = \u03c6\u2082 x := H \u25b8 rfl\nprotected lemma congr_arg (\u03c6 : A \u2192\u2090[R] B) {x y : A} (h : x = y) : \u03c6 x = \u03c6 y := h \u25b8 rfl\n\n@[ext]\ntheorem ext {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x) : \u03c6\u2081 = \u03c6\u2082 :=\ncoe_fn_injective $ funext H\n\ntheorem ext_iff {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : \u03c6\u2081 = \u03c6\u2082 \u2194 \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x :=\n\u27e8alg_hom.congr_fun, ext\u27e9\n\n@[simp] theorem mk_coe {f : A \u2192\u2090[R] B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := ext $ \u03bb _, rfl\n\n@[simp]\ntheorem commutes (r : R) : \u03c6 (algebra_map R A r) = algebra_map R B r := \u03c6.commutes' r\n\ntheorem comp_algebra_map : (\u03c6 : A \u2192+* B).comp (algebra_map R A) = algebra_map R B :=\nring_hom.ext $ \u03c6.commutes\n\n@[simp] lemma map_add (r s : A) : \u03c6 (r + s) = \u03c6 r + \u03c6 s :=\n\u03c6.to_ring_hom.map_add r s\n\n@[simp] lemma map_zero : \u03c6 0 = 0 :=\n\u03c6.to_ring_hom.map_zero\n\n@[simp] lemma map_mul (x y) : \u03c6 (x * y) = \u03c6 x * \u03c6 y :=\n\u03c6.to_ring_hom.map_mul x y\n\n@[simp] lemma map_one : \u03c6 1 = 1 :=\n\u03c6.to_ring_hom.map_one\n\n@[simp] lemma map_smul (r : R) (x : A) : \u03c6 (r \u2022 x) = r \u2022 \u03c6 x :=\nby simp only [algebra.smul_def, map_mul, commutes]\n\n@[simp] lemma map_pow (x : A) (n : \u2115) : \u03c6 (x ^ n) = (\u03c6 x) ^ n :=\n\u03c6.to_ring_hom.map_pow x n\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u2211 x in s, f x) = \u2211 x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.sum g) = f.sum (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_sum _ _\n\n@[simp] lemma map_nat_cast (n : \u2115) : \u03c6 n = n :=\n\u03c6.to_ring_hom.map_nat_cast n\n\n@[simp] lemma map_bit0 (x) : \u03c6 (bit0 x) = bit0 (\u03c6 x) :=\n\u03c6.to_ring_hom.map_bit0 x\n\n@[simp] lemma map_bit1 (x) : \u03c6 (bit1 x) = bit1 (\u03c6 x) :=\n\u03c6.to_ring_hom.map_bit1 x\n\n\/-- If a `ring_hom` is `R`-linear, then it is an `alg_hom`. -\/\ndef mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : A \u2192\u2090[R] B :=\n{ to_fun := f,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, h, f.map_one],\n  .. f }\n\n@[simp] lemma coe_mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : \u21d1(mk' f h) = f := rfl\n\nsection\n\nvariables (R A)\n\/-- Identity map as an `alg_hom`. -\/\nprotected def id : A \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl,\n  ..ring_hom.id A }\n\n@[simp] lemma coe_id : \u21d1(alg_hom.id R A) = id := rfl\n\n@[simp] lemma id_to_ring_hom : (alg_hom.id R A : A \u2192+* A) = ring_hom.id _ := rfl\n\nend\n\nlemma id_apply (p : A) : alg_hom.id R A p = p := rfl\n\n\/-- Composition of algebra homeomorphisms. -\/\ndef comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : A \u2192\u2090[R] C :=\n{ commutes' := \u03bb r : R, by rw [\u2190 \u03c6\u2081.commutes, \u2190 \u03c6\u2082.commutes]; refl,\n  .. \u03c6\u2081.to_ring_hom.comp \u2191\u03c6\u2082 }\n\n@[simp] lemma coe_comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : \u21d1(\u03c6\u2081.comp \u03c6\u2082) = \u03c6\u2081 \u2218 \u03c6\u2082 := rfl\n\nlemma comp_apply (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) (p : A) : \u03c6\u2081.comp \u03c6\u2082 p = \u03c6\u2081 (\u03c6\u2082 p) := rfl\n\nlemma comp_to_ring_hom (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) :\n  \u21d1(\u03c6\u2081.comp \u03c6\u2082 : A \u2192+* C) = (\u03c6\u2081 : B \u2192+* C).comp \u2191\u03c6\u2082 := rfl\n\n@[simp] theorem comp_id : \u03c6.comp (alg_hom.id R A) = \u03c6 :=\next $ \u03bb x, rfl\n\n@[simp] theorem id_comp : (alg_hom.id R B).comp \u03c6 = \u03c6 :=\next $ \u03bb x, rfl\n\ntheorem comp_assoc (\u03c6\u2081 : C \u2192\u2090[R] D) (\u03c6\u2082 : B \u2192\u2090[R] C) (\u03c6\u2083 : A \u2192\u2090[R] B) :\n  (\u03c6\u2081.comp \u03c6\u2082).comp \u03c6\u2083 = \u03c6\u2081.comp (\u03c6\u2082.comp \u03c6\u2083) :=\next $ \u03bb x, rfl\n\n\/-- R-Alg \u2964 R-Mod -\/\ndef to_linear_map : A \u2192\u2097[R] B :=\n{ to_fun := \u03c6,\n  map_add' := \u03c6.map_add,\n  map_smul' := \u03c6.map_smul }\n\n@[simp] lemma to_linear_map_apply (p : A) : \u03c6.to_linear_map p = \u03c6 p := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A \u2192\u2097[R] B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma comp_to_linear_map (f : A \u2192\u2090[R] B) (g : B \u2192\u2090[R] C) :\n  (g.comp f).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_id : to_linear_map (alg_hom.id R A) = linear_map.id :=\nlinear_map.ext $ \u03bb _, rfl\n\n\/-- Promote a `linear_map` to an `alg_hom` by supplying proofs about the behavior on `1` and `*`. -\/\n@[simps]\ndef of_linear_map (f : A \u2192\u2097[R] B) (map_one : f 1 = 1) (map_mul : \u2200 x y, f (x * y) = f x * f y) :\n  A \u2192\u2090[R] B :=\n{ to_fun := f,\n  map_one' := map_one,\n  map_mul' := map_mul,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, f.map_smul, map_one],\n  .. f.to_add_monoid_hom }\n\n@[simp] lemma of_linear_map_to_linear_map (map_one) (map_mul) :\n  of_linear_map \u03c6.to_linear_map map_one map_mul = \u03c6 :=\nby { ext, refl }\n\n@[simp] lemma to_linear_map_of_linear_map (f : A \u2192\u2097[R] B) (map_one) (map_mul) :\n  to_linear_map (of_linear_map f map_one map_mul) = f :=\nby { ext, refl }\n\n@[simp] lemma of_linear_map_id (map_one) (map_mul) :\n  of_linear_map linear_map.id map_one map_mul = alg_hom.id R A :=\next $ \u03bb _, rfl\n\nlemma map_list_prod (s : list A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_list_prod s\n\nsection prod\n\n\/-- First projection as `alg_hom`. -\/\ndef fst : A \u00d7 B \u2192\u2090[R] A :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.fst A B}\n\n\/-- Second projection as `alg_hom`. -\/\ndef snd : A \u00d7 B \u2192\u2090[R] B :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.snd A B}\n\nend prod\n\nlemma algebra_map_eq_apply (f : A \u2192\u2090[R] B) {y : R} {x : A} (h : algebra_map R A y = x) :\n  algebra_map R B y = f x :=\nh \u25b8 (f.commutes _).symm\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A] [comm_semiring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\nlemma map_multiset_prod (s : multiset A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_multiset_prod s\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u220f x in s, f x) = \u220f x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.prod g) = f.prod (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_prod _ _\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_semiring R] [ring A] [ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\n@[simp] lemma map_neg (x) : \u03c6 (-x) = -\u03c6 x :=\n\u03c6.to_ring_hom.map_neg x\n\n@[simp] lemma map_sub (x y) : \u03c6 (x - y) = \u03c6 x - \u03c6 y :=\n\u03c6.to_ring_hom.map_sub x y\n\n@[simp] lemma map_int_cast (n : \u2124) : \u03c6 n = n :=\n\u03c6.to_ring_hom.map_int_cast n\n\nend ring\n\nsection division_ring\n\nvariables [comm_ring R] [division_ring A] [division_ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\n@[simp] lemma map_inv (x) : \u03c6 (x\u207b\u00b9) = (\u03c6 x)\u207b\u00b9 :=\n\u03c6.to_ring_hom.map_inv x\n\n@[simp] lemma map_div (x y) : \u03c6 (x \/ y) = \u03c6 x \/ \u03c6 y :=\n\u03c6.to_ring_hom.map_div x y\n\nend division_ring\n\ntheorem injective_iff {R A B : Type*} [comm_semiring R] [ring A] [semiring B]\n  [algebra R A] [algebra R B] (f : A \u2192\u2090[R] B) :\n  function.injective f \u2194 (\u2200 x, f x = 0 \u2192 x = 0) :=\nring_hom.injective_iff (f : A \u2192+* B)\n\nend alg_hom\n\n@[simp] lemma rat.smul_one_eq_coe {A : Type*} [division_ring A] [algebra \u211a A] (m : \u211a) :\n  m \u2022 (1 : A) = \u2191m :=\nby rw [algebra.smul_def, mul_one, ring_hom.eq_rat_cast]\n\nset_option old_structure_cmd true\n\/-- An equivalence of algebras is an equivalence of rings commuting with the actions of scalars. -\/\nstructure alg_equiv (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B]\n  extends A \u2243 B, A \u2243* B, A \u2243+ B, A \u2243+* B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nattribute [nolint doc_blame] alg_equiv.to_ring_equiv\nattribute [nolint doc_blame] alg_equiv.to_equiv\nattribute [nolint doc_blame] alg_equiv.to_add_equiv\nattribute [nolint doc_blame] alg_equiv.to_mul_equiv\n\nnotation A ` \u2243\u2090[`:50 R `] ` A' := alg_equiv R A A'\n\nnamespace alg_equiv\n\nvariables {R : Type u} {A\u2081 : Type v} {A\u2082 : Type w} {A\u2083 : Type u\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A\u2081] [semiring A\u2082] [semiring A\u2083]\nvariables [algebra R A\u2081] [algebra R A\u2082] [algebra R A\u2083]\nvariables (e : A\u2081 \u2243\u2090[R] A\u2082)\n\ninstance : has_coe_to_fun (A\u2081 \u2243\u2090[R] A\u2082) (\u03bb _, A\u2081 \u2192 A\u2082) := \u27e8alg_equiv.to_fun\u27e9\n\n@[ext]\nlemma ext {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : \u2200 a, f a = g a) : f = g :=\nbegin\n  have h\u2081 : f.to_equiv = g.to_equiv := equiv.ext h,\n  cases f, cases g, congr,\n  { exact (funext h) },\n  { exact congr_arg equiv.inv_fun h\u2081 }\nend\n\nprotected lemma congr_arg {f : A\u2081 \u2243\u2090[R] A\u2082} : \u03a0 {x x' : A\u2081}, x = x' \u2192 f x = f x'\n| _ _ rfl := rfl\n\nprotected lemma congr_fun {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : f = g) (x : A\u2081) : f x = g x := h \u25b8 rfl\n\nlemma ext_iff {f g : A\u2081 \u2243\u2090[R] A\u2082} : f = g \u2194 \u2200 x, f x = g x :=\n\u27e8\u03bb h x, h \u25b8 rfl, ext\u27e9\n\nlemma coe_fun_injective : @function.injective (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192 A\u2082) (\u03bb e, (e : A\u2081 \u2192 A\u2082)) :=\nbegin\n  intros f g w,\n  ext,\n  exact congr_fun w a,\nend\n\ninstance has_coe_to_ring_equiv : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2243+* A\u2082) := \u27e8alg_equiv.to_ring_equiv\u27e9\n\n@[simp] lemma coe_mk {to_fun inv_fun left_inv right_inv map_mul map_add commutes} :\n  \u21d1(\u27e8to_fun, inv_fun, left_inv, right_inv, map_mul, map_add, commutes\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = to_fun :=\nrfl\n\n@[simp] theorem mk_coe (e : A\u2081 \u2243\u2090[R] A\u2082) (e' h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8e, e', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = e := ext $ \u03bb _, rfl\n\n@[simp] lemma to_fun_eq_coe (e : A\u2081 \u2243\u2090[R] A\u2082) : e.to_fun = e := rfl\n\n@[simp] lemma to_ring_equiv_eq_coe : e.to_ring_equiv = e := rfl\n\n@[simp, norm_cast] lemma coe_ring_equiv : ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192 A\u2082) = e := rfl\nlemma coe_ring_equiv' : (e.to_ring_equiv : A\u2081 \u2192 A\u2082) = e := rfl\n\nlemma coe_ring_equiv_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2243+* A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ ring_equiv.congr_fun h\n\n@[simp] lemma map_add : \u2200 x y, e (x + y) = e x + e y := e.to_add_equiv.map_add\n\n@[simp] lemma map_zero : e 0 = 0 := e.to_add_equiv.map_zero\n\n@[simp] lemma map_mul : \u2200 x y, e (x * y) = (e x) * (e y) := e.to_mul_equiv.map_mul\n\n@[simp] lemma map_one : e 1 = 1 := e.to_mul_equiv.map_one\n\n@[simp] lemma commutes : \u2200 (r : R), e (algebra_map R A\u2081 r) = algebra_map R A\u2082 r :=\n  e.commutes'\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u2211 x in s, f x) = \u2211 x in s, e (f x) :=\ne.to_add_equiv.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.sum g) = f.sum (\u03bb i b, e (g i b)) :=\ne.map_sum _ _\n\n\/-- Interpret an algebra equivalence as an algebra homomorphism.\n\nThis definition is included for symmetry with the other `to_*_hom` projections.\nThe `simp` normal form is to use the coercion of the `has_coe_to_alg_hom` instance. -\/\ndef to_alg_hom : A\u2081 \u2192\u2090[R] A\u2082 :=\n{ map_one' := e.map_one, map_zero' := e.map_zero, ..e }\n\ninstance has_coe_to_alg_hom : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192\u2090[R] A\u2082) :=\n\u27e8to_alg_hom\u27e9\n\n@[simp] lemma to_alg_hom_eq_coe : e.to_alg_hom = e := rfl\n\n@[simp, norm_cast] lemma coe_alg_hom : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192 A\u2082) = e :=\nrfl\n\nlemma coe_alg_hom_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2192\u2090[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ alg_hom.congr_fun h\n\n\/-- The two paths coercion can take to a `ring_hom` are equivalent -\/\nlemma coe_ring_hom_commutes : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192+* A\u2082) = ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192+* A\u2082) :=\nrfl\n\n@[simp] lemma map_pow : \u2200 (x : A\u2081) (n : \u2115), e (x ^ n) = (e x) ^ n := e.to_alg_hom.map_pow\n\nlemma injective : function.injective e := e.to_equiv.injective\n\nlemma surjective : function.surjective e := e.to_equiv.surjective\n\nlemma bijective : function.bijective e := e.to_equiv.bijective\n\ninstance : has_one (A\u2081 \u2243\u2090[R] A\u2081) := \u27e8{commutes' := \u03bb r, rfl, ..(1 : A\u2081 \u2243+* A\u2081)}\u27e9\n\ninstance : inhabited (A\u2081 \u2243\u2090[R] A\u2081) := \u27e81\u27e9\n\n\/-- Algebra equivalences are reflexive. -\/\n@[refl]\ndef refl : A\u2081 \u2243\u2090[R] A\u2081 := 1\n\n@[simp] lemma refl_to_alg_hom : \u2191(refl : A\u2081 \u2243\u2090[R] A\u2081) = alg_hom.id R A\u2081 := rfl\n\n@[simp] lemma coe_refl : \u21d1(refl : A\u2081 \u2243\u2090[R] A\u2081) = id := rfl\n\n\/-- Algebra equivalences are symmetric. -\/\n@[symm]\ndef symm (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2243\u2090[R] A\u2081 :=\n{ commutes' := \u03bb r, by { rw \u2190e.to_ring_equiv.symm_apply_apply (algebra_map R A\u2081 r), congr,\n                         change _ = e _, rw e.commutes, },\n  ..e.to_ring_equiv.symm, }\n\n\/-- See Note [custom simps projection] -\/\ndef simps.symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2192 A\u2081 := e.symm\n\ninitialize_simps_projections alg_equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\n\n@[simp] lemma inv_fun_eq_symm {e : A\u2081 \u2243\u2090[R] A\u2082} : e.inv_fun = e.symm := rfl\n\n@[simp] lemma symm_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : e.symm.symm = e :=\nby { ext, refl, }\n\nlemma symm_bijective : function.bijective (symm : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2082 \u2243\u2090[R] A\u2081)) :=\nequiv.bijective \u27e8symm, symm, symm_symm, symm_symm\u27e9\n\n@[simp] lemma mk_coe' (e : A\u2081 \u2243\u2090[R] A\u2082) (f h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, e, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2082 \u2243\u2090[R] A\u2081) = e.symm :=\nsymm_bijective.injective $ ext $ \u03bb x, rfl\n\n@[simp] theorem symm_mk (f f') (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm =\n  { to_fun := f', inv_fun := f,\n    ..(\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm } := rfl\n\n\/-- Algebra equivalences are transitive. -\/\n@[trans]\ndef trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) : A\u2081 \u2243\u2090[R] A\u2083 :=\n{ commutes' := \u03bb r, show e\u2082.to_fun (e\u2081.to_fun _) = _, by rw [e\u2081.commutes', e\u2082.commutes'],\n  ..(e\u2081.to_ring_equiv.trans e\u2082.to_ring_equiv), }\n\n@[simp] lemma apply_symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e (e.symm x) = x :=\n  e.to_equiv.apply_symm_apply\n\n@[simp] lemma symm_apply_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e.symm (e x) = x :=\n  e.to_equiv.symm_apply_apply\n\n@[simp] lemma symm_trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2083) :\n  (e\u2081.trans e\u2082).symm x = e\u2081.symm (e\u2082.symm x) := rfl\n\n@[simp] lemma coe_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  \u21d1(e\u2081.trans e\u2082) = e\u2082 \u2218 e\u2081 := rfl\n\nlemma trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2081) :\n  (e\u2081.trans e\u2082) x = e\u2082 (e\u2081 x) := rfl\n\n@[simp] lemma comp_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp (e : A\u2081 \u2192\u2090[R] A\u2082) \u2191e.symm = alg_hom.id R A\u2082 :=\nby { ext, simp }\n\n@[simp] lemma symm_comp (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp \u2191e.symm (e : A\u2081 \u2192\u2090[R] A\u2082) = alg_hom.id R A\u2081 :=\nby { ext, simp }\n\ntheorem left_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.left_inverse e.symm e := e.left_inv\n\ntheorem right_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.right_inverse e.symm e := e.right_inv\n\n\/-- If `A\u2081` is equivalent to `A\u2081'` and `A\u2082` is equivalent to `A\u2082'`, then the type of maps\n`A\u2081 \u2192\u2090[R] A\u2082` is equivalent to the type of maps `A\u2081' \u2192\u2090[R] A\u2082'`. -\/\ndef arrow_congr {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082'] [algebra R A\u2081'] [algebra R A\u2082']\n  (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') : (A\u2081 \u2192\u2090[R] A\u2082) \u2243 (A\u2081' \u2192\u2090[R] A\u2082') :=\n{ to_fun := \u03bb f, (e\u2082.to_alg_hom.comp f).comp e\u2081.symm.to_alg_hom,\n  inv_fun := \u03bb f, (e\u2082.symm.to_alg_hom.comp f).comp e\u2081.to_alg_hom,\n  left_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, symm_comp],\n    simp only [\u2190alg_hom.comp_assoc, symm_comp, alg_hom.id_comp, alg_hom.comp_id] },\n  right_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, comp_symm],\n    simp only [\u2190alg_hom.comp_assoc, comp_symm, alg_hom.id_comp, alg_hom.comp_id] } }\n\nlemma arrow_congr_comp {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082')\n  (e\u2083 : A\u2083 \u2243\u2090[R] A\u2083') (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2083) :\n  arrow_congr e\u2081 e\u2083 (g.comp f) = (arrow_congr e\u2082 e\u2083 g).comp (arrow_congr e\u2081 e\u2082 f) :=\nby { ext, simp only [arrow_congr, equiv.coe_fn_mk, alg_hom.comp_apply],\n  congr, exact (e\u2082.symm_apply_apply _).symm }\n\n@[simp] lemma arrow_congr_refl :\n  arrow_congr alg_equiv.refl alg_equiv.refl = equiv.refl (A\u2081 \u2192\u2090[R] A\u2082) :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_trans {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2081' : A\u2081' \u2243\u2090[R] A\u2082')\n  (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (e\u2082' : A\u2082' \u2243\u2090[R] A\u2083') :\n  arrow_congr (e\u2081.trans e\u2082) (e\u2081'.trans e\u2082') = (arrow_congr e\u2081 e\u2081').trans (arrow_congr e\u2082 e\u2082') :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_symm {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082']\n  [algebra R A\u2081'] [algebra R A\u2082'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') :\n  (arrow_congr e\u2081 e\u2082).symm = arrow_congr e\u2081.symm e\u2082.symm :=\nby { ext, refl }\n\n\/-- If an algebra morphism has an inverse, it is a algebra isomorphism. -\/\ndef of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 : f.comp g = alg_hom.id R A\u2082)\n  (h\u2082 : g.comp f = alg_hom.id R A\u2081) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun    := f,\n  inv_fun   := g,\n  left_inv  := alg_hom.ext_iff.1 h\u2082,\n  right_inv := alg_hom.ext_iff.1 h\u2081,\n  ..f }\n\nlemma coe_alg_hom_of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  \u2191(of_alg_hom f g h\u2081 h\u2082) = f := alg_hom.ext $ \u03bb _, rfl\n\n@[simp]\nlemma of_alg_hom_coe_alg_hom (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  of_alg_hom \u2191f g h\u2081 h\u2082 = f := ext $ \u03bb _, rfl\n\nlemma of_alg_hom_symm (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  (of_alg_hom f g h\u2081 h\u2082).symm = of_alg_hom g f h\u2082 h\u2081 := rfl\n\n\/-- Promotes a bijective algebra homomorphism to an algebra equivalence. -\/\nnoncomputable def of_bijective (f : A\u2081 \u2192\u2090[R] A\u2082) (hf : function.bijective f) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ .. ring_equiv.of_bijective (f : A\u2081 \u2192+* A\u2082) hf, .. f }\n\n\/-- Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. -\/\n@[simps apply] def to_linear_equiv (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2081 \u2243\u2097[R] A\u2082 :=\n{ to_fun    := e,\n  map_smul' := \u03bb r x, by simp [algebra.smul_def],\n  inv_fun   := e.symm,\n  .. e }\n\n@[simp] lemma to_linear_equiv_refl :\n  (alg_equiv.refl : A\u2081 \u2243\u2090[R] A\u2081).to_linear_equiv = linear_equiv.refl R A\u2081 := rfl\n\n@[simp] lemma to_linear_equiv_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  e.to_linear_equiv.symm = e.symm.to_linear_equiv := rfl\n\n@[simp] lemma to_linear_equiv_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (e\u2081.trans e\u2082).to_linear_equiv = e\u2081.to_linear_equiv.trans e\u2082.to_linear_equiv := rfl\n\ntheorem to_linear_equiv_injective : function.injective (to_linear_equiv : _ \u2192 (A\u2081 \u2243\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_equiv.congr_fun h\n\n\/-- Interpret an algebra equivalence as a linear map. -\/\ndef to_linear_map : A\u2081 \u2192\u2097[R] A\u2082 :=\ne.to_alg_hom.to_linear_map\n\n@[simp] lemma to_alg_hom_to_linear_map :\n  (e : A\u2081 \u2192\u2090[R] A\u2082).to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_equiv_to_linear_map :\n  e.to_linear_equiv.to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_apply (x : A\u2081) : e.to_linear_map x = e x := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A\u2081 \u2192\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma trans_to_linear_map (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2243\u2090[R] A\u2083) :\n  (f.trans g).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\nsection of_linear_equiv\n\nvariables (l : A\u2081 \u2243\u2097[R] A\u2082)\n  (map_mul : \u2200 x y : A\u2081, l (x * y) = l x * l y)\n  (commutes : \u2200 r : R, l (algebra_map R A\u2081 r) = algebra_map R A\u2082 r)\n\n\/--\nUpgrade a linear equivalence to an algebra equivalence,\ngiven that it distributes over multiplication and action of scalars.\n-\/\n@[simps apply]\ndef of_linear_equiv : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun := l,\n  inv_fun := l.symm,\n  map_mul' := map_mul,\n  commutes' := commutes,\n  ..l }\n\n@[simp]\nlemma of_linear_equiv_symm :\n  (of_linear_equiv l map_mul commutes).symm = of_linear_equiv l.symm\n    ((of_linear_equiv l map_mul commutes).symm.map_mul)\n    ((of_linear_equiv l map_mul commutes).symm.commutes) :=\nrfl\n\n@[simp] lemma of_linear_equiv_to_linear_equiv (map_mul) (commutes) :\n  of_linear_equiv e.to_linear_equiv map_mul commutes = e :=\nby { ext, refl }\n\n@[simp] lemma to_linear_equiv_of_linear_equiv :\n  to_linear_equiv (of_linear_equiv l map_mul commutes) = l :=\nby { ext, refl }\n\nend of_linear_equiv\n\ninstance aut : group (A\u2081 \u2243\u2090[R] A\u2081) :=\n{ mul := \u03bb \u03d5 \u03c8, \u03c8.trans \u03d5,\n  mul_assoc := \u03bb \u03d5 \u03c8 \u03c7, rfl,\n  one := 1,\n  one_mul := \u03bb \u03d5, by { ext, refl },\n  mul_one := \u03bb \u03d5, by { ext, refl },\n  inv := symm,\n  mul_left_inv := \u03bb \u03d5, by { ext, exact symm_apply_apply \u03d5 a } }\n\n@[simp] lemma mul_apply (e\u2081 e\u2082 : A\u2081 \u2243\u2090[R] A\u2081) (x : A\u2081) : (e\u2081 * e\u2082) x = e\u2081 (e\u2082 x) := rfl\n\n\/-- An algebra isomorphism induces a group isomorphism between automorphism groups -\/\n@[simps apply]\ndef aut_congr (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (A\u2081 \u2243\u2090[R] A\u2081) \u2243* (A\u2082 \u2243\u2090[R] A\u2082) :=\n{ to_fun := \u03bb \u03c8, \u03d5.symm.trans (\u03c8.trans \u03d5),\n  inv_fun := \u03bb \u03c8, \u03d5.trans (\u03c8.trans \u03d5.symm),\n  left_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, symm_apply_apply] },\n  right_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, apply_symm_apply] },\n  map_mul' := \u03bb \u03c8 \u03c7, by { ext, simp only [mul_apply, trans_apply, symm_apply_apply] } }\n\n@[simp] lemma aut_congr_refl : aut_congr (alg_equiv.refl) = mul_equiv.refl (A\u2081 \u2243\u2090[R] A\u2081) :=\nby { ext, refl }\n\n@[simp] lemma aut_congr_symm (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (aut_congr \u03d5).symm = aut_congr \u03d5.symm := rfl\n\n@[simp] lemma aut_congr_trans (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) (\u03c8 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (aut_congr \u03d5).trans (aut_congr \u03c8) = aut_congr (\u03d5.trans \u03c8) := rfl\n\n\/-- The tautological action by `A\u2081 \u2243\u2090[R] A\u2081` on `A\u2081`.\n\nThis generalizes `function.End.apply_mul_action`. -\/\ninstance apply_mul_semiring_action : mul_semiring_action (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul := ($),\n  smul_zero := alg_equiv.map_zero,\n  smul_add := alg_equiv.map_add,\n  smul_one := alg_equiv.map_one,\n  smul_mul := alg_equiv.map_mul,\n  one_smul := \u03bb _, rfl,\n  mul_smul := \u03bb _ _ _, rfl }\n\n@[simp] protected lemma smul_def (f : A\u2081 \u2243\u2090[R] A\u2081) (a : A\u2081) : f \u2022 a = f a := rfl\n\ninstance apply_has_faithful_scalar : has_faithful_scalar (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n\u27e8\u03bb _ _, alg_equiv.ext\u27e9\n\ninstance apply_smul_comm_class : smul_comm_class R (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul_comm := \u03bb r e a, (e.to_linear_equiv.map_smul r a).symm }\n\ninstance apply_smul_comm_class' : smul_comm_class (A\u2081 \u2243\u2090[R] A\u2081) R A\u2081 :=\n{ smul_comm := \u03bb e r a, (e.to_linear_equiv.map_smul r a) }\n\n@[simp] lemma algebra_map_eq_apply (e : A\u2081 \u2243\u2090[R] A\u2082) {y : R} {x : A\u2081} :\n  (algebra_map R A\u2082 y = e x) \u2194 (algebra_map R A\u2081 y = x) :=\n\u27e8\u03bb h, by simpa using e.symm.to_alg_hom.algebra_map_eq_apply h,\n \u03bb h, e.to_alg_hom.algebra_map_eq_apply h\u27e9\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A\u2081] [comm_semiring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u220f x in s, f x) = \u220f x in s, e (f x) :=\ne.to_alg_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.prod g) = f.prod (\u03bb i a, e (g i a)) :=\ne.to_alg_hom.map_finsupp_prod f g\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_ring R] [ring A\u2081] [ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\n@[simp] lemma map_neg (x) : e (-x) = -e x :=\ne.to_alg_hom.map_neg x\n\n@[simp] lemma map_sub (x y) : e (x - y) = e x - e y :=\ne.to_alg_hom.map_sub x y\n\nend ring\n\nsection division_ring\n\nvariables [comm_ring R] [division_ring A\u2081] [division_ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\n@[simp] lemma map_inv (x) : e (x\u207b\u00b9) = (e x)\u207b\u00b9 :=\ne.to_alg_hom.map_inv x\n\n@[simp] lemma map_div (x y) : e (x \/ y) = e x \/ e y :=\ne.to_alg_hom.map_div x y\n\nend division_ring\n\nend alg_equiv\n\nnamespace mul_semiring_action\n\nvariables {M G : Type*} (R A : Type*) [comm_semiring R] [semiring A] [algebra R A]\n\nsection\nvariables [monoid M] [mul_semiring_action M A] [smul_comm_class M R A]\n\n\/-- Each element of the monoid defines a algebra homomorphism.\n\nThis is a stronger version of `mul_semiring_action.to_ring_hom` and\n`distrib_mul_action.to_linear_map`. -\/\n@[simps]\ndef to_alg_hom (m : M) : A \u2192\u2090[R] A :=\nalg_hom.mk' (mul_semiring_action.to_ring_hom _ _ m) (smul_comm _)\n\ntheorem to_alg_hom_injective [has_faithful_scalar M A] :\n  function.injective (mul_semiring_action.to_alg_hom R A : M \u2192 A \u2192\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_hom.ext_iff.1 h r\n\nend\n\nsection\nvariables [group G] [mul_semiring_action G A] [smul_comm_class G R A]\n\n\/-- Each element of the group defines a algebra equivalence.\n\nThis is a stronger version of `mul_semiring_action.to_ring_equiv` and\n`distrib_mul_action.to_linear_equiv`. -\/\n@[simps]\ndef to_alg_equiv (g : G) : A \u2243\u2090[R] A :=\n{ .. mul_semiring_action.to_ring_equiv _ _ g,\n  .. mul_semiring_action.to_alg_hom R A g }\n\ntheorem to_alg_equiv_injective [has_faithful_scalar G A] :\n  function.injective (mul_semiring_action.to_alg_equiv R A : G \u2192 A \u2243\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_equiv.ext_iff.1 h r\n\nend\n\nend mul_semiring_action\n\nsection nat\n\nvariables {R : Type*} [semiring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2115`-algebras. This is only an issue since `algebra.id` and `algebra_nat` are not yet defeq.\n-- TODO: fix this by adding an `of_nat` field to semirings.\n\/-- Semiring \u2964 \u2115-Alg -\/\n@[priority 99] instance algebra_nat : algebra \u2115 R :=\n{ commutes' := nat.cast_commute,\n  smul_def' := \u03bb _ _, nsmul_eq_mul _ _,\n  to_ring_hom := nat.cast_ring_hom R }\n\ninstance nat_algebra_subsingleton : subsingleton (algebra \u2115 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend nat\n\nnamespace ring_hom\n\nvariables {R S : Type*}\n\n\/-- Reinterpret a `ring_hom` as an `\u2115`-algebra homomorphism. -\/\ndef to_nat_alg_hom [semiring R] [semiring S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2115] S :=\n{ to_fun := f, commutes' := \u03bb n, by simp, .. f }\n\n\/-- Reinterpret a `ring_hom` as a `\u2124`-algebra homomorphism. -\/\ndef to_int_alg_hom [ring R] [ring S] [algebra \u2124 R] [algebra \u2124 S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2124] S :=\n{ commutes' := \u03bb n, by simp, .. f }\n\n@[simp] lemma map_rat_algebra_map [ring R] [ring S] [algebra \u211a R] [algebra \u211a S] (f : R \u2192+* S)\n  (r : \u211a) :\n  f (algebra_map \u211a R r) = algebra_map \u211a S r :=\nring_hom.ext_iff.1 (subsingleton.elim (f.comp (algebra_map \u211a R)) (algebra_map \u211a S)) r\n\n\/-- Reinterpret a `ring_hom` as a `\u211a`-algebra homomorphism. -\/\ndef to_rat_alg_hom [ring R] [ring S] [algebra \u211a R] [algebra \u211a S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u211a] S :=\n{ commutes' := f.map_rat_algebra_map, .. f }\n\nend ring_hom\n\nnamespace rat\n\ninstance algebra_rat {\u03b1} [division_ring \u03b1] [char_zero \u03b1] : algebra \u211a \u03b1 :=\n(rat.cast_hom \u03b1).to_algebra' $ \u03bb r x, r.cast_commute x\n\n@[simp] theorem algebra_map_rat_rat : algebra_map \u211a \u211a = ring_hom.id \u211a :=\nsubsingleton.elim _ _\n\n-- TODO[gh-6025]: make this an instance once safe to do so\nlemma algebra_rat_subsingleton {\u03b1} [semiring \u03b1] :\n  subsingleton (algebra \u211a \u03b1) :=\n\u27e8\u03bb x y, algebra.algebra_ext x y $ ring_hom.congr_fun $ subsingleton.elim _ _\u27e9\n\nend rat\n\nnamespace char_zero\n\nvariables {R : Type*} (S : Type*) [comm_semiring R] [semiring S] [algebra R S]\n\nlemma of_algebra [char_zero S] : char_zero R :=\n\u27e8begin\n  suffices : function.injective (algebra_map R S \u2218 coe),\n  { exact this.of_comp },\n  convert char_zero.cast_injective,\n  ext n,\n  rw [function.comp_app, \u2190 (algebra_map \u2115 _).eq_nat_cast, \u2190 ring_hom.comp_apply,\n      ring_hom.eq_nat_cast],\n  all_goals { apply_instance }\nend\u27e9\n\nend char_zero\n\nnamespace algebra\nopen module\n\nvariables (R : Type u) (A : Type v)\n\nvariables [comm_semiring R] [semiring A] [algebra R A]\n\n\/-- `algebra_map` as an `alg_hom`. -\/\ndef of_id : R \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl, .. algebra_map R A }\nvariables {R}\n\ntheorem of_id_apply (r) : of_id R A r = algebra_map R A r := rfl\n\nend algebra\n\nsection int\n\nvariables (R : Type*) [ring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2124`-algebras. This is only an issue since `algebra.id \u2124` and `algebra_int \u2124` are not yet defeq.\n-- TODO: fix this by adding an `of_int` field to rings.\n\/-- Ring \u2964 \u2124-Alg -\/\n@[priority 99] instance algebra_int : algebra \u2124 R :=\n{ commutes' := int.cast_commute,\n  smul_def' := \u03bb _ _, zsmul_eq_mul _ _,\n  to_ring_hom := int.cast_ring_hom R }\n\nvariables {R}\n\ninstance int_algebra_subsingleton : subsingleton (algebra \u2124 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend int\n\n\/-!\nThe R-algebra structure on `\u03a0 i : I, A i` when each `A i` is an R-algebra.\n\nWe couldn't set this up back in `algebra.pi_instances` because this file imports it.\n-\/\nnamespace pi\n\nvariable {I : Type u}     -- The indexing type\nvariable {R : Type*}      -- The scalar type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\nvariables (I f)\n\ninstance algebra {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] :\n  algebra R (\u03a0 i : I, f i) :=\n{ commutes' := \u03bb a f, begin ext, simp [algebra.commutes], end,\n  smul_def' := \u03bb a f, begin ext, simp [algebra.smul_def], end,\n  ..(pi.ring_hom (\u03bb i, algebra_map R (f i)) : R \u2192+* \u03a0 i : I, f i) }\n\n@[simp] lemma algebra_map_apply {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] (a : R) (i : I) :\n  algebra_map R (\u03a0 i, f i) a i = algebra_map R (f i) a := rfl\n\n-- One could also build a `\u03a0 i, R i`-algebra structure on `\u03a0 i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\n\nvariables {I} (R) (f)\n\n\/-- `function.eval` as an `alg_hom`. The name matches `pi.eval_ring_hom`, `pi.eval_monoid_hom`,\netc. -\/\n@[simps]\ndef eval_alg_hom {r : comm_semiring R} [\u03a0 i, semiring (f i)] [\u03a0 i, algebra R (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192\u2090[R] f i :=\n{ to_fun := \u03bb f, f i, commutes' := \u03bb r, rfl, .. pi.eval_ring_hom f i}\n\nvariables (A B : Type*) [comm_semiring R] [semiring B] [algebra R B]\n\n\/-- `function.const` as an `alg_hom`. The name matches `pi.const_ring_hom`, `pi.const_monoid_hom`,\netc. -\/\n@[simps]\ndef const_alg_hom : B \u2192\u2090[R] (A \u2192 B) :=\n{ to_fun := function.const _,\n  commutes' := \u03bb r, rfl,\n  .. pi.const_ring_hom A B}\n\n\/-- When `R` is commutative and permits an `algebra_map`, `pi.const_ring_hom` is equal to that\nmap. -\/\n@[simp] lemma const_ring_hom_eq_algebra_map : const_ring_hom A R = algebra_map R (A \u2192 R) :=\nrfl\n\n@[simp] lemma const_alg_hom_eq_algebra_of_id : const_alg_hom R A R = algebra.of_id R (A \u2192 R) :=\nrfl\n\nend pi\n\nsection is_scalar_tower\n\nvariables {R : Type*} [comm_semiring R]\nvariables (A : Type*) [semiring A] [algebra R A]\nvariables {M : Type*} [add_comm_monoid M] [module A M] [module R M] [is_scalar_tower R A M]\nvariables {N : Type*} [add_comm_monoid N] [module A N] [module R N] [is_scalar_tower R A N]\n\nlemma algebra_compatible_smul (r : R) (m : M) : r \u2022 m = ((algebra_map R A) r) \u2022 m :=\nby rw [\u2190(one_smul A m), \u2190smul_assoc, algebra.smul_def, mul_one, one_smul]\n\n@[simp] lemma algebra_map_smul (r : R) (m : M) : ((algebra_map R A) r) \u2022 m = r \u2022 m :=\n(algebra_compatible_smul A r m).symm\n\nvariable {A}\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class : smul_comm_class R A M :=\n\u27e8\u03bb r a m, by rw [algebra_compatible_smul A r (a \u2022 m), smul_smul, algebra.commutes, mul_smul,\n  \u2190algebra_compatible_smul]\u27e9\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class' : smul_comm_class A R M :=\nsmul_comm_class.symm _ _ _\n\nlemma smul_algebra_smul_comm (r : R) (a : A) (m : M) : a \u2022 r \u2022 m = r \u2022 a \u2022 m :=\nsmul_comm _ _ _\n\nnamespace linear_map\n\ninstance coe_is_scalar_tower : has_coe (M \u2192\u2097[A] N) (M \u2192\u2097[R] N) :=\n\u27e8restrict_scalars R\u27e9\n\nvariables (R) {A M N}\n\n@[simp, norm_cast squash] lemma coe_restrict_scalars_eq_coe (f : M \u2192\u2097[A] N) :\n  (f.restrict_scalars R : M \u2192 N) = f := rfl\n\n@[simp, norm_cast squash] lemma coe_coe_is_scalar_tower (f : M \u2192\u2097[A] N) :\n  ((f : M \u2192\u2097[R] N) : M \u2192 N) = f := rfl\n\n\/-- `A`-linearly coerce a `R`-linear map from `M` to `A` to a function, given an algebra `A` over\na commutative semiring `R` and `M` a module over `R`. -\/\ndef lto_fun (R : Type u) (M : Type v) (A : Type w)\n  [comm_semiring R] [add_comm_monoid M] [module R M] [comm_ring A] [algebra R A] :\n  (M \u2192\u2097[R] A) \u2192\u2097[A] (M \u2192 A) :=\n{ to_fun := linear_map.to_fun,\n  map_add' := \u03bb f g, rfl,\n  map_smul' := \u03bb c f, rfl }\n\nend linear_map\n\nend is_scalar_tower\n\n\/-! TODO: The following lemmas no longer involve `algebra` at all, and could be moved closer\nto `algebra\/module\/submodule.lean`. Currently this is tricky because `ker`, `range`, `\u22a4`, and `\u22a5`\nare all defined in `linear_algebra\/basic.lean`. -\/\nsection module\nopen module\n\nvariables (R S M N : Type*) [semiring R] [semiring S] [has_scalar R S]\nvariables [add_comm_monoid M] [module R M] [module S M] [is_scalar_tower R S M]\nvariables [add_comm_monoid N] [module R N] [module S N] [is_scalar_tower R S N]\n\nvariables {S M N}\n\n@[simp]\nlemma linear_map.ker_restrict_scalars (f : M \u2192\u2097[S] N) :\n  (f.restrict_scalars R).ker = f.ker.restrict_scalars R :=\nrfl\n\nend module\n\nnamespace submodule\n\nvariables (R A M : Type*)\nvariables [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M]\nvariables [module R M] [module A M] [is_scalar_tower R A M]\n\n\/-- If `A` is an `R`-algebra such that the induced morhpsim `R \u2192+* A` is surjective, then the\n`R`-module generated by a set `X` equals the `A`-module generated by `X`. -\/\nlemma span_eq_restrict_scalars (X : set M) (hsur : function.surjective (algebra_map R A)) :\n  span R X = restrict_scalars R (span A X) :=\nbegin\n  apply (span_le_restrict_scalars R A X).antisymm (\u03bb m hm, _),\n  refine span_induction hm subset_span (zero_mem _) (\u03bb _ _, add_mem _) (\u03bb a m hm, _),\n  obtain \u27e8r, rfl\u27e9 := hsur a,\n  simpa [algebra_map_smul] using smul_mem _ r hm\nend\n\nend submodule\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariables [comm_semiring R] [semiring A] [semiring B]\nvariables [algebra R A] [algebra R B]\n\n\/-- `R`-algebra homomorphism between the function spaces `I \u2192 A` and `I \u2192 B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -\/\n@[simps] protected def comp_left (f : A \u2192\u2090[R] B) (I : Type*) : (I \u2192 A) \u2192\u2090[R] (I \u2192 B) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  commutes' := \u03bb c, by { ext, exact f.commutes' c },\n  .. f.to_ring_hom.comp_left I }\n\nend alg_hom\n","avg_line_length":34.7546907575,"max_line_length":100,"alphanum_fraction":0.6446452851}
-{"size":28075,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2021 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-\/\n\nimport measure_theory.measure.regular\nimport topology.semicontinuous\nimport measure_theory.integral.bochner\nimport topology.instances.ereal\n\n\/-!\n# Vitali-Carath\u00e9odory theorem\n\nVitali-Carath\u00e9odory theorem asserts the following. Consider an integrable function `f : \u03b1 \u2192 \u211d` on\na space with a regular measure. Then there exists a function `g : \u03b1 \u2192 ereal` such that `f x < g x`\neverywhere, `g` is lower semicontinuous, and the integral of `g` is arbitrarily close to that of\n`f`. This theorem is proved in this file, as `exists_lt_lower_semicontinuous_integral_lt`.\n\nSymmetrically, there exists `g < f` which is upper semicontinuous, with integral arbitrarily close\nto that of `f`. It follows from the previous statement applied to `-f`. It is formalized under\nthe name `exists_upper_semicontinuous_lt_integral_gt`.\n\nThe most classical version of Vitali-Carath\u00e9odory theorem only ensures a large inequality\n`f x \u2264 g x`. For applications to the fundamental theorem of calculus, though, the strict inequality\n`f x < g x` is important. Therefore, we prove the stronger version with strict inequalities in this\nfile. There is a price to pay: we require that the measure is `\u03c3`-finite, which is not necessary for\nthe classical Vitali-Carath\u00e9odory theorem. Since this is satisfied in all applications, this is\nnot a real problem.\n\n## Sketch of proof\n\nDecomposing `f` as the difference of its positive and negative parts, it suffices to show that a\npositive function can be bounded from above by a lower semicontinuous function, and from below\nby an upper semicontinuous function, with integrals close to that of `f`.\n\nFor the bound from above, write `f` as a series `\u2211' n, c\u2099 * indicator (s\u2099)` of simple functions.\nThen, approximate `s\u2099` by a larger open set `u\u2099` with measure very close to that of `s\u2099` (this is\npossible by regularity of the measure), and set `g = \u2211' n, c\u2099 * indicator (u\u2099)`. It is\nlower semicontinuous as a series of lower semicontinuous functions, and its integral is arbitrarily\nclose to that of `f`.\n\nFor the bound from below, use finitely many terms in the series, and approximate `s\u2099` from inside by\na closed set `F\u2099`. Then `\u2211 n < N, c\u2099 * indicator (F\u2099)` is bounded from above by `f`, it is\nupper semicontinuous as a finite sum of upper semicontinuous functions, and its integral is\narbitrarily close to that of `f`.\n\nThe main pain point in the implementation is that one needs to jump between the spaces `\u211d`, `\u211d\u22650`,\n`\u211d\u22650\u221e` and `ereal` (and be careful that addition is not well behaved on `ereal`), and between\n`lintegral` and `integral`.\n\nWe first show the bound from above for simple functions and the nonnegative integral\n(this is the main nontrivial mathematical point), then deduce it for general nonnegative functions,\nfirst for the nonnegative integral and then for the Bochner integral.\n\nThen we follow the same steps for the lower bound.\n\nFinally, we glue them together to obtain the main statement\n`exists_lt_lower_semicontinuous_integral_lt`.\n\n## Related results\n\nAre you looking for a result on approximation by continuous functions (not just semicontinuous)?\nSee result `measure_theory.Lp.continuous_map_dense`, in the file\n`measure_theory.continuous_map_dense`.\n\n## References\n\n[Rudin, *Real and Complex Analysis* (Theorem 2.24)][rudin2006real]\n\n-\/\n\nopen_locale ennreal nnreal\n\nopen measure_theory measure_theory.measure\n\nvariables {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1] [borel_space \u03b1] (\u03bc : measure \u03b1)\n  [weakly_regular \u03bc]\n\nnamespace measure_theory\n\nlocal infixr ` \u2192\u209b `:25 := simple_func\n\n\/-! ### Lower semicontinuous upper bound for nonnegative functions -\/\n\n\/-- Given a simple function `f` with values in `\u211d\u22650`, there exists a lower semicontinuous\nfunction `g \u2265 f` with integral arbitrarily close to that of `f`. Formulation in terms of\n`lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma simple_func.exists_le_lower_semicontinuous_lintegral_ge :\n  \u2200 (f : \u03b1 \u2192\u209b \u211d\u22650) {\u03b5 : \u211d\u22650\u221e} (\u03b5pos : 0 < \u03b5),\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, f x \u2264 g x) \u2227 lower_semicontinuous g \u2227\n    (\u222b\u207b x, g x \u2202\u03bc \u2264 \u222b\u207b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  refine simple_func.induction _ _,\n  { assume c s hs \u03b5 \u03b5pos,\n    let f := simple_func.piecewise s hs (simple_func.const \u03b1 c) (simple_func.const \u03b1 0),\n    by_cases h : \u222b\u207b x, f x \u2202\u03bc = \u22a4,\n    { refine \u27e8\u03bb x, c, \u03bb x, _, lower_semicontinuous_const,\n             by simp only [ennreal.top_add, le_top, h]\u27e9,\n      simp only [simple_func.coe_const, simple_func.const_zero, simple_func.coe_zero,\n        set.piecewise_eq_indicator, simple_func.coe_piecewise],\n      exact set.indicator_le_self _ _ _ },\n    by_cases hc : c = 0,\n    { refine \u27e8\u03bb x, 0, _, lower_semicontinuous_const, _\u27e9,\n      { simp only [hc, set.indicator_zero', pi.zero_apply, simple_func.const_zero, implies_true_iff,\n          eq_self_iff_true, simple_func.coe_zero, set.piecewise_eq_indicator,\n          simple_func.coe_piecewise, le_zero_iff] },\n      { simp only [lintegral_const, zero_mul, zero_le, ennreal.coe_zero] } },\n    have : \u03bc s < \u03bc s + \u03b5 \/ c,\n    { have : (0 : \u211d\u22650\u221e) < \u03b5 \/ c := ennreal.div_pos_iff.2 \u27e8\u03b5pos.ne', ennreal.coe_ne_top\u27e9,\n      simpa using (ennreal.add_lt_add_iff_left _).2 this,\n      simpa only [hs, hc, lt_top_iff_ne_top, true_and, simple_func.coe_const, function.const_apply,\n        lintegral_const, ennreal.coe_indicator, set.univ_inter, ennreal.coe_ne_top,\n        measurable_set.univ, with_top.mul_eq_top_iff, simple_func.const_zero, or_false,\n        lintegral_indicator, ennreal.coe_eq_zero, ne.def, not_false_iff, simple_func.coe_zero,\n        set.piecewise_eq_indicator, simple_func.coe_piecewise, false_and, restrict_apply] using h },\n    obtain \u27e8u, u_open, su, \u03bcu\u27e9 : \u2203 u, is_open u \u2227 s \u2286 u \u2227 \u03bc u < \u03bc s + \u03b5 \/ c :=\n      hs.exists_is_open_lt_of_lt _ this,\n    refine \u27e8set.indicator u (\u03bb x, c), \u03bb x, _, u_open.lower_semicontinuous_indicator (zero_le _), _\u27e9,\n    { simp only [simple_func.coe_const, simple_func.const_zero, simple_func.coe_zero,\n        set.piecewise_eq_indicator, simple_func.coe_piecewise],\n      exact set.indicator_le_indicator_of_subset su (\u03bb x, zero_le _) _ },\n    { suffices : (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * \u03bc s + \u03b5, by\n        simpa only [hs, u_open.measurable_set, simple_func.coe_const, function.const_apply,\n          lintegral_const, ennreal.coe_indicator, set.univ_inter, measurable_set.univ,\n          simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,\n          set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply],\n      calc (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * (\u03bc s + \u03b5 \/ c) : ennreal.mul_le_mul (le_refl _) \u03bcu.le\n      ... = c * \u03bc s + \u03b5 :\n        begin\n          simp_rw [mul_add],\n          rw ennreal.mul_div_cancel' _ ennreal.coe_ne_top,\n          simpa using hc,\n        end } },\n  { assume f\u2081 f\u2082 H h\u2081 h\u2082 \u03b5 \u03b5pos,\n    rcases h\u2081 (ennreal.half_pos \u03b5pos) with \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9,\n    rcases h\u2082 (ennreal.half_pos \u03b5pos) with \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9,\n    refine \u27e8\u03bb x, g\u2081 x + g\u2082 x, \u03bb x, add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9,\n    simp only [simple_func.coe_add, ennreal.coe_add, pi.add_apply],\n    rw [lintegral_add f\u2081.measurable.coe_nnreal_ennreal f\u2082.measurable.coe_nnreal_ennreal,\n        lintegral_add g\u2081cont.measurable.coe_nnreal_ennreal g\u2082cont.measurable.coe_nnreal_ennreal],\n    convert add_le_add g\u2081int g\u2082int using 1,\n    conv_lhs { rw \u2190 ennreal.add_halves \u03b5 },\n    abel }\nend\n\nopen simple_func (eapprox_diff tsum_eapprox_diff)\n\n\/-- Given a measurable function `f` with values in `\u211d\u22650`, there exists a lower semicontinuous\nfunction `g \u2265 f` with integral arbitrarily close to that of `f`. Formulation in terms of\n`lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_le_lower_semicontinuous_lintegral_ge\n  (f : \u03b1 \u2192 \u211d\u22650\u221e) (hf : measurable f) {\u03b5 : \u211d\u22650\u221e} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650\u221e, (\u2200 x, f x \u2264 g x) \u2227 lower_semicontinuous g \u2227 (\u222b\u207b x, g x \u2202\u03bc \u2264 \u222b\u207b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  rcases ennreal.exists_pos_sum_of_encodable' \u03b5pos \u2115 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9,\n  have : \u2200 n, \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, simple_func.eapprox_diff f n x \u2264 g x) \u2227 lower_semicontinuous g \u2227\n    (\u222b\u207b x, g x \u2202\u03bc \u2264 \u222b\u207b x, simple_func.eapprox_diff f n x \u2202\u03bc + \u03b4 n) :=\n  \u03bb n, simple_func.exists_le_lower_semicontinuous_lintegral_ge \u03bc\n    (simple_func.eapprox_diff f n) (\u03b4pos n),\n  choose g f_le_g gcont hg using this,\n  refine \u27e8\u03bb x, (\u2211' n, g n x), \u03bb x, _, _, _\u27e9,\n  { rw \u2190 tsum_eapprox_diff f hf,\n    exact ennreal.tsum_le_tsum (\u03bb n, ennreal.coe_le_coe.2 (f_le_g n x)) },\n  { apply lower_semicontinuous_tsum (\u03bb n, _),\n    exact ennreal.continuous_coe.comp_lower_semicontinuous (gcont n)\n      (\u03bb x y hxy, ennreal.coe_le_coe.2 hxy) },\n  { calc \u222b\u207b x, \u2211' (n : \u2115), g n x \u2202\u03bc\n    = \u2211' n, \u222b\u207b x, g n x \u2202\u03bc :\n      by rw lintegral_tsum (\u03bb n, (gcont n).measurable.coe_nnreal_ennreal)\n    ... \u2264 \u2211' n, (\u222b\u207b x, eapprox_diff f n x \u2202\u03bc + \u03b4 n) : ennreal.tsum_le_tsum hg\n    ... = \u2211' n, (\u222b\u207b x, eapprox_diff f n x \u2202\u03bc) + \u2211' n, \u03b4 n : ennreal.tsum_add\n    ... \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 :\n      begin\n        refine add_le_add _ h\u03b4.le,\n        rw [\u2190 lintegral_tsum],\n        { simp_rw [tsum_eapprox_diff f hf, le_refl] },\n        { assume n, exact (simple_func.measurable _).coe_nnreal_ennreal }\n      end }\nend\n\n\/-- Given a measurable function `f` with values in `\u211d\u22650` in a sigma-finite space, there exists a\nlower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.\nFormulation in terms of `lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_lt_lower_semicontinuous_lintegral_ge [sigma_finite \u03bc]\n  (f : \u03b1 \u2192 \u211d\u22650) (fmeas : measurable f) {\u03b5 : \u211d\u22650} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650\u221e, (\u2200 x, (f x : \u211d\u22650\u221e) < g x) \u2227 lower_semicontinuous g \u2227\n    (\u222b\u207b x, g x \u2202\u03bc \u2264 \u222b\u207b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  rcases exists_integrable_pos_of_sigma_finite \u03bc (nnreal.half_pos \u03b5pos) with \u27e8w, wpos, wmeas, wint\u27e9,\n  let f' := \u03bb x, ((f x + w x : \u211d\u22650) : \u211d\u22650\u221e),\n  rcases exists_le_lower_semicontinuous_lintegral_ge \u03bc f' (fmeas.add wmeas).coe_nnreal_ennreal\n    (ennreal.coe_pos.2 (nnreal.half_pos \u03b5pos)) with \u27e8g, le_g, gcont, gint\u27e9,\n  refine \u27e8g, \u03bb x, _, gcont, _\u27e9,\n  { calc (f x : \u211d\u22650\u221e) < f' x : by simpa [\u2190 ennreal.coe_lt_coe] using add_lt_add_left (wpos x) (f x)\n    ... \u2264 g x : le_g x },\n  { calc \u222b\u207b (x : \u03b1), g x \u2202\u03bc\n        \u2264 \u222b\u207b (x : \u03b1), f x + w x \u2202\u03bc + (\u03b5 \/ 2 : \u211d\u22650) : gint\n    ... = \u222b\u207b (x : \u03b1), f x \u2202 \u03bc + \u222b\u207b (x : \u03b1), w x \u2202 \u03bc + (\u03b5 \/ 2 : \u211d\u22650) :\n      by rw lintegral_add fmeas.coe_nnreal_ennreal wmeas.coe_nnreal_ennreal\n    ... \u2264 \u222b\u207b (x : \u03b1), f x \u2202 \u03bc + (\u03b5 \/ 2 : \u211d\u22650) + (\u03b5 \/ 2 : \u211d\u22650) :\n      add_le_add_right (add_le_add_left wint.le _) _\n    ... = \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 : by rw [add_assoc, \u2190 ennreal.coe_add, nnreal.add_halves] },\nend\n\n\/-- Given an almost everywhere measurable function `f` with values in `\u211d\u22650` in a sigma-finite space,\nthere exists a lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.\nFormulation in terms of `lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable [sigma_finite \u03bc]\n  (f : \u03b1 \u2192 \u211d\u22650) (fmeas : ae_measurable f \u03bc) {\u03b5 : \u211d\u22650} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650\u221e, (\u2200 x, (f x : \u211d\u22650\u221e) < g x) \u2227 lower_semicontinuous g \u2227\n    (\u222b\u207b x, g x \u2202\u03bc \u2264 \u222b\u207b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  rcases exists_lt_lower_semicontinuous_lintegral_ge \u03bc (fmeas.mk f) fmeas.measurable_mk\n    (nnreal.half_pos \u03b5pos) with \u27e8g0, f_lt_g0, g0_cont, g0_int\u27e9,\n  rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with \u27e8s, hs, smeas, \u03bcs\u27e9,\n  rcases exists_le_lower_semicontinuous_lintegral_ge \u03bc (s.indicator (\u03bb x, \u221e))\n    (measurable_const.indicator smeas) (ennreal.half_pos (ennreal.coe_pos.2 \u03b5pos)) with\n    \u27e8g1, le_g1, g1_cont, g1_int\u27e9,\n  refine \u27e8\u03bb x, g0 x + g1 x, \u03bb x, _, g0_cont.add g1_cont, _\u27e9,\n  { by_cases h : x \u2208 s,\n    { have := le_g1 x,\n      simp only [h, set.indicator_of_mem, top_le_iff] at this,\n      simp [this] },\n    { have : f x = fmeas.mk f x,\n        by { rw set.compl_subset_comm at hs, exact hs h },\n      rw this,\n      exact (f_lt_g0 x).trans_le le_self_add } },\n  { calc \u222b\u207b x, g0 x + g1 x \u2202\u03bc =  \u222b\u207b x, g0 x \u2202\u03bc + \u222b\u207b x, g1 x \u2202\u03bc :\n      lintegral_add g0_cont.measurable g1_cont.measurable\n    ... \u2264 (\u222b\u207b x, f x \u2202\u03bc + \u03b5 \/ 2) + (0 + \u03b5 \/ 2) :\n      begin\n        refine add_le_add _ _,\n        { convert g0_int using 2,\n          { exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _) },\n          { simp only [ennreal.coe_div, ennreal.coe_one, ennreal.coe_bit0, ne.def, not_false_iff,\n              bit0_eq_zero, one_ne_zero], } },\n        { convert g1_int,\n          simp only [smeas, \u03bcs, lintegral_const, set.univ_inter, measurable_set.univ,\n            lintegral_indicator, mul_zero, restrict_apply] }\n      end\n    ... = \u222b\u207b x, f x \u2202\u03bc + \u03b5 : by simp only [add_assoc, ennreal.add_halves, zero_add] }\nend\n\nvariable {\u03bc}\n\n\/-- Given an integrable function `f` with values in `\u211d\u22650` in a sigma-finite space, there exists a\nlower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.\nFormulation in terms of `integral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_lt_lower_semicontinuous_integral_gt_nnreal [sigma_finite \u03bc] (f : \u03b1 \u2192 \u211d\u22650)\n  (fint : integrable (\u03bb x, (f x : \u211d)) \u03bc) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650\u221e, (\u2200 x, (f x : \u211d\u22650\u221e) < g x) \u2227 lower_semicontinuous g \u2227 (\u2200\u1d50 x \u2202 \u03bc, g x < \u22a4)\n  \u2227 (integrable (\u03bb x, (g x).to_real) \u03bc) \u2227 (\u222b x, (g x).to_real \u2202\u03bc < \u222b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  have fmeas : ae_measurable f \u03bc,\n    by { convert fint.ae_measurable.real_to_nnreal, ext1 x, simp only [real.to_nnreal_coe] },\n  let \u03b4 : \u211d\u22650 := \u27e8\u03b5\/2, (half_pos \u03b5pos).le\u27e9,\n  have \u03b4pos : 0 < \u03b4 := half_pos \u03b5pos,\n  have int_f_lt_top : \u222b\u207b (a : \u03b1), (f a) \u2202\u03bc < \u221e :=\n    has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral,\n  rcases exists_lt_lower_semicontinuous_lintegral_ge_of_ae_measurable \u03bc f fmeas \u03b4pos\n    with \u27e8g, f_lt_g, gcont, gint\u27e9,\n  have gint_lt : \u222b\u207b (x : \u03b1), g x \u2202\u03bc < \u221e := gint.trans_lt (by simpa using int_f_lt_top),\n  have g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_lt,\n  have Ig : \u222b\u207b (a : \u03b1), ennreal.of_real (g a).to_real \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc,\n  { apply lintegral_congr_ae,\n    filter_upwards [g_lt_top],\n    assume x hx,\n    simp only [hx.ne, ennreal.of_real_to_real, ne.def, not_false_iff] },\n  refine \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9,\n  { refine \u27e8gcont.measurable.ennreal_to_real.ae_measurable, _\u27e9,\n    simp [has_finite_integral_iff_norm, real.norm_eq_abs, abs_of_nonneg],\n    convert gint_lt using 1 },\n  { rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae],\n    { calc\n      ennreal.to_real (\u222b\u207b (a : \u03b1), ennreal.of_real (g a).to_real \u2202\u03bc)\n          = ennreal.to_real (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) : by congr' 1\n      ... \u2264 ennreal.to_real (\u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u03b4) :\n        begin\n          apply ennreal.to_real_mono _ gint,\n          simpa using int_f_lt_top.ne,\n        end\n      ... = ennreal.to_real (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) + \u03b4 :\n        by rw [ennreal.to_real_add int_f_lt_top.ne ennreal.coe_ne_top, ennreal.coe_to_real]\n      ... < ennreal.to_real (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) + \u03b5 :\n        add_lt_add_left (by simp [\u03b4, half_lt_self \u03b5pos]) _\n      ... = (\u222b\u207b (a : \u03b1), ennreal.of_real \u2191(f a) \u2202\u03bc).to_real + \u03b5 :\n        by simp },\n    { apply filter.eventually_of_forall (\u03bb x, _), simp },\n    { exact fmeas.coe_nnreal_real, },\n    { apply filter.eventually_of_forall (\u03bb x, _), simp },\n    { apply gcont.measurable.ennreal_to_real.ae_measurable } }\nend\n\n\n\/-! ### Upper semicontinuous lower bound for nonnegative functions -\/\n\n\/-- Given a simple function `f` with values in `\u211d\u22650`, there exists an upper semicontinuous\nfunction `g \u2264 f` with integral arbitrarily close to that of `f`. Formulation in terms of\n`lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma simple_func.exists_upper_semicontinuous_le_lintegral_le :\n  \u2200 (f : \u03b1 \u2192\u209b \u211d\u22650) (int_f : \u222b\u207b x, f x \u2202\u03bc < \u221e) {\u03b5 : \u211d\u22650\u221e} (\u03b5pos : 0 < \u03b5),\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, g x \u2264 f x) \u2227 upper_semicontinuous g \u2227 (\u222b\u207b x, f x \u2202\u03bc \u2264 \u222b\u207b x, g x \u2202\u03bc + \u03b5) :=\nbegin\n  refine simple_func.induction _ _,\n  { assume c s hs int_f \u03b5 \u03b5pos,\n    let f := simple_func.piecewise s hs (simple_func.const \u03b1 c) (simple_func.const \u03b1 0),\n    by_cases hc : c = 0,\n    { refine \u27e8\u03bb x, 0, _, upper_semicontinuous_const, _\u27e9,\n      { simp only [hc, set.indicator_zero', pi.zero_apply, simple_func.const_zero, implies_true_iff,\n          eq_self_iff_true, simple_func.coe_zero, set.piecewise_eq_indicator,\n          simple_func.coe_piecewise, le_zero_iff] },\n      { simp only [hc, set.indicator_zero', lintegral_const, zero_mul, pi.zero_apply,\n         simple_func.const_zero, zero_add, zero_le', simple_func.coe_zero,\n         set.piecewise_eq_indicator, ennreal.coe_zero, simple_func.coe_piecewise, \u03b5pos.le] } },\n    have \u03bcs_lt_top : \u03bc s < \u221e,\n      by simpa only [hs, hc, lt_top_iff_ne_top, true_and, simple_func.coe_const, or_false,\n        lintegral_const, ennreal.coe_indicator, set.univ_inter, ennreal.coe_ne_top, restrict_apply\n        measurable_set.univ, with_top.mul_eq_top_iff, simple_func.const_zero, function.const_apply,\n        lintegral_indicator, ennreal.coe_eq_zero, ne.def, not_false_iff, simple_func.coe_zero,\n        set.piecewise_eq_indicator, simple_func.coe_piecewise, false_and] using int_f,\n    have : (0 : \u211d\u22650\u221e) < \u03b5 \/ c := ennreal.div_pos_iff.2 \u27e8\u03b5pos.ne', ennreal.coe_ne_top\u27e9,\n    obtain \u27e8F, F_closed, Fs, \u03bcF\u27e9 : \u2203 F, is_closed F \u2227 F \u2286 s \u2227 \u03bc s < \u03bc F + \u03b5 \/ c :=\n      hs.exists_lt_is_closed_of_lt_top_of_pos \u03bcs_lt_top this,\n    refine \u27e8set.indicator F (\u03bb x, c), \u03bb x, _,\n      F_closed.upper_semicontinuous_indicator (zero_le _), _\u27e9,\n    { simp only [simple_func.coe_const, simple_func.const_zero, simple_func.coe_zero,\n        set.piecewise_eq_indicator, simple_func.coe_piecewise],\n      exact set.indicator_le_indicator_of_subset Fs (\u03bb x, zero_le _) _ },\n    { suffices : (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * \u03bc F + \u03b5,\n        by simpa only [hs, F_closed.measurable_set, simple_func.coe_const, function.const_apply,\n          lintegral_const, ennreal.coe_indicator, set.univ_inter, measurable_set.univ,\n          simple_func.const_zero, lintegral_indicator, simple_func.coe_zero,\n          set.piecewise_eq_indicator, simple_func.coe_piecewise, restrict_apply],\n      calc (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * (\u03bc F + \u03b5 \/ c) : ennreal.mul_le_mul (le_refl _) \u03bcF.le\n      ... = c * \u03bc F + \u03b5 :\n        begin\n          simp_rw [mul_add],\n          rw ennreal.mul_div_cancel' _ ennreal.coe_ne_top,\n          simpa using hc,\n        end } },\n  { assume f\u2081 f\u2082 H h\u2081 h\u2082 f_int \u03b5 \u03b5pos,\n    have A : \u222b\u207b (x : \u03b1), f\u2081 x \u2202\u03bc + \u222b\u207b (x : \u03b1), f\u2082 x \u2202\u03bc < \u22a4,\n    { rw \u2190 lintegral_add f\u2081.measurable.coe_nnreal_ennreal f\u2082.measurable.coe_nnreal_ennreal,\n      simpa only [simple_func.coe_add, ennreal.coe_add, pi.add_apply] using f_int },\n    rcases h\u2081 (ennreal.add_lt_top.1 A).1 (ennreal.half_pos \u03b5pos) with \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9,\n    rcases h\u2082 (ennreal.add_lt_top.1 A).2 (ennreal.half_pos \u03b5pos) with \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9,\n    refine \u27e8\u03bb x, g\u2081 x + g\u2082 x, \u03bb x, add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9,\n    simp only [simple_func.coe_add, ennreal.coe_add, pi.add_apply],\n    rw [lintegral_add f\u2081.measurable.coe_nnreal_ennreal f\u2082.measurable.coe_nnreal_ennreal,\n        lintegral_add g\u2081cont.measurable.coe_nnreal_ennreal g\u2082cont.measurable.coe_nnreal_ennreal],\n    convert add_le_add g\u2081int g\u2082int using 1,\n    conv_lhs { rw \u2190 ennreal.add_halves \u03b5 },\n    abel }\nend\n\n\/-- Given an integrable function `f` with values in `\u211d\u22650`, there exists an upper semicontinuous\nfunction `g \u2264 f` with integral arbitrarily close to that of `f`. Formulation in terms of\n`lintegral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_upper_semicontinuous_le_lintegral_le\n  (f : \u03b1 \u2192 \u211d\u22650) (int_f : \u222b\u207b x, f x \u2202\u03bc < \u221e) {\u03b5 : \u211d\u22650\u221e} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, g x \u2264 f x) \u2227 upper_semicontinuous g \u2227 (\u222b\u207b x, f x \u2202\u03bc \u2264 \u222b\u207b x, g x \u2202\u03bc + \u03b5) :=\nbegin\n  obtain \u27e8fs, fs_le_f, int_fs\u27e9 : \u2203 (fs : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 x, fs x \u2264 f x) \u2227\n    (\u222b\u207b x, f x \u2202\u03bc \u2264 \u222b\u207b x, fs x \u2202\u03bc + \u03b5\/2) :=\n  begin\n    have := ennreal.lt_add_right int_f (ennreal.half_pos \u03b5pos),\n    conv_rhs at this { rw lintegral_eq_nnreal (\u03bb x, (f x : \u211d\u22650\u221e)) \u03bc },\n    erw ennreal.bsupr_add at this; [skip, exact \u27e80, \u03bb x, by simp\u27e9],\n    simp only [lt_supr_iff] at this,\n    rcases this with \u27e8fs, fs_le_f, int_fs\u27e9,\n    refine \u27e8fs, \u03bb x, by simpa only [ennreal.coe_le_coe] using fs_le_f x, _\u27e9,\n    convert int_fs.le,\n    rw \u2190 simple_func.lintegral_eq_lintegral,\n    refl\n  end,\n  have int_fs_lt_top : \u222b\u207b x, fs x \u2202\u03bc < \u221e,\n  { apply lt_of_le_of_lt (lintegral_mono (\u03bb x, _)) int_f,\n    simpa only [ennreal.coe_le_coe] using fs_le_f x },\n  obtain \u27e8g, g_le_fs, gcont, gint\u27e9 : \u2203 g : \u03b1 \u2192 \u211d\u22650,\n    (\u2200 x, g x \u2264 fs x) \u2227 upper_semicontinuous g \u2227 (\u222b\u207b x, fs x \u2202\u03bc \u2264 \u222b\u207b x, g x \u2202\u03bc + \u03b5\/2) :=\n  fs.exists_upper_semicontinuous_le_lintegral_le int_fs_lt_top (ennreal.half_pos \u03b5pos),\n  refine \u27e8g, \u03bb x, (g_le_fs x).trans (fs_le_f x), gcont, _\u27e9,\n  calc \u222b\u207b x, f x \u2202\u03bc \u2264 \u222b\u207b x, fs x \u2202\u03bc + \u03b5 \/ 2 : int_fs\n  ... \u2264 (\u222b\u207b x, g x \u2202\u03bc + \u03b5 \/ 2) + \u03b5 \/ 2 : add_le_add gint (le_refl _)\n  ... = \u222b\u207b x, g x \u2202\u03bc + \u03b5 : by rw [add_assoc, ennreal.add_halves]\nend\n\n\/-- Given an integrable function `f` with values in `\u211d\u22650`, there exists an upper semicontinuous\nfunction `g \u2264 f` with integral arbitrarily close to that of `f`. Formulation in terms of\n`integral`.\nAuxiliary lemma for Vitali-Carath\u00e9odory theorem `exists_lt_lower_semicontinuous_integral_lt`. -\/\nlemma exists_upper_semicontinuous_le_integral_le (f : \u03b1 \u2192 \u211d\u22650)\n  (fint : integrable (\u03bb x, (f x : \u211d)) \u03bc) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, g x \u2264 f x) \u2227 upper_semicontinuous g \u2227 (integrable (\u03bb x, (g x : \u211d)) \u03bc)\n  \u2227 (\u222b x, (f x : \u211d) \u2202\u03bc - \u03b5 \u2264 \u222b x, g x \u2202\u03bc) :=\nbegin\n  let \u03b4 : \u211d\u22650 := \u27e8\u03b5, \u03b5pos.le\u27e9,\n  have \u03b4pos : (0 : \u211d\u22650\u221e) < \u03b4 := ennreal.coe_lt_coe.2 \u03b5pos,\n  have If : \u222b\u207b x, f x \u2202 \u03bc < \u221e := has_finite_integral_iff_of_nnreal.1 fint.has_finite_integral,\n  rcases exists_upper_semicontinuous_le_lintegral_le f If \u03b4pos with \u27e8g, gf, gcont, gint\u27e9,\n  have Ig : \u222b\u207b x, g x \u2202 \u03bc < \u221e,\n  { apply lt_of_le_of_lt (lintegral_mono (\u03bb x, _)) If,\n    simpa using gf x },\n  refine \u27e8g, gf, gcont, _, _\u27e9,\n  { refine integrable.mono fint gcont.measurable.coe_nnreal_real.ae_measurable _,\n    exact filter.eventually_of_forall (\u03bb x, by simp [gf x]) },\n  { rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae],\n    { rw sub_le_iff_le_add,\n      convert ennreal.to_real_mono _ gint,\n      { simp, },\n      { rw ennreal.to_real_add Ig.ne ennreal.coe_ne_top, simp },\n      { simpa using Ig.ne } },\n    { apply filter.eventually_of_forall, simp },\n    { exact gcont.measurable.coe_nnreal_real.ae_measurable },\n    { apply filter.eventually_of_forall, simp },\n    { exact fint.ae_measurable } }\nend\n\n\/-! ### Vitali-Carath\u00e9odory theorem -\/\n\n\/-- **Vitali-Carath\u00e9odory Theorem**: given an integrable real function `f`, there exists an\nintegrable function `g > f` which is lower semicontinuous, with integral arbitrarily close\nto that of `f`. This function has to be `ereal`-valued in general. -\/\nlemma exists_lt_lower_semicontinuous_integral_lt [sigma_finite \u03bc]\n  (f : \u03b1 \u2192 \u211d) (hf : integrable f \u03bc) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 ereal, (\u2200 x, (f x : ereal) < g x) \u2227 lower_semicontinuous g \u2227\n  (integrable (\u03bb x, ereal.to_real (g x)) \u03bc) \u2227 (\u2200\u1d50 x \u2202 \u03bc, g x < \u22a4) \u2227\n  (\u222b x, ereal.to_real (g x) \u2202\u03bc < \u222b x, f x \u2202\u03bc + \u03b5) :=\nbegin\n  let \u03b4 : \u211d\u22650 := \u27e8\u03b5\/2, (half_pos \u03b5pos).le\u27e9,\n  have \u03b4pos : 0 < \u03b4 := half_pos \u03b5pos,\n  let fp : \u03b1 \u2192 \u211d\u22650 := \u03bb x, real.to_nnreal (f x),\n  have int_fp : integrable (\u03bb x, (fp x : \u211d)) \u03bc := hf.real_to_nnreal,\n  rcases exists_lt_lower_semicontinuous_integral_gt_nnreal fp int_fp \u03b4pos\n    with \u27e8gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint\u27e9,\n  let fm : \u03b1 \u2192 \u211d\u22650 := \u03bb x, real.to_nnreal (-f x),\n  have int_fm : integrable (\u03bb x, (fm x : \u211d)) \u03bc := hf.neg.real_to_nnreal,\n  rcases exists_upper_semicontinuous_le_integral_le fm int_fm \u03b4pos\n    with \u27e8gm, gm_le_fm, gmcont, gm_integrable, gmint\u27e9,\n  let g : \u03b1 \u2192 ereal := \u03bb x, (gp x : ereal) - (gm x),\n  have ae_g : \u2200\u1d50 x \u2202 \u03bc, (g x).to_real = (gp x : ereal).to_real - (gm x : ereal).to_real,\n  { filter_upwards [gp_lt_top],\n    assume x hx,\n    rw ereal.to_real_sub;\n    simp [hx.ne] },\n  refine \u27e8g, _, _, _, _, _\u27e9,\n  show integrable (\u03bb x, ereal.to_real (g x)) \u03bc,\n  { rw integrable_congr ae_g,\n    convert gp_integrable.sub gm_integrable,\n    ext x,\n    simp },\n  show \u222b (x : \u03b1), (g x).to_real \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5, from calc\n    \u222b (x : \u03b1), (g x).to_real \u2202\u03bc = \u222b (x : \u03b1), ereal.to_real (gp x) - ereal.to_real (gm x) \u2202\u03bc :\n      integral_congr_ae ae_g\n    ... = \u222b (x : \u03b1), ereal.to_real (gp x) \u2202 \u03bc - \u222b (x : \u03b1), gm x \u2202\u03bc :\n      begin\n        simp only [ereal.to_real_coe_ennreal, ennreal.coe_to_real, coe_coe],\n        exact integral_sub gp_integrable gm_integrable,\n      end\n    ... < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - \u222b (x : \u03b1), gm x \u2202\u03bc :\n      begin\n        apply sub_lt_sub_right,\n        convert gpint,\n        simp only [ereal.to_real_coe_ennreal],\n      end\n    ... \u2264 \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - (\u222b (x : \u03b1), fm x \u2202\u03bc - \u03b4) :\n      sub_le_sub_left gmint _\n    ... = \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u03b4 :\n      by { simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf, fp, fm], ring }\n    ... = \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 :\n      by { congr' 1, field_simp [\u03b4, mul_comm] },\n  show \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4,\n  { filter_upwards [gp_lt_top],\n    assume x hx,\n    simp [g, ereal.sub_eq_add_neg, lt_top_iff_ne_top, lt_top_iff_ne_top.1 hx] },\n  show \u2200 x, (f x : ereal) < g x,\n  { assume x,\n    rw ereal.coe_real_ereal_eq_coe_to_nnreal_sub_coe_to_nnreal (f x),\n    refine ereal.sub_lt_sub_of_lt_of_le _ _ _ _,\n    { simp only [ereal.coe_ennreal_lt_coe_ennreal_iff, coe_coe], exact (fp_lt_gp x) },\n    { simp only [ennreal.coe_le_coe, ereal.coe_ennreal_le_coe_ennreal_iff, coe_coe],\n      exact (gm_le_fm x) },\n    { simp only [ereal.coe_ennreal_ne_bot, ne.def, not_false_iff, coe_coe] },\n    { simp only [ereal.coe_nnreal_ne_top, ne.def, not_false_iff, coe_coe] } },\n  show lower_semicontinuous g,\n  { apply lower_semicontinuous.add',\n    { exact continuous_coe_ennreal_ereal.comp_lower_semicontinuous gpcont\n        (\u03bb x y hxy, ereal.coe_ennreal_le_coe_ennreal_iff.2 hxy) },\n    { apply ereal.continuous_neg.comp_upper_semicontinuous_antimono _\n        (\u03bb x y hxy, ereal.neg_le_neg_iff.2 hxy),\n      dsimp,\n      apply continuous_coe_ennreal_ereal.comp_upper_semicontinuous _\n        (\u03bb x y hxy, ereal.coe_ennreal_le_coe_ennreal_iff.2 hxy),\n      exact ennreal.continuous_coe.comp_upper_semicontinuous gmcont\n        (\u03bb x y hxy, ennreal.coe_le_coe.2 hxy) },\n    { assume x,\n      exact ereal.continuous_at_add (by simp) (by simp) } }\nend\n\n\/-- **Vitali-Carath\u00e9odory Theorem**: given an integrable real function `f`, there exists an\nintegrable function `g < f` which is upper semicontinuous, with integral arbitrarily close to that\nof `f`. This function has to be `ereal`-valued in general. -\/\nlemma exists_upper_semicontinuous_lt_integral_gt [sigma_finite \u03bc]\n  (f : \u03b1 \u2192 \u211d) (hf : integrable f \u03bc) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n  \u2203 g : \u03b1 \u2192 ereal, (\u2200 x, (g x : ereal) < f x) \u2227 upper_semicontinuous g \u2227\n  (integrable (\u03bb x, ereal.to_real (g x)) \u03bc) \u2227 (\u2200\u1d50 x \u2202\u03bc, \u22a5 < g x) \u2227\n  (\u222b x, f x \u2202\u03bc < \u222b x, ereal.to_real (g x) \u2202\u03bc + \u03b5) :=\nbegin\n  rcases exists_lt_lower_semicontinuous_integral_lt (\u03bb x, - f x) hf.neg \u03b5pos\n    with \u27e8g, g_lt_f, gcont, g_integrable, g_lt_top, gint\u27e9,\n  refine \u27e8\u03bb x, - g x, _, _, _, _, _\u27e9,\n  { exact  \u03bb x, ereal.neg_lt_iff_neg_lt.1 (by simpa only [ereal.coe_neg] using g_lt_f x) },\n  { exact ereal.continuous_neg.comp_lower_semicontinuous_antimono gcont\n      (\u03bb x y hxy, ereal.neg_le_neg_iff.2 hxy) },\n  { convert g_integrable.neg,\n    ext x,\n    simp },\n  { simpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top },\n  { simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint,\n    rw add_comm at gint,\n    simpa [integral_neg] using gint }\nend\n\nend measure_theory\n","avg_line_length":52.6735459662,"max_line_length":100,"alphanum_fraction":0.6700979519}
-{"size":17620,"ext":"lean","lang":"Lean","max_stars_count":332.0,"content":"\/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n-\/\nprelude\n\nnotation `Prop` := Sort 0\nnotation f ` $ `:1 a:0 := f a\n\nuniverses u v w\n\n\/--\nThe kernel definitional equality test (t =?= s) has special support for id_delta applications.\nIt implements the following rules\n\n   1)   (id_delta t) =?= t\n   2)   t =?= (id_delta t)\n   3)   (id_delta t) =?= s  IF (unfold_of t) =?= s\n   4)   t =?= id_delta s    IF t =?= (unfold_of s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use id_delta applications to address performance problems when type checking\nlemmas generated by the equation compiler.\n-\/\n@[inline] def id_delta {\u03b1 : Sort u} (a : \u03b1) : \u03b1 :=\na\n\n\/-- Gadget for optional parameter support. -\/\n@[reducible] def opt_param (\u03b1 : Sort u) (default : \u03b1) : Sort u :=\n\u03b1\n\n\/-- Gadget for marking output parameters in type classes. -\/\n@[reducible] def out_param (\u03b1 : Sort u) : Sort u := \u03b1\n\n\/-\n  id_rhs is an auxiliary declaration used in the equation compiler to address performance\n  issues when proving equational lemmas. The equation compiler uses it as a marker.\n-\/\nabbreviation id_rhs (\u03b1 : Sort u) (a : \u03b1) : \u03b1 := a\n\ninductive punit : Sort u\n| star : punit\n\n\/-- An abbreviation for `punit.{0}`, its most common instantiation.\n    This type should be preferred over `punit` where possible to avoid\n    unnecessary universe parameters. -\/\nabbreviation unit : Type := punit\n\n@[pattern] abbreviation unit.star : unit := punit.star\n\n\/--\nGadget for defining thunks, thunk parameters have special treatment.\nExample: given\n      def f (s : string) (t : thunk nat) : nat\nan application\n     f \"hello\" 10\n is converted into\n     f \"hello\" (\u03bb _, 10)\n-\/\n@[reducible] def thunk (\u03b1 : Type u) : Type u :=\nunit \u2192 \u03b1\n\ninductive true : Prop\n| intro : true\n\ninductive false : Prop\n\ninductive empty : Type\n\n\/--\nLogical not.\n\n`not P`, with notation `\u00ac P`, is the `Prop` which is true if and only if `P` is false. It is\ninternally represented as `P \u2192 false`, so one way to prove a goal `\u22a2 \u00ac P` is to use `intro h`,\nwhich gives you a new hypothesis `h : P` and the goal `\u22a2 false`.\n\nA hypothesis `h : \u00ac P` can be used in term mode as a function, so if `w : P` then `h w : false`.\n\nRelated mathlib tactic: `contrapose`.\n-\/\ndef not (a : Prop) := a \u2192 false\nprefix `\u00ac`:40 := not\n\ninductive eq {\u03b1 : Sort u} (a : \u03b1) : \u03b1 \u2192 Prop\n| refl [] : eq a\n\n\/-\nInitialize the quotient module, which effectively adds the following definitions:\n\nconstant quot {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Sort u\n\nconstant quot.mk {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) : quot r\n\nconstant quot.lift {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  (\u2200 a b : \u03b1, r a b \u2192 eq (f a) (f b)) \u2192 quot r \u2192 \u03b2\n\nconstant quot.ind {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : quot r \u2192 Prop} :\n  (\u2200 a : \u03b1, \u03b2 (quot.mk r a)) \u2192 \u2200 q : quot r, \u03b2 q\n\nAlso the reduction rule:\n\nquot.lift f _ (quot.mk a) ~~> f a\n\n-\/\ninit_quotient\n\n\/--\nHeterogeneous equality.\n\nIts purpose is to write down equalities between terms whose types are not definitionally equal.\nFor example, given `x : vector \u03b1 n` and `y : vector \u03b1 (0+n)`, `x = y` doesn't typecheck but `x == y` does.\n\nIf you have a goal `\u22a2 x == y`,\nyour first instinct should be to ask (either yourself, or on [zulip](https:\/\/leanprover.zulipchat.com\/))\nif something has gone wrong already.\nIf you really do need to follow this route,\nyou may find the lemmas `eq_rec_heq` and `eq_mpr_heq` useful.\n-\/\ninductive heq {\u03b1 : Sort u} (a : \u03b1) : \u03a0 {\u03b2 : Sort u}, \u03b2 \u2192 Prop\n| refl [] : heq a\n\nstructure prod (\u03b1 : Type u) (\u03b2 : Type v) :=\n(fst : \u03b1) (snd : \u03b2)\n\n\/-- Similar to `prod`, but \u03b1 and \u03b2 can be propositions.\n   We use this type internally to automatically generate the brec_on recursor. -\/\nstructure pprod (\u03b1 : Sort u) (\u03b2 : Sort v) :=\n(fst : \u03b1) (snd : \u03b2)\n\n\/--\nLogical and.\n\n`and P Q`, with notation `P \u2227 Q`, is the `Prop` which is true precisely when `P` and `Q` are\nboth true.\n\nTo prove a goal `\u22a2 P \u2227 Q`, you can use the tactic `split`,\nwhich gives two separate goals `\u22a2 P` and `\u22a2 Q`.\n\nGiven a hypothesis `h : P \u2227 Q`, you can use the tactic `cases h with hP hQ`\nto obtain two new hypotheses `hP : P` and `hQ : Q`. See also the `obtain` or `rcases` tactics in\nmathlib.\n-\/\nstructure and (a b : Prop) : Prop :=\nintro :: (left : a) (right : b)\n\nlemma and.elim_left {a b : Prop} (h : and a b) : a := h.1\n\nlemma and.elim_right {a b : Prop} (h : and a b) : b := h.2\n\n\/- eq basic support -\/\n\ninfix ` = `:50 := eq\n\nattribute [refl] eq.refl\n\n\/- This is a `def`, so that it can be used as pattern in the equation compiler. -\/\n@[pattern] def rfl {\u03b1 : Sort u} {a : \u03b1} : a = a := eq.refl a\n\n@[elab_as_eliminator, subst]\nlemma eq.subst {\u03b1 : Sort u} {P : \u03b1 \u2192 Prop} {a b : \u03b1} (h\u2081 : a = b) (h\u2082 : P a) : P b :=\neq.rec h\u2082 h\u2081\n\ninfixr ` \u25b8 `:75 := eq.subst\n\n@[trans] lemma eq.trans {\u03b1 : Sort u} {a b c : \u03b1} (h\u2081 : a = b) (h\u2082 : b = c) : a = c :=\nh\u2082 \u25b8 h\u2081\n\n@[symm] lemma eq.symm {\u03b1 : Sort u} {a b : \u03b1} (h : a = b) : b = a :=\nh \u25b8 rfl\n\ninfix ` == `:50 := heq\n\n\/- This is a `def`, so that it can be used as pattern in the equation compiler. -\/\n@[pattern] def heq.rfl {\u03b1 : Sort u} {a : \u03b1} : a == a := heq.refl a\n\nlemma eq_of_heq {\u03b1 : Sort u} {a a' : \u03b1} (h : a == a') : a = a' :=\nhave \u2200 (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a') (h\u2082 : \u03b1 = \u03b1'), (eq.rec_on h\u2082 a : \u03b1') = a', from\n  \u03bb (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a'), heq.rec_on h\u2081 (\u03bb h\u2082 : \u03b1 = \u03b1, rfl),\nshow (eq.rec_on (eq.refl \u03b1) a : \u03b1) = a', from\n  this \u03b1 a' h (eq.refl \u03b1)\n\n\/- The following four lemmas could not be automatically generated when the\n   structures were declared, so we prove them manually here. -\/\nlemma prod.mk.inj {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 and (x\u2081 = x\u2082) (y\u2081 = y\u2082) :=\n\u03bb h, prod.no_confusion h (\u03bb h\u2081 h\u2082, \u27e8h\u2081, h\u2082\u27e9)\n\nlemma prod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u03a0 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n\u03bb h\u2081 _ h\u2082, prod.no_confusion h\u2081 h\u2082\n\nlemma pprod.mk.inj {\u03b1 : Sort u} {\u03b2 : Sort v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : pprod.mk x\u2081 y\u2081 = pprod.mk x\u2082 y\u2082 \u2192 and (x\u2081 = x\u2082) (y\u2081 = y\u2082) :=\n\u03bb h, pprod.no_confusion h (\u03bb h\u2081 h\u2082, \u27e8h\u2081, h\u2082\u27e9)\n\nlemma pprod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u03a0 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n\u03bb h\u2081 _ h\u2082, prod.no_confusion h\u2081 h\u2082\n\ninductive sum (\u03b1 : Type u) (\u03b2 : Type v)\n| inl (val : \u03b1) : sum\n| inr (val : \u03b2) : sum\n\ninductive psum (\u03b1 : Sort u) (\u03b2 : Sort v)\n| inl (val : \u03b1) : psum\n| inr (val : \u03b2) : psum\n\n\/--\nLogical or.\n\n`or P Q`, with notation `P \u2228 Q`, is the proposition which is true if and only if `P` or `Q` is\ntrue.\n\nTo prove a goal `\u22a2 P \u2228 Q`, if you know which alternative you want to prove,\nyou can use the tactics `left` (which gives the goal `\u22a2 P`)\nor `right` (which gives the goal `\u22a2 Q`).\n\nGiven a hypothesis `h : P \u2228 Q` and goal `\u22a2 R`,\nthe tactic `cases h` will give you two copies of the goal `\u22a2 R`,\nwith the hypothesis `h : P` in the first, and the hypothesis `h : Q` in the second.\n-\/\ninductive or (a b : Prop) : Prop\n| inl (h : a) : or\n| inr (h : b) : or\n\nlemma or.intro_left {a : Prop} (b : Prop) (ha : a) : or a b :=\nor.inl ha\n\nlemma or.intro_right (a : Prop) {b : Prop} (hb : b) : or a b :=\nor.inr hb\n\nstructure sigma {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) :=\nmk :: (fst : \u03b1) (snd : \u03b2 fst)\n\nstructure psigma {\u03b1 : Sort u} (\u03b2 : \u03b1 \u2192 Sort v) :=\nmk :: (fst : \u03b1) (snd : \u03b2 fst)\n\ninductive bool : Type\n| ff : bool\n| tt : bool\n\n\/- Remark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description. -\/\nstructure subtype {\u03b1 : Sort u} (p : \u03b1 \u2192 Prop) :=\n(val : \u03b1) (property : p val)\n\nattribute [pp_using_anonymous_constructor] sigma psigma subtype pprod and\n\nclass inductive decidable (p : Prop)\n| is_false (h : \u00acp) : decidable\n| is_true  (h : p) : decidable\n\n@[reducible]\ndef decidable_pred {\u03b1 : Sort u} (r : \u03b1 \u2192 Prop) :=\n\u03a0 (a : \u03b1), decidable (r a)\n\n@[reducible]\ndef decidable_rel {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\n\u03a0 (a b : \u03b1), decidable (r a b)\n\n@[reducible]\ndef decidable_eq (\u03b1 : Sort u) :=\ndecidable_rel (@eq \u03b1)\n\ninductive option (\u03b1 : Type u)\n| none : option\n| some (val : \u03b1) : option\n\nexport option (none some)\nexport bool (ff tt)\n\ninductive list (T : Type u)\n| nil : list\n| cons (hd : T) (tl : list) : list\n\ninfixr ` :: `:67 := list.cons\nnotation `[` l:(foldr `, ` (h t, list.cons h t) list.nil `]`) := l\n\ninductive nat\n| zero : nat\n| succ (n : nat) : nat\n\nstructure unification_constraint :=\n{\u03b1 : Type u} (lhs : \u03b1) (rhs : \u03b1)\n\ninfix ` \u225f `:50   := unification_constraint.mk\ninfix ` =?= `:50 := unification_constraint.mk\n\nstructure unification_hint :=\n(pattern : unification_constraint)\n(constraints : list unification_constraint)\n\n\/- Declare builtin and reserved notation -\/\n\nclass has_zero     (\u03b1 : Type u) := (zero : \u03b1)\nclass has_one      (\u03b1 : Type u) := (one : \u03b1)\nclass has_add      (\u03b1 : Type u) := (add : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_mul      (\u03b1 : Type u) := (mul : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_inv      (\u03b1 : Type u) := (inv : \u03b1 \u2192 \u03b1)\nclass has_neg      (\u03b1 : Type u) := (neg : \u03b1 \u2192 \u03b1)\nclass has_sub      (\u03b1 : Type u) := (sub : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_div      (\u03b1 : Type u) := (div : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_dvd      (\u03b1 : Type u) := (dvd : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_mod      (\u03b1 : Type u) := (mod : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_le       (\u03b1 : Type u) := (le : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_lt       (\u03b1 : Type u) := (lt : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_append   (\u03b1 : Type u) := (append : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_andthen  (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : out_param $ Type w) := (andthen : \u03b1 \u2192 \u03b2 \u2192 \u03c3)\nclass has_union    (\u03b1 : Type u) := (union : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_inter    (\u03b1 : Type u) := (inter : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_sdiff    (\u03b1 : Type u) := (sdiff : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_equiv    (\u03b1 : Sort u) := (equiv : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_subset   (\u03b1 : Type u) := (subset : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_ssubset  (\u03b1 : Type u) := (ssubset : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\/- Type classes has_emptyc and has_insert are\n   used to implement polymorphic notation for collections.\n   Example: {a, b, c}. -\/\nclass has_emptyc   (\u03b1 : Type u) := (emptyc : \u03b1)\nclass has_insert   (\u03b1 : out_param $ Type u) (\u03b3 : Type v) := (insert : \u03b1 \u2192 \u03b3 \u2192 \u03b3)\nclass has_singleton (\u03b1 : out_param $ Type u) (\u03b2 : Type v) := (singleton : \u03b1 \u2192 \u03b2)\n\/- Type class used to implement the notation { a \u2208 c | p a } -\/\nclass has_sep (\u03b1 : out_param $ Type u) (\u03b3 : Type v) :=\n(sep : (\u03b1 \u2192 Prop) \u2192 \u03b3 \u2192 \u03b3)\n\/- Type class for set-like membership -\/\nclass has_mem (\u03b1 : out_param $ Type u) (\u03b3 : Type v) := (mem : \u03b1 \u2192 \u03b3 \u2192 Prop)\n\nclass has_pow (\u03b1 : Type u) (\u03b2 : Type v) :=\n(pow : \u03b1 \u2192 \u03b2 \u2192 \u03b1)\n\nexport has_andthen (andthen)\nexport has_pow (pow)\n\ninfix ` \u2208 `:50   := has_mem.mem\nnotation a ` \u2209 `:50 s:50 := \u00ac has_mem.mem a s\ninfixl ` + `:65  := has_add.add\ninfixl ` * `:70  := has_mul.mul\ninfixl ` - `:65  := has_sub.sub\ninfixl ` \/ `:70  := has_div.div\ninfix ` \u2223 `:50   := has_dvd.dvd -- Note this is different to `|`.\ninfixl ` % `:70  := has_mod.mod\nprefix `-`:75    := has_neg.neg\ninfix ` <= `:50  := has_le.le\ninfix ` \u2264 `:50   := has_le.le\ninfix ` < `:50   := has_lt.lt\ninfixl ` ++ `:65 := has_append.append\ninfixl `; `:1    := andthen\nnotation `\u2205`     := has_emptyc.emptyc\ninfixl ` \u222a `:65  := has_union.union\ninfixl ` \u2229 `:70  := has_inter.inter\ninfix ` \u2286 `:50   := has_subset.subset\ninfix ` \u2282 `:50   := has_ssubset.ssubset\ninfix ` \\ `:70   := has_sdiff.sdiff\ninfix ` \u2248 `:50   := has_equiv.equiv\ninfixr ` ^ `:80  := has_pow.pow\n\nexport has_append (append)\n\n@[reducible] def ge {\u03b1 : Type u} [has_le \u03b1] (a b : \u03b1) : Prop := has_le.le b a\n@[reducible] def gt {\u03b1 : Type u} [has_lt \u03b1] (a b : \u03b1) : Prop := has_lt.lt b a\n\ninfix ` >= `:50 := ge\ninfix ` \u2265 `:50  := ge\ninfix ` > `:50  := gt\n\n@[reducible] def superset {\u03b1 : Type u} [has_subset \u03b1] (a b : \u03b1) : Prop := has_subset.subset b a\n@[reducible] def ssuperset {\u03b1 : Type u} [has_ssubset \u03b1] (a b : \u03b1) : Prop := has_ssubset.ssubset b a\n\ninfix ` \u2287 `:50 := superset\ninfix ` \u2283 `:50 := ssuperset\n\ndef bit0 {\u03b1 : Type u} [s  : has_add \u03b1] (a  : \u03b1)                 : \u03b1 := a + a\ndef bit1 {\u03b1 : Type u} [s\u2081 : has_one \u03b1] [s\u2082 : has_add \u03b1] (a : \u03b1) : \u03b1 := (bit0 a) + 1\n\nattribute [pattern] has_zero.zero has_one.one bit0 bit1 has_add.add has_neg.neg has_mul.mul\n\nexport has_insert (insert)\n\nclass is_lawful_singleton (\u03b1 : Type u) (\u03b2 : Type v) [has_emptyc \u03b2] [has_insert \u03b1 \u03b2]\n  [has_singleton \u03b1 \u03b2] : Prop :=\n(insert_emptyc_eq : \u2200 (x : \u03b1), (insert x \u2205 : \u03b2) = {x})\n\nexport has_singleton (singleton)\nexport is_lawful_singleton (insert_emptyc_eq)\n\nattribute [simp] insert_emptyc_eq\n\n\/- nat basic instances -\/\n\nnamespace nat\n  protected def add : nat \u2192 nat \u2192 nat\n  | a  zero     := a\n  | a  (succ b) := succ (add a b)\n\n  \/- We mark the following definitions as pattern to make sure they can be used in recursive equations,\n     and reduced by the equation compiler. -\/\n  attribute [pattern] nat.add nat.add._main\nend nat\n\ninstance : has_zero nat := \u27e8nat.zero\u27e9\n\ninstance : has_one nat := \u27e8nat.succ (nat.zero)\u27e9\n\ninstance : has_add nat := \u27e8nat.add\u27e9\n\ndef std.priority.default : nat := 1000\ndef std.priority.max     : nat := 0xFFFFFFFF\n\nnamespace nat\n  protected def prio := std.priority.default + 100\nend nat\n\n\/-\n  Global declarations of right binding strength\n\n  If a module reassigns these, it will be incompatible with other modules that adhere to these\n  conventions.\n\n  When hovering over a symbol, use \"C-c C-k\" to see how to input it.\n-\/\ndef std.prec.max   : nat := 1024 -- the strength of application, identifiers, (, [, etc.\ndef std.prec.arrow : nat := 25\n\n\/-\nThe next def is \"max + 10\". It can be used e.g. for postfix operations that should\nbe stronger than application.\n-\/\n\ndef std.prec.max_plus : nat := std.prec.max + 10\n\npostfix `\u207b\u00b9`:std.prec.max_plus := has_inv.inv  -- input with \\sy or \\-1 or \\inv\n\ninfixr ` \u00d7 `:35 := prod\n-- notation for n-ary tuples\n\n\/- sizeof -\/\n\nclass has_sizeof (\u03b1 : Sort u) :=\n(sizeof : \u03b1 \u2192 nat)\n\ndef sizeof {\u03b1 : Sort u} [s : has_sizeof \u03b1] : \u03b1 \u2192 nat :=\nhas_sizeof.sizeof\n\n\/-\nDeclare sizeof instances and lemmas for types declared before has_sizeof.\nFrom now on, the inductive compiler will automatically generate sizeof instances and lemmas.\n-\/\n\n\/- Every type `\u03b1` has a default has_sizeof instance that just returns 0 for every element of `\u03b1` -\/\nprotected def default.sizeof (\u03b1 : Sort u) : \u03b1 \u2192 nat\n| a := 0\n\ninstance default_has_sizeof (\u03b1 : Sort u) : has_sizeof \u03b1 :=\n\u27e8default.sizeof \u03b1\u27e9\n\nprotected def nat.sizeof : nat \u2192 nat\n| n := n\n\ninstance : has_sizeof nat :=\n\u27e8nat.sizeof\u27e9\n\nprotected def prod.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (prod \u03b1 \u03b2) \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (prod \u03b1 \u03b2) :=\n\u27e8prod.sizeof\u27e9\n\nprotected def sum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (sum \u03b1 \u03b2) \u2192 nat\n| (sum.inl a) := 1 + sizeof a\n| (sum.inr b) := 1 + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (sum \u03b1 \u03b2) :=\n\u27e8sum.sizeof\u27e9\n\nprotected def psum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (psum \u03b1 \u03b2) \u2192 nat\n| (psum.inl a) := 1 + sizeof a\n| (psum.inr b) := 1 + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (psum \u03b1 \u03b2) :=\n\u27e8psum.sizeof\u27e9\n\nprotected def sigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : sigma \u03b2 \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : has_sizeof (sigma \u03b2) :=\n\u27e8sigma.sizeof\u27e9\n\nprotected def psigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : psigma \u03b2 \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : has_sizeof (psigma \u03b2) :=\n\u27e8psigma.sizeof\u27e9\n\nprotected def punit.sizeof : punit \u2192 nat\n| u := 1\n\ninstance : has_sizeof punit := \u27e8punit.sizeof\u27e9\n\nprotected def bool.sizeof : bool \u2192 nat\n| b := 1\n\ninstance : has_sizeof bool := \u27e8bool.sizeof\u27e9\n\nprotected def option.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] : option \u03b1 \u2192 nat\n| none     := 1\n| (some a) := 1 + sizeof a\n\ninstance (\u03b1 : Type u) [has_sizeof \u03b1] : has_sizeof (option \u03b1) :=\n\u27e8option.sizeof\u27e9\n\nprotected def list.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] : list \u03b1 \u2192 nat\n| list.nil        := 1\n| (list.cons a l) := 1 + sizeof a + list.sizeof l\n\ninstance (\u03b1 : Type u) [has_sizeof \u03b1] : has_sizeof (list \u03b1) :=\n\u27e8list.sizeof\u27e9\n\nprotected def subtype.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] {p : \u03b1 \u2192 Prop} : subtype p \u2192 nat\n| \u27e8a, _\u27e9 := sizeof a\n\ninstance {\u03b1 : Type u} [has_sizeof \u03b1] (p : \u03b1 \u2192 Prop) : has_sizeof (subtype p) :=\n\u27e8subtype.sizeof\u27e9\n\nlemma nat_add_zero (n : nat) : n + 0 = n := rfl\n\n\/- Combinator calculus -\/\nnamespace combinator\nuniverses u\u2081 u\u2082 u\u2083\ndef I {\u03b1 : Type u\u2081} (a : \u03b1) := a\ndef K {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} (a : \u03b1) (b : \u03b2) := a\ndef S {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} {\u03b3 : Type u\u2083} (x : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (y : \u03b1 \u2192 \u03b2) (z : \u03b1) := x z (y z)\nend combinator\n\n\/-- Auxiliary datatype for #[ ... ] notation.\n    #[1, 2, 3, 4] is notation for\n\n    bin_tree.node\n      (bin_tree.node (bin_tree.leaf 1) (bin_tree.leaf 2))\n      (bin_tree.node (bin_tree.leaf 3) (bin_tree.leaf 4))\n\n    We use this notation to input long sequences without exhausting the system stack space.\n    Later, we define a coercion from `bin_tree` into `list`.\n-\/\ninductive bin_tree (\u03b1 : Type u)\n| empty : bin_tree\n| leaf (val : \u03b1) : bin_tree\n| node (left right : bin_tree) : bin_tree\n\nattribute [elab_simple] bin_tree.node bin_tree.leaf\n\n\/-- Like `by apply_instance`, but not dependent on the tactic framework. -\/\n@[reducible] def infer_instance {\u03b1 : Sort u} [i : \u03b1] : \u03b1 := i\n","avg_line_length":31.7477477477,"max_line_length":113,"alphanum_fraction":0.6150964813}
-{"size":3986,"ext":"lean","lang":"Lean","max_stars_count":1131.0,"content":"\/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-\/\n\n\/-!\n# Inductive type variant of `fin`\n\n`fin` is defined as a subtype of `\u2115`. This file defines an equivalent type, `fin2`, which is\ndefined inductively. This is useful for its induction principle and different definitional\nequalities.\n\n## Main declarations\n\n* `fin2 n`: Inductive type variant of `fin n`. `fz` corresponds to `0` and `fs n` corresponds to\n  `n`.\n* `to_nat`, `opt_of_nat`, `of_nat'`: Conversions to and from `\u2115`. `of_nat' m` takes a proof that\n  `m < n` through the class `is_lt`.\n* `add k`: Takes `i : fin2 n` to `i + k : fin2 (n + k)`.\n* `left`: Embeds `fin2 n` into `fin2 (n + k)`.\n* `insert_perm a`: Permutation of `fin2 n` which cycles `0, ..., a - 1` and leaves `a, ..., n - 1`\n  unchanged.\n* `remap_left f`: Function `fin2 (m + k) \u2192 fin2 (n + k)` by applying `f : fin m \u2192 fin n` to\n  `0, ..., m - 1` and sending `m + i` to `n + i`.\n-\/\n\nopen nat\nuniverses u\n\n\/-- An alternate definition of `fin n` defined as an inductive type instead of a subtype of `\u2115`. -\/\ninductive fin2 : \u2115 \u2192 Type\n\/-- `0` as a member of `fin (succ n)` (`fin 0` is empty) -\/\n| fz {n} : fin2 (succ n)\n\/-- `n` as a member of `fin (succ n)` -\/\n| fs {n} : fin2 n \u2192 fin2 (succ n)\n\nnamespace fin2\n\n\/-- Define a dependent function on `fin2 (succ n)` by giving its value at\nzero (`H1`) and by giving a dependent function on the rest (`H2`). -\/\n@[elab_as_eliminator]\nprotected def cases' {n} {C : fin2 (succ n) \u2192 Sort u} (H1 : C fz) (H2 : \u03a0 n, C (fs n)) :\n  \u03a0 (i : fin2 (succ n)), C i\n| fz     := H1\n| (fs n) := H2 n\n\n\/-- Ex falso. The dependent eliminator for the empty `fin2 0` type. -\/\ndef elim0 {C : fin2 0 \u2192 Sort u} : \u03a0 (i : fin2 0), C i.\n\n\/-- Converts a `fin2` into a natural. -\/\ndef to_nat : \u03a0 {n}, fin2 n \u2192 \u2115\n| ._ (@fz n)   := 0\n| ._ (@fs n i) := succ (to_nat i)\n\n\/-- Converts a natural into a `fin2` if it is in range -\/\ndef opt_of_nat : \u03a0 {n} (k : \u2115), option (fin2 n)\n| 0 _ := none\n| (succ n) 0 := some fz\n| (succ n) (succ k) := fs <$> @opt_of_nat n k\n\n\/-- `i + k : fin2 (n + k)` when `i : fin2 n` and `k : \u2115` -\/\ndef add {n} (i : fin2 n) : \u03a0 k, fin2 (n + k)\n| 0        := i\n| (succ k) := fs (add k)\n\n\/-- `left k` is the embedding `fin2 n \u2192 fin2 (k + n)` -\/\ndef left (k) : \u03a0 {n}, fin2 n \u2192 fin2 (k + n)\n| ._ (@fz n)   := fz\n| ._ (@fs n i) := fs (left i)\n\n\/-- `insert_perm a` is a permutation of `fin2 n` with the following properties:\n  * `insert_perm a i = i+1` if `i < a`\n  * `insert_perm a a = 0`\n  * `insert_perm a i = i` if `i > a` -\/\ndef insert_perm : \u03a0 {n}, fin2 n \u2192 fin2 n \u2192 fin2 n\n| ._ (@fz n)          (@fz ._)   := fz\n| ._ (@fz n)          (@fs ._ j) := fs j\n| ._ (@fs (succ n) i) (@fz ._)   := fs fz\n| ._ (@fs (succ n) i) (@fs ._ j) := match insert_perm i j with fz := fz | fs k := fs (fs k) end\n\n\/-- `remap_left f k : fin2 (m + k) \u2192 fin2 (n + k)` applies the function\n  `f : fin2 m \u2192 fin2 n` to inputs less than `m`, and leaves the right part\n  on the right (that is, `remap_left f k (m + i) = n + i`). -\/\ndef remap_left {m n} (f : fin2 m \u2192 fin2 n) : \u03a0 k, fin2 (m + k) \u2192 fin2 (n + k)\n| 0        i          := f i\n| (succ k) (@fz ._)   := fz\n| (succ k) (@fs ._ i) := fs (remap_left _ i)\n\n\/-- This is a simple type class inference prover for proof obligations\n  of the form `m < n` where `m n : \u2115`. -\/\nclass is_lt (m n : \u2115) := (h : m < n)\ninstance is_lt.zero (n) : is_lt 0 (succ n) := \u27e8succ_pos _\u27e9\ninstance is_lt.succ (m n) [l : is_lt m n] : is_lt (succ m) (succ n) := \u27e8succ_lt_succ l.h\u27e9\n\n\/-- Use type class inference to infer the boundedness proof, so that we can directly convert a\n`nat` into a `fin2 n`. This supports notation like `&1 : fin 3`. -\/\ndef of_nat' : \u03a0 {n} m [is_lt m n], fin2 n\n| 0        m        \u27e8h\u27e9 := absurd h (nat.not_lt_zero _)\n| (succ n) 0        \u27e8h\u27e9 := fz\n| (succ n) (succ m) \u27e8h\u27e9 := fs (@of_nat' n m \u27e8lt_of_succ_lt_succ h\u27e9)\n\nlocal prefix `&`:max := of_nat'\n\ninstance : inhabited (fin2 1) := \u27e8fz\u27e9\n\nend fin2\n","avg_line_length":36.9074074074,"max_line_length":99,"alphanum_fraction":0.5747616658}
-{"size":26579,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Callum Sutton, Yury Kudryashov\n-\/\nimport algebra.group.type_tags\nimport algebra.group_with_zero.basic\nimport data.pi\n\n\/-!\n# Multiplicative and additive equivs\n\nIn this file we define two extensions of `equiv` called `add_equiv` and `mul_equiv`, which are\ndatatypes representing isomorphisms of `add_monoid`s\/`add_group`s and `monoid`s\/`group`s.\n\n## Notations\n\n* ``infix ` \u2243* `:25 := mul_equiv``\n* ``infix ` \u2243+ `:25 := add_equiv``\n\nThe extended equivs all have coercions to functions, and the coercions are the canonical\nnotation when treating the isomorphisms as maps.\n\n## Implementation notes\n\nThe fields for `mul_equiv`, `add_equiv` now avoid the unbundled `is_mul_hom` and `is_add_hom`, as\nthese are deprecated.\n\n## Tags\n\nequiv, mul_equiv, add_equiv\n-\/\n\nvariables {A : Type*} {B : Type*} {M : Type*} {N : Type*}\n  {P : Type*} {Q : Type*} {G : Type*} {H : Type*}\n\n\/-- Makes a multiplicative inverse from a bijection which preserves multiplication. -\/\n@[to_additive \"Makes an additive inverse from a bijection which preserves addition.\"]\ndef mul_hom.inverse [has_mul M] [has_mul N] (f : mul_hom M N) (g : N \u2192 M)\n  (h\u2081 : function.left_inverse g f) (h\u2082 : function.right_inverse g f) : mul_hom N M :=\n{ to_fun   := g,\n  map_mul' := \u03bb x y,\n    calc g (x * y) = g (f (g x) * f (g y)) : by rw [h\u2082 x, h\u2082 y]\n               ... = g (f (g x * g y)) : by rw f.map_mul\n               ... = g x * g y : h\u2081 _, }\n\n\/-- The inverse of a bijective `monoid_hom` is a `monoid_hom`. -\/\n@[to_additive \"The inverse of a bijective `add_monoid_hom` is an `add_monoid_hom`.\", simps]\ndef monoid_hom.inverse {A B : Type*} [monoid A] [monoid B] (f : A \u2192* B) (g : B \u2192 A)\n  (h\u2081 : function.left_inverse g f) (h\u2082 : function.right_inverse g f) :\n  B \u2192* A :=\n{ to_fun   := g,\n  map_one' := by rw [\u2190 f.map_one, h\u2081],\n  .. (f : mul_hom A B).inverse g h\u2081 h\u2082, }\n\nset_option old_structure_cmd true\n\n\/-- add_equiv \u03b1 \u03b2 is the type of an equiv \u03b1 \u2243 \u03b2 which preserves addition. -\/\n@[ancestor equiv add_hom]\nstructure add_equiv (A B : Type*) [has_add A] [has_add B] extends A \u2243 B, add_hom A B\n\n\/-- `add_equiv_class F A B` states that `F` is a type of addition-preserving morphisms.\nYou should extend this class when you extend `add_equiv`. -\/\nclass add_equiv_class (F A B : Type*) [has_add A] [has_add B]\n  extends equiv_like F A B :=\n(map_add : \u2200 (f : F) a b, f (a + b) = f a + f b)\n\n\/-- The `equiv` underlying an `add_equiv`. -\/\nadd_decl_doc add_equiv.to_equiv\n\/-- The `add_hom` underlying a `add_equiv`. -\/\nadd_decl_doc add_equiv.to_add_hom\n\n\/-- `mul_equiv \u03b1 \u03b2` is the type of an equiv `\u03b1 \u2243 \u03b2` which preserves multiplication. -\/\n@[ancestor equiv mul_hom, to_additive]\nstructure mul_equiv (M N : Type*) [has_mul M] [has_mul N] extends M \u2243 N, mul_hom M N\n\n\/-- The `equiv` underlying a `mul_equiv`. -\/\nadd_decl_doc mul_equiv.to_equiv\n\/-- The `mul_hom` underlying a `mul_equiv`. -\/\nadd_decl_doc mul_equiv.to_mul_hom\n\n\/-- `mul_equiv_class F A B` states that `F` is a type of multiplication-preserving morphisms.\nYou should extend this class when you extend `mul_equiv`. -\/\n@[to_additive]\nclass mul_equiv_class (F A B : Type*) [has_mul A] [has_mul B]\n  extends equiv_like F A B :=\n(map_mul : \u2200 (f : F) a b, f (a * b) = f a * f b)\n\ninfix ` \u2243* `:25 := mul_equiv\ninfix ` \u2243+ `:25 := add_equiv\n\nsection mul_equiv_class\n\nvariables (F : Type*)\n\nnamespace mul_equiv_class\n\n@[priority 100, -- See note [lower instance priority]\n  to_additive]\ninstance [has_mul M] [has_mul N] [h : mul_equiv_class F M N] : mul_hom_class F M N :=\n{ coe := (coe : F \u2192 M \u2192 N),\n  coe_injective' := @fun_like.coe_injective F _ _ _,\n  .. h }\n\n@[priority 100, -- See note [lower instance priority]\n  to_additive]\ninstance [mul_one_class M] [mul_one_class N] [mul_equiv_class F M N] :\n  monoid_hom_class F M N :=\n{ coe := (coe : F \u2192 M \u2192 N),\n  map_one := \u03bb e,\n  calc e 1 = e 1 * 1 : (mul_one _).symm\n       ... = e 1 * e (inv e (1 : N) : M) : congr_arg _ (right_inv e 1).symm\n       ... = e (inv e (1 : N)) : by rw [\u2190 map_mul, one_mul]\n       ... = 1 : right_inv e 1,\n  .. mul_equiv_class.mul_hom_class F }\n\nvariables {F}\n\n@[simp, to_additive]\nlemma map_eq_one_iff {M N} [mul_one_class M] [mul_one_class N] [mul_equiv_class F M N]\n  (h : F) {x : M} :\n  h x = 1 \u2194 x = 1 :=\nby rw [\u2190 map_one h, equiv_like.apply_eq_iff_eq h]\n\n@[to_additive]\nlemma map_ne_one_iff {M N} [mul_one_class M] [mul_one_class N] [mul_equiv_class F M N]\n  (h : F) {x : M} :\n  h x \u2260 1 \u2194 x \u2260 1 :=\nnot_congr (map_eq_one_iff h)\n\nend mul_equiv_class\n\nend mul_equiv_class\n\nnamespace mul_equiv\n\n@[to_additive]\ninstance [has_mul M] [has_mul N] : has_coe_to_fun (M \u2243* N) (\u03bb _, M \u2192 N) := \u27e8mul_equiv.to_fun\u27e9\n\n@[to_additive]\ninstance [has_mul M] [has_mul N] : mul_equiv_class (M \u2243* N) M N :=\n{ coe := to_fun, inv := inv_fun, left_inv := left_inv, right_inv := right_inv,\n  coe_injective' := \u03bb f g h\u2081 h\u2082, by { cases f, cases g, congr' },\n  map_mul := map_mul' }\n\nvariables [has_mul M] [has_mul N] [has_mul P] [has_mul Q]\n\n@[simp, to_additive]\nlemma to_fun_eq_coe {f : M \u2243* N} : f.to_fun = f := rfl\n\n@[simp, to_additive]\nlemma coe_to_equiv {f : M \u2243* N} : \u21d1f.to_equiv = f := rfl\n\n@[simp, to_additive]\nlemma coe_to_mul_hom {f : M \u2243* N} : \u21d1f.to_mul_hom = f := rfl\n\n\/-- A multiplicative isomorphism preserves multiplication (canonical form). -\/\n@[to_additive]\nprotected lemma map_mul (f : M \u2243* N) : \u2200 x y, f (x * y) = f x * f y := map_mul f\n\n\/-- Makes a multiplicative isomorphism from a bijection which preserves multiplication. -\/\n@[to_additive \"Makes an additive isomorphism from a bijection which preserves addition.\"]\ndef mk' (f : M \u2243 N) (h : \u2200 x y, f (x * y) = f x * f y) : M \u2243* N :=\n\u27e8f.1, f.2, f.3, f.4, h\u27e9\n\n@[to_additive]\nprotected lemma bijective (e : M \u2243* N) : function.bijective e := equiv_like.bijective e\n\n@[to_additive]\nprotected lemma injective (e : M \u2243* N) : function.injective e := equiv_like.injective e\n\n@[to_additive]\nprotected lemma surjective (e : M \u2243* N) : function.surjective e := equiv_like.surjective e\n\n\/-- The identity map is a multiplicative isomorphism. -\/\n@[refl, to_additive \"The identity map is an additive isomorphism.\"]\ndef refl (M : Type*) [has_mul M] : M \u2243* M :=\n{ map_mul' := \u03bb _ _, rfl,\n..equiv.refl _}\n\n@[to_additive]\ninstance : inhabited (M \u2243* M) := \u27e8refl M\u27e9\n\n\/-- The inverse of an isomorphism is an isomorphism. -\/\n@[symm, to_additive \"The inverse of an isomorphism is an isomorphism.\"]\ndef symm (h : M \u2243* N) : N \u2243* M :=\n{ map_mul' := (h.to_mul_hom.inverse h.to_equiv.symm h.left_inv h.right_inv).map_mul,\n  .. h.to_equiv.symm}\n\n@[simp, to_additive]\nlemma inv_fun_eq_symm {f : M \u2243* N} : f.inv_fun = f.symm := rfl\n\n\/-- See Note [custom simps projection] -\/\n-- we don't hyperlink the note in the additive version, since that breaks syntax highlighting\n-- in the whole file.\n@[to_additive \"See Note custom simps projection\"]\ndef simps.symm_apply (e : M \u2243* N) : N \u2192 M := e.symm\n\ninitialize_simps_projections add_equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\ninitialize_simps_projections mul_equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\n\n@[simp, to_additive]\ntheorem to_equiv_symm (f : M \u2243* N) : f.symm.to_equiv = f.to_equiv.symm := rfl\n\n@[simp, to_additive]\ntheorem coe_mk (f : M \u2192 N) (g h\u2081 h\u2082 h\u2083) : \u21d1(mul_equiv.mk f g h\u2081 h\u2082 h\u2083) = f := rfl\n\n@[simp, to_additive]\nlemma to_equiv_mk (f : M \u2192 N) (g : N \u2192 M) (h\u2081 h\u2082 h\u2083) :\n  (mk f g h\u2081 h\u2082 h\u2083).to_equiv = \u27e8f, g, h\u2081, h\u2082\u27e9 := rfl\n\n@[simp, to_additive]\nlemma symm_symm : \u2200 (f : M \u2243* N), f.symm.symm = f\n| \u27e8f, g, h\u2081, h\u2082, h\u2083\u27e9 := rfl\n\n@[to_additive]\nlemma symm_bijective : function.bijective (symm : (M \u2243* N) \u2192 (N \u2243* M)) :=\nequiv.bijective \u27e8symm, symm, symm_symm, symm_symm\u27e9\n\n@[simp, to_additive]\ntheorem symm_mk (f : M \u2192 N) (g h\u2081 h\u2082 h\u2083) :\n  (mul_equiv.mk f g h\u2081 h\u2082 h\u2083).symm =\n  { to_fun := g, inv_fun := f, ..(mul_equiv.mk f g h\u2081 h\u2082 h\u2083).symm} := rfl\n\n\/-- Transitivity of multiplication-preserving isomorphisms -\/\n@[trans, to_additive \"Transitivity of addition-preserving isomorphisms\"]\ndef trans (h1 : M \u2243* N) (h2 : N \u2243* P) : (M \u2243* P) :=\n{ map_mul' := \u03bb x y, show h2 (h1 (x * y)) = h2 (h1 x) * h2 (h1 y),\n    by rw [h1.map_mul, h2.map_mul],\n  ..h1.to_equiv.trans h2.to_equiv }\n\n\/-- e.right_inv in canonical form -\/\n@[simp, to_additive]\nlemma apply_symm_apply (e : M \u2243* N) : \u2200 y, e (e.symm y) = y :=\ne.to_equiv.apply_symm_apply\n\n\/-- e.left_inv in canonical form -\/\n@[simp, to_additive]\nlemma symm_apply_apply (e : M \u2243* N) : \u2200 x, e.symm (e x) = x :=\ne.to_equiv.symm_apply_apply\n\n@[simp, to_additive]\ntheorem symm_comp_self (e : M \u2243* N) : e.symm \u2218 e = id := funext e.symm_apply_apply\n\n@[simp, to_additive]\ntheorem self_comp_symm (e : M \u2243* N) : e \u2218 e.symm = id := funext e.apply_symm_apply\n\n@[simp, to_additive]\ntheorem coe_refl : \u21d1(refl M) = id := rfl\n\n@[to_additive]\ntheorem refl_apply (m : M) : refl M m = m := rfl\n\n@[simp, to_additive]\ntheorem coe_trans (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) : \u21d1(e\u2081.trans e\u2082) = e\u2082 \u2218 e\u2081 := rfl\n\n@[to_additive]\ntheorem trans_apply (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) (m : M) : e\u2081.trans e\u2082 m = e\u2082 (e\u2081 m) := rfl\n\n@[simp, to_additive] theorem symm_trans_apply (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) (p : P) :\n  (e\u2081.trans e\u2082).symm p = e\u2081.symm (e\u2082.symm p) := rfl\n\n@[simp, to_additive] theorem apply_eq_iff_eq (e : M \u2243* N) {x y : M} : e x = e y \u2194 x = y :=\ne.injective.eq_iff\n\n@[to_additive]\nlemma apply_eq_iff_symm_apply (e : M \u2243* N) {x : M} {y : N} : e x = y \u2194 x = e.symm y :=\ne.to_equiv.apply_eq_iff_eq_symm_apply\n\n@[to_additive]\nlemma symm_apply_eq (e : M \u2243* N) {x y} : e.symm x = y \u2194 x = e y :=\ne.to_equiv.symm_apply_eq\n\n@[to_additive]\nlemma eq_symm_apply (e : M \u2243* N) {x y} : y = e.symm x \u2194 e y = x :=\ne.to_equiv.eq_symm_apply\n\n\/-- Two multiplicative isomorphisms agree if they are defined by the\n    same underlying function. -\/\n@[ext, to_additive\n  \"Two additive isomorphisms agree if they are defined by the same underlying function.\"]\nlemma ext {f g : mul_equiv M N} (h : \u2200 x, f x = g x) : f = g := fun_like.ext f g h\n\n@[to_additive] lemma ext_iff {f g : mul_equiv M N} : f = g \u2194 \u2200 x, f x = g x := fun_like.ext_iff\n\n@[simp, to_additive] lemma mk_coe (e : M \u2243* N) (e' h\u2081 h\u2082 h\u2083) :\n  (\u27e8e, e', h\u2081, h\u2082, h\u2083\u27e9 : M \u2243* N) = e := ext $ \u03bb _, rfl\n\n@[simp, to_additive] lemma mk_coe' (e : M \u2243* N) (f h\u2081 h\u2082 h\u2083) :\n  (mul_equiv.mk f \u21d1e h\u2081 h\u2082 h\u2083 : N \u2243* M) = e.symm :=\nsymm_bijective.injective $ ext $ \u03bb x, rfl\n\n@[to_additive] protected lemma congr_arg {f : mul_equiv M N} {x x' : M} : x = x' \u2192 f x = f x' :=\nfun_like.congr_arg f\n\n@[to_additive] protected lemma congr_fun {f g : mul_equiv M N} (h : f = g) (x : M) : f x = g x :=\nfun_like.congr_fun h x\n\n\/-- The `mul_equiv` between two monoids with a unique element. -\/\n@[to_additive \"The `add_equiv` between two add_monoids with a unique element.\"]\ndef mul_equiv_of_unique_of_unique {M N}\n  [unique M] [unique N] [has_mul M] [has_mul N] : M \u2243* N :=\n{ map_mul' := \u03bb _ _, subsingleton.elim _ _,\n  ..equiv_of_unique_of_unique }\n\n\/-- There is a unique monoid homomorphism between two monoids with a unique element. -\/\n@[to_additive] instance {M N} [unique M] [unique N] [has_mul M] [has_mul N] : unique (M \u2243* N) :=\n{ default := mul_equiv_of_unique_of_unique ,\n  uniq := \u03bb _, ext $ \u03bb x, subsingleton.elim _ _}\n\n\/-!\n## Monoids\n-\/\n\n\/-- A multiplicative equiv of monoids sends 1 to 1 (and is hence a monoid isomorphism). -\/\n@[to_additive]\nprotected lemma map_one {M N} [mul_one_class M] [mul_one_class N] (h : M \u2243* N) : h 1 = 1 :=\nmap_one h\n\n@[to_additive]\nprotected lemma map_eq_one_iff {M N} [mul_one_class M] [mul_one_class N] (h : M \u2243* N) {x : M} :\n  h x = 1 \u2194 x = 1 :=\nmul_equiv_class.map_eq_one_iff h\n\n@[to_additive]\nlemma map_ne_one_iff {M N} [mul_one_class M] [mul_one_class N] (h : M \u2243* N) {x : M} :\n  h x \u2260 1 \u2194 x \u2260 1 :=\nmul_equiv_class.map_ne_one_iff h\n\n\/-- A bijective `monoid` homomorphism is an isomorphism -\/\n@[to_additive \"A bijective `add_monoid` homomorphism is an isomorphism\"]\nnoncomputable def of_bijective {M N} [mul_one_class M] [mul_one_class N] (f : M \u2192* N)\n  (hf : function.bijective f) : M \u2243* N :=\n{ map_mul' := f.map_mul',\n  ..equiv.of_bijective f hf }\n\n\/--\nExtract the forward direction of a multiplicative equivalence\nas a multiplication-preserving function.\n-\/\n@[to_additive \"Extract the forward direction of an additive equivalence\nas an addition-preserving function.\"]\ndef to_monoid_hom {M N} [mul_one_class M] [mul_one_class N] (h : M \u2243* N) : (M \u2192* N) :=\n{ map_one' := h.map_one, .. h }\n\n@[simp, to_additive]\nlemma coe_to_monoid_hom {M N} [mul_one_class M] [mul_one_class N] (e : M \u2243* N) :\n  \u21d1e.to_monoid_hom = e :=\nrfl\n\n@[to_additive] lemma to_monoid_hom_injective {M N} [mul_one_class M] [mul_one_class N] :\n  function.injective (to_monoid_hom : (M \u2243* N) \u2192 M \u2192* N) :=\n\u03bb f g h, mul_equiv.ext (monoid_hom.ext_iff.1 h)\n\n\n\/--\nA multiplicative analogue of `equiv.arrow_congr`,\nwhere the equivalence between the targets is multiplicative.\n-\/\n@[to_additive \"An additive analogue of `equiv.arrow_congr`,\nwhere the equivalence between the targets is additive.\", simps apply]\ndef arrow_congr {M N P Q : Type*} [mul_one_class P] [mul_one_class Q]\n  (f : M \u2243 N) (g : P \u2243* Q) : (M \u2192 P) \u2243* (N \u2192 Q) :=\n{ to_fun := \u03bb h n, g (h (f.symm n)),\n  inv_fun := \u03bb k m, g.symm (k (f m)),\n  left_inv := \u03bb h, by { ext, simp, },\n  right_inv := \u03bb k, by { ext, simp, },\n  map_mul' := \u03bb h k, by { ext, simp, }, }\n\n\/--\nA multiplicative analogue of `equiv.arrow_congr`,\nfor multiplicative maps from a monoid to a commutative monoid.\n-\/\n@[to_additive \"An additive analogue of `equiv.arrow_congr`,\nfor additive maps from an additive monoid to a commutative additive monoid.\", simps apply]\ndef monoid_hom_congr {M N P Q} [mul_one_class M] [mul_one_class N] [comm_monoid P] [comm_monoid Q]\n  (f : M \u2243* N) (g : P \u2243* Q) : (M \u2192* P) \u2243* (N \u2192* Q) :=\n{ to_fun := \u03bb h,\n  g.to_monoid_hom.comp (h.comp f.symm.to_monoid_hom),\n  inv_fun := \u03bb k,\n  g.symm.to_monoid_hom.comp (k.comp f.to_monoid_hom),\n  left_inv := \u03bb h, by { ext, simp, },\n  right_inv := \u03bb k, by { ext, simp, },\n  map_mul' := \u03bb h k, by { ext, simp, }, }\n\n\/-- A family of multiplicative equivalences `\u03a0 j, (Ms j \u2243* Ns j)` generates a\nmultiplicative equivalence between `\u03a0 j, Ms j` and `\u03a0 j, Ns j`.\n\nThis is the `mul_equiv` version of `equiv.Pi_congr_right`, and the dependent version of\n`mul_equiv.arrow_congr`.\n-\/\n@[to_additive add_equiv.Pi_congr_right \"A family of additive equivalences `\u03a0 j, (Ms j \u2243+ Ns j)`\ngenerates an additive equivalence between `\u03a0 j, Ms j` and `\u03a0 j, Ns j`.\n\nThis is the `add_equiv` version of `equiv.Pi_congr_right`, and the dependent version of\n`add_equiv.arrow_congr`.\", simps apply]\ndef Pi_congr_right {\u03b7 : Type*}\n  {Ms Ns : \u03b7 \u2192 Type*} [\u03a0 j, mul_one_class (Ms j)] [\u03a0 j, mul_one_class (Ns j)]\n  (es : \u2200 j, Ms j \u2243* Ns j) : (\u03a0 j, Ms j) \u2243* (\u03a0 j, Ns j) :=\n{ to_fun := \u03bb x j, es j (x j),\n  inv_fun := \u03bb x j, (es j).symm (x j),\n  map_mul' := \u03bb x y, funext $ \u03bb j, (es j).map_mul (x j) (y j),\n  .. equiv.Pi_congr_right (\u03bb j, (es j).to_equiv) }\n\n@[simp]\nlemma Pi_congr_right_refl {\u03b7 : Type*} {Ms : \u03b7 \u2192 Type*} [\u03a0 j, mul_one_class (Ms j)] :\n  Pi_congr_right (\u03bb j, mul_equiv.refl (Ms j)) = mul_equiv.refl _ := rfl\n\n@[simp]\nlemma Pi_congr_right_symm {\u03b7 : Type*}\n  {Ms Ns : \u03b7 \u2192 Type*} [\u03a0 j, mul_one_class (Ms j)] [\u03a0 j, mul_one_class (Ns j)]\n  (es : \u2200 j, Ms j \u2243* Ns j) : (Pi_congr_right es).symm = (Pi_congr_right $ \u03bb i, (es i).symm) := rfl\n\n@[simp]\nlemma Pi_congr_right_trans {\u03b7 : Type*}\n  {Ms Ns Ps : \u03b7 \u2192 Type*} [\u03a0 j, mul_one_class (Ms j)] [\u03a0 j, mul_one_class (Ns j)]\n  [\u03a0 j, mul_one_class (Ps j)]\n  (es : \u2200 j, Ms j \u2243* Ns j) (fs : \u2200 j, Ns j \u2243* Ps j) :\n  (Pi_congr_right es).trans (Pi_congr_right fs) = (Pi_congr_right $ \u03bb i, (es i).trans (fs i)) := rfl\n\n\/-- A family indexed by a nonempty subsingleton type is equivalent to the element at the single\nindex. -\/\n@[to_additive add_equiv.Pi_subsingleton \"A family indexed by a nonempty subsingleton type is\nequivalent to the element at the single index.\", simps]\ndef Pi_subsingleton\n  {\u03b9 : Type*} (M : \u03b9 \u2192 Type*) [\u03a0 j, has_mul (M j)] [subsingleton \u03b9] (i : \u03b9) :\n  (\u03a0 j, M j) \u2243* M i :=\n{ map_mul' := \u03bb f1 f2, pi.mul_apply _ _ _, ..equiv.Pi_subsingleton M i }\n\n\/-!\n# Groups\n-\/\n\n\/-- A multiplicative equivalence of groups preserves inversion. -\/\n@[to_additive]\nprotected lemma map_inv [group G] [group H] (h : G \u2243* H) (x : G) : h x\u207b\u00b9 = (h x)\u207b\u00b9 :=\nmap_inv h x\n\n\/-- A multiplicative equivalence of groups preserves division. -\/\n@[to_additive]\nprotected lemma map_div [group G] [group H] (h : G \u2243* H) (x y : G) : h (x \/ y) = h x \/ h y :=\nmap_div h x y\n\nend mul_equiv\n\n\/-- Given a pair of monoid homomorphisms `f`, `g` such that `g.comp f = id` and `f.comp g = id`,\nreturns an multiplicative equivalence with `to_fun = f` and `inv_fun = g`.  This constructor is\nuseful if the underlying type(s) have specialized `ext` lemmas for monoid homomorphisms. -\/\n@[to_additive \/-\"Given a pair of additive monoid homomorphisms `f`, `g` such that `g.comp f = id`\nand `f.comp g = id`, returns an additive equivalence with `to_fun = f` and `inv_fun = g`.  This\nconstructor is useful if the underlying type(s) have specialized `ext` lemmas for additive\nmonoid homomorphisms.\"-\/, simps {fully_applied := ff}]\ndef monoid_hom.to_mul_equiv [mul_one_class M] [mul_one_class N] (f : M \u2192* N) (g : N \u2192* M)\n  (h\u2081 : g.comp f = monoid_hom.id _) (h\u2082 : f.comp g = monoid_hom.id _) :\n  M \u2243* N :=\n{ to_fun := f,\n  inv_fun := g,\n  left_inv := monoid_hom.congr_fun h\u2081,\n  right_inv := monoid_hom.congr_fun h\u2082,\n  map_mul' := f.map_mul }\n\n\/-- A group is isomorphic to its group of units. -\/\n@[to_additive to_add_units \"An additive group is isomorphic to its group of additive units\"]\ndef to_units [group G] : G \u2243* G\u02e3 :=\n{ to_fun := \u03bb x, \u27e8x, x\u207b\u00b9, mul_inv_self _, inv_mul_self _\u27e9,\n  inv_fun := coe,\n  left_inv := \u03bb x, rfl,\n  right_inv := \u03bb u, units.ext rfl,\n  map_mul' := \u03bb x y, units.ext rfl }\n\n@[simp, to_additive coe_to_add_units] lemma coe_to_units [group G] (g : G) :\n  (to_units g : G) = g := rfl\n\nprotected lemma group.is_unit {G} [group G] (x : G) : is_unit x := (to_units x).is_unit\n\nnamespace units\n\n@[simp, to_additive] lemma coe_inv [group G] (u : G\u02e3) :\n  \u2191u\u207b\u00b9 = (u\u207b\u00b9 : G) :=\nto_units.symm.map_inv u\n\nvariables [monoid M] [monoid N] [monoid P]\n\n\/-- A multiplicative equivalence of monoids defines a multiplicative equivalence\nof their groups of units. -\/\ndef map_equiv (h : M \u2243* N) : M\u02e3 \u2243* N\u02e3 :=\n{ inv_fun := map h.symm.to_monoid_hom,\n  left_inv := \u03bb u, ext $ h.left_inv u,\n  right_inv := \u03bb u, ext $ h.right_inv u,\n  .. map h.to_monoid_hom }\n\n\/-- Left multiplication by a unit of a monoid is a permutation of the underlying type. -\/\n@[to_additive \"Left addition of an additive unit is a permutation of the underlying type.\",\n  simps apply {fully_applied := ff}]\ndef mul_left (u : M\u02e3) : equiv.perm M :=\n{ to_fun    := \u03bbx, u * x,\n  inv_fun   := \u03bbx, \u2191u\u207b\u00b9 * x,\n  left_inv  := u.inv_mul_cancel_left,\n  right_inv := u.mul_inv_cancel_left }\n\n@[simp, to_additive]\nlemma mul_left_symm (u : M\u02e3) : u.mul_left.symm = u\u207b\u00b9.mul_left :=\nequiv.ext $ \u03bb x, rfl\n\n@[to_additive]\nlemma mul_left_bijective (a : M\u02e3) : function.bijective ((*) a : M \u2192 M) :=\n(mul_left a).bijective\n\n\/-- Right multiplication by a unit of a monoid is a permutation of the underlying type. -\/\n@[to_additive \"Right addition of an additive unit is a permutation of the underlying type.\",\n  simps apply {fully_applied := ff}]\ndef mul_right (u : M\u02e3) : equiv.perm M :=\n{ to_fun    := \u03bbx, x * u,\n  inv_fun   := \u03bbx, x * \u2191u\u207b\u00b9,\n  left_inv  := \u03bb x, mul_inv_cancel_right x u,\n  right_inv := \u03bb x, inv_mul_cancel_right x u }\n\n@[simp, to_additive]\nlemma mul_right_symm (u : M\u02e3) : u.mul_right.symm = u\u207b\u00b9.mul_right :=\nequiv.ext $ \u03bb x, rfl\n\n@[to_additive]\nlemma mul_right_bijective (a : M\u02e3) : function.bijective ((* a) : M \u2192 M) :=\n(mul_right a).bijective\n\nend units\n\nnamespace equiv\n\nsection has_involutive_neg\n\nvariables (G) [has_involutive_inv G]\n\n\/-- Inversion on a `group` or `group_with_zero` is a permutation of the underlying type. -\/\n@[to_additive \"Negation on an `add_group` is a permutation of the underlying type.\",\n  simps apply {fully_applied := ff}]\nprotected def inv : perm G := inv_involutive.to_equiv _\n\nvariable {G}\n\n@[simp, to_additive]\nlemma inv_symm : (equiv.inv G).symm = equiv.inv G := rfl\n\nend has_involutive_neg\n\nsection group\nvariables [group G]\n\n\/-- Left multiplication in a `group` is a permutation of the underlying type. -\/\n@[to_additive \"Left addition in an `add_group` is a permutation of the underlying type.\"]\nprotected def mul_left (a : G) : perm G := (to_units a).mul_left\n\n@[simp, to_additive]\nlemma coe_mul_left (a : G) : \u21d1(equiv.mul_left a) = (*) a := rfl\n\n\/-- extra simp lemma that `dsimp` can use. `simp` will never use this. -\/\n@[simp, nolint simp_nf, to_additive]\nlemma mul_left_symm_apply (a : G) : ((equiv.mul_left a).symm : G \u2192 G) = (*) a\u207b\u00b9 := rfl\n\n@[simp, to_additive]\nlemma mul_left_symm (a : G) : (equiv.mul_left a).symm = equiv.mul_left a\u207b\u00b9 :=\next $ \u03bb x, rfl\n\n@[to_additive]\nlemma _root_.group.mul_left_bijective (a : G) : function.bijective ((*) a) :=\n(equiv.mul_left a).bijective\n\n\/-- Right multiplication in a `group` is a permutation of the underlying type. -\/\n@[to_additive \"Right addition in an `add_group` is a permutation of the underlying type.\"]\nprotected def mul_right (a : G) : perm G := (to_units a).mul_right\n\n@[simp, to_additive]\nlemma coe_mul_right (a : G) : \u21d1(equiv.mul_right a) = \u03bb x, x * a := rfl\n\n@[simp, to_additive]\nlemma mul_right_symm (a : G) : (equiv.mul_right a).symm = equiv.mul_right a\u207b\u00b9 :=\next $ \u03bb x, rfl\n\n\/-- extra simp lemma that `dsimp` can use. `simp` will never use this.  -\/\n@[simp, nolint simp_nf, to_additive]\nlemma mul_right_symm_apply (a : G) : ((equiv.mul_right a).symm : G \u2192 G) = \u03bb x, x * a\u207b\u00b9 := rfl\n\n@[to_additive]\nlemma _root_.group.mul_right_bijective (a : G) : function.bijective (* a) :=\n(equiv.mul_right a).bijective\n\n\/-- A version of `equiv.mul_left a b\u207b\u00b9` that is defeq to `a \/ b`. -\/\n@[to_additive \/-\" A version of `equiv.add_left a (-b)` that is defeq to `a - b`. \"-\/, simps]\nprotected def div_left (a : G) : G \u2243 G :=\n{ to_fun := \u03bb b, a \/ b,\n  inv_fun := \u03bb b, b\u207b\u00b9 * a,\n  left_inv := \u03bb b, by simp [div_eq_mul_inv],\n  right_inv := \u03bb b, by simp [div_eq_mul_inv] }\n\n@[to_additive]\nlemma div_left_eq_inv_trans_mul_left (a : G) :\n  equiv.div_left a = (equiv.inv G).trans (equiv.mul_left a) :=\next $ \u03bb _, div_eq_mul_inv _ _\n\n\/-- A version of `equiv.mul_right a\u207b\u00b9 b` that is defeq to `b \/ a`. -\/\n@[to_additive \/-\" A version of `equiv.add_right (-a) b` that is defeq to `b - a`. \"-\/, simps]\nprotected def div_right (a : G) : G \u2243 G :=\n{ to_fun := \u03bb b, b \/ a,\n  inv_fun := \u03bb b, b * a,\n  left_inv := \u03bb b, by simp [div_eq_mul_inv],\n  right_inv := \u03bb b, by simp [div_eq_mul_inv] }\n\n@[to_additive]\nlemma div_right_eq_mul_right_inv (a : G) : equiv.div_right a = equiv.mul_right a\u207b\u00b9 :=\next $ \u03bb _, div_eq_mul_inv _ _\n\nend group\n\nsection group_with_zero\nvariables [group_with_zero G]\n\n\/-- Left multiplication by a nonzero element in a `group_with_zero` is a permutation of the\nunderlying type. -\/\n@[simps {fully_applied := ff}]\nprotected def mul_left\u2080 (a : G) (ha : a \u2260 0) : perm G :=\n(units.mk0 a ha).mul_left\n\nlemma _root_.mul_left_bijective\u2080 (a : G) (ha : a \u2260 0) :\n  function.bijective ((*) a : G \u2192 G) :=\n(equiv.mul_left\u2080 a ha).bijective\n\n\/-- Right multiplication by a nonzero element in a `group_with_zero` is a permutation of the\nunderlying type. -\/\n@[simps {fully_applied := ff}]\nprotected def mul_right\u2080 (a : G) (ha : a \u2260 0) : perm G :=\n(units.mk0 a ha).mul_right\n\nlemma _root_.mul_right_bijective\u2080 (a : G) (ha : a \u2260 0) :\n  function.bijective ((* a) : G \u2192 G) :=\n(equiv.mul_right\u2080 a ha).bijective\n\nend group_with_zero\n\nend equiv\n\n\/-- When the group is commutative, `equiv.inv` is a `mul_equiv`. There is a variant of this\n`mul_equiv.inv' G : G \u2243* G\u1d50\u1d52\u1d56` for the non-commutative case. -\/\n@[to_additive \"When the `add_group` is commutative, `equiv.neg` is an `add_equiv`.\"]\ndef mul_equiv.inv (G : Type*) [comm_group G] : G \u2243* G :=\n{ to_fun   := has_inv.inv,\n  inv_fun  := has_inv.inv,\n  map_mul' := mul_inv,\n  ..equiv.inv G }\n\n\/-- When the group with zero is commutative, `equiv.inv\u2080` is a `mul_equiv`. -\/\n@[simps apply] def mul_equiv.inv\u2080 (G : Type*) [comm_group_with_zero G] : G \u2243* G :=\n{ to_fun   := has_inv.inv,\n  inv_fun  := has_inv.inv,\n  map_mul' := \u03bb x y, mul_inv\u2080,\n  ..equiv.inv G }\n\n@[simp] lemma mul_equiv.inv\u2080_symm (G : Type*) [comm_group_with_zero G] :\n  (mul_equiv.inv\u2080 G).symm = mul_equiv.inv\u2080 G := rfl\n\nsection type_tags\n\n\/-- Reinterpret `G \u2243+ H` as `multiplicative G \u2243* multiplicative H`. -\/\ndef add_equiv.to_multiplicative [add_zero_class G] [add_zero_class H] :\n  (G \u2243+ H) \u2243 (multiplicative G \u2243* multiplicative H) :=\n{ to_fun := \u03bb f, \u27e8f.to_add_monoid_hom.to_multiplicative,\n                  f.symm.to_add_monoid_hom.to_multiplicative, f.3, f.4, f.5\u27e9,\n  inv_fun := \u03bb f, \u27e8f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5\u27e9,\n  left_inv := \u03bb x, by { ext, refl, },\n  right_inv := \u03bb x, by { ext, refl, }, }\n\n\/-- Reinterpret `G \u2243* H` as `additive G \u2243+ additive H`. -\/\ndef mul_equiv.to_additive [mul_one_class G] [mul_one_class H] :\n  (G \u2243* H) \u2243 (additive G \u2243+ additive H) :=\n{ to_fun := \u03bb f, \u27e8f.to_monoid_hom.to_additive, f.symm.to_monoid_hom.to_additive, f.3, f.4, f.5\u27e9,\n  inv_fun := \u03bb f, \u27e8f.to_add_monoid_hom, f.symm.to_add_monoid_hom, f.3, f.4, f.5\u27e9,\n  left_inv := \u03bb x, by { ext, refl, },\n  right_inv := \u03bb x, by { ext, refl, }, }\n\n\/-- Reinterpret `additive G \u2243+ H` as `G \u2243* multiplicative H`. -\/\ndef add_equiv.to_multiplicative' [mul_one_class G] [add_zero_class H] :\n  (additive G \u2243+ H) \u2243 (G \u2243* multiplicative H) :=\n{ to_fun := \u03bb f, \u27e8f.to_add_monoid_hom.to_multiplicative',\n                  f.symm.to_add_monoid_hom.to_multiplicative'', f.3, f.4, f.5\u27e9,\n  inv_fun := \u03bb f, \u27e8f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5\u27e9,\n  left_inv := \u03bb x, by { ext, refl, },\n  right_inv := \u03bb x, by { ext, refl, }, }\n\n\/-- Reinterpret `G \u2243* multiplicative H` as `additive G \u2243+ H` as. -\/\ndef mul_equiv.to_additive' [mul_one_class G] [add_zero_class H] :\n  (G \u2243* multiplicative H) \u2243 (additive G \u2243+ H) :=\nadd_equiv.to_multiplicative'.symm\n\n\/-- Reinterpret `G \u2243+ additive H` as `multiplicative G \u2243* H`. -\/\ndef add_equiv.to_multiplicative'' [add_zero_class G] [mul_one_class H] :\n  (G \u2243+ additive H) \u2243 (multiplicative G \u2243* H) :=\n{ to_fun := \u03bb f, \u27e8f.to_add_monoid_hom.to_multiplicative'',\n                  f.symm.to_add_monoid_hom.to_multiplicative', f.3, f.4, f.5\u27e9,\n  inv_fun := \u03bb f, \u27e8f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5\u27e9,\n  left_inv := \u03bb x, by { ext, refl, },\n  right_inv := \u03bb x, by { ext, refl, }, }\n\n\/-- Reinterpret `multiplicative G \u2243* H` as `G \u2243+ additive H` as. -\/\ndef mul_equiv.to_additive'' [add_zero_class G] [mul_one_class H] :\n  (multiplicative G \u2243* H) \u2243 (G \u2243+ additive H) :=\nadd_equiv.to_multiplicative''.symm\n\nend type_tags\n","avg_line_length":37.540960452,"max_line_length":100,"alphanum_fraction":0.6573234508}
-{"size":319,"ext":"lean","lang":"Lean","max_stars_count":4.0,"content":"import Lake\nopen Lake DSL\n\npackage scripts\n\n\/--\nDisplay a greeting\n\nUSAGE:\n  lake run greet [name]\n\nGreet the entity with the given name. Otherwise, greet the whole world.\n-\/\nscript greet (args) do\n  if h : 0 < args.length then\n    IO.println s!\"Hello, {args.get 0 h}!\"\n  else\n    IO.println \"Hello, world!\"\n  return 0\n","avg_line_length":15.95,"max_line_length":71,"alphanum_fraction":0.6833855799}
-{"size":594,"ext":"lean","lang":"Lean","max_stars_count":11.0,"content":"import DocGen4.Output.Template\nimport DocGen4.Output.DocString\nimport DocGen4.Process\n\nnamespace DocGen4\nnamespace Output\n\nopen scoped DocGen4.Jsx\n\ndef ctorToHtml (i : Process.NameInfo) : HtmlM Html := do\n  let shortName := i.name.components'.head!.toString\n  let name := i.name.toString\n  pure <li class=\"constructor\" id={name}>{shortName} : [\u2190infoFormatToHtml i.type]<\/li>\n\ndef inductiveToHtml (i : Process.InductiveInfo) : HtmlM (Array Html) := do\n  let constructorsHtml := <ul class=\"constructors\">[\u2190 i.ctors.toArray.mapM ctorToHtml]<\/ul>\n  pure #[constructorsHtml]\n\nend Output\nend DocGen4\n","avg_line_length":28.2857142857,"max_line_length":91,"alphanum_fraction":0.7508417508}
-{"size":1652,"ext":"lean","lang":"Lean","max_stars_count":78.0,"content":"\/-\nCopyright (c) 2021 Yourong Zang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yourong Zang\n-\/\nimport analysis.normed_space.conformal_linear_map\nimport analysis.inner_product_space.basic\n\n\/-!\n# Conformal maps between inner product spaces\n\nIn an inner product space, a map is conformal iff it preserves inner products up to a scalar factor.\n-\/\n\nvariables {E F : Type*} [inner_product_space \u211d E] [inner_product_space \u211d F]\n\nopen linear_isometry continuous_linear_map\nopen_locale real_inner_product_space\n\n\/-- A map between two inner product spaces is a conformal map if and only if it preserves inner\nproducts up to a scalar factor, i.e., there exists a positive `c : \u211d` such that `\u27eaf u, f v\u27eb = c *\n\u27eau, v\u27eb` for all `u`, `v`. -\/\nlemma is_conformal_map_iff (f : E \u2192L[\u211d] F) :\n  is_conformal_map f \u2194 \u2203 (c : \u211d), 0 < c \u2227 \u2200 (u v : E), \u27eaf u, f v\u27eb = c * \u27eau, v\u27eb :=\nbegin\n  split,\n  { rintros \u27e8c\u2081, hc\u2081, li, rfl\u27e9,\n    refine \u27e8c\u2081 * c\u2081, mul_self_pos.2 hc\u2081, \u03bb u v, _\u27e9,\n    simp only [real_inner_smul_left, real_inner_smul_right, mul_assoc, coe_smul',\n      coe_to_continuous_linear_map, pi.smul_apply, inner_map_map] },\n  { rintros \u27e8c\u2081, hc\u2081, huv\u27e9,\n    obtain \u27e8c, hc, rfl\u27e9 : \u2203 c : \u211d, 0 < c \u2227 c\u2081 = c * c,\n      from \u27e8real.sqrt c\u2081, real.sqrt_pos.2 hc\u2081, (real.mul_self_sqrt hc\u2081.le).symm\u27e9,\n    refine \u27e8c, hc.ne', (c\u207b\u00b9 \u2022 f : E \u2192\u2097[\u211d] F).isometry_of_inner (\u03bb u v, _), _\u27e9,\n    { simp only [real_inner_smul_left, real_inner_smul_right, huv, mul_assoc, coe_smul,\n        inv_mul_cancel_left\u2080 hc.ne', linear_map.smul_apply, continuous_linear_map.coe_coe] },\n    { ext1 x,\n      exact (smul_inv_smul\u2080 hc.ne' (f x)).symm } }\nend\n","avg_line_length":41.3,"max_line_length":100,"alphanum_fraction":0.6906779661}
-{"size":11853,"ext":"lean","lang":"Lean","max_stars_count":78.0,"content":"\/-\nCopyright (c) 2020 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa, Jujian Zhang\n-\/\nimport analysis.calculus.mean_value\nimport data.polynomial.denoms_clearable\nimport data.real.irrational\n\/-!\n\n# Liouville's theorem\n\nThis file contains a proof of Liouville's theorem stating that all Liouville numbers are\ntranscendental.\n\nTo obtain this result, there is first a proof that Liouville numbers are irrational and two\ntechnical lemmas.  These lemmas exploit the fact that a polynomial with integer coefficients\ntakes integer values at integers.  When evaluating at a rational number, we can clear denominators\nand obtain precise inequalities that ultimately allow us to prove transcendence of\nLiouville numbers.\n-\/\n\n\/--\nA Liouville number is a real number `x` such that for every natural number `n`, there exist\n`a, b \u2208 \u2124` with `1 < b` such that `0 < |x - a\/b| < 1\/b\u207f`.\nIn the implementation, the condition `x \u2260 a\/b` replaces the traditional equivalent `0 < |x - a\/b|`.\n-\/\ndef liouville (x : \u211d) := \u2200 n : \u2115, \u2203 a b : \u2124, 1 < b \u2227 x \u2260 a \/ b \u2227 |x - a \/ b| < 1 \/ b ^ n\n\nnamespace liouville\n\n@[protected] lemma irrational {x : \u211d} (h : liouville x) : irrational x :=\nbegin\n  -- By contradiction, `x = a \/ b`, with `a \u2208 \u2124`, `0 < b \u2208 \u2115` is a Liouville number,\n  rintros \u27e8\u27e8a, b, bN0, cop\u27e9, rfl\u27e9,\n  -- clear up the mess of constructions of rationals\n  change (liouville (a \/ b)) at h,\n  -- Since `a \/ b` is a Liouville number, there are `p, q \u2208 \u2124`, with `q1 : 1 < q`,\n  -- `a0 : a \/ b \u2260 p \/ q` and `a1 : |a \/ b - p \/ q| < 1 \/ q ^ (b + 1)`\n  rcases h (b + 1) with \u27e8p, q, q1, a0, a1\u27e9,\n  -- A few useful inequalities\n  have qR0 : (0 : \u211d) < q := int.cast_pos.mpr (zero_lt_one.trans q1),\n  have b0 : (b : \u211d) \u2260 0 := ne_of_gt (nat.cast_pos.mpr bN0),\n  have bq0 : (0 : \u211d) < b * q := mul_pos (nat.cast_pos.mpr bN0) qR0,\n  -- At a1, clear denominators...\n  replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q, by\n    rwa [div_sub_div _ _ b0 (ne_of_gt qR0), abs_div, div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0))\n      (pow_pos qR0 _), abs_of_pos bq0, one_mul,\n      -- ... and revert to integers\n       \u2190 int.cast_pow, \u2190 int.cast_mul, \u2190 int.cast_coe_nat, \u2190 int.cast_mul, \u2190 int.cast_mul,\n       \u2190 int.cast_sub, \u2190 int.cast_abs, \u2190 int.cast_mul, int.cast_lt] at a1,\n  -- At a0, clear denominators...\n  replace a0 : \u00aca * q - \u2191b * p = 0, by\n    rwa [ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm \u2191p, \u2190 sub_eq_zero,\n      -- ... and revert to integers\n      \u2190 int.cast_coe_nat, \u2190 int.cast_mul, \u2190 int.cast_mul, \u2190 int.cast_sub, int.cast_eq_zero] at a0,\n  -- Actually, `q` is a natural number\n  lift q to \u2115 using (zero_lt_one.trans q1).le,\n  -- Looks innocuous, but we now have an integer with non-zero absolute value: this is at\n  -- least one away from zero.  The gain here is what gets the proof going.\n  have ap : 0 < |a * \u2191q - \u2191b * p| := abs_pos.mpr a0,\n  -- Actually, the absolute value of an integer is a natural number\n  lift (|a * \u2191q - \u2191b * p|) to \u2115 using (abs_nonneg (a * \u2191q - \u2191b * p)),\n  -- At a1, revert to natural numbers\n  rw [\u2190 int.coe_nat_mul, \u2190 int.coe_nat_pow, \u2190 int.coe_nat_mul, int.coe_nat_lt] at a1,\n  -- Recall this is by contradiction: we obtained the inequality `b * q \u2264 x * q ^ (b + 1)`, so\n  -- we are done.\n  exact not_le.mpr a1 (nat.mul_lt_mul_pow_succ (int.coe_nat_pos.mp ap) (int.coe_nat_lt.mp q1)).le,\nend\n\nopen polynomial metric set real ring_hom\n\n\/-- Let `Z, N` be types, let `R` be a metric space, let `\u03b1 : R` be a point and let\n`j : Z \u2192 N \u2192 R` be a function.  We aim to estimate how close we can get to `\u03b1`, while staying\nin the image of `j`.  The points `j z a` of `R` in the image of `j` come with a \"cost\" equal to\n`d a`.  As we get closer to `\u03b1` while staying in the image of `j`, we are interested in bounding\nthe quantity `d a * dist \u03b1 (j z a)` from below by a strictly positive amount `1 \/ A`: the intuition\nis that approximating well `\u03b1` with the points in the image of `j` should come at a high cost.  The\nhypotheses on the function `f : R \u2192 R` provide us with sufficient conditions to ensure our goal.\nThe first hypothesis is that `f` is Lipschitz at `\u03b1`: this yields a bound on the distance.\nThe second hypothesis is specific to the Liouville argument and provides the missing bound\ninvolving the cost function `d`.\n\nThis lemma collects the properties used in the proof of `exists_pos_real_of_irrational_root`.\nIt is stated in more general form than needed: in the intended application, `Z = \u2124`, `N = \u2115`,\n`R = \u211d`, `d a = (a + 1) ^ f.nat_degree`, `j z a  = z \/ (a + 1)`, `f \u2208 \u2124[x]`, `\u03b1` is an irrational\nroot of `f`, `\u03b5` is small, `M` is a bound on the Lipschitz constant of `f` near `\u03b1`, `n` is\nthe degree of the polynomial `f`.\n-\/\nlemma exists_one_le_pow_mul_dist {Z N R : Type*} [pseudo_metric_space R]\n  {d : N \u2192 \u211d} {j : Z \u2192 N \u2192 R} {f : R \u2192 R} {\u03b1 : R} {\u03b5 M : \u211d}\n-- denominators are positive\n  (d0 : \u2200 (a : N), 1 \u2264 d a)\n  (e0 : 0 < \u03b5)\n-- function is Lipschitz at \u03b1\n  (B : \u2200 \u2983y : R\u2984, y \u2208 closed_ball \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 (dist \u03b1 y) * M)\n-- clear denominators\n  (L : \u2200 \u2983z : Z\u2984, \u2200 \u2983a : N\u2984, j z a \u2208 closed_ball \u03b1 \u03b5 \u2192 1 \u2264 (d a) * dist (f \u03b1) (f (j z a))) :\n  \u2203 A : \u211d, 0 < A \u2227 \u2200 (z : Z), \u2200 (a : N), 1 \u2264 (d a) * (dist \u03b1 (j z a) * A) :=\nbegin\n  -- A useful inequality to keep at hand\n  have me0 : 0 < max (1 \/ \u03b5) M := lt_max_iff.mpr (or.inl (one_div_pos.mpr e0)),\n  -- The maximum between `1 \/ \u03b5` and `M` works\n  refine \u27e8max (1 \/ \u03b5) M, me0, \u03bb z a, _\u27e9,\n  -- First, let's deal with the easy case in which we are far away from `\u03b1`\n  by_cases dm1 : 1 \u2264 (dist \u03b1 (j z a) * max (1 \/ \u03b5) M),\n  { exact one_le_mul_of_one_le_of_one_le (d0 a) dm1 },\n  { -- `j z a = z \/ (a + 1)`: we prove that this ratio is close to `\u03b1`\n    have : j z a \u2208 closed_ball \u03b1 \u03b5,\n    { refine mem_closed_ball'.mp (le_trans _ ((one_div_le me0 e0).mpr (le_max_left _ _))),\n      exact ((le_div_iff me0).mpr (not_le.mp dm1).le) },\n    -- use the \"separation from `1`\" (assumption `L`) for numerators,\n    refine (L this).trans _,\n    -- remove a common factor and use the Lipschitz assumption `B`\n    refine mul_le_mul_of_nonneg_left ((B this).trans _) (zero_le_one.trans (d0 a)),\n    exact mul_le_mul_of_nonneg_left (le_max_right _ M) dist_nonneg }\nend\n\nlemma exists_pos_real_of_irrational_root {\u03b1 : \u211d} (ha : irrational \u03b1)\n  {f : polynomial \u2124} (f0 : f \u2260 0) (fa : eval \u03b1 (map (algebra_map \u2124 \u211d) f) = 0):\n  \u2203 A : \u211d, 0 < A \u2227\n    \u2200 (a : \u2124), \u2200 (b : \u2115), (1 : \u211d) \u2264 (b + 1) ^ f.nat_degree * (|\u03b1 - (a \/ (b + 1))| * A) :=\nbegin\n  -- `fR` is `f` viewed as a polynomial with `\u211d` coefficients.\n  set fR : polynomial \u211d := map (algebra_map \u2124 \u211d) f,\n  -- `fR` is non-zero, since `f` is non-zero.\n  obtain fR0 : fR \u2260 0 := \u03bb fR0, (map_injective (algebra_map \u2124 \u211d) (\u03bb _ _ A, int.cast_inj.mp A)).ne\n    f0 (fR0.trans (polynomial.map_zero _).symm),\n  -- reformulating assumption `fa`: `\u03b1` is a root of `fR`.\n  have ar : \u03b1 \u2208 (fR.roots.to_finset : set \u211d) :=\n    finset.mem_coe.mpr (multiset.mem_to_finset.mpr ((mem_roots fR0).mpr (is_root.def.mpr fa))),\n  -- Since the polynomial `fR` has finitely many roots, there is a closed interval centered at `\u03b1`\n  -- such that `\u03b1` is the only root of `fR` in the interval.\n  obtain \u27e8\u03b6, z0, U\u27e9 : \u2203 \u03b6 > 0, closed_ball \u03b1 \u03b6 \u2229 (fR.roots.to_finset) = {\u03b1} :=\n    @exists_closed_ball_inter_eq_singleton_of_discrete _ _ _ discrete_of_t1_of_finite _ ar,\n  -- Since `fR` is continuous, it is bounded on the interval above.\n  obtain \u27e8xm, -, hM\u27e9 : \u2203 (xm : \u211d) (H : xm \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)), \u2200 (y : \u211d),\n    y \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6) \u2192 |fR.derivative.eval y| \u2264 |fR.derivative.eval xm| :=\n    is_compact.exists_forall_ge is_compact_Icc\n    \u27e8\u03b1, (sub_lt_self \u03b1 z0).le, (lt_add_of_pos_right \u03b1 z0).le\u27e9\n    (continuous_abs.comp fR.derivative.continuous_aeval).continuous_on,\n  -- Use the key lemma `exists_one_le_pow_mul_dist`: we are left to show that ...\n  refine @exists_one_le_pow_mul_dist \u2124 \u2115 \u211d _ _ _ (\u03bb y, fR.eval y) \u03b1 \u03b6\n    (|fR.derivative.eval xm|) _ z0 (\u03bb y hy, _) (\u03bb z a hq, _),\n  -- 1: the denominators are positive -- essentially by definition;\n  { exact \u03bb a, one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _ },\n  -- 2: the polynomial `fR` is Lipschitz at `\u03b1` -- as its derivative continuous;\n  { rw mul_comm,\n    rw real.closed_ball_eq_Icc at hy,\n    -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.\n    refine convex.norm_image_sub_le_of_norm_deriv_le (\u03bb _ _, fR.differentiable_at)\n      (\u03bb y h, by { rw fR.deriv, exact hM _ h }) (convex_Icc _ _) hy (mem_Icc_iff_abs_le.mp _),\n    exact @mem_closed_ball_self \u211d _ \u03b1 \u03b6 (le_of_lt z0) },\n  -- 3: the weird inequality of Liouville type with powers of the denominators.\n  { show 1 \u2264 (a + 1 : \u211d) ^ f.nat_degree * |eval \u03b1 fR - eval (z \/ (a + 1)) fR|,\n    rw [fa, zero_sub, abs_neg],\n    -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,\n    -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction\n    refine one_le_pow_mul_abs_eval_div (int.coe_nat_succ_pos a) (\u03bb hy, _),\n    -- As the evaluation of the polynomial vanishes, we found a root of `fR` that is rational.\n    -- We know that `\u03b1` is the only root of `fR` in our interval, and `\u03b1` is irrational:\n    -- follow your nose.\n    refine (irrational_iff_ne_rational \u03b1).mp ha z (a + 1) ((mem_singleton_iff.mp _).symm),\n    refine U.subset _,\n    refine \u27e8hq, finset.mem_coe.mp (multiset.mem_to_finset.mpr _)\u27e9,\n    exact (mem_roots fR0).mpr (is_root.def.mpr hy) }\nend\n\n\/-- **Liouville's Theorem** -\/\ntheorem transcendental {x : \u211d} (lx : liouville x) :\n  transcendental \u2124 x :=\nbegin\n  -- Proceed by contradiction: if `x` is algebraic, then `x` is the root (`ef0`) of a\n  -- non-zero (`f0`) polynomial `f`\n  rintros \u27e8f : polynomial \u2124, f0, ef0\u27e9,\n  -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.\n  replace ef0 : (f.map (algebra_map \u2124 \u211d)).eval x = 0, { rwa [aeval_def, \u2190 eval_map] at ef0 },\n  -- There is a \"large\" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a \/ (b + 1))| * A`\n  -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.\n  obtain \u27e8A, hA, h\u27e9 : \u2203 (A : \u211d), 0 < A \u2227\n    \u2200 (a : \u2124) (b : \u2115), (1 : \u211d) \u2264 (b.succ) ^ f.nat_degree * (|x - a \/ (b.succ)| * A) :=\n    exists_pos_real_of_irrational_root lx.irrational f0 ef0,\n  -- Since the real numbers are Archimedean, a power of `2` exceeds `A`: `hn : A < 2 ^ r`.\n  rcases pow_unbounded_of_one_lt A (lt_add_one 1) with \u27e8r, hn\u27e9,\n  -- Use the Liouville property, with exponent `r +  deg f`.\n  obtain \u27e8a, b, b1, -, a1\u27e9 : \u2203 (a b : \u2124), 1 < b \u2227 x \u2260 a \/ b \u2227\n    |x - a \/ b| < 1 \/ b ^ (r + f.nat_degree) := lx (r + f.nat_degree),\n  have b0 : (0 : \u211d) < b := zero_lt_one.trans (by { rw \u2190 int.cast_one, exact int.cast_lt.mpr b1 }),\n  -- Prove that `b ^ f.nat_degree * abs (x - a \/ b)` is strictly smaller than itself\n  -- recall, this is a proof by contradiction!\n  refine lt_irrefl ((b : \u211d) ^ f.nat_degree * |x - \u2191a \/ \u2191b|) _,\n  -- clear denominators at `a1`\n  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1,\n  -- split the inequality via `1 \/ A`.\n  refine ((_  : (b : \u211d) ^ f.nat_degree * |x - a \/ b| < 1 \/ A).trans_le _),\n  -- This branch of the proof uses the Liouville condition and the Archimedean property\n  { refine (lt_div_iff' hA).mpr _,\n    refine lt_of_le_of_lt _ a1,\n    refine mul_le_mul_of_nonneg_right _ (mul_nonneg (pow_nonneg b0.le _) (abs_nonneg _)),\n    refine hn.le.trans _,\n    refine pow_le_pow_of_le_left zero_le_two _ _,\n    exact int.cast_two.symm.le.trans (int.cast_le.mpr (int.add_one_le_iff.mpr b1)) },\n  -- this branch of the proof exploits the \"integrality\" of evaluations of polynomials\n  -- at ratios of integers.\n  { lift b to \u2115 using zero_le_one.trans b1.le,\n    specialize h a b.pred,\n    rwa [nat.succ_pred_eq_of_pos (zero_lt_one.trans _), \u2190 mul_assoc, \u2190 (div_le_iff hA)] at h,\n    exact int.coe_nat_lt.mp b1 }\nend\n\nend liouville\n","avg_line_length":55.9103773585,"max_line_length":99,"alphanum_fraction":0.6456593268}
-{"size":36088,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2019 Alexander Bentkamp. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp, Yury Kudriashov\n-\/\n\nimport data.complex.basic\nimport data.set.intervals\nimport tactic.interactive\nimport tactic.linarith\nimport linear_algebra.basic\nimport ring_theory.algebra\nimport algebra.pointwise\n\n\/-!\n# Convex sets and functions on real vector spaces\n\nIn a real vector space, we define the following objects and properties.\n\n* `segment x y` is the closed segment joining `x` and `y`.\n* A set `s` is `convex` if for any two points `x y \u2208 s` it includes `segment x y`;\n* A function `f` is `convex_on` a set `s` if `s` is itself a convex set, and for any two points\n  `x y \u2208 s` the segment joining `(x, f x)` to `(y, f y)` is (non-strictly) above the graph of `f`;\n  equivalently, `convex_on f s` means that the epigraph `{p : E \u00d7 \u211d | p.1 \u2208 s \u2227 f p.1 \u2264 p.2}`\n  is a convex set;\n* Center mass of a finite set of points with prescribed weights.\n* Convex hull of a set `s` is the minimal convex set that includes `s`.\n* Standard simplex `std_simplex \u03b9 [fintype \u03b9]` is the intersection of the positive quadrant with\n  the hyperplane `s.sum = 1` in the space `\u03b9 \u2192 \u211d`.\n\nWe also provide various equivalent versions of the definitions above, prove that some specific sets\nare convex, and prove Jensen's inequality.\n\n## Notations\n\nWe use the following local notations:\n\n* `I = Icc (0:\u211d) 1`;\n* `[x, y] = segment x y`.\n\nThey are defined using `local notation`, so they are not available outside of this file.\n-\/\n\nuniverses u' u v w x\n\nvariables {E : Type u} {F : Type v} {\u03b9 : Type w} {\u03b9' : Type x}\n  [add_comm_group E] [vector_space \u211d E] [add_comm_group F] [vector_space \u211d F]\n  {s : set E}\n\nopen set\nopen_locale classical\n\nlocal notation `I` := (Icc 0 1 : set \u211d)\n\nlocal attribute [instance] set.pointwise_add set.smul_set\n\nsection sets\n\n\/-! ### Segment -\/\n\n\/-- Segments in a vector space -\/\ndef segment (x y : E) : set E :=\n{z : E | \u2203 (a b : \u211d) (ha : 0 \u2264 a) (hb : 0 \u2264 b) (hab : a + b = 1), a \u2022 x + b \u2022 y = z}\nlocal notation `[`x `, ` y `]` := segment x y\n\nlemma segment_symm (x y : E) : [x, y] = [y, x] :=\nset.ext $ \u03bb z,\n\u27e8\u03bb \u27e8a, b, ha, hb, hab, H\u27e9, \u27e8b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H\u27e9,\n  \u03bb \u27e8a, b, ha, hb, hab, H\u27e9, \u27e8b, a, hb, ha, (add_comm _ _).trans hab, (add_comm _ _).trans H\u27e9\u27e9\n\nlemma left_mem_segment (x y : E) : x \u2208 [x, y] :=\n\u27e81, 0, zero_le_one, le_refl 0, add_zero 1, by rw [zero_smul, one_smul, add_zero]\u27e9\n\nlemma right_mem_segment (x y : E) : y \u2208 [x, y] :=\nsegment_symm y x \u25b8 left_mem_segment y x\n\nlemma segment_same (x : E) : [x, x] = {x} :=\nset.ext $ \u03bb z, \u27e8\u03bb \u27e8a, b, ha, hb, hab, hz\u27e9,\n  by simpa only [(add_smul _ _ _).symm, mem_singleton_iff, hab, one_smul, eq_comm] using hz,\n  \u03bb h, mem_singleton_iff.1 h \u25b8 left_mem_segment z z\u27e9\n\nlemma segment_eq_image (x y : E) : segment x y = (\u03bb (\u03b8 : \u211d), (1 - \u03b8) \u2022 x + \u03b8 \u2022 y) '' I :=\nset.ext $ \u03bb z,\n  \u27e8\u03bb \u27e8a, b, ha, hb, hab, hz\u27e9,\n    \u27e8b, \u27e8hb, hab \u25b8 le_add_of_nonneg_left ha\u27e9, hab \u25b8 hz \u25b8 by simp only [add_sub_cancel]\u27e9,\n    \u03bb \u27e8\u03b8, \u27e8h\u03b8\u2080, h\u03b8\u2081\u27e9, hz\u27e9, \u27e81-\u03b8, \u03b8, sub_nonneg.2 h\u03b8\u2081, h\u03b8\u2080, sub_add_cancel _ _, hz\u27e9\u27e9\n\nlemma segment_eq_image' (x y : E) : segment x y = (\u03bb (\u03b8 : \u211d), x + \u03b8 \u2022 (y - x)) '' I :=\nby { convert segment_eq_image x y, ext \u03b8, simp only [smul_sub, sub_smul, one_smul], abel }\n\nlemma segment_eq_image\u2082 (x y : E) :\n  segment x y = (\u03bb p:\u211d\u00d7\u211d, p.1 \u2022 x + p.2 \u2022 y) '' {p | 0 \u2264 p.1 \u2227 0 \u2264 p.2 \u2227 p.1 + p.2 = 1} :=\nby simp only [segment, image, prod.exists, mem_set_of_eq, exists_prop, and_assoc]\n\nlemma segment_eq_Icc {a b : \u211d} (h : a \u2264 b) : [a, b] = Icc a b :=\nbegin\n  rw [segment_eq_image'],\n  show (((+) a) \u2218 (\u03bb t, t * (b - a))) '' Icc 0 1 = Icc a b,\n  rw [image_comp, image_mul_right_Icc (@zero_le_one \u211d _) (sub_nonneg.2 h), image_add_left_Icc],\n  simp\nend\n\nlemma segment_eq_Icc' (a b : \u211d) : [a, b] = Icc (min a b) (max a b) :=\nby cases le_total a b; [skip, rw segment_symm]; simp [segment_eq_Icc, *]\n\nlemma segment_eq_interval (a b : \u211d) : segment a b = interval a b :=\nsegment_eq_Icc' _ _\n\nlemma mem_segment_translate (a : E) {x b c} : a + x \u2208 [a + b, a + c] \u2194 x \u2208 [b, c] :=\nbegin\n  rw [segment_eq_image', segment_eq_image'],\n  refine exists_congr (\u03bb \u03b8, and_congr iff.rfl _),\n  simp only [add_sub_add_left_eq_sub, add_assoc, add_left_inj]\nend\n\nlemma segment_translate_preimage (a b c : E) : (\u03bb x, a + x) \u207b\u00b9' [a + b, a + c] = [b, c] :=\nset.ext $ \u03bb x, mem_segment_translate a\n\nlemma segment_translate_image (a b c: E) : (\u03bbx, a + x) '' [b, c] = [a + b, a + c] :=\nsegment_translate_preimage a b c \u25b8 image_preimage_eq $ add_left_surjective a\n\n\/-! ### Convexity of sets -\/\n\/-- Convexity of sets -\/\ndef convex (s : set E) :=\n\u2200 \u2983x y : E\u2984, x \u2208 s \u2192 y \u2208 s \u2192 \u2200 \u2983a b : \u211d\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192\n  a \u2022 x + b \u2022 y \u2208 s\n\nlemma convex_iff_forall_pos :\n  convex s \u2194 \u2200 \u2983x y\u2984, x \u2208 s \u2192 y \u2208 s \u2192 \u2200 \u2983a b : \u211d\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s :=\nbegin\n  refine \u27e8\u03bb h x y hx hy a b ha hb hab, h hx hy (le_of_lt ha) (le_of_lt hb) hab, _\u27e9,\n  intros h x y hx hy a b ha hb hab,\n  cases eq_or_lt_of_le ha with ha ha,\n  { subst a, rw [zero_add] at hab, simp [hab, hy] },\n  cases eq_or_lt_of_le hb with hb hb,\n  { subst b, rw [add_zero] at hab, simp [hab, hx] },\n  exact h hx hy ha hb hab\nend\n\nlemma convex_iff_segment_subset : convex s \u2194 \u2200 \u2983x y\u2984, x \u2208 s \u2192 y \u2208 s \u2192 [x, y] \u2286 s :=\nby simp only [convex, segment_eq_image\u2082, subset_def, ball_image_iff, prod.forall,\n  mem_set_of_eq, and_imp]\n\nlemma convex.segment_subset (h : convex s) {x y:E} (hx : x \u2208 s) (hy : y \u2208 s) : [x, y] \u2286 s :=\nconvex_iff_segment_subset.1 h hx hy\n\n\/-- Alternative definition of set convexity, in terms of pointwise set operations. -\/\nlemma convex_iff_pointwise_add_subset:\n  convex s \u2194 \u2200 \u2983a b : \u211d\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 s + b \u2022 s \u2286 s :=\niff.intro\n  begin\n    rintros hA a b ha hb hab w \u27e8au, \u27e8u, hu, rfl\u27e9, bv, \u27e8v, hv, rfl\u27e9, rfl\u27e9,\n    exact hA hu hv ha hb hab\n  end\n  (\u03bb h x y hx hy a b ha hb hab,\n    (h ha hb hab) (set.add_mem_pointwise_add \u27e8_, hx, rfl\u27e9 \u27e8_, hy, rfl\u27e9))\n\n\/-- Alternative definition of set convexity, using division -\/\nlemma convex_iff_div:\n  convex s \u2194 \u2200 \u2983x y : E\u2984, x \u2208 s \u2192 y \u2208 s \u2192 \u2200 \u2983a b : \u211d\u2984,\n    0 \u2264 a \u2192 0 \u2264 b \u2192 0 < a + b \u2192 (a\/(a+b)) \u2022 x + (b\/(a+b)) \u2022 y \u2208 s :=\n\u27e8begin\n  assume h x y hx hy a b ha hb hab,\n  apply h hx hy,\n  have ha', from mul_le_mul_of_nonneg_left ha (le_of_lt (inv_pos.2 hab)),\n  rwa [mul_zero, \u2190div_eq_inv_mul'] at ha',\n  have hb', from mul_le_mul_of_nonneg_left hb (le_of_lt (inv_pos.2 hab)),\n  rwa [mul_zero, \u2190div_eq_inv_mul'] at hb',\n  rw [\u2190add_div],\n  exact div_self (ne_of_lt hab).symm\nend,\nbegin\n  assume h x y hx hy a b ha hb hab,\n  have h', from h hx hy ha hb,\n  rw [hab, div_one, div_one] at h',\n  exact h' zero_lt_one\nend\u27e9\n\n\/-! ### Examples of convex sets -\/\n\nlemma convex_empty : convex (\u2205 : set E) :=  by finish\n\nlemma convex_singleton (c : E) : convex ({c} : set E) :=\nbegin\n  intros x y hx hy a b ha hb hab,\n  rw [set.eq_of_mem_singleton hx, set.eq_of_mem_singleton hy, \u2190add_smul, hab, one_smul],\n  exact mem_singleton c\nend\n\nlemma convex_univ : convex (set.univ : set E) := \u03bb _ _ _ _ _ _ _ _ _, trivial\n\nlemma convex.inter {t : set E} (hs: convex s) (ht: convex t) : convex (s \u2229 t) :=\n\u03bb x y (hx : x \u2208 s \u2229 t) (hy : y \u2208 s \u2229 t) a b (ha : 0 \u2264 a) (hb : 0 \u2264 b) (hab : a + b = 1),\n  \u27e8hs hx.left hy.left ha hb hab, ht hx.right hy.right ha hb hab\u27e9\n\nlemma convex_sInter {S : set (set E)} (h : \u2200 s \u2208 S, convex s) : convex (\u22c2\u2080 S) :=\nassume x y hx hy a b ha hb hab s hs,\nh s hs (hx s hs) (hy s hs) ha hb hab\n\nlemma convex_Inter {\u03b9 : Sort*} {s: \u03b9 \u2192 set E} (h: \u2200 i : \u03b9, convex (s i)) : convex (\u22c2 i, s i) :=\n(sInter_range s) \u25b8 convex_sInter $ forall_range_iff.2 h\n\nlemma convex.prod {s : set E} {t : set F} (hs : convex s) (ht : convex t) :\n  convex (s.prod t) :=\nbegin\n  intros x y hx hy a b ha hb hab,\n  apply mem_prod.2,\n  exact \u27e8hs (mem_prod.1 hx).1 (mem_prod.1 hy).1 ha hb hab,\n        ht (mem_prod.1 hx).2 (mem_prod.1 hy).2 ha hb hab\u27e9\nend\n\nlemma convex.is_linear_image (hs : convex s) {f : E \u2192 F} (hf : is_linear_map \u211d f) :\n  convex (f '' s) :=\nbegin\n  rintros _ _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, rfl\u27e9 a b ha hb hab,\n  exact \u27e8a \u2022 x + b \u2022 y, hs hx hy ha hb hab, by simp only [hf.add,hf.smul]\u27e9\nend\n\nlemma convex.linear_image (hs : convex s) (f : E \u2192\u2097[\u211d] F) : convex (image f s) :=\nhs.is_linear_image f.is_linear\n\nlemma convex.is_linear_preimage {s : set F} (hs : convex s) {f : E \u2192 F} (hf : is_linear_map \u211d f) :\n  convex (preimage f s) :=\nbegin\n  intros x y hx hy a b ha hb hab,\n  convert hs hx hy ha hb hab,\n  simp only [mem_preimage, hf.add, hf.smul]\nend\n\nlemma convex.linear_preimage {s : set F} (hs : convex s) (f : E \u2192\u2097[\u211d] F) :\n  convex (preimage f s) :=\nhs.is_linear_preimage f.is_linear\n\nlemma convex.neg (hs : convex s) : convex ((\u03bb z, -z) '' s) :=\nhs.is_linear_image is_linear_map.is_linear_map_neg\n\nlemma convex.neg_preimage (hs : convex s) : convex ((\u03bb z, -z) \u207b\u00b9' s) :=\nhs.is_linear_preimage is_linear_map.is_linear_map_neg\n\nlemma convex.smul (c : \u211d) (hs : convex s) : convex (c \u2022 s) :=\nbegin\n  rw smul_set_eq_image,\n  exact hs.is_linear_image (is_linear_map.is_linear_map_smul c)\nend\n\nlemma convex.smul_preimage (c : \u211d) (hs : convex s) : convex ((\u03bb z, c \u2022 z) \u207b\u00b9' s) :=\nhs.is_linear_preimage (is_linear_map.is_linear_map_smul c)\n\nlemma convex.add {t : set E}  (hs : convex s) (ht : convex t) : convex (s + t) :=\nby { rw pointwise_add_eq_image, exact (hs.prod ht).is_linear_image is_linear_map.is_linear_map_add }\n\nlemma convex.sub {t : set E}  (hs : convex s) (ht : convex t) :\n  convex ((\u03bbx : E \u00d7 E, x.1 - x.2) '' (s.prod t)) :=\n(hs.prod ht).is_linear_image is_linear_map.is_linear_map_sub\n\nlemma convex.translate (hs : convex s) (z : E) : convex ((\u03bbx, z + x) '' s) :=\nbegin\n  convert (convex_singleton z).add hs,\n  ext x,\n  simp [set.mem_image, mem_pointwise_add, eq_comm]\nend\n\nlemma convex.affinity (hs : convex s) (z : E) (c : \u211d) : convex ((\u03bbx, z + c \u2022 x) '' s) :=\nbegin\n  convert (hs.smul c).translate z using 1,\n  erw [smul_set_eq_image, \u2190image_comp]\nend\n\nlemma convex_real_iff {s : set \u211d} :\n  convex s \u2194 \u2200 {x y}, x \u2208 s \u2192 y \u2208 s \u2192 Icc x y \u2286 s :=\nbegin\n  simp only [convex_iff_segment_subset, segment_eq_Icc'],\n  split; intros h x y hx hy,\n  { cases le_or_lt x y with hxy hxy,\n    { simpa [hxy] using h hx hy },\n    { simp [hxy] } },\n  { apply h; cases le_total x y; simp [*] }\nend\n\nlemma convex_Iio (r : \u211d) : convex (Iio r) :=\nconvex_real_iff.2 $ \u03bb x y hx hy z hz, lt_of_le_of_lt hz.2 hy\n\nlemma convex_Ioi (r : \u211d) : convex (Ioi r) :=\nconvex_real_iff.2 $ \u03bb x y hx hy z hz, lt_of_lt_of_le hx hz.1\n\nlemma convex_Iic (r : \u211d) : convex (Iic r) :=\nconvex_real_iff.2 $ \u03bb x y hx hy z hz, le_trans hz.2 hy\n\nlemma convex_Ici (r : \u211d) : convex (Ici r) :=\nconvex_real_iff.2 $ \u03bb x y hx hy z hz, le_trans hx hz.1\n\nlemma convex_Ioo (r : \u211d) (s : \u211d) : convex (Ioo r s) :=\n(convex_Ioi _).inter (convex_Iio _)\n\nlemma convex_Ico (r : \u211d) (s : \u211d) : convex (Ico r s) :=\n(convex_Ici _).inter (convex_Iio _)\n\nlemma convex_Ioc (r : \u211d) (s : \u211d) : convex (Ioc r s) :=\n(convex_Ioi _).inter (convex_Iic _)\n\nlemma convex_Icc (r : \u211d) (s : \u211d) : convex (Icc r s) :=\n(convex_Ici _).inter (convex_Iic _)\n\nlemma convex_segment (a b : E) : convex [a, b] :=\nbegin\n  have : (\u03bb (t : \u211d), a + t \u2022 (b - a)) = (\u03bbz : E, a + z) \u2218 (\u03bbt:\u211d, t \u2022 (b - a)) := rfl,\n  rw [segment_eq_image', this, image_comp],\n  refine ((convex_Icc _ _).is_linear_image _).translate _,\n  exact is_linear_map.is_linear_map_smul' _\nend\n\nlemma convex_halfspace_lt {f : E \u2192 \u211d} (h : is_linear_map \u211d f) (r : \u211d) :\n  convex {w | f w < r} :=\n(convex_Iio r).is_linear_preimage h\n\nlemma convex_halfspace_le {f : E \u2192 \u211d} (h : is_linear_map \u211d f) (r : \u211d) :\n  convex {w | f w \u2264 r} :=\n(convex_Iic r).is_linear_preimage h\n\nlemma convex_halfspace_gt {f : E \u2192 \u211d} (h : is_linear_map \u211d f) (r : \u211d) :\n  convex {w | r < f w} :=\n(convex_Ioi r).is_linear_preimage h\n\nlemma convex_halfspace_ge {f : E \u2192 \u211d} (h : is_linear_map \u211d f) (r : \u211d) :\n  convex {w | r \u2264 f w} :=\n(convex_Ici r).is_linear_preimage h\n\nlemma convex_hyperplane {f : E \u2192 \u211d} (h : is_linear_map \u211d f) (r : \u211d) :\n  convex {w | f w = r} :=\nbegin\n  show convex (f \u207b\u00b9' {p | p = r}),\n  rw set_of_eq_eq_singleton,\n  exact (convex_singleton r).is_linear_preimage h\nend\n\nlemma convex_halfspace_re_lt (r : \u211d) : convex {c : \u2102 | c.re < r} :=\nconvex_halfspace_lt (is_linear_map.mk complex.add_re complex.smul_re) _\n\nlemma convex_halfspace_re_le (r : \u211d) : convex {c : \u2102 | c.re \u2264 r} :=\nconvex_halfspace_le (is_linear_map.mk complex.add_re complex.smul_re) _\n\nlemma convex_halfspace_re_gt (r : \u211d) : convex {c : \u2102 | r < c.re } :=\nconvex_halfspace_gt (is_linear_map.mk complex.add_re complex.smul_re) _\n\nlemma convex_halfspace_re_lge (r : \u211d) : convex {c : \u2102 | r \u2264 c.re} :=\nconvex_halfspace_ge (is_linear_map.mk complex.add_re complex.smul_re) _\n\nlemma convex_halfspace_im_lt (r : \u211d) : convex {c : \u2102 | c.im < r} :=\nconvex_halfspace_lt (is_linear_map.mk complex.add_im complex.smul_im) _\n\nlemma convex_halfspace_im_le (r : \u211d) : convex {c : \u2102 | c.im \u2264 r} :=\nconvex_halfspace_le (is_linear_map.mk complex.add_im complex.smul_im) _\n\nlemma convex_halfspace_im_gt (r : \u211d) : convex {c : \u2102 | r < c.im } :=\nconvex_halfspace_gt (is_linear_map.mk complex.add_im complex.smul_im) _\n\nlemma convex_halfspace_im_lge (r : \u211d) : convex {c : \u2102 | r \u2264 c.im} :=\nconvex_halfspace_ge (is_linear_map.mk complex.add_im complex.smul_im) _\n\nsection submodule\n\nopen submodule\n\nlemma submodule.convex (K : submodule \u211d E) : convex (\u2191K : set E) :=\nby { repeat {intro}, refine add_mem _ (smul_mem _ _ _) (smul_mem _ _ _); assumption }\n\nlemma subspace.convex (K : subspace \u211d E) : convex (\u2191K : set E) := K.convex\n\nend submodule\n\nend sets\n\nsection functions\n\nlocal notation `[`x `, ` y `]` := segment x y\n\n\/-! ### Convex functions -\/\n\n\/-- Convexity of functions -\/\ndef convex_on (s : set E) (f : E \u2192 \u211d) : Prop :=\n  convex s \u2227\n  \u2200 \u2983x y : E\u2984, x \u2208 s \u2192 y \u2208 s \u2192 \u2200 \u2983a b : \u211d\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192\n    f (a \u2022 x + b \u2022 y) \u2264 a * f x + b * f y\n\nvariables {t : set E} {f g : E \u2192 \u211d}\n\nlemma convex_on_iff_div:\n  convex_on s f \u2194 convex s \u2227 \u2200 \u2983x y : E\u2984, x \u2208 s \u2192 y \u2208 s \u2192 \u2200  \u2983a b : \u211d\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 0 < a + b \u2192\n    f ((a\/(a+b)) \u2022 x + (b\/(a+b)) \u2022 y) \u2264 (a\/(a+b)) * f x + (b\/(a+b)) * f y :=\nand_congr iff.rfl\n\u27e8begin\n  intros h x y hx hy a b ha hb hab,\n  apply h hx hy (div_nonneg ha hab) (div_nonneg hb hab),\n  rw [\u2190add_div],\n  exact div_self (ne_of_gt hab)\nend,\nbegin\n  intros h x y hx hy a b ha hb hab,\n  simpa [hab, zero_lt_one] using h hx hy ha hb,\nend\u27e9\n\n\/-- For a function on a convex set in a linear ordered space, in order to prove that it is convex\nit suffices to verify the inequality `f (a \u2022 x + b \u2022 y) \u2264 a * f x + b * f y` only for `x < y`\nand positive `a`, `b`. The main use case is `E = \u211d` however one can apply it, e.g., to `\u211d^n` with\nlexicographic order. -\/\nlemma linear_order.convex_on_of_lt [linear_order E] (hs : convex s)\n  (hf : \u2200 \u2983x y : E\u2984, x \u2208 s \u2192 y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \u211d\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192\n    f (a \u2022 x + b \u2022 y) \u2264 a * f x + b * f y) : convex_on s f :=\nbegin\n  use hs,\n  intros x y hx hy a b ha hb hab,\n  wlog hxy : x<=y using [x y a b, y x b a],\n  { exact le_total _ _ },\n  { cases eq_or_lt_of_le hxy with hxy hxy,\n      by { subst y, rw [\u2190 add_smul, \u2190 add_mul, hab, one_smul, one_mul] },\n    cases eq_or_lt_of_le ha with ha ha,\n      by { subst a, rw [zero_add] at hab, subst b, simp },\n    cases eq_or_lt_of_le hb with hb hb,\n      by { subst b, rw [add_zero] at hab, subst a, simp },\n    exact hf hx hy hxy ha hb hab }\nend\n\n\/-- For a function `f` defined on a convex subset `D` of `\u211d`, if for any three points `x<y<z`\nthe slope of the secant line of `f` on `[x, y]` is less than or equal to the slope\nof the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity\nof a function is used in the proof of convexity of a function with a monotone derivative. -\/\nlemma convex_on_real_of_slope_mono_adjacent {s : set \u211d} (hs : convex s) {f : \u211d \u2192 \u211d}\n  (hf : \u2200 {x y z : \u211d}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192\n    (f y - f x) \/ (y - x) \u2264 (f z - f y) \/ (z - y)) :\n  convex_on s f :=\nlinear_order.convex_on_of_lt hs\nbegin\n  assume x z hx hz hxz a b ha hb hab,\n  let y := a * x + b * z,\n  have hxy : x < y,\n  { rw [\u2190 one_mul x, \u2190 hab, add_mul],\n    exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },\n  have hyz : y < z,\n  { rw [\u2190 one_mul z, \u2190 hab, add_mul],\n    exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ },\n  have : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x),\n    from (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz),\n  have A : z - y + (y - x) = z - x, by abel,\n  have B : 0 < z - x, from sub_pos.2 (lt_trans hxy hyz),\n  rw [sub_mul, sub_mul, sub_le_iff_le_add', \u2190 add_sub_assoc, le_sub_iff_add_le, \u2190 mul_add, A,\n    \u2190 le_div_iff B, add_div, mul_div_assoc, mul_div_assoc,\n    mul_comm (f x), mul_comm (f z)] at this,\n  rw [eq_comm, \u2190 sub_eq_iff_eq_add] at hab; subst a,\n  convert this; symmetry; simp only [div_eq_iff (ne_of_gt B), y]; ring\nend\n\nlemma convex_on.subset (h_convex_on : convex_on t f) (h_subset : s \u2286 t) (h_convex : convex s) :\n  convex_on s f :=\nbegin\n  apply and.intro h_convex,\n  intros x y hx hy,\n  exact h_convex_on.2 (h_subset hx) (h_subset hy),\nend\n\nlemma convex_on.add (hf : convex_on s f) (hg : convex_on s g) : convex_on s (\u03bbx, f x + g x) :=\nbegin\n  apply and.intro hf.1,\n  intros x y hx hy a b ha hb hab,\n  calc\n    f (a \u2022 x + b \u2022 y) + g (a \u2022 x + b \u2022 y) \u2264 (a * f x + b * f y) + (a * g x + b * g y)\n      : add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)\n    ... = a * f x + a * g x + b * f y + b * g y : by linarith\n    ... = a * (f x + g x) + b * (f y + g y) : by simp [mul_add]\nend\n\nlemma convex_on.smul {c : \u211d} (hc : 0 \u2264 c) (hf : convex_on s f) : convex_on s (\u03bbx, c * f x) :=\nbegin\n  apply and.intro hf.1,\n  intros x y hx hy a b ha hb hab,\n  calc\n    c * f (a \u2022 x + b \u2022 y) \u2264 c * (a * f x + b * f y)\n      : mul_le_mul_of_nonneg_left (hf.2 hx hy ha hb hab) hc\n    ... = a * (c * f x) + b * (c * f y) : by rw mul_add; ac_refl\nend\n\nlemma convex_on.le_on_segment' {x y : E} {a b : \u211d}\n  (hf : convex_on s f) (hx : x \u2208 s) (hy : y \u2208 s) (ha : 0 \u2264 a) (hb : 0 \u2264 b) (hab : a + b = 1) :\n  f (a \u2022 x + b \u2022 y) \u2264 max (f x) (f y) :=\ncalc\n  f (a \u2022 x + b \u2022 y) \u2264 a * f x + b * f y : hf.2 hx hy ha hb hab\n  ... \u2264 a * max (f x) (f y) + b * max (f x) (f y) :\n    add_le_add (mul_le_mul_of_nonneg_left (le_max_left _ _) ha) (mul_le_mul_of_nonneg_left (le_max_right _ _) hb)\n  ... \u2264 max (f x) (f y) : by rw [\u2190add_mul, hab, one_mul]\n\nlemma convex_on.le_on_segment (hf : convex_on s f) {x y z : E}\n  (hx : x \u2208 s) (hy : y \u2208 s) (hz : z \u2208 [x, y]) :\n  f z \u2264 max (f x) (f y) :=\nlet \u27e8a, b, ha, hb, hab, hz\u27e9 := hz in hz \u25b8 hf.le_on_segment' hx hy ha hb hab\n\nlemma convex_on.convex_le (hf : convex_on s f) (r : \u211d) : convex {x \u2208 s | f x \u2264 r} :=\nconvex_iff_segment_subset.2 $ \u03bb x y hx hy z hz,\n  \u27e8hf.1.segment_subset hx.1 hy.1 hz,\n    le_trans (hf.le_on_segment hx.1 hy.1 hz) $ max_le hx.2 hy.2\u27e9\n\nlemma convex_on.convex_lt (hf : convex_on s f) (r : \u211d) : convex {x \u2208 s | f x < r} :=\nconvex_iff_segment_subset.2 $ \u03bb x y hx hy z hz,\n  \u27e8hf.1.segment_subset hx.1 hy.1 hz,\n    lt_of_le_of_lt (hf.le_on_segment hx.1 hy.1 hz) $ max_lt hx.2 hy.2\u27e9\n\nlemma convex_on.convex_epigraph (hf : convex_on s f) :\n  convex {p : E \u00d7 \u211d | p.1 \u2208 s \u2227 f p.1 \u2264 p.2} :=\nbegin\n  rintros \u27e8x, r\u27e9 \u27e8y, t\u27e9 \u27e8hx, hr\u27e9 \u27e8hy, ht\u27e9 a b ha hb hab,\n  refine \u27e8hf.1 hx hy ha hb hab, _\u27e9,\n  calc f (a \u2022 x + b \u2022 y) \u2264 a * f x + b * f y : hf.2 hx hy ha hb hab\n  ... \u2264 a * r + b * t : add_le_add (mul_le_mul_of_nonneg_left hr ha)\n    (mul_le_mul_of_nonneg_left ht hb)\nend\n\nlemma convex_on_iff_convex_epigraph : convex_on s f \u2194 convex {p : E \u00d7 \u211d | p.1 \u2208 s \u2227 f p.1 \u2264 p.2} :=\nbegin\n  refine \u27e8convex_on.convex_epigraph, \u03bb h, \u27e8_, _\u27e9\u27e9,\n  { assume x y hx hy a b ha hb hab,\n    exact (@h (x, f x) (y, f y) \u27e8hx, le_refl _\u27e9 \u27e8hy, le_refl _\u27e9 a b ha hb hab).1 },\n  { assume x y hx hy a b ha hb hab,\n    exact (@h (x, f x) (y, f y) \u27e8hx, le_refl _\u27e9 \u27e8hy, le_refl _\u27e9 a b ha hb hab).2 }\nend\n\nend functions\n\nsection center_mass\n\n\/-- Center mass of a finite collection of points with prescribed weights.\nNote that we require neither `0 \u2264 w i` nor `\u2211 w = 1`. -\/\nnoncomputable def finset.center_mass (t : finset \u03b9) (w : \u03b9 \u2192 \u211d) (z : \u03b9 \u2192 E) : E :=\n(t.sum w)\u207b\u00b9 \u2022 (t.sum (\u03bb i, w i \u2022 z i))\n\nvariables (i j : \u03b9) (c : \u211d) (t : finset \u03b9) (w : \u03b9 \u2192 \u211d) (z : \u03b9 \u2192 E)\n\nopen finset (hiding singleton)\n\nlemma finset.center_mass_empty : (\u2205 : finset \u03b9).center_mass w z = 0 :=\nby simp only [center_mass, sum_empty, smul_zero]\n\nlemma finset.center_mass_pair (hne : i \u2260 j) :\n  ({i, j} : finset \u03b9).center_mass w z = (w i \/ (w i + w j)) \u2022 z i + (w j \/ (w i + w j)) \u2022 z j :=\nby simp only [center_mass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul']\n\nvariable {w}\n\nlemma finset.center_mass_insert (ha : i \u2209 t) (hw : t.sum w \u2260 0) :\n  (insert i t).center_mass w z = (w i \/ (w i + t.sum w)) \u2022 z i +\n    (t.sum w \/ (w i + t.sum w)) \u2022 t.center_mass w z :=\nbegin\n  simp only [center_mass, sum_insert ha, smul_add, (mul_smul _ _ _).symm],\n  congr' 2,\n  { apply mul_comm },\n  { rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div_eq_inv] }\nend\n\nlemma finset.center_mass_singleton (hw : w i \u2260 0) : (finset.singleton i).center_mass w z = z i :=\nby rw [center_mass, sum_singleton, sum_singleton, \u2190 mul_smul, inv_mul_cancel hw, one_smul]\n\nlemma finset.center_mass_eq_of_sum_1 (hw : t.sum w = 1) :\n  t.center_mass w z = t.sum (\u03bb i, w i \u2022 z i) :=\nby simp only [finset.center_mass, hw, inv_one, one_smul]\n\nlemma finset.center_mass_smul : t.center_mass w (\u03bb i, c \u2022 z i) = c \u2022 t.center_mass w z :=\nby simp only [finset.center_mass, finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]\n\n\/-- A convex combination of two centers of mass is a center of mass as well. This version\ndeals with two different index types. -\/\nlemma finset.center_mass_segment'\n  (s : finset \u03b9) (t : finset \u03b9') (ws : \u03b9 \u2192 \u211d) (zs : \u03b9 \u2192 E) (wt : \u03b9' \u2192 \u211d) (zt : \u03b9' \u2192 E)\n  (hws : s.sum ws = 1) (hwt : t.sum wt = 1) (a b : \u211d) (hab : a + b = 1):\n  a \u2022 s.center_mass ws zs + b \u2022 t.center_mass wt zt =\n    (s.image sum.inl \u222a t.image sum.inr).center_mass\n      (sum.elim (\u03bb i, a * ws i) (\u03bb j, b * wt j))\n      (sum.elim zs zt) :=\nbegin\n  rw [s.center_mass_eq_of_sum_1 _ hws, t.center_mass_eq_of_sum_1 _ hwt,\n    smul_sum, smul_sum, \u2190 finset.sum_sum_elim, finset.center_mass_eq_of_sum_1],\n  { congr, ext i, cases i; simp only [sum.elim_inl, sum.elim_inr, mul_smul] },\n  { rw [sum_sum_elim, \u2190 mul_sum, \u2190 mul_sum, hws, hwt, mul_one, mul_one, hab] }\nend\n\n\/-- A convex combination of two centers of mass is a center of mass as well. This version\nworks if two centers of mass share the set of original points. -\/\nlemma finset.center_mass_segment\n  (s : finset \u03b9) (w\u2081 w\u2082 : \u03b9 \u2192 \u211d) (z : \u03b9 \u2192 E)\n  (hw\u2081 : s.sum w\u2081 = 1) (hw\u2082 : s.sum w\u2082 = 1) (a b : \u211d) (hab : a + b = 1):\n  a \u2022 s.center_mass w\u2081 z + b \u2022 s.center_mass w\u2082 z =\n    s.center_mass (\u03bb i, a * w\u2081 i + b * w\u2082 i) z :=\nhave hw : s.sum (\u03bb i, a * w\u2081 i + b * w\u2082 i) = 1,\n  by simp only [mul_sum.symm, sum_add_distrib, mul_one, *],\nby simp only [finset.center_mass_eq_of_sum_1, smul_sum, sum_add_distrib, add_smul, mul_smul, *]\n\nlemma finset.center_mass_ite_eq (hi : i \u2208 t) :\n  t.center_mass (\u03bb j, if (i = j) then 1 else 0) z = z i :=\nbegin\n  rw [finset.center_mass_eq_of_sum_1],\n  transitivity t.sum (\u03bb j, if (i = j) then z i else 0),\n  { congr, ext i, split_ifs, exacts [h \u25b8 one_smul _ _, zero_smul _ _] },\n  { rw [sum_ite_eq, if_pos hi] },\n  { rw [sum_ite_eq, if_pos hi] }\nend\n\nvariables {t w}\n\nlemma finset.center_mass_subset {t' : finset \u03b9} (ht : t \u2286 t')\n  (h : \u2200 i \u2208 t', i \u2209 t \u2192 w i = 0) :\n  t.center_mass w z = t'.center_mass w z :=\nbegin\n  rw [center_mass, sum_subset ht h, smul_sum, center_mass, smul_sum],\n  apply sum_subset ht,\n  assume i hit' hit,\n  rw [h i hit' hit, zero_smul, smul_zero]\nend\n\nlemma finset.center_mass_filter_ne_zero :\n  (t.filter (\u03bb i, w i \u2260 0)).center_mass w z = t.center_mass w z :=\nfinset.center_mass_subset z (filter_subset _) $ \u03bb i hit hit',\nby simpa only [hit, mem_filter, true_and, ne.def, not_not] using hit'\n\nvariable {z}\n\n\/-- Center mass of a finite subset of a convex set belongs to the set\nprovided that all weights are non-negative, and the total weight is positive. -\/\nlemma convex.center_mass_mem (hs : convex s) :\n  (\u2200 i \u2208 t, 0 \u2264 w i) \u2192 (0 < t.sum w) \u2192 (\u2200 i \u2208 t, z i \u2208 s) \u2192 t.center_mass w z \u2208 s :=\nbegin\n  refine finset.induction (by simp [lt_irrefl]) (\u03bb i t hi ht h\u2080 hpos hmem, _) t,\n  have zi : z i \u2208 s, from hmem _ (mem_insert_self _ _),\n  have hs\u2080 : \u2200 j \u2208 t, 0 \u2264 w j, from \u03bb j hj, h\u2080 j $ mem_insert_of_mem hj,\n  rw [sum_insert hi] at hpos,\n  by_cases hsum_t : t.sum w = 0,\n  { have ws : \u2200 j \u2208 t, w j = 0, from (sum_eq_zero_iff_of_nonneg hs\u2080).1 hsum_t,\n    have wz : t.sum (\u03bb j, w j \u2022 z j) = 0, from sum_eq_zero (\u03bb i hi, by simp [ws i hi]),\n    simp only [center_mass, sum_insert hi, wz, hsum_t, add_zero],\n    simp only [hsum_t, add_zero] at hpos,\n    rw [\u2190 mul_smul, inv_mul_cancel (ne_of_gt hpos), one_smul],\n    exact zi },\n  { rw [finset.center_mass_insert _ _ _ hi hsum_t],\n    refine convex_iff_div.1 hs zi (ht hs\u2080 _ _) _ (sum_nonneg hs\u2080) hpos,\n    { exact lt_of_le_of_ne (sum_nonneg hs\u2080) (ne.symm hsum_t) },\n    { intros j hj, exact hmem j (mem_insert_of_mem hj) },\n    { exact h\u2080 _ (mem_insert_self _ _) } }\nend\n\nlemma convex.sum_mem (hs : convex s) (h\u2080 : \u2200 i \u2208 t, 0 \u2264 w i) (h\u2081 : t.sum w = 1)\n  (hz : \u2200 i \u2208 t, z i \u2208 s) :\n  t.sum (\u03bb i, w i \u2022 z i) \u2208 s :=\nby simpa only [h\u2081, center_mass, inv_one, one_smul] using\n  hs.center_mass_mem h\u2080 (h\u2081.symm \u25b8 zero_lt_one) hz\n\nlemma convex_iff_sum_mem :\n  convex s \u2194\n    (\u2200 (t : finset E) (w : E \u2192 \u211d),\n      (\u2200 i \u2208 t, 0 \u2264 w i) \u2192 t.sum w = 1 \u2192 (\u2200 x \u2208 t, x \u2208 s) \u2192 t.sum (\u03bbx, w x \u2022 x) \u2208 s ) :=\nbegin\n  refine \u27e8\u03bb hs t w hw\u2080 hw\u2081 hts, hs.sum_mem hw\u2080 hw\u2081 hts, _\u27e9,\n  intros h x y hx hy a b ha hb hab,\n  by_cases h_cases: x = y,\n  { rw [h_cases, \u2190add_smul, hab, one_smul], exact hy },\n  { convert h {x, y} (\u03bb z, if z = y then b else a) _ _ _,\n    { simp only [sum_pair h_cases, if_neg h_cases, if_pos rfl] },\n    { simp_intros i hi,\n      cases hi; subst i; simp [ha, hb, if_neg h_cases] },\n    { simp only [sum_pair h_cases, if_neg h_cases, if_pos rfl, hab] },\n    { simp_intros i hi,\n      cases hi; subst i; simp [hx, hy, if_neg h_cases] } }\nend\n\n\/-- Jensen's inequality, `finset.center_mass` version. -\/\nlemma convex_on.map_center_mass_le {f : E \u2192 \u211d} (hf : convex_on s f)\n  (h\u2080 : \u2200 i \u2208 t, 0 \u2264 w i) (hpos : 0 < t.sum w)\n  (hmem : \u2200 i \u2208 t, z i \u2208 s) : f (t.center_mass w z) \u2264 t.center_mass w (f \u2218 z) :=\nbegin\n  have hmem' : \u2200 i \u2208 t, (z i, (f \u2218 z) i) \u2208 {p : E \u00d7 \u211d | p.1 \u2208 s \u2227 f p.1 \u2264 p.2},\n    from \u03bb i hi, \u27e8hmem i hi, le_refl _\u27e9,\n  convert (hf.convex_epigraph.center_mass_mem h\u2080 hpos hmem').2;\n    simp only [center_mass, function.comp, prod.smul_fst, prod.fst_sum, prod.smul_snd, prod.snd_sum]\nend\n\n\/-- Jensen's inequality, `finset.sum` version. -\/\nlemma convex_on.map_sum_le {f : E \u2192 \u211d} (hf : convex_on s f)\n  (h\u2080 : \u2200 i \u2208 t, 0 \u2264 w i) (h\u2081 : t.sum w = 1)\n  (hmem : \u2200 i \u2208 t, z i \u2208 s) : f (t.sum (\u03bb i, w i \u2022 z i)) \u2264 t.sum (\u03bb i, w i * (f (z i))) :=\nby simpa only [center_mass, h\u2081, inv_one, one_smul]\n  using hf.map_center_mass_le h\u2080 (h\u2081.symm \u25b8 zero_lt_one) hmem\n\n\/-- If a function `f` is convex on `s` takes value `y` at the center mass of some points\n`z i \u2208 s`, then for some `i` we have `y \u2264 f (z i)`. -\/\nlemma convex_on.exists_ge_of_center_mass {f : E \u2192 \u211d} (h : convex_on s f)\n  (hw\u2080 : \u2200 i \u2208 t, 0 \u2264 w i) (hws : 0 < t.sum w) (hz : \u2200 i \u2208 t, z i \u2208 s) :\n  \u2203 i \u2208 t, f (t.center_mass w z) \u2264 f (z i) :=\nbegin\n  set y := t.center_mass w z,\n  have : f y \u2264 t.center_mass w (f \u2218 z) := h.map_center_mass_le hw\u2080 hws hz,\n  rw \u2190 sum_filter_ne_zero at hws,\n  rw [\u2190 finset.center_mass_filter_ne_zero (f \u2218 z), center_mass, smul_eq_mul,\n    \u2190 div_eq_inv_mul', le_div_iff hws, mul_sum] at this,\n  replace : \u2203 i \u2208 t.filter (\u03bb i, w i \u2260 0), f y * w i \u2264 w i \u2022 (f \u2218 z) i :=\n    exists_le_of_sum_le (nonempty_of_sum_ne_zero (ne_of_gt hws)) this,\n  rcases this with \u27e8i, hi, H\u27e9,\n  rw [mem_filter] at hi,\n  use [i, hi.1],\n  simp only [smul_eq_mul, mul_comm (w i)] at H,\n  refine (mul_le_mul_right _).1 H,\n  exact lt_of_le_of_ne (hw\u2080 i hi.1) hi.2.symm\nend\n\nend center_mass\n\nsection convex_hull\n\nvariable {t : set E}\n\n\/-- Convex hull of a set `s` is the minimal convex set that includes `s` -\/\ndef convex_hull (s : set E) : set E :=\n\u22c2 (t : set E) (hst : s \u2286 t) (ht : convex t), t\n\nvariable (s)\n\nlemma subset_convex_hull : s \u2286 convex_hull s :=\nset.subset_Inter $ \u03bb t, set.subset_Inter $ \u03bb hst, set.subset_Inter $ \u03bb ht, hst\n\nlemma convex_convex_hull : convex (convex_hull s) :=\nconvex_Inter $ \u03bb t, convex_Inter $ \u03bb ht, convex_Inter id\n\nvariable {s}\n\nlemma convex_hull_min (hst : s \u2286 t) (ht : convex t) : convex_hull s \u2286 t :=\nset.Inter_subset_of_subset t $ set.Inter_subset_of_subset hst $ set.Inter_subset _ ht\n\nlemma convex_hull_mono (hst : s \u2286 t) : convex_hull s \u2286 convex_hull t :=\nconvex_hull_min (set.subset.trans hst $ subset_convex_hull t) (convex_convex_hull t)\n\nlemma convex.convex_hull_eq {s : set E} (hs : convex s) : convex_hull s = s :=\nset.subset.antisymm (convex_hull_min (set.subset.refl _) hs) (subset_convex_hull s)\n\nlemma is_linear_map.image_convex_hull {f : E \u2192 F} (hf : is_linear_map \u211d f) :\n  f '' (convex_hull s) = convex_hull (f '' s) :=\nbegin\n  refine set.subset.antisymm _ _,\n  { rw [set.image_subset_iff],\n    exact convex_hull_min (set.image_subset_iff.1 $ subset_convex_hull $ f '' s)\n      ((convex_convex_hull (f '' s)).is_linear_preimage hf) },\n  { exact convex_hull_min (set.image_subset _ $ subset_convex_hull s)\n     ((convex_convex_hull s).is_linear_image hf) }\nend\n\nlemma linear_map.image_convex_hull (f : E \u2192\u2097[\u211d] F) :\n  f '' (convex_hull s) = convex_hull (f '' s) :=\nf.is_linear.image_convex_hull\n\nlemma finset.center_mass_mem_convex_hull (t : finset \u03b9) {w : \u03b9 \u2192 \u211d} (hw\u2080 : \u2200 i \u2208 t, 0 \u2264 w i)\n  (hws : 0 < t.sum w) {z : \u03b9 \u2192 E} (hz : \u2200 i \u2208 t, z i \u2208 s) :\n  t.center_mass w z \u2208 convex_hull s :=\n(convex_convex_hull s).center_mass_mem hw\u2080 hws (\u03bb i hi, subset_convex_hull s $ hz i hi)\n\n-- TODO : Do we need other versions of the next lemma?\n\n\/-- Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `z '' t \u2286 s`.\nThis version allows finsets in any type in any universe. -\/\nlemma convex_hull_eq (s : set E) :\n  convex_hull s = {x : E | \u2203 (\u03b9 : Type u') (t : finset \u03b9) (w : \u03b9 \u2192 \u211d) (z : \u03b9 \u2192 E)\n    (hw\u2080 : \u2200 i \u2208 t, 0 \u2264 w i) (hw\u2081 : t.sum w = 1) (hz : \u2200 i \u2208 t, z i \u2208 s) , t.center_mass w z = x} :=\nbegin\n  refine subset.antisymm (convex_hull_min _ _) _,\n  { intros x hx,\n    use [punit, finset.singleton punit.star, \u03bb _, 1, \u03bb _, x, \u03bb _ _, zero_le_one,\n      finset.sum_singleton, \u03bb _ _, hx],\n    simp only [finset.center_mass, finset.sum_singleton, inv_one, one_smul] },\n  { rintros x y \u27e8\u03b9, sx, wx, zx, hwx\u2080, hwx\u2081, hzx, rfl\u27e9 \u27e8\u03b9', sy, wy, zy, hwy\u2080, hwy\u2081, hzy, rfl\u27e9\n      a b ha hb hab,\n    rw [finset.center_mass_segment' _ _ _ _ _ _ hwx\u2081 hwy\u2081 _ _ hab],\n    refine \u27e8_, _, _, _, _, _, _, rfl\u27e9,\n    { rintros i hi,\n      rw [finset.mem_union, finset.mem_image, finset.mem_image] at hi,\n      rcases hi with \u27e8j, hj, rfl\u27e9|\u27e8j, hj, rfl\u27e9;\n        simp only [sum.elim_inl, sum.elim_inr];\n        apply_rules [mul_nonneg, hwx\u2080, hwy\u2080] },\n    { simp [finset.sum_sum_elim, finset.mul_sum.symm, *] },\n    { intros i hi,\n      rw [finset.mem_union, finset.mem_image, finset.mem_image] at hi,\n      rcases hi with \u27e8j, hj, rfl\u27e9|\u27e8j, hj, rfl\u27e9;\n        simp only [sum.elim_inl, sum.elim_inr]; apply_rules [hzx, hzy] } },\n  { rintros _ \u27e8\u03b9, t, w, z, hw\u2080, hw\u2081, hz, rfl\u27e9,\n    exact t.center_mass_mem_convex_hull hw\u2080 (hw\u2081.symm \u25b8 zero_lt_one) hz }\nend\n\n\/-- Maximum principle for convex functions. If a function `f` is convex on the convex hull of `s`,\nthen `f` can't have a maximum on `convex_hull s` outside of `s`. -\/\nlemma convex_on.exists_ge_of_mem_convex_hull {f : E \u2192 \u211d} (hf : convex_on (convex_hull s) f)\n  {x} (hx : x \u2208 convex_hull s) : \u2203 y \u2208 s, f x \u2264 f y :=\nbegin\n  rw convex_hull_eq at hx,\n  rcases hx with \u27e8\u03b1, t, w, z, hw\u2080, hw\u2081, hz, rfl\u27e9,\n  rcases hf.exists_ge_of_center_mass hw\u2080 (hw\u2081.symm \u25b8 zero_lt_one)\n    (\u03bb i hi, subset_convex_hull s (hz i hi)) with \u27e8i, hit, Hi\u27e9,\n  exact \u27e8z i, hz i hit, Hi\u27e9\nend\n\nlemma set.finite.convex_hull_eq {s : set E} (hs : finite s) :\n  convex_hull s = {x : E | \u2203 (w : E \u2192 \u211d) (hw\u2080 : \u2200 y \u2208 s, 0 \u2264 w y) (hw\u2081 : hs.to_finset.sum w = 1),\n    hs.to_finset.center_mass w id = x} :=\nbegin\n  refine subset.antisymm (convex_hull_min _ _) _,\n  { intros x hx,\n    replace hx : x \u2208 hs.to_finset, from finite.mem_to_finset.2 hx,\n    refine \u27e8_, _, _, finset.center_mass_ite_eq _ _ _ hx\u27e9,\n    { intros, split_ifs, exacts [zero_le_one, le_refl 0] },\n    { rw [finset.sum_ite_eq, if_pos hx] } },\n  { rintros x y \u27e8wx, hwx\u2080, hwx\u2081, rfl\u27e9 \u27e8wy, hwy\u2080, hwy\u2081, rfl\u27e9\n      a b ha hb hab,\n    rw [finset.center_mass_segment _ _ _ _ hwx\u2081 hwy\u2081 _ _ hab],\n    refine \u27e8_, _, _, rfl\u27e9,\n    { rintros i hi,\n      apply_rules [add_nonneg, mul_nonneg, hwx\u2080, hwy\u2080], },\n    { simp only [finset.sum_add_distrib, finset.mul_sum.symm, mul_one, *] } },\n  { rintros _ \u27e8w, hw\u2080, hw\u2081, rfl\u27e9,\n    exact hs.to_finset.center_mass_mem_convex_hull (\u03bb x hx, hw\u2080 _ $ finite.mem_to_finset.1 hx)\n      (hw\u2081.symm \u25b8 zero_lt_one) (\u03bb x hx, finite.mem_to_finset.1 hx) }\nend\n\nlemma is_linear_map.convex_hull_image {f : E \u2192 F} (hf : is_linear_map \u211d f) (s : set E) :\n  convex_hull (f '' s) = f '' convex_hull s :=\nset.subset.antisymm (convex_hull_min (image_subset _ (subset_convex_hull s)) $\n  (convex_convex_hull s).is_linear_image hf)\n  (image_subset_iff.2 $ convex_hull_min\n    (image_subset_iff.1 $ subset_convex_hull _)\n    ((convex_convex_hull _).is_linear_preimage hf))\n\nlemma linear_map.convex_hull_image (f : E \u2192\u2097[\u211d] F) (s : set E) :\n  convex_hull (f '' s) = f '' convex_hull s :=\nf.is_linear.convex_hull_image s\n\nend convex_hull\n\n\/-! ### Simplex -\/\n\nsection simplex\n\nvariables (\u03b9) [fintype \u03b9] {f : \u03b9 \u2192 \u211d}\n\n\/-- Standard simplex in the space of functions `\u03b9 \u2192 \u211d` is the set\nof vectors with non-negative coordinates with total sum `1`. -\/\ndef std_simplex (\u03b9 : Type*) [fintype \u03b9] : set (\u03b9 \u2192 \u211d) :=\n{ f | (\u2200 x, 0 \u2264 f x) \u2227 finset.univ.sum f = 1 }\n\nlemma std_simplex_eq_inter :\n  std_simplex \u03b9 = (\u22c2 x, {f | 0 \u2264 f x}) \u2229 {f | finset.univ.sum f = 1} :=\nby { ext f, simp only [std_simplex, set.mem_inter_eq, set.mem_Inter, set.mem_set_of_eq] }\n\nlemma convex_std_simplex : convex (std_simplex \u03b9) :=\nbegin\n  refine \u03bb f g hf hg a b ha hb hab, \u27e8\u03bb x, _, _\u27e9,\n  { apply_rules [add_nonneg, mul_nonneg, hf.1, hg.1] },\n  { erw [finset.sum_add_distrib, \u2190 finset.smul_sum, \u2190 finset.smul_sum, hf.2, hg.2,\n      smul_eq_mul, smul_eq_mul, mul_one, mul_one],\n    exact hab }\nend\n\nvariable {\u03b9}\n\nlemma ite_eq_mem_std_simplex (i : \u03b9) : (\u03bb j, ite (i = j) (1:\u211d) 0) \u2208 std_simplex \u03b9 :=\n\u27e8\u03bb j, by simp only []; split_ifs; norm_num, by rw [finset.sum_ite_eq, if_pos (finset.mem_univ _)] \u27e9\n\n\/-- `std_simplex \u03b9` is the convex hull of the canonical basis in `\u03b9 \u2192 \u211d`. -\/\nlemma convex_hull_basis_eq_std_simplex :\n  convex_hull (range $ \u03bb(i j:\u03b9), if i = j then (1:\u211d) else 0) = std_simplex \u03b9 :=\nbegin\n  refine subset.antisymm (convex_hull_min _ (convex_std_simplex \u03b9)) _,\n  { rintros _ \u27e8i, rfl\u27e9,\n    exact ite_eq_mem_std_simplex i },\n  { rintros w \u27e8hw\u2080, hw\u2081\u27e9,\n    rw [pi_eq_sum_univ w, \u2190 finset.univ.center_mass_eq_of_sum_1 _ hw\u2081],\n    exact finset.univ.center_mass_mem_convex_hull (\u03bb i hi, hw\u2080 i)\n      (hw\u2081.symm \u25b8 zero_lt_one) (\u03bb i hi, mem_range_self i) }\nend\n\nvariable {\u03b9}\n\n\/-- Convex hull of a finite set is the image of the standard simplex in `s \u2192 \u211d`\nunder the linear map sending each function `w` to `s.sum (\u03bb x, w x \u2022 x)`.\n\nSince we have no sums over finite sets, we use sum over `@finset.univ _ hs.fintype`.\nThe map is defined in terms of operations on `(s \u2192 \u211d) \u2192\u2097[\u211d] \u211d` so that later we will not need\nto prove that this map is linear. -\/\nlemma set.finite.convex_hull_eq_image {s : set E} (hs : finite s) :\n  convex_hull s =\n    (\u21d1((@finset.univ _ hs.fintype).sum (\u03bb x, (linear_map.proj x : (s \u2192 \u211d) \u2192\u2097[\u211d] \u211d).smul_right x.1))) ''\n      (@std_simplex _ hs.fintype) :=\nbegin\n  rw [\u2190 convex_hull_basis_eq_std_simplex, \u2190 linear_map.convex_hull_image, \u2190 range_comp, (\u2218)],\n  apply congr_arg,\n  convert (subtype.range_val s).symm,\n  ext x,\n  simp [linear_map.sum_apply, ite_smul, finset.filter_eq]\nend\n\n\/-- All values of a function `f \u2208 std_simplex \u03b9` belong to `[0, 1]`. -\/\nlemma mem_Icc_of_mem_std_simplex (hf : f \u2208 std_simplex \u03b9) (x) :\n  f x \u2208 I :=\n\u27e8hf.1 x, hf.2 \u25b8 finset.single_le_sum (\u03bb y hy, hf.1 y) (finset.mem_univ x)\u27e9\n\nend simplex\n","avg_line_length":39.9203539823,"max_line_length":113,"alphanum_fraction":0.6286023055}
-{"size":523,"ext":"lean","lang":"Lean","max_stars_count":130.0,"content":"import data.list\n\nopen list bool nat\ndefinition filter {A} : list A \u2192 (A \u2192 bool) \u2192 list A\n| filter [] p := []\n| filter (a :: l) p :=\n  match p a with\n  | tt := a :: filter l p\n  | ff := filter l p\n  end\n\nexample : list \u2115 := filter [0, 3, 7, 2, 4, 6, 3, 4]\n    (\u03bb(n : \u2115), begin induction n, exact tt, induction v_0, exact tt, exact ff end)\n\ndefinition foo  : list \u2115 := filter [0, 3, 7, 2, 4, 6, 3, 4]\n    (\u03bb(n : \u2115), begin induction n, exact tt, induction v_0, exact tt, exact ff end)\n\nexample : foo = [0, 2, 4, 6, 4] := rfl\n","avg_line_length":27.5263157895,"max_line_length":82,"alphanum_fraction":0.5659655832}
-{"size":8159,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2020 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-\/\nimport combinatorics.simple_graph.basic\nimport algebra.big_operators.basic\nimport data.nat.parity\nimport data.zmod.parity\n\n\/-!\n# Degree-sum formula and handshaking lemma\n\nThe degree-sum formula is that the sum of the degrees of the vertices in\na finite graph is equal to twice the number of edges.  The handshaking lemma,\na corollary, is that the number of odd-degree vertices is even.\n\n## Main definitions\n\n- A `dart` is a directed edge, consisting of an ordered pair of adjacent vertices,\n  thought of as being a directed edge.\n- `simple_graph.sum_degrees_eq_twice_card_edges` is the degree-sum formula.\n- `simple_graph.even_card_odd_degree_vertices` is the handshaking lemma.\n- `simple_graph.odd_card_odd_degree_vertices_ne` is that the number of odd-degree\n  vertices different from a given odd-degree vertex is odd.\n- `simple_graph.exists_ne_odd_degree_of_exists_odd_degree` is that the existence of an\n  odd-degree vertex implies the existence of another one.\n\n## Implementation notes\n\nWe give a combinatorial proof by using the facts that (1) the map from\ndarts to vertices is such that each fiber has cardinality the degree\nof the corresponding vertex and that (2) the map from darts to edges is 2-to-1.\n\n## Tags\n\nsimple graphs, sums, degree-sum formula, handshaking lemma\n-\/\nopen finset\n\nopen_locale big_operators\n\nnamespace simple_graph\nuniverses u\nvariables {V : Type u} (G : simple_graph V)\n\n\/-- A dart is a directed edge, consisting of an ordered pair of adjacent vertices. -\/\n@[ext, derive decidable_eq]\nstructure dart :=\n(fst snd : V)\n(is_adj : G.adj fst snd)\n\ninstance dart.fintype [fintype V] [decidable_rel G.adj] : fintype G.dart :=\nfintype.of_equiv (\u03a3 v, G.neighbor_set v)\n{ to_fun := \u03bb s, \u27e8s.fst, s.snd, s.snd.property\u27e9,\n  inv_fun := \u03bb d, \u27e8d.fst, d.snd, d.is_adj\u27e9,\n  left_inv := \u03bb s, by ext; simp,\n  right_inv := \u03bb d, by ext; simp }\n\nvariables {G}\n\n\/-- The edge associated to the dart. -\/\ndef dart.edge (d : G.dart) : sym2 V := \u27e6(d.fst, d.snd)\u27e7\n\n@[simp] lemma dart.edge_mem (d : G.dart) : d.edge \u2208 G.edge_set :=\nd.is_adj\n\n\/-- The dart with reversed orientation from a given dart. -\/\ndef dart.rev (d : G.dart) : G.dart :=\n\u27e8d.snd, d.fst, G.sym d.is_adj\u27e9\n\n@[simp] lemma dart.rev_edge (d : G.dart) : d.rev.edge = d.edge :=\nsym2.eq_swap\n\n@[simp] lemma dart.rev_rev (d : G.dart) : d.rev.rev = d :=\ndart.ext _ _ rfl rfl\n\n@[simp] lemma dart.rev_involutive : function.involutive (dart.rev : G.dart \u2192 G.dart) :=\ndart.rev_rev\n\nlemma dart.rev_ne (d : G.dart) : d.rev \u2260 d :=\nbegin\n  cases d with f s h,\n  simp only [dart.rev, not_and, ne.def],\n  rintro rfl,\n  exact false.elim (G.loopless _ h),\nend\n\nlemma dart_edge_eq_iff (d\u2081 d\u2082 : G.dart) :\n  d\u2081.edge = d\u2082.edge \u2194 d\u2081 = d\u2082 \u2228 d\u2081 = d\u2082.rev :=\nbegin\n  cases d\u2081 with s\u2081 t\u2081 h\u2081,\n  cases d\u2082 with s\u2082 t\u2082 h\u2082,\n  simp only [dart.edge, dart.rev_edge, dart.rev],\n  rw sym2.eq_iff,\nend\n\nvariables (G)\n\n\/-- For a given vertex `v`, this is the bijective map from the neighbor set at `v`\nto the darts `d` with `d.fst = v`. --\/\ndef dart_of_neighbor_set (v : V) (w : G.neighbor_set v) : G.dart :=\n\u27e8v, w, w.property\u27e9\n\nlemma dart_of_neighbor_set_injective (v : V) : function.injective (G.dart_of_neighbor_set v) :=\n\u03bb e\u2081 e\u2082 h, by { injection h with h\u2081 h\u2082, exact subtype.ext h\u2082 }\n\ninstance dart.inhabited [inhabited V] [inhabited (G.neighbor_set (default _))] :\n  inhabited G.dart := \u27e8G.dart_of_neighbor_set (default _) (default _)\u27e9\n\nsection degree_sum\nvariables [fintype V] [decidable_rel G.adj]\n\nlemma dart_fst_fiber [decidable_eq V] (v : V) :\n  univ.filter (\u03bb d : G.dart, d.fst = v) = univ.image (G.dart_of_neighbor_set v) :=\nbegin\n  ext d,\n  simp only [mem_image, true_and, mem_filter, set_coe.exists, mem_univ, exists_prop_of_true],\n  split,\n  { rintro rfl,\n    exact \u27e8_, d.is_adj, dart.ext _ _ rfl rfl\u27e9, },\n  { rintro \u27e8e, he, rfl\u27e9,\n    refl, },\nend\n\nlemma dart_fst_fiber_card_eq_degree [decidable_eq V] (v : V) :\n  (univ.filter (\u03bb d : G.dart, d.fst = v)).card = G.degree v :=\nbegin\n  have hh := card_image_of_injective univ (G.dart_of_neighbor_set_injective v),\n  rw [finset.card_univ, card_neighbor_set_eq_degree] at hh,\n  rwa dart_fst_fiber,\nend\n\nlemma dart_card_eq_sum_degrees : fintype.card G.dart = \u2211 v, G.degree v :=\nbegin\n  haveI h : decidable_eq V := by { classical, apply_instance },\n  simp only [\u2190card_univ, \u2190dart_fst_fiber_card_eq_degree],\n  exact card_eq_sum_card_fiberwise (by simp),\nend\n\nvariables {G} [decidable_eq V]\n\nlemma dart.edge_fiber (d : G.dart) :\n  (univ.filter (\u03bb (d' : G.dart), d'.edge = d.edge)) = {d, d.rev} :=\nfinset.ext (\u03bb d', by simpa using dart_edge_eq_iff d' d)\n\nvariables (G)\n\nlemma dart_edge_fiber_card (e : sym2 V) (h : e \u2208 G.edge_set) :\n  (univ.filter (\u03bb (d : G.dart), d.edge = e)).card = 2 :=\nbegin\n  refine quotient.ind (\u03bb p h, _) e h,\n  cases p with v w,\n  let d : G.dart := \u27e8v, w, h\u27e9,\n  convert congr_arg card d.edge_fiber,\n  rw [card_insert_of_not_mem, card_singleton],\n  rw [mem_singleton],\n  exact d.rev_ne.symm,\nend\n\nlemma dart_card_eq_twice_card_edges : fintype.card G.dart = 2 * G.edge_finset.card :=\nbegin\n  rw \u2190card_univ,\n  rw @card_eq_sum_card_fiberwise _ _ _ dart.edge _ G.edge_finset\n    (\u03bb d h, by { rw mem_edge_finset, apply dart.edge_mem }),\n  rw [\u2190mul_comm, sum_const_nat],\n  intros e h,\n  apply G.dart_edge_fiber_card e,\n  rwa \u2190mem_edge_finset,\nend\n\n\/-- The degree-sum formula.  This is also known as the handshaking lemma, which might\nmore specifically refer to `simple_graph.even_card_odd_degree_vertices`. -\/\ntheorem sum_degrees_eq_twice_card_edges : \u2211 v, G.degree v = 2 * G.edge_finset.card :=\nG.dart_card_eq_sum_degrees.symm.trans G.dart_card_eq_twice_card_edges\n\nend degree_sum\n\n\/-- The handshaking lemma.  See also `simple_graph.sum_degrees_eq_twice_card_edges`. -\/\ntheorem even_card_odd_degree_vertices [fintype V] [decidable_rel G.adj] :\n  even (univ.filter (\u03bb v, odd (G.degree v))).card :=\nbegin\n  classical,\n  have h := congr_arg ((\u03bb n, \u2191n) : \u2115 \u2192 zmod 2) G.sum_degrees_eq_twice_card_edges,\n  simp only [zmod.nat_cast_self, zero_mul, nat.cast_mul] at h,\n  rw [sum_nat_cast, \u2190sum_filter_ne_zero] at h,\n  rw @sum_congr _ (zmod 2) _ _ (\u03bb v, (G.degree v : zmod 2)) (\u03bb v, (1 : zmod 2)) _ rfl at h,\n  { simp only [filter_congr_decidable, mul_one, nsmul_eq_mul, sum_const, ne.def] at h,\n    rw \u2190zmod.eq_zero_iff_even,\n    convert h,\n    ext v,\n    rw \u2190zmod.ne_zero_iff_odd,\n    congr' },\n  { intros v,\n    simp only [true_and, mem_filter, mem_univ, ne.def],\n    rw [zmod.eq_zero_iff_even, zmod.eq_one_iff_odd, nat.odd_iff_not_even, imp_self],\n    trivial }\nend\n\nlemma odd_card_odd_degree_vertices_ne [fintype V] [decidable_eq V] [decidable_rel G.adj]\n  (v : V) (h : odd (G.degree v)) :\n  odd (univ.filter (\u03bb w, w \u2260 v \u2227 odd (G.degree w))).card :=\nbegin\n  rcases G.even_card_odd_degree_vertices with \u27e8k, hg\u27e9,\n  have hk : 0 < k,\n  { have hh : (filter (\u03bb (v : V), odd (G.degree v)) univ).nonempty,\n    { use v,\n      simp only [true_and, mem_filter, mem_univ],\n      use h, },\n    rwa [\u2190card_pos, hg, zero_lt_mul_left] at hh,\n    exact zero_lt_two, },\n  have hc : (\u03bb (w : V), w \u2260 v \u2227 odd (G.degree w)) = (\u03bb (w : V), odd (G.degree w) \u2227 w \u2260 v),\n  { ext w,\n    rw and_comm, },\n  simp only [hc, filter_congr_decidable],\n  rw [\u2190filter_filter, filter_ne', card_erase_of_mem],\n  { use k - 1,\n    rw [nat.pred_eq_succ_iff, hg, nat.mul_sub_left_distrib, \u2190 nat.sub_add_comm, eq_comm,\n      \u2190 (nat.sub_eq_iff_eq_add _).symm],\n    { ring },\n    { exact add_le_add_right (zero_le (2 * k)) 2 },\n    { exact nat.mul_le_mul_left _ hk } },\n  { simpa only [true_and, mem_filter, mem_univ] },\nend\n\nlemma exists_ne_odd_degree_of_exists_odd_degree [fintype V] [decidable_rel G.adj]\n  (v : V) (h : odd (G.degree v)) :\n  \u2203 (w : V), w \u2260 v \u2227 odd (G.degree w) :=\nbegin\n  haveI : decidable_eq V := by { classical, apply_instance },\n  rcases G.odd_card_odd_degree_vertices_ne v h with \u27e8k, hg\u27e9,\n  have hg' : (filter (\u03bb (w : V), w \u2260 v \u2227 odd (G.degree w)) univ).card > 0,\n  { rw hg,\n    apply nat.succ_pos, },\n  rcases card_pos.mp hg' with \u27e8w, hw\u27e9,\n  simp only [true_and, mem_filter, mem_univ, ne.def] at hw,\n  exact \u27e8w, hw\u27e9,\nend\n\nend simple_graph\n","avg_line_length":33.8547717842,"max_line_length":95,"alphanum_fraction":0.6960411815}
-{"size":11594,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Guy Leroy. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes H\u00f6lzl\n\nGreatest common divisor (gcd) and least common multiple (lcm) for integers.\n\nThis sets up \u2124 to be a GCD domain, and introduces rules about `int.gcd`, as well as `int.lcm`.\nNOTE: If you add rules to this theory, check that the corresponding rules are available for\n`gcd_domain`.\n-\/\nimport data.int.basic data.nat.prime algebra.gcd_domain\n\n\/- Extended Euclidean algorithm -\/\nnamespace nat\n\ndef xgcd_aux : \u2115 \u2192 \u2124 \u2192 \u2124 \u2192 \u2115 \u2192 \u2124 \u2192 \u2124 \u2192 \u2115 \u00d7 \u2124 \u00d7 \u2124\n| 0          s t r' s' t' := (r', s', t')\n| r@(succ _) s t r' s' t' :=\n  have r' % r < r, from mod_lt _ $ succ_pos _,\n  let q := r' \/ r in xgcd_aux (r' % r) (s' - q * s) (t' - q * t) r s t\n\n@[simp] theorem xgcd_zero_left {s t r' s' t'} : xgcd_aux 0 s t r' s' t' = (r', s', t') :=\nby simp [xgcd_aux]\n\n@[simp] theorem xgcd_aux_rec {r s t r' s' t'} (h : 0 < r) : xgcd_aux r s t r' s' t' = xgcd_aux (r' % r) (s' - (r' \/ r) * s) (t' - (r' \/ r) * t) r s t :=\nby cases r; [exact absurd h (lt_irrefl _), {simp only [xgcd_aux], refl}]\n\n\/-- Use the extended GCD algorithm to generate the `a` and `b` values\n  satisfying `gcd x y = x * a + y * b`. -\/\ndef xgcd (x y : \u2115) : \u2124 \u00d7 \u2124 := (xgcd_aux x 1 0 y 0 1).2\n\n\/-- The extended GCD `a` value in the equation `gcd x y = x * a + y * b`. -\/\ndef gcd_a (x y : \u2115) : \u2124 := (xgcd x y).1\n\n\/-- The extended GCD `b` value in the equation `gcd x y = x * a + y * b`. -\/\ndef gcd_b (x y : \u2115) : \u2124 := (xgcd x y).2\n\n@[simp] theorem xgcd_aux_fst (x y) : \u2200 s t s' t',\n  (xgcd_aux x s t y s' t').1 = gcd x y :=\ngcd.induction x y (by simp) (\u03bb x y h IH s t s' t', by simp [h, IH]; rw \u2190 gcd_rec)\n\ntheorem xgcd_aux_val (x y) : xgcd_aux x 1 0 y 0 1 = (gcd x y, xgcd x y) :=\nby rw [xgcd, \u2190 xgcd_aux_fst x y 1 0 0 1]; cases xgcd_aux x 1 0 y 0 1; refl\n\ntheorem xgcd_val (x y) : xgcd x y = (gcd_a x y, gcd_b x y) :=\nby unfold gcd_a gcd_b; cases xgcd x y; refl\n\nsection\nparameters (a b : \u2115)\n\nprivate def P : \u2115 \u00d7 \u2124 \u00d7 \u2124 \u2192 Prop | (r, s, t) := (r : \u2124) = a * s + b * t\n\ntheorem xgcd_aux_P {r r'} : \u2200 {s t s' t'}, P (r, s, t) \u2192 P (r', s', t') \u2192 P (xgcd_aux r s t r' s' t') :=\ngcd.induction r r' (by simp) $ \u03bb x y h IH s t s' t' p p', begin\n  rw [xgcd_aux_rec h], refine IH _ p, dsimp [P] at *,\n  rw [int.mod_def], generalize : (y \/ x : \u2124) = k,\n  rw [p, p'], simp [mul_add, mul_comm, mul_left_comm]\nend\n\ntheorem gcd_eq_gcd_ab : (gcd a b : \u2124) = a * gcd_a a b + b * gcd_b a b :=\nby have := @xgcd_aux_P a b a b 1 0 0 1 (by simp [P]) (by simp [P]);\n   rwa [xgcd_aux_val, xgcd_val] at this\nend\n\nend nat\n\nnamespace int\n\ntheorem nat_abs_div (a b : \u2124) (H : b \u2223 a) : nat_abs (a \/ b) = (nat_abs a) \/ (nat_abs b) :=\nbegin\n  cases (nat.eq_zero_or_pos (nat_abs b)),\n  rw eq_zero_of_nat_abs_eq_zero h,\n  simp,\n  calc\n  nat_abs (a \/ b) = nat_abs (a \/ b) * 1 : by rw mul_one\n    ... = nat_abs (a \/ b) * (nat_abs b \/ nat_abs b) : by rw nat.div_self h\n    ... = nat_abs (a \/ b) * nat_abs b \/ nat_abs b : by rw (nat.mul_div_assoc _ (dvd_refl _))\n    ... = nat_abs (a \/ b * b) \/ nat_abs b : by rw (nat_abs_mul (a \/ b) b)\n    ... = nat_abs a \/ nat_abs b : by rw int.div_mul_cancel H,\nend\n\ntheorem nat_abs_dvd_abs_iff {i j : \u2124} : i.nat_abs \u2223 j.nat_abs \u2194 i \u2223 j :=\n\u27e8assume (H : i.nat_abs \u2223 j.nat_abs), dvd_nat_abs.mp (nat_abs_dvd.mp (coe_nat_dvd.mpr H)),\nassume H : (i \u2223 j), coe_nat_dvd.mp (dvd_nat_abs.mpr (nat_abs_dvd.mpr H))\u27e9\n\nlemma succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : \u2115} (p_prime : nat.prime p) {m n : \u2124} {k l : \u2115}\n      (hpm : \u2191(p ^ k) \u2223 m)\n      (hpn : \u2191(p ^ l) \u2223 n) (hpmn : \u2191(p ^ (k+l+1)) \u2223 m*n) : \u2191(p ^ (k+1)) \u2223 m \u2228 \u2191(p ^ (l+1)) \u2223 n :=\nhave hpm' : p ^ k \u2223 m.nat_abs, from int.coe_nat_dvd.1 $ int.dvd_nat_abs.2 hpm,\nhave hpn' : p ^ l \u2223 n.nat_abs, from int.coe_nat_dvd.1 $ int.dvd_nat_abs.2 hpn,\nhave hpmn' : (p ^ (k+l+1)) \u2223 m.nat_abs*n.nat_abs,\n  by rw \u2190int.nat_abs_mul; apply (int.coe_nat_dvd.1 $ int.dvd_nat_abs.2 hpmn),\nlet hsd := nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn' in\nhsd.elim\n  (\u03bb hsd1, or.inl begin apply int.dvd_nat_abs.1, apply int.coe_nat_dvd.2 hsd1 end)\n  (\u03bb hsd2, or.inr begin apply int.dvd_nat_abs.1, apply int.coe_nat_dvd.2 hsd2 end)\n\ntheorem dvd_of_mul_dvd_mul_left {i j k : \u2124} (k_non_zero : k \u2260 0) (H : k * i \u2223 k * j) : i \u2223 j :=\ndvd.elim H (\u03bbl H1, by rw mul_assoc at H1; exact \u27e8_, eq_of_mul_eq_mul_left k_non_zero H1\u27e9)\n\ntheorem dvd_of_mul_dvd_mul_right {i j k : \u2124} (k_non_zero : k \u2260 0) (H : i * k \u2223 j * k) : i \u2223 j :=\nby rw [mul_comm i k, mul_comm j k] at H; exact dvd_of_mul_dvd_mul_left k_non_zero H\n\nsection normalization_domain\n\ninstance : normalization_domain \u2124 :=\n{ norm_unit      := \u03bba:\u2124, if 0 \u2264 a then 1 else -1,\n  norm_unit_zero := if_pos (le_refl _),\n  norm_unit_mul  := assume a b hna hnb,\n  begin\n    by_cases ha : 0 \u2264 a; by_cases hb : 0 \u2264 b; simp [ha, hb],\n    exact if_pos (mul_nonneg ha hb),\n    exact if_neg (assume h, hb $ nonneg_of_mul_nonneg_left h $ lt_of_le_of_ne ha hna.symm),\n    exact if_neg (assume h, ha $ nonneg_of_mul_nonneg_right h $ lt_of_le_of_ne hb hnb.symm),\n    exact if_pos (mul_nonneg_of_nonpos_of_nonpos (le_of_not_ge ha) (le_of_not_ge hb))\n  end,\n  norm_unit_coe_units := assume u, (units_eq_one_or u).elim\n    (assume eq, eq.symm \u25b8 if_pos zero_le_one)\n    (assume eq, eq.symm \u25b8 if_neg (not_le_of_gt $ show (-1:\u2124) < 0, by simp [@neg_lt \u2124 _ 1 0])),\n  .. (infer_instance : integral_domain \u2124) }\n\nlemma norm_unit_of_nonneg {z : \u2124} (h : 0 \u2264 z) : norm_unit z = 1 :=\nby unfold norm_unit; exact if_pos h\n\nlemma norm_unit_of_neg {z : \u2124} (h : z < 0) : norm_unit z = -1 :=\nby unfold norm_unit; exact if_neg (not_le_of_gt h)\n\nlemma norm_unit_nat_coe (n : \u2115) : norm_unit (n : \u2124) = 1 :=\nnorm_unit_of_nonneg (coe_nat_le_coe_nat_of_le $ nat.zero_le n)\n\ntheorem coe_nat_abs_eq_mul_norm_unit {z : \u2124} : (z.nat_abs : \u2124) = z * norm_unit z :=\nbegin\n  by_cases 0 \u2264 z,\n  { simp [nat_abs_of_nonneg h, norm_unit_of_nonneg h] },\n  { simp [of_nat_nat_abs_of_nonpos (le_of_not_ge h), norm_unit_of_neg (lt_of_not_ge h)] }\nend\n\nend normalization_domain\n\n\/-- \u2124 specific version of least common multiple. -\/\ndef lcm (i j : \u2124) : \u2115 := nat.lcm (nat_abs i) (nat_abs j)\n\ntheorem lcm_def (i j : \u2124) : lcm i j = nat.lcm (nat_abs i) (nat_abs j) := rfl\n\nsection gcd_domain\n\ntheorem gcd_dvd_left (i j : \u2124) : (gcd i j : \u2124) \u2223 i :=\ndvd_nat_abs.mp $ coe_nat_dvd.mpr $ nat.gcd_dvd_left _ _\n\ntheorem gcd_dvd_right (i j : \u2124) : (gcd i j : \u2124) \u2223 j :=\ndvd_nat_abs.mp $ coe_nat_dvd.mpr $ nat.gcd_dvd_right _ _\n\ntheorem dvd_gcd {i j k : \u2124} (h1 : k \u2223 i) (h2 : k \u2223 j) : k \u2223 gcd i j :=\nnat_abs_dvd.1 $ coe_nat_dvd.2 $ nat.dvd_gcd (nat_abs_dvd_abs_iff.2 h1) (nat_abs_dvd_abs_iff.2 h2)\n\ntheorem gcd_mul_lcm (i j : \u2124) : gcd i j * lcm i j = nat_abs (i * j) :=\nby rw [int.gcd, int.lcm, nat.gcd_mul_lcm, nat_abs_mul]\n\ninstance : gcd_domain \u2124 :=\n{ gcd            := \u03bba b, int.gcd a b,\n  lcm            := \u03bba b, int.lcm a b,\n  gcd_dvd_left   := assume a b, int.gcd_dvd_left _ _,\n  gcd_dvd_right  := assume a b, int.gcd_dvd_right _ _,\n  dvd_gcd        := assume a b c, dvd_gcd,\n  norm_unit_gcd  := assume a b, norm_unit_nat_coe _,\n  gcd_mul_lcm    := by intros; rw [\u2190 int.coe_nat_mul, gcd_mul_lcm, coe_nat_abs_eq_mul_norm_unit],\n  lcm_zero_left  := assume a, coe_nat_eq_zero.2 $ nat.lcm_zero_left _,\n  lcm_zero_right := assume a, coe_nat_eq_zero.2 $ nat.lcm_zero_right _,\n  .. int.normalization_domain }\n\nlemma coe_gcd (i j : \u2124) : \u2191(int.gcd i j) = gcd_domain.gcd i j := rfl\nlemma coe_lcm (i j : \u2124) : \u2191(int.lcm i j) = gcd_domain.lcm i j := rfl\n\nlemma nat_abs_gcd (i j : \u2124) : nat_abs (gcd_domain.gcd i j) = int.gcd i j := rfl\nlemma nat_abs_lcm (i j : \u2124) : nat_abs (gcd_domain.lcm i j) = int.lcm i j := rfl\n\nend gcd_domain\n\ntheorem gcd_comm (i j : \u2124) : gcd i j = gcd j i := nat.gcd_comm _ _\n\ntheorem gcd_assoc (i j k : \u2124) : gcd (gcd i j) k = gcd i (gcd j k) := nat.gcd_assoc _ _ _\n\n@[simp] theorem gcd_self (i : \u2124) : gcd i i = nat_abs i := by simp [gcd]\n\n@[simp] theorem gcd_zero_left (i : \u2124) : gcd 0 i = nat_abs i := by simp [gcd]\n\n@[simp] theorem gcd_zero_right (i : \u2124) : gcd i 0 = nat_abs i := by simp [gcd]\n\n@[simp] theorem gcd_one_left (i : \u2124) : gcd 1 i = 1 := nat.gcd_one_left _\n\n@[simp] theorem gcd_one_right (i : \u2124) : gcd i 1 = 1 := nat.gcd_one_right _\n\ntheorem gcd_mul_left (i j k : \u2124) : gcd (i * j) (i * k) = nat_abs i * gcd j k :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_gcd, coe_nat_abs_eq_mul_norm_unit]\n\ntheorem gcd_mul_right (i j k : \u2124) : gcd (i * j) (k * j) = gcd i k * nat_abs j :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_gcd, coe_nat_abs_eq_mul_norm_unit]\n\ntheorem gcd_pos_of_non_zero_left {i : \u2124} (j : \u2124) (i_non_zero : i \u2260 0) : gcd i j > 0 :=\nnat.gcd_pos_of_pos_left (nat_abs j) (nat_abs_pos_of_ne_zero i_non_zero)\n\ntheorem gcd_pos_of_non_zero_right (i : \u2124) {j : \u2124} (j_non_zero : j \u2260 0) : gcd i j > 0 :=\nnat.gcd_pos_of_pos_right (nat_abs i) (nat_abs_pos_of_ne_zero j_non_zero)\n\ntheorem gcd_eq_zero_iff {i j : \u2124} : gcd i j = 0 \u2194 i = 0 \u2227 j = 0 :=\nby rw [\u2190 int.coe_nat_eq_coe_nat_iff, int.coe_nat_zero, coe_gcd, gcd_eq_zero_iff]\n\ntheorem gcd_div {i j k : \u2124} (H1 : k \u2223 i) (H2 : k \u2223 j) :\n  gcd (i \/ k) (j \/ k) = gcd i j \/ nat_abs k :=\nby rw [gcd, nat_abs_div i k H1, nat_abs_div j k H2];\nexact nat.gcd_div (nat_abs_dvd_abs_iff.mpr H1) (nat_abs_dvd_abs_iff.mpr H2)\n\ntheorem gcd_dvd_gcd_of_dvd_left {i k : \u2124} (j : \u2124) (H : i \u2223 k) : gcd i j \u2223 gcd k j :=\nint.coe_nat_dvd.1 $ dvd_gcd (dvd.trans (gcd_dvd_left i j) H) (gcd_dvd_right i j)\n\ntheorem gcd_dvd_gcd_of_dvd_right {i k : \u2124} (j : \u2124) (H : i \u2223 k) : gcd j i \u2223 gcd j k :=\nint.coe_nat_dvd.1 $ dvd_gcd (gcd_dvd_left j i) (dvd.trans (gcd_dvd_right j i) H)\n\ntheorem gcd_dvd_gcd_mul_left (i j k : \u2124) : gcd i j \u2223 gcd (k * i) j :=\ngcd_dvd_gcd_of_dvd_left _ (dvd_mul_left _ _)\n\ntheorem gcd_dvd_gcd_mul_right (i j k : \u2124) : gcd i j \u2223 gcd (i * k) j :=\ngcd_dvd_gcd_of_dvd_left _ (dvd_mul_right _ _)\n\ntheorem gcd_dvd_gcd_mul_left_right (i j k : \u2124) : gcd i j \u2223 gcd i (k * j) :=\ngcd_dvd_gcd_of_dvd_right _ (dvd_mul_left _ _)\n\ntheorem gcd_dvd_gcd_mul_right_right (i j k : \u2124) : gcd i j \u2223 gcd i (j * k) :=\ngcd_dvd_gcd_of_dvd_right _ (dvd_mul_right _ _)\n\ntheorem gcd_eq_left {i j : \u2124} (H : i \u2223 j) : gcd i j = nat_abs i :=\nnat.dvd_antisymm (by unfold gcd; exact nat.gcd_dvd_left _ _)\n                 (by unfold gcd; exact nat.dvd_gcd (dvd_refl _) (nat_abs_dvd_abs_iff.mpr H))\n\ntheorem gcd_eq_right {i j : \u2124} (H : j \u2223 i) : gcd i j = nat_abs j :=\nby rw [gcd_comm, gcd_eq_left H]\n\n\/- lcm -\/\n\ntheorem lcm_comm (i j : \u2124) : lcm i j = lcm j i :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, lcm_comm]\n\ntheorem lcm_assoc (i j k : \u2124) : lcm (lcm i j) k = lcm i (lcm j k) :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, lcm_assoc]\n\n@[simp] theorem lcm_zero_left (i : \u2124) : lcm 0 i = 0 :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm]\n\n@[simp] theorem lcm_zero_right (i : \u2124) : lcm i 0 = 0 :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm]\n\n@[simp] theorem lcm_one_left (i : \u2124) : lcm 1 i = nat_abs i :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_mul_norm_unit]\n\n@[simp] theorem lcm_one_right (i : \u2124) : lcm i 1 = nat_abs i :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_mul_norm_unit]\n\n@[simp] theorem lcm_self (i : \u2124) : lcm i i = nat_abs i :=\nby simp [(int.coe_nat_eq_coe_nat_iff _ _).symm, coe_lcm, coe_nat_abs_eq_mul_norm_unit]\n\ntheorem dvd_lcm_left (i j : \u2124) : i \u2223 lcm i j :=\nby rw [coe_lcm]; exact dvd_lcm_left _ _\n\ntheorem dvd_lcm_right (i j : \u2124) : j \u2223 lcm i j :=\nby rw [coe_lcm]; exact dvd_lcm_right _ _\n\ntheorem lcm_dvd {i j k : \u2124}  : i \u2223 k \u2192 j \u2223 k \u2192 (lcm i j : \u2124) \u2223 k :=\nby rw [coe_lcm]; exact lcm_dvd\n\nend int\n","avg_line_length":42.625,"max_line_length":152,"alphanum_fraction":0.6446437813}
-{"size":784,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-\/\nprelude\nimport Init.Data.Nat\n\nnamespace Lean\n\ninductive Occurrences\n| all\n| pos (idxs : List Nat)\n| neg (idxs : List Nat)\n\nnamespace Occurrences\n\ninstance : Inhabited Occurrences := \u27e8all\u27e9\n\ndef contains : Occurrences \u2192 Nat \u2192 Bool\n| all,      _   => true\n| pos idxs, idx => idxs.contains idx\n| neg idxs, idx => !idxs.contains idx\n\ndef isAll : Occurrences \u2192 Bool\n| all => true\n| _   => false\n\ndef beq : Occurrences \u2192 Occurrences \u2192 Bool\n| all,     all     => true\n| pos is\u2081, pos is\u2082 => is\u2081 == is\u2082\n| neg is\u2081, neg is\u2082 => is\u2081 == is\u2082\n| _,       _       => false\n\ninstance : HasBeq Occurrences := \u27e8beq\u27e9\n\nend Occurrences\n\nend Lean\n","avg_line_length":19.6,"max_line_length":67,"alphanum_fraction":0.6670918367}
-{"size":200,"ext":"lean","lang":"Lean","max_stars_count":130.0,"content":"inductive func (A B : Type) :=\nmk : (A \u2192 B) \u2192 func A B\n\nsection\n  variables {A B : Type}\n  definition to_function (F : func A B) : A \u2192 B :=\n  func.rec (\u03bbf, f) F\n  attribute to_function [coercion]\nend\n","avg_line_length":20.0,"max_line_length":50,"alphanum_fraction":0.615}
-{"size":13543,"ext":"lean","lang":"Lean","max_stars_count":78.0,"content":"\/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad\n-\/\nimport data.list.perm\n\n\/-!\n# Sorting algorithms on lists\n\nIn this file we define `list.sorted r l` to be an alias for `pairwise r l`. This alias is preferred\nin the case that `r` is a `<` or `\u2264`-like relation. Then we define two sorting algorithms:\n`list.insertion_sort` and `list.merge_sort`, and prove their correctness.\n-\/\n\nopen list.perm\n\nuniverse uu\n\nnamespace list\n\n\/-!\n### The predicate `list.sorted`\n-\/\n\nsection sorted\n\nvariables {\u03b1 : Type uu} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a : \u03b1} {l : list \u03b1}\n\n\/-- `sorted r l` is the same as `pairwise r l`, preferred in the case that `r`\n  is a `<` or `\u2264`-like relation (transitive and antisymmetric or asymmetric) -\/\ndef sorted := @pairwise\n\ninstance decidable_sorted [decidable_rel r] (l : list \u03b1) : decidable (sorted r l) :=\nlist.decidable_pairwise _\n\n@[simp] theorem sorted_nil : sorted r [] := pairwise.nil\n\nlemma sorted.of_cons : sorted r (a :: l) \u2192 sorted r l := pairwise.of_cons\n\ntheorem sorted.tail {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {l : list \u03b1} (h : sorted r l) : sorted r l.tail :=\nh.tail\n\ntheorem rel_of_sorted_cons {a : \u03b1} {l : list \u03b1} : sorted r (a :: l) \u2192\n  \u2200 b \u2208 l, r a b :=\nrel_of_pairwise_cons\n\n@[simp] theorem sorted_cons {a : \u03b1} {l : list \u03b1} :\n  sorted r (a :: l) \u2194 (\u2200 b \u2208 l, r a b) \u2227 sorted r l :=\npairwise_cons\n\nprotected theorem sorted.nodup {r : \u03b1 \u2192 \u03b1 \u2192 Prop} [is_irrefl \u03b1 r] {l : list \u03b1} (h : sorted r l) :\n  nodup l :=\nh.nodup\n\ntheorem eq_of_perm_of_sorted [is_antisymm \u03b1 r]\n  {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) (s\u2081 : sorted r l\u2081) (s\u2082 : sorted r l\u2082) : l\u2081 = l\u2082 :=\nbegin\n  induction s\u2081 with a l\u2081 h\u2081 s\u2081 IH generalizing l\u2082,\n  { exact p.nil_eq },\n  { have : a \u2208 l\u2082 := p.subset (mem_cons_self _ _),\n    rcases mem_split this with \u27e8u\u2082, v\u2082, rfl\u27e9,\n    have p' := (perm_cons a).1 (p.trans perm_middle),\n    obtain rfl := IH p' (s\u2082.sublist $ by simp),\n    change a::u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082), rw \u2190 append_assoc, congr,\n    have : \u2200 (x : \u03b1) (h : x \u2208 u\u2082), x = a := \u03bb x m,\n      antisymm ((pairwise_append.1 s\u2082).2.2 _ m a (mem_cons_self _ _))\n        (h\u2081 _ (by simp [m])),\n    rw [(@eq_repeat _ a (length u\u2082 + 1) (a::u\u2082)).2,\n        (@eq_repeat _ a (length u\u2082 + 1) (u\u2082++[a])).2];\n    split; simp [iff_true_intro this, or_comm] }\nend\n\ntheorem sublist_of_subperm_of_sorted [is_antisymm \u03b1 r]\n  {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 <+~ l\u2082) (s\u2081 : l\u2081.sorted r) (s\u2082 : l\u2082.sorted r) : l\u2081 <+ l\u2082 :=\nlet \u27e8_, h, h'\u27e9 := p in by rwa \u2190eq_of_perm_of_sorted h (s\u2082.sublist h') s\u2081\n\n@[simp] theorem sorted_singleton (a : \u03b1) : sorted r [a] := pairwise_singleton _ _\n\nlemma sorted.rel_nth_le_of_lt {l : list \u03b1}\n  (h : l.sorted r) {a b : \u2115} (ha : a < l.length) (hb : b < l.length) (hab : a < b) :\n  r (l.nth_le a ha) (l.nth_le b hb) :=\nlist.pairwise_iff_nth_le.1 h a b hb hab\n\nlemma sorted.rel_nth_le_of_le [is_refl \u03b1 r] {l : list \u03b1}\n  (h : l.sorted r) {a b : \u2115} (ha : a < l.length) (hb : b < l.length) (hab : a \u2264 b) :\n  r (l.nth_le a ha) (l.nth_le b hb) :=\nbegin\n  cases eq_or_lt_of_le hab with H H,\n  { subst H, exact refl _ },\n  { exact h.rel_nth_le_of_lt _ _ H }\nend\n\nlemma sorted.rel_of_mem_take_of_mem_drop {l : list \u03b1} (h : list.sorted r l)\n  {k : \u2115} {x y : \u03b1} (hx : x \u2208 list.take k l) (hy : y \u2208 list.drop k l) :\n  r x y :=\nbegin\n  obtain \u27e8iy, hiy, rfl\u27e9 := nth_le_of_mem hy,\n  obtain \u27e8ix, hix, rfl\u27e9 := nth_le_of_mem hx,\n  rw [nth_le_take', nth_le_drop'],\n  rw length_take at hix,\n  exact h.rel_nth_le_of_lt _ _ (ix.lt_add_right _ _ (lt_min_iff.mp hix).left)\nend\n\nend sorted\n\nsection sort\nvariables {\u03b1 : Type uu} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [decidable_rel r]\nlocal infix ` \u227c ` : 50 := r\n\n\/-! ### Insertion sort -\/\n\nsection insertion_sort\n\n\/-- `ordered_insert a l` inserts `a` into `l` at such that\n  `ordered_insert a l` is sorted if `l` is. -\/\n@[simp] def ordered_insert (a : \u03b1) : list \u03b1 \u2192 list \u03b1\n| []       := [a]\n| (b :: l) := if a \u227c b then a :: b :: l else b :: ordered_insert l\n\n\/-- `insertion_sort l` returns `l` sorted using the insertion sort algorithm. -\/\n@[simp] def insertion_sort : list \u03b1 \u2192 list \u03b1\n| []       := []\n| (b :: l) := ordered_insert r b (insertion_sort l)\n\n@[simp] lemma ordered_insert_nil (a : \u03b1) : [].ordered_insert r a = [a] := rfl\n\ntheorem ordered_insert_length : \u03a0 (L : list \u03b1) (a : \u03b1), (L.ordered_insert r a).length = L.length + 1\n| [] a := rfl\n| (hd :: tl) a := by { dsimp [ordered_insert], split_ifs; simp [ordered_insert_length], }\n\n\/-- An alternative definition of `ordered_insert` using `take_while` and `drop_while`. -\/\nlemma ordered_insert_eq_take_drop (a : \u03b1) : \u2200 l : list \u03b1,\n  l.ordered_insert r a = l.take_while (\u03bb b, \u00ac(a \u227c b)) ++ (a :: l.drop_while (\u03bb b, \u00ac(a \u227c b)))\n| [] := rfl\n| (b :: l) := by { dsimp only [ordered_insert], split_ifs; simp [take_while, drop_while, *] }\n\nlemma insertion_sort_cons_eq_take_drop (a : \u03b1) (l : list \u03b1) :\n  insertion_sort r (a :: l) = (insertion_sort r l).take_while (\u03bb b, \u00ac(a \u227c b)) ++\n    (a :: (insertion_sort r l).drop_while (\u03bb b, \u00ac(a \u227c b))) :=\nordered_insert_eq_take_drop r a _\n\nsection correctness\nopen perm\n\ntheorem perm_ordered_insert (a) : \u2200 l : list \u03b1, ordered_insert r a l ~ a :: l\n| []       := perm.refl _\n| (b :: l) := by by_cases a \u227c b; [simp [ordered_insert, h],\n  simpa [ordered_insert, h] using\n    ((perm_ordered_insert l).cons _).trans (perm.swap _ _ _)]\n\ntheorem ordered_insert_count [decidable_eq \u03b1] (L : list \u03b1) (a b : \u03b1) :\n  count a (L.ordered_insert r b) = count a L + if (a = b) then 1 else 0 :=\nbegin\n  rw [(L.perm_ordered_insert r b).count_eq, count_cons],\n  split_ifs; simp only [nat.succ_eq_add_one, add_zero],\nend\n\ntheorem perm_insertion_sort : \u2200 l : list \u03b1, insertion_sort r l ~ l\n| []       := perm.nil\n| (b :: l) := by simpa [insertion_sort] using\n  (perm_ordered_insert _ _ _).trans ((perm_insertion_sort l).cons b)\n\nvariable {r}\n\n\/-- If `l` is already `list.sorted` with respect to `r`, then `insertion_sort` does not change\nit. -\/\nlemma sorted.insertion_sort_eq : \u2200 {l : list \u03b1} (h : sorted r l), insertion_sort r l = l\n| [] _ := rfl\n| [a] _ := rfl\n| (a :: b :: l) h :=\n  begin\n    rw [insertion_sort, sorted.insertion_sort_eq, ordered_insert, if_pos],\n    exacts [rel_of_sorted_cons h _ (or.inl rfl), h.tail]\n  end\n\nsection total_and_transitive\nvariables [is_total \u03b1 r] [is_trans \u03b1 r]\n\ntheorem sorted.ordered_insert (a : \u03b1) : \u2200 l, sorted r l \u2192 sorted r (ordered_insert r a l)\n| []       h := sorted_singleton a\n| (b :: l) h := begin\n  by_cases h' : a \u227c b,\n  { simpa [ordered_insert, h', h] using \u03bb b' bm, trans h' (rel_of_sorted_cons h _ bm) },\n  { suffices : \u2200 (b' : \u03b1), b' \u2208 ordered_insert r a l \u2192 r b b',\n    { simpa [ordered_insert, h', h.of_cons.ordered_insert l] },\n    intros b' bm,\n    cases (show b' = a \u2228 b' \u2208 l, by simpa using\n      (perm_ordered_insert _ _ _).subset bm) with be bm,\n    { subst b', exact (total_of r _ _).resolve_left h' },\n    { exact rel_of_sorted_cons h _ bm } }\nend\n\nvariable (r)\n\n\/-- The list `list.insertion_sort r l` is `list.sorted` with respect to `r`. -\/\ntheorem sorted_insertion_sort : \u2200 l, sorted r (insertion_sort r l)\n| []       := sorted_nil\n| (a :: l) := (sorted_insertion_sort l).ordered_insert a _\n\nend total_and_transitive\nend correctness\nend insertion_sort\n\n\/-! ### Merge sort -\/\n\nsection merge_sort\n\n-- TODO(Jeremy): observation: if instead we write (a :: (split l).1, b :: (split l).2), the\n-- equation compiler can't prove the third equation\n\n\/-- Split `l` into two lists of approximately equal length.\n\n     split [1, 2, 3, 4, 5] = ([1, 3, 5], [2, 4]) -\/\n@[simp] def split : list \u03b1 \u2192 list \u03b1 \u00d7 list \u03b1\n| []       := ([], [])\n| (a :: l) := let (l\u2081, l\u2082) := split l in (a :: l\u2082, l\u2081)\n\ntheorem split_cons_of_eq (a : \u03b1) {l l\u2081 l\u2082 : list \u03b1} (h : split l = (l\u2081, l\u2082)) :\n  split (a :: l) = (a :: l\u2082, l\u2081) :=\nby rw [split, h]; refl\n\ntheorem length_split_le : \u2200 {l l\u2081 l\u2082 : list \u03b1},\n  split l = (l\u2081, l\u2082) \u2192 length l\u2081 \u2264 length l \u2227 length l\u2082 \u2264 length l\n| []     ._  ._  rfl := \u27e8nat.le_refl 0, nat.le_refl 0\u27e9\n| (a::l) l\u2081' l\u2082' h   := begin\n  cases e : split l with l\u2081 l\u2082,\n  injection (split_cons_of_eq _ e).symm.trans h, substs l\u2081' l\u2082',\n  cases length_split_le e with h\u2081 h\u2082,\n  exact \u27e8nat.succ_le_succ h\u2082, nat.le_succ_of_le h\u2081\u27e9\nend\n\ntheorem length_split_lt {a b} {l l\u2081 l\u2082 : list \u03b1} (h : split (a::b::l) = (l\u2081, l\u2082)) :\n  length l\u2081 < length (a::b::l) \u2227 length l\u2082 < length (a::b::l) :=\nbegin\n  cases e : split l with l\u2081' l\u2082',\n  injection (split_cons_of_eq _ (split_cons_of_eq _ e)).symm.trans h, substs l\u2081 l\u2082,\n  cases length_split_le e with h\u2081 h\u2082,\n  exact \u27e8nat.succ_le_succ (nat.succ_le_succ h\u2081), nat.succ_le_succ (nat.succ_le_succ h\u2082)\u27e9\nend\n\ntheorem perm_split : \u2200 {l l\u2081 l\u2082 : list \u03b1}, split l = (l\u2081, l\u2082) \u2192 l ~ l\u2081 ++ l\u2082\n| []     ._  ._  rfl := perm.refl _\n| (a::l) l\u2081' l\u2082' h   := begin\n  cases e : split l with l\u2081 l\u2082,\n  injection (split_cons_of_eq _ e).symm.trans h, substs l\u2081' l\u2082',\n  exact ((perm_split e).trans perm_append_comm).cons a,\nend\n\n\/-- Merge two sorted lists into one in linear time.\n\n     merge [1, 2, 4, 5] [0, 1, 3, 4] = [0, 1, 1, 2, 3, 4, 4, 5] -\/\ndef merge : list \u03b1 \u2192 list \u03b1 \u2192 list \u03b1\n| []       l'        := l'\n| l        []        := l\n| (a :: l) (b :: l') := if a \u227c b then a :: merge l (b :: l') else b :: merge (a :: l) l'\n\ninclude r\n\/-- Implementation of a merge sort algorithm to sort a list. -\/\ndef merge_sort : list \u03b1 \u2192 list \u03b1\n| []        := []\n| [a]       := [a]\n| (a::b::l) := begin\n  cases e : split (a::b::l) with l\u2081 l\u2082,\n  cases length_split_lt e with h\u2081 h\u2082,\n  exact merge r (merge_sort l\u2081) (merge_sort l\u2082)\nend\nusing_well_founded\n{ rel_tac := \u03bb_ _, `[exact \u27e8_, inv_image.wf length nat.lt_wf\u27e9],\n  dec_tac := tactic.assumption }\n\ntheorem merge_sort_cons_cons {a b} {l l\u2081 l\u2082 : list \u03b1}\n  (h : split (a::b::l) = (l\u2081, l\u2082)) :\n  merge_sort r (a::b::l) = merge r (merge_sort r l\u2081) (merge_sort r l\u2082) :=\nbegin\n  suffices : \u2200 (L : list \u03b1) h1, @@and.rec\n    (\u03bb a a (_ : length l\u2081 < length l + 1 + 1 \u2227\n      length l\u2082 < length l + 1 + 1), L) h1 h1 = L,\n  { simp [merge_sort, h], apply this },\n  intros, cases h1, refl\nend\n\nsection correctness\n\ntheorem perm_merge : \u2200 (l l' : list \u03b1), merge r l l' ~ l ++ l'\n| []       []        := by simp [merge]\n| []       (b :: l') := by simp [merge]\n| (a :: l) []        := by simp [merge]\n| (a :: l) (b :: l') := begin\n  by_cases a \u227c b,\n  { simpa [merge, h] using perm_merge _ _ },\n  { suffices : b :: merge r (a :: l) l' ~ a :: (l ++ b :: l'), {simpa [merge, h]},\n    exact ((perm_merge _ _).cons _).trans ((swap _ _ _).trans (perm_middle.symm.cons _)) }\nend\n\ntheorem perm_merge_sort : \u2200 l : list \u03b1, merge_sort r l ~ l\n| []        := by simp [merge_sort]\n| [a]       := by simp [merge_sort]\n| (a::b::l) := begin\n  cases e : split (a::b::l) with l\u2081 l\u2082,\n  cases length_split_lt e with h\u2081 h\u2082,\n  rw [merge_sort_cons_cons r e],\n  apply (perm_merge r _ _).trans,\n  exact ((perm_merge_sort l\u2081).append (perm_merge_sort l\u2082)).trans (perm_split e).symm\nend\nusing_well_founded\n{ rel_tac := \u03bb_ _, `[exact \u27e8_, inv_image.wf length nat.lt_wf\u27e9],\n  dec_tac := tactic.assumption }\n\n@[simp] lemma length_merge_sort (l : list \u03b1) : (merge_sort r l).length = l.length :=\n(perm_merge_sort r _).length_eq\n\nsection total_and_transitive\nvariables {r} [is_total \u03b1 r] [is_trans \u03b1 r]\n\ntheorem sorted.merge : \u2200 {l l' : list \u03b1}, sorted r l \u2192 sorted r l' \u2192 sorted r (merge r l l')\n| []       []        h\u2081 h\u2082 := by simp [merge]\n| []       (b :: l') h\u2081 h\u2082 := by simpa [merge] using h\u2082\n| (a :: l) []        h\u2081 h\u2082 := by simpa [merge] using h\u2081\n| (a :: l) (b :: l') h\u2081 h\u2082 := begin\n  by_cases a \u227c b,\n  { suffices : \u2200 (b' : \u03b1) (_ : b' \u2208 merge r l (b :: l')), r a b',\n    { simpa [merge, h, h\u2081.of_cons.merge h\u2082] },\n    intros b' bm,\n    rcases (show b' = b \u2228 b' \u2208 l \u2228 b' \u2208 l', by simpa [or.left_comm] using\n      (perm_merge _ _ _).subset bm) with be | bl | bl',\n    { subst b', assumption },\n    { exact rel_of_sorted_cons h\u2081 _ bl },\n    { exact trans h (rel_of_sorted_cons h\u2082 _ bl') } },\n  { suffices : \u2200 (b' : \u03b1) (_ : b' \u2208 merge r (a :: l) l'), r b b',\n    { simpa [merge, h, h\u2081.merge h\u2082.of_cons] },\n    intros b' bm,\n    have ba : b \u227c a := (total_of r _ _).resolve_left h,\n    rcases (show b' = a \u2228 b' \u2208 l \u2228 b' \u2208 l', by simpa using\n      (perm_merge _ _ _).subset bm) with be | bl | bl',\n    { subst b', assumption },\n    { exact trans ba (rel_of_sorted_cons h\u2081 _ bl) },\n    { exact rel_of_sorted_cons h\u2082 _ bl' } }\nend\n\nvariable (r)\n\ntheorem sorted_merge_sort : \u2200 l : list \u03b1, sorted r (merge_sort r l)\n| []        := by simp [merge_sort]\n| [a]       := by simp [merge_sort]\n| (a::b::l) := begin\n  cases e : split (a::b::l) with l\u2081 l\u2082,\n  cases length_split_lt e with h\u2081 h\u2082,\n  rw [merge_sort_cons_cons r e],\n  exact (sorted_merge_sort l\u2081).merge (sorted_merge_sort l\u2082)\nend\nusing_well_founded\n{ rel_tac := \u03bb_ _, `[exact \u27e8_, inv_image.wf length nat.lt_wf\u27e9],\n  dec_tac := tactic.assumption }\n\ntheorem merge_sort_eq_self [is_antisymm \u03b1 r] {l : list \u03b1} : sorted r l \u2192 merge_sort r l = l :=\neq_of_perm_of_sorted (perm_merge_sort _ _) (sorted_merge_sort _ _)\n\ntheorem merge_sort_eq_insertion_sort [is_antisymm \u03b1 r] (l : list \u03b1) :\n  merge_sort r l = insertion_sort r l :=\neq_of_perm_of_sorted ((perm_merge_sort r l).trans (perm_insertion_sort r l).symm)\n  (sorted_merge_sort r l) (sorted_insertion_sort r l)\n\nend total_and_transitive\nend correctness\n\n@[simp] theorem merge_sort_nil : [].merge_sort r = [] :=\nby rw list.merge_sort\n\n@[simp] theorem merge_sort_singleton (a : \u03b1) : [a].merge_sort r = [a] :=\nby rw list.merge_sort\n\nend merge_sort\nend sort\n\n\/- try them out! -\/\n\n--#eval insertion_sort (\u03bb m n : \u2115, m \u2264 n) [5, 27, 221, 95, 17, 43, 7, 2, 98, 567, 23, 12]\n\n--#eval merge_sort     (\u03bb m n : \u2115, m \u2264 n) [5, 27, 221, 95, 17, 43, 7, 2, 98, 567, 23, 12]\n\nend list\n","avg_line_length":35.3603133159,"max_line_length":100,"alphanum_fraction":0.613822639}
-{"size":515,"ext":"lean","lang":"Lean","max_stars_count":130.0,"content":"import logic\n\ntheorem tst {a b c : Prop} : a \u2192 b \u2192 c \u2192 a \u2227 b :=\nbegin\n  intros [Ha, Hb, Hc],\n  revert Ha,\n  intro Ha2,\n  apply (and.intro Ha2 Hb),\nend\n\ntheorem foo1 {A : Type} (a b c : A) (P : A \u2192 Prop) : P a \u2192 a = b \u2192 P b :=\nbegin\n  intros [Hp, Heq],\n  revert Hp,\n  eapply (eq.rec_on Heq),\n  intro Hpa,\n  apply Hpa\nend\n\ntheorem foo2 {A : Type} (a b c : A) (P : A \u2192 Prop) : P a \u2192 a = b \u2192 P b :=\nbegin\n  intros [Hp, Heq],\n  apply (eq.rec_on Heq Hp)\nend\n\nreveal foo1 foo2\n\nprint definition foo1\nprint definition foo2\n","avg_line_length":17.1666666667,"max_line_length":73,"alphanum_fraction":0.5825242718}
-{"size":1679,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-\/\nprelude\nimport Init.Data.Array.Basic\n\nnamespace Array\n-- TODO: remove the [Inhabited \u03b1] parameters as soon as we have the tactic framework for automating proof generation and using Array.fget\n-- TODO: remove `partial` using well-founded recursion\n\n@[inline] def qpartition {\u03b1 : Type} [Inhabited \u03b1] (as : Array \u03b1) (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (lo hi : Nat) : Nat \u00d7 Array \u03b1 :=\n  let mid := (lo + hi) \/ 2\n  let as  := if lt (as.get! mid) (as.get! lo) then as.swap! lo mid else as\n  let as  := if lt (as.get! hi)  (as.get! lo) then as.swap! lo hi  else as\n  let as  := if lt (as.get! mid) (as.get! hi) then as.swap! mid hi else as\n  let pivot := as.get! hi\n  let rec loop (as : Array \u03b1) (i j : Nat) :=\n    if h : j < hi then\n      if lt (as.get! j) pivot then\n        let as := as.swap! i j\n        loop as (i+1) (j+1)\n      else\n        loop as i (j+1)\n    else\n      let as := as.swap! i hi\n      (i, as)\n  loop as lo lo\ntermination_by _ => hi - j\n\n@[inline] partial def qsort {\u03b1 : Type} [Inhabited \u03b1] (as : Array \u03b1) (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (low := 0) (high := as.size - 1) : Array \u03b1 :=\n  let rec @[specialize] sort (as : Array \u03b1) (low high : Nat) :=\n    if low < high then\n      let p := qpartition as lt low high;\n      -- TODO: fix `partial` support in the equation compiler, it breaks if we use `let (mid, as) := partition as lt low high`\n      let mid := p.1\n      let as  := p.2\n      if mid >= high then as\n      else\n        let as := sort as low mid\n        sort as (mid+1) high\n    else as\n  sort as low high\n\nend Array\n","avg_line_length":35.7234042553,"max_line_length":137,"alphanum_fraction":0.6033353186}
-{"size":81233,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Scott Morrison\n-\/\nimport algebra.group.pi\nimport algebra.big_operators.order\nimport algebra.module.basic\nimport algebra.module.pi\nimport group_theory.submonoid.basic\nimport data.fintype.card\nimport data.finset.preimage\nimport data.multiset.antidiagonal\nimport data.indicator_function\n\n\/-!\n# Type of functions with finite support\n\nFor any type `\u03b1` and a type `M` with zero, we define the type `finsupp \u03b1 M` (notation: `\u03b1 \u2192\u2080 M`)\nof finitely supported functions from `\u03b1` to `M`, i.e. the functions which are zero everywhere\non `\u03b1` except on a finite set.\n\nFunctions with finite support are used (at least) in the following parts of the library:\n\n* `monoid_algebra R M` and `add_monoid_algebra R M` are defined as `M \u2192\u2080 R`;\n\n* polynomials and multivariate polynomials are defined as `add_monoid_algebra`s, hence they use\n  `finsupp` under the hood;\n\n* the linear combination of a family of vectors `v i` with coefficients `f i` (as used, e.g., to\n  define linearly independent family `linear_independent`) is defined as a map\n  `finsupp.total : (\u03b9 \u2192 M) \u2192 (\u03b9 \u2192\u2080 R) \u2192\u2097[R] M`.\n\nSome other constructions are naturally equivalent to `\u03b1 \u2192\u2080 M` with some `\u03b1` and `M` but are defined\nin a different way in the library:\n\n* `multiset \u03b1 \u2243+ \u03b1 \u2192\u2080 \u2115`;\n* `free_abelian_group \u03b1 \u2243+ \u03b1 \u2192\u2080 \u2124`.\n\nMost of the theory assumes that the range is a commutative additive monoid. This gives us the big\nsum operator as a powerful way to construct `finsupp` elements.\n\nMany constructions based on `\u03b1 \u2192\u2080 M` use `semireducible` type tags to avoid reusing unwanted type\ninstances. E.g., `monoid_algebra`, `add_monoid_algebra`, and types based on these two have\nnon-pointwise multiplication.\n\n## Notations\n\nThis file adds `\u03b1 \u2192\u2080 M` as a global notation for `finsupp \u03b1 M`. We also use the following convention\nfor `Type*` variables in this file\n\n* `\u03b1`, `\u03b2`, `\u03b3`: types with no additional structure that appear as the first argument to `finsupp`\n  somewhere in the statement;\n\n* `\u03b9` : an auxiliary index type;\n\n* `M`, `M'`, `N`, `P`: types with `has_zero` or `(add_)(comm_)monoid` structure; `M` is also used\n  for a (semi)module over a (semi)ring.\n\n* `G`, `H`: groups (commutative or not, multiplicative or additive);\n\n* `R`, `S`: (semi)rings.\n\n## TODO\n\n* This file is currently ~2K lines long, so possibly it should be splitted into smaller chunks;\n\n* Add the list of definitions and important lemmas to the module docstring.\n\n## Implementation notes\n\nThis file is a `noncomputable theory` and uses classical logic throughout.\n\n## Notation\n\nThis file defines `\u03b1 \u2192\u2080 \u03b2` as notation for `finsupp \u03b1 \u03b2`.\n\n-\/\n\nnoncomputable theory\nopen_locale classical big_operators\n\nopen finset\n\nvariables {\u03b1 \u03b2 \u03b3 \u03b9 M M' N P G H R S : Type*}\n\n\/-- `finsupp \u03b1 M`, denoted `\u03b1 \u2192\u2080 M`, is the type of functions `f : \u03b1 \u2192 M` such that\n  `f x = 0` for all but finitely many `x`. -\/\nstructure finsupp (\u03b1 : Type*) (M : Type*) [has_zero M] :=\n(support            : finset \u03b1)\n(to_fun             : \u03b1 \u2192 M)\n(mem_support_to_fun : \u2200a, a \u2208 support \u2194 to_fun a \u2260 0)\n\ninfixr ` \u2192\u2080 `:25 := finsupp\n\nnamespace finsupp\n\n\/-! ### Basic declarations about `finsupp` -\/\n\nsection basic\nvariable [has_zero M]\n\ninstance : has_coe_to_fun (\u03b1 \u2192\u2080 M) := \u27e8\u03bb _, \u03b1 \u2192 M, to_fun\u27e9\n\n@[simp] lemma coe_mk (f : \u03b1 \u2192 M) (s : finset \u03b1) (h : \u2200 a, a \u2208 s \u2194 f a \u2260 0) :\n  \u21d1(\u27e8s, f, h\u27e9 : \u03b1 \u2192\u2080 M) = f := rfl\n\ninstance : has_zero (\u03b1 \u2192\u2080 M) := \u27e8\u27e8\u2205, 0, \u03bb _, \u27e8false.elim, \u03bb H, H rfl\u27e9\u27e9\u27e9\n\n@[simp] lemma coe_zero : \u21d1(0 : \u03b1 \u2192\u2080 M) = 0 := rfl\nlemma zero_apply {a : \u03b1} : (0 : \u03b1 \u2192\u2080 M) a = 0 := rfl\n@[simp] lemma support_zero : (0 : \u03b1 \u2192\u2080 M).support = \u2205 := rfl\n\ninstance : inhabited (\u03b1 \u2192\u2080 M) := \u27e80\u27e9\n\n@[simp] lemma mem_support_iff {f : \u03b1 \u2192\u2080 M} : \u2200{a:\u03b1}, a \u2208 f.support \u2194 f a \u2260 0 :=\nf.mem_support_to_fun\n\n@[simp, norm_cast] lemma fun_support_eq (f : \u03b1 \u2192\u2080 M) : function.support f = f.support :=\nset.ext $ \u03bb x, mem_support_iff.symm\n\nlemma not_mem_support_iff {f : \u03b1 \u2192\u2080 M} {a} : a \u2209 f.support \u2194 f a = 0 :=\nnot_iff_comm.1 mem_support_iff.symm\n\nlemma coe_fn_injective : @function.injective (\u03b1 \u2192\u2080 M) (\u03b1 \u2192 M) coe_fn\n| \u27e8s, f, hf\u27e9 \u27e8t, g, hg\u27e9 h :=\n  begin\n    change f = g at h, subst h,\n    have : s = t, { ext a, exact (hf a).trans (hg a).symm },\n    subst this\n  end\n\n@[simp, norm_cast] lemma coe_fn_inj {f g : \u03b1 \u2192\u2080 M} : (f : \u03b1 \u2192 M) = g \u2194 f = g :=\ncoe_fn_injective.eq_iff\n\n@[simp, norm_cast] lemma coe_eq_zero {f : \u03b1 \u2192\u2080 M} : (f : \u03b1 \u2192 M) = 0 \u2194 f = 0 :=\nby rw [\u2190 coe_zero, coe_fn_inj]\n\n@[ext] lemma ext {f g : \u03b1 \u2192\u2080 M} (h : \u2200a, f a = g a) : f = g := coe_fn_injective (funext h)\n\nlemma ext_iff {f g : \u03b1 \u2192\u2080 M} : f = g \u2194 (\u2200a:\u03b1, f a = g a) :=\n\u27e8by rintros rfl a; refl, ext\u27e9\n\nlemma ext_iff' {f g : \u03b1 \u2192\u2080 M} : f = g \u2194 f.support = g.support \u2227 \u2200 x \u2208 f.support, f x = g x :=\n\u27e8\u03bb h, h \u25b8 \u27e8rfl, \u03bb _ _, rfl\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, ext $ \u03bb a,\n  if h : a \u2208 f.support then h\u2082 a h else\n    have hf : f a = 0, from not_mem_support_iff.1 h,\n    have hg : g a = 0, by rwa [h\u2081, not_mem_support_iff] at h,\n    by rw [hf, hg]\u27e9\n\n@[simp] lemma support_eq_empty {f : \u03b1 \u2192\u2080 M} : f.support = \u2205 \u2194 f = 0 :=\nby exact_mod_cast @function.support_eq_empty_iff _ _ _ f\n\nlemma support_nonempty_iff {f : \u03b1 \u2192\u2080 M} : f.support.nonempty \u2194 f \u2260 0 :=\nby simp only [finsupp.support_eq_empty, finset.nonempty_iff_ne_empty, ne.def]\n\nlemma nonzero_iff_exists {f : \u03b1 \u2192\u2080 M} : f \u2260 0 \u2194 \u2203 a : \u03b1, f a \u2260 0 :=\nby simp [\u2190 finsupp.support_eq_empty, finset.eq_empty_iff_forall_not_mem]\n\nlemma card_support_eq_zero {f : \u03b1 \u2192\u2080 M} : card f.support = 0 \u2194 f = 0 :=\nby simp\n\ninstance finsupp.decidable_eq [decidable_eq \u03b1] [decidable_eq M] : decidable_eq (\u03b1 \u2192\u2080 M) :=\nassume f g, decidable_of_iff (f.support = g.support \u2227 (\u2200a\u2208f.support, f a = g a)) ext_iff'.symm\n\nlemma finite_support (f : \u03b1 \u2192\u2080 M) : set.finite (function.support f) :=\nf.fun_support_eq.symm \u25b8 f.support.finite_to_set\n\nlemma support_subset_iff {s : set \u03b1} {f : \u03b1 \u2192\u2080 M} :\n  \u2191f.support \u2286 s \u2194 (\u2200a\u2209s, f a = 0) :=\nby simp only [set.subset_def, mem_coe, mem_support_iff];\n   exact forall_congr (assume a, not_imp_comm)\n\n\/-- Given `fintype \u03b1`, `equiv_fun_on_fintype` is the `equiv` between `\u03b1 \u2192\u2080 \u03b2` and `\u03b1 \u2192 \u03b2`.\n  (All functions on a finite type are finitely supported.) -\/\ndef equiv_fun_on_fintype [fintype \u03b1] : (\u03b1 \u2192\u2080 M) \u2243 (\u03b1 \u2192 M) :=\n\u27e8\u03bbf a, f a, \u03bbf, mk (finset.univ.filter $ \u03bba, f a \u2260 0) f (by simp only [true_and, finset.mem_univ,\n  iff_self, finset.mem_filter, finset.filter_congr_decidable, forall_true_iff]),\n  begin intro f, ext a, refl end,\n  begin intro f, ext a, refl end\u27e9\n\nend basic\n\n\/-! ### Declarations about `single` -\/\n\nsection single\nvariables [has_zero M] {a a' : \u03b1} {b : M}\n\n\/-- `single a b` is the finitely supported function which has\n  value `b` at `a` and zero otherwise. -\/\ndef single (a : \u03b1) (b : M) : \u03b1 \u2192\u2080 M :=\n\u27e8if b = 0 then \u2205 else {a}, \u03bb a', if a = a' then b else 0, \u03bb a', begin\n  by_cases hb : b = 0; by_cases a = a';\n    simp only [hb, h, if_pos, if_false, mem_singleton],\n  { exact \u27e8false.elim, \u03bb H, H rfl\u27e9 },\n  { exact \u27e8false.elim, \u03bb H, H rfl\u27e9 },\n  { exact \u27e8\u03bb _, hb, \u03bb _, rfl\u27e9 },\n  { exact \u27e8\u03bb H _, h H.symm, \u03bb H, (H rfl).elim\u27e9 }\nend\u27e9\n\nlemma single_apply : single a b a' = if a = a' then b else 0 :=\nrfl\n\nlemma single_eq_indicator : \u21d1(single a b) = set.indicator {a} (\u03bb _, b) :=\nby { ext, simp [single_apply, set.indicator, @eq_comm _ a] }\n\n@[simp] lemma single_eq_same : (single a b : \u03b1 \u2192\u2080 M) a = b :=\nif_pos rfl\n\n@[simp] lemma single_eq_of_ne (h : a \u2260 a') : (single a b : \u03b1 \u2192\u2080 M) a' = 0 :=\nif_neg h\n\nlemma single_eq_update : \u21d1(single a b) = function.update 0 a b :=\nby rw [single_eq_indicator, \u2190 set.piecewise_eq_indicator, set.piecewise_singleton]\n\nlemma single_eq_pi_single : \u21d1(single a b) = pi.single a b :=\nsingle_eq_update\n\n@[simp] lemma single_zero : (single a 0 : \u03b1 \u2192\u2080 M) = 0 :=\ncoe_fn_injective $ by simpa only [single_eq_update, coe_zero]\n  using function.update_eq_self a (0 : \u03b1 \u2192 M)\n\nlemma single_of_single_apply (a a' : \u03b1) (b : M) :\n  single a ((single a' b) a) = single a' (single a' b) a :=\nbegin\n  rw [single_apply, single_apply],\n  ext,\n  split_ifs,\n  { rw h, },\n  { rw [zero_apply, single_apply, if_t_t], },\nend\n\nlemma support_single_ne_zero (hb : b \u2260 0) : (single a b).support = {a} :=\nif_neg hb\n\nlemma support_single_subset : (single a b).support \u2286 {a} :=\nshow ite _ _ _ \u2286 _, by split_ifs; [exact empty_subset _, exact subset.refl _]\n\nlemma single_apply_mem (x) : single a b x \u2208 ({0, b} : set M) :=\nby rcases em (a = x) with (rfl|hx); [simp, simp [single_eq_of_ne hx]]\n\nlemma range_single_subset : set.range (single a b) \u2286 {0, b} :=\nset.range_subset_iff.2 single_apply_mem\n\nlemma single_injective (a : \u03b1) : function.injective (single a : M \u2192 \u03b1 \u2192\u2080 M) :=\nassume b\u2081 b\u2082 eq,\nhave (single a b\u2081 : \u03b1 \u2192\u2080 M) a = (single a b\u2082 : \u03b1 \u2192\u2080 M) a, by rw eq,\nby rwa [single_eq_same, single_eq_same] at this\n\nlemma single_apply_eq_zero {a x : \u03b1} {b : M} : single a b x = 0 \u2194 (x = a \u2192 b = 0) :=\nby simp [single_eq_indicator]\n\nlemma mem_support_single (a a' : \u03b1) (b : M) :\n  a \u2208 (single a' b).support \u2194 a = a' \u2227 b \u2260 0 :=\nby simp [single_apply_eq_zero, not_or_distrib]\n\nlemma eq_single_iff {f : \u03b1 \u2192\u2080 M} {a b} : f = single a b \u2194 f.support \u2286 {a} \u2227 f a = b :=\nbegin\n  refine \u27e8\u03bb h, h.symm \u25b8 \u27e8support_single_subset, single_eq_same\u27e9, _\u27e9,\n  rintro \u27e8h, rfl\u27e9,\n  ext x,\n  by_cases hx : a = x; simp only [hx, single_eq_same, single_eq_of_ne, ne.def, not_false_iff],\n  exact not_mem_support_iff.1 (mt (\u03bb hx, (mem_singleton.1 (h hx)).symm) hx)\nend\n\nlemma single_eq_single_iff (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : M) :\n  single a\u2081 b\u2081 = single a\u2082 b\u2082 \u2194 ((a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082) \u2228 (b\u2081 = 0 \u2227 b\u2082 = 0)) :=\nbegin\n  split,\n  { assume eq,\n    by_cases a\u2081 = a\u2082,\n    { refine or.inl \u27e8h, _\u27e9,\n      rwa [h, (single_injective a\u2082).eq_iff] at eq },\n    { rw [ext_iff] at eq,\n      have h\u2081 := eq a\u2081,\n      have h\u2082 := eq a\u2082,\n      simp only [single_eq_same, single_eq_of_ne h, single_eq_of_ne (ne.symm h)] at h\u2081 h\u2082,\n      exact or.inr \u27e8h\u2081, h\u2082.symm\u27e9 } },\n  { rintros (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9),\n    { refl },\n    { rw [single_zero, single_zero] } }\nend\n\nlemma single_left_inj (h : b \u2260 0) :\n  single a b = single a' b \u2194 a = a' :=\n\u27e8\u03bb H, by simpa only [h, single_eq_single_iff,\n  and_false, or_false, eq_self_iff_true, and_true] using H,\n \u03bb H, by rw [H]\u27e9\n\nlemma support_single_ne_bot (i : \u03b1) (h : b \u2260 0) :\n  (single i b).support \u2260 \u22a5 :=\nbegin\n  have : i \u2208 (single i b).support := by simpa using h,\n  intro H,\n  simpa [H]\nend\n\nlemma support_single_disjoint {b' : M} (hb : b \u2260 0) (hb' : b' \u2260 0) {i j : \u03b1} :\n  disjoint (single i b).support (single j b').support \u2194 i \u2260 j :=\nby simpa [support_single_ne_zero, hb, hb'] using ne_comm\n\n@[simp] lemma single_eq_zero : single a b = 0 \u2194 b = 0 :=\nby simp [ext_iff, single_eq_indicator]\n\nlemma single_swap (a\u2081 a\u2082 : \u03b1) (b : M) : single a\u2081 b a\u2082 = single a\u2082 b a\u2081 :=\nby simp only [single_apply]; ac_refl\n\ninstance [nonempty \u03b1] [nontrivial M] : nontrivial (\u03b1 \u2192\u2080 M) :=\nbegin\n  inhabit \u03b1,\n  rcases exists_ne (0 : M) with \u27e8x, hx\u27e9,\n  exact nontrivial_of_ne (single (default \u03b1) x) 0 (mt single_eq_zero.1 hx)\nend\n\nlemma unique_single [unique \u03b1] (x : \u03b1 \u2192\u2080 M) : x = single (default \u03b1) (x (default \u03b1)) :=\next $ unique.forall_iff.2 single_eq_same.symm\n\nlemma unique_ext [unique \u03b1] {f g : \u03b1 \u2192\u2080 M} (h : f (default \u03b1) = g (default \u03b1)) : f = g :=\next $ \u03bb a, by rwa [unique.eq_default a]\n\nlemma unique_ext_iff [unique \u03b1] {f g : \u03b1 \u2192\u2080 M} : f = g \u2194  f (default \u03b1) = g (default \u03b1) :=\n\u27e8\u03bb h, h \u25b8 rfl, unique_ext\u27e9\n\n@[simp] lemma unique_single_eq_iff [unique \u03b1] {b' : M} :\n  single a b = single a' b' \u2194 b = b' :=\nby rw [unique_ext_iff, unique.eq_default a, unique.eq_default a', single_eq_same, single_eq_same]\n\nlemma support_eq_singleton {f : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  f.support = {a} \u2194 f a \u2260 0 \u2227 f = single a (f a) :=\n\u27e8\u03bb h, \u27e8mem_support_iff.1 $ h.symm \u25b8 finset.mem_singleton_self a,\n  eq_single_iff.2 \u27e8subset_of_eq h, rfl\u27e9\u27e9, \u03bb h, h.2.symm \u25b8 support_single_ne_zero h.1\u27e9\n\nlemma support_eq_singleton' {f : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  f.support = {a} \u2194 \u2203 b \u2260 0, f = single a b :=\n\u27e8\u03bb h, let h := support_eq_singleton.1 h in \u27e8_, h.1, h.2\u27e9,\n  \u03bb \u27e8b, hb, hf\u27e9, hf.symm \u25b8 support_single_ne_zero hb\u27e9\n\nlemma card_support_eq_one {f : \u03b1 \u2192\u2080 M} : card f.support = 1 \u2194 \u2203 a, f a \u2260 0 \u2227 f = single a (f a) :=\nby simp only [card_eq_one, support_eq_singleton]\n\nlemma card_support_eq_one' {f : \u03b1 \u2192\u2080 M} : card f.support = 1 \u2194 \u2203 a (b \u2260 0), f = single a b :=\nby simp only [card_eq_one, support_eq_singleton']\n\nend single\n\n\/-! ### Declarations about `on_finset` -\/\n\nsection on_finset\nvariables [has_zero M]\n\n\/-- `on_finset s f hf` is the finsupp function representing `f` restricted to the finset `s`.\n  The function needs to be `0` outside of `s`. Use this when the set needs to be filtered anyways,\n  otherwise a better set representation is often available. -\/\ndef on_finset (s : finset \u03b1) (f : \u03b1 \u2192 M) (hf : \u2200a, f a \u2260 0 \u2192 a \u2208 s) : \u03b1 \u2192\u2080 M :=\n\u27e8s.filter (\u03bba, f a \u2260 0), f, by simpa\u27e9\n\n@[simp] lemma on_finset_apply {s : finset \u03b1} {f : \u03b1 \u2192 M} {hf a} :\n  (on_finset s f hf : \u03b1 \u2192\u2080 M) a = f a :=\nrfl\n\n@[simp] lemma support_on_finset_subset {s : finset \u03b1} {f : \u03b1 \u2192 M} {hf} :\n  (on_finset s f hf).support \u2286 s :=\nfilter_subset _ _\n\n@[simp] lemma mem_support_on_finset\n  {s : finset \u03b1} {f : \u03b1 \u2192 M} (hf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s) {a : \u03b1} :\n  a \u2208 (finsupp.on_finset s f hf).support \u2194 f a \u2260 0 :=\nby rw [finsupp.mem_support_iff, finsupp.on_finset_apply]\n\nlemma support_on_finset\n  {s : finset \u03b1} {f : \u03b1 \u2192 M} (hf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s) :\n  (finsupp.on_finset s f hf).support = s.filter (\u03bb a, f a \u2260 0) :=\nrfl\n\nend on_finset\n\nsection of_support_finite\n\nvariables [has_zero M]\n\n\/-- The natural `finsupp` induced by the function `f` given that it has finite support. -\/\nnoncomputable def of_support_finite\n  (f : \u03b1 \u2192 M) (hf : (function.support f).finite) : \u03b1 \u2192\u2080 M :=\n{ support := hf.to_finset,\n  to_fun := f,\n  mem_support_to_fun := \u03bb _, hf.mem_to_finset }\n\nlemma of_support_finite_coe {f : \u03b1 \u2192 M} {hf : (function.support f).finite} :\n  (of_support_finite f hf : \u03b1 \u2192 M) = f := rfl\n\ninstance : can_lift (\u03b1 \u2192 M) (\u03b1 \u2192\u2080 M) :=\n{ coe := coe_fn,\n  cond := \u03bb f, (function.support f).finite,\n  prf := \u03bb f hf, \u27e8of_support_finite f hf, rfl\u27e9 }\n\nend of_support_finite\n\n\/-! ### Declarations about `map_range` -\/\n\nsection map_range\nvariables [has_zero M] [has_zero N]\n\n\/-- The composition of `f : M \u2192 N` and `g : \u03b1 \u2192\u2080 M` is\n`map_range f hf g : \u03b1 \u2192\u2080 N`, well-defined when `f 0 = 0`. -\/\ndef map_range (f : M \u2192 N) (hf : f 0 = 0) (g : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 N :=\non_finset g.support (f \u2218 g) $\n  assume a, by rw [mem_support_iff, not_imp_not]; exact \u03bb H, (congr_arg f H).trans hf\n\n@[simp] lemma map_range_apply {f : M \u2192 N} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  map_range f hf g a = f (g a) :=\nrfl\n\n@[simp] lemma map_range_zero {f : M \u2192 N} {hf : f 0 = 0} : map_range f hf (0 : \u03b1 \u2192\u2080 M) = 0 :=\next $ \u03bb a, by simp only [hf, zero_apply, map_range_apply]\n\nlemma support_map_range {f : M \u2192 N} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} :\n  (map_range f hf g).support \u2286 g.support :=\nsupport_on_finset_subset\n\n@[simp] lemma map_range_single {f : M \u2192 N} {hf : f 0 = 0} {a : \u03b1} {b : M} :\n  map_range f hf (single a b) = single a (f b) :=\next $ \u03bb a', show f (ite _ _ _) = ite _ _ _, by split_ifs; [refl, exact hf]\n\nend map_range\n\n\/-! ### Declarations about `emb_domain` -\/\n\nsection emb_domain\nvariables [has_zero M] [has_zero N]\n\n\/-- Given `f : \u03b1 \u21aa \u03b2` and `v : \u03b1 \u2192\u2080 M`, `emb_domain f v : \u03b2 \u2192\u2080 M`\nis the finitely supported function whose value at `f a : \u03b2` is `v a`.\nFor a `b : \u03b2` outside the range of `f`, it is zero. -\/\ndef emb_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\nbegin\n  refine \u27e8v.support.map f, \u03bba\u2082,\n    if h : a\u2082 \u2208 v.support.map f then v (v.support.choose (\u03bba\u2081, f a\u2081 = a\u2082) _) else 0, _\u27e9,\n  { rcases finset.mem_map.1 h with \u27e8a, ha, rfl\u27e9,\n    exact exists_unique.intro a \u27e8ha, rfl\u27e9 (assume b \u27e8_, hb\u27e9, f.injective hb) },\n  { assume a\u2082,\n    split_ifs,\n    { simp only [h, true_iff, ne.def],\n      rw [\u2190 not_mem_support_iff, not_not],\n      apply finset.choose_mem },\n    { simp only [h, ne.def, ne_self_iff_false] } }\nend\n\n@[simp] lemma support_emb_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) :\n  (emb_domain f v).support = v.support.map f :=\nrfl\n\n@[simp] lemma emb_domain_zero (f : \u03b1 \u21aa \u03b2) : (emb_domain f 0 : \u03b2 \u2192\u2080 M) = 0 :=\nrfl\n\n@[simp] lemma emb_domain_apply (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  emb_domain f v (f a) = v a :=\nbegin\n  change dite _ _ _ = _,\n  split_ifs; rw [finset.mem_map' f] at h,\n  { refine congr_arg (v : \u03b1 \u2192 M) (f.inj' _),\n    exact finset.choose_property (\u03bba\u2081, f a\u2081 = f a) _ _ },\n  { exact (not_mem_support_iff.1 h).symm }\nend\n\nlemma emb_domain_notin_range (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) (a : \u03b2) (h : a \u2209 set.range f) :\n  emb_domain f v a = 0 :=\nbegin\n  refine dif_neg (mt (assume h, _) h),\n  rcases finset.mem_map.1 h with \u27e8a, h, rfl\u27e9,\n  exact set.mem_range_self a\nend\n\nlemma emb_domain_injective (f : \u03b1 \u21aa \u03b2) :\n  function.injective (emb_domain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)) :=\n\u03bb l\u2081 l\u2082 h, ext $ \u03bb a, by simpa only [emb_domain_apply] using ext_iff.1 h (f a)\n\n@[simp] lemma emb_domain_inj {f : \u03b1 \u21aa \u03b2} {l\u2081 l\u2082 : \u03b1 \u2192\u2080 M} :\n  emb_domain f l\u2081 = emb_domain f l\u2082 \u2194 l\u2081 = l\u2082 :=\n(emb_domain_injective f).eq_iff\n\n@[simp] lemma emb_domain_eq_zero {f : \u03b1 \u21aa \u03b2} {l : \u03b1 \u2192\u2080 M} :\n  emb_domain f l = 0 \u2194 l = 0 :=\n(emb_domain_injective f).eq_iff' $ emb_domain_zero f\n\nlemma emb_domain_map_range\n  (f : \u03b1 \u21aa \u03b2) (g : M \u2192 N) (p : \u03b1 \u2192\u2080 M) (hg : g 0 = 0) :\n  emb_domain f (map_range g hg p) = map_range g hg (emb_domain f p) :=\nbegin\n  ext a,\n  by_cases a \u2208 set.range f,\n  { rcases h with \u27e8a', rfl\u27e9,\n    rw [map_range_apply, emb_domain_apply, emb_domain_apply, map_range_apply] },\n  { rw [map_range_apply, emb_domain_notin_range, emb_domain_notin_range, \u2190 hg]; assumption }\nend\n\nlemma single_of_emb_domain_single\n  (l : \u03b1 \u2192\u2080 M) (f : \u03b1 \u21aa \u03b2) (a : \u03b2) (b : M) (hb : b \u2260 0)\n  (h : l.emb_domain f = single a b) :\n  \u2203 x, l = single x b \u2227 f x = a :=\nbegin\n  have h_map_support : finset.map f (l.support) = {a},\n    by rw [\u2190support_emb_domain, h, support_single_ne_zero hb]; refl,\n  have ha : a \u2208 finset.map f (l.support),\n    by simp only [h_map_support, finset.mem_singleton],\n  rcases finset.mem_map.1 ha with \u27e8c, hc\u2081, hc\u2082\u27e9,\n  use c,\n  split,\n  { ext d,\n    rw [\u2190 emb_domain_apply f l, h],\n    by_cases h_cases : c = d,\n    { simp only [eq.symm h_cases, hc\u2082, single_eq_same] },\n    { rw [single_apply, single_apply, if_neg, if_neg h_cases],\n      by_contra hfd,\n      exact h_cases (f.injective (hc\u2082.trans hfd)) } },\n  { exact hc\u2082 }\nend\n\nend emb_domain\n\n\/-! ### Declarations about `zip_with` -\/\n\nsection zip_with\nvariables [has_zero M] [has_zero N] [has_zero P]\n\n\/-- `zip_with f hf g\u2081 g\u2082` is the finitely supported function satisfying\n  `zip_with f hf g\u2081 g\u2082 a = f (g\u2081 a) (g\u2082 a)`, and it is well-defined when `f 0 0 = 0`. -\/\ndef zip_with (f : M \u2192 N \u2192 P) (hf : f 0 0 = 0) (g\u2081 : \u03b1 \u2192\u2080 M) (g\u2082 : \u03b1 \u2192\u2080 N) : (\u03b1 \u2192\u2080 P) :=\non_finset (g\u2081.support \u222a g\u2082.support) (\u03bba, f (g\u2081 a) (g\u2082 a)) $ \u03bb a H,\nbegin\n  simp only [mem_union, mem_support_iff, ne], rw [\u2190 not_and_distrib],\n  rintro \u27e8h\u2081, h\u2082\u27e9, rw [h\u2081, h\u2082] at H, exact H hf\nend\n\n@[simp] lemma zip_with_apply\n  {f : M \u2192 N \u2192 P} {hf : f 0 0 = 0} {g\u2081 : \u03b1 \u2192\u2080 M} {g\u2082 : \u03b1 \u2192\u2080 N} {a : \u03b1} :\n  zip_with f hf g\u2081 g\u2082 a = f (g\u2081 a) (g\u2082 a) :=\nrfl\n\nlemma support_zip_with [D : decidable_eq \u03b1] {f : M \u2192 N \u2192 P} {hf : f 0 0 = 0}\n  {g\u2081 : \u03b1 \u2192\u2080 M} {g\u2082 : \u03b1 \u2192\u2080 N} : (zip_with f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nby rw subsingleton.elim D; exact support_on_finset_subset\n\nend zip_with\n\n\/-! ### Declarations about `erase` -\/\n\nsection erase\n\nvariables [has_zero M]\n\n\/-- `erase a f` is the finitely supported function equal to `f` except at `a` where it is equal to\n  `0`. -\/\ndef erase (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M :=\n\u27e8f.support.erase a, (\u03bba', if a' = a then 0 else f a'),\n  assume a', by rw [mem_erase, mem_support_iff]; split_ifs;\n    [exact \u27e8\u03bb H _, H.1 h, \u03bb H, (H rfl).elim\u27e9,\n    exact and_iff_right h]\u27e9\n\n@[simp] lemma support_erase {a : \u03b1} {f : \u03b1 \u2192\u2080 M} :\n  (f.erase a).support = f.support.erase a :=\nrfl\n\n@[simp] lemma erase_same {a : \u03b1} {f : \u03b1 \u2192\u2080 M} : (f.erase a) a = 0 :=\nif_pos rfl\n\n@[simp] lemma erase_ne {a a' : \u03b1} {f : \u03b1 \u2192\u2080 M} (h : a' \u2260 a) : (f.erase a) a' = f a' :=\nif_neg h\n\n@[simp] lemma erase_single {a : \u03b1} {b : M} : (erase a (single a b)) = 0 :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, erase_same], refl },\n  { rw [erase_ne hs], exact single_eq_of_ne (ne.symm hs) }\nend\n\nlemma erase_single_ne {a a' : \u03b1} {b : M} (h : a \u2260 a') : (erase a (single a' b)) = single a' b :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, erase_same, single_eq_of_ne (h.symm)] },\n  { rw [erase_ne hs] }\nend\n\n@[simp] lemma erase_zero (a : \u03b1) : erase a (0 : \u03b1 \u2192\u2080 M) = 0 :=\nby rw [\u2190 support_eq_empty, support_erase, support_zero, erase_empty]\n\nend erase\n\n\/-!\n### Declarations about `sum` and `prod`\n\nIn most of this section, the domain `\u03b2` is assumed to be an `add_monoid`.\n-\/\n\nsection sum_prod\n\n-- [to_additive sum] for finsupp.prod doesn't work, the equation lemmas are not generated\n\/-- `sum f g` is the sum of `g a (f a)` over the support of `f`. -\/\ndef sum [has_zero M] [add_comm_monoid N] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : N :=\n\u2211 a in f.support, g a (f a)\n\n\/-- `prod f g` is the product of `g a (f a)` over the support of `f`. -\/\n@[to_additive]\ndef prod [has_zero M] [comm_monoid N] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : N :=\n\u220f a in f.support, g a (f a)\n\nvariables [has_zero M] [has_zero M'] [comm_monoid N]\n\n@[to_additive]\nlemma prod_of_support_subset (f : \u03b1 \u2192\u2080 M) {s : finset \u03b1}\n  (hs : f.support \u2286 s) (g : \u03b1 \u2192 M \u2192 N) (h : \u2200 i \u2208 s, g i 0 = 1) :\n  f.prod g = \u220f x in s, g x (f x) :=\nfinset.prod_subset hs $ \u03bb x hxs hx, h x hxs \u25b8 congr_arg (g x) $ not_mem_support_iff.1 hx\n\n@[to_additive]\nlemma prod_fintype [fintype \u03b1] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) (h : \u2200 i, g i 0 = 1) :\n  f.prod g = \u220f i, g i (f i) :=\nf.prod_of_support_subset (subset_univ _) g (\u03bb x _, h x)\n\n@[simp, to_additive]\nlemma prod_single_index {a : \u03b1} {b : M} {h : \u03b1 \u2192 M \u2192 N} (h_zero : h a 0 = 1) :\n  (single a b).prod h = h a b :=\ncalc (single a b).prod h = \u220f x in {a}, h x (single a b x) :\n  prod_of_support_subset _ support_single_subset h $ \u03bb x hx, (mem_singleton.1 hx).symm \u25b8 h_zero\n... = h a b : by simp\n\n@[to_additive]\nlemma prod_map_range_index {f : M \u2192 M'} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} {h : \u03b1 \u2192 M' \u2192 N}\n  (h0 : \u2200a, h a 0 = 1) : (map_range f hf g).prod h = g.prod (\u03bba b, h a (f b)) :=\nfinset.prod_subset support_map_range $ \u03bb _ _ H,\n  by rw [not_mem_support_iff.1 H, h0]\n\n@[simp, to_additive]\nlemma prod_zero_index {h : \u03b1 \u2192 M \u2192 N} : (0 : \u03b1 \u2192\u2080 M).prod h = 1 := rfl\n\n@[to_additive]\nlemma prod_comm (f : \u03b1 \u2192\u2080 M) (g : \u03b2 \u2192\u2080 M') (h : \u03b1 \u2192 M \u2192 \u03b2 \u2192 M' \u2192 N) :\n  f.prod (\u03bb x v, g.prod (\u03bb x' v', h x v x' v')) = g.prod (\u03bb x' v', f.prod (\u03bb x v, h x v x' v')) :=\nfinset.prod_comm\n\n@[simp, to_additive]\nlemma prod_ite_eq [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 M) (a : \u03b1) (b : \u03b1 \u2192 M \u2192 N) :\n  f.prod (\u03bb x v, ite (a = x) (b x v) 1) = ite (a \u2208 f.support) (b a (f a)) 1 :=\nby { dsimp [finsupp.prod], rw f.support.prod_ite_eq, }\n\n@[simp] lemma sum_ite_self_eq\n  [decidable_eq \u03b1] {N : Type*} [add_comm_monoid N] (f : \u03b1 \u2192\u2080 N) (a : \u03b1) :\n  f.sum (\u03bb x v, ite (a = x) v 0) = f a :=\nby { convert f.sum_ite_eq a (\u03bb x, id), simp [ite_eq_right_iff.2 eq.symm] }\n\n\/-- A restatement of `prod_ite_eq` with the equality test reversed. -\/\n@[simp, to_additive \"A restatement of `sum_ite_eq` with the equality test reversed.\"]\nlemma prod_ite_eq' [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 M) (a : \u03b1) (b : \u03b1 \u2192 M \u2192 N) :\n  f.prod (\u03bb x v, ite (x = a) (b x v) 1) = ite (a \u2208 f.support) (b a (f a)) 1 :=\nby { dsimp [finsupp.prod], rw f.support.prod_ite_eq', }\n\n@[simp] lemma sum_ite_self_eq'\n  [decidable_eq \u03b1] {N : Type*} [add_comm_monoid N] (f : \u03b1 \u2192\u2080 N) (a : \u03b1) :\n  f.sum (\u03bb x v, ite (x = a) v 0) = f a :=\nby { convert f.sum_ite_eq' a (\u03bb x, id), simp [ite_eq_right_iff.2 eq.symm] }\n\n@[simp] lemma prod_pow [fintype \u03b1] (f : \u03b1 \u2192\u2080 \u2115) (g : \u03b1 \u2192 N) :\n  f.prod (\u03bb a b, g a ^ b) = \u220f a, g a ^ (f a) :=\nf.prod_fintype _ $ \u03bb a, pow_zero _\n\n\/-- If `g` maps a second argument of 0 to 1, then multiplying it over the\nresult of `on_finset` is the same as multiplying it over the original\n`finset`. -\/\n@[to_additive \"If `g` maps a second argument of 0 to 0, summing it over the\nresult of `on_finset` is the same as summing it over the original\n`finset`.\"]\nlemma on_finset_prod {s : finset \u03b1} {f : \u03b1 \u2192 M} {g : \u03b1 \u2192 M \u2192 N}\n    (hf : \u2200a, f a \u2260 0 \u2192 a \u2208 s) (hg : \u2200 a, g a 0 = 1) :\n  (on_finset s f hf).prod g = \u220f a in s, g a (f a) :=\nfinset.prod_subset support_on_finset_subset $ by simp [*] { contextual := tt }\n\nend sum_prod\n\n\/-!\n### Additive monoid structure on `\u03b1 \u2192\u2080 M`\n-\/\n\nsection add_monoid\n\nvariables [add_monoid M]\n\ninstance : has_add (\u03b1 \u2192\u2080 M) := \u27e8zip_with (+) (add_zero 0)\u27e9\n\n@[simp] lemma coe_add (f g : \u03b1 \u2192\u2080 M) : \u21d1(f + g) = f + g := rfl\nlemma add_apply (g\u2081 g\u2082 : \u03b1 \u2192\u2080 M) (a : \u03b1) : (g\u2081 + g\u2082) a = g\u2081 a + g\u2082 a := rfl\n\nlemma support_add [decidable_eq \u03b1] {g\u2081 g\u2082 : \u03b1 \u2192\u2080 M} :\n  (g\u2081 + g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nsupport_zip_with\n\nlemma support_add_eq [decidable_eq \u03b1] {g\u2081 g\u2082 : \u03b1 \u2192\u2080 M} (h : disjoint g\u2081.support g\u2082.support) :\n  (g\u2081 + g\u2082).support = g\u2081.support \u222a g\u2082.support :=\nle_antisymm support_zip_with $ assume a ha,\n(finset.mem_union.1 ha).elim\n  (assume ha, have a \u2209 g\u2082.support, from disjoint_left.1 h ha,\n    by simp only [mem_support_iff, not_not] at *;\n    simpa only [add_apply, this, add_zero])\n  (assume ha, have a \u2209 g\u2081.support, from disjoint_right.1 h ha,\n    by simp only [mem_support_iff, not_not] at *;\n    simpa only [add_apply, this, zero_add])\n\n@[simp] lemma single_add {a : \u03b1} {b\u2081 b\u2082 : M} : single a (b\u2081 + b\u2082) = single a b\u2081 + single a b\u2082 :=\next $ assume a',\nbegin\n  by_cases h : a = a',\n  { rw [h, add_apply, single_eq_same, single_eq_same, single_eq_same] },\n  { rw [add_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h, zero_add] }\nend\n\ninstance : add_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_monoid .\n  zero      := 0,\n  add       := (+),\n  add_assoc := assume \u27e8s, f, hf\u27e9 \u27e8t, g, hg\u27e9 \u27e8u, h, hh\u27e9, ext $ assume a, add_assoc _ _ _,\n  zero_add  := assume \u27e8s, f, hf\u27e9, ext $ assume a, zero_add _,\n  add_zero  := assume \u27e8s, f, hf\u27e9, ext $ assume a, add_zero _ }\n\n\/-- `finsupp.single` as an `add_monoid_hom`.\n\nSee `finsupp.lsingle` for the stronger version as a linear map.\n-\/\n@[simps] def single_add_hom (a : \u03b1) : M \u2192+ \u03b1 \u2192\u2080 M :=\n\u27e8single a, single_zero, \u03bb _ _, single_add\u27e9\n\n\/-- Evaluation of a function `f : \u03b1 \u2192\u2080 M` at a point as an additive monoid homomorphism.\n\nSee `finsupp.lapply` for the stronger version as a linear map. -\/\n@[simps apply]\ndef apply_add_hom (a : \u03b1) : (\u03b1 \u2192\u2080 M) \u2192+ M := \u27e8\u03bb g, g a, zero_apply, \u03bb _ _, add_apply _ _ _\u27e9\n\nlemma single_add_erase (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : single a (f a) + f.erase a = f :=\next $ \u03bb a',\nif h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, add_zero]\nelse by simp only [add_apply, single_eq_of_ne h, zero_add, erase_ne (ne.symm h)]\n\nlemma erase_add_single (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : f.erase a + single a (f a) = f :=\next $ \u03bb a',\nif h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, zero_add]\nelse by simp only [add_apply, single_eq_of_ne h, add_zero, erase_ne (ne.symm h)]\n\n@[simp] lemma erase_add (a : \u03b1) (f f' : \u03b1 \u2192\u2080 M) : erase a (f + f') = erase a f + erase a f' :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, add_apply, erase_same, erase_same, erase_same, add_zero] },\n  rw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply],\nend\n\n@[elab_as_eliminator]\nprotected theorem induction {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (ha : \u2200a b (f : \u03b1 \u2192\u2080 M), a \u2209 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)) :\n  p f :=\nsuffices \u2200s (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f, from this _ _ rfl,\nassume s, finset.induction_on s (\u03bb f hf, by rwa [support_eq_empty.1 hf]) $\nassume a s has ih f hf,\nsuffices p (single a (f a) + f.erase a), by rwa [single_add_erase] at this,\nbegin\n  apply ha,\n  { rw [support_erase, mem_erase], exact \u03bb H, H.1 rfl },\n  { rw [\u2190 mem_support_iff, hf], exact mem_insert_self _ _ },\n  { apply ih _ _,\n    rw [support_erase, hf, finset.erase_insert has] }\nend\n\nlemma induction\u2082 {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (ha : \u2200a b (f : \u03b1 \u2192\u2080 M), a \u2209 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)) :\n  p f :=\nsuffices \u2200s (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f, from this _ _ rfl,\nassume s, finset.induction_on s (\u03bb f hf, by rwa [support_eq_empty.1 hf]) $\nassume a s has ih f hf,\nsuffices p (f.erase a + single a (f a)), by rwa [erase_add_single] at this,\nbegin\n  apply ha,\n  { rw [support_erase, mem_erase], exact \u03bb H, H.1 rfl },\n  { rw [\u2190 mem_support_iff, hf], exact mem_insert_self _ _ },\n  { apply ih _ _,\n    rw [support_erase, hf, finset.erase_insert has] }\nend\n\nlemma induction_linear {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (hadd : \u2200 f g : \u03b1 \u2192\u2080 M, p f \u2192 p g \u2192 p (f + g)) (hsingle : \u2200 a b, p (single a b)) :\n  p f :=\ninduction\u2082 f h0 (\u03bb a b f _ _ w, hadd _ _ w (hsingle _ _))\n\n@[simp] lemma add_closure_Union_range_single :\n  add_submonoid.closure (\u22c3 a : \u03b1, set.range (single a : M \u2192 \u03b1 \u2192\u2080 M)) = \u22a4 :=\ntop_unique $ \u03bb x hx, finsupp.induction x (add_submonoid.zero_mem _) $\n  \u03bb a b f ha hb hf, add_submonoid.add_mem _\n    (add_submonoid.subset_closure $ set.mem_Union.2 \u27e8a, set.mem_range_self _\u27e9) hf\n\n\/-- If two additive homomorphisms from `\u03b1 \u2192\u2080 M` are equal on each `single a b`, then\nthey are equal. -\/\nlemma add_hom_ext [add_monoid N] \u2983f g : (\u03b1 \u2192\u2080 M) \u2192+ N\u2984\n  (H : \u2200 x y, f (single x y) = g (single x y)) :\n  f = g :=\nbegin\n  refine add_monoid_hom.eq_of_eq_on_mdense add_closure_Union_range_single (\u03bb f hf, _),\n  simp only [set.mem_Union, set.mem_range] at hf,\n  rcases hf with \u27e8x, y, rfl\u27e9,\n  apply H\nend\n\n\/-- If two additive homomorphisms from `\u03b1 \u2192\u2080 M` are equal on each `single a b`, then\nthey are equal.\n\nWe formulate this using equality of `add_monoid_hom`s so that `ext` tactic can apply a type-specific\nextensionality lemma after this one.  E.g., if the fiber `M` is `\u2115` or `\u2124`, then it suffices to\nverify `f (single a 1) = g (single a 1)`. -\/\n@[ext] lemma add_hom_ext' [add_monoid N] \u2983f g : (\u03b1 \u2192\u2080 M) \u2192+ N\u2984\n  (H : \u2200 x, f.comp (single_add_hom x) = g.comp (single_add_hom x)) :\n  f = g :=\nadd_hom_ext $ \u03bb x, add_monoid_hom.congr_fun (H x)\n\nlemma mul_hom_ext [monoid N] \u2983f g : multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\u2984\n  (H : \u2200 x y, f (multiplicative.of_add $ single x y) = g (multiplicative.of_add $ single x y)) :\n  f = g :=\nmonoid_hom.ext $ add_monoid_hom.congr_fun $\n  @add_hom_ext \u03b1 M (additive N) _ _ f.to_additive'' g.to_additive'' H\n\n@[ext] lemma mul_hom_ext' [monoid N] {f g : multiplicative (\u03b1 \u2192\u2080 M) \u2192* N}\n  (H : \u2200 x, f.comp (single_add_hom x).to_multiplicative =\n    g.comp (single_add_hom x).to_multiplicative) :\n  f = g :=\nmul_hom_ext $ \u03bb x, monoid_hom.congr_fun (H x)\n\nlemma map_range_add [add_monoid N]\n  {f : M \u2192 N} {hf : f 0 = 0} (hf' : \u2200 x y, f (x + y) = f x + f y) (v\u2081 v\u2082 : \u03b1 \u2192\u2080 M) :\n  map_range f hf (v\u2081 + v\u2082) = map_range f hf v\u2081 + map_range f hf v\u2082 :=\next $ \u03bb a, by simp only [hf', add_apply, map_range_apply]\n\nend add_monoid\n\nend finsupp\n\n@[to_additive]\nlemma mul_equiv.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]\n  (h : N \u2243* P) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\n@[to_additive]\nlemma monoid_hom.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]\n  (h : N \u2192* P) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\nlemma ring_hom.map_finsupp_sum [has_zero M] [semiring R] [semiring S]\n  (h : R \u2192+* S) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 R) : h (f.sum g) = f.sum (\u03bb a b, h (g a b)) :=\nh.map_sum _ _\n\nlemma ring_hom.map_finsupp_prod [has_zero M] [comm_semiring R] [comm_semiring S]\n  (h : R \u2192+* S) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 R) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\n@[to_additive]\nlemma monoid_hom.coe_finsupp_prod [has_zero \u03b2] [monoid N] [comm_monoid P]\n  (f : \u03b1 \u2192\u2080 \u03b2) (g : \u03b1 \u2192 \u03b2 \u2192 N \u2192* P) :\n  \u21d1(f.prod g) = f.prod (\u03bb i fi, g i fi) :=\nmonoid_hom.coe_prod _ _\n\n@[simp, to_additive]\nlemma monoid_hom.finsupp_prod_apply [has_zero \u03b2] [monoid N] [comm_monoid P]\n  (f : \u03b1 \u2192\u2080 \u03b2) (g : \u03b1 \u2192 \u03b2 \u2192 N \u2192* P) (x : N) :\n  f.prod g x = f.prod (\u03bb i fi, g i fi x) :=\nmonoid_hom.finset_prod_apply _ _ _\n\nnamespace finsupp\n\nsection nat_sub\ninstance nat_sub : has_sub (\u03b1 \u2192\u2080 \u2115) := \u27e8zip_with (\u03bb m n, m - n) (nat.sub_zero 0)\u27e9\n\n@[simp] lemma coe_nat_sub (g\u2081 g\u2082 : \u03b1 \u2192\u2080 \u2115) : \u21d1(g\u2081 - g\u2082) = g\u2081 - g\u2082 := rfl\nlemma nat_sub_apply (g\u2081 g\u2082 : \u03b1 \u2192\u2080 \u2115) (a : \u03b1) : (g\u2081 - g\u2082) a = g\u2081 a - g\u2082 a := rfl\n\n@[simp] lemma single_sub {a : \u03b1} {n\u2081 n\u2082 : \u2115} : single a (n\u2081 - n\u2082) = single a n\u2081 - single a n\u2082 :=\nbegin\n  ext f,\n  by_cases h : (a = f),\n  { rw [h, nat_sub_apply, single_eq_same, single_eq_same, single_eq_same] },\n  rw [nat_sub_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h]\nend\n\n-- These next two lemmas are used in developing\n-- the partial derivative on `mv_polynomial`.\n\nlemma sub_single_one_add {a : \u03b1} {u u' : \u03b1 \u2192\u2080 \u2115} (h : u a \u2260 0) :\n  u - single a 1 + u' = u + u' - single a 1 :=\nbegin\n  ext b,\n  rw [add_apply, nat_sub_apply, nat_sub_apply, add_apply],\n  by_cases h : a = b,\n  { rw [\u2190h, single_eq_same], cases (u a), { contradiction }, { simp }, },\n  { simp [h], }\nend\n\nlemma add_sub_single_one {a : \u03b1} {u u' : \u03b1 \u2192\u2080 \u2115} (h : u' a \u2260 0) :\n  u + (u' - single a 1) = u + u' - single a 1 :=\nbegin\n  ext b,\n  rw [add_apply, nat_sub_apply, nat_sub_apply, add_apply],\n  by_cases h : a = b,\n  { rw [\u2190h, single_eq_same], cases (u' a), { contradiction }, { simp }, },\n  { simp [h], }\nend\n\n@[simp] lemma nat_zero_sub (f : \u03b1 \u2192\u2080 \u2115) : 0 - f = 0 := ext $ \u03bb x, nat.zero_sub _\n\nend nat_sub\n\ninstance [add_comm_monoid M] : add_comm_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_comm := assume \u27e8s, f, _\u27e9 \u27e8t, g, _\u27e9, ext $ assume a, add_comm _ _,\n  .. finsupp.add_monoid }\n\ninstance [add_group G] : has_sub (\u03b1 \u2192\u2080 G) := \u27e8zip_with has_sub.sub (sub_zero _)\u27e9\n\ninstance [add_group G] : add_group (\u03b1 \u2192\u2080 G) :=\n{ neg            := map_range (has_neg.neg) neg_zero,\n  sub            := has_sub.sub,\n  sub_eq_add_neg := \u03bb x y, ext (\u03bb i, sub_eq_add_neg _ _),\n  add_left_neg   := assume \u27e8s, f, _\u27e9, ext $ assume x, add_left_neg _,\n  .. finsupp.add_monoid }\n\ninstance [add_comm_group G] : add_comm_group (\u03b1 \u2192\u2080 G) :=\n{ add_comm := add_comm, ..finsupp.add_group }\n\nlemma single_multiset_sum [add_comm_monoid M] (s : multiset M) (a : \u03b1) :\n  single a s.sum = (s.map (single a)).sum :=\nmultiset.induction_on s single_zero $ \u03bb a s ih,\nby rw [multiset.sum_cons, single_add, ih, multiset.map_cons, multiset.sum_cons]\n\nlemma single_finset_sum [add_comm_monoid M] (s : finset \u03b9) (f : \u03b9 \u2192 M) (a : \u03b1) :\n  single a (\u2211 b in s, f b) = \u2211 b in s, single a (f b) :=\nbegin\n  transitivity,\n  apply single_multiset_sum,\n  rw [multiset.map_map],\n  refl\nend\n\nlemma single_sum [has_zero M] [add_comm_monoid N] (s : \u03b9 \u2192\u2080 M) (f : \u03b9 \u2192 M \u2192 N) (a : \u03b1) :\n  single a (s.sum f) = s.sum (\u03bbd c, single a (f d c)) :=\nsingle_finset_sum _ _ _\n\n@[to_additive]\nlemma prod_neg_index [add_group G] [comm_monoid M] {g : \u03b1 \u2192\u2080 G} {h : \u03b1 \u2192 G \u2192 M}\n  (h0 : \u2200a, h a 0 = 1) :\n  (-g).prod h = g.prod (\u03bba b, h a (- b)) :=\nprod_map_range_index h0\n\n@[simp] lemma coe_neg [add_group G] (g : \u03b1 \u2192\u2080 G) : \u21d1(-g) = -g := rfl\nlemma neg_apply [add_group G] (g : \u03b1 \u2192\u2080 G) (a : \u03b1) : (- g) a = - g a := rfl\n\n@[simp] lemma coe_sub [add_group G] (g\u2081 g\u2082 : \u03b1 \u2192\u2080 G) : \u21d1(g\u2081 - g\u2082) = g\u2081 - g\u2082 := rfl\nlemma sub_apply [add_group G] (g\u2081 g\u2082 : \u03b1 \u2192\u2080 G) (a : \u03b1) : (g\u2081 - g\u2082) a = g\u2081 a - g\u2082 a := rfl\n\n@[simp] lemma support_neg [add_group G] {f : \u03b1 \u2192\u2080 G} : support (-f) = support f :=\nfinset.subset.antisymm\n  support_map_range\n  (calc support f = support (- (- f)) : congr_arg support (neg_neg _).symm\n     ... \u2286 support (- f) : support_map_range)\n\n@[simp] lemma sum_apply [has_zero M] [add_comm_monoid N]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 \u03b2 \u2192\u2080 N} {a\u2082 : \u03b2} :\n  (f.sum g) a\u2082 = f.sum (\u03bba\u2081 b, g a\u2081 b a\u2082) :=\n(apply_add_hom a\u2082 : (\u03b2 \u2192\u2080 N) \u2192+ _).map_sum _ _\n\nlemma support_sum [decidable_eq \u03b2] [has_zero M] [add_comm_monoid N]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 (\u03b2 \u2192\u2080 N)} :\n  (f.sum g).support \u2286 f.support.bUnion (\u03bba, (g a (f a)).support) :=\nhave \u2200 c, f.sum (\u03bb a b, g a b c) \u2260 0 \u2192 (\u2203 a, f a \u2260 0 \u2227 \u00ac (g a (f a)) c = 0),\n  from assume a\u2081 h,\n  let \u27e8a, ha, ne\u27e9 := finset.exists_ne_zero_of_sum_ne_zero h in\n  \u27e8a, mem_support_iff.mp ha, ne\u27e9,\nby simpa only [finset.subset_iff, mem_support_iff, finset.mem_bUnion, sum_apply, exists_prop]\n\n@[simp] lemma sum_zero [has_zero M] [add_comm_monoid N] {f : \u03b1 \u2192\u2080 M} :\n  f.sum (\u03bba b, (0 : N)) = 0 :=\nfinset.sum_const_zero\n\n@[simp, to_additive]\nlemma prod_mul  [has_zero M] [comm_monoid N] {f : \u03b1 \u2192\u2080 M} {h\u2081 h\u2082 : \u03b1 \u2192 M \u2192 N} :\n  f.prod (\u03bba b, h\u2081 a b * h\u2082 a b) = f.prod h\u2081 * f.prod h\u2082 :=\nfinset.prod_mul_distrib\n\n@[simp, to_additive]\nlemma prod_inv [has_zero M] [comm_group G] {f : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 G} : f.prod (\u03bba b, (h a b)\u207b\u00b9) = (f.prod h)\u207b\u00b9 :=\n(((monoid_hom.id G)\u207b\u00b9).map_prod _ _).symm\n\n@[simp] lemma sum_sub [has_zero M] [add_comm_group G] {f : \u03b1 \u2192\u2080 M}\n  {h\u2081 h\u2082 : \u03b1 \u2192 M \u2192 G} :\n  f.sum (\u03bba b, h\u2081 a b - h\u2082 a b) = f.sum h\u2081 - f.sum h\u2082 :=\nfinset.sum_sub_distrib\n\n@[to_additive]\nlemma prod_add_index [add_comm_monoid M] [comm_monoid N] {f g : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  (f + g).prod h = f.prod h * g.prod h :=\nhave hf : f.prod h = \u220f a in f.support \u222a g.support, h a (f a),\n  from f.prod_of_support_subset (subset_union_left _ _) _ $ \u03bb a ha, h_zero a,\nhave hg : g.prod h = \u220f a in f.support \u222a g.support, h a (g a),\n  from g.prod_of_support_subset (subset_union_right _ _) _ $ \u03bb a ha, h_zero a,\nhave hfg : (f + g).prod h = \u220f a in f.support \u222a g.support, h a ((f + g) a),\n  from (f + g).prod_of_support_subset support_add _ $ \u03bb a ha, h_zero a,\nby simp only [*, add_apply, prod_mul_distrib]\n\n@[simp]\nlemma sum_add_index' [add_comm_monoid M] [add_comm_monoid N] {f g : \u03b1 \u2192\u2080 M} (h : \u03b1 \u2192 M \u2192+ N) :\n  (f + g).sum (\u03bb x, h x) = f.sum (\u03bb x, h x) + g.sum (\u03bb x, h x) :=\nsum_add_index (\u03bb a, (h a).map_zero) (\u03bb a, (h a).map_add)\n\n@[simp]\nlemma prod_add_index' [add_comm_monoid M] [comm_monoid N] {f g : \u03b1 \u2192\u2080 M}\n  (h : \u03b1 \u2192 multiplicative M \u2192* N) :\n  (f + g).prod (\u03bb a b, h a (multiplicative.of_add b)) =\n    f.prod (\u03bb a b, h a (multiplicative.of_add b)) * g.prod (\u03bb a b, h a (multiplicative.of_add b)) :=\nprod_add_index (\u03bb a, (h a).map_one) (\u03bb a, (h a).map_mul)\n\n\/-- The canonical isomorphism between families of additive monoid homomorphisms `\u03b1 \u2192 (M \u2192+ N)`\nand monoid homomorphisms `(\u03b1 \u2192\u2080 M) \u2192+ N`. -\/\ndef lift_add_hom [add_comm_monoid M] [add_comm_monoid N] : (\u03b1 \u2192 M \u2192+ N) \u2243+ ((\u03b1 \u2192\u2080 M) \u2192+ N) :=\n{ to_fun := \u03bb F,\n  { to_fun := \u03bb f, f.sum (\u03bb x, F x),\n    map_zero' := finset.sum_empty,\n    map_add' := \u03bb _ _, sum_add_index (\u03bb x, (F x).map_zero) (\u03bb x, (F x).map_add) },\n  inv_fun := \u03bb F x, F.comp $ single_add_hom x,\n  left_inv := \u03bb F, by { ext, simp },\n  right_inv := \u03bb F, by { ext, simp },\n  map_add' := \u03bb F G, by { ext, simp } }\n\n@[simp] lemma lift_add_hom_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : \u03b1 \u2192 M \u2192+ N) (f : \u03b1 \u2192\u2080 M) :\n  lift_add_hom F f = f.sum (\u03bb x, F x) :=\nrfl\n\n@[simp] lemma lift_add_hom_symm_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : (\u03b1 \u2192\u2080 M) \u2192+ N) (x : \u03b1) :\n  lift_add_hom.symm F x = F.comp (single_add_hom x) :=\nrfl\n\nlemma lift_add_hom_symm_apply_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : (\u03b1 \u2192\u2080 M) \u2192+ N) (x : \u03b1) (y : M) :\n  lift_add_hom.symm F x y = F (single x y) :=\nrfl\n\n@[simp] lemma lift_add_hom_single_add_hom [add_comm_monoid M] :\n  lift_add_hom (single_add_hom : \u03b1 \u2192 M \u2192+ \u03b1 \u2192\u2080 M) = add_monoid_hom.id _ :=\nlift_add_hom.to_equiv.apply_eq_iff_eq_symm_apply.2 rfl\n\n@[simp] lemma sum_single [add_comm_monoid M] (f : \u03b1 \u2192\u2080 M) :\n  f.sum single = f :=\nadd_monoid_hom.congr_fun lift_add_hom_single_add_hom f\n\n@[simp] lemma lift_add_hom_apply_single [add_comm_monoid M] [add_comm_monoid N]\n  (f : \u03b1 \u2192 M \u2192+ N) (a : \u03b1) (b : M) :\n  lift_add_hom f (single a b) = f a b :=\nsum_single_index (f a).map_zero\n\n@[simp] lemma lift_add_hom_comp_single [add_comm_monoid M] [add_comm_monoid N] (f : \u03b1 \u2192 M \u2192+ N)\n  (a : \u03b1) :\n  (lift_add_hom f).comp (single_add_hom a) = f a :=\nadd_monoid_hom.ext $ \u03bb b, lift_add_hom_apply_single f a b\n\nlemma comp_lift_add_hom [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n  (g : N \u2192+ P) (f : \u03b1 \u2192 M \u2192+ N) :\n  g.comp (lift_add_hom f) = lift_add_hom (\u03bb a, g.comp (f a)) :=\nlift_add_hom.symm_apply_eq.1 $ funext $ \u03bb a,\n  by rw [lift_add_hom_symm_apply, add_monoid_hom.comp_assoc, lift_add_hom_comp_single]\n\nlemma sum_sub_index [add_comm_group \u03b2] [add_comm_group \u03b3] {f g : \u03b1 \u2192\u2080 \u03b2}\n  {h : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (h_sub : \u2200a b\u2081 b\u2082, h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082) :\n  (f - g).sum h = f.sum h - g.sum h :=\n(lift_add_hom (\u03bb a, add_monoid_hom.of_map_sub (h a) (h_sub a))).map_sub f g\n\n@[to_additive]\nlemma prod_emb_domain [has_zero M] [comm_monoid N] {v : \u03b1 \u2192\u2080 M} {f : \u03b1 \u21aa \u03b2} {g : \u03b2 \u2192 M \u2192 N} :\n  (v.emb_domain f).prod g = v.prod (\u03bb a b, g (f a) b) :=\nbegin\n  rw [prod, prod, support_emb_domain, finset.prod_map],\n  simp_rw emb_domain_apply,\nend\n\n@[to_additive]\nlemma prod_finset_sum_index [add_comm_monoid M] [comm_monoid N]\n  {s : finset \u03b9} {g : \u03b9 \u2192 \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  \u220f i in s, (g i).prod h = (\u2211 i in s, g i).prod h :=\nfinset.induction_on s rfl $ \u03bb a s has ih,\nby rw [prod_insert has, ih, sum_insert has, prod_add_index h_zero h_add]\n\n@[to_additive]\nlemma prod_sum_index\n  [add_comm_monoid M] [add_comm_monoid N] [comm_monoid P]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 \u03b2 \u2192\u2080 N}\n  {h : \u03b2 \u2192 N \u2192 P} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  (f.sum g).prod h = f.prod (\u03bba b, (g a b).prod h) :=\n(prod_finset_sum_index h_zero h_add).symm\n\nlemma multiset_sum_sum_index\n  [add_comm_monoid M] [add_comm_monoid N]\n  (f : multiset (\u03b1 \u2192\u2080 M)) (h : \u03b1 \u2192 M \u2192 N)\n  (h\u2080 : \u2200a, h a 0 = 0) (h\u2081 : \u2200 (a : \u03b1) (b\u2081 b\u2082 : M), h a (b\u2081 + b\u2082) = h a b\u2081 + h a b\u2082) :\n  (f.sum.sum h) = (f.map $ \u03bbg:\u03b1 \u2192\u2080 M, g.sum h).sum :=\nmultiset.induction_on f rfl $ assume a s ih,\nby rw [multiset.sum_cons, multiset.map_cons, multiset.sum_cons, sum_add_index h\u2080 h\u2081, ih]\n\nlemma support_sum_eq_bUnion {\u03b1 : Type*} {\u03b9 : Type*} {M : Type*} [add_comm_monoid M]\n  {g : \u03b9 \u2192 \u03b1 \u2192\u2080 M} (s : finset \u03b9) (h : \u2200 i\u2081 i\u2082, i\u2081 \u2260 i\u2082 \u2192 disjoint (g i\u2081).support (g i\u2082).support) :\n  (\u2211 i in s, g i).support = s.bUnion (\u03bb i, (g i).support) :=\nbegin\n  apply finset.induction_on s,\n  { simp },\n  { intros i s hi,\n    simp only [hi, sum_insert, not_false_iff, bUnion_insert],\n    intro hs,\n    rw [finsupp.support_add_eq, hs],\n    rw [hs],\n    intros x hx,\n    simp only [mem_bUnion, exists_prop, inf_eq_inter, ne.def, mem_inter] at hx,\n    obtain \u27e8hxi, j, hj, hxj\u27e9 := hx,\n    have hn : i \u2260 j := \u03bb H, hi (H.symm \u25b8 hj),\n    apply h _ _ hn,\n    simp [hxi, hxj] }\nend\n\nlemma multiset_map_sum [has_zero M] {f : \u03b1 \u2192\u2080 M} {m : \u03b2 \u2192 \u03b3} {h : \u03b1 \u2192 M \u2192 multiset \u03b2} :\n  multiset.map m (f.sum h) = f.sum (\u03bba b, (h a b).map m) :=\n(f.support.sum_hom _).symm\n\nlemma multiset_sum_sum [has_zero M] [add_comm_monoid N] {f : \u03b1 \u2192\u2080 M} {h : \u03b1 \u2192 M \u2192 multiset N} :\n  multiset.sum (f.sum h) = f.sum (\u03bba b, multiset.sum (h a b)) :=\n(f.support.sum_hom multiset.sum).symm\n\nsection map_range\nvariables\n  [add_comm_monoid M] [add_comm_monoid N] (f : M \u2192+ N)\n\n\/--\nComposition with a fixed additive homomorphism is itself an additive homomorphism on functions.\n-\/\ndef map_range.add_monoid_hom : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b1 \u2192\u2080 N) :=\n{ to_fun := (map_range f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 (\u03b1 \u2192\u2080 N)),\n  map_zero' := map_range_zero,\n  map_add' := \u03bb a b, map_range_add f.map_add _ _ }\n\nlemma map_range_multiset_sum (m : multiset (\u03b1 \u2192\u2080 M)) :\n  map_range f f.map_zero m.sum = (m.map $ \u03bbx, map_range f f.map_zero x).sum :=\n(m.sum_hom (map_range.add_monoid_hom f)).symm\n\nlemma map_range_finset_sum (s : finset \u03b9) (g : \u03b9 \u2192 (\u03b1 \u2192\u2080 M))  :\n  map_range f f.map_zero (\u2211 x in s, g x) = \u2211 x in s, map_range f f.map_zero (g x) :=\nby rw [finset.sum.equations._eqn_1, map_range_multiset_sum, multiset.map_map]; refl\n\nend map_range\n\n\/-! ### Declarations about `map_domain` -\/\n\nsection map_domain\nvariables [add_comm_monoid M] {v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M}\n\n\/-- Given `f : \u03b1 \u2192 \u03b2` and `v : \u03b1 \u2192\u2080 M`, `map_domain f v : \u03b2 \u2192\u2080 M`\n  is the finitely supported function whose value at `a : \u03b2` is the sum\n  of `v x` over all `x` such that `f x = a`. -\/\ndef map_domain (f : \u03b1 \u2192 \u03b2) (v : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\nv.sum $ \u03bba, single (f a)\n\nlemma map_domain_apply {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) (x : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  map_domain f x (f a) = x a :=\nbegin\n  rw [map_domain, sum_apply, sum, finset.sum_eq_single a, single_eq_same],\n  { assume b _ hba, exact single_eq_of_ne (hf.ne hba) },\n  { assume h, rw [not_mem_support_iff.1 h, single_zero, zero_apply] }\nend\n\nlemma map_domain_notin_range {f : \u03b1 \u2192 \u03b2} (x : \u03b1 \u2192\u2080 M) (a : \u03b2) (h : a \u2209 set.range f) :\n  map_domain f x a = 0 :=\nbegin\n  rw [map_domain, sum_apply, sum],\n  exact finset.sum_eq_zero\n    (assume a' h', single_eq_of_ne $ assume eq, h $ eq \u25b8 set.mem_range_self _)\nend\n\nlemma map_domain_id : map_domain id v = v :=\nsum_single _\n\nlemma map_domain_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} :\n  map_domain (g \u2218 f) v = map_domain g (map_domain f v) :=\nbegin\n  refine ((sum_sum_index _ _).trans _).symm,\n  { intros, exact single_zero },\n  { intros, exact single_add },\n  refine sum_congr rfl (\u03bb _ _, sum_single_index _),\n  { exact single_zero }\nend\n\nlemma map_domain_single {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : M} : map_domain f (single a b) = single (f a) b :=\nsum_single_index single_zero\n\n@[simp] lemma map_domain_zero {f : \u03b1 \u2192 \u03b2} : map_domain f (0 : \u03b1 \u2192\u2080 M) = (0 : \u03b2 \u2192\u2080 M) :=\nsum_zero_index\n\nlemma map_domain_congr {f g : \u03b1 \u2192 \u03b2} (h : \u2200x\u2208v.support, f x = g x) :\n  v.map_domain f = v.map_domain g :=\nfinset.sum_congr rfl $ \u03bb _ H, by simp only [h _ H]\n\nlemma map_domain_add {f : \u03b1 \u2192 \u03b2} : map_domain f (v\u2081 + v\u2082) = map_domain f v\u2081 + map_domain f v\u2082 :=\nsum_add_index (\u03bb _, single_zero) (\u03bb _ _ _, single_add)\n\nlemma map_domain_finset_sum {f : \u03b1 \u2192 \u03b2} {s : finset \u03b9} {v : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n  map_domain f (\u2211 i in s, v i) = \u2211 i in s, map_domain f (v i) :=\neq.symm $ sum_finset_sum_index (\u03bb _, single_zero) (\u03bb _ _ _, single_add)\n\nlemma map_domain_sum [has_zero N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 N} {v : \u03b1 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n  map_domain f (s.sum v) = s.sum (\u03bba b, map_domain f (v a b)) :=\neq.symm $ sum_finset_sum_index (\u03bb _, single_zero) (\u03bb _ _ _, single_add)\n\nlemma map_domain_support [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M} :\n  (s.map_domain f).support \u2286 s.support.image f :=\nfinset.subset.trans support_sum $\n  finset.subset.trans (finset.bUnion_mono $ assume a ha, support_single_subset) $\n  by rw [finset.bUnion_singleton]; exact subset.refl _\n\n@[to_additive]\nlemma prod_map_domain_index [comm_monoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M}\n  {h : \u03b2 \u2192 M \u2192 N} (h_zero : \u2200b, h b 0 = 1) (h_add : \u2200b m\u2081 m\u2082, h b (m\u2081 + m\u2082) = h b m\u2081 * h b m\u2082) :\n  (map_domain f s).prod h = s.prod (\u03bba m, h (f a) m) :=\n(prod_sum_index h_zero h_add).trans $ prod_congr rfl $ \u03bb _ _, prod_single_index (h_zero _)\n\n\/--\nA version of `sum_map_domain_index` that takes a bundled `add_monoid_hom`,\nrather than separate linearity hypotheses.\n-\/\n-- Note that in `prod_map_domain_index`, `M` is still an additive monoid,\n-- so there is no analogous version in terms of `monoid_hom`.\n@[simp]\nlemma sum_map_domain_index_add_monoid_hom [add_comm_monoid N] {f : \u03b1 \u2192 \u03b2}\n  {s : \u03b1 \u2192\u2080 M} (h : \u03b2 \u2192 M \u2192+ N) :\n  (map_domain f s).sum (\u03bb b m, h b m) = s.sum (\u03bb a m, h (f a) m) :=\n@sum_map_domain_index _ _ _ _ _ _ _ _\n  (\u03bb b m, h b m)\n  (\u03bb b, (h b).map_zero)\n  (\u03bb b m\u2081 m\u2082, (h b).map_add _ _)\n\nlemma emb_domain_eq_map_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) :\n  emb_domain f v = map_domain f v :=\nbegin\n  ext a,\n  by_cases a \u2208 set.range f,\n  { rcases h with \u27e8a, rfl\u27e9,\n    rw [map_domain_apply f.injective, emb_domain_apply] },\n  { rw [map_domain_notin_range, emb_domain_notin_range]; assumption }\nend\n\n@[to_additive]\nlemma prod_map_domain_index_inj [comm_monoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M}\n  {h : \u03b2 \u2192 M \u2192 N} (hf : function.injective f) :\n  (s.map_domain f).prod h = s.prod (\u03bba b, h (f a) b) :=\nby rw [\u2190function.embedding.coe_fn_mk f hf, \u2190emb_domain_eq_map_domain, prod_emb_domain]\n\nlemma map_domain_injective {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  function.injective (map_domain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)) :=\nbegin\n  assume v\u2081 v\u2082 eq, ext a,\n  have : map_domain f v\u2081 (f a) = map_domain f v\u2082 (f a), { rw eq },\n  rwa [map_domain_apply hf, map_domain_apply hf] at this,\nend\n\nend map_domain\n\n\/-! ### Declarations about `comap_domain` -\/\n\nsection comap_domain\n\n\/-- Given `f : \u03b1 \u2192 \u03b2`, `l : \u03b2 \u2192\u2080 M` and a proof `hf` that `f` is injective on\nthe preimage of `l.support`, `comap_domain f l hf` is the finitely supported function\nfrom `\u03b1` to `M` given by composing `l` with `f`. -\/\ndef comap_domain [has_zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : set.inj_on f (f \u207b\u00b9' \u2191l.support)) :\n  \u03b1 \u2192\u2080 M :=\n{ support := l.support.preimage f hf,\n  to_fun := (\u03bb a, l (f a)),\n  mem_support_to_fun :=\n    begin\n      intros a,\n      simp only [finset.mem_def.symm, finset.mem_preimage],\n      exact l.mem_support_to_fun (f a),\n    end }\n\n@[simp]\nlemma comap_domain_apply [has_zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M)\n  (hf : set.inj_on f (f \u207b\u00b9' \u2191l.support)) (a : \u03b1) :\n  comap_domain f l hf a = l (f a) :=\nrfl\n\nlemma sum_comap_domain [has_zero M] [add_comm_monoid N]\n  (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (g : \u03b2 \u2192 M \u2192 N)\n  (hf : set.bij_on f (f \u207b\u00b9' \u2191l.support) \u2191l.support) :\n  (comap_domain f l hf.inj_on).sum (g \u2218 f) = l.sum g :=\nbegin\n  simp only [sum, comap_domain_apply, (\u2218)],\n  simp [comap_domain, finset.sum_preimage_of_bij f _ _ (\u03bb x, g x (l x))],\nend\n\nlemma eq_zero_of_comap_domain_eq_zero [add_comm_monoid M]\n  (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : set.bij_on f (f \u207b\u00b9' \u2191l.support) \u2191l.support) :\n   comap_domain f l hf.inj_on = 0 \u2192 l = 0 :=\nbegin\n  rw [\u2190 support_eq_empty, \u2190 support_eq_empty, comap_domain],\n  simp only [finset.ext_iff, finset.not_mem_empty, iff_false, mem_preimage],\n  assume h a ha,\n  cases hf.2.2 ha with b hb,\n  exact h b (hb.2.symm \u25b8 ha)\nend\n\nlemma map_domain_comap_domain [add_comm_monoid M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M)\n  (hf : function.injective f) (hl : \u2191l.support \u2286 set.range f):\n  map_domain f (comap_domain f l (hf.inj_on _)) = l :=\nbegin\n  ext a,\n  by_cases h_cases: a \u2208 set.range f,\n  { rcases set.mem_range.1 h_cases with \u27e8b, hb\u27e9,\n    rw [hb.symm, map_domain_apply hf, comap_domain_apply] },\n  { rw map_domain_notin_range _ _ h_cases,\n    by_contra h_contr,\n    apply h_cases (hl $ finset.mem_coe.2 $ mem_support_iff.2 $ \u03bb h, h_contr h.symm) }\nend\n\nend comap_domain\n\n\/-! ### Declarations about `filter` -\/\n\nsection filter\nsection has_zero\nvariables [has_zero M] (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M)\n\n\/-- `filter p f` is the function which is `f a` if `p a` is true and 0 otherwise. -\/\ndef filter (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M :=\n{ to_fun := \u03bb a, if p a then f a else 0,\n  support := f.support.filter (\u03bb a, p a),\n  mem_support_to_fun := \u03bb a, by split_ifs; { simp only [h, mem_filter, mem_support_iff], tauto } }\n\nlemma filter_apply (a : \u03b1) [D : decidable (p a)] : f.filter p a = if p a then f a else 0 :=\nby rw subsingleton.elim D; refl\n\nlemma filter_eq_indicator : \u21d1(f.filter p) = set.indicator {x | p x} f := rfl\n\n@[simp] lemma filter_apply_pos {a : \u03b1} (h : p a) : f.filter p a = f a :=\nif_pos h\n\n@[simp] lemma filter_apply_neg {a : \u03b1} (h : \u00ac p a) : f.filter p a = 0 :=\nif_neg h\n\n@[simp] lemma support_filter [D : decidable_pred p] : (f.filter p).support = f.support.filter p :=\nby rw subsingleton.elim D; refl\n\nlemma filter_zero : (0 : \u03b1 \u2192\u2080 M).filter p = 0 :=\nby rw [\u2190 support_eq_empty, support_filter, support_zero, finset.filter_empty]\n\n@[simp] lemma filter_single_of_pos\n  {a : \u03b1} {b : M} (h : p a) : (single a b).filter p = single a b :=\ncoe_fn_injective $ by simp [filter_eq_indicator, set.subset_def, mem_support_single, h]\n\n@[simp] lemma filter_single_of_neg\n  {a : \u03b1} {b : M} (h : \u00ac p a) : (single a b).filter p = 0 :=\next $ by simp [filter_eq_indicator, single_apply_eq_zero, @imp.swap (p _), h]\n\nend has_zero\n\nlemma filter_pos_add_filter_neg [add_monoid M] (f : \u03b1 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n  f.filter p + f.filter (\u03bba, \u00ac p a) = f :=\ncoe_fn_injective $ set.indicator_self_add_compl {x | p x} f\n\nend filter\n\n\/-! ### Declarations about `frange` -\/\n\nsection frange\nvariables [has_zero M]\n\n\/-- `frange f` is the image of `f` on the support of `f`. -\/\ndef frange (f : \u03b1 \u2192\u2080 M) : finset M := finset.image f f.support\n\ntheorem mem_frange {f : \u03b1 \u2192\u2080 M} {y : M} :\n  y \u2208 f.frange \u2194 y \u2260 0 \u2227 \u2203 x, f x = y :=\nfinset.mem_image.trans\n\u27e8\u03bb \u27e8x, hx1, hx2\u27e9, \u27e8hx2 \u25b8 mem_support_iff.1 hx1, x, hx2\u27e9,\n\u03bb \u27e8hy, x, hx\u27e9, \u27e8x, mem_support_iff.2 (hx.symm \u25b8 hy), hx\u27e9\u27e9\n\ntheorem zero_not_mem_frange {f : \u03b1 \u2192\u2080 M} : (0:M) \u2209 f.frange :=\n\u03bb H, (mem_frange.1 H).1 rfl\n\ntheorem frange_single {x : \u03b1} {y : M} : frange (single x y) \u2286 {y} :=\n\u03bb r hr, let \u27e8t, ht1, ht2\u27e9 := mem_frange.1 hr in ht2 \u25b8\n  (by rw single_apply at ht2 \u22a2; split_ifs at ht2 \u22a2; [exact finset.mem_singleton_self _, cc])\n\nend frange\n\n\/-! ### Declarations about `subtype_domain` -\/\n\nsection subtype_domain\n\n\nsection zero\n\nvariables [has_zero M] {p : \u03b1 \u2192 Prop}\n\n\/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -\/\ndef subtype_domain (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : (subtype p \u2192\u2080 M) :=\n\u27e8f.support.subtype p, f \u2218 coe, \u03bb a, by simp only [mem_subtype, mem_support_iff]\u27e9\n\n@[simp] lemma support_subtype_domain [D : decidable_pred p] {f : \u03b1 \u2192\u2080 M} :\n  (subtype_domain p f).support = f.support.subtype p :=\nby rw subsingleton.elim D; refl\n\n@[simp] lemma subtype_domain_apply {a : subtype p} {v : \u03b1 \u2192\u2080 M} :\n  (subtype_domain p v) a = v (a.val) :=\nrfl\n\n@[simp] lemma subtype_domain_zero : subtype_domain p (0 : \u03b1 \u2192\u2080 M) = 0 :=\nrfl\n\nlemma subtype_domain_eq_zero_iff' {f : \u03b1 \u2192\u2080 M} :\n  f.subtype_domain p = 0 \u2194 \u2200 x, p x \u2192 f x = 0 :=\nby simp_rw [\u2190 support_eq_empty, support_subtype_domain, subtype_eq_empty, not_mem_support_iff]\n\nlemma subtype_domain_eq_zero_iff {f : \u03b1 \u2192\u2080 M} (hf : \u2200 x \u2208 f.support , p x) :\n  f.subtype_domain p = 0 \u2194 f = 0 :=\nsubtype_domain_eq_zero_iff'.trans \u27e8\u03bb H, ext $ \u03bb x,\n  if hx : p x then H x hx else not_mem_support_iff.1 $ mt (hf x) hx, \u03bb H x _, by simp [H]\u27e9\n\n@[to_additive]\nlemma prod_subtype_domain_index [comm_monoid N] {v : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (hp : \u2200x\u2208v.support, p x) :\n  (v.subtype_domain p).prod (\u03bba b, h a b) = v.prod h :=\nprod_bij (\u03bbp _, p.val)\n  (\u03bb _, mem_subtype.1)\n  (\u03bb _ _, rfl)\n  (\u03bb _ _ _ _, subtype.eq)\n  (\u03bb b hb, \u27e8\u27e8b, hp b hb\u27e9, mem_subtype.2 hb, rfl\u27e9)\n\nend zero\n\nsection monoid\nvariables [add_monoid M] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 M}\n\n@[simp] lemma subtype_domain_add {v v' : \u03b1 \u2192\u2080 M} :\n  (v + v').subtype_domain p = v.subtype_domain p + v'.subtype_domain p :=\next $ \u03bb _, rfl\n\ninstance subtype_domain.is_add_monoid_hom :\n  is_add_monoid_hom (subtype_domain p : (\u03b1 \u2192\u2080 M) \u2192 subtype p \u2192\u2080 M) :=\n{ map_add := \u03bb _ _, subtype_domain_add, map_zero := subtype_domain_zero }\n\n\/-- `finsupp.filter` as an `add_monoid_hom`. -\/\ndef filter_add_hom (p : \u03b1 \u2192 Prop) : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b1 \u2192\u2080 M) :=\n{ to_fun := filter p,\n  map_zero' := filter_zero p,\n  map_add' := \u03bb f g, coe_fn_injective $ set.indicator_add {x | p x} f g }\n\n@[simp] lemma filter_add {v v' : \u03b1 \u2192\u2080 M} : (v + v').filter p = v.filter p + v'.filter p :=\n(filter_add_hom p).map_add v v'\n\nend monoid\n\nsection comm_monoid\nvariables [add_comm_monoid M] {p : \u03b1 \u2192 Prop}\n\nlemma subtype_domain_sum {s : finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n  (\u2211 c in s, h c).subtype_domain p = \u2211 c in s, (h c).subtype_domain p :=\neq.symm (s.sum_hom _)\n\nlemma subtype_domain_finsupp_sum [has_zero N] {s : \u03b2 \u2192\u2080 N} {h : \u03b2 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n  (s.sum h).subtype_domain p = s.sum (\u03bbc d, (h c d).subtype_domain p) :=\nsubtype_domain_sum\n\nlemma filter_sum (s : finset \u03b9) (f : \u03b9 \u2192 \u03b1 \u2192\u2080 M) :\n  (\u2211 a in s, f a).filter p = \u2211 a in s, filter p (f a) :=\n(filter_add_hom p : (\u03b1 \u2192\u2080 M) \u2192+ _).map_sum f s\n\nlemma filter_eq_sum (p : \u03b1 \u2192 Prop) [D : decidable_pred p] (f : \u03b1 \u2192\u2080 M) :\n  f.filter p = \u2211 i in f.support.filter p, single i (f i) :=\n(f.filter p).sum_single.symm.trans $ finset.sum_congr (by rw subsingleton.elim D; refl) $\n  \u03bb x hx, by rw [filter_apply_pos _ _ (mem_filter.1 hx).2]\n\nend comm_monoid\n\nsection group\nvariables [add_group G] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 G}\n\n@[simp] lemma subtype_domain_neg : (- v).subtype_domain p = - v.subtype_domain p :=\next $ \u03bb _, rfl\n\n@[simp] lemma subtype_domain_sub :\n  (v - v').subtype_domain p = v.subtype_domain p - v'.subtype_domain p :=\next $ \u03bb _, rfl\n\nend group\n\nend subtype_domain\n\n\/-! ### Declarations relating `finsupp` to `multiset` -\/\n\nsection multiset\n\n\/-- Given `f : \u03b1 \u2192\u2080 \u2115`, `f.to_multiset` is the multiset with multiplicities given by the values of\n`f` on the elements of `\u03b1`. We define this function as an `add_equiv`. -\/\ndef to_multiset : (\u03b1 \u2192\u2080 \u2115) \u2243+ multiset \u03b1 :=\n{ to_fun := \u03bb f, f.sum (\u03bba n, n \u2022\u2115 {a}),\n  inv_fun := \u03bb s, \u27e8s.to_finset, \u03bb a, s.count a, \u03bb a, by simp\u27e9,\n  left_inv := \u03bb f, ext $ \u03bb a,\n    suffices (if f a = 0 then 0 else f a) = f a,\n    by simpa [finsupp.sum, multiset.count_sum', multiset.count_cons],\n    by split_ifs with h; [rw h, refl],\n  right_inv := \u03bb s, by simp [finsupp.sum],\n  map_add' := \u03bb f g, sum_add_index (\u03bb a, zero_nsmul _) (\u03bb a, add_nsmul _) }\n\nlemma to_multiset_zero : (0 : \u03b1 \u2192\u2080 \u2115).to_multiset = 0 :=\nrfl\n\nlemma to_multiset_add (m n : \u03b1 \u2192\u2080 \u2115) :\n  (m + n).to_multiset = m.to_multiset + n.to_multiset :=\nto_multiset.map_add m n\n\nlemma to_multiset_apply (f : \u03b1 \u2192\u2080 \u2115) : f.to_multiset = f.sum (\u03bb a n, n \u2022\u2115 {a}) := rfl\n\n@[simp] lemma to_multiset_single (a : \u03b1) (n : \u2115) : to_multiset (single a n) = n \u2022\u2115 {a} :=\nby rw [to_multiset_apply, sum_single_index]; apply zero_nsmul\n\nlemma to_multiset_sum {\u03b9 : Type*} {f : \u03b9 \u2192 \u03b1 \u2192\u2080 \u2115} (s : finset \u03b9) :\n  finsupp.to_multiset (\u2211 i in s, f i) = \u2211 i in s, finsupp.to_multiset (f i) :=\nbegin\n  apply finset.induction_on s,\n  { simp },\n  { intros i s hi,\n    simp [hi] }\nend\n\nlemma to_multiset_sum_single {\u03b9 : Type*} (s : finset \u03b9) (n : \u2115) :\n  finsupp.to_multiset (\u2211 i in s, single i n) = n \u2022\u2115 s.val :=\nby simp_rw [to_multiset_sum, finsupp.to_multiset_single, multiset.singleton_eq_singleton,\n            sum_nsmul, sum_multiset_singleton]\n\nlemma card_to_multiset (f : \u03b1 \u2192\u2080 \u2115) : f.to_multiset.card = f.sum (\u03bba, id) :=\nby simp [to_multiset_apply, add_monoid_hom.map_finsupp_sum, function.id_def]\n\nlemma to_multiset_map (f : \u03b1 \u2192\u2080 \u2115) (g : \u03b1 \u2192 \u03b2) :\n  f.to_multiset.map g = (f.map_domain g).to_multiset :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.map_zero, map_domain_zero, to_multiset_zero] },\n  { assume a n f _ _ ih,\n    rw [to_multiset_add, multiset.map_add, ih, map_domain_add, map_domain_single,\n      to_multiset_single, to_multiset_add, to_multiset_single,\n      is_add_monoid_hom.map_nsmul (multiset.map g)],\n    refl }\nend\n\n@[simp] lemma prod_to_multiset [comm_monoid M] (f : M \u2192\u2080 \u2115) :\n  f.to_multiset.prod = f.prod (\u03bba n, a ^ n) :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.prod_zero, finsupp.prod_zero_index] },\n  { assume a n f _ _ ih,\n    rw [to_multiset_add, multiset.prod_add, ih, to_multiset_single, finsupp.prod_add_index,\n      finsupp.prod_single_index, multiset.prod_nsmul, multiset.singleton_eq_singleton,\n      multiset.prod_singleton],\n    { exact pow_zero a },\n    { exact pow_zero },\n    { exact pow_add  } }\nend\n\n@[simp] lemma to_finset_to_multiset [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 \u2115) :\n  f.to_multiset.to_finset = f.support :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.to_finset_zero, support_zero] },\n  { assume a n f ha hn ih,\n    rw [to_multiset_add, multiset.to_finset_add, ih, to_multiset_single, support_add_eq,\n      support_single_ne_zero hn, multiset.to_finset_nsmul _ _ hn,\n      multiset.singleton_eq_singleton, multiset.to_finset_cons, multiset.to_finset_zero],\n    refl,\n    refine disjoint.mono_left support_single_subset _,\n    rwa [finset.singleton_disjoint] }\nend\n\n@[simp] lemma count_to_multiset [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 \u2115) (a : \u03b1) :\n  f.to_multiset.count a = f a :=\ncalc f.to_multiset.count a = f.sum (\u03bbx n, (n \u2022\u2115 {x} : multiset \u03b1).count a) :\n    (f.support.sum_hom $ multiset.count a).symm\n  ... = f.sum (\u03bbx n, n * ({x} : multiset \u03b1).count a) : by simp only [multiset.count_nsmul]\n  ... = f.sum (\u03bbx n, n * (x ::\u2098 0 : multiset \u03b1).count a) : rfl\n  ... = f a * (a ::\u2098 0 : multiset \u03b1).count a : sum_eq_single _\n    (\u03bb a' _ H, by simp only [multiset.count_cons_of_ne (ne.symm H), multiset.count_zero, mul_zero])\n    (\u03bb H, by simp only [not_mem_support_iff.1 H, zero_mul])\n  ... = f a : by simp only [multiset.count_singleton, mul_one]\n\nlemma mem_support_multiset_sum [add_comm_monoid M]\n  {s : multiset (\u03b1 \u2192\u2080 M)} (a : \u03b1) :\n  a \u2208 s.sum.support \u2192 \u2203f\u2208s, a \u2208 (f : \u03b1 \u2192\u2080 M).support :=\nmultiset.induction_on s false.elim\n  begin\n    assume f s ih ha,\n    by_cases a \u2208 f.support,\n    { exact \u27e8f, multiset.mem_cons_self _ _, h\u27e9 },\n    { simp only [multiset.sum_cons, mem_support_iff, add_apply,\n        not_mem_support_iff.1 h, zero_add] at ha,\n      rcases ih (mem_support_iff.2 ha) with \u27e8f', h\u2080, h\u2081\u27e9,\n      exact \u27e8f', multiset.mem_cons_of_mem h\u2080, h\u2081\u27e9 }\n  end\n\nlemma mem_support_finset_sum [add_comm_monoid M]\n  {s : finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} (a : \u03b1) (ha : a \u2208 (\u2211 c in s, h c).support) :\n  \u2203 c \u2208 s, a \u2208 (h c).support :=\nlet \u27e8f, hf, hfa\u27e9 := mem_support_multiset_sum a ha in\nlet \u27e8c, hc, eq\u27e9 := multiset.mem_map.1 hf in\n\u27e8c, hc, eq.symm \u25b8 hfa\u27e9\n\n@[simp] lemma mem_to_multiset (f : \u03b1 \u2192\u2080 \u2115) (i : \u03b1) :\n  i \u2208 f.to_multiset \u2194 i \u2208 f.support :=\nby rw [\u2190 multiset.count_ne_zero, finsupp.count_to_multiset, finsupp.mem_support_iff]\n\nend multiset\n\n\/-! ### Declarations about `curry` and `uncurry` -\/\n\nsection curry_uncurry\n\nvariables [add_comm_monoid M] [add_comm_monoid N]\n\n\/-- Given a finitely supported function `f` from a product type `\u03b1 \u00d7 \u03b2` to `\u03b3`,\n`curry f` is the \"curried\" finitely supported function from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `\u03b3`. -\/\nprotected def curry (f : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M) : \u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M) :=\nf.sum $ \u03bbp c, single p.1 (single p.2 c)\n\nlemma sum_curry_index (f : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M) (g : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N)\n  (hg\u2080 : \u2200 a b, g a b 0 = 0) (hg\u2081 : \u2200a b c\u2080 c\u2081, g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081) :\n  f.curry.sum (\u03bba f, f.sum (g a)) = f.sum (\u03bbp c, g p.1 p.2 c) :=\nbegin\n  rw [finsupp.curry],\n  transitivity,\n  { exact sum_sum_index (assume a, sum_zero_index)\n      (assume a b\u2080 b\u2081, sum_add_index (assume a, hg\u2080 _ _) (assume c d\u2080 d\u2081, hg\u2081 _ _ _ _)) },\n  congr, funext p c,\n  transitivity,\n  { exact sum_single_index sum_zero_index },\n  exact sum_single_index (hg\u2080 _ _)\nend\n\n\/-- Given a finitely supported function `f` from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `M`,\n`uncurry f` is the \"uncurried\" finitely supported function from `\u03b1 \u00d7 \u03b2` to `M`. -\/\nprotected def uncurry (f : \u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M)) : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M :=\nf.sum $ \u03bba g, g.sum $ \u03bbb c, single (a, b) c\n\n\/-- `finsupp_prod_equiv` defines the `equiv` between `((\u03b1 \u00d7 \u03b2) \u2192\u2080 M)` and `(\u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M))` given by\ncurrying and uncurrying. -\/\ndef finsupp_prod_equiv : ((\u03b1 \u00d7 \u03b2) \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M)) :=\nby refine \u27e8finsupp.curry, finsupp.uncurry, \u03bb f, _, \u03bb f, _\u27e9; simp only [\n  finsupp.curry, finsupp.uncurry, sum_sum_index, sum_zero_index, sum_add_index,\n  sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff,\n  forall_3_true_iff, prod.mk.eta, (single_sum _ _ _).symm, sum_single]\n\nlemma filter_curry (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n  (f.filter (\u03bba:\u03b1\u00d7\u03b2, p a.1)).curry = f.curry.filter p :=\nbegin\n  rw [finsupp.curry, finsupp.curry, finsupp.sum, finsupp.sum, filter_sum, support_filter,\n    sum_filter],\n  refine finset.sum_congr rfl _,\n  rintros \u27e8a\u2081, a\u2082\u27e9 ha,\n  dsimp only,\n  split_ifs,\n  { rw [filter_apply_pos, filter_single_of_pos]; exact h },\n  { rwa [filter_single_of_neg] }\nend\n\nlemma support_curry [decidable_eq \u03b1] (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) :\n  f.curry.support \u2286 f.support.image prod.fst :=\nbegin\n  rw \u2190 finset.bUnion_singleton,\n  refine finset.subset.trans support_sum _,\n  refine finset.bUnion_mono (assume a _, support_single_subset)\nend\n\nend curry_uncurry\n\nsection\nvariables [group G] [mul_action G \u03b1] [add_comm_monoid M]\n\n\/--\nScalar multiplication by a group element g,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain.\n-\/\ndef comap_has_scalar : has_scalar G (\u03b1 \u2192\u2080 M) :=\n{ smul := \u03bb g f, f.comap_domain (\u03bb a, g\u207b\u00b9 \u2022 a)\n  (\u03bb a a' m m' h, by simpa [\u2190mul_smul] using (congr_arg (\u03bb a, g \u2022 a) h)) }\n\nlocal attribute [instance] comap_has_scalar\n\n\/--\nScalar multiplication by a group element,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain,\nis multiplicative in g.\n-\/\ndef comap_mul_action : mul_action G (\u03b1 \u2192\u2080 M) :=\n{ one_smul := \u03bb f, by { ext, dsimp [(\u2022)], simp, },\n  mul_smul := \u03bb g g' f, by { ext, dsimp [(\u2022)], simp [mul_smul], }, }\n\nlocal attribute [instance] comap_mul_action\n\n\/--\nScalar multiplication by a group element,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain,\nis additive in the second argument.\n-\/\ndef comap_distrib_mul_action :\n  distrib_mul_action G (\u03b1 \u2192\u2080 M) :=\n{ smul_zero := \u03bb g, by { ext, dsimp [(\u2022)], simp, },\n  smul_add := \u03bb g f f', by { ext, dsimp [(\u2022)], simp, }, }\n\n\/--\nScalar multiplication by a group element on finitely supported functions on a group,\ngiven by precomposition with the action of g\u207b\u00b9. -\/\ndef comap_distrib_mul_action_self :\n  distrib_mul_action G (G \u2192\u2080 M) :=\n@finsupp.comap_distrib_mul_action G M G _ (monoid.to_mul_action G) _\n\n@[simp]\nlemma comap_smul_single (g : G) (a : \u03b1) (b : M) :\n  g \u2022 single a b = single (g \u2022 a) b :=\nbegin\n  ext a',\n  dsimp [(\u2022)],\n  by_cases h : g \u2022 a = a',\n  { subst h, simp [\u2190mul_smul], },\n  { simp [single_eq_of_ne h], rw [single_eq_of_ne],\n    rintro rfl, simpa [\u2190mul_smul] using h, }\nend\n\n@[simp]\nlemma comap_smul_apply (g : G) (f : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  (g \u2022 f) a = f (g\u207b\u00b9 \u2022 a) := rfl\n\nend\n\nsection\ninstance [semiring R] [add_comm_monoid M] [semimodule R M] : has_scalar R (\u03b1 \u2192\u2080 M) :=\n\u27e8\u03bba v, v.map_range ((\u2022) a) (smul_zero _)\u27e9\n\n\/-!\nThroughout this section, some `semiring` arguments are specified with `{}` instead of `[]`.\nSee note [implicit instance arguments].\n-\/\n\n@[simp] lemma coe_smul {_ : semiring R} [add_comm_monoid M] [semimodule R M]\n  (b : R) (v : \u03b1 \u2192\u2080 M) : \u21d1(b \u2022 v) = b \u2022 v := rfl\nlemma smul_apply {_ : semiring R} [add_comm_monoid M] [semimodule R M]\n  (b : R) (v : \u03b1 \u2192\u2080 M) (a : \u03b1) : (b \u2022 v) a = b \u2022 (v a) := rfl\n\nvariables (\u03b1 M)\n\ninstance [semiring R] [add_comm_monoid M] [semimodule R M] : semimodule R (\u03b1 \u2192\u2080 M) :=\n{ smul      := (\u2022),\n  smul_add  := \u03bb a x y, ext $ \u03bb _, smul_add _ _ _,\n  add_smul  := \u03bb a x y, ext $ \u03bb _, add_smul _ _ _,\n  one_smul  := \u03bb x, ext $ \u03bb _, one_smul _ _,\n  mul_smul  := \u03bb r s x, ext $ \u03bb _, mul_smul _ _ _,\n  zero_smul := \u03bb x, ext $ \u03bb _, zero_smul _ _,\n  smul_zero := \u03bb x, ext $ \u03bb _, smul_zero _ }\n\ninstance [semiring R] [semiring S] [add_comm_monoid M] [semimodule R M] [semimodule S M]\n  [has_scalar R S] [is_scalar_tower R S M] :\n  is_scalar_tower R S (\u03b1 \u2192\u2080 M) :=\n{ smul_assoc := \u03bb r s a, ext $ \u03bb _, smul_assoc _ _ _ }\n\ninstance [semiring R] [semiring S] [add_comm_monoid M] [semimodule R M] [semimodule S M]\n  [smul_comm_class R S M] :\n  smul_comm_class R S (\u03b1 \u2192\u2080 M) :=\n{ smul_comm := \u03bb r s a, ext $ \u03bb _, smul_comm _ _ _ }\n\nvariables {\u03b1 M} {R}\n\nlemma support_smul {_ : semiring R} [add_comm_monoid M] [semimodule R M] {b : R} {g : \u03b1 \u2192\u2080 M} :\n  (b \u2022 g).support \u2286 g.support :=\n\u03bb a, by simp only [smul_apply, mem_support_iff, ne.def]; exact mt (\u03bb h, h.symm \u25b8 smul_zero _)\n\nsection\n\nvariables {p : \u03b1 \u2192 Prop}\n\n@[simp] lemma filter_smul {_ : semiring R} [add_comm_monoid M] [semimodule R M]\n  {b : R} {v : \u03b1 \u2192\u2080 M} : (b \u2022 v).filter p = b \u2022 v.filter p :=\ncoe_fn_injective $ set.indicator_smul {x | p x} b v\n\nend\n\nlemma map_domain_smul {_ : semiring R} [add_comm_monoid M] [semimodule R M]\n   {f : \u03b1 \u2192 \u03b2} (b : R) (v : \u03b1 \u2192\u2080 M) : map_domain f (b \u2022 v) = b \u2022 map_domain f v :=\nbegin\n  change map_domain f (map_range _ _ _) = map_range _ _ _,\n  apply finsupp.induction v, { simp only [map_domain_zero, map_range_zero] },\n  intros a b v' hv\u2081 hv\u2082 IH,\n  rw [map_range_add, map_domain_add, IH, map_domain_add, map_range_add,\n    map_range_single, map_domain_single, map_domain_single, map_range_single];\n  apply smul_add\nend\n\n@[simp] lemma smul_single {_ : semiring R} [add_comm_monoid M] [semimodule R M]\n  (c : R) (a : \u03b1) (b : M) : c \u2022 finsupp.single a b = finsupp.single a (c \u2022 b) :=\nmap_range_single\n\n@[simp] lemma smul_single' {_ : semiring R}\n  (c : R) (a : \u03b1) (b : R) : c \u2022 finsupp.single a b = finsupp.single a (c * b) :=\nsmul_single _ _ _\n\nlemma smul_single_one [semiring R] (a : \u03b1) (b : R) : b \u2022 single a 1 = single a b :=\nby rw [smul_single, smul_eq_mul, mul_one]\n\nend\n\nlemma sum_smul_index [semiring R] [add_comm_monoid M] {g : \u03b1 \u2192\u2080 R} {b : R} {h : \u03b1 \u2192 R \u2192 M}\n  (h0 : \u2200i, h i 0 = 0) : (b \u2022 g).sum h = g.sum (\u03bbi a, h i (b * a)) :=\nfinsupp.sum_map_range_index h0\n\nlemma sum_smul_index' [semiring R] [add_comm_monoid M] [semimodule R M] [add_comm_monoid N]\n  {g : \u03b1 \u2192\u2080 M} {b : R} {h : \u03b1 \u2192 M \u2192 N} (h0 : \u2200i, h i 0 = 0) :\n  (b \u2022 g).sum h = g.sum (\u03bbi c, h i (b \u2022 c)) :=\nfinsupp.sum_map_range_index h0\n\n\/-- A version of `finsupp.sum_smul_index'` for bundled additive maps. -\/\nlemma sum_smul_index_add_monoid_hom\n  [semiring R] [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n  {g : \u03b1 \u2192\u2080 M} {b : R} {h : \u03b1 \u2192 M \u2192+ N} :\n  (b \u2022 g).sum (\u03bb a, h a) = g.sum (\u03bb i c, h i (b \u2022 c)) :=\nsum_map_range_index (\u03bb i, (h i).map_zero)\n\ninstance [semiring R] [add_comm_monoid M] [semimodule R M] {\u03b9 : Type*}\n  [no_zero_smul_divisors R M] : no_zero_smul_divisors R (\u03b9 \u2192\u2080 M) :=\n\u27e8\u03bb c f h, or_iff_not_imp_left.mpr (\u03bb hc, finsupp.ext\n  (\u03bb i, (smul_eq_zero.mp (finsupp.ext_iff.mp h i)).resolve_left hc))\u27e9\n\nsection\nvariables [semiring R] [semiring S]\n\nlemma sum_mul (b : S) (s : \u03b1 \u2192\u2080 R) {f : \u03b1 \u2192 R \u2192 S} :\n  (s.sum f) * b = s.sum (\u03bb a c, (f a c) * b) :=\nby simp only [finsupp.sum, finset.sum_mul]\n\nlemma mul_sum (b : S) (s : \u03b1 \u2192\u2080 R) {f : \u03b1 \u2192 R \u2192 S} :\n  b * (s.sum f) = s.sum (\u03bb a c, b * (f a c)) :=\nby simp only [finsupp.sum, finset.mul_sum]\n\ninstance unique_of_right [subsingleton R] : unique (\u03b1 \u2192\u2080 R) :=\n{ uniq := \u03bb l, ext $ \u03bb i, subsingleton.elim _ _,\n  .. finsupp.inhabited }\n\nend\n\n\/-- Given an `add_comm_monoid M` and `s : set \u03b1`, `restrict_support_equiv s M` is the `equiv`\nbetween the subtype of finitely supported functions with support contained in `s` and\nthe type of finitely supported functions from `s`. -\/\ndef restrict_support_equiv (s : set \u03b1) (M : Type*) [add_comm_monoid M] :\n  {f : \u03b1 \u2192\u2080 M \/\/ \u2191f.support \u2286 s } \u2243 (s \u2192\u2080 M) :=\nbegin\n  refine \u27e8\u03bbf, subtype_domain (\u03bbx, x \u2208 s) f.1, \u03bb f, \u27e8f.map_domain subtype.val, _\u27e9, _, _\u27e9,\n  { refine set.subset.trans (finset.coe_subset.2 map_domain_support) _,\n    rw [finset.coe_image, set.image_subset_iff],\n    exact assume x hx, x.2 },\n  { rintros \u27e8f, hf\u27e9,\n    apply subtype.eq,\n    ext a,\n    dsimp only,\n    refine classical.by_cases (assume h : a \u2208 set.range (subtype.val : s \u2192 \u03b1), _) (assume h, _),\n    { rcases h with \u27e8x, rfl\u27e9,\n      rw [map_domain_apply subtype.val_injective, subtype_domain_apply] },\n    { convert map_domain_notin_range _ _ h,\n      rw [\u2190 not_mem_support_iff],\n      refine mt _ h,\n      exact assume ha, \u27e8\u27e8a, hf ha\u27e9, rfl\u27e9 } },\n  { assume f,\n    ext \u27e8a, ha\u27e9,\n    dsimp only,\n    rw [subtype_domain_apply, map_domain_apply subtype.val_injective] }\nend\n\n\/-- Given `add_comm_monoid M` and `e : \u03b1 \u2243 \u03b2`, `dom_congr e` is the corresponding `equiv` between\n`\u03b1 \u2192\u2080 M` and `\u03b2 \u2192\u2080 M`. -\/\nprotected def dom_congr [add_comm_monoid M] (e : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192\u2080 M) \u2243+ (\u03b2 \u2192\u2080 M) :=\n{ to_fun := map_domain e,\n  inv_fun := map_domain e.symm,\n  left_inv := begin\n    assume v,\n    simp only [map_domain_comp.symm, (\u2218), equiv.symm_apply_apply],\n    exact map_domain_id\n  end,\n  right_inv := begin\n    assume v,\n    simp only [map_domain_comp.symm, (\u2218), equiv.apply_symm_apply],\n    exact map_domain_id\n  end,\n  map_add' := \u03bb a b, map_domain_add, }\n\nend finsupp\n\nnamespace finsupp\n\n\/-! ### Declarations about sigma types -\/\n\nsection sigma\n\nvariables {\u03b1s : \u03b9 \u2192 Type*} [has_zero M] (l : (\u03a3 i, \u03b1s i) \u2192\u2080 M)\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `M` and\nan index element `i : \u03b9`, `split l i` is the `i`th component of `l`,\na finitely supported function from `as i` to `M`. -\/\ndef split (i : \u03b9) : \u03b1s i \u2192\u2080 M :=\nl.comap_domain (sigma.mk i) (\u03bb x1 x2 _ _ hx, heq_iff_eq.1 (sigma.mk.inj hx).2)\n\nlemma split_apply (i : \u03b9) (x : \u03b1s i) : split l i x = l \u27e8i, x\u27e9 :=\nbegin\n  dunfold split,\n  rw comap_domain_apply\nend\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `\u03b2`,\n`split_support l` is the finset of indices in `\u03b9` that appear in the support of `l`. -\/\ndef split_support : finset \u03b9 := l.support.image sigma.fst\n\nlemma mem_split_support_iff_nonzero (i : \u03b9) :\n  i \u2208 split_support l \u2194 split l i \u2260 0 :=\nbegin\n  rw [split_support, mem_image, ne.def, \u2190 support_eq_empty, \u2190 ne.def,\n    \u2190 finset.nonempty_iff_ne_empty, split, comap_domain, finset.nonempty],\n  simp only [exists_prop, finset.mem_preimage, exists_and_distrib_right, exists_eq_right,\n    mem_support_iff, sigma.exists, ne.def]\nend\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 i, \u03b1s i` to `\u03b2` and\nan `\u03b9`-indexed family `g` of functions from `(\u03b1s i \u2192\u2080 \u03b2)` to `\u03b3`, `split_comp` defines a\nfinitely supported function from the index type `\u03b9` to `\u03b3` given by composing `g i` with\n`split l i`. -\/\ndef split_comp [has_zero N] (g : \u03a0 i, (\u03b1s i \u2192\u2080 M) \u2192 N)\n  (hg : \u2200 i x, x = 0 \u2194 g i x = 0) : \u03b9 \u2192\u2080 N :=\n{ support := split_support l,\n  to_fun := \u03bb i, g i (split l i),\n  mem_support_to_fun :=\n  begin\n    intros i,\n    rw [mem_split_support_iff_nonzero, not_iff_not, hg],\n  end }\n\nlemma sigma_support : l.support = l.split_support.sigma (\u03bb i, (l.split i).support) :=\nby simp only [finset.ext_iff, split_support, split, comap_domain, mem_image,\n  mem_preimage, sigma.forall, mem_sigma]; tauto\n\nlemma sigma_sum [add_comm_monoid N] (f : (\u03a3 (i : \u03b9), \u03b1s i) \u2192 M \u2192 N) :\n  l.sum f = \u2211 i in split_support l, (split l i).sum (\u03bb (a : \u03b1s i) b, f \u27e8i, a\u27e9 b) :=\nby simp only [sum, sigma_support, sum_sigma, split_apply]\n\nend sigma\n\nend finsupp\n\n\/-! ### Declarations relating `multiset` to `finsupp` -\/\n\nnamespace multiset\n\n\/-- Given a multiset `s`, `s.to_finsupp` returns the finitely supported function on `\u2115` given by\nthe multiplicities of the elements of `s`. -\/\ndef to_finsupp : multiset \u03b1 \u2243+ (\u03b1 \u2192\u2080 \u2115) := finsupp.to_multiset.symm\n\n@[simp] lemma to_finsupp_support [D : decidable_eq \u03b1] (s : multiset \u03b1) :\n  s.to_finsupp.support = s.to_finset :=\nby rw subsingleton.elim D; refl\n\n@[simp] lemma to_finsupp_apply [D : decidable_eq \u03b1] (s : multiset \u03b1) (a : \u03b1) :\n  to_finsupp s a = s.count a :=\nby rw subsingleton.elim D; refl\n\nlemma to_finsupp_zero : to_finsupp (0 : multiset \u03b1) = 0 := add_equiv.map_zero _\n\nlemma to_finsupp_add (s t : multiset \u03b1) :\n  to_finsupp (s + t) = to_finsupp s + to_finsupp t :=\nto_finsupp.map_add s t\n\n@[simp] lemma to_finsupp_singleton (a : \u03b1) : to_finsupp (a ::\u2098 0) = finsupp.single a 1 :=\nfinsupp.to_multiset.symm_apply_eq.2 $ by simp\n\n@[simp] lemma to_finsupp_to_multiset (s : multiset \u03b1) :\n  s.to_finsupp.to_multiset = s :=\nfinsupp.to_multiset.apply_symm_apply s\n\nlemma to_finsupp_eq_iff {s : multiset \u03b1} {f : \u03b1 \u2192\u2080 \u2115} : s.to_finsupp = f \u2194 s = f.to_multiset :=\nfinsupp.to_multiset.symm_apply_eq\n\nend multiset\n\n@[simp] lemma finsupp.to_multiset_to_finsupp (f : \u03b1 \u2192\u2080 \u2115) :\n  f.to_multiset.to_finsupp = f :=\nfinsupp.to_multiset.symm_apply_apply f\n\n\/-! ### Declarations about order(ed) instances on `finsupp` -\/\n\nnamespace finsupp\n\ninstance [preorder M] [has_zero M] : preorder (\u03b1 \u2192\u2080 M) :=\n{ le := \u03bb f g, \u2200 s, f s \u2264 g s,\n  le_refl := \u03bb f s, le_refl _,\n  le_trans := \u03bb f g h Hfg Hgh s, le_trans (Hfg s) (Hgh s) }\n\ninstance [partial_order M] [has_zero M] : partial_order (\u03b1 \u2192\u2080 M) :=\n{ le_antisymm := \u03bb f g hfg hgf, ext $ \u03bb s, le_antisymm (hfg s) (hgf s),\n  .. finsupp.preorder }\n\ninstance [ordered_cancel_add_comm_monoid M] : add_left_cancel_semigroup (\u03b1 \u2192\u2080 M) :=\n{ add_left_cancel := \u03bb a b c h, ext $ \u03bb s,\n  by { rw ext_iff at h, exact add_left_cancel (h s) },\n  .. finsupp.add_monoid }\n\ninstance [ordered_cancel_add_comm_monoid M] : add_right_cancel_semigroup (\u03b1 \u2192\u2080 M) :=\n{ add_right_cancel := \u03bb a b c h, ext $ \u03bb s,\n  by { rw ext_iff at h, exact add_right_cancel (h s) },\n  .. finsupp.add_monoid }\n\ninstance [ordered_cancel_add_comm_monoid M] : ordered_cancel_add_comm_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_le_add_left := \u03bb a b h c s, add_le_add_left (h s) (c s),\n  le_of_add_le_add_left := \u03bb a b c h s, le_of_add_le_add_left (h s),\n  .. finsupp.add_comm_monoid, .. finsupp.partial_order,\n  .. finsupp.add_left_cancel_semigroup, .. finsupp.add_right_cancel_semigroup }\n\nlemma le_def [preorder M] [has_zero M] {f g : \u03b1 \u2192\u2080 M} : f \u2264 g \u2194 \u2200 x, f x \u2264 g x := iff.rfl\n\nlemma le_iff [canonically_ordered_add_monoid M] (f g : \u03b1 \u2192\u2080 M) :\n  f \u2264 g \u2194 \u2200 s \u2208 f.support, f s \u2264 g s :=\n\u27e8\u03bb h s hs, h s,\n\u03bb h s, if H : s \u2208 f.support then h s H else (not_mem_support_iff.1 H).symm \u25b8 zero_le (g s)\u27e9\n\n@[simp] lemma add_eq_zero_iff [canonically_ordered_add_monoid M] (f g : \u03b1 \u2192\u2080 M) :\n  f + g = 0 \u2194 f = 0 \u2227 g = 0 :=\nby simp [ext_iff, forall_and_distrib]\n\n\/-- `finsupp.to_multiset` as an order isomorphism. -\/\ndef order_iso_multiset : (\u03b1 \u2192\u2080 \u2115) \u2243o multiset \u03b1 :=\n{ to_equiv := to_multiset.to_equiv,\n  map_rel_iff' := \u03bb f g, by simp [multiset.le_iff_count, le_def] }\n\n@[simp] lemma coe_order_iso_multiset : \u21d1(@order_iso_multiset \u03b1) = to_multiset := rfl\n\n@[simp] lemma coe_order_iso_multiset_symm :\n  \u21d1(@order_iso_multiset \u03b1).symm = multiset.to_finsupp := rfl\n\nlemma to_multiset_strict_mono : strict_mono (@to_multiset \u03b1) :=\norder_iso_multiset.strict_mono\n\nlemma sum_id_lt_of_lt (m n : \u03b1 \u2192\u2080 \u2115) (h : m < n) :\n  m.sum (\u03bb _, id) < n.sum (\u03bb _, id) :=\nbegin\n  rw [\u2190 card_to_multiset, \u2190 card_to_multiset],\n  apply multiset.card_lt_of_lt,\n  exact to_multiset_strict_mono h\nend\n\nvariable (\u03b1)\n\n\/-- The order on `\u03c3 \u2192\u2080 \u2115` is well-founded.-\/\nlemma lt_wf : well_founded (@has_lt.lt (\u03b1 \u2192\u2080 \u2115) _) :=\nsubrelation.wf (sum_id_lt_of_lt) $ inv_image.wf _ nat.lt_wf\n\ninstance decidable_le : decidable_rel (@has_le.le (\u03b1 \u2192\u2080 \u2115) _) :=\n\u03bb m n, by rw le_iff; apply_instance\n\nvariable {\u03b1}\n\n@[simp] lemma nat_add_sub_cancel (f g : \u03b1 \u2192\u2080 \u2115) : f + g - g = f :=\next $ \u03bb a, nat.add_sub_cancel _ _\n\n@[simp] lemma nat_add_sub_cancel_left (f g : \u03b1 \u2192\u2080 \u2115) : f + g - f = g :=\next $ \u03bb a, nat.add_sub_cancel_left _ _\n\nlemma nat_add_sub_of_le {f g : \u03b1 \u2192\u2080 \u2115} (h : f \u2264 g) : f + (g - f) = g :=\next $ \u03bb a, nat.add_sub_of_le (h a)\n\nlemma nat_sub_add_cancel {f g : \u03b1 \u2192\u2080 \u2115} (h : f \u2264 g) : g - f + f = g :=\next $ \u03bb a, nat.sub_add_cancel (h a)\n\ninstance : canonically_ordered_add_monoid (\u03b1 \u2192\u2080 \u2115) :=\n{ bot := 0,\n  bot_le := \u03bb f s, zero_le (f s),\n  le_iff_exists_add := \u03bb f g, \u27e8\u03bb H, \u27e8g - f, (nat_add_sub_of_le H).symm\u27e9,\n    \u03bb \u27e8c, hc\u27e9, hc.symm \u25b8 \u03bb x, by simp\u27e9,\n .. (infer_instance : ordered_add_comm_monoid (\u03b1 \u2192\u2080 \u2115)) }\n\n\/-- The `finsupp` counterpart of `multiset.antidiagonal`: the antidiagonal of\n`s : \u03b1 \u2192\u2080 \u2115` consists of all pairs `(t\u2081, t\u2082) : (\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)` such that `t\u2081 + t\u2082 = s`.\nThe finitely supported function `antidiagonal s` is equal to the multiplicities of these pairs. -\/\ndef antidiagonal (f : \u03b1 \u2192\u2080 \u2115) : ((\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)) \u2192\u2080 \u2115 :=\n(f.to_multiset.antidiagonal.map (prod.map multiset.to_finsupp multiset.to_finsupp)).to_finsupp\n\n@[simp] lemma mem_antidiagonal_support {f : \u03b1 \u2192\u2080 \u2115} {p : (\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)} :\n  p \u2208 (antidiagonal f).support \u2194 p.1 + p.2 = f :=\nbegin\n  rcases p with \u27e8p\u2081, p\u2082\u27e9,\n  simp [antidiagonal, \u2190 and.assoc, \u2190 finsupp.to_multiset.apply_eq_iff_eq]\nend\n\nlemma swap_mem_antidiagonal_support {n : \u03b1 \u2192\u2080 \u2115} {f : (\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)} :\n  f.swap \u2208 (antidiagonal n).support \u2194 f \u2208 (antidiagonal n).support :=\nby simp only [mem_antidiagonal_support, add_comm, prod.swap]\n\nlemma antidiagonal_support_filter_fst_eq (f g : \u03b1 \u2192\u2080 \u2115)\n  [D : \u03a0 (p : (\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)), decidable (p.1 = g)] :\n  (antidiagonal f).support.filter (\u03bb p, p.1 = g) = if g \u2264 f then {(g, f - g)} else \u2205 :=\nbegin\n  ext \u27e8a, b\u27e9,\n  suffices : a = g \u2192 (a + b = f \u2194 g \u2264 f \u2227 b = f - g),\n  { simpa [apply_ite ((\u2208) (a, b)), \u2190 and.assoc, @and.right_comm _ (a = _), and.congr_left_iff] },\n  unfreezingI {rintro rfl}, split,\n  { rintro rfl, exact \u27e8le_add_right le_rfl, (nat_add_sub_cancel_left _ _).symm\u27e9 },\n  { rintro \u27e8h, rfl\u27e9, exact nat_add_sub_of_le h }\nend\n\nlemma antidiagonal_support_filter_snd_eq (f g : \u03b1 \u2192\u2080 \u2115)\n  [D : \u03a0 (p : (\u03b1 \u2192\u2080 \u2115) \u00d7 (\u03b1 \u2192\u2080 \u2115)), decidable (p.2 = g)] :\n  (antidiagonal f).support.filter (\u03bb p, p.2 = g) = if g \u2264 f then {(f - g, g)} else \u2205 :=\nbegin\n  ext \u27e8a, b\u27e9,\n  suffices : b = g \u2192 (a + b = f \u2194 g \u2264 f \u2227 a = f - g),\n  { simpa [apply_ite ((\u2208) (a, b)), \u2190 and.assoc, and.congr_left_iff] },\n  unfreezingI {rintro rfl}, split,\n  { rintro rfl, exact \u27e8le_add_left le_rfl, (nat_add_sub_cancel _ _).symm\u27e9 },\n  { rintro \u27e8h, rfl\u27e9, exact nat_sub_add_cancel h }\nend\n\n@[simp] lemma antidiagonal_zero : antidiagonal (0 : \u03b1 \u2192\u2080 \u2115) = single (0,0) 1 :=\nby rw [\u2190 multiset.to_finsupp_singleton]; refl\n\n@[to_additive]\nlemma prod_antidiagonal_support_swap {M : Type*} [comm_monoid M] (n : \u03b1 \u2192\u2080 \u2115)\n  (f : (\u03b1 \u2192\u2080 \u2115) \u2192 (\u03b1 \u2192\u2080 \u2115) \u2192 M) :\n  \u220f p in (antidiagonal n).support, f p.1 p.2 = \u220f p in (antidiagonal n).support, f p.2 p.1 :=\nfinset.prod_bij (\u03bb p hp, p.swap) (\u03bb p, swap_mem_antidiagonal_support.2) (\u03bb p hp, rfl)\n  (\u03bb p\u2081 p\u2082 _ _ h, prod.swap_injective h)\n  (\u03bb p hp, \u27e8p.swap, swap_mem_antidiagonal_support.2 hp, p.swap_swap.symm\u27e9)\n\n\/-- The set `{m : \u03b1 \u2192\u2080 \u2115 | m \u2264 n}` as a `finset`. -\/\ndef Iic_finset (n : \u03b1 \u2192\u2080 \u2115) : finset (\u03b1 \u2192\u2080 \u2115) :=\n(antidiagonal n).support.image prod.fst\n\n@[simp] lemma mem_Iic_finset {m n : \u03b1 \u2192\u2080 \u2115} : m \u2208 Iic_finset n \u2194 m \u2264 n :=\nby simp [Iic_finset, le_iff_exists_add, eq_comm]\n\n@[simp] lemma coe_Iic_finset (n : \u03b1 \u2192\u2080 \u2115) : \u2191(Iic_finset n) = set.Iic n :=\nby { ext, simp }\n\n\/-- Let `n : \u03b1 \u2192\u2080 \u2115` be a finitely supported function.\nThe set of `m : \u03b1 \u2192\u2080 \u2115` that are coordinatewise less than or equal to `n`,\nis a finite set. -\/\nlemma finite_le_nat (n : \u03b1 \u2192\u2080 \u2115) : set.finite {m | m \u2264 n} :=\nby simpa using (Iic_finset n).finite_to_set\n\n\/-- Let `n : \u03b1 \u2192\u2080 \u2115` be a finitely supported function.\nThe set of `m : \u03b1 \u2192\u2080 \u2115` that are coordinatewise less than or equal to `n`,\nbut not equal to `n` everywhere, is a finite set. -\/\nlemma finite_lt_nat (n : \u03b1 \u2192\u2080 \u2115) : set.finite {m | m < n} :=\n(finite_le_nat n).subset $ \u03bb m, le_of_lt\n\nend finsupp\n\nnamespace multiset\n\nlemma to_finsuppstrict_mono : strict_mono (@to_finsupp \u03b1) :=\nfinsupp.order_iso_multiset.symm.strict_mono\n\nend multiset\n","avg_line_length":37.3313419118,"max_line_length":100,"alphanum_fraction":0.6357884111}
-{"size":7689,"ext":"lean","lang":"Lean","max_stars_count":1131.0,"content":"\/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-\/\nimport category_theory.limits.has_limits\nimport category_theory.discrete_category\n\n\/-!\n# Categorical (co)products\n\nThis file defines (co)products as special cases of (co)limits.\n\nA product is the categorical generalization of the object `\u03a0 i, f i` where `f : \u03b9 \u2192 C`. It is a\nlimit cone over the diagram formed by `f`, implemented by converting `f` into a functor\n`discrete \u03b9 \u2964 C`.\n\nA coproduct is the dual concept.\n\n## Main definitions\n\n* a `fan` is a cone over a discrete category\n* `fan.mk` constructs a fan from an indexed collection of maps\n* a `pi` is a `limit (discrete.functor f)`\n\nEach of these has a dual.\n\n## Implementation notes\nAs with the other special shapes in the limits library, all the definitions here are given as\n`abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about\ngeneral limits can be used.\n-\/\n\nnoncomputable theory\n\nuniverses v u u\u2082\n\nopen category_theory\n\nnamespace category_theory.limits\n\nvariables {\u03b2 : Type v}\nvariables {C : Type u} [category.{v} C]\n\n-- We don't need an analogue of `pair` (for binary products), `parallel_pair` (for equalizers),\n-- or `(co)span`, since we already have `discrete.functor`.\n\n\/-- A fan over `f : \u03b2 \u2192 C` consists of a collection of maps from an object `P` to every `f b`. -\/\nabbreviation fan (f : \u03b2 \u2192 C) := cone (discrete.functor f)\n\/-- A cofan over `f : \u03b2 \u2192 C` consists of a collection of maps from every `f b` to an object `P`. -\/\nabbreviation cofan (f : \u03b2 \u2192 C) := cocone (discrete.functor f)\n\n\/-- A fan over `f : \u03b2 \u2192 C` consists of a collection of maps from an object `P` to every `f b`. -\/\n@[simps]\ndef fan.mk {f : \u03b2 \u2192 C} (P : C) (p : \u03a0 b, P \u27f6 f b) : fan f :=\n{ X := P,\n  \u03c0 := { app := p } }\n\n\/-- A cofan over `f : \u03b2 \u2192 C` consists of a collection of maps from every `f b` to an object `P`. -\/\n@[simps]\ndef cofan.mk {f : \u03b2 \u2192 C} (P : C) (p : \u03a0 b, f b \u27f6 P) : cofan f :=\n{ X := P,\n  \u03b9 := { app := p } }\n\n\/-- An abbreviation for `has_limit (discrete.functor f)`. -\/\nabbreviation has_product (f : \u03b2 \u2192 C) := has_limit (discrete.functor f)\n\n\/-- An abbreviation for `has_colimit (discrete.functor f)`. -\/\nabbreviation has_coproduct (f : \u03b2 \u2192 C) := has_colimit (discrete.functor f)\n\nsection\nvariables (C)\n\n\/-- An abbreviation for `has_limits_of_shape (discrete f)`. -\/\nabbreviation has_products_of_shape (\u03b2 : Type v) := has_limits_of_shape.{v} (discrete \u03b2)\n\/-- An abbreviation for `has_colimits_of_shape (discrete f)`. -\/\nabbreviation has_coproducts_of_shape (\u03b2 : Type v) := has_colimits_of_shape.{v} (discrete \u03b2)\nend\n\n\/-- `pi_obj f` computes the product of a family of elements `f`.\n(It is defined as an abbreviation for `limit (discrete.functor f)`,\nso for most facts about `pi_obj f`, you will just use general facts about limits.) -\/\nabbreviation pi_obj (f : \u03b2 \u2192 C) [has_product f] := limit (discrete.functor f)\n\/-- `sigma_obj f` computes the coproduct of a family of elements `f`.\n(It is defined as an abbreviation for `colimit (discrete.functor f)`,\nso for most facts about `sigma_obj f`, you will just use general facts about colimits.) -\/\nabbreviation sigma_obj (f : \u03b2 \u2192 C) [has_coproduct f] := colimit (discrete.functor f)\n\nnotation `\u220f ` f:20 := pi_obj f\nnotation `\u2210 ` f:20 := sigma_obj f\n\n\/-- The `b`-th projection from the pi object over `f` has the form `\u220f f \u27f6 f b`. -\/\nabbreviation pi.\u03c0 (f : \u03b2 \u2192 C) [has_product f] (b : \u03b2) : \u220f f \u27f6 f b :=\nlimit.\u03c0 (discrete.functor f) b\n\/-- The `b`-th inclusion into the sigma object over `f` has the form `f b \u27f6 \u2210 f`. -\/\nabbreviation sigma.\u03b9 (f : \u03b2 \u2192 C) [has_coproduct f] (b : \u03b2) : f b \u27f6 \u2210 f :=\ncolimit.\u03b9 (discrete.functor f) b\n\n\/-- The fan constructed of the projections from the product is limiting. -\/\ndef product_is_product (f : \u03b2 \u2192 C) [has_product f] :\n  is_limit (fan.mk _ (pi.\u03c0 f)) :=\nis_limit.of_iso_limit (limit.is_limit (discrete.functor f)) (cones.ext (iso.refl _) (by tidy))\n\n\/-- The cofan constructed of the inclusions from the coproduct is colimiting. -\/\ndef coproduct_is_coproduct (f : \u03b2 \u2192 C) [has_coproduct f] :\n  is_colimit (cofan.mk _ (sigma.\u03b9 f)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit (discrete.functor f)) (cocones.ext (iso.refl _)\n  (by tidy))\n\n\/-- A collection of morphisms `P \u27f6 f b` induces a morphism `P \u27f6 \u220f f`. -\/\nabbreviation pi.lift {f : \u03b2 \u2192 C} [has_product f] {P : C} (p : \u03a0 b, P \u27f6 f b) : P \u27f6 \u220f f :=\nlimit.lift _ (fan.mk P p)\n\/-- A collection of morphisms `f b \u27f6 P` induces a morphism `\u2210 f \u27f6 P`. -\/\nabbreviation sigma.desc {f : \u03b2 \u2192 C} [has_coproduct f] {P : C} (p : \u03a0 b, f b \u27f6 P) : \u2210 f \u27f6 P :=\ncolimit.desc _ (cofan.mk P p)\n\n\/--\nConstruct a morphism between categorical products (indexed by the same type)\nfrom a family of morphisms between the factors.\n-\/\nabbreviation pi.map {f g : \u03b2 \u2192 C} [has_product f] [has_product g]\n  (p : \u03a0 b, f b \u27f6 g b) : \u220f f \u27f6 \u220f g :=\nlim_map (discrete.nat_trans p)\n\/--\nConstruct an isomorphism between categorical products (indexed by the same type)\nfrom a family of isomorphisms between the factors.\n-\/\nabbreviation pi.map_iso {f g : \u03b2 \u2192 C} [has_products_of_shape \u03b2 C]\n  (p : \u03a0 b, f b \u2245 g b) : \u220f f \u2245 \u220f g :=\nlim.map_iso (discrete.nat_iso p)\n\/--\nConstruct a morphism between categorical coproducts (indexed by the same type)\nfrom a family of morphisms between the factors.\n-\/\nabbreviation sigma.map {f g : \u03b2 \u2192 C} [has_coproduct f] [has_coproduct g]\n  (p : \u03a0 b, f b \u27f6 g b) : \u2210 f \u27f6 \u2210 g :=\ncolim_map (discrete.nat_trans p)\n\/--\nConstruct an isomorphism between categorical coproducts (indexed by the same type)\nfrom a family of isomorphisms between the factors.\n-\/\nabbreviation sigma.map_iso {f g : \u03b2 \u2192 C} [has_coproducts_of_shape \u03b2 C]\n  (p : \u03a0 b, f b \u2245 g b) : \u2210 f \u2245 \u2210 g :=\ncolim.map_iso (discrete.nat_iso p)\n\nsection comparison\n\nvariables {D : Type u\u2082} [category.{v} D] (G : C \u2964 D)\nvariables (f : \u03b2 \u2192 C)\n\n\/-- The comparison morphism for the product of `f`. This is an iso iff `G` preserves the product\nof `f`, see `preserves_product.of_iso_comparison`. -\/\ndef pi_comparison [has_product f] [has_product (\u03bb b, G.obj (f b))] :\n  G.obj (\u220f f) \u27f6 \u220f (\u03bb b, G.obj (f b)) :=\npi.lift (\u03bb b, G.map (pi.\u03c0 f b))\n\n@[simp, reassoc]\nlemma pi_comparison_comp_\u03c0 [has_product f] [has_product (\u03bb b, G.obj (f b))] (b : \u03b2) :\n  pi_comparison G f \u226b pi.\u03c0 _ b = G.map (pi.\u03c0 f b) :=\nlimit.lift_\u03c0 _ b\n\n@[simp, reassoc]\nlemma map_lift_pi_comparison [has_product f] [has_product (\u03bb b, G.obj (f b))]\n  (P : C) (g : \u03a0 j, P \u27f6 f j) :\n  G.map (pi.lift g) \u226b pi_comparison G f = pi.lift (\u03bb j, G.map (g j)) :=\nby { ext, simp [\u2190 G.map_comp] }\n\n\/-- The comparison morphism for the coproduct of `f`. This is an iso iff `G` preserves the coproduct\nof `f`, see `preserves_coproduct.of_iso_comparison`. -\/\ndef sigma_comparison [has_coproduct f] [has_coproduct (\u03bb b, G.obj (f b))] :\n  \u2210 (\u03bb b, G.obj (f b)) \u27f6 G.obj (\u2210 f) :=\nsigma.desc (\u03bb b, G.map (sigma.\u03b9 f b))\n\n@[simp, reassoc]\nlemma \u03b9_comp_sigma_comparison [has_coproduct f] [has_coproduct (\u03bb b, G.obj (f b))] (b : \u03b2) :\n  sigma.\u03b9 _ b \u226b sigma_comparison G f = G.map (sigma.\u03b9 f b) :=\ncolimit.\u03b9_desc _ b\n\n@[simp, reassoc]\nlemma sigma_comparison_map_desc [has_coproduct f] [has_coproduct (\u03bb b, G.obj (f b))]\n  (P : C) (g : \u03a0 j, f j \u27f6 P) :\n  sigma_comparison G f \u226b G.map (sigma.desc g) = sigma.desc (\u03bb j, G.map (g j)) :=\nby { ext, simp [\u2190 G.map_comp] }\n\nend comparison\n\nvariables (C)\n\n\/-- An abbreviation for `\u03a0 J, has_limits_of_shape (discrete J) C` -\/\nabbreviation has_products := \u03a0 (J : Type v), has_limits_of_shape (discrete J) C\n\/-- An abbreviation for `\u03a0 J, has_colimits_of_shape (discrete J) C` -\/\nabbreviation has_coproducts := \u03a0 (J : Type v), has_colimits_of_shape (discrete J) C\n\nend category_theory.limits\n","avg_line_length":39.4307692308,"max_line_length":100,"alphanum_fraction":0.6744700221}
-{"size":248,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2021 OpenAI. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kunhao Zheng\n-\/\nimport data.complex.basic\n\nexample (a : \u2102) : (a - 10) * (a + 11) = a ^ 2 + a - 110 :=\nbegin\n  ring,\nend\n","avg_line_length":20.6666666667,"max_line_length":67,"alphanum_fraction":0.6653225806}
-{"size":19532,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n\nA proof of Hensel's lemma on \u2124_p, roughly following Keith Conrad's writeup:\nhttp:\/\/www.math.uconn.edu\/~kconrad\/blurbs\/gradnumthy\/hensel.pdf\n-\/\n\nimport data.padics.padic_integers data.polynomial topology.metric_space.cau_seq_filter\nimport analysis.specific_limits topology.instances.polynomial\nimport tactic.basic\n\nnoncomputable theory\n\nlocal attribute [instance] classical.prop_decidable\n\nlemma padic_polynomial_dist {p : \u2115} [p.prime] (F : polynomial \u2124_[p]) (x y : \u2124_[p]) :\n  \u2225F.eval x - F.eval y\u2225 \u2264 \u2225x - y\u2225 :=\nlet \u27e8z, hz\u27e9 := F.eval_sub_factor x y in calc\n  \u2225F.eval x - F.eval y\u2225 = \u2225z\u2225 * \u2225x - y\u2225 : by simp [hz]\n    ... \u2264 1 * \u2225x - y\u2225 : mul_le_mul_of_nonneg_right (padic_norm_z.le_one _) (norm_nonneg _)\n    ... = \u2225x - y\u2225 : by simp\n\nopen filter metric\n\nprivate lemma comp_tendsto_lim {p : \u2115} [p.prime] {F : polynomial \u2124_[p]} (ncs : cau_seq \u2124_[p] norm) :\n  tendsto (\u03bb i, F.eval (ncs i)) at_top (nhds (F.eval ncs.lim)) :=\n@tendsto.comp _ _ _ ncs\n  (\u03bb k, F.eval k)\n  _ _ _\n  (continuous_iff_continuous_at.1 F.continuous_eval _) (tendsto_limit ncs)\n\nsection\nparameters {p : \u2115} [nat.prime p] {ncs : cau_seq \u2124_[p] norm} {F : polynomial \u2124_[p]} {a : \u2124_[p]}\n           (ncs_der_val : \u2200 n, \u2225F.derivative.eval (ncs n)\u2225 = \u2225F.derivative.eval a\u2225)\ninclude ncs_der_val\n\nprivate lemma ncs_tendsto_const :\n  tendsto (\u03bb i, \u2225F.derivative.eval (ncs i)\u2225) at_top (nhds \u2225F.derivative.eval a\u2225) :=\nby convert tendsto_const_nhds; ext; rw ncs_der_val\n\nprivate lemma ncs_tendsto_lim :\n  tendsto (\u03bb i, \u2225F.derivative.eval (ncs i)\u2225) at_top (nhds (\u2225F.derivative.eval ncs.lim\u2225)) :=\ntendsto.comp (continuous_iff_continuous_at.1 continuous_norm _) (comp_tendsto_lim _)\n\nprivate lemma norm_deriv_eq : \u2225F.derivative.eval ncs.lim\u2225 = \u2225F.derivative.eval a\u2225 :=\ntendsto_nhds_unique at_top_ne_bot ncs_tendsto_lim ncs_tendsto_const\n\nend\n\nsection\nparameters {p : \u2115} [nat.prime p] {ncs : cau_seq \u2124_[p] norm} {F : polynomial \u2124_[p]}\n           (hnorm : tendsto (\u03bb i, \u2225F.eval (ncs i)\u2225) at_top (nhds 0))\ninclude hnorm\n\nprivate lemma tendsto_zero_of_norm_tendsto_zero : tendsto (\u03bb i, F.eval (ncs i)) at_top (nhds 0) :=\ntendsto_iff_norm_tendsto_zero.2 (by simpa using hnorm)\n\nlemma limit_zero_of_norm_tendsto_zero : F.eval ncs.lim = 0 :=\ntendsto_nhds_unique at_top_ne_bot (comp_tendsto_lim _) tendsto_zero_of_norm_tendsto_zero\n\nend\n\nsection hensel\nopen nat\n\nparameters {p : \u2115} [nat.prime p] {F : polynomial \u2124_[p]} {a : \u2124_[p]}\n           (hnorm : \u2225F.eval a\u2225 < \u2225F.derivative.eval a\u2225^2) (hnsol : F.eval a \u2260 0)\ninclude hnorm\n\nprivate def T : \u211d := \u2225(F.eval a).val \/ ((F.derivative.eval a).val)^2\u2225\n\nprivate lemma deriv_sq_norm_pos : 0 < \u2225F.derivative.eval a\u2225 ^ 2 :=\nlt_of_le_of_lt (norm_nonneg _) hnorm\n\nprivate lemma deriv_sq_norm_ne_zero : \u2225F.derivative.eval a\u2225^2 \u2260 0 := ne_of_gt deriv_sq_norm_pos\n\nprivate lemma deriv_norm_ne_zero : \u2225F.derivative.eval a\u2225 \u2260 0 :=\n\u03bb h, deriv_sq_norm_ne_zero (by simp [*, _root_.pow_two])\n\nprivate lemma deriv_norm_pos : 0 < \u2225F.derivative.eval a\u2225 :=\nlt_of_le_of_ne (norm_nonneg _) (ne.symm deriv_norm_ne_zero)\n\nprivate lemma deriv_ne_zero : F.derivative.eval a \u2260 0 := mt (norm_eq_zero _).2 deriv_norm_ne_zero\n\nprivate lemma T_def : T = \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225^2 :=\ncalc T = \u2225(F.eval a).val\u2225 \/ \u2225((F.derivative.eval a).val)^2\u2225 : norm_div _ _\n   ... = \u2225F.eval a\u2225 \/ \u2225(F.derivative.eval a)^2\u2225 : by simp [norm, padic_norm_z]\n   ... = \u2225F.eval a\u2225 \/ \u2225(F.derivative.eval a)\u2225^2 : by simp [pow, monoid.pow]\n\nprivate lemma T_lt_one : T < 1 :=\nlet h := (div_lt_one_iff_lt deriv_sq_norm_pos).2 hnorm in\nby rw T_def; apply h\n\nprivate lemma T_pow {n : \u2115} (hn : n > 0) : T ^ n < 1 :=\nhave T ^ n \u2264 T ^ 1, from pow_le_pow_of_le_one (norm_nonneg _) (le_of_lt T_lt_one) (succ_le_of_lt hn),\nlt_of_le_of_lt (by simpa) T_lt_one\n\nprivate lemma T_pow' (n : \u2115) : T ^ (2 ^ n) < 1 := (T_pow (nat.pow_pos (by norm_num) _))\n\nprivate lemma T_pow_nonneg (n : \u2115) : T ^ n \u2265 0 := pow_nonneg (norm_nonneg _) _\n\nprivate def ih (n : \u2115) (z : \u2124_[p]) : Prop :=\n\u2225F.derivative.eval z\u2225 = \u2225F.derivative.eval a\u2225 \u2227 \u2225F.eval z\u2225 \u2264 \u2225F.derivative.eval a\u2225^2 * T ^ (2^n)\n\nprivate lemma ih_0 : ih 0 a :=\n\u27e8 rfl, by simp [T_def, mul_div_cancel' _ (ne_of_gt (deriv_sq_norm_pos hnorm))] \u27e9\n\nprivate lemma calc_norm_le_one {n : \u2115} {z : \u2124_[p]} (hz : ih n z) :\n         \u2225(\u2191(F.eval z) : \u211a_[p]) \/ \u2191(F.derivative.eval z)\u2225 \u2264 1 :=\ncalc \u2225(\u2191(F.eval z) : \u211a_[p]) \/ \u2191(F.derivative.eval z)\u2225\n    = \u2225(\u2191(F.eval z) : \u211a_[p])\u2225 \/ \u2225(\u2191(F.derivative.eval z) : \u211a_[p])\u2225 : norm_div _ _\n... = \u2225F.eval z\u2225 \/ \u2225F.derivative.eval a\u2225 : by simp [hz.1]\n... \u2264 \u2225F.derivative.eval a\u2225^2 * T^(2^n) \/ \u2225F.derivative.eval a\u2225 :\n  (div_le_div_right deriv_norm_pos).2 hz.2\n... = \u2225F.derivative.eval a\u2225 * T^(2^n) : div_sq_cancel (ne_of_gt deriv_norm_pos) _\n... \u2264 1 : mul_le_one (padic_norm_z.le_one _) (T_pow_nonneg _) (le_of_lt (T_pow' _))\n\nprivate lemma calc_deriv_dist {z z' z1 : \u2124_[p]} (hz' : z' = z - z1)\n  (hz1 : \u2225z1\u2225 = \u2225F.eval z\u2225 \/ \u2225F.derivative.eval a\u2225) {n} (hz : ih n z) :\n  \u2225F.derivative.eval z' - F.derivative.eval z\u2225 < \u2225F.derivative.eval a\u2225 :=\ncalc\n  \u2225F.derivative.eval z' - F.derivative.eval z\u2225\n    \u2264 \u2225z' - z\u2225 : padic_polynomial_dist _ _ _\n... = \u2225z1\u2225 : by simp [hz']\n... = \u2225F.eval z\u2225 \/ \u2225F.derivative.eval a\u2225 : hz1\n... \u2264 \u2225F.derivative.eval a\u2225^2 * T^(2^n) \/ \u2225F.derivative.eval a\u2225 : (div_le_div_right deriv_norm_pos).2 hz.2\n... = \u2225F.derivative.eval a\u2225 * T^(2^n) : div_sq_cancel deriv_norm_ne_zero _\n... < \u2225F.derivative.eval a\u2225 : (mul_lt_iff_lt_one_right deriv_norm_pos).2 (T_pow (pow_pos (by norm_num) _))\n\nprivate def calc_eval_z'  {z z' z1 : \u2124_[p]} (hz' : z' = z - z1) {n} (hz : ih n z)\n  (h1 : \u2225(\u2191(F.eval z) : \u211a_[p]) \/ \u2191(F.derivative.eval z)\u2225 \u2264 1) (hzeq : z1 = \u27e8_, h1\u27e9) :\n  {q : \u2124_[p] \/\/ F.eval z' = q * z1^2} :=\nhave hdzne' : (\u2191(F.derivative.eval z) : \u211a_[p]) \u2260 0, from\n  have hdzne : F.derivative.eval z \u2260 0,\n    from mt (norm_eq_zero _).2 (by rw hz.1; apply deriv_norm_ne_zero; assumption),\n  \u03bb h, hdzne $ subtype.ext.2 h,\nlet \u27e8q, hq\u27e9 := F.binom_expansion z (-z1) in\nhave \u2225(\u2191(F.derivative.eval z) * (\u2191(F.eval z) \/ \u2191(F.derivative.eval z)) : \u211a_[p])\u2225 \u2264 1,\n  by {rw padic_norm_e.mul, apply mul_le_one, apply padic_norm_z.le_one, apply norm_nonneg, apply h1},\nhave F.derivative.eval z * (-z1) = -F.eval z, from calc\n  F.derivative.eval z * (-z1)\n    = (F.derivative.eval z) * -\u27e8\u2191(F.eval z) \/ \u2191(F.derivative.eval z), h1\u27e9 : by rw [hzeq]\n... = -((F.derivative.eval z) * \u27e8\u2191(F.eval z) \/ \u2191(F.derivative.eval z), h1\u27e9) : by simp\n... = -(\u27e8\u2191(F.derivative.eval z) * (\u2191(F.eval z) \/ \u2191(F.derivative.eval z)), this\u27e9) : subtype.ext.2 $ by simp\n... = -(F.eval z) : by simp [mul_div_cancel' _ hdzne'],\nhave heq : F.eval z' = q * z1^2, by simpa [this, hz'] using hq,\n\u27e8q, heq\u27e9\n\nprivate def calc_eval_z'_norm {z z' z1 : \u2124_[p]} {n} (hz : ih n z) {q}\n  (heq : F.eval z' = q * z1^2) (h1 : \u2225(\u2191(F.eval z) : \u211a_[p]) \/ \u2191(F.derivative.eval z)\u2225 \u2264 1)\n  (hzeq : z1 = \u27e8_, h1\u27e9) : \u2225F.eval z'\u2225 \u2264 \u2225F.derivative.eval a\u2225^2 * T^(2^(n+1)) :=\ncalc \u2225F.eval z'\u2225\n    = \u2225q\u2225 * \u2225z1\u2225^2 : by simp [heq]\n... \u2264 1 * \u2225z1\u2225^2 : mul_le_mul_of_nonneg_right (padic_norm_z.le_one _) (pow_nonneg (norm_nonneg _) _)\n... = \u2225F.eval z\u2225^2 \/ \u2225F.derivative.eval a\u2225^2 :\n  by simp [hzeq, hz.1, div_pow _ (deriv_norm_ne_zero hnorm)]\n... \u2264 (\u2225F.derivative.eval a\u2225^2 * T^(2^n))^2 \/ \u2225F.derivative.eval a\u2225^2 :\n  (div_le_div_right deriv_sq_norm_pos).2 (pow_le_pow_of_le_left (norm_nonneg _) hz.2 _)\n... = (\u2225F.derivative.eval a\u2225^2)^2 * (T^(2^n))^2 \/ \u2225F.derivative.eval a\u2225^2 : by simp only [_root_.mul_pow]\n... = \u2225F.derivative.eval a\u2225^2 * (T^(2^n))^2 : div_sq_cancel deriv_sq_norm_ne_zero _\n... = \u2225F.derivative.eval a\u2225^2 * T^(2^(n + 1)) : by rw [\u2190pow_mul]; refl\n\nset_option eqn_compiler.zeta true\n\n-- we need (ih k) in order to construct the value for k+1, otherwise it might not be an integer.\nprivate def ih_n {n : \u2115} {z : \u2124_[p]} (hz : ih n z) : {z' : \u2124_[p] \/\/ ih (n+1) z'} :=\nhave h1 : \u2225(\u2191(F.eval z) : \u211a_[p]) \/ \u2191(F.derivative.eval z)\u2225 \u2264 1, from calc_norm_le_one hz,\nlet z1 : \u2124_[p] := \u27e8_, h1\u27e9,\n    z' : \u2124_[p] := z - z1 in\n\u27e8 z',\n  have hdist : \u2225F.derivative.eval z' - F.derivative.eval z\u2225 < \u2225F.derivative.eval a\u2225,\n    from calc_deriv_dist rfl (by simp [z1, hz.1]) hz,\n  have hfeq : \u2225F.derivative.eval z'\u2225 = \u2225F.derivative.eval a\u2225,\n    begin\n      rw [sub_eq_add_neg, \u2190 hz.1, \u2190norm_neg (F.derivative.eval z)] at hdist,\n      have := padic_norm_z.eq_of_norm_add_lt_right hdist,\n      rwa [norm_neg, hz.1] at this\n    end,\n  let \u27e8q, heq\u27e9 := calc_eval_z' rfl hz h1 rfl in\n  have hnle : \u2225F.eval z'\u2225 \u2264 \u2225F.derivative.eval a\u2225^2 * T^(2^(n+1)),\n    from calc_eval_z'_norm hz heq h1 rfl,\n  \u27e8hfeq, hnle\u27e9\u27e9\n\nset_option eqn_compiler.zeta false\n\n-- why doesn't \"noncomputable theory\" stick here?\nprivate noncomputable def newton_seq_aux : \u03a0 n : \u2115, {z : \u2124_[p] \/\/ ih n z}\n| 0 := \u27e8a, ih_0\u27e9\n| (k+1) := ih_n (newton_seq_aux k).2\n\nprivate def newton_seq (n : \u2115) : \u2124_[p] := (newton_seq_aux n).1\n\nprivate lemma newton_seq_deriv_norm (n : \u2115) :\n  \u2225F.derivative.eval (newton_seq n)\u2225 = \u2225F.derivative.eval a\u2225 :=\n(newton_seq_aux n).2.1\n\nprivate lemma newton_seq_norm_le (n : \u2115) :\n  \u2225F.eval (newton_seq n)\u2225 \u2264 \u2225F.derivative.eval a\u2225^2 * T ^ (2^n) :=\n(newton_seq_aux n).2.2\n\nprivate lemma newton_seq_norm_eq (n : \u2115) :\n  \u2225newton_seq (n+1) - newton_seq n\u2225 = \u2225F.eval (newton_seq n)\u2225 \/ \u2225F.derivative.eval (newton_seq n)\u2225 :=\nby induction n; simp [newton_seq, newton_seq_aux, ih_n]\n\nprivate lemma newton_seq_succ_dist (n : \u2115) :\n  \u2225newton_seq (n+1) - newton_seq n\u2225 \u2264 \u2225F.derivative.eval a\u2225 * T^(2^n) :=\ncalc \u2225newton_seq (n+1) - newton_seq n\u2225\n    = \u2225F.eval (newton_seq n)\u2225 \/ \u2225F.derivative.eval (newton_seq n)\u2225 : newton_seq_norm_eq _\n... = \u2225F.eval (newton_seq n)\u2225 \/ \u2225F.derivative.eval a\u2225 : by rw newton_seq_deriv_norm\n... \u2264 \u2225F.derivative.eval a\u2225^2 * T ^ (2^n) \/ \u2225F.derivative.eval a\u2225 :\n  (div_le_div_right deriv_norm_pos).2 (newton_seq_norm_le _)\n... = \u2225F.derivative.eval a\u2225 * T^(2^n) : div_sq_cancel (ne_of_gt deriv_norm_pos) _\n\ninclude hnsol\nprivate lemma T_pos : T > 0 :=\nbegin\n  rw T_def,\n  apply div_pos_of_pos_of_pos,\n  { apply (norm_pos_iff _).2,\n    apply hnsol },\n  { exact deriv_sq_norm_pos hnorm }\nend\n\nprivate lemma newton_seq_succ_dist_weak (n : \u2115) :\n  \u2225newton_seq (n+2) - newton_seq (n+1)\u2225 < \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225 :=\nhave 2 \u2264 2^(n+1),\n  from have _, from pow_le_pow (by norm_num : 1 \u2264 2) (nat.le_add_left _ _ : 1 \u2264 n + 1),\n    by simpa using this,\ncalc \u2225newton_seq (n+2) - newton_seq (n+1)\u2225\n    \u2264 \u2225F.derivative.eval a\u2225 * T^(2^(n+1)) : newton_seq_succ_dist _\n... \u2264 \u2225F.derivative.eval a\u2225 * T^2 : mul_le_mul_of_nonneg_left\n                                      (pow_le_pow_of_le_one (norm_nonneg _) (le_of_lt T_lt_one) this)\n                                      (norm_nonneg _)\n... < \u2225F.derivative.eval a\u2225 * T^1 : mul_lt_mul_of_pos_left (pow_lt_pow_of_lt_one T_pos T_lt_one (by norm_num))\n                                                           deriv_norm_pos\n... = \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225 :\n  begin\n    rw [T, _root_.pow_two, _root_.pow_one, norm_div, \u2190mul_div_assoc, padic_norm_e.mul],\n    apply mul_div_mul_left',\n    apply deriv_norm_ne_zero; assumption\n  end\n\nprivate lemma newton_seq_dist_aux (n : \u2115) :\n  \u2200 k : \u2115, \u2225newton_seq (n + k) - newton_seq n\u2225 \u2264 \u2225F.derivative.eval a\u2225 * T^(2^n)\n| 0 := begin simp, apply mul_nonneg, {apply norm_nonneg}, {apply T_pow_nonneg} end\n| (k+1) :=\n  have 2^n \u2264 2^(n+k),\n    by {rw [\u2190nat.pow_eq_pow, \u2190nat.pow_eq_pow], apply pow_le_pow, norm_num, apply nat.le_add_right},\n  calc\n  \u2225newton_seq (n + (k + 1)) - newton_seq n\u2225\n    = \u2225newton_seq ((n + k) + 1) - newton_seq n\u2225 : by simp\n... = \u2225(newton_seq ((n + k) + 1) - newton_seq (n+k)) + (newton_seq (n+k) - newton_seq n)\u2225 : by rw \u2190sub_add_sub_cancel\n... \u2264 max (\u2225newton_seq ((n + k) + 1) - newton_seq (n+k)\u2225) (\u2225newton_seq (n+k) - newton_seq n\u2225) : padic_norm_z.nonarchimedean _ _\n... \u2264 max (\u2225F.derivative.eval a\u2225 * T^(2^((n + k)))) (\u2225F.derivative.eval a\u2225 * T^(2^n)) :\n  max_le_max (newton_seq_succ_dist _) (newton_seq_dist_aux _)\n... = \u2225F.derivative.eval a\u2225 * T^(2^n) :\n  max_eq_right $ mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (norm_nonneg _) (le_of_lt T_lt_one) this) (norm_nonneg _)\n\nprivate lemma newton_seq_dist {n k : \u2115} (hnk : n \u2264 k) :\n  \u2225newton_seq k - newton_seq n\u2225 \u2264 \u2225F.derivative.eval a\u2225 * T^(2^n) :=\nhave hex : \u2203 m, k = n + m, from exists_eq_add_of_le hnk,\n--  \u27e8k - n, by rw [\u2190nat.add_sub_assoc hnk, add_comm, nat.add_sub_assoc (le_refl n), nat.sub_self, nat.add_zero]\u27e9,\nlet \u27e8_, hex'\u27e9 := hex in\nby rw hex'; apply newton_seq_dist_aux; assumption\n\nprivate lemma newton_seq_dist_to_a : \u2200 n : \u2115, 0 < n \u2192 \u2225newton_seq n - a\u2225 = \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225\n| 1 h := by simp [newton_seq, newton_seq_aux, ih_n]; apply norm_div\n| (k+2) h :=\n  have hlt : \u2225newton_seq (k+2) - newton_seq (k+1)\u2225 < \u2225newton_seq (k+1) - a\u2225,\n    by rw newton_seq_dist_to_a (k+1) (succ_pos _); apply newton_seq_succ_dist_weak; assumption,\n  have hne' : \u2225newton_seq (k + 2) - newton_seq (k+1)\u2225 \u2260 \u2225newton_seq (k+1) - a\u2225, from ne_of_lt hlt,\n  calc  \u2225newton_seq (k + 2) - a\u2225\n    = \u2225(newton_seq (k + 2) - newton_seq (k+1)) + (newton_seq (k+1) - a)\u2225 : by rw \u2190sub_add_sub_cancel\n... = max (\u2225newton_seq (k + 2) - newton_seq (k+1)\u2225) (\u2225newton_seq (k+1) - a\u2225) : padic_norm_z.add_eq_max_of_ne hne'\n... = \u2225newton_seq (k+1) - a\u2225 : max_eq_right_of_lt hlt\n... = \u2225polynomial.eval a F\u2225 \/ \u2225polynomial.eval a (polynomial.derivative F)\u2225 : newton_seq_dist_to_a (k+1) (succ_pos _)\n\nprivate lemma bound' : tendsto (\u03bb n : \u2115, \u2225F.derivative.eval a\u2225 * T^(2^n)) at_top (nhds 0) :=\nbegin\n  rw \u2190mul_zero (\u2225F.derivative.eval a\u2225),\n  exact tendsto_mul (tendsto_const_nhds)\n                    (tendsto.comp\n                      (tendsto_pow_at_top_nhds_0_of_lt_1 (norm_nonneg _) (T_lt_one hnorm))\n                      (tendsto_pow_at_top_at_top_of_gt_1_nat (by norm_num)))\nend\n\nprivate lemma bound : \u2200 {\u03b5}, \u03b5 > 0 \u2192 \u2203 N : \u2115, \u2200 {n}, n \u2265 N \u2192 \u2225F.derivative.eval a\u2225 * T^(2^n) < \u03b5 :=\nhave mtn : \u2200 n : \u2115, \u2225polynomial.eval a (polynomial.derivative F)\u2225 * T ^ (2 ^ n) \u2265 0,\n  from \u03bb n, mul_nonneg (norm_nonneg _) (T_pow_nonneg _),\nbegin\n  have := bound' hnorm hnsol,\n  simp [tendsto, nhds] at this,\n  intros \u03b5 h\u03b5,\n  cases this (ball 0 \u03b5) (mem_ball_self h\u03b5) (is_open_ball) with N hN,\n  existsi N, intros n hn,\n  simpa [normed_field.norm_mul, real.norm_eq_abs, abs_of_nonneg (mtn n)] using hN _ hn\nend\n\nprivate lemma bound'_sq : tendsto (\u03bb n : \u2115, \u2225F.derivative.eval a\u2225^2 * T^(2^n)) at_top (nhds 0) :=\nbegin\n  rw [\u2190mul_zero (\u2225F.derivative.eval a\u2225), _root_.pow_two],\n  simp only [mul_assoc],\n  apply tendsto_mul,\n  { apply tendsto_const_nhds },\n  { apply bound', assumption }\nend\n\nprivate theorem newton_seq_is_cauchy : is_cau_seq norm newton_seq :=\nbegin\n  intros \u03b5 h\u03b5,\n  cases bound hnorm hnsol h\u03b5 with N hN,\n  existsi N,\n  intros j hj,\n  apply lt_of_le_of_lt,\n  { apply newton_seq_dist _ _ hj, assumption },\n  { apply hN, apply le_refl }\nend\n\nprivate def newton_cau_seq : cau_seq \u2124_[p] norm := \u27e8_, newton_seq_is_cauchy\u27e9\n\nprivate def soln : \u2124_[p] := newton_cau_seq.lim\n\nprivate lemma soln_spec {\u03b5 : \u211d} (h\u03b5 : \u03b5 > 0) :\n  \u2203 (N : \u2115), \u2200 {i : \u2115}, i \u2265 N \u2192 \u2225soln - newton_cau_seq i\u2225 < \u03b5 :=\nsetoid.symm (cau_seq.equiv_lim newton_cau_seq) _ h\u03b5\n\nprivate lemma soln_deriv_norm : \u2225F.derivative.eval soln\u2225 = \u2225F.derivative.eval a\u2225 :=\nnorm_deriv_eq newton_seq_deriv_norm\n\nprivate lemma newton_seq_norm_tendsto_zero : tendsto (\u03bb i, \u2225F.eval (newton_cau_seq i)\u2225) at_top (nhds 0) :=\nsqueeze_zero (\u03bb _, norm_nonneg _) newton_seq_norm_le bound'_sq\n\nprivate lemma newton_seq_dist_tendsto :\n  tendsto (\u03bb n, \u2225newton_cau_seq n - a\u2225) at_top (nhds (\u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225)) :=\ntendsto.congr'\n  (suffices \u2203 k, \u2200 n \u2265 k,  \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225 = \u2225newton_cau_seq n - a\u2225, by simpa,\n    \u27e81, \u03bb _ hx, (newton_seq_dist_to_a _ hx).symm\u27e9)\n  (tendsto_const_nhds)\n\nprivate lemma newton_seq_dist_tendsto' :\n  tendsto (\u03bb n, \u2225newton_cau_seq n - a\u2225) at_top (nhds \u2225soln - a\u2225) :=\ntendsto.comp (continuous_iff_continuous_at.1 continuous_norm _)\n             (tendsto_sub (tendsto_limit _) tendsto_const_nhds)\n\n\nprivate lemma soln_dist_to_a : \u2225soln - a\u2225 = \u2225F.eval a\u2225 \/ \u2225F.derivative.eval a\u2225 :=\ntendsto_nhds_unique at_top_ne_bot newton_seq_dist_tendsto' newton_seq_dist_tendsto\n\nprivate lemma soln_dist_to_a_lt_deriv : \u2225soln - a\u2225 < \u2225F.derivative.eval a\u2225 :=\nbegin\n  rw soln_dist_to_a,\n  apply div_lt_of_mul_lt_of_pos,\n  { apply deriv_norm_pos; assumption },\n  { rwa _root_.pow_two at hnorm }\nend\n\nprivate lemma eval_soln : F.eval soln = 0 :=\nlimit_zero_of_norm_tendsto_zero newton_seq_norm_tendsto_zero\n\nprivate lemma soln_unique (z : \u2124_[p]) (hev : F.eval z = 0) (hnlt : \u2225z - a\u2225 < \u2225F.derivative.eval a\u2225) :\n  z = soln :=\nhave soln_dist : \u2225z - soln\u2225 < \u2225F.derivative.eval a\u2225, from calc\n  \u2225z - soln\u2225 = \u2225(z - a) + (a - soln)\u2225 : by rw sub_add_sub_cancel\n        ... \u2264 max (\u2225z - a\u2225) (\u2225a - soln\u2225) : padic_norm_z.nonarchimedean _ _\n        ... < \u2225F.derivative.eval a\u2225 : max_lt hnlt (by rw norm_sub_rev; apply soln_dist_to_a_lt_deriv),\nlet h := z - soln,\n    \u27e8q, hq\u27e9 := F.binom_expansion soln h in\nhave (F.derivative.eval soln + q * h) * h = 0, from eq.symm (calc\n  0 = F.eval (soln + h) : by simp [hev, h]\n... = F.derivative.eval soln * h + q * h^2 : by rw [hq, eval_soln, zero_add]\n... = (F.derivative.eval soln + q * h) * h : by rw [_root_.pow_two, right_distrib, mul_assoc]),\nhave h = 0, from by_contradiction $ \u03bb hne,\n  have F.derivative.eval soln + q * h = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne,\n  have F.derivative.eval soln = (-q) * h, by simpa using eq_neg_of_add_eq_zero this,\n  lt_irrefl \u2225F.derivative.eval soln\u2225 (calc\n  \u2225F.derivative.eval soln\u2225 = \u2225(-q) * h\u2225 : by rw this\n... \u2264 1 * \u2225h\u2225 : by rw [padic_norm_z.mul]; exact mul_le_mul_of_nonneg_right (padic_norm_z.le_one _) (norm_nonneg _)\n... = \u2225z - soln\u2225 : by simp [h]\n... < \u2225F.derivative.eval soln\u2225 : by rw soln_deriv_norm; apply soln_dist),\neq_of_sub_eq_zero (by rw \u2190this; refl)\n\nend hensel\n\nvariables {p : \u2115} [nat.prime p] {F : polynomial \u2124_[p]} {a : \u2124_[p]}\n\nprivate lemma a_soln_is_unique (ha : F.eval a = 0) (z' : \u2124_[p]) (hz' : F.eval z' = 0)\n  (hnormz' : \u2225z' - a\u2225 < \u2225F.derivative.eval a\u2225) : z' = a :=\nlet h := z' - a,\n    \u27e8q, hq\u27e9 := F.binom_expansion a h in\nhave (F.derivative.eval a + q * h) * h = 0, from eq.symm (calc\n  0 = F.eval (a + h) : show 0 = F.eval (a + (z' - a)), by rw add_comm; simp [hz']\n... = F.derivative.eval a * h + q * h^2 : by rw [hq, ha, zero_add]\n... = (F.derivative.eval a + q * h) * h : by rw [_root_.pow_two, right_distrib, mul_assoc]),\nhave h = 0, from by_contradiction $ \u03bb hne,\n  have F.derivative.eval a + q * h = 0, from (eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne,\n  have F.derivative.eval a = (-q) * h, by simpa using eq_neg_of_add_eq_zero this,\n  lt_irrefl \u2225F.derivative.eval a\u2225 (calc\n    \u2225F.derivative.eval a\u2225 = \u2225q\u2225*\u2225h\u2225 : by simp [this]\n    ... \u2264 1*\u2225h\u2225 : mul_le_mul_of_nonneg_right (padic_norm_z.le_one _) (norm_nonneg _)\n    ... < \u2225F.derivative.eval a\u2225 : by simpa [h]),\neq_of_sub_eq_zero (by rw \u2190this; refl)\n\nvariable (hnorm : \u2225F.eval a\u2225 < \u2225F.derivative.eval a\u2225^2)\ninclude hnorm\n\nprivate lemma a_is_soln (ha : F.eval a = 0) :\n  F.eval a = 0 \u2227 \u2225a - a\u2225 < \u2225F.derivative.eval a\u2225 \u2227 \u2225F.derivative.eval a\u2225 = \u2225F.derivative.eval a\u2225 \u2227\n  \u2200 z', F.eval z' = 0 \u2192 \u2225z' - a\u2225 < \u2225F.derivative.eval a\u2225 \u2192 z' = a :=\n\u27e8ha, by simp; apply deriv_norm_pos; apply hnorm, rfl, a_soln_is_unique ha\u27e9\n\nlemma hensels_lemma : \u2203 z : \u2124_[p], F.eval z = 0 \u2227 \u2225z - a\u2225 < \u2225F.derivative.eval a\u2225 \u2227\n  \u2225F.derivative.eval z\u2225 = \u2225F.derivative.eval a\u2225 \u2227\n  \u2200 z', F.eval z' = 0 \u2192 \u2225z' - a\u2225 < \u2225F.derivative.eval a\u2225 \u2192 z' = z :=\nif ha : F.eval a = 0 then \u27e8a, a_is_soln hnorm ha\u27e9 else\nby refine \u27e8soln _ _, eval_soln _ _, soln_dist_to_a_lt_deriv _ _, soln_deriv_norm _ _, soln_unique _ _\u27e9;\n   assumption\n","avg_line_length":46.61575179,"max_line_length":127,"alphanum_fraction":0.6463239812}
-{"size":4540,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import analysis.analytic.composition\nimport analysis.normed_space.real_inner_product\nimport topology.metric_space.pi_Lp\nimport analysis.calculus.iterated_deriv\nimport analysis.calculus.mean_value\nimport analysis.calculus.implicit\nimport measure_theory.bochner_integration\nimport measure_theory.lebesgue_measure\n\n\nnamespace lftcm\n\nnoncomputable theory\n\nopen real\nopen_locale topological_space filter classical\n\n\/-!\n# Derivatives\n\nLean can automatically compute some simple derivatives using `simp` tactic.\n-\/\n\nexample : deriv (\u03bb x : \u211d, x^5) 6 = 5 * 6^4 := sorry\n\nexample (x\u2080 : \u211d) (h\u2080 : x\u2080 \u2260 0) : deriv (\u03bb x:\u211d, 1 \/ x) x\u2080 = -1 \/ x\u2080^2 := sorry\n\nexample : deriv sin pi = -1 := sorry\n\n-- Sometimes you need `ring` and\/or `field_simp` after `simp`\nexample (x\u2080 : \u211d) (h : x\u2080 \u2260 0) :\n  deriv (\u03bb x : \u211d, exp(x^2) \/ x^5) x\u2080 = (2 * x\u2080^2 - 5) * exp (x\u2080^2) \/ x\u2080^6 :=\nbegin\n  have : x\u2080^5 \u2260 0, { sorry },\n  simp [this],\n  sorry\nend\n\nexample (a b x\u2080 : \u211d) (h : x\u2080 \u2260 1) :\n  deriv (\u03bb x, (a * x + b) \/ (x - 1)) x\u2080 = -(a + b) \/ (x\u2080 - 1)^2 :=\nbegin\n  sorry\nend\n\n-- Currently `simp` is unable to solve the next example.\n-- A PR that will make this example provable `by simp` would be very welcome!\nexample : iterated_deriv 7 (\u03bb x, sin (tan x) - tan (sin x)) 0 = -168 := sorry\n\nvariables (m n : Type) [fintype m] [fintype n]\n\n-- Generalizations of the next two instances should go to `analysis\/normed_space\/basic`\ninstance : normed_group (matrix m n \u211d) := pi.normed_group\ninstance : normed_space \u211d (matrix m n \u211d) := pi.normed_space\n\n\/-- Trace of a matrix as a continuous linear map. -\/\ndef matrix.trace_clm : matrix n n \u211d \u2192L[\u211d] \u211d :=\n(matrix.trace n \u211d \u211d).mk_continuous (fintype.card n)\nbegin\n  sorry\nend\n\n-- Another hard exercise that would make a very good PR\nexample :\n  has_fderiv_at (\u03bb m : matrix n n \u211d, m.det) (matrix.trace_clm n) 1 :=\nbegin\n  sorry\nend\n\nend lftcm\n\n\n#check deriv\n\n#check has_fderiv_at\n\n\nexample (y : \u211d) : has_deriv_at (\u03bb x : \u211d, 2 * x + 5) 2 y :=\nbegin\n  have := ((has_deriv_at_id y).const_mul 2).add_const 5,\n  rwa [mul_one] at this,\nend\n\nexample (y : \u211d) : deriv (\u03bb x : \u211d, 2 * x + 5) y = 2 := by simp\n\n#check exists_has_deriv_at_eq_slope\n\n#check exists_deriv_eq_slope\n\n\nopen set topological_space\n\nnamespace measure_theory\n\nvariables {\u03b1 E : Type*} [measurable_space \u03b1] [normed_group E] [normed_space \u211d E]\n  [measurable_space E] [borel_space E] [complete_space E] [second_countable_topology E]\n  {\u03bc : measure \u03b1} {f : \u03b1 \u2192 E}\n\n#check integral\n\n#check \u222b x : \u211d, x ^ 2\n\n#check \u222b x in Icc (0:\u211d) 1, x^2\n\n#check \u222b x, f x \u2202\u03bc\n\n#check integral_add\n\n#check integral_add_meas\n\nlemma integral_union (f : \u03b1 \u2192 E) (hfm : measurable f) {s t : set \u03b1}\n  (hs : is_measurable s) (ht : is_measurable t) (hst : disjoint s t)\n  (hfi : integrable f $ \u03bc.restrict (s \u222a t)) :\n  \u222b x in s \u222a t, f x \u2202\u03bc = \u222b x in s, f x \u2202\u03bc + \u222b x in t, f x \u2202\u03bc :=\nbegin\n  rw [measure.restrict_union, integral_add_meas]; try { assumption },\n  rwa [\u2190 measure.restrict_union]; assumption\nend\n\nlemma integral_sdiff (f : \u03b1 \u2192 E) (hfm : measurable f) {s t : set \u03b1}\n  (hs : is_measurable s) (ht : is_measurable t) (hst : s \u2286 t)\n  (hfi : integrable f $ \u03bc.restrict t) :\n  \u222b x in t \\ s, f x \u2202\u03bc = \u222b x in t, f x \u2202\u03bc - \u222b x in s, f x \u2202\u03bc :=\nbegin\n  -- hint: apply `integral_union` to `s` and `t \\ s`\nend\n\nlemma integral_Icc_sub_Icc_of_le [linear_order \u03b1] [topological_space \u03b1] [order_topology \u03b1]\n  [borel_space \u03b1] {x y z : \u03b1} (hxy : x \u2264 y) (hyz : y \u2264 z)\n  {f : \u03b1 \u2192 \u211d} (hfm : measurable f) (hfi : integrable f (\u03bc.restrict $ Icc x z)) :\n  \u222b a in Icc x z, f a \u2202\u03bc - \u222b a in Icc x y, f a \u2202\u03bc = \u222b a in Ioc y z, f a \u2202\u03bc :=\nbegin\n  rw [sub_eq_iff_eq_add', \u2190 integral_union, Icc_union_Ioc_eq_Icc],\n  sorry\nend\n\nlemma set_integral_const (c : E) {s : set \u03b1} (hs : is_measurable s) :\n  \u222b a in s, c \u2202\u03bc = (\u03bc s).to_real \u2022 c :=\nby simp\n\nend measure_theory\n\nopen measure_theory\n\ntheorem FTC {f : \u211d \u2192 \u211d} {x y : \u211d} (hy : continuous_at f y) (h : x < y)\n  (hfm : measurable f)\n  (hfi : integrable f (volume.restrict $ Icc x y)) :\n  has_deriv_at (\u03bb z, \u222b a in Icc x z, f a) (f y) y :=\nbegin\n  have A : has_deriv_within_at (\u03bb z, \u222b a in Icc x z, f a) (f y) (Ici y) y,\n  { rw [has_deriv_within_at_iff_tendsto, metric.tendsto_nhds_within_nhds],\n    intros \u03b5 \u03b50,\n    rw [metric.continuous_at_iff] at hy,\n    rcases hy \u03b5 \u03b50 with \u27e8\u03b4, \u03b40, h\u03b4\u27e9,\n    use [\u03b4, \u03b40],\n    intros z hyz hz\u03b4,\n    rw [integral_Icc_sub_Icc_of_le, dist_zero_right, real.norm_eq_abs, abs_mul, abs_of_nonneg, abs_of_nonneg],\n    all_goals {sorry } },\n  have B : has_deriv_within_at (\u03bb z, \u222b a in Icc x z, f a) (f y) (Iic y) y,\n  { sorry },\n  have := B.union A,\n  simpa using this\nend\n","avg_line_length":27.8527607362,"max_line_length":110,"alphanum_fraction":0.654185022}
-{"size":867,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"open smt_tactic\n\nconstant p : nat \u2192 nat \u2192 Prop\nconstant f : nat \u2192 nat\naxiom pf (a : nat) : p (f a) (f a) \u2192 p a a\n\nlemma ex1 (a b c : nat) : a = b + 0 \u2192 a + c = b + c :=\nby using_smt $ do\n  pr \u2190 tactic.to_expr ```(add_zero b),\n  note `h none pr,\n  trace_state, return ()\n\nlemma ex2(a b c : nat) : a = b \u2192 p (f a) (f b) \u2192 p a b :=\nby using_smt $ do\n  intros,\n  t \u2190 tactic.to_expr ```(p (f a) (f a)),\n  assert `h t,  -- assert automatically closed the new goal\n  trace_state,\n  tactic.trace \"-----\",\n  pr \u2190 tactic.to_expr ```(pf _ h),\n  note `h2 none pr,\n  trace_state,\n  return ()\n\ndef foo := 0\nlemma fooex : foo = 0 := rfl\n\nlemma ex3 (a b c : nat) : a = b + foo \u2192 a + c = b + c :=\nbegin [smt]\n  intros,\n  add_fact fooex,\n  ematch\nend\n\nlemma ex4 (a b c : nat) : a = b \u2192 p (f a) (f b) \u2192 p a b :=\nbegin [smt]\n  intros,\n  assert h : p (f a) (f a),\n  add_fact (pf _ h)\nend\n","avg_line_length":21.1463414634,"max_line_length":59,"alphanum_fraction":0.5490196078}
-{"size":7955,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-\/\nimport Lean.Meta.Transform\nimport Lean.Meta.Inductive\nimport Lean.Elab.Deriving.Basic\nimport Lean.Elab.Deriving.Util\n\nnamespace Lean.Elab.Deriving.DecEq\nopen Lean.Parser.Term\nopen Meta\n\ndef mkDecEqHeader (ctx : Context) (indVal : InductiveVal) : TermElabM Header := do\n  mkHeader ctx `DecidableEq 2 indVal\n\ndef mkMatch (ctx : Context) (header : Header) (indVal : InductiveVal) (auxFunName : Name) (argNames : Array Name) : TermElabM Syntax := do\n  let discrs \u2190 mkDiscrs header indVal\n  let alts \u2190 mkAlts\n  `(match $[$discrs],* with $alts:matchAlt*)\nwhere\n  mkSameCtorRhs : List (Syntax \u00d7 Syntax \u00d7 Bool) \u2192 TermElabM Syntax\n    | [] => ``(isTrue rfl)\n    | (a, b, recField) :: todo => withFreshMacroScope do\n      let rhs \u2190\n        `(if h : $a = $b then\n           by subst h; exact $(\u2190 mkSameCtorRhs todo):term\n          else\n           isFalse (by intro n; injection n; apply h _; assumption))\n      if recField then\n        -- add local instance for `a = b` using the function being defined `auxFunName`\n        `(let inst := $(mkIdent auxFunName) $a $b; $rhs)\n      else\n        return rhs\n\n  mkAlts : TermElabM (Array Syntax) := do\n    let mut alts := #[]\n    for ctorName\u2081 in indVal.ctors do\n      let ctorInfo \u2190 getConstInfoCtor ctorName\u2081\n      for ctorName\u2082 in indVal.ctors do\n        let mut patterns := #[]\n        -- add `_` pattern for indices\n        for i in [:indVal.numIndices] do\n          patterns := patterns.push (\u2190 `(_))\n        if ctorName\u2081 == ctorName\u2082 then\n          let alt \u2190 forallTelescopeReducing ctorInfo.type fun xs type => do\n            let type \u2190 Core.betaReduce type -- we 'beta-reduce' to eliminate \"artificial\" dependencies\n            let mut patterns  := patterns\n            let mut ctorArgs1 := #[]\n            let mut ctorArgs2 := #[]\n            -- add `_` for inductive parameters, they are inaccessible\n            for i in [:indVal.numParams] do\n              ctorArgs1 := ctorArgs1.push (\u2190 `(_))\n              ctorArgs2 := ctorArgs2.push (\u2190 `(_))\n            let mut todo := #[]\n            for i in [:ctorInfo.numFields] do\n              let x := xs[indVal.numParams + i]\n              if type.containsFVar x.fvarId! then\n                -- If resulting type depends on this field, we don't need to compare\n                ctorArgs1 := ctorArgs1.push (\u2190 `(_))\n                ctorArgs2 := ctorArgs2.push (\u2190 `(_))\n              else\n                let a := mkIdent (\u2190 mkFreshUserName `a)\n                let b := mkIdent (\u2190 mkFreshUserName `b)\n                ctorArgs1 := ctorArgs1.push a\n                ctorArgs2 := ctorArgs2.push b\n                let recField  := (\u2190 inferType x).isAppOf indVal.name\n                todo := todo.push (a, b, recField)\n            patterns := patterns.push (\u2190 `(@$(mkIdent ctorName\u2081):ident $ctorArgs1:term*))\n            patterns := patterns.push (\u2190 `(@$(mkIdent ctorName\u2081):ident $ctorArgs2:term*))\n            let rhs \u2190 mkSameCtorRhs todo.toList\n            `(matchAltExpr| | $[$patterns:term],* => $rhs:term)\n          alts := alts.push alt\n        else if (\u2190 compatibleCtors ctorName\u2081 ctorName\u2082) then\n          patterns := patterns ++ #[(\u2190 `($(mkIdent ctorName\u2081) ..)), (\u2190 `($(mkIdent ctorName\u2082) ..))]\n          let rhs \u2190 `(isFalse (by intro h; injection h))\n          alts := alts.push (\u2190 `(matchAltExpr| | $[$patterns:term],* => $rhs:term))\n    return alts\n\ndef mkAuxFunction (ctx : Context) : TermElabM Syntax := do\n  let auxFunName := ctx.auxFunNames[0]\n  let indVal     :=ctx.typeInfos[0]\n  let header     \u2190 mkDecEqHeader ctx indVal\n  let mut body   \u2190 mkMatch ctx header indVal auxFunName header.argNames\n  let binders    := header.binders\n  let type       \u2190 `(Decidable ($(mkIdent header.targetNames[0]) = $(mkIdent header.targetNames[1])))\n  `(private def $(mkIdent auxFunName):ident $binders:explicitBinder* : $type:term := $body:term)\n\ndef mkDecEqCmds (indVal : InductiveVal) : TermElabM (Array Syntax) := do\n  let ctx \u2190 mkContext \"decEq\" indVal.name\n  let cmds := #[\u2190 mkAuxFunction ctx] ++ (\u2190 mkInstanceCmds ctx `DecidableEq #[indVal.name] (useAnonCtor := false))\n  trace[Elab.Deriving.decEq] \"\\n{cmds}\"\n  return cmds\n\nopen Command\n\ndef mkDecEq (declName : Name) : CommandElabM Bool := do\n  let indVal \u2190 getConstInfoInduct declName\n  if indVal.isNested then\n    return false -- nested inductive types are not supported yet\n  else\n    let cmds \u2190 liftTermElabM none <| mkDecEqCmds indVal\n    cmds.forM elabCommand\n    return true\n\npartial def mkEnumOfNat (declName : Name) : MetaM Unit := do\n  let indVal \u2190 getConstInfoInduct declName\n  let enumType := mkConst declName\n  let ctors := indVal.ctors.toArray\n  withLocalDeclD `n (mkConst ``Nat) fun n => do\n    let cond := mkConst ``cond [levelZero]\n    let rec mkDecTree (low high : Nat) : Expr :=\n      if low + 1 == high then\n        mkConst ctors[low]\n      else if low + 2 == high then\n        mkApp4 cond enumType (mkApp2 (mkConst ``Nat.beq) n (mkRawNatLit low)) (mkConst ctors[low]) (mkConst ctors[low+1])\n      else\n        let mid := (low + high)\/2\n        let lowBranch := mkDecTree low mid\n        let highBranch := mkDecTree mid high\n        mkApp4 cond enumType (mkApp2 (mkConst ``Nat.ble) (mkRawNatLit mid) n) highBranch lowBranch\n    let value \u2190 mkLambdaFVars #[n] (mkDecTree 0 ctors.size)\n    let type \u2190 mkArrow (mkConst ``Nat) enumType\n    addAndCompile <| Declaration.defnDecl {\n      name := Name.mkStr declName \"ofNat\"\n      levelParams := []\n      safety := DefinitionSafety.safe\n      hints  := ReducibilityHints.abbrev\n      value, type\n    }\n\ndef mkEnumOfNatThm (declName : Name) : MetaM Unit := do\n  let indVal \u2190 getConstInfoInduct declName\n  let toCtorIdx := mkConst (Name.mkStr declName \"toCtorIdx\")\n  let ofNat     := mkConst (Name.mkStr declName \"ofNat\")\n  let enumType  := mkConst declName\n  let eqEnum    := mkApp (mkConst ``Eq [levelOne]) enumType\n  let rflEnum   := mkApp (mkConst ``Eq.refl [levelOne]) enumType\n  let ctors := indVal.ctors\n  withLocalDeclD `x enumType fun x => do\n    let resultType := mkApp2 eqEnum (mkApp ofNat (mkApp toCtorIdx x)) x\n    let motive     \u2190 mkLambdaFVars #[x] resultType\n    let casesOn    := mkConst (mkCasesOnName declName) [levelZero]\n    let mut value  := mkApp2 casesOn motive x\n    for ctor in ctors do\n      value := mkApp value (mkApp rflEnum (mkConst ctor))\n    value \u2190 mkLambdaFVars #[x] value\n    let type \u2190 mkForallFVars #[x] resultType\n    addAndCompile <| Declaration.thmDecl {\n      name := Name.mkStr declName \"ofNat_toCtorIdx\"\n      levelParams := []\n      value, type\n    }\n\ndef mkDecEqEnum (declName : Name) : CommandElabM Unit := do\n  liftTermElabM none <| mkEnumOfNat declName\n  liftTermElabM none <| mkEnumOfNatThm declName\n  let ofNatIdent  := mkIdent (Name.mkStr declName \"ofNat\")\n  let auxThmIdent := mkIdent (Name.mkStr declName \"ofNat_toCtorIdx\")\n  let indVal \u2190 getConstInfoInduct declName\n  let cmd \u2190 `(\n    instance : DecidableEq $(mkIdent declName) :=\n      fun x y =>\n        if h : x.toCtorIdx = y.toCtorIdx then\n          -- We use `rfl` in the following proof because the first script fails for unit-like datatypes due to etaStruct.\n          isTrue (by first | have aux := congrArg $ofNatIdent h; rw [$auxThmIdent:ident, $auxThmIdent:ident] at aux; assumption | rfl)\n        else\n          isFalse fun h => by subst h; contradiction\n  )\n  trace[Elab.Deriving.decEq] \"\\n{cmd}\"\n  elabCommand cmd\n\ndef mkDecEqInstanceHandler (declNames : Array Name) : CommandElabM Bool := do\n  if declNames.size != 1 then\n    return false -- mutually inductive types are not supported yet\n  else if (\u2190 isEnumType declNames[0]) then\n    mkDecEqEnum declNames[0]\n    return true\n  else\n    mkDecEq declNames[0]\n\nbuiltin_initialize\n  registerBuiltinDerivingHandler `DecidableEq mkDecEqInstanceHandler\n  registerTraceClass `Elab.Deriving.decEq\n\nend Lean.Elab.Deriving.DecEq\n","avg_line_length":42.3138297872,"max_line_length":138,"alphanum_fraction":0.6457573853}
-{"size":125352,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-\/\nimport analysis.calculus.mean_value\nimport analysis.calculus.formal_multilinear_series\n\n\/-!\n# Higher differentiability\n\nA function is `C^1` on a domain if it is differentiable there, and its derivative is continuous.\nBy induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1` there or,\nequivalently, if it is `C^1` and its derivative is `C^{n-1}`.\nFinally, it is `C^\u221e` if it is `C^n` for all n.\n\nWe formalize these notions by defining iteratively the `n+1`-th derivative of a function as the\nderivative of the `n`-th derivative. It is called `iterated_fderiv \ud835\udd5c n f x` where `\ud835\udd5c` is the\nfield, `n` is the number of iterations, `f` is the function and `x` is the point, and it is given\nas an `n`-multilinear map. We also define a version `iterated_fderiv_within` relative to a domain,\nas well as predicates `times_cont_diff_within_at`, `times_cont_diff_at`, `times_cont_diff_on` and\n`times_cont_diff` saying that the function is `C^n` within a set at a point, at a point, on a set\nand on the whole space respectively.\n\nTo avoid the issue of choice when choosing a derivative in sets where the derivative is not\nnecessarily unique, `times_cont_diff_on` is not defined directly in terms of the\nregularity of the specific choice `iterated_fderiv_within \ud835\udd5c n f s` inside `s`, but in terms of the\nexistence of a nice sequence of derivatives, expressed with a predicate\n`has_ftaylor_series_up_to_on`.\n\nWe prove basic properties of these notions.\n\n## Main definitions and results\nLet `f : E \u2192 F` be a map between normed vector spaces over a nondiscrete normed field `\ud835\udd5c`.\n\n* `has_ftaylor_series_up_to n f p`: expresses that the formal multilinear series `p` is a sequence\n  of iterated derivatives of `f`, up to the `n`-th term (where `n` is a natural number or `\u221e`).\n* `has_ftaylor_series_up_to_on n f p s`: same thing, but inside a set `s`. The notion of derivative\n  is now taken inside `s`. In particular, derivatives don't have to be unique.\n* `times_cont_diff \ud835\udd5c n f`: expresses that `f` is `C^n`, i.e., it admits a Taylor series up to\n  rank `n`.\n* `times_cont_diff_on \ud835\udd5c n f s`: expresses that `f` is `C^n` in `s`.\n* `times_cont_diff_at \ud835\udd5c n f x`: expresses that `f` is `C^n` around `x`.\n* `times_cont_diff_within_at \ud835\udd5c n f s x`: expresses that `f` is `C^n` around `x` within the set `s`.\n* `iterated_fderiv_within \ud835\udd5c n f s x` is an `n`-th derivative of `f` over the field `\ud835\udd5c` on the\n  set `s` at the point `x`. It is a continuous multilinear map from `E^n` to `F`, defined as a\n  derivative within `s` of `iterated_fderiv_within \ud835\udd5c (n-1) f s` if one exists, and `0` otherwise.\n* `iterated_fderiv \ud835\udd5c n f x` is the `n`-th derivative of `f` over the field `\ud835\udd5c` at the point `x`.\n  It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of\n  `iterated_fderiv \ud835\udd5c (n-1) f` if one exists, and `0` otherwise.\n\nIn sets of unique differentiability, `times_cont_diff_on \ud835\udd5c n f s` can be expressed in terms of the\nproperties of `iterated_fderiv_within \ud835\udd5c m f s` for `m \u2264 n`. In the whole space,\n`times_cont_diff \ud835\udd5c n f` can be expressed in terms of the properties of `iterated_fderiv \ud835\udd5c m f`\nfor `m \u2264 n`.\n\nWe also prove that the usual operations (addition, multiplication, difference, composition, and\nso on) preserve `C^n` functions.\n\n## Implementation notes\n\nThe definitions in this file are designed to work on any field `\ud835\udd5c`. They are sometimes slightly more\ncomplicated than the naive definitions one would guess from the intuition over the real or complex\nnumbers, but they are designed to circumvent the lack of gluing properties and partitions of unity\nin general. In the usual situations, they coincide with the usual definitions.\n\n### Definition of `C^n` functions in domains\n\nOne could define `C^n` functions in a domain `s` by fixing an arbitrary choice of derivatives (this\nis what we do with `iterated_fderiv_within`) and requiring that all these derivatives up to `n` are\ncontinuous. If the derivative is not unique, this could lead to strange behavior like two `C^n`\nfunctions `f` and `g` on `s` whose sum is not `C^n`. A better definition is thus to say that a\nfunction is `C^n` inside `s` if it admits a sequence of derivatives up to `n` inside `s`.\n\nThis definition still has the problem that a function which is locally `C^n` would not need to\nbe `C^n`, as different choices of sequences of derivatives around different points might possibly\nnot be glued together to give a globally defined sequence of derivatives. (Note that this issue\ncan not happen over reals, thanks to partition of unity, but the behavior over a general field is\nnot so clear, and we want a definition for general fields). Also, there are locality\nproblems for the order parameter: one could image a function which, for each `n`, has a nice\nsequence of derivatives up to order `n`, but they do not coincide for varying `n` and can therefore\nnot be  glued to give rise to an infinite sequence of derivatives. This would give a function\nwhich is `C^n` for all `n`, but not `C^\u221e`. We solve this issue by putting locality conditions\nin space and order in our definition of `times_cont_diff_within_at` and `times_cont_diff_on`.\nThe resulting definition is slightly more complicated to work with (in fact not so much), but it\ngives rise to completely satisfactory theorems.\n\nFor instance, with this definition, a real function which is `C^m` (but not better) on `(-1\/m, 1\/m)`\nfor each natural `m` is by definition `C^\u221e` at `0`.\n\nThere is another issue with the definition of `times_cont_diff_within_at \ud835\udd5c n f s x`. We can\nrequire the existence and good behavior of derivatives up to order `n` on a neighborhood of `x`\nwithin `s`. However, this does not imply continuity or differentiability within `s` of the function\nat `x` when `x` does not belong to `s`. Therefore, we require such existence and good behavior on\na neighborhood of `x` within `s \u222a {x}` (which appears as `insert x s` in this file).\n\n### Side of the composition, and universe issues\n\nWith a na\u00efve direct definition, the `n`-th derivative of a function belongs to the space\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] (E ... F)...)))` where there are n iterations of `E \u2192L[\ud835\udd5c]`. This space\nmay also be seen as the space of continuous multilinear functions on `n` copies of `E` with\nvalues in `F`, by uncurrying. This is the point of view that is usually adopted in textbooks,\nand that we also use. This means that the definition and the first proofs are slightly involved,\nas one has to keep track of the uncurrying operation. The uncurrying can be done from the\nleft or from the right, amounting to defining the `n+1`-th derivative either as the derivative of\nthe `n`-th derivative, or as the `n`-th derivative of the derivative.\nFor proofs, it would be more convenient to use the latter approach (from the right),\nas it means to prove things at the `n+1`-th step we only need to understand well enough the\nderivative in `E \u2192L[\ud835\udd5c] F` (contrary to the approach from the left, where one would need to know\nenough on the `n`-th derivative to deduce things on the `n+1`-th derivative).\n\nHowever, the definition from the right leads to a universe polymorphism problem: if we define\n`iterated_fderiv \ud835\udd5c (n + 1) f x = iterated_fderiv \ud835\udd5c n (fderiv \ud835\udd5c f) x` by induction, we need to\ngeneralize over all spaces (as `f` and `fderiv \ud835\udd5c f` don't take values in the same space). It is\nonly possible to generalize over all spaces in some fixed universe in an inductive definition.\nFor `f : E \u2192 F`, then `fderiv \ud835\udd5c f` is a map `E \u2192 (E \u2192L[\ud835\udd5c] F)`. Therefore, the definition will only\nwork if `F` and `E \u2192L[\ud835\udd5c] F` are in the same universe.\n\nThis issue does not appear with the definition from the left, where one does not need to generalize\nover all spaces. Therefore, we use the definition from the left. This means some proofs later on\nbecome a little bit more complicated: to prove that a function is `C^n`, the most efficient approach\nis to exhibit a formula for its `n`-th derivative and prove it is continuous (contrary to the\ninductive approach where one would prove smoothness statements without giving a formula for the\nderivative). In the end, this approach is still satisfactory as it is good to have formulas for the\niterated derivatives in various constructions.\n\nOne point where we depart from this explicit approach is in the proof of smoothness of a\ncomposition: there is a formula for the `n`-th derivative of a composition (Fa\u00e0 di Bruno's formula),\nbut it is very complicated and barely usable, while the inductive proof is very simple. Thus, we\ngive the inductive proof. As explained above, it works by generalizing over the target space, hence\nit only works well if all spaces belong to the same universe. To get the general version, we lift\nthings to a common universe using a trick.\n\n### Variables management\n\nThe textbook definitions and proofs use various identifications and abuse of notations, for instance\nwhen saying that the natural space in which the derivative lives, i.e.,\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] ( ... \u2192L[\ud835\udd5c] F))`, is the same as a space of multilinear maps. When doing things\nformally, we need to provide explicit maps for these identifications, and chase some diagrams to see\neverything is compatible with the identifications. In particular, one needs to check that taking the\nderivative and then doing the identification, or first doing the identification and then taking the\nderivative, gives the same result. The key point for this is that taking the derivative commutes\nwith continuous linear equivalences. Therefore, we need to implement all our identifications with\ncontinuous linear equivs.\n\n## Notations\n\nWe use the notation `E [\u00d7n]\u2192L[\ud835\udd5c] F` for the space of continuous multilinear maps on `E^n` with\nvalues in `F`. This is the space in which the `n`-th derivative of a function from `E` to `F` lives.\n\nIn this file, we denote `\u22a4 : with_top \u2115` with `\u221e`.\n\n## Tags\n\nderivative, differentiability, higher derivative, `C^n`, multilinear, Taylor series, formal series\n-\/\n\nnoncomputable theory\nopen_locale classical big_operators\n\nlocal notation `\u221e` := (\u22a4 : with_top \u2115)\n\nuniverses u v w\n\nlocal attribute [instance, priority 1001]\nnormed_group.to_add_comm_group normed_space.to_semimodule add_comm_group.to_add_comm_monoid\n\nopen set fin\nopen_locale topological_space\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n{E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n{F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n{G : Type*} [normed_group G] [normed_space \ud835\udd5c G]\n{s s\u2081 t u : set E} {f f\u2081 : E \u2192 F} {g : F \u2192 G} {x : E} {c : F}\n{b : E \u00d7 F \u2192 G}\n\n\/-! ### Functions with a Taylor series on a domain -\/\n\nvariable {p : E \u2192 formal_multilinear_series \ud835\udd5c E F}\n\n\/-- `has_ftaylor_series_up_to_on n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`has_fderiv_within_at` but for higher order derivatives. -\/\nstructure has_ftaylor_series_up_to_on (n : with_top \u2115)\n  (f : E \u2192 F) (p : E \u2192 formal_multilinear_series \ud835\udd5c E F) (s : set E) : Prop :=\n(zero_eq       : \u2200 x \u2208 s, (p x 0).uncurry0 = f x)\n(fderiv_within : \u2200 (m : \u2115) (hm : (m : with_top \u2115) < n), \u2200 x \u2208 s,\n   has_fderiv_within_at (\u03bb y, p y m) (p x m.succ).curry_left s x)\n(cont          : \u2200 (m : \u2115) (hm : (m : with_top \u2115) \u2264 n), continuous_on (\u03bb x, p x m) s)\n\nlemma has_ftaylor_series_up_to_on.zero_eq' {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) {x : E} (hx : x \u2208 s) :\n  p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x) :=\nby { rw \u2190 h.zero_eq x hx, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ }\n\n\/-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a\nTaylor series for the second one. -\/\nlemma has_ftaylor_series_up_to_on.congr {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  has_ftaylor_series_up_to_on n f\u2081 p s :=\nbegin\n  refine \u27e8\u03bb x hx, _, h.fderiv_within, h.cont\u27e9,\n  rw h\u2081 x hx,\n  exact h.zero_eq x hx\nend\n\nlemma has_ftaylor_series_up_to_on.mono {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) {t : set E} (hst : t \u2286 s) :\n  has_ftaylor_series_up_to_on n f p t :=\n\u27e8\u03bb x hx, h.zero_eq x (hst hx),\n\u03bb m hm x hx, (h.fderiv_within m hm x (hst hx)).mono hst,\n\u03bb m hm, (h.cont m hm).mono hst\u27e9\n\nlemma has_ftaylor_series_up_to_on.of_le {m n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hmn : m \u2264 n) :\n  has_ftaylor_series_up_to_on m f p s :=\n\u27e8h.zero_eq,\n\u03bb k hk x hx, h.fderiv_within k (lt_of_lt_of_le hk hmn) x hx,\n\u03bb k hk, h.cont k (le_trans hk hmn)\u27e9\n\nlemma has_ftaylor_series_up_to_on.continuous_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) : continuous_on f s :=\nbegin\n  have := (h.cont 0 bot_le).congr (\u03bb x hx, (h.zero_eq' hx).symm),\n  rwa continuous_linear_equiv.comp_continuous_on_iff at this\nend\n\nlemma has_ftaylor_series_up_to_on_zero_iff :\n  has_ftaylor_series_up_to_on 0 f p s \u2194 continuous_on f s \u2227 (\u2200 x \u2208 s, (p x 0).uncurry0 = f x) :=\nbegin\n  refine \u27e8\u03bb H, \u27e8H.continuous_on, H.zero_eq\u27e9,\n          \u03bb H, \u27e8H.2, \u03bb m hm, false.elim (not_le.2 hm bot_le), _\u27e9\u27e9,\n  assume m hm,\n  have : (m : with_top \u2115) = ((0 : \u2115) : with_bot \u2115) := le_antisymm hm bot_le,\n  rw with_top.coe_eq_coe at this,\n  rw this,\n  have : \u2200 x \u2208 s, p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x),\n    by { assume x hx, rw \u2190 H.2 x hx, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ },\n  rw [continuous_on_congr this, continuous_linear_equiv.comp_continuous_on_iff],\n  exact H.1\nend\n\nlemma has_ftaylor_series_up_to_on_top_iff :\n  (has_ftaylor_series_up_to_on \u221e f p s) \u2194 (\u2200 (n : \u2115), has_ftaylor_series_up_to_on n f p s) :=\nbegin\n  split,\n  { assume H n, exact H.of_le le_top },\n  { assume H,\n    split,\n    { exact (H 0).zero_eq },\n    { assume m hm,\n      apply (H m.succ).fderiv_within m (with_top.coe_lt_coe.2 (lt_add_one m)) },\n    { assume m hm,\n      apply (H m).cont m (le_refl _) } }\nend\n\n\/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -\/\nlemma has_ftaylor_series_up_to_on.has_fderiv_within_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hx : x \u2208 s) :\n  has_fderiv_within_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) s x :=\nbegin\n  have A : \u2200 y \u2208 s, f y = (continuous_multilinear_curry_fin0 \ud835\udd5c E F) (p y 0),\n  { assume y hy, rw \u2190 h.zero_eq y hy, refl },\n  suffices H : has_fderiv_within_at\n      (\u03bb y, continuous_multilinear_curry_fin0 \ud835\udd5c E F (p y 0))\n      (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) s x,\n    by exact H.congr A (A x hx),\n  rw continuous_linear_equiv.comp_has_fderiv_within_at_iff',\n  have : ((0 : \u2115) : with_top \u2115) < n :=\n    lt_of_lt_of_le (with_top.coe_lt_coe.2 nat.zero_lt_one) hn,\n  convert h.fderiv_within _ this x hx,\n  ext y v,\n  change (p x 1) (snoc 0 y) = (p x 1) (cons y v),\n  unfold_coes,\n  congr' with i,\n  rw unique.eq_default i,\n  refl\nend\n\nlemma has_ftaylor_series_up_to_on.differentiable_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) : differentiable_on \ud835\udd5c f s :=\n\u03bb x hx, (h.has_fderiv_within_at hn hx).differentiable_within_at\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and\n`p (n + 1)` is a derivative of `p n`. -\/\ntheorem has_ftaylor_series_up_to_on_succ_iff_left {n : \u2115} :\n  has_ftaylor_series_up_to_on (n + 1) f p s \u2194\n  has_ftaylor_series_up_to_on n f p s\n  \u2227 (\u2200 x \u2208 s, has_fderiv_within_at (\u03bb y, p y n) (p x n.succ).curry_left s x)\n  \u2227 continuous_on (\u03bb x, p x (n + 1)) s :=\nbegin\n  split,\n  { assume h,\n    exact \u27e8h.of_le (with_top.coe_le_coe.2 (nat.le_succ n)),\n           h.fderiv_within _ (with_top.coe_lt_coe.2 (lt_add_one n)),\n           h.cont (n + 1) (le_refl _)\u27e9 },\n  { assume h,\n    split,\n    { exact h.1.zero_eq },\n    { assume m hm,\n      by_cases h' : m < n,\n      { exact h.1.fderiv_within m (with_top.coe_lt_coe.2 h') },\n      { have : m = n := nat.eq_of_lt_succ_of_not_lt (with_top.coe_lt_coe.1 hm) h',\n        rw this,\n        exact h.2.1 } },\n    { assume m hm,\n      by_cases h' : m \u2264 n,\n      { apply h.1.cont m (with_top.coe_le_coe.2 h') },\n      { have : m = (n + 1) := le_antisymm (with_top.coe_le_coe.1 hm) (not_le.1 h'),\n        rw this,\n        exact h.2.2 } } }\nend\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -\/\ntheorem has_ftaylor_series_up_to_on_succ_iff_right {n : \u2115} :\n  has_ftaylor_series_up_to_on ((n + 1) : \u2115) f p s \u2194\n  (\u2200 x \u2208 s, (p x 0).uncurry0 = f x)\n  \u2227 (\u2200 x \u2208 s, has_fderiv_within_at (\u03bb y, p y 0) (p x 1).curry_left s x)\n  \u2227 has_ftaylor_series_up_to_on n\n    (\u03bb x, continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) (\u03bb x, (p x).shift) s :=\nbegin\n  split,\n  { assume H,\n    refine \u27e8H.zero_eq, H.fderiv_within 0 (with_top.coe_lt_coe.2 (nat.succ_pos n)), _\u27e9,\n    split,\n    { assume x hx, refl },\n    { assume m (hm : (m : with_top \u2115) < n) x (hx : x \u2208 s),\n      have A : (m.succ : with_top \u2115) < n.succ,\n        by { rw with_top.coe_lt_coe at \u22a2 hm, exact nat.lt_succ_iff.mpr hm },\n      change has_fderiv_within_at\n        ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ))\n        (p x m.succ.succ).curry_right.curry_left s x,\n      rw continuous_linear_equiv.comp_has_fderiv_within_at_iff',\n      convert H.fderiv_within _ A x hx,\n      ext y v,\n      change (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))\n        = (p x (nat.succ (nat.succ m))) (cons y v),\n      rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self] },\n    { assume m (hm : (m : with_top \u2115) \u2264 n),\n      have A : (m.succ : with_top \u2115) \u2264 n.succ,\n        by { rw with_top.coe_le_coe at \u22a2 hm, exact nat.pred_le_iff.mp hm },\n      change continuous_on ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) s,\n      rw continuous_linear_equiv.comp_continuous_on_iff,\n      exact H.cont _ A } },\n  { rintros \u27e8Hzero_eq, Hfderiv_zero, Htaylor\u27e9,\n    split,\n    { exact Hzero_eq },\n    { assume m (hm : (m : with_top \u2115) < n.succ) x (hx : x \u2208 s),\n      cases m,\n      { exact Hfderiv_zero x hx },\n      { have A : (m : with_top \u2115) < n,\n          by { rw with_top.coe_lt_coe at hm \u22a2, exact nat.lt_of_succ_lt_succ hm },\n        have : has_fderiv_within_at ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) ((p x).shift m.succ).curry_left s x :=\n          Htaylor.fderiv_within _ A x hx,\n        rw continuous_linear_equiv.comp_has_fderiv_within_at_iff' at this,\n        convert this,\n        ext y v,\n        change (p x (nat.succ (nat.succ m))) (cons y v)\n          = (p x m.succ.succ) (snoc (cons y (init v)) (v (last _))),\n        rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self] } },\n    { assume m (hm : (m : with_top \u2115) \u2264 n.succ),\n      cases m,\n      { have : differentiable_on \ud835\udd5c (\u03bb x, p x 0) s :=\n          \u03bb x hx, (Hfderiv_zero x hx).differentiable_within_at,\n        exact this.continuous_on },\n      { have A : (m : with_top \u2115) \u2264 n,\n          by { rw with_top.coe_le_coe at hm \u22a2, exact nat.lt_succ_iff.mp hm },\n        have : continuous_on ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) s :=\n        Htaylor.cont _ A,\n        rwa continuous_linear_equiv.comp_continuous_on_iff at this } } }\nend\n\n\/-! ### Smooth functions within a set around a point -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to order `n` within a set `s` at a point `x` if\nit admits continuous derivatives up to order `n` in a neighborhood of `x` in `s \u222a {x}`.\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n\nFor instance, a real function which is `C^m` on `(-1\/m, 1\/m)` for each natural `m`, but not\nbetter, is `C^\u221e` at `0` within `univ`.\n-\/\ndef times_cont_diff_within_at (n : with_top \u2115) (f : E \u2192 F) (s : set E) (x : E) :=\n\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F,\n    has_ftaylor_series_up_to_on m f p u\n\nvariable {\ud835\udd5c}\n\nlemma times_cont_diff_within_at_nat {n : \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F,\n  has_ftaylor_series_up_to_on n f p u :=\n\u27e8\u03bb H, H n (le_refl _), \u03bb \u27e8u, hu, p, hp\u27e9 m hm, \u27e8u, hu, p, hp.of_le hm\u27e9\u27e9\n\nlemma times_cont_diff_within_at.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hmn : m \u2264 n) :\n  times_cont_diff_within_at \ud835\udd5c m f s x :=\n\u03bb k hk, h k (le_trans hk hmn)\n\nlemma times_cont_diff_within_at_iff_forall_nat_le {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 times_cont_diff_within_at \ud835\udd5c m f s x :=\n\u27e8\u03bb H m hm, H.of_le hm, \u03bb H m hm, H m hm _ le_rfl\u27e9\n\nlemma times_cont_diff_within_at_top :\n  times_cont_diff_within_at \ud835\udd5c \u221e f s x \u2194 \u2200 (n : \u2115), times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_iff_forall_nat_le.trans $ by simp only [forall_prop_of_true, le_top]\n\nlemma times_cont_diff_within_at.continuous_within_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) : continuous_within_at f s x :=\nbegin\n  rcases h 0 bot_le with \u27e8u, hu, p, H\u27e9,\n  rw [mem_nhds_within_insert] at hu,\n  exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2\nend\n\nlemma times_cont_diff_within_at.congr_of_eventually_eq {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\n\u03bb m hm, let \u27e8u, hu, p, H\u27e9 := h m hm in\n\u27e8{x \u2208 u | f\u2081 x = f x}, filter.inter_mem_sets hu (mem_nhds_within_insert.2 \u27e8hx, h\u2081\u27e9), p,\n  (H.mono (sep_subset _ _)).congr (\u03bb _, and.right)\u27e9\n\nlemma times_cont_diff_within_at.congr_of_eventually_eq' {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\nh.congr_of_eventually_eq h\u2081 $ h\u2081.self_of_nhds_within hx\n\nlemma filter.eventually_eq.times_cont_diff_within_at_iff {n : with_top \u2115}\n  (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\n\u27e8\u03bb H, times_cont_diff_within_at.congr_of_eventually_eq H h\u2081.symm hx.symm,\n\u03bb H, H.congr_of_eventually_eq h\u2081 hx\u27e9\n\nlemma times_cont_diff_within_at.congr {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\nh.congr_of_eventually_eq (filter.eventually_eq_of_mem self_mem_nhds_within h\u2081) hx\n\nlemma times_cont_diff_within_at.mono_of_mem {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : s \u2208 \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nbegin\n  assume m hm,\n  rcases h m hm with \u27e8u, hu, p, H\u27e9,\n  exact \u27e8u, nhds_within_le_of_mem (insert_mem_nhds_within_insert hst) hu, p, H\u27e9\nend\n\nlemma times_cont_diff_within_at.mono {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : t \u2286 s) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nh.mono_of_mem $ filter.mem_sets_of_superset self_mem_nhds_within hst\n\nlemma times_cont_diff_within_at.congr_nhds {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nh.mono_of_mem $ hst \u25b8 self_mem_nhds_within\n\nlemma times_cont_diff_within_at_congr_nhds {n : with_top \u2115} {t : set E} (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194 times_cont_diff_within_at \ud835\udd5c n f t x :=\n\u27e8\u03bb h, h.congr_nhds hst, \u03bb h, h.congr_nhds hst.symm\u27e9\n\nlemma times_cont_diff_within_at_inter' {n : with_top \u2115} (h : t \u2208 \ud835\udcdd[s] x) :\n  times_cont_diff_within_at \ud835\udd5c n f (s \u2229 t) x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_congr_nhds $ eq.symm $ nhds_within_restrict'' _ h\n\nlemma times_cont_diff_within_at_inter {n : with_top \u2115} (h : t \u2208 \ud835\udcdd x) :\n  times_cont_diff_within_at \ud835\udd5c n f (s \u2229 t) x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_inter' (mem_nhds_within_of_mem_nhds h)\n\n\/-- If a function is `C^n` within a set at a point, with `n \u2265 1`, then it is differentiable\nwithin this set at this point. -\/\nlemma times_cont_diff_within_at.differentiable_within_at' {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n  differentiable_within_at \ud835\udd5c f (insert x s) x :=\nbegin\n  rcases h 1 hn with \u27e8u, hu, p, H\u27e9,\n  rcases mem_nhds_within.1 hu with \u27e8t, t_open, xt, tu\u27e9,\n  rw inter_comm at tu,\n  have := ((H.mono tu).differentiable_on (le_refl _)) x \u27e8mem_insert x s, xt\u27e9,\n  exact (differentiable_within_at_inter (mem_nhds_sets t_open xt)).1 this,\nend\n\nlemma times_cont_diff_within_at.differentiable_within_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n  differentiable_within_at \ud835\udd5c f s x :=\n(h.differentiable_within_at' hn).mono  (subset_insert x s)\n\n\/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_within_at_succ_iff_has_fderiv_within_at {n : \u2115} :\n  times_cont_diff_within_at \ud835\udd5c ((n + 1) : \u2115) f s x\n  \u2194 \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F),\n    (\u2200 x \u2208 u, has_fderiv_within_at f (f' x) u x) \u2227 (times_cont_diff_within_at \ud835\udd5c n f' u x) :=\nbegin\n  split,\n  { assume h,\n    rcases h n.succ (le_refl _) with \u27e8u, hu, p, Hp\u27e9,\n    refine \u27e8u, hu, \u03bb y, (continuous_multilinear_curry_fin1 \ud835\udd5c E F) (p y 1),\n      \u03bb y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _\u27e9,\n    assume m hm,\n    refine \u27e8u, _, \u03bb (y : E), (p y).shift, _\u27e9,\n    { convert self_mem_nhds_within,\n      have : x \u2208 insert x s, by simp,\n      exact (insert_eq_of_mem (mem_of_mem_nhds_within this hu)) },\n    { rw has_ftaylor_series_up_to_on_succ_iff_right at Hp,\n      exact Hp.2.2.of_le hm } },\n  { rintros \u27e8u, hu, f', f'_eq_deriv, Hf'\u27e9,\n    rw times_cont_diff_within_at_nat,\n    rcases Hf' n (le_refl _) with \u27e8v, hv, p', Hp'\u27e9,\n    refine \u27e8v \u2229 u, _, \u03bb x, (p' x).unshift (f x), _\u27e9,\n    { apply filter.inter_mem_sets _ hu,\n      apply nhds_within_le_of_mem hu,\n      exact nhds_within_mono _ (subset_insert x u) hv },\n    { rw has_ftaylor_series_up_to_on_succ_iff_right,\n      refine \u27e8\u03bb y hy, rfl, \u03bb y hy, _, _\u27e9,\n      { change has_fderiv_within_at (\u03bb (z : E), (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f z))\n          ((formal_multilinear_series.unshift (p' y) (f y) 1).curry_left) (v \u2229 u) y,\n        rw continuous_linear_equiv.comp_has_fderiv_within_at_iff',\n        convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u),\n        rw \u2190 Hp'.zero_eq y hy.1,\n        ext z,\n        change ((p' y 0) (init (@cons 0 (\u03bb i, E) z 0))) (@cons 0 (\u03bb i, E) z 0 (last 0))\n          = ((p' y 0) 0) z,\n        unfold_coes,\n        congr },\n      { convert (Hp'.mono (inter_subset_left v u)).congr (\u03bb x hx, Hp'.zero_eq x hx.1),\n        { ext x y,\n          change p' x 0 (init (@snoc 0 (\u03bb i : fin 1, E) 0 y)) y = p' x 0 0 y,\n          rw init_snoc },\n        { ext x k v y,\n          change p' x k (init (@snoc k (\u03bb i : fin k.succ, E) v y))\n            (@snoc k (\u03bb i : fin k.succ, E) v y (last k)) = p' x k v y,\n          rw [snoc_last, init_snoc] } } } }\nend\n\n\/-! ### Smooth functions within a set -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` on `s` if, for any point `x` in `s`, it\nadmits continuous derivatives up to order `n` on a neighborhood of `x` in `s`.\n\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n-\/\ndefinition times_cont_diff_on (n : with_top \u2115) (f : E \u2192 F) (s : set E) :=\n\u2200 x \u2208 s, times_cont_diff_within_at \ud835\udd5c n f s x\n\nvariable {\ud835\udd5c}\n\nlemma times_cont_diff_on.times_cont_diff_within_at {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nh x hx\n\nlemma times_cont_diff_within_at.times_cont_diff_on {n : with_top \u2115} {m : \u2115}\n  (hm : (m : with_top \u2115) \u2264 n) (h : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, u \u2286 insert x s \u2227 times_cont_diff_on \ud835\udd5c m f u :=\nbegin\n  rcases h m hm with \u27e8u, u_nhd, p, hp\u27e9,\n  refine \u27e8u \u2229 insert x s, filter.inter_mem_sets u_nhd self_mem_nhds_within,\n    inter_subset_right _ _, _\u27e9,\n  assume y hy m' hm',\n  refine \u27e8u \u2229 insert x s, _, p, (hp.mono (inter_subset_left _ _)).of_le hm'\u27e9,\n  convert self_mem_nhds_within,\n  exact insert_eq_of_mem hy\nend\n\nlemma times_cont_diff_on.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : m \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m f s :=\n\u03bb x hx, (h x hx).of_le hmn\n\nlemma times_cont_diff_on_iff_forall_nat_le {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f s \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 times_cont_diff_on \ud835\udd5c m f s :=\n\u27e8\u03bb H m hm, H.of_le hm, \u03bb H x hx m hm, H m hm x hx m le_rfl\u27e9\n\nlemma times_cont_diff_on_top :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194 \u2200 (n : \u2115), times_cont_diff_on \ud835\udd5c n f s :=\ntimes_cont_diff_on_iff_forall_nat_le.trans $ by simp only [le_top, forall_prop_of_true]\n\nlemma times_cont_diff_on_all_iff_nat :\n  (\u2200 n, times_cont_diff_on \ud835\udd5c n f s) \u2194 (\u2200 n : \u2115, times_cont_diff_on \ud835\udd5c n f s) :=\nbegin\n  refine \u27e8\u03bb H n, H n, _\u27e9,\n  rintro H (_|n),\n  exacts [times_cont_diff_on_top.2 H, H n]\nend\n\nlemma times_cont_diff_on.continuous_on {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) : continuous_on f s :=\n\u03bb x hx, (h x hx).continuous_within_at\n\nlemma times_cont_diff_on.congr {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s :=\n\u03bb x hx, (h x hx).congr h\u2081 (h\u2081 x hx)\n\nlemma times_cont_diff_on_congr {n : with_top \u2115} (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s \u2194 times_cont_diff_on \ud835\udd5c n f s :=\n\u27e8\u03bb H, H.congr (\u03bb x hx, (h\u2081 x hx).symm), \u03bb H, H.congr h\u2081\u27e9\n\nlemma times_cont_diff_on.mono {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) {t : set E} (hst : t \u2286 s) :\n  times_cont_diff_on \ud835\udd5c n f t :=\n\u03bb x hx, (h x (hst hx)).mono hst\n\nlemma times_cont_diff_on.congr_mono {n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s\u2081, f\u2081 x = f x) (hs : s\u2081 \u2286 s) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s\u2081 :=\n(hf.mono hs).congr h\u2081\n\n\/-- If a function is `C^n` on a set with `n \u2265 1`, then it is differentiable there. -\/\nlemma times_cont_diff_on.differentiable_on {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hn : 1 \u2264 n) : differentiable_on \ud835\udd5c f s :=\n\u03bb x hx, (h x hx).differentiable_within_at hn\n\n\/-- If a function is `C^n` around each point in a set, then it is `C^n` on the set. -\/\nlemma times_cont_diff_on_of_locally_times_cont_diff_on {n : with_top \u2115}\n  (h : \u2200 x \u2208 s, \u2203u, is_open u \u2227 x \u2208 u \u2227 times_cont_diff_on \ud835\udd5c n f (s \u2229 u)) :\n  times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  assume x xs,\n  rcases h x xs with \u27e8u, u_open, xu, hu\u27e9,\n  apply (times_cont_diff_within_at_inter _).1 (hu x \u27e8xs, xu\u27e9),\n  exact mem_nhds_sets u_open xu\nend\n\n\/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_has_fderiv_within_at {n : \u2115} :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s\n  \u2194 \u2200 x \u2208 s, \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F),\n    (\u2200 x \u2208 u, has_fderiv_within_at f (f' x) u x) \u2227 (times_cont_diff_on \ud835\udd5c n f' u) :=\nbegin\n  split,\n  { assume h x hx,\n    rcases (h x hx) n.succ (le_refl _) with \u27e8u, hu, p, Hp\u27e9,\n    refine \u27e8u, hu, \u03bb y, (continuous_multilinear_curry_fin1 \ud835\udd5c E F) (p y 1),\n      \u03bb y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _\u27e9,\n    rw has_ftaylor_series_up_to_on_succ_iff_right at Hp,\n    assume z hz m hm,\n    refine \u27e8u, _, \u03bb (x : E), (p x).shift, Hp.2.2.of_le hm\u27e9,\n    convert self_mem_nhds_within,\n    exact insert_eq_of_mem hz, },\n  { assume h x hx,\n    rw times_cont_diff_within_at_succ_iff_has_fderiv_within_at,\n    rcases h x hx with \u27e8u, u_nhbd, f', hu, hf'\u27e9,\n    have : x \u2208 u := mem_of_mem_nhds_within (mem_insert _ _) u_nhbd,\n    exact \u27e8u, u_nhbd, f', hu, hf' x this\u27e9 }\nend\n\n\/-! ### Iterated derivative within a set -\/\nvariable (\ud835\udd5c)\n\n\/--\nThe `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th\nderivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with\nan uncurrying step to see it as a multilinear map in `n+1` variables..\n-\/\nnoncomputable def iterated_fderiv_within (n : \u2115) (f : E \u2192 F) (s : set E) :\n  E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F) :=\nnat.rec_on n\n  (\u03bb x, continuous_multilinear_map.curry0 \ud835\udd5c E (f x))\n  (\u03bb n rec x, continuous_linear_map.uncurry_left (fderiv_within \ud835\udd5c rec s x))\n\n\/-- Formal Taylor series associated to a function within a set. -\/\ndef ftaylor_series_within (f : E \u2192 F) (s : set E) (x : E) : formal_multilinear_series \ud835\udd5c E F :=\n\u03bb n, iterated_fderiv_within \ud835\udd5c n f s x\n\nvariable {\ud835\udd5c}\n\n@[simp] lemma iterated_fderiv_within_zero_apply (m : (fin 0) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c 0 f s x : ((fin 0) \u2192  E) \u2192 F) m = f x := rfl\n\nlemma iterated_fderiv_within_zero_eq_comp :\n  iterated_fderiv_within \ud835\udd5c 0 f s = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm \u2218 f := rfl\n\nlemma iterated_fderiv_within_succ_apply_left {n : \u2115} (m : fin (n + 1) \u2192 E):\n  (iterated_fderiv_within \ud835\udd5c (n + 1) f s x : (fin (n + 1) \u2192 E) \u2192 F) m\n  = (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n f s) s x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) := rfl\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -\/\nlemma iterated_fderiv_within_succ_eq_comp_left {n : \u2115} :\n  iterated_fderiv_within \ud835\udd5c (n + 1) f s =\n  (continuous_multilinear_curry_left_equiv \ud835\udd5c (\u03bb(i : fin (n + 1)), E) F)\n    \u2218 (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n f s) s) := rfl\n\ntheorem iterated_fderiv_within_succ_apply_right {n : \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) (m : fin (n + 1) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c (n + 1) f s x : (fin (n + 1) \u2192 E) \u2192 F) m\n    = iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s x (init m) (m (last n)) :=\nbegin\n  induction n with n IH generalizing x,\n  { rw [iterated_fderiv_within_succ_eq_comp_left, iterated_fderiv_within_zero_eq_comp,\n        iterated_fderiv_within_zero_apply,\n        function.comp_apply, continuous_linear_equiv.comp_fderiv_within _ (hs x hx)],\n    refl },\n  { let I := continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F,\n    have A : \u2200 y \u2208 s, iterated_fderiv_within \ud835\udd5c n.succ f s y\n        = (I \u2218 (iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s)) y,\n      by { assume y hy, ext m, rw @IH m y hy, refl },\n    calc\n    (iterated_fderiv_within \ud835\udd5c (n+2) f s x : (fin (n+2) \u2192 E) \u2192 F) m =\n    (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n.succ f s) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) : rfl\n    ... = (fderiv_within \ud835\udd5c (I \u2218 (iterated_fderiv_within \ud835\udd5c n (fderiv_within \ud835\udd5c f s) s)) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) :\n      by rw fderiv_within_congr (hs x hx) A (A x hx)\n    ... = (I \u2218 fderiv_within \ud835\udd5c ((iterated_fderiv_within \ud835\udd5c n (fderiv_within \ud835\udd5c f s) s)) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) :\n      by { rw continuous_linear_equiv.comp_fderiv_within _ (hs x hx), refl }\n    ... = (fderiv_within \ud835\udd5c ((iterated_fderiv_within \ud835\udd5c n (\u03bb y, fderiv_within \ud835\udd5c f s y) s)) s x\n              : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] F))) (m 0) (init (tail m)) ((tail m) (last n)) : rfl\n    ... = iterated_fderiv_within \ud835\udd5c (nat.succ n) (\u03bb y, fderiv_within \ud835\udd5c f s y) s x\n              (init m) (m (last (n + 1))) :\n      by { rw [iterated_fderiv_within_succ_apply_left, tail_init_eq_init_tail], refl } }\nend\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the `n`-th derivative of the derivative. -\/\nlemma iterated_fderiv_within_succ_eq_comp_right {n : \u2115} (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c (n + 1) f s x =\n  ((continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F)\n    \u2218 (iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s)) x :=\nby { ext m, rw iterated_fderiv_within_succ_apply_right hs hx, refl }\n\n@[simp] lemma iterated_fderiv_within_one_apply\n  (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) (m : (fin 1) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c 1 f s x : ((fin 1) \u2192 E) \u2192 F) m\n  = (fderiv_within \ud835\udd5c f s x : E \u2192 F) (m 0) :=\nby { rw [iterated_fderiv_within_succ_apply_right hs hx, iterated_fderiv_within_zero_apply], refl }\n\n\/-- If two functions coincide on a set `s` of unique differentiability, then their iterated\ndifferentials within this set coincide. -\/\nlemma iterated_fderiv_within_congr {n : \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) (hL : \u2200y\u2208s, f\u2081 y = f y) (hx : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f\u2081 s x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  induction n with n IH generalizing x,\n  { ext m, simp [hL x hx] },\n  { have : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f\u2081 s y) s x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) s x :=\n      fderiv_within_congr (hs x hx) (\u03bb y hy, IH hy) (IH hx),\n    ext m,\n    rw [iterated_fderiv_within_succ_apply_left, iterated_fderiv_within_succ_apply_left, this] }\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with an open set containing `x`. -\/\nlemma iterated_fderiv_within_inter_open {n : \u2115} (hu : is_open u)\n  (hs : unique_diff_on \ud835\udd5c (s \u2229 u)) (hx : x \u2208 s \u2229 u) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  induction n with n IH generalizing x,\n  { ext m, simp },\n  { have A : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) y) (s \u2229 u) x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) (s \u2229 u) x :=\n      fderiv_within_congr (hs x hx) (\u03bb y hy, IH hy) (IH hx),\n    have B : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) (s \u2229 u) x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) s x :=\n      fderiv_within_inter (mem_nhds_sets hu hx.2)\n        ((unique_diff_within_at_inter (mem_nhds_sets hu hx.2)).1 (hs x hx)),\n    ext m,\n    rw [iterated_fderiv_within_succ_apply_left, iterated_fderiv_within_succ_apply_left, A, B] }\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x` within `s`. -\/\nlemma iterated_fderiv_within_inter' {n : \u2115}\n  (hu : u \u2208 \ud835\udcdd[s] x) (hs : unique_diff_on \ud835\udd5c s) (xs : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  obtain \u27e8v, v_open, xv, vu\u27e9 : \u2203 v, is_open v \u2227 x \u2208 v \u2227 v \u2229 s \u2286 u := mem_nhds_within.1 hu,\n  have A : (s \u2229 u) \u2229 v = s \u2229 v,\n  { apply subset.antisymm (inter_subset_inter (inter_subset_left _ _) (subset.refl _)),\n    exact \u03bb y \u27e8ys, yv\u27e9, \u27e8\u27e8ys, vu \u27e8yv, ys\u27e9\u27e9, yv\u27e9 },\n  have : iterated_fderiv_within \ud835\udd5c n f (s \u2229 v) x = iterated_fderiv_within \ud835\udd5c n f s x :=\n    iterated_fderiv_within_inter_open v_open (hs.inter v_open) \u27e8xs, xv\u27e9,\n  rw \u2190 this,\n  have : iterated_fderiv_within \ud835\udd5c n f ((s \u2229 u) \u2229 v) x = iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x,\n  { refine iterated_fderiv_within_inter_open v_open  _ \u27e8\u27e8xs, vu \u27e8xv, xs\u27e9\u27e9, xv\u27e9,\n    rw A,\n    exact hs.inter v_open },\n  rw A at this,\n  rw \u2190 this\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x`. -\/\nlemma iterated_fderiv_within_inter {n : \u2115}\n  (hu : u \u2208 \ud835\udcdd x) (hs : unique_diff_on \ud835\udd5c s) (xs : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\niterated_fderiv_within_inter' (mem_nhds_within_of_mem_nhds hu) hs xs\n\n@[simp] lemma times_cont_diff_on_zero :\n  times_cont_diff_on \ud835\udd5c 0 f s \u2194 continuous_on f s :=\nbegin\n  refine \u27e8\u03bb H, H.continuous_on, \u03bb H, _\u27e9,\n  assume x hx m hm,\n  have : (m : with_top \u2115) = 0 := le_antisymm hm bot_le,\n  rw this,\n  refine \u27e8insert x s, self_mem_nhds_within, ftaylor_series_within \ud835\udd5c f s, _\u27e9,\n  rw has_ftaylor_series_up_to_on_zero_iff,\n  exact \u27e8by rwa insert_eq_of_mem hx, \u03bb x hx, by simp [ftaylor_series_within]\u27e9\nend\n\nlemma times_cont_diff_within_at_zero (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c 0 f s x \u2194 \u2203 u \u2208 \ud835\udcdd[s] x, continuous_on f (s \u2229 u) :=\nbegin\n  split,\n  { intros h,\n    obtain \u27e8u, H, p, hp\u27e9 := h 0 (by norm_num),\n    refine \u27e8u, _, _\u27e9,\n    { simpa [hx] using H },\n    { simp only [with_top.coe_zero, has_ftaylor_series_up_to_on_zero_iff] at hp,\n      exact hp.1.mono (inter_subset_right s u) } },\n  { rintros \u27e8u, H, hu\u27e9,\n    rw \u2190 times_cont_diff_within_at_inter' H,\n    have h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhds_within hx H\u27e9,\n    exact (times_cont_diff_on_zero.mpr hu).times_cont_diff_within_at h' }\nend\n\n\/-- On a set with unique differentiability, any choice of iterated differential has to coincide\nwith the one we have chosen in `iterated_fderiv_within \ud835\udd5c m f s`. -\/\ntheorem has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s)\n  {m : \u2115} (hmn : (m : with_top \u2115) \u2264 n) (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) :\n  p x m = iterated_fderiv_within \ud835\udd5c m f s x :=\nbegin\n  induction m with m IH generalizing x,\n  { rw [h.zero_eq' hx, iterated_fderiv_within_zero_eq_comp] },\n  { have A : (m : with_top \u2115) < n := lt_of_lt_of_le (with_top.coe_lt_coe.2 (lt_add_one m)) hmn,\n    have : has_fderiv_within_at (\u03bb (y : E), iterated_fderiv_within \ud835\udd5c m f s y)\n      (continuous_multilinear_map.curry_left (p x (nat.succ m))) s x :=\n    (h.fderiv_within m A x hx).congr (\u03bb y hy, (IH (le_of_lt A) hy).symm) (IH (le_of_lt A) hx).symm,\n    rw [iterated_fderiv_within_succ_eq_comp_left, function.comp_apply, this.fderiv_within (hs x hx)],\n    exact (continuous_multilinear_map.uncurry_curry_left _).symm }\nend\n\n\/-- When a function is `C^n` in a set `s` of unique differentiability, it admits\n`ftaylor_series_within \ud835\udd5c f s` as a Taylor series up to order `n` in `s`. -\/\ntheorem times_cont_diff_on.ftaylor_series_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) :\n  has_ftaylor_series_up_to_on n f (ftaylor_series_within \ud835\udd5c f s) s :=\nbegin\n  split,\n  { assume x hx,\n    simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,\n               iterated_fderiv_within_zero_apply] },\n  { assume m hm x hx,\n    rcases (h x hx) m.succ (with_top.add_one_le_of_lt hm) with \u27e8u, hu, p, Hp\u27e9,\n    rw insert_eq_of_mem hx at hu,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw inter_comm at ho,\n    have : p x m.succ = ftaylor_series_within \ud835\udd5c f s x m.succ,\n    { change p x m.succ = iterated_fderiv_within \ud835\udd5c m.succ f s x,\n      rw \u2190 iterated_fderiv_within_inter (mem_nhds_sets o_open xo) hs hx,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (le_refl _)\n        (hs.inter o_open) \u27e8hx, xo\u27e9 },\n    rw [\u2190 this, \u2190 has_fderiv_within_at_inter (mem_nhds_sets o_open xo)],\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylor_series_within \ud835\udd5c f s y m,\n    { rintros y \u27e8hy, yo\u27e9,\n      change p y m = iterated_fderiv_within \ud835\udd5c m f s y,\n      rw \u2190 iterated_fderiv_within_inter (mem_nhds_sets o_open yo) hs hy,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (with_top.coe_le_coe.2 (nat.le_succ m))\n        (hs.inter o_open) \u27e8hy, yo\u27e9 },\n    exact ((Hp.mono ho).fderiv_within m (with_top.coe_lt_coe.2 (lt_add_one m)) x \u27e8hx, xo\u27e9).congr\n      (\u03bb y hy, (A y hy).symm) (A x \u27e8hx, xo\u27e9).symm },\n  { assume m hm,\n    apply continuous_on_of_locally_continuous_on,\n    assume x hx,\n    rcases h x hx m hm with \u27e8u, hu, p, Hp\u27e9,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw insert_eq_of_mem hx at ho,\n    rw inter_comm at ho,\n    refine \u27e8o, o_open, xo, _\u27e9,\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylor_series_within \ud835\udd5c f s y m,\n    { rintros y \u27e8hy, yo\u27e9,\n      change p y m = iterated_fderiv_within \ud835\udd5c m f s y,\n      rw \u2190 iterated_fderiv_within_inter (mem_nhds_sets o_open yo) hs hy,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (le_refl _)\n        (hs.inter o_open) \u27e8hy, yo\u27e9 },\n    exact ((Hp.mono ho).cont m (le_refl _)).congr (\u03bb y hy, (A y hy).symm) }\nend\n\nlemma times_cont_diff_on_of_continuous_on_differentiable_on {n : with_top \u2115}\n  (Hcont : \u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n    continuous_on (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s)\n  (Hdiff : \u2200 (m : \u2115), (m : with_top \u2115) < n \u2192\n    differentiable_on \ud835\udd5c (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  assume x hx m hm,\n  rw insert_eq_of_mem hx,\n  refine \u27e8s, self_mem_nhds_within, ftaylor_series_within \ud835\udd5c f s, _\u27e9,\n  split,\n  { assume y hy,\n    simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,\n                iterated_fderiv_within_zero_apply] },\n  { assume k hk y hy,\n    convert (Hdiff k (lt_of_lt_of_le hk hm) y hy).has_fderiv_within_at,\n    simp only [ftaylor_series_within, iterated_fderiv_within_succ_eq_comp_left,\n                continuous_linear_equiv.coe_apply, function.comp_app, coe_fn_coe_base],\n    exact continuous_linear_map.curry_uncurry_left _ },\n  { assume k hk,\n    exact Hcont k (le_trans hk hm) }\nend\n\nlemma times_cont_diff_on_of_differentiable_on {n : with_top \u2115}\n  (h : \u2200(m : \u2115), (m : with_top \u2115) \u2264 n \u2192 differentiable_on \ud835\udd5c (iterated_fderiv_within \ud835\udd5c m f s) s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\ntimes_cont_diff_on_of_continuous_on_differentiable_on\n  (\u03bb m hm, (h m hm).continuous_on) (\u03bb m hm, (h m (le_of_lt hm)))\n\nlemma times_cont_diff_on.continuous_on_iterated_fderiv_within {n : with_top \u2115} {m : \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : (m : with_top \u2115) \u2264 n) (hs : unique_diff_on \ud835\udd5c s) :\n  continuous_on (iterated_fderiv_within \ud835\udd5c m f s) s :=\n(h.ftaylor_series_within hs).cont m hmn\n\nlemma times_cont_diff_on.differentiable_on_iterated_fderiv_within {n : with_top \u2115} {m : \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : (m : with_top \u2115) < n) (hs : unique_diff_on \ud835\udd5c s) :\n  differentiable_on \ud835\udd5c (iterated_fderiv_within \ud835\udd5c m f s) s :=\n\u03bb x hx, ((h.ftaylor_series_within hs).fderiv_within m hmn x hx).differentiable_within_at\n\nlemma times_cont_diff_on_iff_continuous_on_differentiable_on {n : with_top \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c n f s \u2194\n  (\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n    continuous_on (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s)\n  \u2227 (\u2200 (m : \u2115), (m : with_top \u2115) < n \u2192\n    differentiable_on \ud835\udd5c (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s) :=\nbegin\n  split,\n  { assume h,\n    split,\n    { assume m hm, exact h.continuous_on_iterated_fderiv_within hm hs },\n    { assume m hm, exact h.differentiable_on_iterated_fderiv_within hm hs } },\n  { assume h,\n    exact times_cont_diff_on_of_continuous_on_differentiable_on h.1 h.2 }\nend\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative (expressed with `fderiv_within`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_fderiv_within {n : \u2115} (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c n (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  split,\n  { assume H,\n    refine \u27e8H.differentiable_on (with_top.coe_le_coe.2 (nat.le_add_left 1 n)), \u03bb x hx, _\u27e9,\n    rcases times_cont_diff_within_at_succ_iff_has_fderiv_within_at.1 (H x hx)\n      with \u27e8u, hu, f', hff', hf'\u27e9,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw [inter_comm, insert_eq_of_mem hx] at ho,\n    have := hf'.mono ho,\n    rw times_cont_diff_within_at_inter' (mem_nhds_within_of_mem_nhds (mem_nhds_sets o_open xo))\n      at this,\n    apply this.congr_of_eventually_eq' _ hx,\n    have : o \u2229 s \u2208 \ud835\udcdd[s] x := mem_nhds_within.2 \u27e8o, o_open, xo, subset.refl _\u27e9,\n    rw inter_comm at this,\n    apply filter.eventually_eq_of_mem this (\u03bb y hy, _),\n    have A : fderiv_within \ud835\udd5c f (s \u2229 o) y = f' y :=\n      ((hff' y (ho hy)).mono ho).fderiv_within (hs.inter o_open y hy),\n    rwa fderiv_within_inter (mem_nhds_sets o_open hy.2) (hs y hy.1) at A, },\n  { rintros \u27e8hdiff, h\u27e9 x hx,\n    rw [times_cont_diff_within_at_succ_iff_has_fderiv_within_at, insert_eq_of_mem hx],\n    exact \u27e8s, self_mem_nhds_within, fderiv_within \ud835\udd5c f s,\n      \u03bb y hy, (hdiff y hy).has_fderiv_within_at, h x hx\u27e9 }\nend\n\n\/-- A function is `C^(n + 1)` on an open domain if and only if it is\ndifferentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_fderiv_of_open {n : \u2115} (hs : is_open s) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c n (\u03bb y, fderiv \ud835\udd5c f y) s :=\nbegin\n  rw times_cont_diff_on_succ_iff_fderiv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact fderiv_within_of_open hs hx\nend\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative (expressed with `fderiv_within`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_fderiv_within (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c \u221e (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  split,\n  { assume h,\n    refine \u27e8h.differentiable_on le_top, _\u27e9,\n    apply times_cont_diff_on_top.2 (\u03bb n, ((times_cont_diff_on_succ_iff_fderiv_within hs).1 _).2),\n    exact h.of_le le_top },\n  { assume h,\n    refine times_cont_diff_on_top.2 (\u03bb n, _),\n    have A : (n : with_top \u2115) \u2264 \u221e := le_top,\n    apply ((times_cont_diff_on_succ_iff_fderiv_within hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le,\n    exact with_top.coe_le_coe.2 (nat.le_succ n) }\nend\n\n\/-- A function is `C^\u221e` on an open domain if and only if it is differentiable there, and its\nderivative (expressed with `fderiv`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_fderiv_of_open (hs : is_open s) :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c \u221e (\u03bb y, fderiv \ud835\udd5c f y) s :=\nbegin\n  rw times_cont_diff_on_top_iff_fderiv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact fderiv_within_of_open hs hx\nend\n\nlemma times_cont_diff_on.fderiv_within {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  cases m,\n  { change \u221e + 1 \u2264 n at hmn,\n    have : n = \u221e, by simpa using hmn,\n    rw this at hf,\n    exact ((times_cont_diff_on_top_iff_fderiv_within hs).1 hf).2 },\n  { change (m.succ : with_top \u2115) \u2264 n at hmn,\n    exact ((times_cont_diff_on_succ_iff_fderiv_within hs).1 (hf.of_le hmn)).2 }\nend\n\nlemma times_cont_diff_on.fderiv_of_open {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : is_open s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bb y, fderiv \ud835\udd5c f y) s :=\n(hf.fderiv_within hs.unique_diff_on hmn).congr (\u03bb x hx, (fderiv_within_of_open hs hx).symm)\n\nlemma times_cont_diff_on.continuous_on_fderiv_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bb x, fderiv_within \ud835\udd5c f s x) s :=\n((times_cont_diff_on_succ_iff_fderiv_within hs).1 (h.of_le hn)).2.continuous_on\n\nlemma times_cont_diff_on.continuous_on_fderiv_of_open {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : is_open s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bb x, fderiv \ud835\udd5c f x) s :=\n((times_cont_diff_on_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.continuous_on\n\n\/-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is\ncontinuous. -\/\nlemma times_cont_diff_on.continuous_on_fderiv_within_apply\n  {n : with_top \u2115} (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1 : E \u2192 F) p.2) (set.prod s univ) :=\nbegin\n  have A : continuous (\u03bbq : (E \u2192L[\ud835\udd5c] F) \u00d7 E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous,\n  have B : continuous_on (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1, p.2)) (set.prod s univ),\n  { apply continuous_on.prod _ continuous_snd.continuous_on,\n    exact continuous_on.comp (h.continuous_on_fderiv_within hs hn) continuous_fst.continuous_on\n      (prod_subset_preimage_fst _ _) },\n  exact A.comp_continuous_on B\nend\n\n\/-! ### Functions with a Taylor series on the whole space -\/\n\n\/-- `has_ftaylor_series_up_to n f p` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`has_fderiv_at` but for higher order derivatives. -\/\nstructure has_ftaylor_series_up_to (n : with_top \u2115)\n  (f : E \u2192 F) (p : E \u2192 formal_multilinear_series \ud835\udd5c E F) : Prop :=\n(zero_eq : \u2200 x, (p x 0).uncurry0 = f x)\n(fderiv  : \u2200 (m : \u2115) (hm : (m : with_top \u2115) < n), \u2200 x,\n    has_fderiv_at (\u03bb y, p y m) (p x m.succ).curry_left x)\n(cont    : \u2200 (m : \u2115) (hm : (m : with_top \u2115) \u2264 n), continuous (\u03bb x, p x m))\n\nlemma has_ftaylor_series_up_to.zero_eq' {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (x : E) :\n  p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x) :=\nby { rw \u2190 h.zero_eq x, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ }\n\nlemma has_ftaylor_series_up_to_on_univ_iff {n : with_top \u2115} :\n  has_ftaylor_series_up_to_on n f p univ \u2194 has_ftaylor_series_up_to n f p :=\nbegin\n  split,\n  { assume H,\n    split,\n    { exact \u03bb x, H.zero_eq x (mem_univ x) },\n    { assume m hm x,\n      rw \u2190 has_fderiv_within_at_univ,\n      exact H.fderiv_within m hm x (mem_univ x) },\n    { assume m hm,\n      rw continuous_iff_continuous_on_univ,\n      exact H.cont m hm } },\n  { assume H,\n    split,\n    { exact \u03bb x hx, H.zero_eq x },\n    { assume m hm x hx,\n      rw has_fderiv_within_at_univ,\n      exact H.fderiv m hm x },\n    { assume m hm,\n      rw \u2190 continuous_iff_continuous_on_univ,\n      exact H.cont m hm } }\nend\n\nlemma has_ftaylor_series_up_to.has_ftaylor_series_up_to_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (s : set E) :\n  has_ftaylor_series_up_to_on n f p s :=\n(has_ftaylor_series_up_to_on_univ_iff.2 h).mono (subset_univ _)\n\nlemma has_ftaylor_series_up_to.of_le {m n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hmn : m \u2264 n) :\n  has_ftaylor_series_up_to m f p :=\nby { rw \u2190 has_ftaylor_series_up_to_on_univ_iff at h \u22a2, exact h.of_le hmn }\n\nlemma has_ftaylor_series_up_to.continuous {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) : continuous f :=\nbegin\n  rw \u2190 has_ftaylor_series_up_to_on_univ_iff at h,\n  rw continuous_iff_continuous_on_univ,\n  exact h.continuous_on\nend\n\nlemma has_ftaylor_series_up_to_zero_iff :\n  has_ftaylor_series_up_to 0 f p \u2194 continuous f \u2227 (\u2200 x, (p x 0).uncurry0 = f x) :=\nby simp [has_ftaylor_series_up_to_on_univ_iff.symm, continuous_iff_continuous_on_univ,\n         has_ftaylor_series_up_to_on_zero_iff]\n\n\/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -\/\nlemma has_ftaylor_series_up_to.has_fderiv_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hn : 1 \u2264 n) (x : E) :\n  has_fderiv_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) x :=\nbegin\n  rw [\u2190 has_fderiv_within_at_univ],\n  exact (has_ftaylor_series_up_to_on_univ_iff.2 h).has_fderiv_within_at hn (mem_univ _)\nend\n\nlemma has_ftaylor_series_up_to.differentiable {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hn : 1 \u2264 n) : differentiable \ud835\udd5c f :=\n\u03bb x, (h.has_fderiv_at hn x).differentiable_at\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -\/\ntheorem has_ftaylor_series_up_to_succ_iff_right {n : \u2115} :\n  has_ftaylor_series_up_to ((n + 1) : \u2115) f p \u2194\n  (\u2200 x, (p x 0).uncurry0 = f x)\n  \u2227 (\u2200 x, has_fderiv_at (\u03bb y, p y 0) (p x 1).curry_left x)\n  \u2227 has_ftaylor_series_up_to n\n    (\u03bb x, continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) (\u03bb x, (p x).shift) :=\nby simp [has_ftaylor_series_up_to_on_succ_iff_right, has_ftaylor_series_up_to_on_univ_iff.symm,\n         -add_comm, -with_zero.coe_add]\n\n\/-! ### Smooth functions at a point -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` at a point `x` if, for any integer `k \u2264 n`,\nthere is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous.\n-\/\ndef times_cont_diff_at (n : with_top \u2115) (f : E \u2192 F) (x : E) :=\ntimes_cont_diff_within_at \ud835\udd5c n f univ x\n\nvariable {\ud835\udd5c}\n\ntheorem times_cont_diff_within_at_univ {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f univ x \u2194 times_cont_diff_at \ud835\udd5c n f x :=\niff.rfl\n\nlemma times_cont_diff_at_top :\n  times_cont_diff_at \ud835\udd5c \u221e f x \u2194 \u2200 (n : \u2115), times_cont_diff_at \ud835\udd5c n f x :=\nby simp [\u2190 times_cont_diff_within_at_univ, times_cont_diff_within_at_top]\n\nlemma times_cont_diff_at.times_cont_diff_within_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) : times_cont_diff_within_at \ud835\udd5c n f s x :=\nh.mono (subset_univ _)\n\nlemma times_cont_diff_within_at.times_cont_diff_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hx : s \u2208 \ud835\udcdd x) :\n  times_cont_diff_at \ud835\udd5c n f x :=\nby rwa [times_cont_diff_at, \u2190 times_cont_diff_within_at_inter hx, univ_inter]\n\nlemma times_cont_diff_at.congr_of_eventually_eq {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hg : f\u2081 =\u1da0[\ud835\udcdd x] f) :\n  times_cont_diff_at \ud835\udd5c n f\u2081 x :=\nh.congr_of_eventually_eq' (by rwa nhds_within_univ) (mem_univ x)\n\nlemma times_cont_diff_at.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hmn : m \u2264 n) :\n  times_cont_diff_at \ud835\udd5c m f x :=\nh.of_le hmn\n\nlemma times_cont_diff_at.continuous_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) : continuous_at f x :=\nby simpa [continuous_within_at_univ] using h.continuous_within_at\n\n\/-- If a function is `C^n` with `n \u2265 1` at a point, then it is differentiable there. -\/\nlemma times_cont_diff_at.differentiable_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hn : 1 \u2264 n) : differentiable_at \ud835\udd5c f x :=\nby simpa [hn, differentiable_within_at_univ] using h.differentiable_within_at\n\n\/-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_at_succ_iff_has_fderiv_at {n : \u2115} :\n  times_cont_diff_at \ud835\udd5c ((n + 1) : \u2115) f x\n  \u2194 (\u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F), (\u2203 u \u2208 \ud835\udcdd x, (\u2200 x \u2208 u, has_fderiv_at f (f' x) x))\n      \u2227 (times_cont_diff_at \ud835\udd5c n f' x)) :=\nbegin\n  rw [\u2190 times_cont_diff_within_at_univ, times_cont_diff_within_at_succ_iff_has_fderiv_within_at],\n  simp only [nhds_within_univ, exists_prop, mem_univ, insert_eq_of_mem],\n  split,\n  { rintros \u27e8u, H, f', h_fderiv, h_times_cont_diff\u27e9,\n    rcases mem_nhds_sets_iff.mp H with \u27e8t, htu, ht, hxt\u27e9,\n    refine \u27e8f', \u27e8t, _\u27e9, h_times_cont_diff.times_cont_diff_at H\u27e9,\n    refine \u27e8mem_nhds_sets_iff.mpr \u27e8t, subset.rfl, ht, hxt\u27e9, _\u27e9,\n    intros y hyt,\n    refine (h_fderiv y (htu hyt)).has_fderiv_at _,\n    exact mem_nhds_sets_iff.mpr \u27e8t, htu, ht, hyt\u27e9 },\n  { rintros \u27e8f', \u27e8u, H, h_fderiv\u27e9, h_times_cont_diff\u27e9,\n    refine \u27e8u, H, f', _, h_times_cont_diff.times_cont_diff_within_at\u27e9,\n    intros x hxu,\n    exact (h_fderiv x hxu).has_fderiv_within_at }\nend\n\n\/-! ### Smooth functions -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` if it admits derivatives up to\norder `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives\nmight not be unique) we do not need to localize the definition in space or time.\n-\/\ndefinition times_cont_diff (n : with_top \u2115) (f : E \u2192 F)  :=\n\u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F, has_ftaylor_series_up_to n f p\n\nvariable {\ud835\udd5c}\n\ntheorem times_cont_diff_on_univ {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f univ \u2194 times_cont_diff \ud835\udd5c n f :=\nbegin\n  split,\n  { assume H,\n    use ftaylor_series_within \ud835\udd5c f univ,\n    rw \u2190 has_ftaylor_series_up_to_on_univ_iff,\n    exact H.ftaylor_series_within unique_diff_on_univ },\n  { rintros \u27e8p, hp\u27e9 x hx m hm,\n    exact \u27e8univ, filter.univ_sets _, p, (hp.has_ftaylor_series_up_to_on univ).of_le hm\u27e9 }\nend\n\nlemma times_cont_diff_iff_times_cont_diff_at {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194 \u2200 x, times_cont_diff_at \ud835\udd5c n f x :=\nby simp [\u2190 times_cont_diff_on_univ, times_cont_diff_on, times_cont_diff_at]\n\nlemma times_cont_diff.times_cont_diff_at {n : with_top \u2115} (h : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff_at \ud835\udd5c n f x :=\ntimes_cont_diff_iff_times_cont_diff_at.1 h x\n\nlemma times_cont_diff.times_cont_diff_within_at {n : with_top \u2115} (h : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nh.times_cont_diff_at.times_cont_diff_within_at\n\nlemma times_cont_diff_top :\n  times_cont_diff \ud835\udd5c \u221e f \u2194 \u2200 (n : \u2115), times_cont_diff \ud835\udd5c n f :=\nby simp [times_cont_diff_on_univ.symm, times_cont_diff_on_top]\n\nlemma times_cont_diff_all_iff_nat :\n  (\u2200 n, times_cont_diff \ud835\udd5c n f) \u2194 (\u2200 n : \u2115, times_cont_diff \ud835\udd5c n f) :=\nby simp only [\u2190 times_cont_diff_on_univ, times_cont_diff_on_all_iff_nat]\n\nlemma times_cont_diff.times_cont_diff_on {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) : times_cont_diff_on \ud835\udd5c n f s :=\n(times_cont_diff_on_univ.2 h).mono (subset_univ _)\n\n@[simp] lemma times_cont_diff_zero :\n  times_cont_diff \ud835\udd5c 0 f \u2194 continuous f :=\nbegin\n  rw [\u2190 times_cont_diff_on_univ, continuous_iff_continuous_on_univ],\n  exact times_cont_diff_on_zero\nend\n\nlemma times_cont_diff_at_zero :\n  times_cont_diff_at \ud835\udd5c 0 f x \u2194 \u2203 u \u2208 \ud835\udcdd x, continuous_on f u :=\nby { rw \u2190 times_cont_diff_within_at_univ, simp [times_cont_diff_within_at_zero, nhds_within_univ] }\n\nlemma times_cont_diff.of_le {m n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hmn : m \u2264 n) :\n  times_cont_diff \ud835\udd5c m f :=\ntimes_cont_diff_on_univ.1 $ (times_cont_diff_on_univ.2 h).of_le hmn\n\nlemma times_cont_diff.continuous {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) : continuous f :=\ntimes_cont_diff_zero.1 (h.of_le bot_le)\n\n\/-- If a function is `C^n` with `n \u2265 1`, then it is differentiable. -\/\nlemma times_cont_diff.differentiable {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) : differentiable \ud835\udd5c f :=\ndifferentiable_on_univ.1 $ (times_cont_diff_on_univ.2 h).differentiable_on hn\n\n\n\/-! ### Iterated derivative -\/\n\nvariable (\ud835\udd5c)\n\n\/-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -\/\nnoncomputable def iterated_fderiv (n : \u2115) (f : E \u2192 F) :\n  E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F) :=\nnat.rec_on n\n  (\u03bb x, continuous_multilinear_map.curry0 \ud835\udd5c E (f x))\n  (\u03bb n rec x, continuous_linear_map.uncurry_left (fderiv \ud835\udd5c rec x))\n\n\/-- Formal Taylor series associated to a function within a set. -\/\ndef ftaylor_series (f : E \u2192 F) (x : E) : formal_multilinear_series \ud835\udd5c E F :=\n\u03bb n, iterated_fderiv \ud835\udd5c n f x\n\nvariable {\ud835\udd5c}\n\n@[simp] lemma iterated_fderiv_zero_apply (m : (fin 0) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c 0 f x : ((fin 0) \u2192  E) \u2192 F) m = f x := rfl\n\nlemma iterated_fderiv_zero_eq_comp :\n  iterated_fderiv \ud835\udd5c 0 f = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm \u2218 f := rfl\n\nlemma iterated_fderiv_succ_apply_left {n : \u2115} (m : fin (n + 1) \u2192 E):\n  (iterated_fderiv \ud835\udd5c (n + 1) f x : (fin (n + 1) \u2192 E) \u2192 F) m\n  = (fderiv \ud835\udd5c (iterated_fderiv \ud835\udd5c n f) x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) := rfl\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -\/\nlemma iterated_fderiv_succ_eq_comp_left {n : \u2115} :\n  iterated_fderiv \ud835\udd5c (n + 1) f =\n  (continuous_multilinear_curry_left_equiv \ud835\udd5c (\u03bb(i : fin (n + 1)), E) F)\n    \u2218 (fderiv \ud835\udd5c (iterated_fderiv \ud835\udd5c n f)) := rfl\n\nlemma iterated_fderiv_within_univ {n : \u2115} :\n  iterated_fderiv_within \ud835\udd5c n f univ = iterated_fderiv \ud835\udd5c n f :=\nbegin\n  induction n with n IH,\n  { ext x, simp },\n  { ext x m,\n    rw [iterated_fderiv_succ_apply_left, iterated_fderiv_within_succ_apply_left, IH,\n        fderiv_within_univ] }\nend\n\nlemma ftaylor_series_within_univ :\n  ftaylor_series_within \ud835\udd5c f univ = ftaylor_series \ud835\udd5c f :=\nbegin\n  ext1 x, ext1 n,\n  change iterated_fderiv_within \ud835\udd5c n f univ x = iterated_fderiv \ud835\udd5c n f x,\n  rw iterated_fderiv_within_univ\nend\n\ntheorem iterated_fderiv_succ_apply_right {n : \u2115} (m : fin (n + 1) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c (n + 1) f x : (fin (n + 1) \u2192 E) \u2192 F) m\n    = iterated_fderiv \ud835\udd5c n (\u03bby, fderiv \ud835\udd5c f y) x (init m) (m (last n)) :=\nbegin\n  rw [\u2190 iterated_fderiv_within_univ, \u2190 iterated_fderiv_within_univ, \u2190 fderiv_within_univ],\n  exact iterated_fderiv_within_succ_apply_right unique_diff_on_univ (mem_univ _) _\nend\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the `n`-th derivative of the derivative. -\/\nlemma iterated_fderiv_succ_eq_comp_right {n : \u2115} :\n  iterated_fderiv \ud835\udd5c (n + 1) f x =\n  ((continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F)\n    \u2218 (iterated_fderiv \ud835\udd5c n (\u03bby, fderiv \ud835\udd5c f y))) x :=\nby { ext m, rw iterated_fderiv_succ_apply_right, refl }\n\n@[simp] lemma iterated_fderiv_one_apply (m : (fin 1) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c 1 f x : ((fin 1) \u2192 E) \u2192 F) m\n  = (fderiv \ud835\udd5c f x : E \u2192 F) (m 0) :=\nby { rw [iterated_fderiv_succ_apply_right, iterated_fderiv_zero_apply], refl }\n\n\/-- When a function is `C^n` in a set `s` of unique differentiability, it admits\n`ftaylor_series_within \ud835\udd5c f s` as a Taylor series up to order `n` in `s`. -\/\ntheorem times_cont_diff_on_iff_ftaylor_series {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194 has_ftaylor_series_up_to n f (ftaylor_series \ud835\udd5c f) :=\nbegin\n  split,\n  { rw [\u2190 times_cont_diff_on_univ, \u2190 has_ftaylor_series_up_to_on_univ_iff,\n        \u2190 ftaylor_series_within_univ],\n    exact \u03bb h, times_cont_diff_on.ftaylor_series_within h unique_diff_on_univ },\n  { assume h, exact \u27e8ftaylor_series \ud835\udd5c f, h\u27e9 }\nend\n\nlemma times_cont_diff_iff_continuous_differentiable {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194\n  (\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192 continuous (\u03bb x, iterated_fderiv \ud835\udd5c m f x))\n  \u2227 (\u2200 (m : \u2115), (m : with_top \u2115) < n \u2192 differentiable \ud835\udd5c (\u03bb x, iterated_fderiv \ud835\udd5c m f x)) :=\nby simp [times_cont_diff_on_univ.symm, continuous_iff_continuous_on_univ,\n    differentiable_on_univ.symm, iterated_fderiv_within_univ,\n    times_cont_diff_on_iff_continuous_on_differentiable_on unique_diff_on_univ]\n\nlemma times_cont_diff_of_differentiable_iterated_fderiv {n : with_top \u2115}\n  (h : \u2200(m : \u2115), (m : with_top \u2115) \u2264 n \u2192 differentiable \ud835\udd5c (iterated_fderiv \ud835\udd5c m f)) :\n  times_cont_diff \ud835\udd5c n f :=\ntimes_cont_diff_iff_continuous_differentiable.2\n\u27e8\u03bb m hm, (h m hm).continuous, \u03bb m hm, (h m (le_of_lt hm))\u27e9\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative is `C^n`. -\/\ntheorem times_cont_diff_succ_iff_fderiv {n : \u2115} :\n  times_cont_diff \ud835\udd5c ((n + 1) : \u2115) f \u2194\n  differentiable \ud835\udd5c f \u2227 times_cont_diff \ud835\udd5c n (\u03bb y, fderiv \ud835\udd5c f y) :=\nby simp [times_cont_diff_on_univ.symm, differentiable_on_univ.symm, fderiv_within_univ.symm,\n         - fderiv_within_univ, times_cont_diff_on_succ_iff_fderiv_within unique_diff_on_univ,\n         -with_zero.coe_add, -add_comm]\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative is `C^\u221e`. -\/\ntheorem times_cont_diff_top_iff_fderiv :\n  times_cont_diff \ud835\udd5c \u221e f \u2194\n  differentiable \ud835\udd5c f \u2227 times_cont_diff \ud835\udd5c \u221e (\u03bb y, fderiv \ud835\udd5c f y) :=\nbegin\n  simp [times_cont_diff_on_univ.symm, differentiable_on_univ.symm, fderiv_within_univ.symm,\n        - fderiv_within_univ],\n  rw times_cont_diff_on_top_iff_fderiv_within unique_diff_on_univ,\nend\n\nlemma times_cont_diff.continuous_fderiv {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) :\n  continuous (\u03bb x, fderiv \ud835\udd5c f x) :=\n((times_cont_diff_succ_iff_fderiv).1 (h.of_le hn)).2.continuous\n\n\/-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is\ncontinuous. -\/\nlemma times_cont_diff.continuous_fderiv_apply {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) :\n  continuous (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1 : E \u2192 F) p.2) :=\nbegin\n  have A : continuous (\u03bbq : (E \u2192L[\ud835\udd5c] F) \u00d7 E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous,\n  have B : continuous (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1, p.2)),\n  { apply continuous.prod_mk _ continuous_snd,\n    exact continuous.comp (h.continuous_fderiv hn) continuous_fst },\n  exact A.comp B\nend\n\n\/-! ### Constants -\/\n\nlemma iterated_fderiv_within_zero_fun {n : \u2115} :\n  iterated_fderiv \ud835\udd5c n (\u03bb x : E, (0 : F)) = 0 :=\nbegin\n  induction n with n IH,\n  { ext m, simp },\n  { ext x m,\n    rw [iterated_fderiv_succ_apply_left, IH],\n    change (fderiv \ud835\udd5c (\u03bb (x : E), (0 : (E [\u00d7n]\u2192L[\ud835\udd5c] F))) x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) = _,\n    rw fderiv_const,\n    refl }\nend\n\nlemma times_cont_diff_zero_fun {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb x : E, (0 : F)) :=\nbegin\n  apply times_cont_diff_of_differentiable_iterated_fderiv (\u03bbm hm, _),\n  rw iterated_fderiv_within_zero_fun,\n  apply differentiable_const (0 : (E [\u00d7m]\u2192L[\ud835\udd5c] F))\nend\n\n\/--\nConstants are `C^\u221e`.\n-\/\nlemma times_cont_diff_const {n : with_top \u2115} {c : F} : times_cont_diff \ud835\udd5c n (\u03bbx : E, c) :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e (\u03bbx : E, c), by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8differentiable_const c, _\u27e9,\n  rw fderiv_const,\n  exact times_cont_diff_zero_fun\nend\n\nlemma times_cont_diff_on_const {n : with_top \u2115} {c : F} {s : set E} :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx : E, c) s :=\ntimes_cont_diff_const.times_cont_diff_on\n\nlemma times_cont_diff_at_const {n : with_top \u2115} {c : F} :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx : E, c) x :=\ntimes_cont_diff_const.times_cont_diff_at\n\nlemma times_cont_diff_within_at_const {n : with_top \u2115} {c : F} :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx : E, c) s x :=\ntimes_cont_diff_at_const.times_cont_diff_within_at\n\n@[nontriviality] lemma times_cont_diff_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_const }\n\n@[nontriviality] lemma times_cont_diff_at_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n f x :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_at_const }\n\n@[nontriviality] lemma times_cont_diff_within_at_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_within_at_const }\n\n@[nontriviality] lemma times_cont_diff_on_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f s :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_on_const }\n\n\/-! ### Linear functions -\/\n\n\/--\nUnbundled bounded linear functions are `C^\u221e`.\n-\/\nlemma is_bounded_linear_map.times_cont_diff {n : with_top \u2115} (hf : is_bounded_linear_map \ud835\udd5c f) :\n  times_cont_diff \ud835\udd5c n f :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e f, by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8hf.differentiable, _\u27e9,\n  simp [hf.fderiv],\n  exact times_cont_diff_const\nend\n\nlemma continuous_linear_map.times_cont_diff {n : with_top \u2115} (f : E \u2192L[\ud835\udd5c] F) :\n  times_cont_diff \ud835\udd5c n f :=\nf.is_bounded_linear_map.times_cont_diff\n\n\/--\nThe first projection in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_fst {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) :=\nis_bounded_linear_map.times_cont_diff is_bounded_linear_map.fst\n\n\/--\nThe first projection on a domain in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_on_fst {s : set (E\u00d7F)} {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) s :=\ntimes_cont_diff.times_cont_diff_on times_cont_diff_fst\n\n\/--\nThe first projection at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_at_fst {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) p :=\ntimes_cont_diff_fst.times_cont_diff_at\n\n\/--\nThe first projection within a domain at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_within_at_fst {s : set (E \u00d7 F)} {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) s p :=\ntimes_cont_diff_fst.times_cont_diff_within_at\n\n\/--\nThe second projection in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_snd {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) :=\nis_bounded_linear_map.times_cont_diff is_bounded_linear_map.snd\n\n\/--\nThe second projection on a domain in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_on_snd {s : set (E\u00d7F)} {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) s :=\ntimes_cont_diff.times_cont_diff_on times_cont_diff_snd\n\n\/--\nThe second projection at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_at_snd {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) p :=\ntimes_cont_diff_snd.times_cont_diff_at\n\n\/--\nThe second projection within a domain at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_within_at_snd {s : set (E \u00d7 F)} {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) s p :=\ntimes_cont_diff_snd.times_cont_diff_within_at\n\n\/--\nThe identity is `C^\u221e`.\n-\/\nlemma times_cont_diff_id {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (id : E \u2192 E) :=\nis_bounded_linear_map.id.times_cont_diff\n\nlemma times_cont_diff_within_at_id {n : with_top \u2115} {s x} :\n  times_cont_diff_within_at \ud835\udd5c n (id : E \u2192 E) s x :=\ntimes_cont_diff_id.times_cont_diff_within_at\n\nlemma times_cont_diff_at_id {n : with_top \u2115} {x} :\n  times_cont_diff_at \ud835\udd5c n (id : E \u2192 E) x :=\ntimes_cont_diff_id.times_cont_diff_at\n\nlemma times_cont_diff_on_id {n : with_top \u2115} {s} :\n  times_cont_diff_on \ud835\udd5c n (id : E \u2192 E) s :=\ntimes_cont_diff_id.times_cont_diff_on\n\n\/--\nBilinear functions are `C^\u221e`.\n-\/\nlemma is_bounded_bilinear_map.times_cont_diff {n : with_top \u2115} (hb : is_bounded_bilinear_map \ud835\udd5c b) :\n  times_cont_diff \ud835\udd5c n b :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e b, by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8hb.differentiable, _\u27e9,\n  simp [hb.fderiv],\n  exact hb.is_bounded_linear_map_deriv.times_cont_diff\nend\n\n\/-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `g \u2218 f` admits a Taylor\nseries whose `k`-th term is given by `g \u2218 (p k)`. -\/\nlemma has_ftaylor_series_up_to_on.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : has_ftaylor_series_up_to_on n f p s) :\n  has_ftaylor_series_up_to_on n (g \u2218 f) (\u03bb x k, g.comp_continuous_multilinear_map (p x k)) s :=\nbegin\n  split,\n  { assume x hx, simp [(hf.zero_eq x hx).symm] },\n  { assume m hm x hx,\n    let A : (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u2192 (E [\u00d7m]\u2192L[\ud835\udd5c] G) := \u03bb f, g.comp_continuous_multilinear_map f,\n    have hA : is_bounded_linear_map \ud835\udd5c A :=\n      is_bounded_bilinear_map_comp_multilinear.is_bounded_linear_map_right _,\n    have := hf.fderiv_within m hm x hx,\n    convert has_fderiv_at.comp_has_fderiv_within_at x (hA.has_fderiv_at) this },\n  { assume m hm,\n    let A : (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u2192 (E [\u00d7m]\u2192L[\ud835\udd5c] G) :=\n      \u03bb f, g.comp_continuous_multilinear_map f,\n    have hA : is_bounded_linear_map \ud835\udd5c A :=\n      is_bounded_bilinear_map_comp_multilinear.is_bounded_linear_map_right _,\n    exact hA.continuous.comp_continuous_on (hf.cont m hm) }\nend\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain\nat a point. -\/\nlemma times_cont_diff_within_at.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  exact \u27e8u, hu, _, hp.continuous_linear_map_comp g\u27e9,\nend\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain\nat a point. -\/\nlemma times_cont_diff_at.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\ntimes_cont_diff_within_at.continuous_linear_map_comp g hf\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions on domains. -\/\nlemma times_cont_diff_on.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\n\u03bb x hx, (hf x hx).continuous_linear_map_comp g\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions. -\/\nlemma times_cont_diff.continuous_linear_map_comp {n : with_top \u2115} {f : E \u2192 F} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff \ud835\udd5c n f) : times_cont_diff \ud835\udd5c n (\u03bbx, g (f x)) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.continuous_linear_map_comp\n  _ (times_cont_diff_on_univ.2 hf)\n\n\/-- Composition by continuous linear equivs on the left respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.comp_times_cont_diff_within_at_iff\n  {n : with_top \u2115} (e : F \u2243L[\ud835\udd5c] G) :\n  times_cont_diff_within_at \ud835\udd5c n (e \u2218 f) s x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\nbegin\n  split,\n  { assume H,\n    have : f = e.symm \u2218 (e \u2218 f),\n      by { ext y, simp only [function.comp_app], rw e.symm_apply_apply (f y) },\n    rw this,\n    exact H.continuous_linear_map_comp _ },\n  { assume H,\n    exact H.continuous_linear_map_comp _ }\nend\n\n\/-- Composition by continuous linear equivs on the left respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.comp_times_cont_diff_on_iff\n  {n : with_top \u2115} (e : F \u2243L[\ud835\udd5c] G) :\n  times_cont_diff_on \ud835\udd5c n (e \u2218 f) s \u2194 times_cont_diff_on \ud835\udd5c n f s :=\nby simp [times_cont_diff_on, e.comp_times_cont_diff_within_at_iff]\n\n\/-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `f \u2218 g` admits a Taylor\nseries in `g \u207b\u00b9' s`, whose `k`-th term is given by `p k (g v\u2081, ..., g v\u2096)` . -\/\nlemma has_ftaylor_series_up_to_on.comp_continuous_linear_map {n : with_top \u2115}\n  (hf : has_ftaylor_series_up_to_on n f p s) (g : G \u2192L[\ud835\udd5c] E) :\n  has_ftaylor_series_up_to_on n (f \u2218 g)\n    (\u03bb x k, (p (g x) k).comp_continuous_linear_map (\u03bb _, g)) (g \u207b\u00b9' s) :=\nbegin\n  let A : \u03a0 m : \u2115, (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u2192 (G [\u00d7m]\u2192L[\ud835\udd5c] F) :=\n    \u03bb m h, h.comp_continuous_linear_map (\u03bb _, g),\n  have hA : \u2200 m, is_bounded_linear_map \ud835\udd5c (A m) :=\n    \u03bb m, is_bounded_linear_map_continuous_multilinear_map_comp_linear g,\n  split,\n  { assume x hx,\n    simp only [(hf.zero_eq (g x) hx).symm, function.comp_app],\n    change p (g x) 0 (\u03bb (i : fin 0), g 0) = p (g x) 0 0,\n    rw continuous_linear_map.map_zero,\n    refl },\n  { assume m hm x hx,\n    convert ((hA m).has_fderiv_at).comp_has_fderiv_within_at x\n      ((hf.fderiv_within m hm (g x) hx).comp x (g.has_fderiv_within_at) (subset.refl _)),\n    ext y v,\n    change p (g x) (nat.succ m) (g \u2218 (cons y v)) = p (g x) m.succ (cons (g y) (g \u2218 v)),\n    rw comp_cons },\n  { assume m hm,\n    exact (hA m).continuous.comp_continuous_on\n      ((hf.cont m hm).comp g.continuous.continuous_on (subset.refl _)) }\nend\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions at a point on\na domain. -\/\nlemma times_cont_diff_within_at.comp_continuous_linear_map {n : with_top \u2115} {x : G}\n  (g : G \u2192L[\ud835\udd5c] E) (hf : times_cont_diff_within_at \ud835\udd5c n f s (g x)) :\n  times_cont_diff_within_at \ud835\udd5c n (f \u2218 g) (g \u207b\u00b9' s) x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  refine \u27e8g \u207b\u00b9' u, _, _, hp.comp_continuous_linear_map g\u27e9,\n  apply continuous_within_at.preimage_mem_nhds_within',\n  { exact g.continuous.continuous_within_at },\n  { apply nhds_within_mono (g x) _ hu,\n    rw image_insert_eq,\n    exact insert_subset_insert (image_preimage_subset g s) }\nend\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions on domains. -\/\nlemma times_cont_diff_on.comp_continuous_linear_map {n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (g : G \u2192L[\ud835\udd5c] E) :\n  times_cont_diff_on \ud835\udd5c n (f \u2218 g) (g \u207b\u00b9' s) :=\n\u03bb x hx, (hf (g x) hx).comp_continuous_linear_map g\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions. -\/\nlemma times_cont_diff.comp_continuous_linear_map {n : with_top \u2115} {f : E \u2192 F} {g : G \u2192L[\ud835\udd5c] E}\n  (hf : times_cont_diff \ud835\udd5c n f) : times_cont_diff \ud835\udd5c n (f \u2218 g) :=\ntimes_cont_diff_on_univ.1 $\ntimes_cont_diff_on.comp_continuous_linear_map (times_cont_diff_on_univ.2 hf) _\n\n\/-- Composition by continuous linear equivs on the right respects higher differentiability at a\npoint in a domain. -\/\nlemma continuous_linear_equiv.times_cont_diff_within_at_comp_iff {n : with_top \u2115} (e : G \u2243L[\ud835\udd5c] E) :\n  times_cont_diff_within_at \ud835\udd5c n (f \u2218 e) (e \u207b\u00b9' s) (e.symm x) \u2194\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nbegin\n  split,\n  { assume H,\n    have A : f = (f \u2218 e) \u2218 e.symm,\n      by { ext y, simp only [function.comp_app], rw e.apply_symm_apply y },\n    have B : e.symm \u207b\u00b9' (e \u207b\u00b9' s) = s,\n      by { rw [\u2190 preimage_comp, e.self_comp_symm], refl },\n    rw [A, \u2190 B],\n    exact H.comp_continuous_linear_map _},\n  { assume H,\n    have : x = e (e.symm x), by simp,\n    rw this at H,\n    exact H.comp_continuous_linear_map _ },\nend\n\n\n\/-- Composition by continuous linear equivs on the right respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.times_cont_diff_on_comp_iff {n : with_top \u2115} (e : G \u2243L[\ud835\udd5c] E) :\n  times_cont_diff_on \ud835\udd5c n (f \u2218 e) (e \u207b\u00b9' s) \u2194 times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  refine \u27e8\u03bb H, _, \u03bb H, H.comp_continuous_linear_map _\u27e9,\n  have A : f = (f \u2218 e) \u2218 e.symm,\n    by { ext y, simp only [function.comp_app], rw e.apply_symm_apply y },\n  have B : e.symm \u207b\u00b9' (e \u207b\u00b9' s) = s,\n    by { rw [\u2190 preimage_comp, e.self_comp_symm], refl },\n  rw [A, \u2190 B],\n  exact H.comp_continuous_linear_map _\nend\n\n\/-- If two functions `f` and `g` admit Taylor series `p` and `q` in a set `s`, then the cartesian\nproduct of `f` and `g` admits the cartesian product of `p` and `q` as a Taylor series. -\/\nlemma has_ftaylor_series_up_to_on.prod {n : with_top \u2115} (hf : has_ftaylor_series_up_to_on n f p s)\n  {g : E \u2192 G} {q : E \u2192 formal_multilinear_series \ud835\udd5c E G} (hg : has_ftaylor_series_up_to_on n g q s) :\n  has_ftaylor_series_up_to_on n (\u03bb y, (f y, g y)) (\u03bb y k, (p y k).prod (q y k)) s :=\nbegin\n  split,\n  { assume x hx, rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx], refl },\n  { assume m hm x hx,\n    let A : (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u00d7 (E [\u00d7m]\u2192L[\ud835\udd5c] G) \u2192 (E [\u00d7m]\u2192L[\ud835\udd5c] (F \u00d7 G)) := \u03bb p, p.1.prod p.2,\n    have hA : is_bounded_linear_map \ud835\udd5c A := is_bounded_linear_map_prod_multilinear,\n    convert hA.has_fderiv_at.comp_has_fderiv_within_at x\n      ((hf.fderiv_within m hm x hx).prod (hg.fderiv_within m hm x hx)) },\n  { assume m hm,\n    let A : (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u00d7 (E [\u00d7m]\u2192L[\ud835\udd5c] G) \u2192 (E [\u00d7m]\u2192L[\ud835\udd5c] (F \u00d7 G)) := \u03bb p, p.1.prod p.2,\n    have hA : is_bounded_linear_map \ud835\udd5c A := is_bounded_linear_map_prod_multilinear,\n    exact hA.continuous.comp_continuous_on ((hf.cont m hm).prod (hg.cont m hm)) }\nend\n\n\/-- The cartesian product of `C^n` functions at a point in a domain is `C^n`. -\/\nlemma times_cont_diff_within_at.prod {n : with_top \u2115} {s : set E} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx:E, (f x, g x)) s x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  rcases hg m hm with \u27e8v, hv, q, hq\u27e9,\n  exact \u27e8u \u2229 v, filter.inter_mem_sets hu hv, _,\n        (hp.mono (inter_subset_left u v)).prod (hq.mono (inter_subset_right u v))\u27e9\nend\n\n\/-- The cartesian product of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.prod {n : with_top \u2115} {s : set E} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx:E, (f x, g x)) s :=\n\u03bb x hx, (hf x hx).prod (hg x hx)\n\n\/-- The cartesian product of `C^n` functions at a point is `C^n`. -\/\nlemma times_cont_diff_at.prod {n : with_top \u2115} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx:E, (f x, g x)) x :=\ntimes_cont_diff_within_at_univ.1 $ times_cont_diff_within_at.prod (times_cont_diff_within_at_univ.2 hf)\n  (times_cont_diff_within_at_univ.2 hg)\n\n\/--\nThe cartesian product of `C^n` functions is `C^n`.\n-\/\nlemma times_cont_diff.prod {n : with_top \u2115} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bbx:E, (f x, g x)) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.prod (times_cont_diff_on_univ.2 hf)\n  (times_cont_diff_on_univ.2 hg)\n\n\/-!\n### Composition of `C^n` functions\n\nWe show that the composition of `C^n` functions is `C^n`. One way to prove it would be to write\nthe `n`-th derivative of the composition (this is Fa\u00e0 di Bruno's formula) and check its continuity,\nbut this is very painful. Instead, we go for a simple inductive proof. Assume it is done for `n`.\nThen, to check it for `n+1`, one needs to check that the derivative of `g \u2218 f` is `C^n`, i.e.,\nthat `Dg(f x) \u2b1d Df(x)` is `C^n`. The term `Dg (f x)` is the composition of two `C^n` functions, so\nit is `C^n` by the inductive assumption. The term `Df(x)` is also `C^n`. Then, the matrix\nmultiplication is the application of a bilinear map (which is `C^\u221e`, and therefore `C^n`) to\n`x \u21a6 (Dg(f x), Df x)`. As the composition of two `C^n` maps, it is again `C^n`, and we are done.\n\nThere is a subtlety in this argument: we apply the inductive assumption to functions on other Banach\nspaces. In maths, one would say: prove by induction over `n` that, for all `C^n` maps between all\npairs of Banach spaces, their composition is `C^n`. In Lean, this is fine as long as the spaces\nstay in the same universe. This is not the case in the above argument: if `E` lives in universe `u`\nand `F` lives in universe `v`, then linear maps from `E` to `F` (to which the derivative of `f`\nbelongs) is in universe `max u v`. If one could quantify over finitely many universes, the above\nproof would work fine, but this is not the case. One could still write the proof considering spaces\nin any universe in `u, v, w, max u v, max v w, max u v w`, but it would be extremely tedious and\nlead to a lot of duplication. Instead, we formulate the above proof when all spaces live in the same\nuniverse (where everything is fine), and then we deduce the general result by lifting all our spaces\nto a common universe. We use the trick that any space `H` is isomorphic through a continuous linear\nequiv to `continuous_multilinear_map (\u03bb (i : fin 0), E \u00d7 F \u00d7 G) H` to change the universe level,\nand then argue that composing with such a linear equiv does not change the fact of being `C^n`,\nwhich we have already proved previously.\n-\/\n\n\/-- Auxiliary lemma proving that the composition of `C^n` functions on domains is `C^n` when all\nspaces live in the same universe. Use instead `times_cont_diff_on.comp` which removes the universe\nassumption (but is deduced from this one). -\/\nprivate lemma times_cont_diff_on.comp_same_univ\n  {Eu : Type u} [normed_group Eu] [normed_space \ud835\udd5c Eu]\n  {Fu : Type u} [normed_group Fu] [normed_space \ud835\udd5c Fu]\n  {Gu : Type u} [normed_group Gu] [normed_space \ud835\udd5c Gu]\n  {n : with_top \u2115} {s : set Eu} {t : set Fu} {g : Fu \u2192 Gu} {f : Eu \u2192 Fu}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\nbegin\n  unfreezingI { induction n using with_top.nat_induction with n IH Itop generalizing Eu Fu Gu },\n  { rw times_cont_diff_on_zero at hf hg \u22a2,\n    exact continuous_on.comp hg hf st },\n  { rw times_cont_diff_on_succ_iff_has_fderiv_within_at at hg \u22a2,\n    assume x hx,\n    rcases (times_cont_diff_on_succ_iff_has_fderiv_within_at.1 hf) x hx\n      with \u27e8u, hu, f', hf', f'_diff\u27e9,\n    rcases hg (f x) (st hx) with \u27e8v, hv, g', hg', g'_diff\u27e9,\n    rw insert_eq_of_mem hx at hu \u22a2,\n    have xu : x \u2208 u := mem_of_mem_nhds_within hx hu,\n    let w := s \u2229 (u \u2229 f\u207b\u00b9' v),\n    have wv : w \u2286 f \u207b\u00b9' v := \u03bb y hy, hy.2.2,\n    have wu : w \u2286 u := \u03bb y hy, hy.2.1,\n    have ws : w \u2286 s := \u03bb y hy, hy.1,\n    refine \u27e8w, _, \u03bb y, (g' (f y)).comp (f' y), _, _\u27e9,\n    show w \u2208 \ud835\udcdd[s] x,\n    { apply filter.inter_mem_sets self_mem_nhds_within,\n      apply filter.inter_mem_sets hu,\n      apply continuous_within_at.preimage_mem_nhds_within',\n      { rw \u2190 continuous_within_at_inter' hu,\n        exact (hf' x xu).differentiable_within_at.continuous_within_at.mono\n          (inter_subset_right _ _) },\n      { apply nhds_within_mono _ _ hv,\n        exact subset.trans (image_subset_iff.mpr st) (subset_insert (f x) t) } },\n    show \u2200 y \u2208 w,\n      has_fderiv_within_at (g \u2218 f) ((g' (f y)).comp (f' y)) w y,\n    { rintros y \u27e8ys, yu, yv\u27e9,\n      exact (hg' (f y) yv).comp y ((hf' y yu).mono wu) wv },\n    show times_cont_diff_on \ud835\udd5c n (\u03bb y, (g' (f y)).comp (f' y)) w,\n    { have A : times_cont_diff_on \ud835\udd5c n (\u03bb y, g' (f y)) w :=\n        IH g'_diff ((hf.of_le (with_top.coe_le_coe.2 (nat.le_succ n))).mono ws) wv,\n      have B : times_cont_diff_on \ud835\udd5c n f' w := f'_diff.mono wu,\n      have C : times_cont_diff_on \ud835\udd5c n (\u03bb y, (f' y, g' (f y))) w :=\n        times_cont_diff_on.prod B A,\n      have D : times_cont_diff_on \ud835\udd5c n (\u03bb(p : (Eu \u2192L[\ud835\udd5c] Fu) \u00d7 (Fu \u2192L[\ud835\udd5c] Gu)), p.2.comp p.1) univ :=\n        is_bounded_bilinear_map_comp.times_cont_diff.times_cont_diff_on,\n      exact IH D C (subset_univ _) } },\n  { rw times_cont_diff_on_top at hf hg \u22a2,\n    assume n,\n    apply Itop n (hg n) (hf n) st }\nend\n\n\/-- The composition of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.comp\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\nbegin\n  \/- we lift all the spaces to a common universe, as we have already proved the result in this\n  situation. For the lift, we use the trick that `H` is isomorphic through a\n  continuous linear equiv to `continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) H`, and\n  continuous linear equivs respect smoothness classes. -\/\n  let Eu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) E,\n  letI : normed_group Eu := by apply_instance,\n  letI : normed_space \ud835\udd5c Eu := by apply_instance,\n  let Fu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) F,\n  letI : normed_group Fu := by apply_instance,\n  letI : normed_space \ud835\udd5c Fu := by apply_instance,\n  let Gu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) G,\n  letI : normed_group Gu := by apply_instance,\n  letI : normed_space \ud835\udd5c Gu := by apply_instance,\n  -- declare the isomorphisms\n  let isoE : Eu \u2243L[\ud835\udd5c] E := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) E,\n  let isoF : Fu \u2243L[\ud835\udd5c] F := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) F,\n  let isoG : Gu \u2243L[\ud835\udd5c] G := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) G,\n  -- lift the functions to the new spaces, check smoothness there, and then go back.\n  let fu : Eu \u2192 Fu := (isoF.symm \u2218 f) \u2218 isoE,\n  have fu_diff : times_cont_diff_on \ud835\udd5c n fu (isoE \u207b\u00b9' s),\n    by rwa [isoE.times_cont_diff_on_comp_iff, isoF.symm.comp_times_cont_diff_on_iff],\n  let gu : Fu \u2192 Gu := (isoG.symm \u2218 g) \u2218 isoF,\n  have gu_diff : times_cont_diff_on \ud835\udd5c n gu (isoF \u207b\u00b9' t),\n    by rwa [isoF.times_cont_diff_on_comp_iff, isoG.symm.comp_times_cont_diff_on_iff],\n  have main : times_cont_diff_on \ud835\udd5c n (gu \u2218 fu) (isoE \u207b\u00b9' s),\n  { apply times_cont_diff_on.comp_same_univ gu_diff fu_diff,\n    assume y hy,\n    simp only [fu, continuous_linear_equiv.coe_apply, function.comp_app, mem_preimage],\n    rw isoF.apply_symm_apply (f (isoE y)),\n    exact st hy },\n  have : gu \u2218 fu = (isoG.symm \u2218 (g \u2218 f)) \u2218 isoE,\n  { ext y,\n    simp only [function.comp_apply, gu, fu],\n    rw isoF.apply_symm_apply (f (isoE y)) },\n  rwa [this, isoE.times_cont_diff_on_comp_iff, isoG.symm.comp_times_cont_diff_on_iff] at main\nend\n\n\/-- The composition of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.comp'\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) (s \u2229 f\u207b\u00b9' t) :=\nhg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)\n\n\/-- The composition of a `C^n` function on a domain with a `C^n` function is `C^n`. -\/\nlemma times_cont_diff.comp_times_cont_diff_on {n : with_top \u2115} {s : set E} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff \ud835\udd5c n g) (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\n(times_cont_diff_on_univ.2 hg).comp hf subset_preimage_univ\n\n\/-- The composition of `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.comp {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff \ud835\udd5c n g) (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (g \u2218 f) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.comp (times_cont_diff_on_univ.2 hg)\n  (times_cont_diff_on_univ.2 hf) (subset_univ _)\n\n\/-- The composition of `C^n` functions at points in domains is `C^n`. -\/\nlemma times_cont_diff_within_at.comp\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff_within_at \ud835\udd5c n g t (f x))\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nbegin\n  assume m hm,\n  rcases hg.times_cont_diff_on hm with \u27e8u, u_nhd, ut, hu\u27e9,\n  rcases hf.times_cont_diff_on hm with \u27e8v, v_nhd, vs, hv\u27e9,\n  have xmem : x \u2208 f \u207b\u00b9' u \u2229 v :=\n    \u27e8(mem_of_mem_nhds_within (mem_insert (f x) _) u_nhd : _),\n    mem_of_mem_nhds_within (mem_insert x s) v_nhd\u27e9,\n  have : f \u207b\u00b9' u \u2208 \ud835\udcdd[insert x s] x,\n  { apply hf.continuous_within_at.insert_self.preimage_mem_nhds_within',\n    apply nhds_within_mono _ _ u_nhd,\n    rw image_insert_eq,\n    exact insert_subset_insert (image_subset_iff.mpr st) },\n  have Z := ((hu.comp (hv.mono (inter_subset_right (f \u207b\u00b9' u) v)) (inter_subset_left _ _))\n    .times_cont_diff_within_at) xmem m (le_refl _),\n  have : \ud835\udcdd[f \u207b\u00b9' u \u2229 v] x = \ud835\udcdd[insert x s] x,\n  { have A : f \u207b\u00b9' u \u2229 v = (insert x s) \u2229 (f \u207b\u00b9' u \u2229 v),\n    { apply subset.antisymm _ (inter_subset_right _ _),\n      rintros y \u27e8hy1, hy2\u27e9,\n      simp [hy1, hy2, vs hy2] },\n    rw [A, \u2190 nhds_within_restrict''],\n    exact filter.inter_mem_sets this v_nhd },\n  rwa [insert_eq_of_mem xmem, this] at Z,\nend\n\n\/-- The composition of `C^n` functions at points in domains is `C^n`. -\/\nlemma times_cont_diff_within_at.comp' {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G}\n  {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff_within_at \ud835\udd5c n g t (f x)) (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) (s \u2229 f\u207b\u00b9' t) x :=\nhg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)\n\nlemma times_cont_diff_at.comp_times_cont_diff_within_at {n} (x : E)\n  (hg : times_cont_diff_at \ud835\udd5c n g (f x)) (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nhg.comp x hf (maps_to_univ _ _)\n\n\/-- The composition of `C^n` functions at points is `C^n`. -\/\nlemma times_cont_diff_at.comp {n : with_top \u2115} (x : E)\n  (hg : times_cont_diff_at \ud835\udd5c n g (f x))\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\nhg.comp x hf subset_preimage_univ\n\nlemma times_cont_diff.comp_times_cont_diff_within_at\n  {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F} (h : times_cont_diff \ud835\udd5c n g)\n  (hf : times_cont_diff_within_at \ud835\udd5c n f t x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) t x :=\nbegin\n  have : times_cont_diff_within_at \ud835\udd5c n g univ (f x) :=\n    h.times_cont_diff_at.times_cont_diff_within_at,\n  exact this.comp x hf (subset_univ _),\nend\n\nlemma times_cont_diff.comp_times_cont_diff_at\n  {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff \ud835\udd5c n g)\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\nhg.comp_times_cont_diff_within_at hf\n\n\/-- The bundled derivative of a `C^{n+1}` function is `C^n`. -\/\nlemma times_cont_diff_on_fderiv_within_apply {m n : with_top  \u2115} {s : set E}\n  {f : E \u2192 F} (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1 : E \u2192L[\ud835\udd5c] F) p.2)\n  (set.prod s (univ : set E)) :=\nbegin\n  have A : times_cont_diff \ud835\udd5c m (\u03bbp : (E \u2192L[\ud835\udd5c] F) \u00d7 E, p.1 p.2),\n  { apply is_bounded_bilinear_map.times_cont_diff,\n    exact is_bounded_bilinear_map_apply },\n  have B : times_cont_diff_on \ud835\udd5c m\n    (\u03bb (p : E \u00d7 E), ((fderiv_within \ud835\udd5c f s p.fst), p.snd)) (set.prod s univ),\n  { apply times_cont_diff_on.prod _ _,\n    { have I : times_cont_diff_on \ud835\udd5c m (\u03bb (x : E), fderiv_within \ud835\udd5c f s x) s :=\n        hf.fderiv_within hs hmn,\n      have J : times_cont_diff_on \ud835\udd5c m (\u03bb (x : E \u00d7 E), x.1) (set.prod s univ) :=\n        times_cont_diff_fst.times_cont_diff_on,\n      exact times_cont_diff_on.comp I J (prod_subset_preimage_fst _ _) },\n    { apply times_cont_diff.times_cont_diff_on _ ,\n      apply is_bounded_linear_map.snd.times_cont_diff } },\n  exact A.comp_times_cont_diff_on B\nend\n\n\/-- The bundled derivative of a `C^{n+1}` function is `C^n`. -\/\nlemma times_cont_diff.times_cont_diff_fderiv_apply {n m : with_top \u2115} {f : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hmn : m + 1 \u2264 n) :\n  times_cont_diff \ud835\udd5c m (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1 : E \u2192L[\ud835\udd5c] F) p.2) :=\nbegin\n  rw \u2190 times_cont_diff_on_univ at \u22a2 hf,\n  rw [\u2190 fderiv_within_univ, \u2190 univ_prod_univ],\n  exact times_cont_diff_on_fderiv_within_apply hf unique_diff_on_univ hmn\nend\n\n\/-! ### Sum of two functions -\/\n\n\/- The sum is smooth. -\/\nlemma times_cont_diff_add {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bbp : F \u00d7 F, p.1 + p.2) :=\n(is_bounded_linear_map.fst.add is_bounded_linear_map.snd).times_cont_diff\n\n\/-- The sum of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.add {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx, f x + g x) s x :=\ntimes_cont_diff_add.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The sum of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.add {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx, f x + g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.add hg\n\n\/-- The sum of two `C^n`functions is `C^n`. -\/\nlemma times_cont_diff.add {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bbx, f x + g x) :=\ntimes_cont_diff_add.comp (hf.prod hg)\n\n\/-- The sum of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.add {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx, f x + g x) s :=\n\u03bb x hx, (hf x hx).add (hg x hx)\n\n\/-! ### Negative -\/\n\n\/- The negative is smooth. -\/\nlemma times_cont_diff_neg {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bbp : F, -p) :=\nis_bounded_linear_map.id.neg.times_cont_diff\n\n\/-- The negative of a `C^n` function within a domain at a point is `C^n` within this domain at\nthis point. -\/\nlemma times_cont_diff_within_at.neg {n : with_top \u2115} {s : set E} {f : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) : times_cont_diff_within_at \ud835\udd5c n (\u03bbx, -f x) s x :=\ntimes_cont_diff_neg.times_cont_diff_within_at.comp x hf subset_preimage_univ\n\n\/-- The negative of a `C^n` function at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.neg {n : with_top \u2115} {f : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) : times_cont_diff_at \ud835\udd5c n (\u03bbx, -f x) x :=\nby rw \u2190 times_cont_diff_within_at_univ at *; exact hf.neg\n\n\/-- The negative of a `C^n`function is `C^n`. -\/\nlemma times_cont_diff.neg {n : with_top \u2115} {f : E \u2192 F} (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (\u03bbx, -f x) :=\ntimes_cont_diff_neg.comp hf\n\n\/-- The negative of a `C^n` function on a domain is `C^n`. -\/\nlemma times_cont_diff_on.neg {n : with_top \u2115} {s : set E} {f : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) : times_cont_diff_on \ud835\udd5c n (\u03bbx, -f x) s :=\n\u03bb x hx, (hf x hx).neg\n\n\/-! ### Subtraction -\/\n\n\/-- The difference of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.sub {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx, f x - g x) s x :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.sub {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx, f x - g x) x :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.sub {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx, f x - g x) s :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.sub {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) : times_cont_diff \ud835\udd5c n (\u03bbx, f x - g x) :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-! ### Sum of finitely many functions -\/\n\nlemma times_cont_diff_within_at.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {t : set E} {x : E}\n  (h : \u2200 i \u2208 s, times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f i x) t x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) t x :=\nbegin\n  classical,\n  induction s using finset.induction_on with i s is IH,\n  { simp [times_cont_diff_within_at_const] },\n  { simp only [is, finset.sum_insert, not_false_iff],\n    exact (h _ (finset.mem_insert_self i s)).add (IH (\u03bb j hj, h _ (finset.mem_insert_of_mem hj))) }\nend\n\nlemma times_cont_diff_at.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {x : E}\n  (h : \u2200 i \u2208 s, times_cont_diff_at \ud835\udd5c n (\u03bb x, f i x) x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact times_cont_diff_within_at.sum h\n\nlemma times_cont_diff_on.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {t : set E}\n  (h : \u2200 i \u2208 s, times_cont_diff_on \ud835\udd5c n (\u03bb x, f i x) t) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) t :=\n\u03bb x hx, times_cont_diff_within_at.sum (\u03bb i hi, h i hi x hx)\n\nlemma times_cont_diff.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115}\n  (h : \u2200 i \u2208 s, times_cont_diff \ud835\udd5c n (\u03bb x, f i x)) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) :=\nby simp [\u2190 times_cont_diff_on_univ] at *; exact times_cont_diff_on.sum h\n\n\/-! ### Product of two functions -\/\n\n\/- The product is smooth. -\/\nlemma times_cont_diff_mul {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb p : \ud835\udd5c \u00d7 \ud835\udd5c, p.1 * p.2) :=\nis_bounded_bilinear_map_mul.times_cont_diff\n\n\/-- The product of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.mul {n : with_top \u2115} {s : set E} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x * g x) s x :=\ntimes_cont_diff_mul.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The product of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.mul {n : with_top \u2115} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x * g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.mul hg\n\n\/-- The product of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.mul {n : with_top \u2115} {s : set E} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x * g x) s :=\n\u03bb x hx, (hf x hx).mul (hg x hx)\n\n\/-- The product of two `C^n`functions is `C^n`. -\/\nlemma times_cont_diff.mul {n : with_top \u2115} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x * g x) :=\ntimes_cont_diff_mul.comp (hf.prod hg)\n\nlemma times_cont_diff_within_at.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \/ c) s x :=\nhf.mul times_cont_diff_within_at_const\n\nlemma times_cont_diff_at.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \/ c) x :=\nhf.mul times_cont_diff_at_const\n\nlemma times_cont_diff_on.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x \/ c) s :=\nhf.mul times_cont_diff_on_const\n\nlemma times_cont_diff.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \/ c) :=\nhf.mul times_cont_diff_const\n\nlemma times_cont_diff.pow {n : with_top \u2115} {f : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff \ud835\udd5c n f) :\n  \u2200 m : \u2115, times_cont_diff \ud835\udd5c n (\u03bb x, (f x) ^ m)\n| 0 := by simpa using times_cont_diff_const\n| (m + 1) := hf.mul (times_cont_diff.pow m)\n\n\/-! ### Scalar multiplication -\/\n\n\/- The scalar multiplication is smooth. -\/\nlemma times_cont_diff_smul {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb p : \ud835\udd5c \u00d7 F, p.1 \u2022 p.2) :=\nis_bounded_bilinear_map_smul.times_cont_diff\n\n\/-- The scalar multiplication of two `C^n` functions within a set at a point is `C^n` within this\nset at this point. -\/\nlemma times_cont_diff_within_at.smul {n : with_top \u2115} {s : set E} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \u2022 g x) s x :=\ntimes_cont_diff_smul.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The scalar multiplication of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.smul {n : with_top \u2115} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \u2022 g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.smul hg\n\n\/-- The scalar multiplication of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.smul {n : with_top \u2115} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \u2022 g x) :=\ntimes_cont_diff_smul.comp (hf.prod hg)\n\n\/-- The scalar multiplication of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.smul {n : with_top \u2115} {s : set E} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x \u2022 g x) s :=\n\u03bb x hx, (hf x hx).smul (hg x hx)\n\n\/-! ### Cartesian product of two functions-\/\n\nsection prod_map\nvariables {E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n{F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n{n : with_top \u2115}\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_within_at.prod_map'\n  {s : set E} {t : set E'} {f : E \u2192 F} {g : E' \u2192 F'} {p : E \u00d7 E'}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s p.1) (hg : times_cont_diff_within_at \ud835\udd5c n g t p.2) :\n  times_cont_diff_within_at \ud835\udd5c n (prod.map f g) (set.prod s t) p :=\n(hf.comp p times_cont_diff_within_at_fst (prod_subset_preimage_fst _ _)).prod\n  (hg.comp p times_cont_diff_within_at_snd (prod_subset_preimage_snd _ _))\n\nlemma times_cont_diff_within_at.prod_map\n  {s : set E} {t : set E'} {f : E \u2192 F} {g : E' \u2192 F'} {x : E} {y : E'}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g t y) :\n  times_cont_diff_within_at \ud835\udd5c n (prod.map f g) (set.prod s t) (x, y) :=\ntimes_cont_diff_within_at.prod_map' hf hg\n\n\/-- The product map of two `C^n` functions on a set is `C^n` on the product set. -\/\nlemma times_cont_diff_on.prod_map {E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n  {F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n  {s : set E} {t : set E'} {n : with_top \u2115} {f : E \u2192 F} {g : E' \u2192 F'}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g t) :\n  times_cont_diff_on \ud835\udd5c n (prod.map f g) (set.prod s t) :=\n(hf.comp times_cont_diff_on_fst (prod_subset_preimage_fst _ _)).prod\n  (hg.comp (times_cont_diff_on_snd) (prod_subset_preimage_snd _ _))\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_at.prod_map {f : E \u2192 F} {g : E' \u2192 F'} {x : E} {y : E'}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g y) :\n  times_cont_diff_at \ud835\udd5c n (prod.map f g) (x, y) :=\nbegin\n  rw times_cont_diff_at at *,\n  convert hf.prod_map hg,\n  simp only [univ_prod_univ]\nend\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_at.prod_map' {f : E \u2192 F} {g : E' \u2192 F'} {p : E \u00d7 E'}\n  (hf : times_cont_diff_at \ud835\udd5c n f p.1) (hg : times_cont_diff_at \ud835\udd5c n g p.2) :\n  times_cont_diff_at \ud835\udd5c n (prod.map f g) p :=\nbegin\n  rcases p,\n  exact times_cont_diff_at.prod_map hf hg\nend\n\n\/-- The product map of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.prod_map\n  {f : E \u2192 F} {g : E' \u2192 F'}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (prod.map f g) :=\nbegin\n  rw times_cont_diff_iff_times_cont_diff_at at *,\n  exact \u03bb \u27e8x, y\u27e9, (hf x).prod_map (hg y)\nend\n\nend prod_map\n\n\/-! ### Inversion in a complete normed algebra -\/\n\nsection algebra_inverse\nvariables (\ud835\udd5c) {R : Type*} [normed_ring R] [normed_algebra \ud835\udd5c R]\nopen normed_ring continuous_linear_map ring\n\n\/-- In a complete normed algebra, the operation of inversion is `C^n`, for all `n`, at each\ninvertible element.  The proof is by induction, bootstrapping using an identity expressing the\nderivative of inversion as a bilinear map of inversion itself. -\/\nlemma times_cont_diff_at_ring_inverse [complete_space R] {n : with_top \u2115} (x : units R) :\n  times_cont_diff_at \ud835\udd5c n ring.inverse (x : R) :=\nbegin\n  induction n using with_top.nat_induction with n IH Itop,\n  { intros m hm,\n    refine \u27e8{y : R | is_unit y}, _, _\u27e9,\n    { simp [nhds_within_univ],\n      exact x.nhds },\n    { use (ftaylor_series_within \ud835\udd5c inverse univ),\n      rw [le_antisymm hm bot_le, has_ftaylor_series_up_to_on_zero_iff],\n      split,\n      { rintros _ \u27e8x', hx'\u27e9,\n        rw \u2190 hx',\n        exact (inverse_continuous_at x').continuous_within_at },\n      { simp [ftaylor_series_within] } } },\n  { apply times_cont_diff_at_succ_iff_has_fderiv_at.mpr,\n    refine \u27e8\u03bb (x : R), - lmul_left_right \ud835\udd5c R (inverse x, inverse x), _, _\u27e9,\n    { refine \u27e8{y : R | is_unit y}, x.nhds, _\u27e9,\n      intros y hy,\n      cases mem_set_of_eq.mp hy with y' hy',\n      rw [\u2190 hy', inverse_unit],\n      exact @has_fderiv_at_ring_inverse \ud835\udd5c _ _ _ _ _ y' },\n    { exact (lmul_left_right_is_bounded_bilinear \ud835\udd5c R).times_cont_diff.neg.comp_times_cont_diff_at\n        (x : R) (IH.prod IH) } },\n  { exact times_cont_diff_at_top.mpr Itop }\nend\n\nvariables (\ud835\udd5c) {\ud835\udd5c' : Type*} [normed_field \ud835\udd5c'] [normed_algebra \ud835\udd5c \ud835\udd5c'] [complete_space \ud835\udd5c']\n\nlemma times_cont_diff_at_inv {x : \ud835\udd5c'} (hx : x \u2260 0) {n} :\n  times_cont_diff_at \ud835\udd5c n has_inv.inv x :=\nby simpa only [inverse_eq_has_inv] using times_cont_diff_at_ring_inverse \ud835\udd5c (units.mk0 x hx)\n\nlemma times_cont_diff_on_inv {n} : times_cont_diff_on \ud835\udd5c n (has_inv.inv : \ud835\udd5c' \u2192 \ud835\udd5c') {0}\u1d9c :=\n\u03bb x hx, (times_cont_diff_at_inv \ud835\udd5c hx).times_cont_diff_within_at\n\nvariable {\ud835\udd5c}\n\n-- TODO: the next few lemmas don't need `\ud835\udd5c` or `\ud835\udd5c'` to be complete\n-- A good way to show this is to generalize `times_cont_diff_at_ring_inverse` to the setting\n-- of a function `f` such that `\u2200\u1da0 x in \ud835\udcdd a, x * f x = 1`.\n\nlemma times_cont_diff_within_at.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff_within_at \ud835\udd5c n f s x)\n  (hx : f x \u2260 0) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) s x :=\n(times_cont_diff_at_inv \ud835\udd5c hx).comp_times_cont_diff_within_at x hf\n\nlemma times_cont_diff_at.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff_at \ud835\udd5c n f x) (hx : f x \u2260 0) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) x :=\nhf.inv hx\n\n-- TODO: generalize to `f g : E \u2192 \ud835\udd5c'`\nlemma times_cont_diff_within_at.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x)\n  (hx : g x \u2260 0) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \/ g x) s x :=\nhf.mul (hg.inv hx)\n\nlemma times_cont_diff_at.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x)\n  (hx : g x \u2260 0) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \/ g x) x :=\nhf.div hg hx\n\nlemma times_cont_diff.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g)\n  (h0 : \u2200 x, g x \u2260 0) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \/ g x) :=\nbegin\n  simp only [times_cont_diff_iff_times_cont_diff_at] at *,\n  exact \u03bb x, (hf x).div (hg x) (h0 x)\nend\n\nend algebra_inverse\n\n\/-! ### Inversion of continuous linear maps between Banach spaces -\/\n\nsection map_inverse\nopen continuous_linear_map\n\n\/-- At a continuous linear equivalence `e : E \u2243L[\ud835\udd5c] F` between Banach spaces, the operation of\ninversion is `C^n`, for all `n`. -\/\nlemma times_cont_diff_at_map_inverse [complete_space E] {n : with_top \u2115} (e : E \u2243L[\ud835\udd5c] F) :\n  times_cont_diff_at \ud835\udd5c n inverse (e : E \u2192L[\ud835\udd5c] F) :=\nbegin\n  nontriviality E,\n  -- first, we use the lemma `to_ring_inverse` to rewrite in terms of `ring.inverse` in the ring\n  -- `E \u2192L[\ud835\udd5c] E`\n  let O\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 (F \u2192L[\ud835\udd5c] E) := \u03bb f, f.comp (e.symm : (F \u2192L[\ud835\udd5c] E)),\n  let O\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 (E \u2192L[\ud835\udd5c] E) := \u03bb f, (e.symm : (F \u2192L[\ud835\udd5c] E)).comp f,\n  have : continuous_linear_map.inverse = O\u2081 \u2218 ring.inverse \u2218 O\u2082 :=\n    funext (to_ring_inverse e),\n  rw this,\n  -- `O\u2081` and `O\u2082` are `times_cont_diff`, so we reduce to proving that `ring.inverse` is `times_cont_diff`\n  have h\u2081 : times_cont_diff \ud835\udd5c n O\u2081,\n    from is_bounded_bilinear_map_comp.times_cont_diff.comp\n      (times_cont_diff_const.prod times_cont_diff_id),\n  have h\u2082 : times_cont_diff \ud835\udd5c n O\u2082,\n    from is_bounded_bilinear_map_comp.times_cont_diff.comp\n      (times_cont_diff_id.prod times_cont_diff_const),\n  refine h\u2081.times_cont_diff_at.comp _ (times_cont_diff_at.comp _ _ h\u2082.times_cont_diff_at),\n  convert times_cont_diff_at_ring_inverse \ud835\udd5c (1 : units (E \u2192L[\ud835\udd5c] E)),\n  simp [O\u2082, one_def]\nend\n\nend map_inverse\n\nsection function_inverse\nopen continuous_linear_map\n\n\/-- If `f` is a local homeomorphism and the point `a` is in its target, and if `f` is `n` times\ncontinuously differentiable at `f.symm a`, and if the derivative at `f.symm a` is a continuous linear\nequivalence, then `f.symm` is `n` times continuously differentiable at the point `a`.\n\nThis is one of the easy parts of the inverse function theorem: it assumes that we already have\nan inverse function. -\/\ntheorem local_homeomorph.times_cont_diff_at_symm [complete_space E] {n : with_top \u2115}\n  (f : local_homeomorph E F) {f\u2080' : E \u2243L[\ud835\udd5c] F} {a : F} (ha : a \u2208 f.target)\n  (hf\u2080' : has_fderiv_at f (f\u2080' : E \u2192L[\ud835\udd5c] F) (f.symm a)) (hf : times_cont_diff_at \ud835\udd5c n f (f.symm a)) :\n  times_cont_diff_at \ud835\udd5c n f.symm a :=\nbegin\n  -- We prove this by induction on `n`\n  induction n using with_top.nat_induction with n IH Itop,\n  { rw times_cont_diff_at_zero,\n    exact \u27e8f.target, mem_nhds_sets f.open_target ha, f.continuous_inv_fun\u27e9 },\n  { obtain \u27e8f', \u27e8u, hu, hff'\u27e9, hf'\u27e9 := times_cont_diff_at_succ_iff_has_fderiv_at.mp hf,\n    apply times_cont_diff_at_succ_iff_has_fderiv_at.mpr,\n    -- For showing `n.succ` times continuous differentiability (the main inductive step), it\n    -- suffices to produce the derivative and show that it is `n` times continuously differentiable\n    have eq_f\u2080' : f' (f.symm a) = f\u2080',\n    { exact has_fderiv_at_unique (hff' (f.symm a) (mem_of_nhds hu)) hf\u2080' },\n    -- This follows by a bootstrapping formula expressing the derivative as a function of `f` itself\n    refine \u27e8inverse \u2218 f' \u2218 f.symm, _, _\u27e9,\n    { -- We first check that the derivative of `f` is that formula\n      have h_nhds : {y : E | \u2203 (e : E \u2243L[\ud835\udd5c] F), \u2191e = f' y} \u2208 \ud835\udcdd ((f.symm) a),\n      { have hf\u2080' := f\u2080'.nhds,\n        rw \u2190 eq_f\u2080' at hf\u2080',\n        exact hf'.continuous_at.preimage_mem_nhds hf\u2080' },\n      obtain \u27e8t, htu, ht, htf\u27e9 := mem_nhds_sets_iff.mp (filter.inter_mem_sets hu h_nhds),\n      use f.target \u2229 (f.symm) \u207b\u00b9' t,\n      refine \u27e8mem_nhds_sets _ _, _\u27e9,\n      { exact f.preimage_open_of_open_symm ht },\n      { exact mem_inter ha (mem_preimage.mpr htf) },\n      intros x hx,\n      obtain \u27e8hxu, e, he\u27e9 := htu hx.2,\n      have h_deriv : has_fderiv_at f \u2191e ((f.symm) x),\n      { rw he,\n        exact hff' (f.symm x) hxu },\n      convert f.has_fderiv_at_symm hx.1 h_deriv,\n      simp [\u2190 he] },\n    { -- Then we check that the formula, being a composition of `times_cont_diff` pieces, is\n      -- itself `times_cont_diff`\n      have h_deriv\u2081 : times_cont_diff_at \ud835\udd5c n inverse (f' (f.symm a)),\n      { rw eq_f\u2080',\n        exact times_cont_diff_at_map_inverse _ },\n      have h_deriv\u2082 : times_cont_diff_at \ud835\udd5c n f.symm a,\n      { refine IH (hf.of_le _),\n        norm_cast,\n        exact nat.le_succ n },\n      exact (h_deriv\u2081.comp _ hf').comp _ h_deriv\u2082 } },\n  { refine times_cont_diff_at_top.mpr _,\n    intros n,\n    exact Itop n (times_cont_diff_at_top.mp hf n) }\nend\n\nend function_inverse\n\nsection real\n\/-!\n### Results over `\u211d`\n  The results in this section rely on the Mean Value Theorem, and therefore hold only over `\u211d` (and\n  its extension fields such as `\u2102`).\n-\/\n\nvariables\n{E' : Type*} [normed_group E'] [normed_space \u211d E']\n{F' : Type*} [normed_group F'] [normed_space \u211d F']\n\n\/-- If a function has a Taylor series at order at least 1, then at points in the interior of the\n    domain of definition, the term of order 1 of this series is a strict derivative of `f`. -\/\nlemma has_ftaylor_series_up_to_on.has_strict_fderiv_at\n  {s : set E'} {f : E' \u2192 F'} {x : E'} {p : E' \u2192 formal_multilinear_series \u211d E' F'} {n : with_top \u2115}\n  (hf : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hs : s \u2208 \ud835\udcdd x) :\n  has_strict_fderiv_at f ((continuous_multilinear_curry_fin1 \u211d E' F') (p x 1)) x :=\nbegin\n  let f' := \u03bb x, (continuous_multilinear_curry_fin1 \u211d E' F') (p x 1),\n  have hf' : \u2200 x, x \u2208 s \u2192 has_fderiv_within_at f (f' x) s x :=\n    \u03bb x, has_ftaylor_series_up_to_on.has_fderiv_within_at hf hn,\n  have hcont : continuous_on f' s :=\n    (continuous_multilinear_curry_fin1 \u211d E' F').continuous.comp_continuous_on (hf.cont 1 hn),\n  exact strict_fderiv_of_cont_diff hf' hcont hs,\nend\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, then the derivative of `f` at this point\nis also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_fderiv_at {f : E' \u2192 F'} {x : E'} {n : with_top \u2115}\n  (hf : times_cont_diff_at \u211d n f x) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f (fderiv \u211d f x) x :=\nbegin\n  rcases hf 1 hn with \u27e8u, H, p, hp\u27e9,\n  simp only [nhds_within_univ, mem_univ, insert_eq_of_mem] at H,\n  have := hp.has_strict_fderiv_at (by norm_num) H,\n  convert this,\n  exact this.has_fderiv_at.fderiv\nend\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, and its derivative at that point is given to\nus as `f'`, then `f'` is also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_fderiv_at'\n  {f : E' \u2192 F'} {f' : E' \u2192L[\u211d] F'} {x : E'}\n  {n : with_top \u2115} (hf : times_cont_diff_at \u211d n f x) (hf' : has_fderiv_at f f' x) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f f' x :=\nby simpa only [hf'.fderiv] using hf.has_strict_fderiv_at hn\n\n\/-- If a function is `C^n` with `1 \u2264 n`, then the derivative of `f` is also a strict derivative. -\/\nlemma times_cont_diff.has_strict_fderiv_at\n  {f : E' \u2192 F'} {x : E'} {n : with_top \u2115} (hf : times_cont_diff \u211d n f) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f (fderiv \u211d f x) x :=\nhf.times_cont_diff_at.has_strict_fderiv_at hn\n\nend real\n\nsection deriv\n\/-!\n### One dimension\n\nAll results up to now have been expressed in terms of the general Fr\u00e9chet derivative `fderiv`. For\nmaps defined on the field, the one-dimensional derivative `deriv` is often easier to use. In this\nparagraph, we reformulate some higher smoothness results in terms of `deriv`.\n-\/\n\nvariables {f\u2082 : \ud835\udd5c \u2192 F} {s\u2082 : set \ud835\udd5c}\nopen continuous_linear_map (smul_right)\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative (formulated with `deriv_within`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_deriv_within {n : \u2115} (hs : unique_diff_on \ud835\udd5c s\u2082) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c n (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_succ_iff_fderiv_within hs,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  split,\n  { assume h,\n    have : deriv_within f\u2082 s\u2082 = (\u03bb u : \ud835\udd5c \u2192L[\ud835\udd5c] F, u 1) \u2218 (fderiv_within \ud835\udd5c f\u2082 s\u2082),\n      by { ext x, refl },\n    simp only [this],\n    apply times_cont_diff.comp_times_cont_diff_on _ h,\n    exact (is_bounded_bilinear_map_apply.is_bounded_linear_map_left _).times_cont_diff },\n  { assume h,\n    have : fderiv_within \ud835\udd5c f\u2082 s\u2082 = (\u03bb u, smul_right 1 u) \u2218 (\u03bb x, deriv_within f\u2082 s\u2082 x),\n      by { ext x, simp [deriv_within] },\n    simp only [this],\n    apply times_cont_diff.comp_times_cont_diff_on _ h,\n    exact (is_bounded_bilinear_map_smul_right.is_bounded_linear_map_right _).times_cont_diff }\nend\n\n\/-- A function is `C^(n + 1)` on an open domain if and only if it is\ndifferentiable there, and its derivative (formulated with `deriv`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_deriv_of_open {n : \u2115} (hs : is_open s\u2082) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c n (deriv f\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_succ_iff_deriv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact deriv_within_of_open hs hx\nend\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative (formulated with `deriv_within`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_deriv_within (hs : unique_diff_on \ud835\udd5c s\u2082) :\n  times_cont_diff_on \ud835\udd5c \u221e f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c \u221e (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  split,\n  { assume h,\n    refine \u27e8h.differentiable_on le_top, _\u27e9,\n    apply times_cont_diff_on_top.2 (\u03bb n, ((times_cont_diff_on_succ_iff_deriv_within hs).1 _).2),\n    exact h.of_le le_top },\n  { assume h,\n    refine times_cont_diff_on_top.2 (\u03bb n, _),\n    have A : (n : with_top \u2115) \u2264 \u221e := le_top,\n    apply ((times_cont_diff_on_succ_iff_deriv_within hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le,\n    exact with_top.coe_le_coe.2 (nat.le_succ n) }\nend\n\n\/-- A function is `C^\u221e` on an open domain if and only if it is differentiable\nthere, and its derivative (formulated with `deriv`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_deriv_of_open (hs : is_open s\u2082) :\n  times_cont_diff_on \ud835\udd5c \u221e f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c \u221e (deriv f\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_top_iff_deriv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact deriv_within_of_open hs hx\nend\n\nlemma times_cont_diff_on.deriv_within {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : unique_diff_on \ud835\udd5c s\u2082) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  cases m,\n  { change \u221e + 1 \u2264 n at hmn,\n    have : n = \u221e, by simpa using hmn,\n    rw this at hf,\n    exact ((times_cont_diff_on_top_iff_deriv_within hs).1 hf).2 },\n  { change (m.succ : with_top \u2115) \u2264 n at hmn,\n    exact ((times_cont_diff_on_succ_iff_deriv_within hs).1 (hf.of_le hmn)).2 }\nend\n\nlemma times_cont_diff_on.deriv_of_open {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : is_open s\u2082) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (deriv f\u2082) s\u2082 :=\n(hf.deriv_within hs.unique_diff_on hmn).congr (\u03bb x hx, (deriv_within_of_open hs hx).symm)\n\nlemma times_cont_diff_on.continuous_on_deriv_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : unique_diff_on \ud835\udd5c s\u2082) (hn : 1 \u2264 n) :\n  continuous_on (deriv_within f\u2082 s\u2082) s\u2082 :=\n((times_cont_diff_on_succ_iff_deriv_within hs).1 (h.of_le hn)).2.continuous_on\n\nlemma times_cont_diff_on.continuous_on_deriv_of_open {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : is_open s\u2082) (hn : 1 \u2264 n) :\n  continuous_on (deriv f\u2082) s\u2082 :=\n((times_cont_diff_on_succ_iff_deriv_of_open hs).1 (h.of_le hn)).2.continuous_on\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative is `C^n`. -\/\ntheorem times_cont_diff_succ_iff_deriv {n : \u2115} :\n  times_cont_diff \ud835\udd5c ((n + 1) : \u2115) f\u2082 \u2194\n    differentiable \ud835\udd5c f\u2082 \u2227 times_cont_diff \ud835\udd5c n (deriv f\u2082) :=\nby simp only [\u2190 times_cont_diff_on_univ, times_cont_diff_on_succ_iff_deriv_of_open, is_open_univ,\n  differentiable_on_univ]\n\nend deriv\n\nsection restrict_scalars\n\/-!\n### Restricting from `\u2102` to `\u211d`, or generally from `\ud835\udd5c'` to `\ud835\udd5c`\n\nIf a function is `n` times continuously differentiable over `\u2102`, then it is `n` times continuously\ndifferentiable over `\u211d`. In this paragraph, we give variants of this statement, in the general\nsituation where `\u2102` and `\u211d` are replaced respectively by `\ud835\udd5c'` and `\ud835\udd5c` where `\ud835\udd5c'` is a normed algebra\nover `\ud835\udd5c`.\n-\/\n\nvariables (\ud835\udd5c) {\ud835\udd5c' : Type*} [nondiscrete_normed_field \ud835\udd5c'] [normed_algebra \ud835\udd5c \ud835\udd5c']\nvariables [normed_space \ud835\udd5c' E] [is_scalar_tower \ud835\udd5c \ud835\udd5c' E]\nvariables [normed_space \ud835\udd5c' F] [is_scalar_tower \ud835\udd5c \ud835\udd5c' F]\nvariables {p' : E \u2192 formal_multilinear_series \ud835\udd5c' E F} {n : with_top \u2115}\n\nlemma has_ftaylor_series_up_to_on.restrict_scalars\n  (h : has_ftaylor_series_up_to_on n f p' s) :\n  has_ftaylor_series_up_to_on n f (\u03bb x, (p' x).restrict_scalars \ud835\udd5c) s :=\n{ zero_eq := \u03bb x hx, h.zero_eq x hx,\n  fderiv_within :=\n    begin\n      intros m hm x hx,\n      convert ((continuous_multilinear_map.restrict_scalars_linear \ud835\udd5c).has_fderiv_at)\n        .comp_has_fderiv_within_at _ ((h.fderiv_within m hm x hx).restrict_scalars \ud835\udd5c),\n    end,\n  cont := \u03bb m hm, continuous_multilinear_map.continuous_restrict_scalars.comp_continuous_on\n    (h.cont m hm) }\n\nlemma times_cont_diff_within_at.restrict_scalars (h : times_cont_diff_within_at \ud835\udd5c' n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nbegin\n  intros m hm,\n  rcases h m hm with \u27e8u, u_mem, p', hp'\u27e9,\n  exact \u27e8u, u_mem, _, hp'.restrict_scalars _\u27e9\nend\n\nlemma times_cont_diff_on.restrict_scalars (h : times_cont_diff_on \ud835\udd5c' n f s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\n\u03bb x hx, (h x hx).restrict_scalars _\n\nlemma times_cont_diff_at.restrict_scalars (h : times_cont_diff_at \ud835\udd5c' n f x) :\n  times_cont_diff_at \ud835\udd5c n f x :=\ntimes_cont_diff_within_at_univ.1 $ h.times_cont_diff_within_at.restrict_scalars _\n\nlemma times_cont_diff.restrict_scalars (h : times_cont_diff \ud835\udd5c' n f) :\n  times_cont_diff \ud835\udd5c n f :=\ntimes_cont_diff_iff_times_cont_diff_at.2 $ \u03bb x, h.times_cont_diff_at.restrict_scalars _\n\nend restrict_scalars\n","avg_line_length":45.8157894737,"max_line_length":106,"alphanum_fraction":0.6872885953}
-{"size":6851,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Floris van Doorn\n-\/\nimport topology.bases\nimport topology.homeomorph\n\/-!\n# Open sets\n\n## Summary\n\nWe define the subtype of open sets in a topological space.\n\n## Main Definitions\n\n- `opens \u03b1` is the type of open subsets of a topological space `\u03b1`.\n- `open_nhds_of x` is the type of open subsets of a topological space `\u03b1` containing `x : \u03b1`.\n-\n-\/\n\nopen filter set\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n  [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\nnamespace topological_space\nvariable (\u03b1)\n\/-- The type of open subsets of a topological space. -\/\ndef opens := {s : set \u03b1 \/\/ is_open s}\n\nvariable {\u03b1}\nnamespace opens\ninstance : has_coe (opens \u03b1) (set \u03b1) := { coe := subtype.val }\n\nlemma val_eq_coe (U : opens \u03b1) : U.1 = \u2191U := rfl\n\ninstance : has_subset (opens \u03b1) :=\n{ subset := \u03bb U V, (U : set \u03b1) \u2286 V }\n\ninstance : has_mem \u03b1 (opens \u03b1) :=\n{ mem := \u03bb a U, a \u2208 (U : set \u03b1) }\n\n@[ext] lemma ext {U V : opens \u03b1} (h : (U : set \u03b1) = V) : U = V := subtype.ext_iff.mpr h\n\n@[ext] lemma ext_iff {U V : opens \u03b1} : (U : set \u03b1) = V \u2194 U = V :=\n\u27e8opens.ext, congr_arg coe\u27e9\n\ninstance : partial_order (opens \u03b1) := subtype.partial_order _\n\n\/-- The interior of a set, as an element of `opens`. -\/\ndef interior (s : set \u03b1) : opens \u03b1 := \u27e8interior s, is_open_interior\u27e9\n\nlemma gc : galois_connection (coe : opens \u03b1 \u2192 set \u03b1) interior :=\n\u03bb U s, \u27e8\u03bb h, interior_maximal h U.property, \u03bb h, le_trans h interior_subset\u27e9\n\n\/-- The galois insertion between sets and opens, but ordered by reverse inclusion. -\/\ndef gi : @galois_insertion (order_dual (set \u03b1)) (order_dual (opens \u03b1)) _ _ interior subtype.val :=\n{ choice := \u03bb s hs, \u27e8s, interior_eq_iff_open.mp $ le_antisymm interior_subset hs\u27e9,\n  gc := gc.dual,\n  le_l_u := \u03bb _, interior_subset,\n  choice_eq := \u03bb s hs, le_antisymm interior_subset hs }\n\n@[simp] lemma gi_choice_val {s : order_dual (set \u03b1)} {hs} : (gi.choice s hs).val = s := rfl\n\ninstance : complete_lattice (opens \u03b1) :=\ncomplete_lattice.copy\n  (@order_dual.complete_lattice _ (galois_insertion.lift_complete_lattice (@gi \u03b1 _)))\n\/- le  -\/ (\u03bb U V, U \u2286 V) rfl\n\/- top -\/ \u27e8set.univ, is_open_univ\u27e9 (subtype.ext_iff_val.mpr interior_univ.symm)\n\/- bot -\/ \u27e8\u2205, is_open_empty\u27e9 rfl\n\/- sup -\/ (\u03bb U V, \u27e8\u2191U \u222a \u2191V, is_open_union U.2 V.2\u27e9) rfl\n\/- inf -\/ (\u03bb U V, \u27e8\u2191U \u2229 \u2191V, is_open_inter U.2 V.2\u27e9)\nbegin\n  funext,\n  apply subtype.ext_iff_val.mpr,\n  symmetry,\n  apply interior_eq_of_open,\n  exact (is_open_inter U.2 V.2),\nend\n\/- Sup -\/ (\u03bb Us, \u27e8\u22c3\u2080 (coe '' Us), is_open_sUnion $ \u03bb U hU,\nby { rcases hU with \u27e8\u27e8V, hV\u27e9, h, h'\u27e9, dsimp at h', subst h', exact hV}\u27e9)\nbegin\n  funext,\n  apply subtype.ext_iff_val.mpr,\n  simp [Sup_range],\n  refl,\nend\n\/- Inf -\/ _ rfl\n\ninstance : has_inter (opens \u03b1) := \u27e8\u03bb U V, U \u2293 V\u27e9\ninstance : has_union (opens \u03b1) := \u27e8\u03bb U V, U \u2294 V\u27e9\ninstance : has_emptyc (opens \u03b1) := \u27e8\u22a5\u27e9\ninstance : inhabited (opens \u03b1) := \u27e8\u2205\u27e9\n\n@[simp] lemma inter_eq (U V : opens \u03b1) : U \u2229 V = U \u2293 V := rfl\n@[simp] lemma union_eq (U V : opens \u03b1) : U \u222a V = U \u2294 V := rfl\n@[simp] lemma empty_eq : (\u2205 : opens \u03b1) = \u22a5 := rfl\n\n@[simp] lemma Sup_s {Us : set (opens \u03b1)} : \u2191(Sup Us) = \u22c3\u2080 ((coe : _ \u2192 set \u03b1) '' Us) :=\nbegin\n  rw [@galois_connection.l_Sup (opens \u03b1) (set \u03b1) _ _ (coe : opens \u03b1 \u2192 set \u03b1) interior gc Us],\n  rw [set.sUnion_image]\nend\n\nlemma supr_def {\u03b9} (s : \u03b9 \u2192 opens \u03b1) : (\u2a06 i, s i) = \u27e8\u22c3 i, s i, is_open_Union $ \u03bb i, (s i).2\u27e9 :=\nby { ext, simp only [supr, opens.Sup_s, sUnion_image, bUnion_range], refl }\n\ndef is_basis (B : set (opens \u03b1)) : Prop := is_topological_basis ((coe : _ \u2192 set \u03b1) '' B)\n\nlemma is_basis_iff_nbhd {B : set (opens \u03b1)} :\n  is_basis B \u2194 \u2200 {U : opens \u03b1} {x}, x \u2208 U \u2192 \u2203 U' \u2208 B, x \u2208 U' \u2227 U' \u2286 U :=\nbegin\n  split; intro h,\n  { rintros \u27e8sU, hU\u27e9 x hx,\n    rcases (mem_nhds_of_is_topological_basis h).mp (mem_nhds_sets hU hx) with \u27e8sV, \u27e8\u27e8V, H\u2081, H\u2082\u27e9, hsV\u27e9\u27e9,\n    refine \u27e8V, H\u2081, _\u27e9,\n    cases V, dsimp at H\u2082, subst H\u2082, exact hsV },\n  { refine is_topological_basis_of_open_of_nhds _ _,\n    { rintros sU \u27e8U, \u27e8H\u2081, H\u2082\u27e9\u27e9, subst H\u2082, exact U.property },\n    { intros x sU hx hsU,\n      rcases @h (\u27e8sU, hsU\u27e9 : opens \u03b1) x hx with \u27e8V, hV, H\u27e9,\n      exact \u27e8V, \u27e8V, hV, rfl\u27e9, H\u27e9 } }\nend\n\nlemma is_basis_iff_cover {B : set (opens \u03b1)} :\n  is_basis B \u2194 \u2200 U : opens \u03b1, \u2203 Us \u2286 B, U = Sup Us :=\nbegin\n  split,\n  { intros hB U,\n    rcases sUnion_basis_of_is_open hB U.prop with \u27e8sUs, H, hU\u27e9,\n    existsi {U : opens \u03b1 | U \u2208 B \u2227 \u2191U \u2208 sUs},\n    split,\n    { intros U hU, exact hU.left },\n    { apply ext,\n      rw [Sup_s, hU],\n      congr,\n      ext s; split; intro hs,\n      { rcases H hs with \u27e8V, hV\u27e9,\n        rw \u2190 hV.right at hs,\n        refine \u27e8V, \u27e8\u27e8hV.left, hs\u27e9, hV.right\u27e9\u27e9 },\n      { rcases hs with \u27e8V, \u27e8\u27e8H\u2081, H\u2082\u27e9, H\u2083\u27e9\u27e9,\n        subst H\u2083, exact H\u2082 } } },\n  { intro h,\n    rw is_basis_iff_nbhd,\n    intros U x hx,\n    rcases h U with \u27e8Us, hUs, H\u27e9,\n    replace H := congr_arg (coe : _ \u2192 set \u03b1) H,\n    rw Sup_s at H,\n    change x \u2208 \u2191U at hx,\n    rw H at hx,\n    rcases set.mem_sUnion.mp hx with \u27e8sV, \u27e8\u27e8V, H\u2081, H\u2082\u27e9, hsV\u27e9\u27e9,\n    refine \u27e8V,hUs H\u2081,_\u27e9,\n    cases V with V hV,\n    dsimp at H\u2082, subst H\u2082,\n    refine \u27e8hsV,_\u27e9,\n    change V \u2286 U, rw H,\n    exact set.subset_sUnion_of_mem \u27e8\u27e8V, _\u27e9, \u27e8H\u2081, rfl\u27e9\u27e9 }\nend\n\n\/-- The preimage of an open set, as an open set. -\/\ndef comap {f : \u03b1 \u2192 \u03b2} (hf : continuous f) (V : opens \u03b2) : opens \u03b1 :=\n\u27e8f \u207b\u00b9' V.1, hf V.1 V.2\u27e9\n\n@[simp] lemma comap_id (U : opens \u03b1) : U.comap continuous_id = U := by { ext, refl }\n\nlemma comap_mono {f : \u03b1 \u2192 \u03b2} (hf : continuous f) {V W : opens \u03b2} (hVW : V \u2286 W) :\n  V.comap hf \u2286 W.comap hf :=\n\u03bb _ h, hVW h\n\n@[simp] lemma coe_comap {f : \u03b1 \u2192 \u03b2} (hf : continuous f) (U : opens \u03b2) :\n  \u2191(U.comap hf) = f \u207b\u00b9' U := rfl\n\n@[simp] lemma comap_val {f : \u03b1 \u2192 \u03b2} (hf : continuous f) (U : opens \u03b2) :\n  (U.comap hf).1 = f \u207b\u00b9' U := rfl\n\nprotected lemma comap_comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : continuous g) (hf : continuous f)\n  (U : opens \u03b3) : U.comap (hg.comp hf) = (U.comap hg).comap hf :=\nby { ext1, simp only [coe_comap, preimage_preimage] }\n\n\/-- A homeomorphism induces an equivalence on open sets, by taking comaps. -\/\n@[simp] protected def equiv (f : \u03b1 \u2243\u209c \u03b2) : opens \u03b1 \u2243 opens \u03b2 :=\n{ to_fun := opens.comap f.symm.continuous,\n  inv_fun := opens.comap f.continuous,\n  left_inv := by { intro U, ext1,\n    simp only [coe_comap, \u2190 preimage_comp, f.symm_comp_self, preimage_id] },\n  right_inv := by { intro U, ext1,\n    simp only [coe_comap, \u2190 preimage_comp, f.self_comp_symm, preimage_id] } }\n\nend opens\n\n\/-- The open neighborhoods of a point. See also `opens` or `nhds`. -\/\ndef open_nhds_of (x : \u03b1) : Type* := { s : set \u03b1 \/\/ is_open s \u2227 x \u2208 s }\n\ninstance open_nhds_of.inhabited {\u03b1 : Type*} [topological_space \u03b1] (x : \u03b1) :\n  inhabited (open_nhds_of x) := \u27e8\u27e8set.univ, is_open_univ, set.mem_univ _\u27e9\u27e9\n\nend topological_space\n\nnamespace continuous\nopen topological_space\n\n\nend continuous\n","avg_line_length":33.5833333333,"max_line_length":103,"alphanum_fraction":0.6188877536}
-{"size":4031,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/- Copyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Seul Baek\n\nLinear integer arithmetic formulas in pre-normalized form. -\/\n\nimport tactic.omega.int.preterm\n\nnamespace omega\nnamespace int\n\n\/-- Intermediate shadow syntax for LNA formulas that includes unreified exprs -\/\nmeta inductive exprform\n| eq  : exprterm \u2192 exprterm \u2192 exprform\n| le  : exprterm \u2192 exprterm \u2192 exprform\n| not : exprform \u2192 exprform\n| or  : exprform \u2192 exprform \u2192 exprform\n| and : exprform \u2192 exprform \u2192 exprform\n\n\/-- Intermediate shadow syntax for LIA formulas that includes non-canonical terms -\/\n@[derive has_reflect, derive inhabited]\ninductive preform\n| eq  : preterm \u2192 preterm \u2192 preform\n| le  : preterm \u2192 preterm \u2192 preform\n| not : preform \u2192 preform\n| or  : preform \u2192 preform \u2192 preform\n| and : preform \u2192 preform \u2192 preform\n\nlocalized \"notation x ` =* ` y := omega.int.preform.eq x y\" in omega.int\nlocalized \"notation x ` \u2264* ` y := omega.int.preform.le x y\" in omega.int\nlocalized \"notation `\u00ac* ` p   := omega.int.preform.not p\" in omega.int\nlocalized \"notation p ` \u2228* ` q := omega.int.preform.or p q\" in omega.int\nlocalized \"notation p ` \u2227* ` q := omega.int.preform.and p q\" in omega.int\n\nnamespace preform\n\n\/-- Evaluate a preform into prop using the valuation v. -\/\n@[simp] def holds (v : nat \u2192 int) : preform \u2192 Prop\n| (t =* s) := t.val v = s.val v\n| (t \u2264* s) := t.val v \u2264 s.val v\n| (\u00ac* p)   := \u00ac p.holds\n| (p \u2228* q) := p.holds \u2228 q.holds\n| (p \u2227* q) := p.holds \u2227 q.holds\n\nend preform\n\n\/-- univ_close p n := p closed by prepending n universal quantifiers -\/\n@[simp] def univ_close (p : preform) : (nat \u2192 int) \u2192 nat \u2192 Prop\n| v 0     := p.holds v\n| v (k+1) := \u2200 i : int, univ_close (update_zero i v) k\n\nnamespace preform\n\n\/-- Fresh de Brujin index not used by any variable in argument -\/\ndef fresh_index : preform \u2192 nat\n| (t =* s) := max t.fresh_index s.fresh_index\n| (t \u2264* s) := max t.fresh_index s.fresh_index\n| (\u00ac* p)   := p.fresh_index\n| (p \u2228* q) := max p.fresh_index q.fresh_index\n| (p \u2227* q) := max p.fresh_index q.fresh_index\n\n\/-- All valuations satisfy argument -\/\ndef valid (p : preform) : Prop :=\n\u2200 v, holds v p\n\n\/-- There exists some valuation that satisfies argument -\/\ndef sat (p : preform) : Prop :=\n\u2203 v, holds v p\n\n\/-- implies p q := under any valuation, q holds if p holds -\/\ndef implies (p q : preform) : Prop :=\n\u2200 v, (holds v p \u2192 holds v q)\n\n\/-- equiv p q := under any valuation, p holds iff q holds -\/\ndef equiv (p q : preform) : Prop :=\n\u2200 v, (holds v p \u2194 holds v q)\n\nlemma sat_of_implies_of_sat {p q : preform} :\n  implies p q \u2192 sat p \u2192 sat q :=\nbegin intros h1 h2, apply exists_imp_exists h1 h2 end\n\nlemma sat_or {p q : preform} :\n  sat (p \u2228* q) \u2194 sat p \u2228 sat q :=\nbegin\n  constructor; intro h1,\n  { cases h1 with v h1, cases h1 with h1 h1;\n    [left,right]; refine \u27e8v,_\u27e9; assumption },\n  { cases h1 with h1 h1; cases h1 with v h1;\n    refine \u27e8v,_\u27e9; [left,right]; assumption }\nend\n\n\/-- There does not exist any valuation that satisfies argument -\/\ndef unsat (p : preform) : Prop := \u00ac sat p\n\ndef repr : preform \u2192 string\n| (t =* s) := \"(\" ++ t.repr ++ \" = \" ++ s.repr ++ \")\"\n| (t \u2264* s) := \"(\" ++ t.repr ++ \" \u2264 \" ++ s.repr ++ \")\"\n| (\u00ac* p)   := \"\u00ac\" ++ p.repr\n| (p \u2228* q) := \"(\" ++ p.repr ++ \" \u2228 \" ++ q.repr ++ \")\"\n| (p \u2227* q) := \"(\" ++ p.repr ++ \" \u2227 \" ++ q.repr ++ \")\"\n\ninstance has_repr : has_repr preform := \u27e8repr\u27e9\nmeta instance has_to_format : has_to_format preform := \u27e8\u03bb x, x.repr\u27e9\n\nend preform\n\nlemma univ_close_of_valid {p : preform} :\n  \u2200 {m v}, p.valid \u2192 univ_close p v m\n| 0 v h1     := h1 _\n| (m+1) v h1 := \u03bb i, univ_close_of_valid h1\n\nlemma valid_of_unsat_not {p : preform} : (\u00ac*p).unsat \u2192 p.valid :=\nbegin\n  simp only [preform.sat, preform.unsat, preform.valid, preform.holds],\n  rw classical.not_exists_not, intro h, assumption\nend\n\n\/-- Tactic for setting up proof by induction over preforms. -\/\nmeta def preform.induce (t : tactic unit := tactic.skip) : tactic unit :=\n`[ intro p, induction p with t s t s p ih p q ihp ihq p q ihp ihq; t]\n\nend int\n\nend omega\n","avg_line_length":32.248,"max_line_length":84,"alphanum_fraction":0.6375589184}
-{"size":24377,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-\/\n\nimport field_theory.finite\nimport data.zmod.basic\nimport data.nat.parity\n\n\/-!\n# Quadratic reciprocity.\n\nThis file contains results about quadratic residues modulo a prime number.\n\nThe main results are the law of quadratic reciprocity, `quadratic_reciprocity`, as well as the\ninterpretations in terms of existence of square roots depending on the congruence mod 4,\n`exists_pow_two_eq_prime_iff_of_mod_four_eq_one`, and\n`exists_pow_two_eq_prime_iff_of_mod_four_eq_three`.\n\nAlso proven are conditions for `-1` and `2` to be a square modulo a prime,\n`exists_pow_two_eq_neg_one_iff_mod_four_ne_three` and\n`exists_pow_two_eq_two_iff`\n\n## Implementation notes\n\nThe proof of quadratic reciprocity implemented uses Gauss' lemma and Eisenstein's lemma\n-\/\n\nopen function finset nat finite_field zmod\nopen_locale big_operators\n\nnamespace zmod\n\nvariables (p q : \u2115) [fact p.prime] [fact q.prime]\n\n@[simp] lemma card_units : fintype.card (units (zmod p)) = p - 1 :=\nby rw [card_units, card]\n\n\/-- Fermat's Little Theorem: for every unit `a` of `zmod p`, we have `a ^ (p - 1) = 1`. -\/\ntheorem fermat_little_units {p : \u2115} [fact p.prime] (a : units (zmod p)) :\n  a ^ (p - 1) = 1 :=\nby rw [\u2190 card_units p, pow_card_eq_one]\n\n\/-- Fermat's Little Theorem: for all nonzero `a : zmod p`, we have `a ^ (p - 1) = 1`. -\/\ntheorem fermat_little {a : zmod p} (ha : a \u2260 0) : a ^ (p - 1) = 1 :=\nbegin\n  have := fermat_little_units (units.mk0 a ha),\n  apply_fun (coe : units (zmod p) \u2192 zmod p) at this,\n  simpa,\nend\n\n\/-- Euler's Criterion: A unit `x` of `zmod p` is a square if and only if `x ^ (p \/ 2) = 1`. -\/\nlemma euler_criterion_units (x : units (zmod p)) :\n  (\u2203 y : units (zmod p), y ^ 2 = x) \u2194 x ^ (p \/ 2) = 1 :=\nbegin\n  cases nat.prime.eq_two_or_odd \u2039p.prime\u203a with hp2 hp_odd,\n  { resetI, subst p, refine iff_of_true \u27e81, _\u27e9 _; apply subsingleton.elim  },\n  obtain \u27e8g, hg\u27e9 := is_cyclic.exists_generator (units (zmod p)),\n  obtain \u27e8n, hn\u27e9 : x \u2208 powers g, { rw powers_eq_gpowers, apply hg },\n  split,\n  { rintro \u27e8y, rfl\u27e9, rw [\u2190 pow_mul, two_mul_odd_div_two hp_odd, fermat_little_units], },\n  { subst x, assume h,\n    have key : 2 * (p \/ 2) \u2223 n * (p \/ 2),\n    { rw [\u2190 pow_mul] at h,\n      rw [two_mul_odd_div_two hp_odd, \u2190 card_units, \u2190 order_of_eq_card_of_forall_mem_gpowers hg],\n      apply order_of_dvd_of_pow_eq_one h },\n    have : 0 < p \/ 2 := nat.div_pos (show fact (1 < p), by apply_instance) dec_trivial,\n    obtain \u27e8m, rfl\u27e9 := dvd_of_mul_dvd_mul_right this key,\n    refine \u27e8g ^ m, _\u27e9,\n    rw [mul_comm, pow_mul], },\nend\n\n\/-- Euler's Criterion: a nonzero `a : zmod p` is a square if and only if `x ^ (p \/ 2) = 1`. -\/\nlemma euler_criterion {a : zmod p} (ha : a \u2260 0) :\n  (\u2203 y : zmod p, y ^ 2 = a) \u2194 a ^ (p \/ 2) = 1 :=\nbegin\n  apply (iff_congr _ (by simp [units.ext_iff])).mp (euler_criterion_units p (units.mk0 a ha)),\n  simp only [units.ext_iff, _root_.pow_two, units.coe_mk0, units.coe_mul],\n  split, { rintro \u27e8y, hy\u27e9, exact \u27e8y, hy\u27e9 },\n  { rintro \u27e8y, rfl\u27e9,\n    have hy : y \u2260 0, { rintro rfl, simpa [_root_.zero_pow] using ha, },\n    refine \u27e8units.mk0 y hy, _\u27e9, simp, }\nend\n\nlemma exists_pow_two_eq_neg_one_iff_mod_four_ne_three :\n  (\u2203 y : zmod p, y ^ 2 = -1) \u2194 p % 4 \u2260 3 :=\nbegin\n  cases nat.prime.eq_two_or_odd \u2039p.prime\u203a with hp2 hp_odd,\n  { resetI, subst p, exact dec_trivial },\n  change fact (p % 2 = 1) at hp_odd, resetI,\n  have neg_one_ne_zero : (-1 : zmod p) \u2260 0, from mt neg_eq_zero.1 one_ne_zero,\n  rw [euler_criterion p neg_one_ne_zero, neg_one_pow_eq_pow_mod_two],\n  cases mod_two_eq_zero_or_one (p \/ 2) with p_half_even p_half_odd,\n  { rw [p_half_even, _root_.pow_zero, eq_self_iff_true, true_iff],\n    contrapose! p_half_even with hp,\n    rw [\u2190 nat.mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp],\n    exact dec_trivial },\n  { rw [p_half_odd, _root_.pow_one,\n        iff_false_intro (ne_neg_self p one_ne_zero).symm, false_iff, not_not],\n    rw [\u2190 nat.mod_mul_right_div_self, show 2 * 2 = 4, from rfl] at p_half_odd,\n    rw [_root_.fact, \u2190 nat.mod_mul_left_mod _ 2, show 2 * 2 = 4, from rfl] at hp_odd,\n    have hp : p % 4 < 4, from nat.mod_lt _ dec_trivial,\n    revert hp hp_odd p_half_odd,\n    generalize : p % 4 = k, revert k, exact dec_trivial }\nend\n\nlemma pow_div_two_eq_neg_one_or_one {a : zmod p} (ha : a \u2260 0) :\n  a ^ (p \/ 2) = 1 \u2228 a ^ (p \/ 2) = -1 :=\nbegin\n  cases nat.prime.eq_two_or_odd \u2039p.prime\u203a with hp2 hp_odd,\n  { resetI, subst p, revert a ha, exact dec_trivial },\n  rw [\u2190 mul_self_eq_one_iff, \u2190 _root_.pow_add, \u2190 two_mul, two_mul_odd_div_two hp_odd],\n  exact fermat_little p ha\nend\n\n\/-- Wilson's Lemma: the product of `1`, ..., `p-1` is `-1` modulo `p`. -\/\n@[simp] lemma wilsons_lemma : (nat.fact (p - 1) : zmod p) = -1 :=\nbegin\n  refine\n  calc (nat.fact (p - 1) : zmod p) = (\u220f x in Ico 1 (succ (p - 1)), x) :\n    by rw [\u2190 finset.prod_Ico_id_eq_fact, prod_nat_cast]\n                               ... = (\u220f x : units (zmod p), x) : _\n                               ... = -1 :\n    by rw [prod_hom _ (coe : units (zmod p) \u2192 zmod p),\n           prod_univ_units_id_eq_neg_one, units.coe_neg, units.coe_one],\n  have hp : 0 < p := nat.prime.pos \u2039p.prime\u203a,\n  symmetry,\n  refine prod_bij (\u03bb a _, (a : zmod p).val) _ _ _ _,\n  { intros a ha,\n    rw [Ico.mem, \u2190 nat.succ_sub hp, nat.succ_sub_one],\n    split,\n    { apply nat.pos_of_ne_zero, rw \u2190 @val_zero p,\n      assume h, apply units.coe_ne_zero a (val_injective p h) },\n    { exact val_lt _ } },\n  { intros a ha, simp only [cast_id, nat_cast_val], },\n  { intros _ _ _ _ h, rw units.ext_iff, exact val_injective p h },\n  { intros b hb,\n    rw [Ico.mem, nat.succ_le_iff, \u2190 succ_sub hp, succ_sub_one, nat.pos_iff_ne_zero] at hb,\n    refine \u27e8units.mk0 b _, finset.mem_univ _, _\u27e9,\n    { assume h, apply hb.1, apply_fun val at h,\n      simpa only [val_cast_of_lt hb.right, val_zero] using h },\n    { simp only [val_cast_of_lt hb.right, units.coe_mk0], } }\nend\n\n@[simp] lemma prod_Ico_one_prime : (\u220f x in Ico 1 p, (x : zmod p)) = -1 :=\nbegin\n  conv in (Ico 1 p) { rw [\u2190 succ_sub_one p, succ_sub (nat.prime.pos \u2039p.prime\u203a)] },\n  rw [\u2190 prod_nat_cast, finset.prod_Ico_id_eq_fact, wilsons_lemma]\nend\n\nend zmod\n\n\/-- The image of the map sending a non zero natural number `x \u2264 p \/ 2` to the absolute value\n  of the element of interger in the interval `(-p\/2, p\/2]` congruent to `a * x` mod p is the set\n  of non zero natural numbers `x` such that `x \u2264 p \/ 2` -\/\nlemma Ico_map_val_min_abs_nat_abs_eq_Ico_map_id\n  (p : \u2115) [hp : fact p.prime] (a : zmod p) (hap : a \u2260 0) :\n  (Ico 1 (p \/ 2).succ).1.map (\u03bb x, (a * x).val_min_abs.nat_abs) =\n  (Ico 1 (p \/ 2).succ).1.map (\u03bb a, a) :=\nbegin\n  have he : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 x \u2260 0 \u2227 x \u2264 p \/ 2,\n    by simp [nat.lt_succ_iff, nat.succ_le_iff, nat.pos_iff_ne_zero] {contextual := tt},\n  have hep : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 x < p,\n    from \u03bb x hx, lt_of_le_of_lt (he hx).2 (nat.div_lt_self hp.pos dec_trivial),\n  have hpe : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 \u00ac p \u2223 x,\n    from \u03bb x hx hpx, not_lt_of_ge (le_of_dvd (nat.pos_of_ne_zero (he hx).1) hpx) (hep hx),\n  have hmem : \u2200 (x : \u2115) (hx : x \u2208 Ico 1 (p \/ 2).succ),\n    (a * x : zmod p).val_min_abs.nat_abs \u2208 Ico 1 (p \/ 2).succ,\n  { assume x hx,\n    simp [hap, char_p.cast_eq_zero_iff (zmod p) p, hpe hx, lt_succ_iff, succ_le_iff,\n        nat.pos_iff_ne_zero, nat_abs_val_min_abs_le _], },\n  have hsurj : \u2200 (b : \u2115) (hb : b \u2208 Ico 1 (p \/ 2).succ),\n    \u2203 x \u2208 Ico 1 (p \/ 2).succ, b = (a * x : zmod p).val_min_abs.nat_abs,\n  { assume b hb,\n    refine \u27e8(b \/ a : zmod p).val_min_abs.nat_abs, Ico.mem.mpr \u27e8_, _\u27e9, _\u27e9,\n    { apply nat.pos_of_ne_zero,\n      simp only [div_eq_mul_inv, hap, char_p.cast_eq_zero_iff (zmod p) p, hpe hb, not_false_iff,\n        val_min_abs_eq_zero, inv_eq_zero, int.nat_abs_eq_zero, ne.def, mul_eq_zero_iff', or_self] },\n      { apply lt_succ_of_le, apply nat_abs_val_min_abs_le },\n      { rw cast_nat_abs_val_min_abs,\n        split_ifs,\n        { erw [mul_div_cancel' _ hap, val_min_abs_def_pos, val_cast_of_lt (hep hb),\n            if_pos (le_of_lt_succ (Ico.mem.1 hb).2), int.nat_abs_of_nat], },\n        { erw [mul_neg_eq_neg_mul_symm, mul_div_cancel' _ hap, nat_abs_val_min_abs_neg,\n            val_min_abs_def_pos, val_cast_of_lt (hep hb), if_pos (le_of_lt_succ (Ico.mem.1 hb).2),\n            int.nat_abs_of_nat] } } },\n  exact multiset.map_eq_map_of_bij_of_nodup _ _ (finset.nodup _) (finset.nodup _)\n    (\u03bb x _, (a * x : zmod p).val_min_abs.nat_abs) hmem (\u03bb _ _, rfl)\n    (inj_on_of_surj_on_of_card_le _ hmem hsurj (le_refl _)) hsurj\nend\n\nprivate lemma gauss_lemma_aux\u2081 (p : \u2115) [hp : fact p.prime] [hp2 : fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  (a^(p \/ 2) * (p \/ 2).fact : zmod p) =\n  (-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card * (p \/ 2).fact :=\ncalc (a ^ (p \/ 2) * (p \/ 2).fact : zmod p) =\n    (\u220f x in Ico 1 (p \/ 2).succ, a * x) :\n  by rw [prod_mul_distrib, \u2190 prod_nat_cast, \u2190 prod_nat_cast, prod_Ico_id_eq_fact,\n      prod_const, Ico.card, succ_sub_one]; simp\n... = (\u220f x in Ico 1 (p \/ 2).succ, (a * x : zmod p).val) : by simp\n... = (\u220f x in Ico 1 (p \/ 2).succ,\n    (if (a * x : zmod p).val \u2264 p \/ 2 then 1 else -1) *\n      (a * x : zmod p).val_min_abs.nat_abs) :\n  prod_congr rfl $ \u03bb _ _, begin\n    simp only [cast_nat_abs_val_min_abs],\n    split_ifs; simp\n  end\n... = (-1)^((Ico 1 (p \/ 2).succ).filter\n      (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card *\n    (\u220f x in Ico 1 (p \/ 2).succ, (a * x : zmod p).val_min_abs.nat_abs) :\n  have (\u220f x in Ico 1 (p \/ 2).succ,\n        if (a * x : zmod p).val \u2264 p \/ 2 then (1 : zmod p) else -1) =\n      (\u220f x in (Ico 1 (p \/ 2).succ).filter\n        (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2), -1),\n    from prod_bij_ne_one (\u03bb x _ _, x)\n      (\u03bb x, by split_ifs; simp * at * {contextual := tt})\n      (\u03bb _ _ _ _ _ _, id)\n      (\u03bb b h _, \u27e8b, by simp [-not_le, *] at *\u27e9)\n      (by intros; split_ifs at *; simp * at *),\n  by rw [prod_mul_distrib, this]; simp\n... = (-1)^((Ico 1 (p \/ 2).succ).filter\n      (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card * (p \/ 2).fact :\n  by rw [\u2190 prod_nat_cast, finset.prod_eq_multiset_prod,\n      Ico_map_val_min_abs_nat_abs_eq_Ico_map_id p a hap,\n      \u2190 finset.prod_eq_multiset_prod, prod_Ico_id_eq_fact]\n\nprivate lemma gauss_lemma_aux\u2082 (p : \u2115) [hp : fact p.prime] [hp2 : fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  (a^(p \/ 2) : zmod p) = (-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card :=\n(domain.mul_left_inj\n    (show ((p \/ 2).fact : zmod p) \u2260 0,\n      by rw [ne.def, char_p.cast_eq_zero_iff (zmod p) p, hp.dvd_fact, not_le];\n          exact nat.div_lt_self hp.pos dec_trivial)).1 $\n  by simpa using gauss_lemma_aux\u2081 p hap\n\nprivate lemma eisenstein_lemma_aux\u2081 (p : \u2115) [hp : fact p.prime] [hp2 : fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  (((Ico 1 (p \/ 2).succ).sum (\u03bb x, a * x) : \u2115) : zmod 2) =\n    ((Ico 1 (p \/ 2).succ).filter\n      ((\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val))).card +\n      (Ico 1 (p \/ 2).succ).sum (\u03bb x, x)\n    + ((Ico 1 (p \/ 2).succ).sum (\u03bb x, (a * x) \/ p) : \u2115) :=\nhave hp2 : (p : zmod 2) = (1 : \u2115), from (eq_iff_modeq_nat _).2 hp2,\ncalc (((Ico 1 (p \/ 2).succ).sum (\u03bb x, a * x) : \u2115) : zmod 2)\n    = (((Ico 1 (p \/ 2).succ).sum (\u03bb x, (a * x) % p + p * ((a * x) \/ p)) : \u2115) : zmod 2) :\n  by simp only [mod_add_div]\n... = ((Ico 1 (p \/ 2).succ).sum (\u03bb x, ((a * x : \u2115) : zmod p).val) : \u2115) +\n    ((Ico 1 (p \/ 2).succ).sum (\u03bb x, (a * x) \/ p) : \u2115) :\n  by simp only [val_cast_nat];\n    simp [sum_add_distrib, mul_sum.symm, nat.cast_add, nat.cast_mul, sum_nat_cast, hp2]\n... = _ : congr_arg2 (+)\n  (calc (((Ico 1 (p \/ 2).succ).sum (\u03bb x, ((a * x : \u2115) : zmod p).val) : \u2115) : zmod 2)\n      = (Ico 1 (p \/ 2).succ).sum\n          (\u03bb x, ((((a * x : zmod p).val_min_abs +\n            (if (a * x : zmod p).val \u2264 p \/ 2 then 0 else p)) : \u2124) : zmod 2)) :\n        by simp only [(val_eq_ite_val_min_abs _).symm]; simp [sum_nat_cast]\n  ... = ((Ico 1 (p \/ 2).succ).filter\n        (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card +\n      (((Ico 1 (p \/ 2).succ).sum (\u03bb x, (a * x : zmod p).val_min_abs.nat_abs)) : \u2115) :\n    by { simp [ite_cast, add_comm, sum_add_distrib, finset.sum_ite, hp2, sum_nat_cast], }\n  ... = _ : by rw [finset.sum_eq_multiset_sum,\n      Ico_map_val_min_abs_nat_abs_eq_Ico_map_id p a hap,\n      \u2190 finset.sum_eq_multiset_sum];\n    simp [sum_nat_cast]) rfl\n\nprivate lemma eisenstein_lemma_aux\u2082 (p : \u2115) [hp : fact p.prime] [hp2 : fact (p % 2 = 1)]\n  {a : \u2115} (ha2 : a % 2 = 1) (hap : (a : zmod p) \u2260 0) :\n  ((Ico 1 (p \/ 2).succ).filter\n    ((\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val))).card\n  \u2261 (Ico 1 (p \/ 2).succ).sum (\u03bb x, (x * a) \/ p) [MOD 2] :=\nhave ha2 : (a : zmod 2) = (1 : \u2115), from (eq_iff_modeq_nat _).2 ha2,\n(eq_iff_modeq_nat 2).1 $ sub_eq_zero.1 $\n  by simpa [add_left_comm, sub_eq_add_neg, finset.mul_sum.symm, mul_comm, ha2, sum_nat_cast,\n            add_neg_eq_iff_eq_add.symm, neg_eq_self_mod_two, add_assoc]\n    using eq.symm (eisenstein_lemma_aux\u2081 p hap)\n\nlemma div_eq_filter_card {a b c : \u2115} (hb0 : 0 < b) (hc : a \/ b \u2264 c) : a \/ b =\n  ((Ico 1 c.succ).filter (\u03bb x, x * b \u2264 a)).card :=\ncalc a \/ b = (Ico 1 (a \/ b).succ).card : by simp\n... = ((Ico 1 c.succ).filter (\u03bb x, x * b \u2264 a)).card :\n  congr_arg _$ finset.ext.2 $ \u03bb x,\n    have x * b \u2264 a \u2192 x \u2264 c,\n      from \u03bb h, le_trans (by rwa [le_div_iff_mul_le _ _ hb0]) hc,\n    by simp [lt_succ_iff, le_div_iff_mul_le _ _ hb0]; tauto\n\n\/-- The given sum is the number of integer points in the triangle formed by the diagonal of the\n  rectangle `(0, p\/2) \u00d7 (0, q\/2)`  -\/\nprivate lemma sum_Ico_eq_card_lt {p q : \u2115} :\n  (Ico 1 (p \/ 2).succ).sum (\u03bb a, (a * q) \/ p) =\n  (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n  (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q)).card :=\nif hp0 : p = 0 then by simp [hp0, finset.ext]\nelse\n  calc (Ico 1 (p \/ 2).succ).sum (\u03bb a, (a * q) \/ p) =\n    (Ico 1 (p \/ 2).succ).sum (\u03bb a,\n      ((Ico 1 (q \/ 2).succ).filter (\u03bb x, x * p \u2264 a * q)).card) :\n    finset.sum_congr rfl $ \u03bb x hx,\n      div_eq_filter_card (nat.pos_of_ne_zero hp0)\n        (calc x * q \/ p \u2264 (p \/ 2) * q \/ p :\n            nat.div_le_div_right (mul_le_mul_of_nonneg_right\n              (le_of_lt_succ $ by finish)\n              (nat.zero_le _))\n          ... \u2264 _ : nat.div_mul_div_le_div _ _ _)\n  ... = _ : by rw [\u2190 card_sigma];\n    exact card_congr (\u03bb a _, \u27e8a.1, a.2\u27e9)\n      (by simp {contextual := tt})\n      (\u03bb \u27e8_, _\u27e9 \u27e8_, _\u27e9, by simp {contextual := tt})\n      (\u03bb \u27e8b\u2081, b\u2082\u27e9 h, \u27e8\u27e8b\u2081, b\u2082\u27e9,\n        by revert h; simp {contextual := tt}\u27e9)\n\n\/-- Each of the sums in this lemma is the cardinality of the set integer points in each of the\n  two triangles formed by the diagonal of the rectangle `(0, p\/2) \u00d7 (0, q\/2)`. Adding them\n  gives the number of points in the rectangle. -\/\nprivate lemma sum_mul_div_add_sum_mul_div_eq_mul (p q : \u2115) [hp : fact p.prime]\n  (hq0 : (q : zmod p) \u2260 0) :\n  (Ico 1 (p \/ 2).succ).sum (\u03bb a, (a * q) \/ p) +\n  (Ico 1 (q \/ 2).succ).sum (\u03bb a, (a * p) \/ q) =\n  (p \/ 2) * (q \/ 2) :=\nhave hswap : (((Ico 1 (q \/ 2).succ).product (Ico 1 (p \/ 2).succ)).filter\n    (\u03bb x : \u2115 \u00d7 \u2115, x.2 * q \u2264 x.1 * p)).card =\n  (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n    (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p)).card :=\n  card_congr (\u03bb x _, prod.swap x)\n    (\u03bb \u27e8_, _\u27e9, by simp {contextual := tt})\n    (\u03bb \u27e8_, _\u27e9 \u27e8_, _\u27e9, by simp {contextual := tt})\n    (\u03bb \u27e8x\u2081, x\u2082\u27e9 h, \u27e8\u27e8x\u2082, x\u2081\u27e9, by revert h; simp {contextual := tt}\u27e9),\nhave hdisj : disjoint\n    (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q))\n    (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p)),\n  from disjoint_filter.2 $ \u03bb x hx hpq hqp,\n  have hxp : x.1 < p, from lt_of_le_of_lt\n    (show x.1 \u2264 p \/ 2, by simp [*, nat.lt_succ_iff] at *; tauto)\n    (nat.div_lt_self hp.pos dec_trivial),\n  begin\n    have : (x.1 : zmod p) = 0,\n    { simpa [hq0] using congr_arg (coe : \u2115 \u2192 zmod p) (le_antisymm hpq hqp) },\n    apply_fun zmod.val at this,\n    rw [val_cast_of_lt hxp, val_zero] at this,\n    simp * at *\n  end,\nhave hunion : ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q) \u222a\n    ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p) =\n    ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)),\n  from finset.ext.2 $ \u03bb x, by have := le_total (x.2 * p) (x.1 * q); simp; tauto,\nby rw [sum_Ico_eq_card_lt, sum_Ico_eq_card_lt, hswap, \u2190 card_disjoint_union hdisj, hunion,\n    card_product];\n  simp\n\nvariables (p q : \u2115) [fact p.prime] [fact q.prime]\n\nnamespace zmod\n\n\/-- The Legendre symbol of `a` and `p` is an integer defined as\n\n* `0` if `a` is `0` modulo `p`;\n* `1` if `a ^ (p \/ 2)` is `1` modulo `p`\n   (by `euler_criterion` this is equivalent to \u201c`a` is a square modulo `p`\u201d);\n* `-1` otherwise.\n\n-\/\ndef legendre_sym (a p : \u2115) : \u2124 :=\nif      (a : zmod p) = 0           then  0\nelse if (a : zmod p) ^ (p \/ 2) = 1 then  1\n                                   else -1\n\nlemma legendre_sym_eq_pow (a p : \u2115) [hp : fact p.prime] :\n  (legendre_sym a p : zmod p) = (a ^ (p \/ 2)) :=\nbegin\n  rw legendre_sym,\n  by_cases ha : (a : zmod p) = 0,\n  { simp only [if_pos, ha, _root_.zero_pow (nat.div_pos (hp.two_le) (succ_pos 1)), int.cast_zero] },\n  cases hp.eq_two_or_odd with hp2 hp_odd,\n  { resetI, subst p,\n    have : \u2200 (a : zmod 2),\n      ((if a = 0 then 0 else if a ^ (2 \/ 2) = 1 then 1 else -1 : \u2124) : zmod 2) = a ^ (2 \/ 2),\n    by exact dec_trivial,\n    exact this a },\n  { change fact (p % 2 = 1) at hp_odd, resetI,\n    rw if_neg ha,\n    have : (-1 : zmod p) \u2260 1, from (ne_neg_self p one_ne_zero).symm,\n    cases pow_div_two_eq_neg_one_or_one p ha with h h,\n    { rw [if_pos h, h, int.cast_one], },\n    { rw [h, if_neg this, int.cast_neg, int.cast_one], } }\nend\n\nlemma legendre_sym_eq_one_or_neg_one (a p : \u2115) (ha : (a : zmod p) \u2260 0) :\n  legendre_sym a p = -1 \u2228 legendre_sym a p = 1 :=\nby unfold legendre_sym; split_ifs; simp only [*, eq_self_iff_true, or_true, true_or] at *\n\nlemma legendre_sym_eq_zero_iff (a p : \u2115) :\n  legendre_sym a p = 0 \u2194 (a : zmod p) = 0 :=\nbegin\n  split,\n  { classical, contrapose,\n    assume ha, cases legendre_sym_eq_one_or_neg_one a p ha with h h,\n    all_goals { rw h, norm_num } },\n  { assume ha, rw [legendre_sym, if_pos ha] }\nend\n\n\/-- Gauss' lemma. The legendre symbol can be computed by considering the number of naturals less\n  than `p\/2` such that `(a * x) % p > p \/ 2` -\/\nlemma gauss_lemma {a : \u2115} [hp1 : fact (p % 2 = 1)] (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = (-1) ^ ((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card :=\nhave (legendre_sym a p : zmod p) = (((-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card : \u2124) : zmod p),\n  by rw [legendre_sym_eq_pow, gauss_lemma_aux\u2082 p ha0]; simp,\nbegin\n  cases legendre_sym_eq_one_or_neg_one a p ha0;\n  cases @neg_one_pow_eq_or \u2124 _  ((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card;\n  simp [*, ne_neg_self p one_ne_zero, (ne_neg_self p one_ne_zero).symm] at *\nend\n\nlemma legendre_sym_eq_one_iff {a : \u2115} (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = 1 \u2194 (\u2203 b : zmod p, b ^ 2 = a) :=\nbegin\n  rw [euler_criterion p ha0, legendre_sym, if_neg ha0],\n  split_ifs,\n  { simp only [h, eq_self_iff_true] },\n  finish -- this is quite slow. I'm actually surprised that it can close the goal at all!\nend\n\nlemma eisenstein_lemma [hp1 : fact (p % 2 = 1)] {a : \u2115} (ha1 : a % 2 = 1) (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = (-1)^(Ico 1 (p \/ 2).succ).sum (\u03bb x, (x * a) \/ p) :=\nby rw [neg_one_pow_eq_pow_mod_two, gauss_lemma p ha0, neg_one_pow_eq_pow_mod_two,\n    show _ = _, from eisenstein_lemma_aux\u2082 p ha1 ha0]\n\ntheorem quadratic_reciprocity [hp1 : fact (p % 2 = 1)] [hq1 : fact (q % 2 = 1)] (hpq : p \u2260 q) :\n  legendre_sym p q * legendre_sym q p = (-1) ^ ((p \/ 2) * (q \/ 2)) :=\nhave hpq0 : (p : zmod q) \u2260 0, from prime_ne_zero q p hpq.symm,\nhave hqp0 : (q : zmod p) \u2260 0, from prime_ne_zero p q hpq,\nby rw [eisenstein_lemma q hp1 hpq0, eisenstein_lemma p hq1 hqp0,\n  \u2190 _root_.pow_add, sum_mul_div_add_sum_mul_div_eq_mul q p hpq0, mul_comm]\n\n-- move this\ninstance fact_prime_two : fact (nat.prime 2) := nat.prime_two\n\nlemma legendre_sym_two [hp1 : fact (p % 2 = 1)] : legendre_sym 2 p = (-1) ^ (p \/ 4 + p \/ 2) :=\nhave hp2 : p \u2260 2, from mt (congr_arg (% 2)) (by simpa using hp1),\nhave hp22 : p \/ 2 \/ 2 = _ := div_eq_filter_card (show 0 < 2, from dec_trivial)\n  (nat.div_le_self (p \/ 2) 2),\nhave hcard : (Ico 1 (p \/ 2).succ).card = p \/ 2, by simp,\nhave hx2 : \u2200 x \u2208 Ico 1 (p \/ 2).succ, (2 * x : zmod p).val = 2 * x,\n  from \u03bb x hx, have h2xp : 2 * x < p,\n      from calc 2 * x \u2264 2 * (p \/ 2) : mul_le_mul_of_nonneg_left\n        (le_of_lt_succ $ by finish) dec_trivial\n      ... < _ : by conv_rhs {rw [\u2190 mod_add_div p 2, add_comm, show p % 2 = 1, from hp1]}; exact lt_succ_self _,\n    by rw [\u2190 nat.cast_two, \u2190 nat.cast_mul, val_cast_of_lt h2xp],\nhave hdisj : disjoint\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, p \/ 2 < ((2 : \u2115) * x : zmod p).val))\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, x * 2 \u2264 p \/ 2)),\n  from disjoint_filter.2 (\u03bb x hx, by simp [hx2 _ hx, mul_comm]),\nhave hunion :\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, p \/ 2 < ((2 : \u2115) * x : zmod p).val)) \u222a\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, x * 2 \u2264 p \/ 2)) =\n    Ico 1 (p \/ 2).succ,\n  begin\n    rw [filter_union_right],\n    conv_rhs {rw [\u2190 @filter_true _ (Ico 1 (p \/ 2).succ)]},\n    exact filter_congr (\u03bb x hx, by simp [hx2 _ hx, lt_or_le, mul_comm])\n  end,\nbegin\n  rw [gauss_lemma p (prime_ne_zero p 2 hp2),\n    neg_one_pow_eq_pow_mod_two, @neg_one_pow_eq_pow_mod_two _ _ (p \/ 4 + p \/ 2)],\n  refine congr_arg2 _ rfl ((eq_iff_modeq_nat 2).1 _),\n  rw [show 4 = 2 * 2, from rfl, \u2190 nat.div_div_eq_div_mul, hp22, nat.cast_add,\n    \u2190 sub_eq_iff_eq_add', sub_eq_add_neg, neg_eq_self_mod_two,\n    \u2190 nat.cast_add, \u2190 card_disjoint_union hdisj, hunion, hcard]\nend\n\nlemma exists_pow_two_eq_two_iff [hp1 : fact (p % 2 = 1)] :\n  (\u2203 a : zmod p, a ^ 2 = 2) \u2194 p % 8 = 1 \u2228 p % 8 = 7 :=\nhave hp2 : ((2 : \u2115) : zmod p) \u2260 0,\n  from prime_ne_zero p 2 (\u03bb h, by simpa [h] using hp1),\nhave hpm4 : p % 4 = p % 8 % 4, from (nat.mod_mul_left_mod p 2 4).symm,\nhave hpm2 : p % 2 = p % 8 % 2, from (nat.mod_mul_left_mod p 4 2).symm,\nbegin\n  rw [show (2 : zmod p) = (2 : \u2115), by simp, \u2190 legendre_sym_eq_one_iff p hp2,\n    legendre_sym_two p, neg_one_pow_eq_one_iff_even (show (-1 : \u2124) \u2260 1, from dec_trivial),\n    even_add, even_div, even_div],\n  have := nat.mod_lt p (show 0 < 8, from dec_trivial),\n  resetI, rw _root_.fact at hp1,\n  revert this hp1,\n  erw [hpm4, hpm2],\n  generalize hm : p % 8 = m,\n  clear hm,\n  revert m,\n  exact dec_trivial\nend\n\nlemma exists_pow_two_eq_prime_iff_of_mod_four_eq_one (hp1 : p % 4 = 1) [hq1 : fact (q % 2 = 1)] :\n  (\u2203 a : zmod p, a ^ 2 = q) \u2194 \u2203 b : zmod q, b ^ 2 = p :=\nif hpq : p = q then by resetI; subst hpq else\nhave h1 : ((p \/ 2) * (q \/ 2)) % 2 = 0,\n  from (dvd_iff_mod_eq_zero _ _).1\n    (dvd_mul_of_dvd_left ((dvd_iff_mod_eq_zero _ _).2 $\n    by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp1]; refl) _),\nbegin\n  haveI hp_odd : fact (p % 2 = 1) := odd_of_mod_four_eq_one hp1,\n  have hpq0 : (p : zmod q) \u2260 0 := prime_ne_zero q p (ne.symm hpq),\n  have hqp0 : (q : zmod p) \u2260 0 := prime_ne_zero p q hpq,\n  have := quadratic_reciprocity p q hpq,\n  rw [neg_one_pow_eq_pow_mod_two, h1, legendre_sym, legendre_sym,\n    if_neg hqp0, if_neg hpq0] at this,\n  rw [euler_criterion q hpq0, euler_criterion p hqp0],\n  split_ifs at this; simp *; contradiction,\nend\n\nlemma exists_pow_two_eq_prime_iff_of_mod_four_eq_three (hp3 : p % 4 = 3)\n  (hq3 : q % 4 = 3) (hpq : p \u2260 q) : (\u2203 a : zmod p, a ^ 2 = q) \u2194 \u00ac\u2203 b : zmod q, b ^ 2 = p :=\nhave h1 : ((p \/ 2) * (q \/ 2)) % 2 = 1,\n  from nat.odd_mul_odd\n    (by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp3]; refl)\n    (by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hq3]; refl),\nbegin\n  haveI hp_odd : fact (p % 2 = 1) := odd_of_mod_four_eq_three hp3,\n  haveI hq_odd : fact (q % 2 = 1) := odd_of_mod_four_eq_three hq3,\n  have hpq0 : (p : zmod q) \u2260 0 := prime_ne_zero q p (ne.symm hpq),\n  have hqp0 : (q : zmod p) \u2260 0 := prime_ne_zero p q hpq,\n  have := quadratic_reciprocity p q hpq,\n  rw [neg_one_pow_eq_pow_mod_two, h1, legendre_sym, legendre_sym,\n    if_neg hpq0, if_neg hqp0] at this,\n  rw [euler_criterion q hpq0, euler_criterion p hqp0],\n  split_ifs at this; simp *; contradiction\nend\n\nend zmod\n","avg_line_length":45.4794776119,"max_line_length":111,"alphanum_fraction":0.5998687287}
-{"size":491,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2021 Joscha Mennicken. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Joscha Mennicken\n-\/\n\nimport Lean.Data.Lsp.Basic\nimport Lean.Data.Json\n\nnamespace Lean\nnamespace Lsp\n\nopen Json\n\nstructure Registration where\n  id : String\n  method : String\n  registerOptions : Option Json\n  deriving ToJson, FromJson\n\nstructure RegistrationParams where\n  registrations : Array Registration\n  deriving ToJson, FromJson\n\nend Lsp\nend Lean\n","avg_line_length":17.5357142857,"max_line_length":67,"alphanum_fraction":0.7861507128}
-{"size":8947,"ext":"lean","lang":"Lean","max_stars_count":78.0,"content":"\/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-\/\nimport model_theory.substructures\n\n\/-!\n# Finitely Generated First-Order Structures\nThis file defines what it means for a first-order (sub)structure to be finitely or countably\ngenerated, similarly to other finitely-generated objects in the algebra library.\n\n## Main Definitions\n* `first_order.language.substructure.fg` indicates that a substructure is finitely generated.\n* `first_order.language.Structure.fg` indicates that a structure is finitely generated.\n* `first_order.language.substructure.cg` indicates that a substructure is countably generated.\n* `first_order.language.Structure.cg` indicates that a structure is countably generated.\n\n\n## TODO\nDevelop a more unified definition of finite generation using the theory of closure operators, or use\nthis definition of finite generation to define the others.\n\n-\/\n\nopen_locale first_order\nopen set\n\nnamespace first_order\nnamespace language\nopen Structure\n\nvariables {L : language} {M : Type*} [L.Structure M]\n\nnamespace substructure\n\n\/-- A substructure of `M` is finitely generated if it is the closure of a finite subset of `M`. -\/\ndef fg (N : L.substructure M) : Prop := \u2203 S : finset M, closure L \u2191S = N\n\ntheorem fg_def {N : L.substructure M} :\n  N.fg \u2194 \u2203 S : set M, S.finite \u2227 closure L S = N :=\n\u27e8\u03bb \u27e8t, h\u27e9, \u27e8_, finset.finite_to_set t, h\u27e9, begin\n  rintro \u27e8t', h, rfl\u27e9,\n  rcases finite.exists_finset_coe h with \u27e8t, rfl\u27e9,\n  exact \u27e8t, rfl\u27e9\nend\u27e9\n\nlemma fg_iff_exists_fin_generating_family {N : L.substructure M} :\n  N.fg \u2194 \u2203 (n : \u2115) (s : fin n \u2192 M), closure L (range s) = N :=\nbegin\n  rw fg_def,\n  split,\n  { rintros \u27e8S, Sfin, hS\u27e9,\n    obtain \u27e8n, f, rfl\u27e9 := Sfin.fin_embedding,\n    exact \u27e8n, f, hS\u27e9, },\n  { rintros \u27e8n, s, hs\u27e9,\n    refine \u27e8range s, finite_range s, hs\u27e9 },\nend\n\ntheorem fg_bot : (\u22a5 : L.substructure M).fg :=\n\u27e8\u2205, by rw [finset.coe_empty, closure_empty]\u27e9\n\ntheorem fg_closure {s : set M} (hs : s.finite) : fg (closure L s) :=\n\u27e8hs.to_finset, by rw [hs.coe_to_finset]\u27e9\n\ntheorem fg_closure_singleton (x : M) : fg (closure L ({x} : set M)) :=\nfg_closure (finite_singleton x)\n\ntheorem fg.sup {N\u2081 N\u2082 : L.substructure M}\n  (hN\u2081 : N\u2081.fg) (hN\u2082 : N\u2082.fg) : (N\u2081 \u2294 N\u2082).fg :=\nlet \u27e8t\u2081, ht\u2081\u27e9 := fg_def.1 hN\u2081, \u27e8t\u2082, ht\u2082\u27e9 := fg_def.1 hN\u2082 in\nfg_def.2 \u27e8t\u2081 \u222a t\u2082, ht\u2081.1.union ht\u2082.1, by rw [closure_union, ht\u2081.2, ht\u2082.2]\u27e9\n\ntheorem fg.map {N : Type*} [L.Structure N] (f : M \u2192[L] N) {s : L.substructure M} (hs : s.fg) :\n  (s.map f).fg :=\nlet \u27e8t, ht\u27e9 := fg_def.1 hs in fg_def.2 \u27e8f '' t, ht.1.image _, by rw [closure_image, ht.2]\u27e9\n\ntheorem fg.of_map_embedding {N : Type*} [L.Structure N] (f : M \u21aa[L] N) {s : L.substructure M}\n  (hs : (s.map f.to_hom).fg) : s.fg :=\nbegin\n  rcases hs with \u27e8t, h\u27e9,\n  rw fg_def,\n  refine \u27e8f \u207b\u00b9' t, t.finite_to_set.preimage (f.injective.inj_on _), _\u27e9,\n  have hf : function.injective f.to_hom := f.injective,\n  refine map_injective_of_injective hf _,\n  rw [\u2190 h, map_closure, embedding.coe_to_hom, image_preimage_eq_of_subset],\n  intros x hx,\n  have h' := subset_closure hx,\n  rw h at h',\n  exact hom.map_le_range h'\nend\n\n\/-- A substructure of `M` is countably generated if it is the closure of a countable subset of `M`.\n-\/\ndef cg (N : L.substructure M) : Prop := \u2203 S : set M, S.countable \u2227 closure L S = N\n\ntheorem cg_def {N : L.substructure M} :\n  N.cg \u2194 \u2203 S : set M, S.countable \u2227 closure L S = N := iff.refl _\n\ntheorem fg.cg {N : L.substructure M} (h : N.fg) : N.cg :=\nbegin\n  obtain \u27e8s, hf, rfl\u27e9 := fg_def.1 h,\n  refine \u27e8s, hf.countable, rfl\u27e9,\nend\n\nlemma cg_iff_empty_or_exists_nat_generating_family {N : L.substructure M} :\n  N.cg \u2194 (\u2191N = (\u2205 : set M)) \u2228 \u2203 (s : \u2115 \u2192 M), closure L (range s) = N :=\nbegin\n  rw cg_def,\n  split,\n  { rintros \u27e8S, Scount, hS\u27e9,\n    cases eq_empty_or_nonempty \u2191N with h h,\n    { exact or.intro_left _ h },\n    obtain \u27e8f, h'\u27e9 := (Scount.union (set.countable_singleton h.some)).exists_surjective\n      (singleton_nonempty h.some).inr,\n    refine or.intro_right _ \u27e8f, _\u27e9,\n    rw [\u2190 h', closure_union, hS, sup_eq_left, closure_le],\n    exact singleton_subset_iff.2 h.some_mem },\n  { intro h,\n    cases h with h h,\n    { refine \u27e8\u2205, countable_empty, closure_eq_of_le (empty_subset _) _\u27e9,\n      rw [\u2190 set_like.coe_subset_coe, h],\n      exact empty_subset _ },\n    { obtain \u27e8f, rfl\u27e9 := h,\n      exact \u27e8range f, countable_range _, rfl\u27e9 } },\nend\n\ntheorem cg_bot : (\u22a5 : L.substructure M).cg := fg_bot.cg\n\ntheorem cg_closure {s : set M} (hs : s.countable) : cg (closure L s) :=\n\u27e8s, hs, rfl\u27e9\n\ntheorem cg_closure_singleton (x : M) : cg (closure L ({x} : set M)) := (fg_closure_singleton x).cg\n\ntheorem cg.sup {N\u2081 N\u2082 : L.substructure M}\n  (hN\u2081 : N\u2081.cg) (hN\u2082 : N\u2082.cg) : (N\u2081 \u2294 N\u2082).cg :=\nlet \u27e8t\u2081, ht\u2081\u27e9 := cg_def.1 hN\u2081, \u27e8t\u2082, ht\u2082\u27e9 := cg_def.1 hN\u2082 in\ncg_def.2 \u27e8t\u2081 \u222a t\u2082, ht\u2081.1.union ht\u2082.1, by rw [closure_union, ht\u2081.2, ht\u2082.2]\u27e9\n\ntheorem cg.map {N : Type*} [L.Structure N] (f : M \u2192[L] N) {s : L.substructure M} (hs : s.cg) :\n  (s.map f).cg :=\nlet \u27e8t, ht\u27e9 := cg_def.1 hs in cg_def.2 \u27e8f '' t, ht.1.image _, by rw [closure_image, ht.2]\u27e9\n\ntheorem cg.of_map_embedding {N : Type*} [L.Structure N] (f : M \u21aa[L] N) {s : L.substructure M}\n  (hs : (s.map f.to_hom).cg) : s.cg :=\nbegin\n  rcases hs with \u27e8t, h1, h2\u27e9,\n  rw cg_def,\n  refine \u27e8f \u207b\u00b9' t, h1.preimage f.injective, _\u27e9,\n  have hf : function.injective f.to_hom := f.injective,\n  refine map_injective_of_injective hf _,\n  rw [\u2190 h2, map_closure, embedding.coe_to_hom, image_preimage_eq_of_subset],\n  intros x hx,\n  have h' := subset_closure hx,\n  rw h2 at h',\n  exact hom.map_le_range h'\nend\n\ntheorem cg_iff_countable [L.countable_functions] {s : L.substructure M} :\n  s.cg \u2194 nonempty (encodable s) :=\nbegin\n  refine \u27e8_, \u03bb h, \u27e8s, h, s.closure_eq\u27e9\u27e9,\n  rintro \u27e8s, h, rfl\u27e9,\n  exact h.substructure_closure L\nend\n\nend substructure\n\nopen substructure\n\nnamespace Structure\n\nvariables (L) (M)\n\n\/-- A structure is finitely generated if it is the closure of a finite subset. -\/\nclass fg : Prop := (out : (\u22a4 : L.substructure M).fg)\n\n\/-- A structure is countably generated if it is the closure of a countable subset. -\/\nclass cg : Prop := (out : (\u22a4 : L.substructure M).cg)\n\nvariables {L M}\n\nlemma fg_def : fg L M \u2194 (\u22a4 : L.substructure M).fg := \u27e8\u03bb h, h.1, \u03bb h, \u27e8h\u27e9\u27e9\n\n\/-- An equivalent expression of `Structure.fg` in terms of `set.finite` instead of `finset`. -\/\nlemma fg_iff : fg L M \u2194 \u2203 S : set M, S.finite \u2227 closure L S = (\u22a4 : L.substructure M) :=\nby rw [fg_def, substructure.fg_def]\n\nlemma fg.range {N : Type*} [L.Structure N] (h : fg L M) (f : M \u2192[L] N) :\n  f.range.fg :=\nbegin\n  rw [hom.range_eq_map],\n  exact (fg_def.1 h).map f,\nend\n\nlemma fg.map_of_surjective {N : Type*} [L.Structure N] (h : fg L M) (f : M \u2192[L] N)\n  (hs : function.surjective f) :\n  fg L N :=\nbegin\n  rw \u2190 hom.range_eq_top at hs,\n  rw [fg_def, \u2190 hs],\n  exact h.range f,\nend\n\nlemma cg_def : cg L M \u2194 (\u22a4 : L.substructure M).cg := \u27e8\u03bb h, h.1, \u03bb h, \u27e8h\u27e9\u27e9\n\n\/-- An equivalent expression of `Structure.cg`. -\/\nlemma cg_iff : cg L M \u2194 \u2203 S : set M, S.countable \u2227 closure L S = (\u22a4 : L.substructure M) :=\nby rw [cg_def, substructure.cg_def]\n\nlemma cg.range {N : Type*} [L.Structure N] (h : cg L M) (f : M \u2192[L] N) :\n  f.range.cg :=\nbegin\n  rw [hom.range_eq_map],\n  exact (cg_def.1 h).map f,\nend\n\nlemma cg.map_of_surjective {N : Type*} [L.Structure N] (h : cg L M) (f : M \u2192[L] N)\n  (hs : function.surjective f) :\n  cg L N :=\nbegin\n  rw \u2190 hom.range_eq_top at hs,\n  rw [cg_def, \u2190 hs],\n  exact h.range f,\nend\n\nlemma cg_iff_countable [L.countable_functions] :\n  cg L M \u2194 nonempty (encodable M) :=\nby rw [cg_def, cg_iff_countable, cardinal.encodable_iff, cardinal.encodable_iff,\n  top_equiv.to_equiv.cardinal_eq]\n\nlemma fg.cg (h : fg L M) : cg L M :=\ncg_def.2 (fg_def.1 h).cg\n\n@[priority 100] instance cg_of_fg [h : fg L M] : cg L M := h.cg\n\nend Structure\n\nlemma equiv.fg_iff {N : Type*} [L.Structure N] (f : M \u2243[L] N) :\n  Structure.fg L M \u2194 Structure.fg L N :=\n\u27e8\u03bb h, h.map_of_surjective f.to_hom f.to_equiv.surjective,\n  \u03bb h, h.map_of_surjective f.symm.to_hom f.to_equiv.symm.surjective\u27e9\n\nlemma substructure.fg_iff_Structure_fg (S : L.substructure M) :\n  S.fg \u2194 Structure.fg L S :=\nbegin\n  rw Structure.fg_def,\n  refine \u27e8\u03bb h, fg.of_map_embedding S.subtype _, \u03bb h, _\u27e9,\n  { rw [\u2190 hom.range_eq_map, range_subtype],\n    exact h },\n  { have h := h.map S.subtype.to_hom,\n    rw [\u2190 hom.range_eq_map, range_subtype] at h,\n    exact h }\nend\n\nlemma equiv.cg_iff {N : Type*} [L.Structure N] (f : M \u2243[L] N) :\n  Structure.cg L M \u2194 Structure.cg L N :=\n\u27e8\u03bb h, h.map_of_surjective f.to_hom f.to_equiv.surjective,\n  \u03bb h, h.map_of_surjective f.symm.to_hom f.to_equiv.symm.surjective\u27e9\n\nlemma substructure.cg_iff_Structure_cg (S : L.substructure M) :\n  S.cg \u2194 Structure.cg L S :=\nbegin\n  rw Structure.cg_def,\n  refine \u27e8\u03bb h, cg.of_map_embedding S.subtype _, \u03bb h, _\u27e9,\n  { rw [\u2190 hom.range_eq_map, range_subtype],\n    exact h },\n  { have h := h.map S.subtype.to_hom,\n    rw [\u2190 hom.range_eq_map, range_subtype] at h,\n    exact h }\nend\n\nend language\nend first_order\n","avg_line_length":32.5345454545,"max_line_length":100,"alphanum_fraction":0.6595506874}
-{"size":137021,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-\/\nimport analysis.calculus.mean_value\nimport analysis.normed_space.multilinear\nimport analysis.calculus.formal_multilinear_series\n\n\/-!\n# Higher differentiability\n\nA function is `C^1` on a domain if it is differentiable there, and its derivative is continuous.\nBy induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1` there or,\nequivalently, if it is `C^1` and its derivative is `C^{n-1}`.\nFinally, it is `C^\u221e` if it is `C^n` for all n.\n\nWe formalize these notions by defining iteratively the `n+1`-th derivative of a function as the\nderivative of the `n`-th derivative. It is called `iterated_fderiv \ud835\udd5c n f x` where `\ud835\udd5c` is the\nfield, `n` is the number of iterations, `f` is the function and `x` is the point, and it is given\nas an `n`-multilinear map. We also define a version `iterated_fderiv_within` relative to a domain,\nas well as predicates `times_cont_diff_within_at`, `times_cont_diff_at`, `times_cont_diff_on` and\n`times_cont_diff` saying that the function is `C^n` within a set at a point, at a point, on a set\nand on the whole space respectively.\n\nTo avoid the issue of choice when choosing a derivative in sets where the derivative is not\nnecessarily unique, `times_cont_diff_on` is not defined directly in terms of the\nregularity of the specific choice `iterated_fderiv_within \ud835\udd5c n f s` inside `s`, but in terms of the\nexistence of a nice sequence of derivatives, expressed with a predicate\n`has_ftaylor_series_up_to_on`.\n\nWe prove basic properties of these notions.\n\n## Main definitions and results\nLet `f : E \u2192 F` be a map between normed vector spaces over a nondiscrete normed field `\ud835\udd5c`.\n\n* `has_ftaylor_series_up_to n f p`: expresses that the formal multilinear series `p` is a sequence\n  of iterated derivatives of `f`, up to the `n`-th term (where `n` is a natural number or `\u221e`).\n* `has_ftaylor_series_up_to_on n f p s`: same thing, but inside a set `s`. The notion of derivative\n  is now taken inside `s`. In particular, derivatives don't have to be unique.\n* `times_cont_diff \ud835\udd5c n f`: expresses that `f` is `C^n`, i.e., it admits a Taylor series up to\n  rank `n`.\n* `times_cont_diff_on \ud835\udd5c n f s`: expresses that `f` is `C^n` in `s`.\n* `times_cont_diff_at \ud835\udd5c n f x`: expresses that `f` is `C^n` around `x`.\n* `times_cont_diff_within_at \ud835\udd5c n f s x`: expresses that `f` is `C^n` around `x` within the set `s`.\n* `iterated_fderiv_within \ud835\udd5c n f s x` is an `n`-th derivative of `f` over the field `\ud835\udd5c` on the\n  set `s` at the point `x`. It is a continuous multilinear map from `E^n` to `F`, defined as a\n  derivative within `s` of `iterated_fderiv_within \ud835\udd5c (n-1) f s` if one exists, and `0` otherwise.\n* `iterated_fderiv \ud835\udd5c n f x` is the `n`-th derivative of `f` over the field `\ud835\udd5c` at the point `x`.\n  It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of\n  `iterated_fderiv \ud835\udd5c (n-1) f` if one exists, and `0` otherwise.\n\nIn sets of unique differentiability, `times_cont_diff_on \ud835\udd5c n f s` can be expressed in terms of the\nproperties of `iterated_fderiv_within \ud835\udd5c m f s` for `m \u2264 n`. In the whole space,\n`times_cont_diff \ud835\udd5c n f` can be expressed in terms of the properties of `iterated_fderiv \ud835\udd5c m f`\nfor `m \u2264 n`.\n\nWe also prove that the usual operations (addition, multiplication, difference, composition, and\nso on) preserve `C^n` functions.\n\n## Implementation notes\n\nThe definitions in this file are designed to work on any field `\ud835\udd5c`. They are sometimes slightly more\ncomplicated than the naive definitions one would guess from the intuition over the real or complex\nnumbers, but they are designed to circumvent the lack of gluing properties and partitions of unity\nin general. In the usual situations, they coincide with the usual definitions.\n\n### Definition of `C^n` functions in domains\n\nOne could define `C^n` functions in a domain `s` by fixing an arbitrary choice of derivatives (this\nis what we do with `iterated_fderiv_within`) and requiring that all these derivatives up to `n` are\ncontinuous. If the derivative is not unique, this could lead to strange behavior like two `C^n`\nfunctions `f` and `g` on `s` whose sum is not `C^n`. A better definition is thus to say that a\nfunction is `C^n` inside `s` if it admits a sequence of derivatives up to `n` inside `s`.\n\nThis definition still has the problem that a function which is locally `C^n` would not need to\nbe `C^n`, as different choices of sequences of derivatives around different points might possibly\nnot be glued together to give a globally defined sequence of derivatives. (Note that this issue\ncan not happen over reals, thanks to partition of unity, but the behavior over a general field is\nnot so clear, and we want a definition for general fields). Also, there are locality\nproblems for the order parameter: one could image a function which, for each `n`, has a nice\nsequence of derivatives up to order `n`, but they do not coincide for varying `n` and can therefore\nnot be  glued to give rise to an infinite sequence of derivatives. This would give a function\nwhich is `C^n` for all `n`, but not `C^\u221e`. We solve this issue by putting locality conditions\nin space and order in our definition of `times_cont_diff_within_at` and `times_cont_diff_on`.\nThe resulting definition is slightly more complicated to work with (in fact not so much), but it\ngives rise to completely satisfactory theorems.\n\nFor instance, with this definition, a real function which is `C^m` (but not better) on `(-1\/m, 1\/m)`\nfor each natural `m` is by definition `C^\u221e` at `0`.\n\nThere is another issue with the definition of `times_cont_diff_within_at \ud835\udd5c n f s x`. We can\nrequire the existence and good behavior of derivatives up to order `n` on a neighborhood of `x`\nwithin `s`. However, this does not imply continuity or differentiability within `s` of the function\nat `x` when `x` does not belong to `s`. Therefore, we require such existence and good behavior on\na neighborhood of `x` within `s \u222a {x}` (which appears as `insert x s` in this file).\n\n### Side of the composition, and universe issues\n\nWith a na\u00efve direct definition, the `n`-th derivative of a function belongs to the space\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] (E ... F)...)))` where there are n iterations of `E \u2192L[\ud835\udd5c]`. This space\nmay also be seen as the space of continuous multilinear functions on `n` copies of `E` with\nvalues in `F`, by uncurrying. This is the point of view that is usually adopted in textbooks,\nand that we also use. This means that the definition and the first proofs are slightly involved,\nas one has to keep track of the uncurrying operation. The uncurrying can be done from the\nleft or from the right, amounting to defining the `n+1`-th derivative either as the derivative of\nthe `n`-th derivative, or as the `n`-th derivative of the derivative.\nFor proofs, it would be more convenient to use the latter approach (from the right),\nas it means to prove things at the `n+1`-th step we only need to understand well enough the\nderivative in `E \u2192L[\ud835\udd5c] F` (contrary to the approach from the left, where one would need to know\nenough on the `n`-th derivative to deduce things on the `n+1`-th derivative).\n\nHowever, the definition from the right leads to a universe polymorphism problem: if we define\n`iterated_fderiv \ud835\udd5c (n + 1) f x = iterated_fderiv \ud835\udd5c n (fderiv \ud835\udd5c f) x` by induction, we need to\ngeneralize over all spaces (as `f` and `fderiv \ud835\udd5c f` don't take values in the same space). It is\nonly possible to generalize over all spaces in some fixed universe in an inductive definition.\nFor `f : E \u2192 F`, then `fderiv \ud835\udd5c f` is a map `E \u2192 (E \u2192L[\ud835\udd5c] F)`. Therefore, the definition will only\nwork if `F` and `E \u2192L[\ud835\udd5c] F` are in the same universe.\n\nThis issue does not appear with the definition from the left, where one does not need to generalize\nover all spaces. Therefore, we use the definition from the left. This means some proofs later on\nbecome a little bit more complicated: to prove that a function is `C^n`, the most efficient approach\nis to exhibit a formula for its `n`-th derivative and prove it is continuous (contrary to the\ninductive approach where one would prove smoothness statements without giving a formula for the\nderivative). In the end, this approach is still satisfactory as it is good to have formulas for the\niterated derivatives in various constructions.\n\nOne point where we depart from this explicit approach is in the proof of smoothness of a\ncomposition: there is a formula for the `n`-th derivative of a composition (Fa\u00e0 di Bruno's formula),\nbut it is very complicated and barely usable, while the inductive proof is very simple. Thus, we\ngive the inductive proof. As explained above, it works by generalizing over the target space, hence\nit only works well if all spaces belong to the same universe. To get the general version, we lift\nthings to a common universe using a trick.\n\n### Variables management\n\nThe textbook definitions and proofs use various identifications and abuse of notations, for instance\nwhen saying that the natural space in which the derivative lives, i.e.,\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] ( ... \u2192L[\ud835\udd5c] F))`, is the same as a space of multilinear maps. When doing things\nformally, we need to provide explicit maps for these identifications, and chase some diagrams to see\neverything is compatible with the identifications. In particular, one needs to check that taking the\nderivative and then doing the identification, or first doing the identification and then taking the\nderivative, gives the same result. The key point for this is that taking the derivative commutes\nwith continuous linear equivalences. Therefore, we need to implement all our identifications with\ncontinuous linear equivs.\n\n## Notations\n\nWe use the notation `E [\u00d7n]\u2192L[\ud835\udd5c] F` for the space of continuous multilinear maps on `E^n` with\nvalues in `F`. This is the space in which the `n`-th derivative of a function from `E` to `F` lives.\n\nIn this file, we denote `\u22a4 : with_top \u2115` with `\u221e`.\n\n## Tags\n\nderivative, differentiability, higher derivative, `C^n`, multilinear, Taylor series, formal series\n-\/\n\nnoncomputable theory\nopen_locale classical big_operators nnreal\n\nlocal notation `\u221e` := (\u22a4 : with_top \u2115)\n\nuniverses u v w\n\nlocal attribute [instance, priority 1001]\nnormed_group.to_add_comm_group normed_space.to_module add_comm_group.to_add_comm_monoid\n\nopen set fin filter\nopen_locale topological_space\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n{E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n{F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n{G : Type*} [normed_group G] [normed_space \ud835\udd5c G]\n{s s\u2081 t u : set E} {f f\u2081 : E \u2192 F} {g : F \u2192 G} {x : E} {c : F}\n{b : E \u00d7 F \u2192 G}\n\n\/-! ### Functions with a Taylor series on a domain -\/\n\nvariable {p : E \u2192 formal_multilinear_series \ud835\udd5c E F}\n\n\/-- `has_ftaylor_series_up_to_on n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`has_fderiv_within_at` but for higher order derivatives. -\/\nstructure has_ftaylor_series_up_to_on (n : with_top \u2115)\n  (f : E \u2192 F) (p : E \u2192 formal_multilinear_series \ud835\udd5c E F) (s : set E) : Prop :=\n(zero_eq       : \u2200 x \u2208 s, (p x 0).uncurry0 = f x)\n(fderiv_within : \u2200 (m : \u2115) (hm : (m : with_top \u2115) < n), \u2200 x \u2208 s,\n   has_fderiv_within_at (\u03bb y, p y m) (p x m.succ).curry_left s x)\n(cont          : \u2200 (m : \u2115) (hm : (m : with_top \u2115) \u2264 n), continuous_on (\u03bb x, p x m) s)\n\nlemma has_ftaylor_series_up_to_on.zero_eq' {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) {x : E} (hx : x \u2208 s) :\n  p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x) :=\nby { rw \u2190 h.zero_eq x hx, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ }\n\n\/-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a\nTaylor series for the second one. -\/\nlemma has_ftaylor_series_up_to_on.congr {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  has_ftaylor_series_up_to_on n f\u2081 p s :=\nbegin\n  refine \u27e8\u03bb x hx, _, h.fderiv_within, h.cont\u27e9,\n  rw h\u2081 x hx,\n  exact h.zero_eq x hx\nend\n\nlemma has_ftaylor_series_up_to_on.mono {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) {t : set E} (hst : t \u2286 s) :\n  has_ftaylor_series_up_to_on n f p t :=\n\u27e8\u03bb x hx, h.zero_eq x (hst hx),\n\u03bb m hm x hx, (h.fderiv_within m hm x (hst hx)).mono hst,\n\u03bb m hm, (h.cont m hm).mono hst\u27e9\n\nlemma has_ftaylor_series_up_to_on.of_le {m n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hmn : m \u2264 n) :\n  has_ftaylor_series_up_to_on m f p s :=\n\u27e8h.zero_eq,\n\u03bb k hk x hx, h.fderiv_within k (lt_of_lt_of_le hk hmn) x hx,\n\u03bb k hk, h.cont k (le_trans hk hmn)\u27e9\n\nlemma has_ftaylor_series_up_to_on.continuous_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) : continuous_on f s :=\nbegin\n  have := (h.cont 0 bot_le).congr (\u03bb x hx, (h.zero_eq' hx).symm),\n  rwa linear_isometry_equiv.comp_continuous_on_iff at this\nend\n\nlemma has_ftaylor_series_up_to_on_zero_iff :\n  has_ftaylor_series_up_to_on 0 f p s \u2194 continuous_on f s \u2227 (\u2200 x \u2208 s, (p x 0).uncurry0 = f x) :=\nbegin\n  refine \u27e8\u03bb H, \u27e8H.continuous_on, H.zero_eq\u27e9,\n          \u03bb H, \u27e8H.2, \u03bb m hm, false.elim (not_le.2 hm bot_le), _\u27e9\u27e9,\n  assume m hm,\n  obtain rfl : m = 0, by exact_mod_cast (hm.antisymm (zero_le _)),\n  have : \u2200 x \u2208 s, p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x),\n    by { assume x hx, rw \u2190 H.2 x hx, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ },\n  rw [continuous_on_congr this, linear_isometry_equiv.comp_continuous_on_iff],\n  exact H.1\nend\n\nlemma has_ftaylor_series_up_to_on_top_iff :\n  (has_ftaylor_series_up_to_on \u221e f p s) \u2194 (\u2200 (n : \u2115), has_ftaylor_series_up_to_on n f p s) :=\nbegin\n  split,\n  { assume H n, exact H.of_le le_top },\n  { assume H,\n    split,\n    { exact (H 0).zero_eq },\n    { assume m hm,\n      apply (H m.succ).fderiv_within m (with_top.coe_lt_coe.2 (lt_add_one m)) },\n    { assume m hm,\n      apply (H m).cont m (le_refl _) } }\nend\n\n\/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -\/\nlemma has_ftaylor_series_up_to_on.has_fderiv_within_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hx : x \u2208 s) :\n  has_fderiv_within_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) s x :=\nbegin\n  have A : \u2200 y \u2208 s, f y = (continuous_multilinear_curry_fin0 \ud835\udd5c E F) (p y 0),\n  { assume y hy, rw \u2190 h.zero_eq y hy, refl },\n  suffices H : has_fderiv_within_at\n      (\u03bb y, continuous_multilinear_curry_fin0 \ud835\udd5c E F (p y 0))\n      (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) s x,\n    by exact H.congr A (A x hx),\n  rw linear_isometry_equiv.comp_has_fderiv_within_at_iff',\n  have : ((0 : \u2115) : with_top \u2115) < n :=\n    lt_of_lt_of_le (with_top.coe_lt_coe.2 nat.zero_lt_one) hn,\n  convert h.fderiv_within _ this x hx,\n  ext y v,\n  change (p x 1) (snoc 0 y) = (p x 1) (cons y v),\n  unfold_coes,\n  congr' with i,\n  rw unique.eq_default i,\n  refl\nend\n\nlemma has_ftaylor_series_up_to_on.differentiable_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) : differentiable_on \ud835\udd5c f s :=\n\u03bb x hx, (h.has_fderiv_within_at hn hx).differentiable_within_at\n\n\/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term\nof order `1` of this series is a derivative of `f` at `x`. -\/\nlemma has_ftaylor_series_up_to_on.has_fderiv_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hx : s \u2208 \ud835\udcdd x) :\n  has_fderiv_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) x :=\n(h.has_fderiv_within_at hn (mem_of_mem_nhds hx)).has_fderiv_at hx\n\n\/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then\nin a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -\/\nlemma has_ftaylor_series_up_to_on.eventually_has_fderiv_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hx : s \u2208 \ud835\udcdd x) :\n  \u2200\u1da0 y in \ud835\udcdd x, has_fderiv_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p y 1)) y :=\n(eventually_eventually_nhds.2 hx).mono $ \u03bb y hy, h.has_fderiv_at hn hy\n\n\/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then\nit is differentiable at `x`. -\/\nlemma has_ftaylor_series_up_to_on.differentiable_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hx : s \u2208 \ud835\udcdd x) :\n  differentiable_at \ud835\udd5c f x :=\n(h.has_fderiv_at hn hx).differentiable_at\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and\n`p (n + 1)` is a derivative of `p n`. -\/\ntheorem has_ftaylor_series_up_to_on_succ_iff_left {n : \u2115} :\n  has_ftaylor_series_up_to_on (n + 1) f p s \u2194\n  has_ftaylor_series_up_to_on n f p s\n  \u2227 (\u2200 x \u2208 s, has_fderiv_within_at (\u03bb y, p y n) (p x n.succ).curry_left s x)\n  \u2227 continuous_on (\u03bb x, p x (n + 1)) s :=\nbegin\n  split,\n  { assume h,\n    exact \u27e8h.of_le (with_top.coe_le_coe.2 (nat.le_succ n)),\n           h.fderiv_within _ (with_top.coe_lt_coe.2 (lt_add_one n)),\n           h.cont (n + 1) (le_refl _)\u27e9 },\n  { assume h,\n    split,\n    { exact h.1.zero_eq },\n    { assume m hm,\n      by_cases h' : m < n,\n      { exact h.1.fderiv_within m (with_top.coe_lt_coe.2 h') },\n      { have : m = n := nat.eq_of_lt_succ_of_not_lt (with_top.coe_lt_coe.1 hm) h',\n        rw this,\n        exact h.2.1 } },\n    { assume m hm,\n      by_cases h' : m \u2264 n,\n      { apply h.1.cont m (with_top.coe_le_coe.2 h') },\n      { have : m = (n + 1) := le_antisymm (with_top.coe_le_coe.1 hm) (not_le.1 h'),\n        rw this,\n        exact h.2.2 } } }\nend\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -\/\ntheorem has_ftaylor_series_up_to_on_succ_iff_right {n : \u2115} :\n  has_ftaylor_series_up_to_on ((n + 1) : \u2115) f p s \u2194\n  (\u2200 x \u2208 s, (p x 0).uncurry0 = f x)\n  \u2227 (\u2200 x \u2208 s, has_fderiv_within_at (\u03bb y, p y 0) (p x 1).curry_left s x)\n  \u2227 has_ftaylor_series_up_to_on n\n    (\u03bb x, continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) (\u03bb x, (p x).shift) s :=\nbegin\n  split,\n  { assume H,\n    refine \u27e8H.zero_eq, H.fderiv_within 0 (with_top.coe_lt_coe.2 (nat.succ_pos n)), _\u27e9,\n    split,\n    { assume x hx, refl },\n    { assume m (hm : (m : with_top \u2115) < n) x (hx : x \u2208 s),\n      have A : (m.succ : with_top \u2115) < n.succ,\n        by { rw with_top.coe_lt_coe at \u22a2 hm, exact nat.lt_succ_iff.mpr hm },\n      change has_fderiv_within_at\n        ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ))\n        (p x m.succ.succ).curry_right.curry_left s x,\n      rw linear_isometry_equiv.comp_has_fderiv_within_at_iff',\n      convert H.fderiv_within _ A x hx,\n      ext y v,\n      change (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))\n        = (p x (nat.succ (nat.succ m))) (cons y v),\n      rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self] },\n    { assume m (hm : (m : with_top \u2115) \u2264 n),\n      have A : (m.succ : with_top \u2115) \u2264 n.succ,\n        by { rw with_top.coe_le_coe at \u22a2 hm, exact nat.pred_le_iff.mp hm },\n      change continuous_on ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) s,\n      rw linear_isometry_equiv.comp_continuous_on_iff,\n      exact H.cont _ A } },\n  { rintros \u27e8Hzero_eq, Hfderiv_zero, Htaylor\u27e9,\n    split,\n    { exact Hzero_eq },\n    { assume m (hm : (m : with_top \u2115) < n.succ) x (hx : x \u2208 s),\n      cases m,\n      { exact Hfderiv_zero x hx },\n      { have A : (m : with_top \u2115) < n,\n          by { rw with_top.coe_lt_coe at hm \u22a2, exact nat.lt_of_succ_lt_succ hm },\n        have : has_fderiv_within_at ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) ((p x).shift m.succ).curry_left s x :=\n          Htaylor.fderiv_within _ A x hx,\n        rw linear_isometry_equiv.comp_has_fderiv_within_at_iff' at this,\n        convert this,\n        ext y v,\n        change (p x (nat.succ (nat.succ m))) (cons y v)\n          = (p x m.succ.succ) (snoc (cons y (init v)) (v (last _))),\n        rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self] } },\n    { assume m (hm : (m : with_top \u2115) \u2264 n.succ),\n      cases m,\n      { have : differentiable_on \ud835\udd5c (\u03bb x, p x 0) s :=\n          \u03bb x hx, (Hfderiv_zero x hx).differentiable_within_at,\n        exact this.continuous_on },\n      { have A : (m : with_top \u2115) \u2264 n,\n          by { rw with_top.coe_le_coe at hm \u22a2, exact nat.lt_succ_iff.mp hm },\n        have : continuous_on ((continuous_multilinear_curry_right_equiv' \ud835\udd5c m E F).symm\n           \u2218 (\u03bb (y : E), p y m.succ)) s :=\n        Htaylor.cont _ A,\n        rwa linear_isometry_equiv.comp_continuous_on_iff at this } } }\nend\n\n\/-! ### Smooth functions within a set around a point -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to order `n` within a set `s` at a point `x` if\nit admits continuous derivatives up to order `n` in a neighborhood of `x` in `s \u222a {x}`.\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n\nFor instance, a real function which is `C^m` on `(-1\/m, 1\/m)` for each natural `m`, but not\nbetter, is `C^\u221e` at `0` within `univ`.\n-\/\ndef times_cont_diff_within_at (n : with_top \u2115) (f : E \u2192 F) (s : set E) (x : E) :=\n\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F,\n    has_ftaylor_series_up_to_on m f p u\n\nvariable {\ud835\udd5c}\n\nlemma times_cont_diff_within_at_nat {n : \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F,\n  has_ftaylor_series_up_to_on n f p u :=\n\u27e8\u03bb H, H n (le_refl _), \u03bb \u27e8u, hu, p, hp\u27e9 m hm, \u27e8u, hu, p, hp.of_le hm\u27e9\u27e9\n\nlemma times_cont_diff_within_at.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hmn : m \u2264 n) :\n  times_cont_diff_within_at \ud835\udd5c m f s x :=\n\u03bb k hk, h k (le_trans hk hmn)\n\nlemma times_cont_diff_within_at_iff_forall_nat_le {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 times_cont_diff_within_at \ud835\udd5c m f s x :=\n\u27e8\u03bb H m hm, H.of_le hm, \u03bb H m hm, H m hm _ le_rfl\u27e9\n\nlemma times_cont_diff_within_at_top :\n  times_cont_diff_within_at \ud835\udd5c \u221e f s x \u2194 \u2200 (n : \u2115), times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_iff_forall_nat_le.trans $ by simp only [forall_prop_of_true, le_top]\n\nlemma times_cont_diff_within_at.continuous_within_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) : continuous_within_at f s x :=\nbegin\n  rcases h 0 bot_le with \u27e8u, hu, p, H\u27e9,\n  rw [mem_nhds_within_insert] at hu,\n  exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2\nend\n\nlemma times_cont_diff_within_at.congr_of_eventually_eq {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\n\u03bb m hm, let \u27e8u, hu, p, H\u27e9 := h m hm in\n\u27e8{x \u2208 u | f\u2081 x = f x}, filter.inter_mem hu (mem_nhds_within_insert.2 \u27e8hx, h\u2081\u27e9), p,\n  (H.mono (sep_subset _ _)).congr (\u03bb _, and.right)\u27e9\n\nlemma times_cont_diff_within_at.congr_of_eventually_eq' {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\nh.congr_of_eventually_eq h\u2081 $ h\u2081.self_of_nhds_within hx\n\nlemma filter.eventually_eq.times_cont_diff_within_at_iff {n : with_top \u2115}\n  (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\n\u27e8\u03bb H, times_cont_diff_within_at.congr_of_eventually_eq H h\u2081.symm hx.symm,\n\u03bb H, H.congr_of_eventually_eq h\u2081 hx\u27e9\n\nlemma times_cont_diff_within_at.congr {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y) (hx : f\u2081 x = f x) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\nh.congr_of_eventually_eq (filter.eventually_eq_of_mem self_mem_nhds_within h\u2081) hx\n\nlemma times_cont_diff_within_at.congr' {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y) (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c n f\u2081 s x :=\nh.congr h\u2081 (h\u2081 _ hx)\n\nlemma times_cont_diff_within_at.mono_of_mem {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : s \u2208 \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nbegin\n  assume m hm,\n  rcases h m hm with \u27e8u, hu, p, H\u27e9,\n  exact \u27e8u, nhds_within_le_of_mem (insert_mem_nhds_within_insert hst) hu, p, H\u27e9\nend\n\nlemma times_cont_diff_within_at.mono {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : t \u2286 s) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nh.mono_of_mem $ filter.mem_of_superset self_mem_nhds_within hst\n\nlemma times_cont_diff_within_at.congr_nhds {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) {t : set E} (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f t x :=\nh.mono_of_mem $ hst \u25b8 self_mem_nhds_within\n\nlemma times_cont_diff_within_at_congr_nhds {n : with_top \u2115} {t : set E} (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) :\n  times_cont_diff_within_at \ud835\udd5c n f s x \u2194 times_cont_diff_within_at \ud835\udd5c n f t x :=\n\u27e8\u03bb h, h.congr_nhds hst, \u03bb h, h.congr_nhds hst.symm\u27e9\n\nlemma times_cont_diff_within_at_inter' {n : with_top \u2115} (h : t \u2208 \ud835\udcdd[s] x) :\n  times_cont_diff_within_at \ud835\udd5c n f (s \u2229 t) x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_congr_nhds $ eq.symm $ nhds_within_restrict'' _ h\n\nlemma times_cont_diff_within_at_inter {n : with_top \u2115} (h : t \u2208 \ud835\udcdd x) :\n  times_cont_diff_within_at \ud835\udd5c n f (s \u2229 t) x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\ntimes_cont_diff_within_at_inter' (mem_nhds_within_of_mem_nhds h)\n\n\/-- If a function is `C^n` within a set at a point, with `n \u2265 1`, then it is differentiable\nwithin this set at this point. -\/\nlemma times_cont_diff_within_at.differentiable_within_at' {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n  differentiable_within_at \ud835\udd5c f (insert x s) x :=\nbegin\n  rcases h 1 hn with \u27e8u, hu, p, H\u27e9,\n  rcases mem_nhds_within.1 hu with \u27e8t, t_open, xt, tu\u27e9,\n  rw inter_comm at tu,\n  have := ((H.mono tu).differentiable_on (le_refl _)) x \u27e8mem_insert x s, xt\u27e9,\n  exact (differentiable_within_at_inter (is_open.mem_nhds t_open xt)).1 this,\nend\n\nlemma times_cont_diff_within_at.differentiable_within_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n  differentiable_within_at \ud835\udd5c f s x :=\n(h.differentiable_within_at' hn).mono  (subset_insert x s)\n\n\/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_within_at_succ_iff_has_fderiv_within_at {n : \u2115} :\n  times_cont_diff_within_at \ud835\udd5c ((n + 1) : \u2115) f s x\n  \u2194 \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F),\n    (\u2200 x \u2208 u, has_fderiv_within_at f (f' x) u x) \u2227 (times_cont_diff_within_at \ud835\udd5c n f' u x) :=\nbegin\n  split,\n  { assume h,\n    rcases h n.succ (le_refl _) with \u27e8u, hu, p, Hp\u27e9,\n    refine \u27e8u, hu, \u03bb y, (continuous_multilinear_curry_fin1 \ud835\udd5c E F) (p y 1),\n      \u03bb y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _\u27e9,\n    assume m hm,\n    refine \u27e8u, _, \u03bb (y : E), (p y).shift, _\u27e9,\n    { convert self_mem_nhds_within,\n      have : x \u2208 insert x s, by simp,\n      exact (insert_eq_of_mem (mem_of_mem_nhds_within this hu)) },\n    { rw has_ftaylor_series_up_to_on_succ_iff_right at Hp,\n      exact Hp.2.2.of_le hm } },\n  { rintros \u27e8u, hu, f', f'_eq_deriv, Hf'\u27e9,\n    rw times_cont_diff_within_at_nat,\n    rcases Hf' n (le_refl _) with \u27e8v, hv, p', Hp'\u27e9,\n    refine \u27e8v \u2229 u, _, \u03bb x, (p' x).unshift (f x), _\u27e9,\n    { apply filter.inter_mem _ hu,\n      apply nhds_within_le_of_mem hu,\n      exact nhds_within_mono _ (subset_insert x u) hv },\n    { rw has_ftaylor_series_up_to_on_succ_iff_right,\n      refine \u27e8\u03bb y hy, rfl, \u03bb y hy, _, _\u27e9,\n      { change has_fderiv_within_at (\u03bb z, (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f z))\n          ((formal_multilinear_series.unshift (p' y) (f y) 1).curry_left) (v \u2229 u) y,\n        rw linear_isometry_equiv.comp_has_fderiv_within_at_iff',\n        convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u),\n        rw \u2190 Hp'.zero_eq y hy.1,\n        ext z,\n        change ((p' y 0) (init (@cons 0 (\u03bb i, E) z 0))) (@cons 0 (\u03bb i, E) z 0 (last 0))\n          = ((p' y 0) 0) z,\n        unfold_coes,\n        congr },\n      { convert (Hp'.mono (inter_subset_left v u)).congr (\u03bb x hx, Hp'.zero_eq x hx.1),\n        { ext x y,\n          change p' x 0 (init (@snoc 0 (\u03bb i : fin 1, E) 0 y)) y = p' x 0 0 y,\n          rw init_snoc },\n        { ext x k v y,\n          change p' x k (init (@snoc k (\u03bb i : fin k.succ, E) v y))\n            (@snoc k (\u03bb i : fin k.succ, E) v y (last k)) = p' x k v y,\n          rw [snoc_last, init_snoc] } } } }\nend\n\n\/-! ### Smooth functions within a set -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` on `s` if, for any point `x` in `s`, it\nadmits continuous derivatives up to order `n` on a neighborhood of `x` in `s`.\n\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n-\/\ndefinition times_cont_diff_on (n : with_top \u2115) (f : E \u2192 F) (s : set E) :=\n\u2200 x \u2208 s, times_cont_diff_within_at \ud835\udd5c n f s x\n\nvariable {\ud835\udd5c}\n\nlemma times_cont_diff_on.times_cont_diff_within_at {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nh x hx\n\nlemma times_cont_diff_within_at.times_cont_diff_on {n : with_top \u2115} {m : \u2115}\n  (hm : (m : with_top \u2115) \u2264 n) (h : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  \u2203 u \u2208 \ud835\udcdd[insert x s] x, u \u2286 insert x s \u2227 times_cont_diff_on \ud835\udd5c m f u :=\nbegin\n  rcases h m hm with \u27e8u, u_nhd, p, hp\u27e9,\n  refine \u27e8u \u2229 insert x s, filter.inter_mem u_nhd self_mem_nhds_within,\n    inter_subset_right _ _, _\u27e9,\n  assume y hy m' hm',\n  refine \u27e8u \u2229 insert x s, _, p, (hp.mono (inter_subset_left _ _)).of_le hm'\u27e9,\n  convert self_mem_nhds_within,\n  exact insert_eq_of_mem hy\nend\n\nprotected lemma times_cont_diff_within_at.eventually {n : \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  \u2200\u1da0 y in \ud835\udcdd[insert x s] x, times_cont_diff_within_at \ud835\udd5c n f s y :=\nbegin\n  rcases h.times_cont_diff_on le_rfl with \u27e8u, hu, hu_sub, hd\u27e9,\n  have : \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u,\n    from (eventually_nhds_within_nhds_within.2 hu).and hu,\n  refine this.mono (\u03bb y hy, (hd y hy.2).mono_of_mem _),\n  exact nhds_within_mono y (subset_insert _ _) hy.1\nend\n\nlemma times_cont_diff_on.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : m \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m f s :=\n\u03bb x hx, (h x hx).of_le hmn\n\nlemma times_cont_diff_on_iff_forall_nat_le {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f s \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 times_cont_diff_on \ud835\udd5c m f s :=\n\u27e8\u03bb H m hm, H.of_le hm, \u03bb H x hx m hm, H m hm x hx m le_rfl\u27e9\n\nlemma times_cont_diff_on_top :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194 \u2200 (n : \u2115), times_cont_diff_on \ud835\udd5c n f s :=\ntimes_cont_diff_on_iff_forall_nat_le.trans $ by simp only [le_top, forall_prop_of_true]\n\nlemma times_cont_diff_on_all_iff_nat :\n  (\u2200 n, times_cont_diff_on \ud835\udd5c n f s) \u2194 (\u2200 n : \u2115, times_cont_diff_on \ud835\udd5c n f s) :=\nbegin\n  refine \u27e8\u03bb H n, H n, _\u27e9,\n  rintro H (_|n),\n  exacts [times_cont_diff_on_top.2 H, H n]\nend\n\nlemma times_cont_diff_on.continuous_on {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) : continuous_on f s :=\n\u03bb x hx, (h x hx).continuous_within_at\n\nlemma times_cont_diff_on.congr {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s :=\n\u03bb x hx, (h x hx).congr h\u2081 (h\u2081 x hx)\n\nlemma times_cont_diff_on_congr {n : with_top \u2115} (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s \u2194 times_cont_diff_on \ud835\udd5c n f s :=\n\u27e8\u03bb H, H.congr (\u03bb x hx, (h\u2081 x hx).symm), \u03bb H, H.congr h\u2081\u27e9\n\nlemma times_cont_diff_on.mono {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) {t : set E} (hst : t \u2286 s) :\n  times_cont_diff_on \ud835\udd5c n f t :=\n\u03bb x hx, (h x (hst hx)).mono hst\n\nlemma times_cont_diff_on.congr_mono {n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s\u2081, f\u2081 x = f x) (hs : s\u2081 \u2286 s) :\n  times_cont_diff_on \ud835\udd5c n f\u2081 s\u2081 :=\n(hf.mono hs).congr h\u2081\n\n\/-- If a function is `C^n` on a set with `n \u2265 1`, then it is differentiable there. -\/\nlemma times_cont_diff_on.differentiable_on {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hn : 1 \u2264 n) : differentiable_on \ud835\udd5c f s :=\n\u03bb x hx, (h x hx).differentiable_within_at hn\n\n\/-- If a function is `C^n` around each point in a set, then it is `C^n` on the set. -\/\nlemma times_cont_diff_on_of_locally_times_cont_diff_on {n : with_top \u2115}\n  (h : \u2200 x \u2208 s, \u2203u, is_open u \u2227 x \u2208 u \u2227 times_cont_diff_on \ud835\udd5c n f (s \u2229 u)) :\n  times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  assume x xs,\n  rcases h x xs with \u27e8u, u_open, xu, hu\u27e9,\n  apply (times_cont_diff_within_at_inter _).1 (hu x \u27e8xs, xu\u27e9),\n  exact is_open.mem_nhds u_open xu\nend\n\n\/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_has_fderiv_within_at {n : \u2115} :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s\n  \u2194 \u2200 x \u2208 s, \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F),\n    (\u2200 x \u2208 u, has_fderiv_within_at f (f' x) u x) \u2227 (times_cont_diff_on \ud835\udd5c n f' u) :=\nbegin\n  split,\n  { assume h x hx,\n    rcases (h x hx) n.succ (le_refl _) with \u27e8u, hu, p, Hp\u27e9,\n    refine \u27e8u, hu, \u03bb y, (continuous_multilinear_curry_fin1 \ud835\udd5c E F) (p y 1),\n      \u03bb y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _\u27e9,\n    rw has_ftaylor_series_up_to_on_succ_iff_right at Hp,\n    assume z hz m hm,\n    refine \u27e8u, _, \u03bb (x : E), (p x).shift, Hp.2.2.of_le hm\u27e9,\n    convert self_mem_nhds_within,\n    exact insert_eq_of_mem hz, },\n  { assume h x hx,\n    rw times_cont_diff_within_at_succ_iff_has_fderiv_within_at,\n    rcases h x hx with \u27e8u, u_nhbd, f', hu, hf'\u27e9,\n    have : x \u2208 u := mem_of_mem_nhds_within (mem_insert _ _) u_nhbd,\n    exact \u27e8u, u_nhbd, f', hu, hf' x this\u27e9 }\nend\n\n\/-! ### Iterated derivative within a set -\/\nvariable (\ud835\udd5c)\n\n\/--\nThe `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th\nderivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with\nan uncurrying step to see it as a multilinear map in `n+1` variables..\n-\/\nnoncomputable def iterated_fderiv_within (n : \u2115) (f : E \u2192 F) (s : set E) :\n  E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F) :=\nnat.rec_on n\n  (\u03bb x, continuous_multilinear_map.curry0 \ud835\udd5c E (f x))\n  (\u03bb n rec x, continuous_linear_map.uncurry_left (fderiv_within \ud835\udd5c rec s x))\n\n\/-- Formal Taylor series associated to a function within a set. -\/\ndef ftaylor_series_within (f : E \u2192 F) (s : set E) (x : E) : formal_multilinear_series \ud835\udd5c E F :=\n\u03bb n, iterated_fderiv_within \ud835\udd5c n f s x\n\nvariable {\ud835\udd5c}\n\n@[simp] lemma iterated_fderiv_within_zero_apply (m : (fin 0) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c 0 f s x : ((fin 0) \u2192  E) \u2192 F) m = f x := rfl\n\nlemma iterated_fderiv_within_zero_eq_comp :\n  iterated_fderiv_within \ud835\udd5c 0 f s = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm \u2218 f := rfl\n\nlemma iterated_fderiv_within_succ_apply_left {n : \u2115} (m : fin (n + 1) \u2192 E):\n  (iterated_fderiv_within \ud835\udd5c (n + 1) f s x : (fin (n + 1) \u2192 E) \u2192 F) m\n  = (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n f s) s x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F))\n      (m 0) (tail m) := rfl\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -\/\nlemma iterated_fderiv_within_succ_eq_comp_left {n : \u2115} :\n  iterated_fderiv_within \ud835\udd5c (n + 1) f s =\n  (continuous_multilinear_curry_left_equiv \ud835\udd5c (\u03bb(i : fin (n + 1)), E) F)\n    \u2218 (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n f s) s) := rfl\n\ntheorem iterated_fderiv_within_succ_apply_right {n : \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) (m : fin (n + 1) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c (n + 1) f s x : (fin (n + 1) \u2192 E) \u2192 F) m\n    = iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s x (init m) (m (last n)) :=\nbegin\n  induction n with n IH generalizing x,\n  { rw [iterated_fderiv_within_succ_eq_comp_left, iterated_fderiv_within_zero_eq_comp,\n        iterated_fderiv_within_zero_apply,\n        function.comp_apply, linear_isometry_equiv.comp_fderiv_within _ (hs x hx)],\n    refl },\n  { let I := continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F,\n    have A : \u2200 y \u2208 s, iterated_fderiv_within \ud835\udd5c n.succ f s y\n        = (I \u2218 (iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s)) y,\n      by { assume y hy, ext m, rw @IH m y hy, refl },\n    calc\n    (iterated_fderiv_within \ud835\udd5c (n+2) f s x : (fin (n+2) \u2192 E) \u2192 F) m =\n    (fderiv_within \ud835\udd5c (iterated_fderiv_within \ud835\udd5c n.succ f s) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) : rfl\n    ... = (fderiv_within \ud835\udd5c (I \u2218 (iterated_fderiv_within \ud835\udd5c n (fderiv_within \ud835\udd5c f s) s)) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) :\n      by rw fderiv_within_congr (hs x hx) A (A x hx)\n    ... = (I \u2218 fderiv_within \ud835\udd5c ((iterated_fderiv_within \ud835\udd5c n (fderiv_within \ud835\udd5c f s) s)) s x\n              : E \u2192 (E [\u00d7(n + 1)]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) :\n      by { rw linear_isometry_equiv.comp_fderiv_within _ (hs x hx), refl }\n    ... = (fderiv_within \ud835\udd5c ((iterated_fderiv_within \ud835\udd5c n (\u03bb y, fderiv_within \ud835\udd5c f s y) s)) s x\n              : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] F))) (m 0) (init (tail m)) ((tail m) (last n)) : rfl\n    ... = iterated_fderiv_within \ud835\udd5c (nat.succ n) (\u03bb y, fderiv_within \ud835\udd5c f s y) s x\n              (init m) (m (last (n + 1))) :\n      by { rw [iterated_fderiv_within_succ_apply_left, tail_init_eq_init_tail], refl } }\nend\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the `n`-th derivative of the derivative. -\/\nlemma iterated_fderiv_within_succ_eq_comp_right {n : \u2115} (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c (n + 1) f s x =\n  ((continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F)\n    \u2218 (iterated_fderiv_within \ud835\udd5c n (\u03bby, fderiv_within \ud835\udd5c f s y) s)) x :=\nby { ext m, rw iterated_fderiv_within_succ_apply_right hs hx, refl }\n\n@[simp] lemma iterated_fderiv_within_one_apply\n  (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) (m : (fin 1) \u2192 E) :\n  (iterated_fderiv_within \ud835\udd5c 1 f s x : ((fin 1) \u2192 E) \u2192 F) m\n  = (fderiv_within \ud835\udd5c f s x : E \u2192 F) (m 0) :=\nby { rw [iterated_fderiv_within_succ_apply_right hs hx, iterated_fderiv_within_zero_apply], refl }\n\n\/-- If two functions coincide on a set `s` of unique differentiability, then their iterated\ndifferentials within this set coincide. -\/\nlemma iterated_fderiv_within_congr {n : \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) (hL : \u2200y\u2208s, f\u2081 y = f y) (hx : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f\u2081 s x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  induction n with n IH generalizing x,\n  { ext m, simp [hL x hx] },\n  { have : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f\u2081 s y) s x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) s x :=\n      fderiv_within_congr (hs x hx) (\u03bb y hy, IH hy) (IH hx),\n    ext m,\n    rw [iterated_fderiv_within_succ_apply_left, iterated_fderiv_within_succ_apply_left, this] }\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with an open set containing `x`. -\/\nlemma iterated_fderiv_within_inter_open {n : \u2115} (hu : is_open u)\n  (hs : unique_diff_on \ud835\udd5c (s \u2229 u)) (hx : x \u2208 s \u2229 u) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  induction n with n IH generalizing x,\n  { ext m, simp },\n  { have A : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) y) (s \u2229 u) x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) (s \u2229 u) x :=\n      fderiv_within_congr (hs x hx) (\u03bb y hy, IH hy) (IH hx),\n    have B : fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) (s \u2229 u) x\n           = fderiv_within \ud835\udd5c (\u03bb y, iterated_fderiv_within \ud835\udd5c n f s y) s x :=\n      fderiv_within_inter (is_open.mem_nhds hu hx.2)\n        ((unique_diff_within_at_inter (is_open.mem_nhds hu hx.2)).1 (hs x hx)),\n    ext m,\n    rw [iterated_fderiv_within_succ_apply_left, iterated_fderiv_within_succ_apply_left, A, B] }\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x` within `s`. -\/\nlemma iterated_fderiv_within_inter' {n : \u2115}\n  (hu : u \u2208 \ud835\udcdd[s] x) (hs : unique_diff_on \ud835\udd5c s) (xs : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\nbegin\n  obtain \u27e8v, v_open, xv, vu\u27e9 : \u2203 v, is_open v \u2227 x \u2208 v \u2227 v \u2229 s \u2286 u := mem_nhds_within.1 hu,\n  have A : (s \u2229 u) \u2229 v = s \u2229 v,\n  { apply subset.antisymm (inter_subset_inter (inter_subset_left _ _) (subset.refl _)),\n    exact \u03bb y \u27e8ys, yv\u27e9, \u27e8\u27e8ys, vu \u27e8yv, ys\u27e9\u27e9, yv\u27e9 },\n  have : iterated_fderiv_within \ud835\udd5c n f (s \u2229 v) x = iterated_fderiv_within \ud835\udd5c n f s x :=\n    iterated_fderiv_within_inter_open v_open (hs.inter v_open) \u27e8xs, xv\u27e9,\n  rw \u2190 this,\n  have : iterated_fderiv_within \ud835\udd5c n f ((s \u2229 u) \u2229 v) x = iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x,\n  { refine iterated_fderiv_within_inter_open v_open  _ \u27e8\u27e8xs, vu \u27e8xv, xs\u27e9\u27e9, xv\u27e9,\n    rw A,\n    exact hs.inter v_open },\n  rw A at this,\n  rw \u2190 this\nend\n\n\/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x`. -\/\nlemma iterated_fderiv_within_inter {n : \u2115}\n  (hu : u \u2208 \ud835\udcdd x) (hs : unique_diff_on \ud835\udd5c s) (xs : x \u2208 s) :\n  iterated_fderiv_within \ud835\udd5c n f (s \u2229 u) x = iterated_fderiv_within \ud835\udd5c n f s x :=\niterated_fderiv_within_inter' (mem_nhds_within_of_mem_nhds hu) hs xs\n\n@[simp] lemma times_cont_diff_on_zero :\n  times_cont_diff_on \ud835\udd5c 0 f s \u2194 continuous_on f s :=\nbegin\n  refine \u27e8\u03bb H, H.continuous_on, \u03bb H, _\u27e9,\n  assume x hx m hm,\n  have : (m : with_top \u2115) = 0 := le_antisymm hm bot_le,\n  rw this,\n  refine \u27e8insert x s, self_mem_nhds_within, ftaylor_series_within \ud835\udd5c f s, _\u27e9,\n  rw has_ftaylor_series_up_to_on_zero_iff,\n  exact \u27e8by rwa insert_eq_of_mem hx, \u03bb x hx, by simp [ftaylor_series_within]\u27e9\nend\n\nlemma times_cont_diff_within_at_zero (hx : x \u2208 s) :\n  times_cont_diff_within_at \ud835\udd5c 0 f s x \u2194 \u2203 u \u2208 \ud835\udcdd[s] x, continuous_on f (s \u2229 u) :=\nbegin\n  split,\n  { intros h,\n    obtain \u27e8u, H, p, hp\u27e9 := h 0 (by norm_num),\n    refine \u27e8u, _, _\u27e9,\n    { simpa [hx] using H },\n    { simp only [with_top.coe_zero, has_ftaylor_series_up_to_on_zero_iff] at hp,\n      exact hp.1.mono (inter_subset_right s u) } },\n  { rintros \u27e8u, H, hu\u27e9,\n    rw \u2190 times_cont_diff_within_at_inter' H,\n    have h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhds_within hx H\u27e9,\n    exact (times_cont_diff_on_zero.mpr hu).times_cont_diff_within_at h' }\nend\n\n\/-- On a set with unique differentiability, any choice of iterated differential has to coincide\nwith the one we have chosen in `iterated_fderiv_within \ud835\udd5c m f s`. -\/\ntheorem has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to_on n f p s)\n  {m : \u2115} (hmn : (m : with_top \u2115) \u2264 n) (hs : unique_diff_on \ud835\udd5c s) (hx : x \u2208 s) :\n  p x m = iterated_fderiv_within \ud835\udd5c m f s x :=\nbegin\n  induction m with m IH generalizing x,\n  { rw [h.zero_eq' hx, iterated_fderiv_within_zero_eq_comp] },\n  { have A : (m : with_top \u2115) < n := lt_of_lt_of_le (with_top.coe_lt_coe.2 (lt_add_one m)) hmn,\n    have : has_fderiv_within_at (\u03bb (y : E), iterated_fderiv_within \ud835\udd5c m f s y)\n      (continuous_multilinear_map.curry_left (p x (nat.succ m))) s x :=\n    (h.fderiv_within m A x hx).congr (\u03bb y hy, (IH (le_of_lt A) hy).symm) (IH (le_of_lt A) hx).symm,\n    rw [iterated_fderiv_within_succ_eq_comp_left, function.comp_apply,\n      this.fderiv_within (hs x hx)],\n    exact (continuous_multilinear_map.uncurry_curry_left _).symm }\nend\n\n\/-- When a function is `C^n` in a set `s` of unique differentiability, it admits\n`ftaylor_series_within \ud835\udd5c f s` as a Taylor series up to order `n` in `s`. -\/\ntheorem times_cont_diff_on.ftaylor_series_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) :\n  has_ftaylor_series_up_to_on n f (ftaylor_series_within \ud835\udd5c f s) s :=\nbegin\n  split,\n  { assume x hx,\n    simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,\n               iterated_fderiv_within_zero_apply] },\n  { assume m hm x hx,\n    rcases (h x hx) m.succ (with_top.add_one_le_of_lt hm) with \u27e8u, hu, p, Hp\u27e9,\n    rw insert_eq_of_mem hx at hu,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw inter_comm at ho,\n    have : p x m.succ = ftaylor_series_within \ud835\udd5c f s x m.succ,\n    { change p x m.succ = iterated_fderiv_within \ud835\udd5c m.succ f s x,\n      rw \u2190 iterated_fderiv_within_inter (is_open.mem_nhds o_open xo) hs hx,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (le_refl _)\n        (hs.inter o_open) \u27e8hx, xo\u27e9 },\n    rw [\u2190 this, \u2190 has_fderiv_within_at_inter (is_open.mem_nhds o_open xo)],\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylor_series_within \ud835\udd5c f s y m,\n    { rintros y \u27e8hy, yo\u27e9,\n      change p y m = iterated_fderiv_within \ud835\udd5c m f s y,\n      rw \u2190 iterated_fderiv_within_inter (is_open.mem_nhds o_open yo) hs hy,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (with_top.coe_le_coe.2 (nat.le_succ m))\n        (hs.inter o_open) \u27e8hy, yo\u27e9 },\n    exact ((Hp.mono ho).fderiv_within m (with_top.coe_lt_coe.2 (lt_add_one m)) x \u27e8hx, xo\u27e9).congr\n      (\u03bb y hy, (A y hy).symm) (A x \u27e8hx, xo\u27e9).symm },\n  { assume m hm,\n    apply continuous_on_of_locally_continuous_on,\n    assume x hx,\n    rcases h x hx m hm with \u27e8u, hu, p, Hp\u27e9,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw insert_eq_of_mem hx at ho,\n    rw inter_comm at ho,\n    refine \u27e8o, o_open, xo, _\u27e9,\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylor_series_within \ud835\udd5c f s y m,\n    { rintros y \u27e8hy, yo\u27e9,\n      change p y m = iterated_fderiv_within \ud835\udd5c m f s y,\n      rw \u2190 iterated_fderiv_within_inter (is_open.mem_nhds o_open yo) hs hy,\n      exact (Hp.mono ho).eq_ftaylor_series_of_unique_diff_on (le_refl _)\n        (hs.inter o_open) \u27e8hy, yo\u27e9 },\n    exact ((Hp.mono ho).cont m (le_refl _)).congr (\u03bb y hy, (A y hy).symm) }\nend\n\nlemma times_cont_diff_on_of_continuous_on_differentiable_on {n : with_top \u2115}\n  (Hcont : \u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n    continuous_on (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s)\n  (Hdiff : \u2200 (m : \u2115), (m : with_top \u2115) < n \u2192\n    differentiable_on \ud835\udd5c (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  assume x hx m hm,\n  rw insert_eq_of_mem hx,\n  refine \u27e8s, self_mem_nhds_within, ftaylor_series_within \ud835\udd5c f s, _\u27e9,\n  split,\n  { assume y hy,\n    simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,\n                iterated_fderiv_within_zero_apply] },\n  { assume k hk y hy,\n    convert (Hdiff k (lt_of_lt_of_le hk hm) y hy).has_fderiv_within_at,\n    simp only [ftaylor_series_within, iterated_fderiv_within_succ_eq_comp_left,\n                continuous_linear_equiv.coe_apply, function.comp_app, coe_fn_coe_base],\n    exact continuous_linear_map.curry_uncurry_left _ },\n  { assume k hk,\n    exact Hcont k (le_trans hk hm) }\nend\n\nlemma times_cont_diff_on_of_differentiable_on {n : with_top \u2115}\n  (h : \u2200(m : \u2115), (m : with_top \u2115) \u2264 n \u2192 differentiable_on \ud835\udd5c (iterated_fderiv_within \ud835\udd5c m f s) s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\ntimes_cont_diff_on_of_continuous_on_differentiable_on\n  (\u03bb m hm, (h m hm).continuous_on) (\u03bb m hm, (h m (le_of_lt hm)))\n\nlemma times_cont_diff_on.continuous_on_iterated_fderiv_within {n : with_top \u2115} {m : \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : (m : with_top \u2115) \u2264 n) (hs : unique_diff_on \ud835\udd5c s) :\n  continuous_on (iterated_fderiv_within \ud835\udd5c m f s) s :=\n(h.ftaylor_series_within hs).cont m hmn\n\nlemma times_cont_diff_on.differentiable_on_iterated_fderiv_within {n : with_top \u2115} {m : \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hmn : (m : with_top \u2115) < n) (hs : unique_diff_on \ud835\udd5c s) :\n  differentiable_on \ud835\udd5c (iterated_fderiv_within \ud835\udd5c m f s) s :=\n\u03bb x hx, ((h.ftaylor_series_within hs).fderiv_within m hmn x hx).differentiable_within_at\n\nlemma times_cont_diff_on_iff_continuous_on_differentiable_on {n : with_top \u2115}\n  (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c n f s \u2194\n  (\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192\n    continuous_on (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s)\n  \u2227 (\u2200 (m : \u2115), (m : with_top \u2115) < n \u2192\n    differentiable_on \ud835\udd5c (\u03bb x, iterated_fderiv_within \ud835\udd5c m f s x) s) :=\nbegin\n  split,\n  { assume h,\n    split,\n    { assume m hm, exact h.continuous_on_iterated_fderiv_within hm hs },\n    { assume m hm, exact h.differentiable_on_iterated_fderiv_within hm hs } },\n  { assume h,\n    exact times_cont_diff_on_of_continuous_on_differentiable_on h.1 h.2 }\nend\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative (expressed with `fderiv_within`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_fderiv_within {n : \u2115} (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c n (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  split,\n  { assume H,\n    refine \u27e8H.differentiable_on (with_top.coe_le_coe.2 (nat.le_add_left 1 n)), \u03bb x hx, _\u27e9,\n    rcases times_cont_diff_within_at_succ_iff_has_fderiv_within_at.1 (H x hx)\n      with \u27e8u, hu, f', hff', hf'\u27e9,\n    rcases mem_nhds_within.1 hu with \u27e8o, o_open, xo, ho\u27e9,\n    rw [inter_comm, insert_eq_of_mem hx] at ho,\n    have := hf'.mono ho,\n    rw times_cont_diff_within_at_inter' (mem_nhds_within_of_mem_nhds (is_open.mem_nhds o_open xo))\n      at this,\n    apply this.congr_of_eventually_eq' _ hx,\n    have : o \u2229 s \u2208 \ud835\udcdd[s] x := mem_nhds_within.2 \u27e8o, o_open, xo, subset.refl _\u27e9,\n    rw inter_comm at this,\n    apply filter.eventually_eq_of_mem this (\u03bb y hy, _),\n    have A : fderiv_within \ud835\udd5c f (s \u2229 o) y = f' y :=\n      ((hff' y (ho hy)).mono ho).fderiv_within (hs.inter o_open y hy),\n    rwa fderiv_within_inter (is_open.mem_nhds o_open hy.2) (hs y hy.1) at A, },\n  { rintros \u27e8hdiff, h\u27e9 x hx,\n    rw [times_cont_diff_within_at_succ_iff_has_fderiv_within_at, insert_eq_of_mem hx],\n    exact \u27e8s, self_mem_nhds_within, fderiv_within \ud835\udd5c f s,\n      \u03bb y hy, (hdiff y hy).has_fderiv_within_at, h x hx\u27e9 }\nend\n\n\/-- A function is `C^(n + 1)` on an open domain if and only if it is\ndifferentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_fderiv_of_open {n : \u2115} (hs : is_open s) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c n (\u03bb y, fderiv \ud835\udd5c f y) s :=\nbegin\n  rw times_cont_diff_on_succ_iff_fderiv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact fderiv_within_of_open hs hx\nend\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative (expressed with `fderiv_within`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_fderiv_within (hs : unique_diff_on \ud835\udd5c s) :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c \u221e (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  split,\n  { assume h,\n    refine \u27e8h.differentiable_on le_top, _\u27e9,\n    apply times_cont_diff_on_top.2 (\u03bb n, ((times_cont_diff_on_succ_iff_fderiv_within hs).1 _).2),\n    exact h.of_le le_top },\n  { assume h,\n    refine times_cont_diff_on_top.2 (\u03bb n, _),\n    have A : (n : with_top \u2115) \u2264 \u221e := le_top,\n    apply ((times_cont_diff_on_succ_iff_fderiv_within hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le,\n    exact with_top.coe_le_coe.2 (nat.le_succ n) }\nend\n\n\/-- A function is `C^\u221e` on an open domain if and only if it is differentiable there, and its\nderivative (expressed with `fderiv`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_fderiv_of_open (hs : is_open s) :\n  times_cont_diff_on \ud835\udd5c \u221e f s \u2194\n  differentiable_on \ud835\udd5c f s \u2227 times_cont_diff_on \ud835\udd5c \u221e (\u03bb y, fderiv \ud835\udd5c f y) s :=\nbegin\n  rw times_cont_diff_on_top_iff_fderiv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact fderiv_within_of_open hs hx\nend\n\nlemma times_cont_diff_on.fderiv_within {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bb y, fderiv_within \ud835\udd5c f s y) s :=\nbegin\n  cases m,\n  { change \u221e + 1 \u2264 n at hmn,\n    have : n = \u221e, by simpa using hmn,\n    rw this at hf,\n    exact ((times_cont_diff_on_top_iff_fderiv_within hs).1 hf).2 },\n  { change (m.succ : with_top \u2115) \u2264 n at hmn,\n    exact ((times_cont_diff_on_succ_iff_fderiv_within hs).1 (hf.of_le hmn)).2 }\nend\n\nlemma times_cont_diff_on.fderiv_of_open {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : is_open s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bb y, fderiv \ud835\udd5c f y) s :=\n(hf.fderiv_within hs.unique_diff_on hmn).congr (\u03bb x hx, (fderiv_within_of_open hs hx).symm)\n\nlemma times_cont_diff_on.continuous_on_fderiv_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bb x, fderiv_within \ud835\udd5c f s x) s :=\n((times_cont_diff_on_succ_iff_fderiv_within hs).1 (h.of_le hn)).2.continuous_on\n\nlemma times_cont_diff_on.continuous_on_fderiv_of_open {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f s) (hs : is_open s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bb x, fderiv \ud835\udd5c f x) s :=\n((times_cont_diff_on_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.continuous_on\n\n\/-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is\ncontinuous. -\/\nlemma times_cont_diff_on.continuous_on_fderiv_within_apply\n  {n : with_top \u2115} (h : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hn : 1 \u2264 n) :\n  continuous_on (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1 : E \u2192 F) p.2) (set.prod s univ) :=\nbegin\n  have A : continuous (\u03bbq : (E \u2192L[\ud835\udd5c] F) \u00d7 E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous,\n  have B : continuous_on (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1, p.2)) (set.prod s univ),\n  { apply continuous_on.prod _ continuous_snd.continuous_on,\n    exact continuous_on.comp (h.continuous_on_fderiv_within hs hn) continuous_fst.continuous_on\n      (prod_subset_preimage_fst _ _) },\n  exact A.comp_continuous_on B\nend\n\n\/-! ### Functions with a Taylor series on the whole space -\/\n\n\/-- `has_ftaylor_series_up_to n f p` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`has_fderiv_at` but for higher order derivatives. -\/\nstructure has_ftaylor_series_up_to (n : with_top \u2115)\n  (f : E \u2192 F) (p : E \u2192 formal_multilinear_series \ud835\udd5c E F) : Prop :=\n(zero_eq : \u2200 x, (p x 0).uncurry0 = f x)\n(fderiv  : \u2200 (m : \u2115) (hm : (m : with_top \u2115) < n), \u2200 x,\n    has_fderiv_at (\u03bb y, p y m) (p x m.succ).curry_left x)\n(cont    : \u2200 (m : \u2115) (hm : (m : with_top \u2115) \u2264 n), continuous (\u03bb x, p x m))\n\nlemma has_ftaylor_series_up_to.zero_eq' {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (x : E) :\n  p x 0 = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm (f x) :=\nby { rw \u2190 h.zero_eq x, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ }\n\nlemma has_ftaylor_series_up_to_on_univ_iff {n : with_top \u2115} :\n  has_ftaylor_series_up_to_on n f p univ \u2194 has_ftaylor_series_up_to n f p :=\nbegin\n  split,\n  { assume H,\n    split,\n    { exact \u03bb x, H.zero_eq x (mem_univ x) },\n    { assume m hm x,\n      rw \u2190 has_fderiv_within_at_univ,\n      exact H.fderiv_within m hm x (mem_univ x) },\n    { assume m hm,\n      rw continuous_iff_continuous_on_univ,\n      exact H.cont m hm } },\n  { assume H,\n    split,\n    { exact \u03bb x hx, H.zero_eq x },\n    { assume m hm x hx,\n      rw has_fderiv_within_at_univ,\n      exact H.fderiv m hm x },\n    { assume m hm,\n      rw \u2190 continuous_iff_continuous_on_univ,\n      exact H.cont m hm } }\nend\n\nlemma has_ftaylor_series_up_to.has_ftaylor_series_up_to_on {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (s : set E) :\n  has_ftaylor_series_up_to_on n f p s :=\n(has_ftaylor_series_up_to_on_univ_iff.2 h).mono (subset_univ _)\n\nlemma has_ftaylor_series_up_to.of_le {m n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hmn : m \u2264 n) :\n  has_ftaylor_series_up_to m f p :=\nby { rw \u2190 has_ftaylor_series_up_to_on_univ_iff at h \u22a2, exact h.of_le hmn }\n\nlemma has_ftaylor_series_up_to.continuous {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) : continuous f :=\nbegin\n  rw \u2190 has_ftaylor_series_up_to_on_univ_iff at h,\n  rw continuous_iff_continuous_on_univ,\n  exact h.continuous_on\nend\n\nlemma has_ftaylor_series_up_to_zero_iff :\n  has_ftaylor_series_up_to 0 f p \u2194 continuous f \u2227 (\u2200 x, (p x 0).uncurry0 = f x) :=\nby simp [has_ftaylor_series_up_to_on_univ_iff.symm, continuous_iff_continuous_on_univ,\n         has_ftaylor_series_up_to_on_zero_iff]\n\n\/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -\/\nlemma has_ftaylor_series_up_to.has_fderiv_at {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hn : 1 \u2264 n) (x : E) :\n  has_fderiv_at f (continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) x :=\nbegin\n  rw [\u2190 has_fderiv_within_at_univ],\n  exact (has_ftaylor_series_up_to_on_univ_iff.2 h).has_fderiv_within_at hn (mem_univ _)\nend\n\nlemma has_ftaylor_series_up_to.differentiable {n : with_top \u2115}\n  (h : has_ftaylor_series_up_to n f p) (hn : 1 \u2264 n) : differentiable \ud835\udd5c f :=\n\u03bb x, (h.has_fderiv_at hn x).differentiable_at\n\n\/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -\/\ntheorem has_ftaylor_series_up_to_succ_iff_right {n : \u2115} :\n  has_ftaylor_series_up_to ((n + 1) : \u2115) f p \u2194\n  (\u2200 x, (p x 0).uncurry0 = f x)\n  \u2227 (\u2200 x, has_fderiv_at (\u03bb y, p y 0) (p x 1).curry_left x)\n  \u2227 has_ftaylor_series_up_to n\n    (\u03bb x, continuous_multilinear_curry_fin1 \ud835\udd5c E F (p x 1)) (\u03bb x, (p x).shift) :=\nby simp [has_ftaylor_series_up_to_on_succ_iff_right, has_ftaylor_series_up_to_on_univ_iff.symm,\n         -add_comm, -with_zero.coe_add]\n\n\/-! ### Smooth functions at a point -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` at a point `x` if, for any integer `k \u2264 n`,\nthere is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous.\n-\/\ndef times_cont_diff_at (n : with_top \u2115) (f : E \u2192 F) (x : E) :=\ntimes_cont_diff_within_at \ud835\udd5c n f univ x\n\nvariable {\ud835\udd5c}\n\ntheorem times_cont_diff_within_at_univ {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f univ x \u2194 times_cont_diff_at \ud835\udd5c n f x :=\niff.rfl\n\nlemma times_cont_diff_at_top :\n  times_cont_diff_at \ud835\udd5c \u221e f x \u2194 \u2200 (n : \u2115), times_cont_diff_at \ud835\udd5c n f x :=\nby simp [\u2190 times_cont_diff_within_at_univ, times_cont_diff_within_at_top]\n\nlemma times_cont_diff_at.times_cont_diff_within_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) : times_cont_diff_within_at \ud835\udd5c n f s x :=\nh.mono (subset_univ _)\n\nlemma times_cont_diff_within_at.times_cont_diff_at {n : with_top \u2115}\n  (h : times_cont_diff_within_at \ud835\udd5c n f s x) (hx : s \u2208 \ud835\udcdd x) :\n  times_cont_diff_at \ud835\udd5c n f x :=\nby rwa [times_cont_diff_at, \u2190 times_cont_diff_within_at_inter hx, univ_inter]\n\nlemma times_cont_diff_at.congr_of_eventually_eq {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hg : f\u2081 =\u1da0[\ud835\udcdd x] f) :\n  times_cont_diff_at \ud835\udd5c n f\u2081 x :=\nh.congr_of_eventually_eq' (by rwa nhds_within_univ) (mem_univ x)\n\nlemma times_cont_diff_at.of_le {m n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hmn : m \u2264 n) :\n  times_cont_diff_at \ud835\udd5c m f x :=\nh.of_le hmn\n\nlemma times_cont_diff_at.continuous_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) : continuous_at f x :=\nby simpa [continuous_within_at_univ] using h.continuous_within_at\n\n\/-- If a function is `C^n` with `n \u2265 1` at a point, then it is differentiable there. -\/\nlemma times_cont_diff_at.differentiable_at {n : with_top \u2115}\n  (h : times_cont_diff_at \ud835\udd5c n f x) (hn : 1 \u2264 n) : differentiable_at \ud835\udd5c f x :=\nby simpa [hn, differentiable_within_at_univ] using h.differentiable_within_at\n\n\/-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -\/\ntheorem times_cont_diff_at_succ_iff_has_fderiv_at {n : \u2115} :\n  times_cont_diff_at \ud835\udd5c ((n + 1) : \u2115) f x\n  \u2194 (\u2203 f' : E \u2192 (E \u2192L[\ud835\udd5c] F), (\u2203 u \u2208 \ud835\udcdd x, (\u2200 x \u2208 u, has_fderiv_at f (f' x) x))\n      \u2227 (times_cont_diff_at \ud835\udd5c n f' x)) :=\nbegin\n  rw [\u2190 times_cont_diff_within_at_univ, times_cont_diff_within_at_succ_iff_has_fderiv_within_at],\n  simp only [nhds_within_univ, exists_prop, mem_univ, insert_eq_of_mem],\n  split,\n  { rintros \u27e8u, H, f', h_fderiv, h_times_cont_diff\u27e9,\n    rcases mem_nhds_iff.mp H with \u27e8t, htu, ht, hxt\u27e9,\n    refine \u27e8f', \u27e8t, _\u27e9, h_times_cont_diff.times_cont_diff_at H\u27e9,\n    refine \u27e8mem_nhds_iff.mpr \u27e8t, subset.rfl, ht, hxt\u27e9, _\u27e9,\n    intros y hyt,\n    refine (h_fderiv y (htu hyt)).has_fderiv_at _,\n    exact mem_nhds_iff.mpr \u27e8t, htu, ht, hyt\u27e9 },\n  { rintros \u27e8f', \u27e8u, H, h_fderiv\u27e9, h_times_cont_diff\u27e9,\n    refine \u27e8u, H, f', _, h_times_cont_diff.times_cont_diff_within_at\u27e9,\n    intros x hxu,\n    exact (h_fderiv x hxu).has_fderiv_within_at }\nend\n\nprotected theorem times_cont_diff_at.eventually {n : \u2115} (h : times_cont_diff_at \ud835\udd5c n f x) :\n  \u2200\u1da0 y in \ud835\udcdd x, times_cont_diff_at \ud835\udd5c n f y :=\nby simpa [nhds_within_univ] using h.eventually\n\n\/-! ### Smooth functions -\/\n\nvariable (\ud835\udd5c)\n\n\/-- A function is continuously differentiable up to `n` if it admits derivatives up to\norder `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives\nmight not be unique) we do not need to localize the definition in space or time.\n-\/\ndefinition times_cont_diff (n : with_top \u2115) (f : E \u2192 F)  :=\n\u2203 p : E \u2192 formal_multilinear_series \ud835\udd5c E F, has_ftaylor_series_up_to n f p\n\nvariable {\ud835\udd5c}\n\ntheorem times_cont_diff_on_univ {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f univ \u2194 times_cont_diff \ud835\udd5c n f :=\nbegin\n  split,\n  { assume H,\n    use ftaylor_series_within \ud835\udd5c f univ,\n    rw \u2190 has_ftaylor_series_up_to_on_univ_iff,\n    exact H.ftaylor_series_within unique_diff_on_univ },\n  { rintros \u27e8p, hp\u27e9 x hx m hm,\n    exact \u27e8univ, filter.univ_sets _, p, (hp.has_ftaylor_series_up_to_on univ).of_le hm\u27e9 }\nend\n\nlemma times_cont_diff_iff_times_cont_diff_at {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194 \u2200 x, times_cont_diff_at \ud835\udd5c n f x :=\nby simp [\u2190 times_cont_diff_on_univ, times_cont_diff_on, times_cont_diff_at]\n\nlemma times_cont_diff.times_cont_diff_at {n : with_top \u2115} (h : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff_at \ud835\udd5c n f x :=\ntimes_cont_diff_iff_times_cont_diff_at.1 h x\n\nlemma times_cont_diff.times_cont_diff_within_at {n : with_top \u2115} (h : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nh.times_cont_diff_at.times_cont_diff_within_at\n\nlemma times_cont_diff_top :\n  times_cont_diff \ud835\udd5c \u221e f \u2194 \u2200 (n : \u2115), times_cont_diff \ud835\udd5c n f :=\nby simp [times_cont_diff_on_univ.symm, times_cont_diff_on_top]\n\nlemma times_cont_diff_all_iff_nat :\n  (\u2200 n, times_cont_diff \ud835\udd5c n f) \u2194 (\u2200 n : \u2115, times_cont_diff \ud835\udd5c n f) :=\nby simp only [\u2190 times_cont_diff_on_univ, times_cont_diff_on_all_iff_nat]\n\nlemma times_cont_diff.times_cont_diff_on {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) : times_cont_diff_on \ud835\udd5c n f s :=\n(times_cont_diff_on_univ.2 h).mono (subset_univ _)\n\n@[simp] lemma times_cont_diff_zero :\n  times_cont_diff \ud835\udd5c 0 f \u2194 continuous f :=\nbegin\n  rw [\u2190 times_cont_diff_on_univ, continuous_iff_continuous_on_univ],\n  exact times_cont_diff_on_zero\nend\n\nlemma times_cont_diff_at_zero :\n  times_cont_diff_at \ud835\udd5c 0 f x \u2194 \u2203 u \u2208 \ud835\udcdd x, continuous_on f u :=\nby { rw \u2190 times_cont_diff_within_at_univ, simp [times_cont_diff_within_at_zero, nhds_within_univ] }\n\nlemma times_cont_diff.of_le {m n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hmn : m \u2264 n) :\n  times_cont_diff \ud835\udd5c m f :=\ntimes_cont_diff_on_univ.1 $ (times_cont_diff_on_univ.2 h).of_le hmn\n\nlemma times_cont_diff.continuous {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) : continuous f :=\ntimes_cont_diff_zero.1 (h.of_le bot_le)\n\n\/-- If a function is `C^n` with `n \u2265 1`, then it is differentiable. -\/\nlemma times_cont_diff.differentiable {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) : differentiable \ud835\udd5c f :=\ndifferentiable_on_univ.1 $ (times_cont_diff_on_univ.2 h).differentiable_on hn\n\n\n\/-! ### Iterated derivative -\/\n\nvariable (\ud835\udd5c)\n\n\/-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -\/\nnoncomputable def iterated_fderiv (n : \u2115) (f : E \u2192 F) :\n  E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F) :=\nnat.rec_on n\n  (\u03bb x, continuous_multilinear_map.curry0 \ud835\udd5c E (f x))\n  (\u03bb n rec x, continuous_linear_map.uncurry_left (fderiv \ud835\udd5c rec x))\n\n\/-- Formal Taylor series associated to a function within a set. -\/\ndef ftaylor_series (f : E \u2192 F) (x : E) : formal_multilinear_series \ud835\udd5c E F :=\n\u03bb n, iterated_fderiv \ud835\udd5c n f x\n\nvariable {\ud835\udd5c}\n\n@[simp] lemma iterated_fderiv_zero_apply (m : (fin 0) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c 0 f x : ((fin 0) \u2192  E) \u2192 F) m = f x := rfl\n\nlemma iterated_fderiv_zero_eq_comp :\n  iterated_fderiv \ud835\udd5c 0 f = (continuous_multilinear_curry_fin0 \ud835\udd5c E F).symm \u2218 f := rfl\n\nlemma iterated_fderiv_succ_apply_left {n : \u2115} (m : fin (n + 1) \u2192 E):\n  (iterated_fderiv \ud835\udd5c (n + 1) f x : (fin (n + 1) \u2192 E) \u2192 F) m\n  = (fderiv \ud835\udd5c (iterated_fderiv \ud835\udd5c n f) x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) := rfl\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -\/\nlemma iterated_fderiv_succ_eq_comp_left {n : \u2115} :\n  iterated_fderiv \ud835\udd5c (n + 1) f =\n  (continuous_multilinear_curry_left_equiv \ud835\udd5c (\u03bb(i : fin (n + 1)), E) F)\n    \u2218 (fderiv \ud835\udd5c (iterated_fderiv \ud835\udd5c n f)) := rfl\n\nlemma iterated_fderiv_within_univ {n : \u2115} :\n  iterated_fderiv_within \ud835\udd5c n f univ = iterated_fderiv \ud835\udd5c n f :=\nbegin\n  induction n with n IH,\n  { ext x, simp },\n  { ext x m,\n    rw [iterated_fderiv_succ_apply_left, iterated_fderiv_within_succ_apply_left, IH,\n        fderiv_within_univ] }\nend\n\nlemma ftaylor_series_within_univ :\n  ftaylor_series_within \ud835\udd5c f univ = ftaylor_series \ud835\udd5c f :=\nbegin\n  ext1 x, ext1 n,\n  change iterated_fderiv_within \ud835\udd5c n f univ x = iterated_fderiv \ud835\udd5c n f x,\n  rw iterated_fderiv_within_univ\nend\n\ntheorem iterated_fderiv_succ_apply_right {n : \u2115} (m : fin (n + 1) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c (n + 1) f x : (fin (n + 1) \u2192 E) \u2192 F) m\n    = iterated_fderiv \ud835\udd5c n (\u03bby, fderiv \ud835\udd5c f y) x (init m) (m (last n)) :=\nbegin\n  rw [\u2190 iterated_fderiv_within_univ, \u2190 iterated_fderiv_within_univ, \u2190 fderiv_within_univ],\n  exact iterated_fderiv_within_succ_apply_right unique_diff_on_univ (mem_univ _) _\nend\n\n\/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the `n`-th derivative of the derivative. -\/\nlemma iterated_fderiv_succ_eq_comp_right {n : \u2115} :\n  iterated_fderiv \ud835\udd5c (n + 1) f x =\n  ((continuous_multilinear_curry_right_equiv' \ud835\udd5c n E F)\n    \u2218 (iterated_fderiv \ud835\udd5c n (\u03bby, fderiv \ud835\udd5c f y))) x :=\nby { ext m, rw iterated_fderiv_succ_apply_right, refl }\n\n@[simp] lemma iterated_fderiv_one_apply (m : (fin 1) \u2192 E) :\n  (iterated_fderiv \ud835\udd5c 1 f x : ((fin 1) \u2192 E) \u2192 F) m\n  = (fderiv \ud835\udd5c f x : E \u2192 F) (m 0) :=\nby { rw [iterated_fderiv_succ_apply_right, iterated_fderiv_zero_apply], refl }\n\n\/-- When a function is `C^n` in a set `s` of unique differentiability, it admits\n`ftaylor_series_within \ud835\udd5c f s` as a Taylor series up to order `n` in `s`. -\/\ntheorem times_cont_diff_on_iff_ftaylor_series {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194 has_ftaylor_series_up_to n f (ftaylor_series \ud835\udd5c f) :=\nbegin\n  split,\n  { rw [\u2190 times_cont_diff_on_univ, \u2190 has_ftaylor_series_up_to_on_univ_iff,\n        \u2190 ftaylor_series_within_univ],\n    exact \u03bb h, times_cont_diff_on.ftaylor_series_within h unique_diff_on_univ },\n  { assume h, exact \u27e8ftaylor_series \ud835\udd5c f, h\u27e9 }\nend\n\nlemma times_cont_diff_iff_continuous_differentiable {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f \u2194\n  (\u2200 (m : \u2115), (m : with_top \u2115) \u2264 n \u2192 continuous (\u03bb x, iterated_fderiv \ud835\udd5c m f x))\n  \u2227 (\u2200 (m : \u2115), (m : with_top \u2115) < n \u2192 differentiable \ud835\udd5c (\u03bb x, iterated_fderiv \ud835\udd5c m f x)) :=\nby simp [times_cont_diff_on_univ.symm, continuous_iff_continuous_on_univ,\n    differentiable_on_univ.symm, iterated_fderiv_within_univ,\n    times_cont_diff_on_iff_continuous_on_differentiable_on unique_diff_on_univ]\n\nlemma times_cont_diff_of_differentiable_iterated_fderiv {n : with_top \u2115}\n  (h : \u2200(m : \u2115), (m : with_top \u2115) \u2264 n \u2192 differentiable \ud835\udd5c (iterated_fderiv \ud835\udd5c m f)) :\n  times_cont_diff \ud835\udd5c n f :=\ntimes_cont_diff_iff_continuous_differentiable.2\n\u27e8\u03bb m hm, (h m hm).continuous, \u03bb m hm, (h m (le_of_lt hm))\u27e9\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if\nit is differentiable there, and its derivative is `C^n`. -\/\ntheorem times_cont_diff_succ_iff_fderiv {n : \u2115} :\n  times_cont_diff \ud835\udd5c ((n + 1) : \u2115) f \u2194\n  differentiable \ud835\udd5c f \u2227 times_cont_diff \ud835\udd5c n (\u03bb y, fderiv \ud835\udd5c f y) :=\nby simp [times_cont_diff_on_univ.symm, differentiable_on_univ.symm, fderiv_within_univ.symm,\n         - fderiv_within_univ, times_cont_diff_on_succ_iff_fderiv_within unique_diff_on_univ,\n         -with_zero.coe_add, -add_comm]\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative is `C^\u221e`. -\/\ntheorem times_cont_diff_top_iff_fderiv :\n  times_cont_diff \ud835\udd5c \u221e f \u2194\n  differentiable \ud835\udd5c f \u2227 times_cont_diff \ud835\udd5c \u221e (\u03bb y, fderiv \ud835\udd5c f y) :=\nbegin\n  simp [times_cont_diff_on_univ.symm, differentiable_on_univ.symm, fderiv_within_univ.symm,\n        - fderiv_within_univ],\n  rw times_cont_diff_on_top_iff_fderiv_within unique_diff_on_univ,\nend\n\nlemma times_cont_diff.continuous_fderiv {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) :\n  continuous (\u03bb x, fderiv \ud835\udd5c f x) :=\n((times_cont_diff_succ_iff_fderiv).1 (h.of_le hn)).2.continuous\n\n\/-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is\ncontinuous. -\/\nlemma times_cont_diff.continuous_fderiv_apply {n : with_top \u2115}\n  (h : times_cont_diff \ud835\udd5c n f) (hn : 1 \u2264 n) :\n  continuous (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1 : E \u2192 F) p.2) :=\nbegin\n  have A : continuous (\u03bbq : (E \u2192L[\ud835\udd5c] F) \u00d7 E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous,\n  have B : continuous (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1, p.2)),\n  { apply continuous.prod_mk _ continuous_snd,\n    exact continuous.comp (h.continuous_fderiv hn) continuous_fst },\n  exact A.comp B\nend\n\n\/-! ### Constants -\/\n\nlemma iterated_fderiv_within_zero_fun {n : \u2115} :\n  iterated_fderiv \ud835\udd5c n (\u03bb x : E, (0 : F)) = 0 :=\nbegin\n  induction n with n IH,\n  { ext m, simp },\n  { ext x m,\n    rw [iterated_fderiv_succ_apply_left, IH],\n    change (fderiv \ud835\udd5c (\u03bb (x : E), (0 : (E [\u00d7n]\u2192L[\ud835\udd5c] F))) x : E \u2192 (E [\u00d7n]\u2192L[\ud835\udd5c] F)) (m 0) (tail m) = _,\n    rw fderiv_const,\n    refl }\nend\n\nlemma times_cont_diff_zero_fun {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb x : E, (0 : F)) :=\nbegin\n  apply times_cont_diff_of_differentiable_iterated_fderiv (\u03bbm hm, _),\n  rw iterated_fderiv_within_zero_fun,\n  apply differentiable_const (0 : (E [\u00d7m]\u2192L[\ud835\udd5c] F))\nend\n\n\/--\nConstants are `C^\u221e`.\n-\/\nlemma times_cont_diff_const {n : with_top \u2115} {c : F} : times_cont_diff \ud835\udd5c n (\u03bbx : E, c) :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e (\u03bbx : E, c), by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8differentiable_const c, _\u27e9,\n  rw fderiv_const,\n  exact times_cont_diff_zero_fun\nend\n\nlemma times_cont_diff_on_const {n : with_top \u2115} {c : F} {s : set E} :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx : E, c) s :=\ntimes_cont_diff_const.times_cont_diff_on\n\nlemma times_cont_diff_at_const {n : with_top \u2115} {c : F} :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx : E, c) x :=\ntimes_cont_diff_const.times_cont_diff_at\n\nlemma times_cont_diff_within_at_const {n : with_top \u2115} {c : F} :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx : E, c) s x :=\ntimes_cont_diff_at_const.times_cont_diff_within_at\n\n@[nontriviality] lemma times_cont_diff_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n f :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_const }\n\n@[nontriviality] lemma times_cont_diff_at_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n f x :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_at_const }\n\n@[nontriviality] lemma times_cont_diff_within_at_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_within_at_const }\n\n@[nontriviality] lemma times_cont_diff_on_of_subsingleton [subsingleton F] {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n f s :=\nby { rw [subsingleton.elim f (\u03bb _, 0)], exact times_cont_diff_on_const }\n\n\/-! ### Linear functions -\/\n\n\/--\nUnbundled bounded linear functions are `C^\u221e`.\n-\/\nlemma is_bounded_linear_map.times_cont_diff {n : with_top \u2115} (hf : is_bounded_linear_map \ud835\udd5c f) :\n  times_cont_diff \ud835\udd5c n f :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e f, by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8hf.differentiable, _\u27e9,\n  simp [hf.fderiv],\n  exact times_cont_diff_const\nend\n\nlemma continuous_linear_map.times_cont_diff {n : with_top \u2115} (f : E \u2192L[\ud835\udd5c] F) :\n  times_cont_diff \ud835\udd5c n f :=\nf.is_bounded_linear_map.times_cont_diff\n\nlemma continuous_linear_equiv.times_cont_diff {n : with_top \u2115} (f : E \u2243L[\ud835\udd5c] F) :\n  times_cont_diff \ud835\udd5c n f :=\n(f : E \u2192L[\ud835\udd5c] F).times_cont_diff\n\nlemma linear_isometry.times_cont_diff {n : with_top \u2115} (f : E \u2192\u2097\u1d62[\ud835\udd5c] F) :\n  times_cont_diff \ud835\udd5c n f :=\nf.to_continuous_linear_map.times_cont_diff\n\nlemma linear_isometry_equiv.times_cont_diff {n : with_top \u2115} (f : E \u2243\u2097\u1d62[\ud835\udd5c] F) :\n  times_cont_diff \ud835\udd5c n f :=\n(f : E \u2192L[\ud835\udd5c] F).times_cont_diff\n\n\/--\nThe first projection in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_fst {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) :=\nis_bounded_linear_map.times_cont_diff is_bounded_linear_map.fst\n\n\/--\nThe first projection on a domain in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_on_fst {s : set (E\u00d7F)} {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) s :=\ntimes_cont_diff.times_cont_diff_on times_cont_diff_fst\n\n\/--\nThe first projection at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_at_fst {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) p :=\ntimes_cont_diff_fst.times_cont_diff_at\n\n\/--\nThe first projection within a domain at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_within_at_fst {s : set (E \u00d7 F)} {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n (prod.fst : E \u00d7 F \u2192 E) s p :=\ntimes_cont_diff_fst.times_cont_diff_within_at\n\n\/--\nThe second projection in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_snd {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) :=\nis_bounded_linear_map.times_cont_diff is_bounded_linear_map.snd\n\n\/--\nThe second projection on a domain in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_on_snd {s : set (E\u00d7F)} {n : with_top \u2115} :\n  times_cont_diff_on \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) s :=\ntimes_cont_diff.times_cont_diff_on times_cont_diff_snd\n\n\/--\nThe second projection at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_at_snd {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_at \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) p :=\ntimes_cont_diff_snd.times_cont_diff_at\n\n\/--\nThe second projection within a domain at a point in a product is `C^\u221e`.\n-\/\nlemma times_cont_diff_within_at_snd {s : set (E \u00d7 F)} {p : E \u00d7 F} {n : with_top \u2115} :\n  times_cont_diff_within_at \ud835\udd5c n (prod.snd : E \u00d7 F \u2192 F) s p :=\ntimes_cont_diff_snd.times_cont_diff_within_at\n\n\/--\nThe natural equivalence `(E \u00d7 F) \u00d7 G \u2243 E \u00d7 (F \u00d7 G)` is smooth.\n\nWarning: if you think you need this lemma, it is likely that you can simplify your proof by\nreformulating the lemma that you're applying next using the tips in\nNote [continuity lemma statement]\n-\/\nlemma times_cont_diff_prod_assoc : times_cont_diff \ud835\udd5c \u22a4 $ equiv.prod_assoc E F G :=\n(linear_isometry_equiv.prod_assoc \ud835\udd5c E F G).times_cont_diff\n\n\/--\nThe natural equivalence `E \u00d7 (F \u00d7 G) \u2243 (E \u00d7 F) \u00d7 G` is smooth.\n\nWarning: see remarks attached to `times_cont_diff_prod_assoc`\n-\/\nlemma times_cont_diff_prod_assoc_symm : times_cont_diff \ud835\udd5c \u22a4 $ (equiv.prod_assoc E F G).symm :=\n(linear_isometry_equiv.prod_assoc \ud835\udd5c E F G).symm.times_cont_diff\n\n\/--\nThe identity is `C^\u221e`.\n-\/\nlemma times_cont_diff_id {n : with_top \u2115} : times_cont_diff \ud835\udd5c n (id : E \u2192 E) :=\nis_bounded_linear_map.id.times_cont_diff\n\nlemma times_cont_diff_within_at_id {n : with_top \u2115} {s x} :\n  times_cont_diff_within_at \ud835\udd5c n (id : E \u2192 E) s x :=\ntimes_cont_diff_id.times_cont_diff_within_at\n\nlemma times_cont_diff_at_id {n : with_top \u2115} {x} :\n  times_cont_diff_at \ud835\udd5c n (id : E \u2192 E) x :=\ntimes_cont_diff_id.times_cont_diff_at\n\nlemma times_cont_diff_on_id {n : with_top \u2115} {s} :\n  times_cont_diff_on \ud835\udd5c n (id : E \u2192 E) s :=\ntimes_cont_diff_id.times_cont_diff_on\n\n\/--\nBilinear functions are `C^\u221e`.\n-\/\nlemma is_bounded_bilinear_map.times_cont_diff {n : with_top \u2115} (hb : is_bounded_bilinear_map \ud835\udd5c b) :\n  times_cont_diff \ud835\udd5c n b :=\nbegin\n  suffices h : times_cont_diff \ud835\udd5c \u221e b, by exact h.of_le le_top,\n  rw times_cont_diff_top_iff_fderiv,\n  refine \u27e8hb.differentiable, _\u27e9,\n  simp [hb.fderiv],\n  exact hb.is_bounded_linear_map_deriv.times_cont_diff\nend\n\n\/-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `g \u2218 f` admits a Taylor\nseries whose `k`-th term is given by `g \u2218 (p k)`. -\/\nlemma has_ftaylor_series_up_to_on.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : has_ftaylor_series_up_to_on n f p s) :\n  has_ftaylor_series_up_to_on n (g \u2218 f) (\u03bb x k, g.comp_continuous_multilinear_map (p x k)) s :=\nbegin\n  set L : \u03a0 m : \u2115, (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u2192L[\ud835\udd5c] (E [\u00d7m]\u2192L[\ud835\udd5c] G) :=\n    \u03bb m, continuous_linear_map.comp_continuous_multilinear_mapL g,\n  split,\n  { exact \u03bb x hx, congr_arg g (hf.zero_eq x hx) },\n  { intros m hm x hx,\n    convert (L m).has_fderiv_at.comp_has_fderiv_within_at x (hf.fderiv_within m hm x hx) },\n  { intros m hm,\n    convert (L m).continuous.comp_continuous_on (hf.cont m hm) }\nend\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain\nat a point. -\/\nlemma times_cont_diff_within_at.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  exact \u27e8u, hu, _, hp.continuous_linear_map_comp g\u27e9,\nend\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain\nat a point. -\/\nlemma times_cont_diff_at.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\ntimes_cont_diff_within_at.continuous_linear_map_comp g hf\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions on domains. -\/\nlemma times_cont_diff_on.continuous_linear_map_comp {n : with_top \u2115} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\n\u03bb x hx, (hf x hx).continuous_linear_map_comp g\n\n\/-- Composition by continuous linear maps on the left preserves `C^n` functions. -\/\nlemma times_cont_diff.continuous_linear_map_comp {n : with_top \u2115} {f : E \u2192 F} (g : F \u2192L[\ud835\udd5c] G)\n  (hf : times_cont_diff \ud835\udd5c n f) : times_cont_diff \ud835\udd5c n (\u03bbx, g (f x)) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.continuous_linear_map_comp\n  _ (times_cont_diff_on_univ.2 hf)\n\n\/-- Composition by continuous linear equivs on the left respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.comp_times_cont_diff_within_at_iff\n  {n : with_top \u2115} (e : F \u2243L[\ud835\udd5c] G) :\n  times_cont_diff_within_at \ud835\udd5c n (e \u2218 f) s x \u2194 times_cont_diff_within_at \ud835\udd5c n f s x :=\n\u27e8\u03bb H, by simpa only [(\u2218), e.symm.coe_coe, e.symm_apply_apply]\n  using H.continuous_linear_map_comp (e.symm : G \u2192L[\ud835\udd5c] F),\n  \u03bb H, H.continuous_linear_map_comp (e : F \u2192L[\ud835\udd5c] G)\u27e9\n\n\/-- Composition by continuous linear equivs on the left respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.comp_times_cont_diff_on_iff\n  {n : with_top \u2115} (e : F \u2243L[\ud835\udd5c] G) :\n  times_cont_diff_on \ud835\udd5c n (e \u2218 f) s \u2194 times_cont_diff_on \ud835\udd5c n f s :=\nby simp [times_cont_diff_on, e.comp_times_cont_diff_within_at_iff]\n\n\/-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `f \u2218 g` admits a Taylor\nseries in `g \u207b\u00b9' s`, whose `k`-th term is given by `p k (g v\u2081, ..., g v\u2096)` . -\/\nlemma has_ftaylor_series_up_to_on.comp_continuous_linear_map {n : with_top \u2115}\n  (hf : has_ftaylor_series_up_to_on n f p s) (g : G \u2192L[\ud835\udd5c] E) :\n  has_ftaylor_series_up_to_on n (f \u2218 g)\n    (\u03bb x k, (p (g x) k).comp_continuous_linear_map (\u03bb _, g)) (g \u207b\u00b9' s) :=\nbegin\n  let A : \u03a0 m : \u2115, (E [\u00d7m]\u2192L[\ud835\udd5c] F) \u2192 (G [\u00d7m]\u2192L[\ud835\udd5c] F) :=\n    \u03bb m h, h.comp_continuous_linear_map (\u03bb _, g),\n  have hA : \u2200 m, is_bounded_linear_map \ud835\udd5c (A m) :=\n    \u03bb m, is_bounded_linear_map_continuous_multilinear_map_comp_linear g,\n  split,\n  { assume x hx,\n    simp only [(hf.zero_eq (g x) hx).symm, function.comp_app],\n    change p (g x) 0 (\u03bb (i : fin 0), g 0) = p (g x) 0 0,\n    rw continuous_linear_map.map_zero,\n    refl },\n  { assume m hm x hx,\n    convert ((hA m).has_fderiv_at).comp_has_fderiv_within_at x\n      ((hf.fderiv_within m hm (g x) hx).comp x (g.has_fderiv_within_at) (subset.refl _)),\n    ext y v,\n    change p (g x) (nat.succ m) (g \u2218 (cons y v)) = p (g x) m.succ (cons (g y) (g \u2218 v)),\n    rw comp_cons },\n  { assume m hm,\n    exact (hA m).continuous.comp_continuous_on\n      ((hf.cont m hm).comp g.continuous.continuous_on (subset.refl _)) }\nend\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions at a point on\na domain. -\/\nlemma times_cont_diff_within_at.comp_continuous_linear_map {n : with_top \u2115} {x : G}\n  (g : G \u2192L[\ud835\udd5c] E) (hf : times_cont_diff_within_at \ud835\udd5c n f s (g x)) :\n  times_cont_diff_within_at \ud835\udd5c n (f \u2218 g) (g \u207b\u00b9' s) x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  refine \u27e8g \u207b\u00b9' u, _, _, hp.comp_continuous_linear_map g\u27e9,\n  apply continuous_within_at.preimage_mem_nhds_within',\n  { exact g.continuous.continuous_within_at },\n  { apply nhds_within_mono (g x) _ hu,\n    rw image_insert_eq,\n    exact insert_subset_insert (image_preimage_subset g s) }\nend\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions on domains. -\/\nlemma times_cont_diff_on.comp_continuous_linear_map {n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (g : G \u2192L[\ud835\udd5c] E) :\n  times_cont_diff_on \ud835\udd5c n (f \u2218 g) (g \u207b\u00b9' s) :=\n\u03bb x hx, (hf (g x) hx).comp_continuous_linear_map g\n\n\/-- Composition by continuous linear maps on the right preserves `C^n` functions. -\/\nlemma times_cont_diff.comp_continuous_linear_map {n : with_top \u2115} {f : E \u2192 F} {g : G \u2192L[\ud835\udd5c] E}\n  (hf : times_cont_diff \ud835\udd5c n f) : times_cont_diff \ud835\udd5c n (f \u2218 g) :=\ntimes_cont_diff_on_univ.1 $\ntimes_cont_diff_on.comp_continuous_linear_map (times_cont_diff_on_univ.2 hf) _\n\n\/-- Composition by continuous linear equivs on the right respects higher differentiability at a\npoint in a domain. -\/\nlemma continuous_linear_equiv.times_cont_diff_within_at_comp_iff {n : with_top \u2115} (e : G \u2243L[\ud835\udd5c] E) :\n  times_cont_diff_within_at \ud835\udd5c n (f \u2218 e) (e \u207b\u00b9' s) (e.symm x) \u2194\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nbegin\n  split,\n  { assume H,\n    simpa [\u2190 preimage_comp, (\u2218)] using H.comp_continuous_linear_map (e.symm : E \u2192L[\ud835\udd5c] G) },\n  { assume H,\n    rw [\u2190 e.apply_symm_apply x, \u2190 e.coe_coe] at H,\n    exact H.comp_continuous_linear_map _ },\nend\n\n\/-- Composition by continuous linear equivs on the right respects higher differentiability on\ndomains. -\/\nlemma continuous_linear_equiv.times_cont_diff_on_comp_iff {n : with_top \u2115} (e : G \u2243L[\ud835\udd5c] E) :\n  times_cont_diff_on \ud835\udd5c n (f \u2218 e) (e \u207b\u00b9' s) \u2194 times_cont_diff_on \ud835\udd5c n f s :=\nbegin\n  refine \u27e8\u03bb H, _, \u03bb H, H.comp_continuous_linear_map (e : G \u2192L[\ud835\udd5c] E)\u27e9,\n  have A : f = (f \u2218 e) \u2218 e.symm,\n    by { ext y, simp only [function.comp_app], rw e.apply_symm_apply y },\n  have B : e.symm \u207b\u00b9' (e \u207b\u00b9' s) = s,\n    by { rw [\u2190 preimage_comp, e.self_comp_symm], refl },\n  rw [A, \u2190 B],\n  exact H.comp_continuous_linear_map (e.symm : E \u2192L[\ud835\udd5c] G)\nend\n\n\/-- If two functions `f` and `g` admit Taylor series `p` and `q` in a set `s`, then the cartesian\nproduct of `f` and `g` admits the cartesian product of `p` and `q` as a Taylor series. -\/\nlemma has_ftaylor_series_up_to_on.prod {n : with_top \u2115} (hf : has_ftaylor_series_up_to_on n f p s)\n  {g : E \u2192 G} {q : E \u2192 formal_multilinear_series \ud835\udd5c E G} (hg : has_ftaylor_series_up_to_on n g q s) :\n  has_ftaylor_series_up_to_on n (\u03bb y, (f y, g y)) (\u03bb y k, (p y k).prod (q y k)) s :=\nbegin\n  set L := \u03bb m, continuous_multilinear_map.prodL \ud835\udd5c (\u03bb i : fin m, E) F G,\n  split,\n  { assume x hx, rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx], refl },\n  { assume m hm x hx,\n    convert (L m).has_fderiv_at.comp_has_fderiv_within_at x\n      ((hf.fderiv_within m hm x hx).prod (hg.fderiv_within m hm x hx)) },\n  { assume m hm,\n    exact (L m).continuous.comp_continuous_on ((hf.cont m hm).prod (hg.cont m hm)) }\nend\n\n\/-- The cartesian product of `C^n` functions at a point in a domain is `C^n`. -\/\nlemma times_cont_diff_within_at.prod {n : with_top \u2115} {s : set E} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx:E, (f x, g x)) s x :=\nbegin\n  assume m hm,\n  rcases hf m hm with \u27e8u, hu, p, hp\u27e9,\n  rcases hg m hm with \u27e8v, hv, q, hq\u27e9,\n  exact \u27e8u \u2229 v, filter.inter_mem hu hv, _,\n        (hp.mono (inter_subset_left u v)).prod (hq.mono (inter_subset_right u v))\u27e9\nend\n\n\/-- The cartesian product of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.prod {n : with_top \u2115} {s : set E} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx:E, (f x, g x)) s :=\n\u03bb x hx, (hf x hx).prod (hg x hx)\n\n\/-- The cartesian product of `C^n` functions at a point is `C^n`. -\/\nlemma times_cont_diff_at.prod {n : with_top \u2115} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx:E, (f x, g x)) x :=\ntimes_cont_diff_within_at_univ.1 $ times_cont_diff_within_at.prod\n  (times_cont_diff_within_at_univ.2 hf)\n  (times_cont_diff_within_at_univ.2 hg)\n\n\/--\nThe cartesian product of `C^n` functions is `C^n`.\n-\/\nlemma times_cont_diff.prod {n : with_top \u2115} {f : E \u2192 F} {g : E \u2192 G}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bbx:E, (f x, g x)) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.prod (times_cont_diff_on_univ.2 hf)\n  (times_cont_diff_on_univ.2 hg)\n\n\/-!\n### Smoothness of functions `f : E \u2192 \u03a0 i, F' i`\n-\/\n\nsection pi\n\nvariables {\u03b9 : Type*} [fintype \u03b9] {F' : \u03b9 \u2192 Type*} [\u03a0 i, normed_group (F' i)]\n  [\u03a0 i, normed_space \ud835\udd5c (F' i)] {\u03c6 : \u03a0 i, E \u2192 F' i}\n  {p' : \u03a0 i, E \u2192 formal_multilinear_series \ud835\udd5c E (F' i)}\n  {\u03a6 : E \u2192 \u03a0 i, F' i} {P' : E \u2192 formal_multilinear_series \ud835\udd5c E (\u03a0 i, F' i)}\n  {n : with_top \u2115}\n\nlemma has_ftaylor_series_up_to_on_pi :\n  has_ftaylor_series_up_to_on n (\u03bb x i, \u03c6 i x)\n    (\u03bb x m, continuous_multilinear_map.pi (\u03bb i, p' i x m)) s \u2194\n    \u2200 i, has_ftaylor_series_up_to_on n (\u03c6 i) (p' i) s :=\nbegin\n  set pr := @continuous_linear_map.proj \ud835\udd5c _ \u03b9 F' _ _ _,\n  letI : \u03a0 (m : \u2115) (i : \u03b9), normed_space \ud835\udd5c (E [\u00d7m]\u2192L[\ud835\udd5c] (F' i)) := \u03bb m i, infer_instance,\n  set L : \u03a0 m : \u2115, (\u03a0 i, E [\u00d7m]\u2192L[\ud835\udd5c] (F' i)) \u2243\u2097\u1d62[\ud835\udd5c] (E [\u00d7m]\u2192L[\ud835\udd5c] (\u03a0 i, F' i)) :=\n    \u03bb m, continuous_multilinear_map.pi\u2097\u1d62 _ _,\n  refine \u27e8\u03bb h i, _, \u03bb h, \u27e8\u03bb x hx, _, _, _\u27e9\u27e9,\n  { convert h.continuous_linear_map_comp (pr i),\n    ext, refl },\n  { ext1 i,\n    exact (h i).zero_eq x hx },\n  { intros m hm x hx,\n    have := has_fderiv_within_at_pi.2 (\u03bb i, (h i).fderiv_within m hm x hx),\n    -- TODO: lean can't find the instance without this: If we remove this `letI`, we have to add\n    -- `local attribute [-instance] punit.mul_action` instead!\n    letI : normed_space \ud835\udd5c (E [\u00d7m]\u2192L[\ud835\udd5c] (\u03a0 i, F' i)) := infer_instance,\n    convert (L m).has_fderiv_at.comp_has_fderiv_within_at x this },\n  { intros m hm,\n    have := continuous_on_pi.2 (\u03bb i, (h i).cont m hm),\n    convert (L m).continuous.comp_continuous_on this }\nend\n\n@[simp] lemma has_ftaylor_series_up_to_on_pi' :\n  has_ftaylor_series_up_to_on n \u03a6 P' s \u2194\n    \u2200 i, has_ftaylor_series_up_to_on n (\u03bb x, \u03a6 x i)\n      (\u03bb x m, (@continuous_linear_map.proj \ud835\udd5c _ \u03b9 F' _ _ _ i).comp_continuous_multilinear_map\n        (P' x m)) s :=\nby { convert has_ftaylor_series_up_to_on_pi, ext, refl }\n\nlemma times_cont_diff_within_at_pi :\n  times_cont_diff_within_at \ud835\udd5c n \u03a6 s x \u2194\n    \u2200 i, times_cont_diff_within_at \ud835\udd5c n (\u03bb x, \u03a6 x i) s x :=\nbegin\n  set pr := @continuous_linear_map.proj \ud835\udd5c _ \u03b9 F' _ _ _,\n  refine \u27e8\u03bb h i, h.continuous_linear_map_comp (pr i), \u03bb h m hm, _\u27e9,\n  choose u hux p hp using \u03bb i, h i m hm,\n  exact \u27e8\u22c2 i, u i, filter.Inter_mem.2 hux, _,\n    has_ftaylor_series_up_to_on_pi.2 (\u03bb i, (hp i).mono $ Inter_subset _ _)\u27e9,\nend\n\nlemma times_cont_diff_on_pi :\n  times_cont_diff_on \ud835\udd5c n \u03a6 s \u2194 \u2200 i, times_cont_diff_on \ud835\udd5c n (\u03bb x, \u03a6 x i) s :=\n\u27e8\u03bb h i x hx, times_cont_diff_within_at_pi.1 (h x hx) _,\n  \u03bb h x hx, times_cont_diff_within_at_pi.2 (\u03bb i, h i x hx)\u27e9\n\nlemma times_cont_diff_at_pi :\n  times_cont_diff_at \ud835\udd5c n \u03a6 x \u2194 \u2200 i, times_cont_diff_at \ud835\udd5c n (\u03bb x, \u03a6 x i) x :=\ntimes_cont_diff_within_at_pi\n\nlemma times_cont_diff_pi :\n  times_cont_diff \ud835\udd5c n \u03a6 \u2194 \u2200 i, times_cont_diff \ud835\udd5c n (\u03bb x, \u03a6 x i) :=\nby simp only [\u2190 times_cont_diff_on_univ, times_cont_diff_on_pi]\n\nend pi\n\n\/-!\n### Composition of `C^n` functions\n\nWe show that the composition of `C^n` functions is `C^n`. One way to prove it would be to write\nthe `n`-th derivative of the composition (this is Fa\u00e0 di Bruno's formula) and check its continuity,\nbut this is very painful. Instead, we go for a simple inductive proof. Assume it is done for `n`.\nThen, to check it for `n+1`, one needs to check that the derivative of `g \u2218 f` is `C^n`, i.e.,\nthat `Dg(f x) \u2b1d Df(x)` is `C^n`. The term `Dg (f x)` is the composition of two `C^n` functions, so\nit is `C^n` by the inductive assumption. The term `Df(x)` is also `C^n`. Then, the matrix\nmultiplication is the application of a bilinear map (which is `C^\u221e`, and therefore `C^n`) to\n`x \u21a6 (Dg(f x), Df x)`. As the composition of two `C^n` maps, it is again `C^n`, and we are done.\n\nThere is a subtlety in this argument: we apply the inductive assumption to functions on other Banach\nspaces. In maths, one would say: prove by induction over `n` that, for all `C^n` maps between all\npairs of Banach spaces, their composition is `C^n`. In Lean, this is fine as long as the spaces\nstay in the same universe. This is not the case in the above argument: if `E` lives in universe `u`\nand `F` lives in universe `v`, then linear maps from `E` to `F` (to which the derivative of `f`\nbelongs) is in universe `max u v`. If one could quantify over finitely many universes, the above\nproof would work fine, but this is not the case. One could still write the proof considering spaces\nin any universe in `u, v, w, max u v, max v w, max u v w`, but it would be extremely tedious and\nlead to a lot of duplication. Instead, we formulate the above proof when all spaces live in the same\nuniverse (where everything is fine), and then we deduce the general result by lifting all our spaces\nto a common universe. We use the trick that any space `H` is isomorphic through a continuous linear\nequiv to `continuous_multilinear_map (\u03bb (i : fin 0), E \u00d7 F \u00d7 G) H` to change the universe level,\nand then argue that composing with such a linear equiv does not change the fact of being `C^n`,\nwhich we have already proved previously.\n-\/\n\n\/-- Auxiliary lemma proving that the composition of `C^n` functions on domains is `C^n` when all\nspaces live in the same universe. Use instead `times_cont_diff_on.comp` which removes the universe\nassumption (but is deduced from this one). -\/\nprivate lemma times_cont_diff_on.comp_same_univ\n  {Eu : Type u} [normed_group Eu] [normed_space \ud835\udd5c Eu]\n  {Fu : Type u} [normed_group Fu] [normed_space \ud835\udd5c Fu]\n  {Gu : Type u} [normed_group Gu] [normed_space \ud835\udd5c Gu]\n  {n : with_top \u2115} {s : set Eu} {t : set Fu} {g : Fu \u2192 Gu} {f : Eu \u2192 Fu}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\nbegin\n  unfreezingI { induction n using with_top.nat_induction with n IH Itop generalizing Eu Fu Gu },\n  { rw times_cont_diff_on_zero at hf hg \u22a2,\n    exact continuous_on.comp hg hf st },\n  { rw times_cont_diff_on_succ_iff_has_fderiv_within_at at hg \u22a2,\n    assume x hx,\n    rcases (times_cont_diff_on_succ_iff_has_fderiv_within_at.1 hf) x hx\n      with \u27e8u, hu, f', hf', f'_diff\u27e9,\n    rcases hg (f x) (st hx) with \u27e8v, hv, g', hg', g'_diff\u27e9,\n    rw insert_eq_of_mem hx at hu \u22a2,\n    have xu : x \u2208 u := mem_of_mem_nhds_within hx hu,\n    let w := s \u2229 (u \u2229 f\u207b\u00b9' v),\n    have wv : w \u2286 f \u207b\u00b9' v := \u03bb y hy, hy.2.2,\n    have wu : w \u2286 u := \u03bb y hy, hy.2.1,\n    have ws : w \u2286 s := \u03bb y hy, hy.1,\n    refine \u27e8w, _, \u03bb y, (g' (f y)).comp (f' y), _, _\u27e9,\n    show w \u2208 \ud835\udcdd[s] x,\n    { apply filter.inter_mem self_mem_nhds_within,\n      apply filter.inter_mem hu,\n      apply continuous_within_at.preimage_mem_nhds_within',\n      { rw \u2190 continuous_within_at_inter' hu,\n        exact (hf' x xu).differentiable_within_at.continuous_within_at.mono\n          (inter_subset_right _ _) },\n      { apply nhds_within_mono _ _ hv,\n        exact subset.trans (image_subset_iff.mpr st) (subset_insert (f x) t) } },\n    show \u2200 y \u2208 w,\n      has_fderiv_within_at (g \u2218 f) ((g' (f y)).comp (f' y)) w y,\n    { rintros y \u27e8ys, yu, yv\u27e9,\n      exact (hg' (f y) yv).comp y ((hf' y yu).mono wu) wv },\n    show times_cont_diff_on \ud835\udd5c n (\u03bb y, (g' (f y)).comp (f' y)) w,\n    { have A : times_cont_diff_on \ud835\udd5c n (\u03bb y, g' (f y)) w :=\n        IH g'_diff ((hf.of_le (with_top.coe_le_coe.2 (nat.le_succ n))).mono ws) wv,\n      have B : times_cont_diff_on \ud835\udd5c n f' w := f'_diff.mono wu,\n      have C : times_cont_diff_on \ud835\udd5c n (\u03bb y, (f' y, g' (f y))) w :=\n        times_cont_diff_on.prod B A,\n      have D : times_cont_diff_on \ud835\udd5c n (\u03bb(p : (Eu \u2192L[\ud835\udd5c] Fu) \u00d7 (Fu \u2192L[\ud835\udd5c] Gu)), p.2.comp p.1) univ :=\n        is_bounded_bilinear_map_comp.times_cont_diff.times_cont_diff_on,\n      exact IH D C (subset_univ _) } },\n  { rw times_cont_diff_on_top at hf hg \u22a2,\n    assume n,\n    apply Itop n (hg n) (hf n) st }\nend\n\n\/-- The composition of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.comp\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\nbegin\n  \/- we lift all the spaces to a common universe, as we have already proved the result in this\n  situation. For the lift, we use the trick that `H` is isomorphic through a\n  continuous linear equiv to `continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) H`, and\n  continuous linear equivs respect smoothness classes. -\/\n  let Eu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) E,\n  letI : normed_group Eu := by apply_instance,\n  letI : normed_space \ud835\udd5c Eu := by apply_instance,\n  let Fu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) F,\n  letI : normed_group Fu := by apply_instance,\n  letI : normed_space \ud835\udd5c Fu := by apply_instance,\n  let Gu := continuous_multilinear_map \ud835\udd5c (\u03bb (i : fin 0), (E \u00d7 F \u00d7 G)) G,\n  letI : normed_group Gu := by apply_instance,\n  letI : normed_space \ud835\udd5c Gu := by apply_instance,\n  -- declare the isomorphisms\n  let isoE : Eu \u2243L[\ud835\udd5c] E := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) E,\n  let isoF : Fu \u2243L[\ud835\udd5c] F := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) F,\n  let isoG : Gu \u2243L[\ud835\udd5c] G := continuous_multilinear_curry_fin0 \ud835\udd5c (E \u00d7 F \u00d7 G) G,\n  -- lift the functions to the new spaces, check smoothness there, and then go back.\n  let fu : Eu \u2192 Fu := (isoF.symm \u2218 f) \u2218 isoE,\n  have fu_diff : times_cont_diff_on \ud835\udd5c n fu (isoE \u207b\u00b9' s),\n    by rwa [isoE.times_cont_diff_on_comp_iff, isoF.symm.comp_times_cont_diff_on_iff],\n  let gu : Fu \u2192 Gu := (isoG.symm \u2218 g) \u2218 isoF,\n  have gu_diff : times_cont_diff_on \ud835\udd5c n gu (isoF \u207b\u00b9' t),\n    by rwa [isoF.times_cont_diff_on_comp_iff, isoG.symm.comp_times_cont_diff_on_iff],\n  have main : times_cont_diff_on \ud835\udd5c n (gu \u2218 fu) (isoE \u207b\u00b9' s),\n  { apply times_cont_diff_on.comp_same_univ gu_diff fu_diff,\n    assume y hy,\n    simp only [fu, continuous_linear_equiv.coe_apply, function.comp_app, mem_preimage],\n    rw isoF.apply_symm_apply (f (isoE y)),\n    exact st hy },\n  have : gu \u2218 fu = (isoG.symm \u2218 (g \u2218 f)) \u2218 isoE,\n  { ext y,\n    simp only [function.comp_apply, gu, fu],\n    rw isoF.apply_symm_apply (f (isoE y)) },\n  rwa [this, isoE.times_cont_diff_on_comp_iff, isoG.symm.comp_times_cont_diff_on_iff] at main\nend\n\n\/-- The composition of `C^n` functions on domains is `C^n`. -\/\nlemma times_cont_diff_on.comp'\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff_on \ud835\udd5c n g t) (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) (s \u2229 f\u207b\u00b9' t) :=\nhg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)\n\n\/-- The composition of a `C^n` function on a domain with a `C^n` function is `C^n`. -\/\nlemma times_cont_diff.comp_times_cont_diff_on {n : with_top \u2115} {s : set E} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff \ud835\udd5c n g) (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (g \u2218 f) s :=\n(times_cont_diff_on_univ.2 hg).comp hf subset_preimage_univ\n\n\/-- The composition of `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.comp {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F}\n  (hg : times_cont_diff \ud835\udd5c n g) (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (g \u2218 f) :=\ntimes_cont_diff_on_univ.1 $ times_cont_diff_on.comp (times_cont_diff_on_univ.2 hg)\n  (times_cont_diff_on_univ.2 hf) (subset_univ _)\n\n\/-- The composition of `C^n` functions at points in domains is `C^n`. -\/\nlemma times_cont_diff_within_at.comp\n  {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G} {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff_within_at \ud835\udd5c n g t (f x))\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (st : s \u2286 f \u207b\u00b9' t) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nbegin\n  assume m hm,\n  rcases hg.times_cont_diff_on hm with \u27e8u, u_nhd, ut, hu\u27e9,\n  rcases hf.times_cont_diff_on hm with \u27e8v, v_nhd, vs, hv\u27e9,\n  have xmem : x \u2208 f \u207b\u00b9' u \u2229 v :=\n    \u27e8(mem_of_mem_nhds_within (mem_insert (f x) _) u_nhd : _),\n    mem_of_mem_nhds_within (mem_insert x s) v_nhd\u27e9,\n  have : f \u207b\u00b9' u \u2208 \ud835\udcdd[insert x s] x,\n  { apply hf.continuous_within_at.insert_self.preimage_mem_nhds_within',\n    apply nhds_within_mono _ _ u_nhd,\n    rw image_insert_eq,\n    exact insert_subset_insert (image_subset_iff.mpr st) },\n  have Z := ((hu.comp (hv.mono (inter_subset_right (f \u207b\u00b9' u) v)) (inter_subset_left _ _))\n    .times_cont_diff_within_at) xmem m (le_refl _),\n  have : \ud835\udcdd[f \u207b\u00b9' u \u2229 v] x = \ud835\udcdd[insert x s] x,\n  { have A : f \u207b\u00b9' u \u2229 v = (insert x s) \u2229 (f \u207b\u00b9' u \u2229 v),\n    { apply subset.antisymm _ (inter_subset_right _ _),\n      rintros y \u27e8hy1, hy2\u27e9,\n      simp [hy1, hy2, vs hy2] },\n    rw [A, \u2190 nhds_within_restrict''],\n    exact filter.inter_mem this v_nhd },\n  rwa [insert_eq_of_mem xmem, this] at Z,\nend\n\n\/-- The composition of `C^n` functions at points in domains is `C^n`. -\/\nlemma times_cont_diff_within_at.comp' {n : with_top \u2115} {s : set E} {t : set F} {g : F \u2192 G}\n  {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff_within_at \ud835\udd5c n g t (f x)) (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) (s \u2229 f\u207b\u00b9' t) x :=\nhg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)\n\nlemma times_cont_diff_at.comp_times_cont_diff_within_at {n} (x : E)\n  (hg : times_cont_diff_at \ud835\udd5c n g (f x)) (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) s x :=\nhg.comp x hf (maps_to_univ _ _)\n\n\/-- The composition of `C^n` functions at points is `C^n`. -\/\nlemma times_cont_diff_at.comp {n : with_top \u2115} (x : E)\n  (hg : times_cont_diff_at \ud835\udd5c n g (f x))\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\nhg.comp x hf subset_preimage_univ\n\nlemma times_cont_diff.comp_times_cont_diff_within_at\n  {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F} (h : times_cont_diff \ud835\udd5c n g)\n  (hf : times_cont_diff_within_at \ud835\udd5c n f t x) :\n  times_cont_diff_within_at \ud835\udd5c n (g \u2218 f) t x :=\nbegin\n  have : times_cont_diff_within_at \ud835\udd5c n g univ (f x) :=\n    h.times_cont_diff_at.times_cont_diff_within_at,\n  exact this.comp x hf (subset_univ _),\nend\n\nlemma times_cont_diff.comp_times_cont_diff_at\n  {n : with_top \u2115} {g : F \u2192 G} {f : E \u2192 F} (x : E)\n  (hg : times_cont_diff \ud835\udd5c n g)\n  (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (g \u2218 f) x :=\nhg.comp_times_cont_diff_within_at hf\n\n\/-- The bundled derivative of a `C^{n+1}` function is `C^n`. -\/\nlemma times_cont_diff_on_fderiv_within_apply {m n : with_top  \u2115} {s : set E}\n  {f : E \u2192 F} (hf : times_cont_diff_on \ud835\udd5c n f s) (hs : unique_diff_on \ud835\udd5c s) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (\u03bbp : E \u00d7 E, (fderiv_within \ud835\udd5c f s p.1 : E \u2192L[\ud835\udd5c] F) p.2)\n  (set.prod s (univ : set E)) :=\nbegin\n  have A : times_cont_diff \ud835\udd5c m (\u03bbp : (E \u2192L[\ud835\udd5c] F) \u00d7 E, p.1 p.2),\n  { apply is_bounded_bilinear_map.times_cont_diff,\n    exact is_bounded_bilinear_map_apply },\n  have B : times_cont_diff_on \ud835\udd5c m\n    (\u03bb (p : E \u00d7 E), ((fderiv_within \ud835\udd5c f s p.fst), p.snd)) (set.prod s univ),\n  { apply times_cont_diff_on.prod _ _,\n    { have I : times_cont_diff_on \ud835\udd5c m (\u03bb (x : E), fderiv_within \ud835\udd5c f s x) s :=\n        hf.fderiv_within hs hmn,\n      have J : times_cont_diff_on \ud835\udd5c m (\u03bb (x : E \u00d7 E), x.1) (set.prod s univ) :=\n        times_cont_diff_fst.times_cont_diff_on,\n      exact times_cont_diff_on.comp I J (prod_subset_preimage_fst _ _) },\n    { apply times_cont_diff.times_cont_diff_on _ ,\n      apply is_bounded_linear_map.snd.times_cont_diff } },\n  exact A.comp_times_cont_diff_on B\nend\n\n\/-- The bundled derivative of a `C^{n+1}` function is `C^n`. -\/\nlemma times_cont_diff.times_cont_diff_fderiv_apply {n m : with_top \u2115} {f : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hmn : m + 1 \u2264 n) :\n  times_cont_diff \ud835\udd5c m (\u03bbp : E \u00d7 E, (fderiv \ud835\udd5c f p.1 : E \u2192L[\ud835\udd5c] F) p.2) :=\nbegin\n  rw \u2190 times_cont_diff_on_univ at \u22a2 hf,\n  rw [\u2190 fderiv_within_univ, \u2190 univ_prod_univ],\n  exact times_cont_diff_on_fderiv_within_apply hf unique_diff_on_univ hmn\nend\n\n\/-! ### Sum of two functions -\/\n\n\/- The sum is smooth. -\/\nlemma times_cont_diff_add {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bbp : F \u00d7 F, p.1 + p.2) :=\n(is_bounded_linear_map.fst.add is_bounded_linear_map.snd).times_cont_diff\n\n\/-- The sum of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.add {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx, f x + g x) s x :=\ntimes_cont_diff_add.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The sum of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.add {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx, f x + g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.add hg\n\n\/-- The sum of two `C^n`functions is `C^n`. -\/\nlemma times_cont_diff.add {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bbx, f x + g x) :=\ntimes_cont_diff_add.comp (hf.prod hg)\n\n\/-- The sum of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.add {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx, f x + g x) s :=\n\u03bb x hx, (hf x hx).add (hg x hx)\n\n\/-! ### Negative -\/\n\n\/- The negative is smooth. -\/\nlemma times_cont_diff_neg {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bbp : F, -p) :=\nis_bounded_linear_map.id.neg.times_cont_diff\n\n\/-- The negative of a `C^n` function within a domain at a point is `C^n` within this domain at\nthis point. -\/\nlemma times_cont_diff_within_at.neg {n : with_top \u2115} {s : set E} {f : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) : times_cont_diff_within_at \ud835\udd5c n (\u03bbx, -f x) s x :=\ntimes_cont_diff_neg.times_cont_diff_within_at.comp x hf subset_preimage_univ\n\n\/-- The negative of a `C^n` function at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.neg {n : with_top \u2115} {f : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) : times_cont_diff_at \ud835\udd5c n (\u03bbx, -f x) x :=\nby rw \u2190 times_cont_diff_within_at_univ at *; exact hf.neg\n\n\/-- The negative of a `C^n`function is `C^n`. -\/\nlemma times_cont_diff.neg {n : with_top \u2115} {f : E \u2192 F} (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (\u03bbx, -f x) :=\ntimes_cont_diff_neg.comp hf\n\n\/-- The negative of a `C^n` function on a domain is `C^n`. -\/\nlemma times_cont_diff_on.neg {n : with_top \u2115} {s : set E} {f : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) : times_cont_diff_on \ud835\udd5c n (\u03bbx, -f x) s :=\n\u03bb x hx, (hf x hx).neg\n\n\/-! ### Subtraction -\/\n\n\/-- The difference of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.sub {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bbx, f x - g x) s x :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.sub {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bbx, f x - g x) x :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.sub {n : with_top \u2115} {s : set E} {f g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bbx, f x - g x) s :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-- The difference of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.sub {n : with_top \u2115} {f g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) : times_cont_diff \ud835\udd5c n (\u03bbx, f x - g x) :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\n\/-! ### Sum of finitely many functions -\/\n\nlemma times_cont_diff_within_at.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {t : set E} {x : E}\n  (h : \u2200 i \u2208 s, times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f i x) t x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) t x :=\nbegin\n  classical,\n  induction s using finset.induction_on with i s is IH,\n  { simp [times_cont_diff_within_at_const] },\n  { simp only [is, finset.sum_insert, not_false_iff],\n    exact (h _ (finset.mem_insert_self i s)).add (IH (\u03bb j hj, h _ (finset.mem_insert_of_mem hj))) }\nend\n\nlemma times_cont_diff_at.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {x : E}\n  (h : \u2200 i \u2208 s, times_cont_diff_at \ud835\udd5c n (\u03bb x, f i x) x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact times_cont_diff_within_at.sum h\n\nlemma times_cont_diff_on.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115} {t : set E}\n  (h : \u2200 i \u2208 s, times_cont_diff_on \ud835\udd5c n (\u03bb x, f i x) t) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) t :=\n\u03bb x hx, times_cont_diff_within_at.sum (\u03bb i hi, h i hi x hx)\n\nlemma times_cont_diff.sum\n  {\u03b9 : Type*} {f : \u03b9 \u2192 E \u2192 F} {s : finset \u03b9} {n : with_top \u2115}\n  (h : \u2200 i \u2208 s, times_cont_diff \ud835\udd5c n (\u03bb x, f i x)) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, (\u2211 i in s, f i x)) :=\nby simp [\u2190 times_cont_diff_on_univ] at *; exact times_cont_diff_on.sum h\n\n\/-! ### Product of two functions -\/\n\n\/- The product is smooth. -\/\nlemma times_cont_diff_mul {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb p : \ud835\udd5c \u00d7 \ud835\udd5c, p.1 * p.2) :=\nis_bounded_bilinear_map_mul.times_cont_diff\n\n\/-- The product of two `C^n` functions within a set at a point is `C^n` within this set\nat this point. -\/\nlemma times_cont_diff_within_at.mul {n : with_top \u2115} {s : set E} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x * g x) s x :=\ntimes_cont_diff_mul.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The product of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.mul {n : with_top \u2115} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x * g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.mul hg\n\n\/-- The product of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.mul {n : with_top \u2115} {s : set E} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x * g x) s :=\n\u03bb x hx, (hf x hx).mul (hg x hx)\n\n\/-- The product of two `C^n`functions is `C^n`. -\/\nlemma times_cont_diff.mul {n : with_top \u2115} {f g : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x * g x) :=\ntimes_cont_diff_mul.comp (hf.prod hg)\n\nlemma times_cont_diff_within_at.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \/ c) s x :=\nby simpa only [div_eq_mul_inv] using hf.mul times_cont_diff_within_at_const\n\nlemma times_cont_diff_at.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff_at \ud835\udd5c n f x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \/ c) x :=\nby simpa only [div_eq_mul_inv] using hf.mul times_cont_diff_at_const\n\nlemma times_cont_diff_on.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff_on \ud835\udd5c n f s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x \/ c) s :=\nby simpa only [div_eq_mul_inv] using hf.mul times_cont_diff_on_const\n\nlemma times_cont_diff.div_const {f : E \u2192 \ud835\udd5c} {n} {c : \ud835\udd5c} (hf : times_cont_diff \ud835\udd5c n f) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \/ c) :=\nby simpa only [div_eq_mul_inv] using hf.mul times_cont_diff_const\n\nlemma times_cont_diff.pow {n : with_top \u2115} {f : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff \ud835\udd5c n f) :\n  \u2200 m : \u2115, times_cont_diff \ud835\udd5c n (\u03bb x, (f x) ^ m)\n| 0       := by simpa using times_cont_diff_const\n| (m + 1) := by simpa [pow_succ] using hf.mul (times_cont_diff.pow m)\n\nlemma times_cont_diff_at.pow {n : with_top \u2115} {f : E \u2192 \ud835\udd5c} (hf : times_cont_diff_at \ud835\udd5c n f x)\n  (m : \u2115) : times_cont_diff_at \ud835\udd5c n (\u03bb y, f y ^ m) x :=\n(times_cont_diff_id.pow m).times_cont_diff_at.comp x hf\n\nlemma times_cont_diff_within_at.pow {n : with_top \u2115} {f : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (m : \u2115) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb y, f y ^ m) s x :=\n(times_cont_diff_id.pow m).times_cont_diff_at.comp_times_cont_diff_within_at x hf\n\nlemma times_cont_diff_on.pow {n : with_top \u2115} {f : E \u2192 \ud835\udd5c}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (m : \u2115) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb y, f y ^ m) s :=\n\u03bb y hy, (hf y hy).pow m\n\n\/-! ### Scalar multiplication -\/\n\n\/- The scalar multiplication is smooth. -\/\nlemma times_cont_diff_smul {n : with_top \u2115} :\n  times_cont_diff \ud835\udd5c n (\u03bb p : \ud835\udd5c \u00d7 F, p.1 \u2022 p.2) :=\nis_bounded_bilinear_map_smul.times_cont_diff\n\n\/-- The scalar multiplication of two `C^n` functions within a set at a point is `C^n` within this\nset at this point. -\/\nlemma times_cont_diff_within_at.smul {n : with_top \u2115} {s : set E} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \u2022 g x) s x :=\ntimes_cont_diff_smul.times_cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ\n\n\/-- The scalar multiplication of two `C^n` functions at a point is `C^n` at this point. -\/\nlemma times_cont_diff_at.smul {n : with_top \u2115} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \u2022 g x) x :=\nby rw [\u2190 times_cont_diff_within_at_univ] at *; exact hf.smul hg\n\n\/-- The scalar multiplication of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.smul {n : with_top \u2115} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \u2022 g x) :=\ntimes_cont_diff_smul.comp (hf.prod hg)\n\n\/-- The scalar multiplication of two `C^n` functions on a domain is `C^n`. -\/\nlemma times_cont_diff_on.smul {n : with_top \u2115} {s : set E} {f : E \u2192 \ud835\udd5c} {g : E \u2192 F}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, f x \u2022 g x) s :=\n\u03bb x hx, (hf x hx).smul (hg x hx)\n\n\/-! ### Cartesian product of two functions-\/\n\nsection prod_map\nvariables {E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n{F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n{n : with_top \u2115}\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_within_at.prod_map'\n  {s : set E} {t : set E'} {f : E \u2192 F} {g : E' \u2192 F'} {p : E \u00d7 E'}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s p.1) (hg : times_cont_diff_within_at \ud835\udd5c n g t p.2) :\n  times_cont_diff_within_at \ud835\udd5c n (prod.map f g) (set.prod s t) p :=\n(hf.comp p times_cont_diff_within_at_fst (prod_subset_preimage_fst _ _)).prod\n  (hg.comp p times_cont_diff_within_at_snd (prod_subset_preimage_snd _ _))\n\nlemma times_cont_diff_within_at.prod_map\n  {s : set E} {t : set E'} {f : E \u2192 F} {g : E' \u2192 F'} {x : E} {y : E'}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g t y) :\n  times_cont_diff_within_at \ud835\udd5c n (prod.map f g) (set.prod s t) (x, y) :=\ntimes_cont_diff_within_at.prod_map' hf hg\n\n\/-- The product map of two `C^n` functions on a set is `C^n` on the product set. -\/\nlemma times_cont_diff_on.prod_map {E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n  {F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n  {s : set E} {t : set E'} {n : with_top \u2115} {f : E \u2192 F} {g : E' \u2192 F'}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g t) :\n  times_cont_diff_on \ud835\udd5c n (prod.map f g) (set.prod s t) :=\n(hf.comp times_cont_diff_on_fst (prod_subset_preimage_fst _ _)).prod\n  (hg.comp (times_cont_diff_on_snd) (prod_subset_preimage_snd _ _))\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_at.prod_map {f : E \u2192 F} {g : E' \u2192 F'} {x : E} {y : E'}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g y) :\n  times_cont_diff_at \ud835\udd5c n (prod.map f g) (x, y) :=\nbegin\n  rw times_cont_diff_at at *,\n  convert hf.prod_map hg,\n  simp only [univ_prod_univ]\nend\n\n\/-- The product map of two `C^n` functions within a set at a point is `C^n`\nwithin the product set at the product point. -\/\nlemma times_cont_diff_at.prod_map' {f : E \u2192 F} {g : E' \u2192 F'} {p : E \u00d7 E'}\n  (hf : times_cont_diff_at \ud835\udd5c n f p.1) (hg : times_cont_diff_at \ud835\udd5c n g p.2) :\n  times_cont_diff_at \ud835\udd5c n (prod.map f g) p :=\nbegin\n  rcases p,\n  exact times_cont_diff_at.prod_map hf hg\nend\n\n\/-- The product map of two `C^n` functions is `C^n`. -\/\nlemma times_cont_diff.prod_map\n  {f : E \u2192 F} {g : E' \u2192 F'}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g) :\n  times_cont_diff \ud835\udd5c n (prod.map f g) :=\nbegin\n  rw times_cont_diff_iff_times_cont_diff_at at *,\n  exact \u03bb \u27e8x, y\u27e9, (hf x).prod_map (hg y)\nend\n\nend prod_map\n\n\/-! ### Inversion in a complete normed algebra -\/\n\nsection algebra_inverse\nvariables (\ud835\udd5c) {R : Type*} [normed_ring R] [normed_algebra \ud835\udd5c R]\nopen normed_ring continuous_linear_map ring\n\n\/-- In a complete normed algebra, the operation of inversion is `C^n`, for all `n`, at each\ninvertible element.  The proof is by induction, bootstrapping using an identity expressing the\nderivative of inversion as a bilinear map of inversion itself. -\/\nlemma times_cont_diff_at_ring_inverse [complete_space R] {n : with_top \u2115} (x : R\u02e3) :\n  times_cont_diff_at \ud835\udd5c n ring.inverse (x : R) :=\nbegin\n  induction n using with_top.nat_induction with n IH Itop,\n  { intros m hm,\n    refine \u27e8{y : R | is_unit y}, _, _\u27e9,\n    { simp [nhds_within_univ],\n      exact x.nhds },\n    { use (ftaylor_series_within \ud835\udd5c inverse univ),\n      rw [le_antisymm hm bot_le, has_ftaylor_series_up_to_on_zero_iff],\n      split,\n      { rintros _ \u27e8x', rfl\u27e9,\n        exact (inverse_continuous_at x').continuous_within_at },\n      { simp [ftaylor_series_within] } } },\n  { apply times_cont_diff_at_succ_iff_has_fderiv_at.mpr,\n    refine \u27e8\u03bb (x : R), - lmul_left_right \ud835\udd5c R (inverse x) (inverse x), _, _\u27e9,\n    { refine \u27e8{y : R | is_unit y}, x.nhds, _\u27e9,\n      rintros _ \u27e8y, rfl\u27e9,\n      rw [inverse_unit],\n      exact has_fderiv_at_ring_inverse y },\n    { convert (lmul_left_right_is_bounded_bilinear \ud835\udd5c R).times_cont_diff.neg.comp_times_cont_diff_at\n        (x : R) (IH.prod IH) } },\n  { exact times_cont_diff_at_top.mpr Itop }\nend\n\nvariables (\ud835\udd5c) {\ud835\udd5c' : Type*} [normed_field \ud835\udd5c'] [normed_algebra \ud835\udd5c \ud835\udd5c'] [complete_space \ud835\udd5c']\n\nlemma times_cont_diff_at_inv {x : \ud835\udd5c'} (hx : x \u2260 0) {n} :\n  times_cont_diff_at \ud835\udd5c n has_inv.inv x :=\nby simpa only [ring.inverse_eq_inv'] using times_cont_diff_at_ring_inverse \ud835\udd5c (units.mk0 x hx)\n\nlemma times_cont_diff_on_inv {n} : times_cont_diff_on \ud835\udd5c n (has_inv.inv : \ud835\udd5c' \u2192 \ud835\udd5c') {0}\u1d9c :=\n\u03bb x hx, (times_cont_diff_at_inv \ud835\udd5c hx).times_cont_diff_within_at\n\nvariable {\ud835\udd5c}\n\n-- TODO: the next few lemmas don't need `\ud835\udd5c` or `\ud835\udd5c'` to be complete\n-- A good way to show this is to generalize `times_cont_diff_at_ring_inverse` to the setting\n-- of a function `f` such that `\u2200\u1da0 x in \ud835\udcdd a, x * f x = 1`.\n\nlemma times_cont_diff_within_at.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff_within_at \ud835\udd5c n f s x)\n  (hx : f x \u2260 0) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) s x :=\n(times_cont_diff_at_inv \ud835\udd5c hx).comp_times_cont_diff_within_at x hf\n\nlemma times_cont_diff_on.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff_on \ud835\udd5c n f s)\n  (h : \u2200 x \u2208 s, f x \u2260 0) :\n  times_cont_diff_on \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) s :=\n\u03bb x hx, (hf.times_cont_diff_within_at hx).inv (h x hx)\n\nlemma times_cont_diff_at.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff_at \ud835\udd5c n f x) (hx : f x \u2260 0) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) x :=\nhf.inv hx\n\nlemma times_cont_diff.inv {f : E \u2192 \ud835\udd5c'} {n} (hf : times_cont_diff \ud835\udd5c n f) (h : \u2200 x, f x \u2260 0) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, (f x)\u207b\u00b9) :=\nby { rw times_cont_diff_iff_times_cont_diff_at, exact \u03bb x, hf.times_cont_diff_at.inv (h x) }\n\n-- TODO: generalize to `f g : E \u2192 \ud835\udd5c'`\nlemma times_cont_diff_within_at.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff_within_at \ud835\udd5c n f s x) (hg : times_cont_diff_within_at \ud835\udd5c n g s x)\n  (hx : g x \u2260 0) :\n  times_cont_diff_within_at \ud835\udd5c n (\u03bb x, f x \/ g x) s x :=\nby simpa only [div_eq_mul_inv] using hf.mul (hg.inv hx)\n\nlemma times_cont_diff_on.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff_on \ud835\udd5c n f s) (hg : times_cont_diff_on \ud835\udd5c n g s) (h\u2080 : \u2200 x \u2208 s, g x \u2260 0) :\n  times_cont_diff_on \ud835\udd5c n (f \/ g) s :=\n\u03bb x hx, (hf x hx).div (hg x hx) (h\u2080 x hx)\n\nlemma times_cont_diff_at.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff_at \ud835\udd5c n f x) (hg : times_cont_diff_at \ud835\udd5c n g x)\n  (hx : g x \u2260 0) :\n  times_cont_diff_at \ud835\udd5c n (\u03bb x, f x \/ g x) x :=\nhf.div hg hx\n\nlemma times_cont_diff.div [complete_space \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n}\n  (hf : times_cont_diff \ud835\udd5c n f) (hg : times_cont_diff \ud835\udd5c n g)\n  (h0 : \u2200 x, g x \u2260 0) :\n  times_cont_diff \ud835\udd5c n (\u03bb x, f x \/ g x) :=\nbegin\n  simp only [times_cont_diff_iff_times_cont_diff_at] at *,\n  exact \u03bb x, (hf x).div (hg x) (h0 x)\nend\n\nend algebra_inverse\n\n\/-! ### Inversion of continuous linear maps between Banach spaces -\/\n\nsection map_inverse\nopen continuous_linear_map\n\n\/-- At a continuous linear equivalence `e : E \u2243L[\ud835\udd5c] F` between Banach spaces, the operation of\ninversion is `C^n`, for all `n`. -\/\nlemma times_cont_diff_at_map_inverse [complete_space E] {n : with_top \u2115} (e : E \u2243L[\ud835\udd5c] F) :\n  times_cont_diff_at \ud835\udd5c n inverse (e : E \u2192L[\ud835\udd5c] F) :=\nbegin\n  nontriviality E,\n  -- first, we use the lemma `to_ring_inverse` to rewrite in terms of `ring.inverse` in the ring\n  -- `E \u2192L[\ud835\udd5c] E`\n  let O\u2081 : (E \u2192L[\ud835\udd5c] E) \u2192 (F \u2192L[\ud835\udd5c] E) := \u03bb f, f.comp (e.symm : (F \u2192L[\ud835\udd5c] E)),\n  let O\u2082 : (E \u2192L[\ud835\udd5c] F) \u2192 (E \u2192L[\ud835\udd5c] E) := \u03bb f, (e.symm : (F \u2192L[\ud835\udd5c] E)).comp f,\n  have : continuous_linear_map.inverse = O\u2081 \u2218 ring.inverse \u2218 O\u2082 :=\n    funext (to_ring_inverse e),\n  rw this,\n  -- `O\u2081` and `O\u2082` are `times_cont_diff`,\n  -- so we reduce to proving that `ring.inverse` is `times_cont_diff`\n  have h\u2081 : times_cont_diff \ud835\udd5c n O\u2081,\n    from is_bounded_bilinear_map_comp.times_cont_diff.comp\n      (times_cont_diff_const.prod times_cont_diff_id),\n  have h\u2082 : times_cont_diff \ud835\udd5c n O\u2082,\n    from is_bounded_bilinear_map_comp.times_cont_diff.comp\n      (times_cont_diff_id.prod times_cont_diff_const),\n  refine h\u2081.times_cont_diff_at.comp _ (times_cont_diff_at.comp _ _ h\u2082.times_cont_diff_at),\n  convert times_cont_diff_at_ring_inverse \ud835\udd5c (1 : (E \u2192L[\ud835\udd5c] E)\u02e3),\n  simp [O\u2082, one_def]\nend\n\nend map_inverse\n\nsection function_inverse\nopen continuous_linear_map\n\n\/-- If `f` is a local homeomorphism and the point `a` is in its target,\nand if `f` is `n` times continuously differentiable at `f.symm a`,\nand if the derivative at `f.symm a` is a continuous linear equivalence,\nthen `f.symm` is `n` times continuously differentiable at the point `a`.\n\nThis is one of the easy parts of the inverse function theorem: it assumes that we already have\nan inverse function. -\/\ntheorem local_homeomorph.times_cont_diff_at_symm [complete_space E] {n : with_top \u2115}\n  (f : local_homeomorph E F) {f\u2080' : E \u2243L[\ud835\udd5c] F} {a : F} (ha : a \u2208 f.target)\n  (hf\u2080' : has_fderiv_at f (f\u2080' : E \u2192L[\ud835\udd5c] F) (f.symm a)) (hf : times_cont_diff_at \ud835\udd5c n f (f.symm a)) :\n  times_cont_diff_at \ud835\udd5c n f.symm a :=\nbegin\n  -- We prove this by induction on `n`\n  induction n using with_top.nat_induction with n IH Itop,\n  { rw times_cont_diff_at_zero,\n    exact \u27e8f.target, is_open.mem_nhds f.open_target ha, f.continuous_inv_fun\u27e9 },\n  { obtain \u27e8f', \u27e8u, hu, hff'\u27e9, hf'\u27e9 := times_cont_diff_at_succ_iff_has_fderiv_at.mp hf,\n    apply times_cont_diff_at_succ_iff_has_fderiv_at.mpr,\n    -- For showing `n.succ` times continuous differentiability (the main inductive step), it\n    -- suffices to produce the derivative and show that it is `n` times continuously differentiable\n    have eq_f\u2080' : f' (f.symm a) = f\u2080',\n    { exact (hff' (f.symm a) (mem_of_mem_nhds hu)).unique hf\u2080' },\n    -- This follows by a bootstrapping formula expressing the derivative as a function of `f` itself\n    refine \u27e8inverse \u2218 f' \u2218 f.symm, _, _\u27e9,\n    { -- We first check that the derivative of `f` is that formula\n      have h_nhds : {y : E | \u2203 (e : E \u2243L[\ud835\udd5c] F), \u2191e = f' y} \u2208 \ud835\udcdd ((f.symm) a),\n      { have hf\u2080' := f\u2080'.nhds,\n        rw \u2190 eq_f\u2080' at hf\u2080',\n        exact hf'.continuous_at.preimage_mem_nhds hf\u2080' },\n      obtain \u27e8t, htu, ht, htf\u27e9 := mem_nhds_iff.mp (filter.inter_mem hu h_nhds),\n      use f.target \u2229 (f.symm) \u207b\u00b9' t,\n      refine \u27e8is_open.mem_nhds _ _, _\u27e9,\n      { exact f.preimage_open_of_open_symm ht },\n      { exact mem_inter ha (mem_preimage.mpr htf) },\n      intros x hx,\n      obtain \u27e8hxu, e, he\u27e9 := htu hx.2,\n      have h_deriv : has_fderiv_at f \u2191e ((f.symm) x),\n      { rw he,\n        exact hff' (f.symm x) hxu },\n      convert f.has_fderiv_at_symm hx.1 h_deriv,\n      simp [\u2190 he] },\n    { -- Then we check that the formula, being a composition of `times_cont_diff` pieces, is\n      -- itself `times_cont_diff`\n      have h_deriv\u2081 : times_cont_diff_at \ud835\udd5c n inverse (f' (f.symm a)),\n      { rw eq_f\u2080',\n        exact times_cont_diff_at_map_inverse _ },\n      have h_deriv\u2082 : times_cont_diff_at \ud835\udd5c n f.symm a,\n      { refine IH (hf.of_le _),\n        norm_cast,\n        exact nat.le_succ n },\n      exact (h_deriv\u2081.comp _ hf').comp _ h_deriv\u2082 } },\n  { refine times_cont_diff_at_top.mpr _,\n    intros n,\n    exact Itop n (times_cont_diff_at_top.mp hf n) }\nend\n\n\/-- Let `f` be a local homeomorphism of a nondiscrete normed field, let `a` be a point in its\ntarget. if `f` is `n` times continuously differentiable at `f.symm a`, and if the derivative at\n`f.symm a` is nonzero, then `f.symm` is `n` times continuously differentiable at the point `a`.\n\nThis is one of the easy parts of the inverse function theorem: it assumes that we already have\nan inverse function. -\/\ntheorem local_homeomorph.times_cont_diff_at_symm_deriv [complete_space \ud835\udd5c] {n : with_top \u2115}\n  (f : local_homeomorph \ud835\udd5c \ud835\udd5c) {f\u2080' a : \ud835\udd5c} (h\u2080 : f\u2080' \u2260 0) (ha : a \u2208 f.target)\n  (hf\u2080' : has_deriv_at f f\u2080' (f.symm a)) (hf : times_cont_diff_at \ud835\udd5c n f (f.symm a)) :\n  times_cont_diff_at \ud835\udd5c n f.symm a :=\nf.times_cont_diff_at_symm ha (hf\u2080'.has_fderiv_at_equiv h\u2080) hf\n\nend function_inverse\n\nsection real\n\/-!\n### Results over `\u211d` or `\u2102`\n  The results in this section rely on the Mean Value Theorem, and therefore hold only over `\u211d` (and\n  its extension fields such as `\u2102`).\n-\/\n\nvariables\n{\ud835\udd42 : Type*} [is_R_or_C \ud835\udd42]\n{E' : Type*} [normed_group E'] [normed_space \ud835\udd42 E']\n{F' : Type*} [normed_group F'] [normed_space \ud835\udd42 F']\n\n\/-- If a function has a Taylor series at order at least 1, then at points in the interior of the\n    domain of definition, the term of order 1 of this series is a strict derivative of `f`. -\/\nlemma has_ftaylor_series_up_to_on.has_strict_fderiv_at\n  {s : set E'} {f : E' \u2192 F'} {x : E'} {p : E' \u2192 formal_multilinear_series \ud835\udd42 E' F'} {n : with_top \u2115}\n  (hf : has_ftaylor_series_up_to_on n f p s) (hn : 1 \u2264 n) (hs : s \u2208 \ud835\udcdd x) :\n  has_strict_fderiv_at f ((continuous_multilinear_curry_fin1 \ud835\udd42 E' F') (p x 1)) x :=\nhas_strict_fderiv_at_of_has_fderiv_at_of_continuous_at (hf.eventually_has_fderiv_at hn hs) $\n  (continuous_multilinear_curry_fin1 \ud835\udd42 E' F').continuous_at.comp $\n    (hf.cont 1 hn).continuous_at hs\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, and its derivative at that point is given to\nus as `f'`, then `f'` is also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_fderiv_at'\n  {f : E' \u2192 F'} {f' : E' \u2192L[\ud835\udd42] F'} {x : E'}\n  {n : with_top \u2115} (hf : times_cont_diff_at \ud835\udd42 n f x) (hf' : has_fderiv_at f f' x) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f f' x :=\nbegin\n  rcases hf 1 hn with \u27e8u, H, p, hp\u27e9,\n  simp only [nhds_within_univ, mem_univ, insert_eq_of_mem] at H,\n  have := hp.has_strict_fderiv_at le_rfl H,\n  rwa hf'.unique this.has_fderiv_at\nend\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, and its derivative at that point is given to\nus as `f'`, then `f'` is also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_deriv_at' {f : \ud835\udd42 \u2192 F'} {f' : F'} {x : \ud835\udd42}\n  {n : with_top \u2115} (hf : times_cont_diff_at \ud835\udd42 n f x) (hf' : has_deriv_at f f' x) (hn : 1 \u2264 n) :\n  has_strict_deriv_at f f' x :=\nhf.has_strict_fderiv_at' hf' hn\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, then the derivative of `f` at this point\nis also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_fderiv_at {f : E' \u2192 F'} {x : E'} {n : with_top \u2115}\n  (hf : times_cont_diff_at \ud835\udd42 n f x) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f (fderiv \ud835\udd42 f x) x :=\nhf.has_strict_fderiv_at' (hf.differentiable_at hn).has_fderiv_at hn\n\n\/-- If a function is `C^n` with `1 \u2264 n` around a point, then the derivative of `f` at this point\nis also a strict derivative. -\/\nlemma times_cont_diff_at.has_strict_deriv_at {f : \ud835\udd42 \u2192 F'} {x : \ud835\udd42} {n : with_top \u2115}\n  (hf : times_cont_diff_at \ud835\udd42 n f x) (hn : 1 \u2264 n) :\n  has_strict_deriv_at f (deriv f x) x :=\n(hf.has_strict_fderiv_at hn).has_strict_deriv_at\n\n\/-- If a function is `C^n` with `1 \u2264 n`, then the derivative of `f` is also a strict derivative. -\/\nlemma times_cont_diff.has_strict_fderiv_at\n  {f : E' \u2192 F'} {x : E'} {n : with_top \u2115} (hf : times_cont_diff \ud835\udd42 n f) (hn : 1 \u2264 n) :\n  has_strict_fderiv_at f (fderiv \ud835\udd42 f x) x :=\nhf.times_cont_diff_at.has_strict_fderiv_at hn\n\n\/-- If a function is `C^n` with `1 \u2264 n`, then the derivative of `f` is also a strict derivative. -\/\nlemma times_cont_diff.has_strict_deriv_at\n  {f : \ud835\udd42 \u2192 F'} {x : \ud835\udd42} {n : with_top \u2115} (hf : times_cont_diff \ud835\udd42 n f) (hn : 1 \u2264 n) :\n  has_strict_deriv_at f (deriv f x) x :=\nhf.times_cont_diff_at.has_strict_deriv_at hn\n\n\/-- If `f` has a formal Taylor series `p` up to order `1` on `{x} \u222a s`, where `s` is a convex set,\nand `\u2225p x 1\u2225\u208a < K`, then `f` is `K`-Lipschitz in a neighborhood of `x` within `s`. -\/\nlemma has_ftaylor_series_up_to_on.exists_lipschitz_on_with_of_nnnorm_lt {E F : Type*}\n  [normed_group E] [normed_space \u211d E] [normed_group F] [normed_space \u211d F] {f : E \u2192 F}\n  {p : E \u2192 formal_multilinear_series \u211d E F} {s : set E} {x : E}\n  (hf : has_ftaylor_series_up_to_on 1 f p (insert x s)) (hs : convex \u211d s) (K : \u211d\u22650)\n  (hK : \u2225p x 1\u2225\u208a < K) :\n  \u2203 t \u2208 \ud835\udcdd[s] x, lipschitz_on_with K f t :=\nbegin\n  set f' := \u03bb y, continuous_multilinear_curry_fin1 \u211d E F (p y 1),\n  have hder : \u2200 y \u2208 s, has_fderiv_within_at f (f' y) s y,\n    from \u03bb y hy, (hf.has_fderiv_within_at le_rfl (subset_insert x s hy)).mono (subset_insert x s),\n  have hcont : continuous_within_at f' s x,\n    from (continuous_multilinear_curry_fin1 \u211d E F).continuous_at.comp_continuous_within_at\n      ((hf.cont _ le_rfl _ (mem_insert _ _)).mono (subset_insert x s)),\n  replace hK : \u2225f' x\u2225\u208a < K, by simpa only [linear_isometry_equiv.nnnorm_map],\n  exact hs.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt\n    (eventually_nhds_within_iff.2 $ eventually_of_forall hder) hcont K hK\nend\n\n\/-- If `f` has a formal Taylor series `p` up to order `1` on `{x} \u222a s`, where `s` is a convex set,\nthen `f` is Lipschitz in a neighborhood of `x` within `s`. -\/\nlemma has_ftaylor_series_up_to_on.exists_lipschitz_on_with {E F : Type*}\n  [normed_group E] [normed_space \u211d E] [normed_group F] [normed_space \u211d F] {f : E \u2192 F}\n  {p : E \u2192 formal_multilinear_series \u211d E F} {s : set E} {x : E}\n  (hf : has_ftaylor_series_up_to_on 1 f p (insert x s)) (hs : convex \u211d s) :\n  \u2203 K (t \u2208 \ud835\udcdd[s] x), lipschitz_on_with K f t :=\n(no_top _).imp $ hf.exists_lipschitz_on_with_of_nnnorm_lt hs\n\n\/-- If `f` is `C^1` within a conves set `s` at `x`, then it is Lipschitz on a neighborhood of `x`\nwithin `s`. -\/\nlemma times_cont_diff_within_at.exists_lipschitz_on_with {E F : Type*} [normed_group E]\n  [normed_space \u211d E] [normed_group F] [normed_space \u211d F] {f : E \u2192 F} {s : set E}\n  {x : E} (hf : times_cont_diff_within_at \u211d 1 f s x) (hs : convex \u211d s) :\n  \u2203 (K : \u211d\u22650) (t \u2208 \ud835\udcdd[s] x), lipschitz_on_with K f t :=\nbegin\n  rcases hf 1 le_rfl with \u27e8t, hst, p, hp\u27e9,\n  rcases metric.mem_nhds_within_iff.mp hst with \u27e8\u03b5, \u03b50, h\u03b5\u27e9,\n  replace hp : has_ftaylor_series_up_to_on 1 f p (metric.ball x \u03b5 \u2229 insert x s) := hp.mono h\u03b5,\n  clear hst h\u03b5 t,\n  rw [\u2190 insert_eq_of_mem (metric.mem_ball_self \u03b50), \u2190 insert_inter] at hp,\n  rcases hp.exists_lipschitz_on_with ((convex_ball _ _).inter hs) with \u27e8K, t, hst, hft\u27e9,\n  rw [inter_comm, \u2190 nhds_within_restrict' _ (metric.ball_mem_nhds _ \u03b50)] at hst,\n  exact \u27e8K, t, hst, hft\u27e9\nend\n\n\/-- If `f` is `C^1` at `x` and `K > \u2225fderiv \ud835\udd42 f x\u2225`, then `f` is `K`-Lipschitz in a neighborhood of\n`x`. -\/\nlemma times_cont_diff_at.exists_lipschitz_on_with_of_nnnorm_lt {f : E' \u2192 F'} {x : E'}\n  (hf : times_cont_diff_at \ud835\udd42 1 f x) (K : \u211d\u22650) (hK : \u2225fderiv \ud835\udd42 f x\u2225\u208a < K) :\n  \u2203 t \u2208 \ud835\udcdd x, lipschitz_on_with K f t :=\n(hf.has_strict_fderiv_at le_rfl).exists_lipschitz_on_with_of_nnnorm_lt K hK\n\n\/-- If `f` is `C^1` at `x`, then `f` is Lipschitz in a neighborhood of `x`. -\/\nlemma times_cont_diff_at.exists_lipschitz_on_with {f : E' \u2192 F'} {x : E'}\n  (hf : times_cont_diff_at \ud835\udd42 1 f x) :\n  \u2203 K (t \u2208 \ud835\udcdd x), lipschitz_on_with K f t :=\n(hf.has_strict_fderiv_at le_rfl).exists_lipschitz_on_with\n\nend real\n\nsection deriv\n\/-!\n### One dimension\n\nAll results up to now have been expressed in terms of the general Fr\u00e9chet derivative `fderiv`. For\nmaps defined on the field, the one-dimensional derivative `deriv` is often easier to use. In this\nparagraph, we reformulate some higher smoothness results in terms of `deriv`.\n-\/\n\nvariables {f\u2082 : \ud835\udd5c \u2192 F} {s\u2082 : set \ud835\udd5c}\nopen continuous_linear_map (smul_right)\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative (formulated with `deriv_within`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_deriv_within {n : \u2115} (hs : unique_diff_on \ud835\udd5c s\u2082) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c n (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_succ_iff_fderiv_within hs,\n  congr' 2,\n  apply le_antisymm,\n  { assume h,\n    have : deriv_within f\u2082 s\u2082 = (\u03bb u : \ud835\udd5c \u2192L[\ud835\udd5c] F, u 1) \u2218 (fderiv_within \ud835\udd5c f\u2082 s\u2082),\n      by { ext x, refl },\n    simp only [this],\n    apply times_cont_diff.comp_times_cont_diff_on _ h,\n    exact (is_bounded_bilinear_map_apply.is_bounded_linear_map_left _).times_cont_diff },\n  { assume h,\n    have : fderiv_within \ud835\udd5c f\u2082 s\u2082 = smul_right (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) \u2218 deriv_within f\u2082 s\u2082,\n      by { ext x, simp [deriv_within] },\n    simp only [this],\n    apply times_cont_diff.comp_times_cont_diff_on _ h,\n    exact (is_bounded_bilinear_map_smul_right.is_bounded_linear_map_right _).times_cont_diff }\nend\n\n\/-- A function is `C^(n + 1)` on an open domain if and only if it is\ndifferentiable there, and its derivative (formulated with `deriv`) is `C^n`. -\/\ntheorem times_cont_diff_on_succ_iff_deriv_of_open {n : \u2115} (hs : is_open s\u2082) :\n  times_cont_diff_on \ud835\udd5c ((n + 1) : \u2115) f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c n (deriv f\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_succ_iff_deriv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact deriv_within_of_open hs hx\nend\n\n\/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative (formulated with `deriv_within`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_deriv_within (hs : unique_diff_on \ud835\udd5c s\u2082) :\n  times_cont_diff_on \ud835\udd5c \u221e f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c \u221e (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  split,\n  { assume h,\n    refine \u27e8h.differentiable_on le_top, _\u27e9,\n    apply times_cont_diff_on_top.2 (\u03bb n, ((times_cont_diff_on_succ_iff_deriv_within hs).1 _).2),\n    exact h.of_le le_top },\n  { assume h,\n    refine times_cont_diff_on_top.2 (\u03bb n, _),\n    have A : (n : with_top \u2115) \u2264 \u221e := le_top,\n    apply ((times_cont_diff_on_succ_iff_deriv_within hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le,\n    exact with_top.coe_le_coe.2 (nat.le_succ n) }\nend\n\n\/-- A function is `C^\u221e` on an open domain if and only if it is differentiable\nthere, and its derivative (formulated with `deriv`) is `C^\u221e`. -\/\ntheorem times_cont_diff_on_top_iff_deriv_of_open (hs : is_open s\u2082) :\n  times_cont_diff_on \ud835\udd5c \u221e f\u2082 s\u2082 \u2194\n  differentiable_on \ud835\udd5c f\u2082 s\u2082 \u2227 times_cont_diff_on \ud835\udd5c \u221e (deriv f\u2082) s\u2082 :=\nbegin\n  rw times_cont_diff_on_top_iff_deriv_within hs.unique_diff_on,\n  congr' 2,\n  rw \u2190 iff_iff_eq,\n  apply times_cont_diff_on_congr,\n  assume x hx,\n  exact deriv_within_of_open hs hx\nend\n\nlemma times_cont_diff_on.deriv_within {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : unique_diff_on \ud835\udd5c s\u2082) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (deriv_within f\u2082 s\u2082) s\u2082 :=\nbegin\n  cases m,\n  { change \u221e + 1 \u2264 n at hmn,\n    have : n = \u221e, by simpa using hmn,\n    rw this at hf,\n    exact ((times_cont_diff_on_top_iff_deriv_within hs).1 hf).2 },\n  { change (m.succ : with_top \u2115) \u2264 n at hmn,\n    exact ((times_cont_diff_on_succ_iff_deriv_within hs).1 (hf.of_le hmn)).2 }\nend\n\nlemma times_cont_diff_on.deriv_of_open {m n : with_top \u2115}\n  (hf : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : is_open s\u2082) (hmn : m + 1 \u2264 n) :\n  times_cont_diff_on \ud835\udd5c m (deriv f\u2082) s\u2082 :=\n(hf.deriv_within hs.unique_diff_on hmn).congr (\u03bb x hx, (deriv_within_of_open hs hx).symm)\n\nlemma times_cont_diff_on.continuous_on_deriv_within {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : unique_diff_on \ud835\udd5c s\u2082) (hn : 1 \u2264 n) :\n  continuous_on (deriv_within f\u2082 s\u2082) s\u2082 :=\n((times_cont_diff_on_succ_iff_deriv_within hs).1 (h.of_le hn)).2.continuous_on\n\nlemma times_cont_diff_on.continuous_on_deriv_of_open {n : with_top \u2115}\n  (h : times_cont_diff_on \ud835\udd5c n f\u2082 s\u2082) (hs : is_open s\u2082) (hn : 1 \u2264 n) :\n  continuous_on (deriv f\u2082) s\u2082 :=\n((times_cont_diff_on_succ_iff_deriv_of_open hs).1 (h.of_le hn)).2.continuous_on\n\n\/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative is `C^n`. -\/\ntheorem times_cont_diff_succ_iff_deriv {n : \u2115} :\n  times_cont_diff \ud835\udd5c ((n + 1) : \u2115) f\u2082 \u2194\n    differentiable \ud835\udd5c f\u2082 \u2227 times_cont_diff \ud835\udd5c n (deriv f\u2082) :=\nby simp only [\u2190 times_cont_diff_on_univ, times_cont_diff_on_succ_iff_deriv_of_open, is_open_univ,\n  differentiable_on_univ]\n\nend deriv\n\nsection restrict_scalars\n\/-!\n### Restricting from `\u2102` to `\u211d`, or generally from `\ud835\udd5c'` to `\ud835\udd5c`\n\nIf a function is `n` times continuously differentiable over `\u2102`, then it is `n` times continuously\ndifferentiable over `\u211d`. In this paragraph, we give variants of this statement, in the general\nsituation where `\u2102` and `\u211d` are replaced respectively by `\ud835\udd5c'` and `\ud835\udd5c` where `\ud835\udd5c'` is a normed algebra\nover `\ud835\udd5c`.\n-\/\n\nvariables (\ud835\udd5c) {\ud835\udd5c' : Type*} [nondiscrete_normed_field \ud835\udd5c'] [normed_algebra \ud835\udd5c \ud835\udd5c']\nvariables [normed_space \ud835\udd5c' E] [is_scalar_tower \ud835\udd5c \ud835\udd5c' E]\nvariables [normed_space \ud835\udd5c' F] [is_scalar_tower \ud835\udd5c \ud835\udd5c' F]\nvariables {p' : E \u2192 formal_multilinear_series \ud835\udd5c' E F} {n : with_top \u2115}\n\nlemma has_ftaylor_series_up_to_on.restrict_scalars\n  (h : has_ftaylor_series_up_to_on n f p' s) :\n  has_ftaylor_series_up_to_on n f (\u03bb x, (p' x).restrict_scalars \ud835\udd5c) s :=\n{ zero_eq := \u03bb x hx, h.zero_eq x hx,\n  fderiv_within :=\n    begin\n      intros m hm x hx,\n      convert ((continuous_multilinear_map.restrict_scalars_linear \ud835\udd5c).has_fderiv_at)\n        .comp_has_fderiv_within_at _ ((h.fderiv_within m hm x hx).restrict_scalars \ud835\udd5c),\n    end,\n  cont := \u03bb m hm, continuous_multilinear_map.continuous_restrict_scalars.comp_continuous_on\n    (h.cont m hm) }\n\nlemma times_cont_diff_within_at.restrict_scalars (h : times_cont_diff_within_at \ud835\udd5c' n f s x) :\n  times_cont_diff_within_at \ud835\udd5c n f s x :=\nbegin\n  intros m hm,\n  rcases h m hm with \u27e8u, u_mem, p', hp'\u27e9,\n  exact \u27e8u, u_mem, _, hp'.restrict_scalars _\u27e9\nend\n\nlemma times_cont_diff_on.restrict_scalars (h : times_cont_diff_on \ud835\udd5c' n f s) :\n  times_cont_diff_on \ud835\udd5c n f s :=\n\u03bb x hx, (h x hx).restrict_scalars _\n\nlemma times_cont_diff_at.restrict_scalars (h : times_cont_diff_at \ud835\udd5c' n f x) :\n  times_cont_diff_at \ud835\udd5c n f x :=\ntimes_cont_diff_within_at_univ.1 $ h.times_cont_diff_within_at.restrict_scalars _\n\nlemma times_cont_diff.restrict_scalars (h : times_cont_diff \ud835\udd5c' n f) :\n  times_cont_diff \ud835\udd5c n f :=\ntimes_cont_diff_iff_times_cont_diff_at.2 $ \u03bb x, h.times_cont_diff_at.restrict_scalars _\n\nend restrict_scalars\n","avg_line_length":46.1505557427,"max_line_length":100,"alphanum_fraction":0.6871793375}
-{"size":8517,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-\/\nimport logic.basic tactic.solve_by_elim tactic.hint\n\nnamespace tactic\n\nopen expr\nopen tactic.interactive ( casesm constructor_matching )\n\n\/--\n  find all assumptions of the shape `\u00ac (p \u2227 q)` or `\u00ac (p \u2228 q)` and\n  replace them using de Morgan's law.\n-\/\nmeta def distrib_not : tactic unit :=\ndo hs \u2190 local_context,\n   hs.for_each $ \u03bb h,\n    all_goals $\n    iterate_at_most 3 $\n      do h \u2190 get_local h.local_pp_name,\n         e \u2190 infer_type h,\n         match e with\n         | `(\u00ac _ = _) := replace h.local_pp_name ``(mt iff.to_eq %%h)\n         | `(_ \u2260 _)   := replace h.local_pp_name ``(mt iff.to_eq %%h)\n         | `(_ = _)   := replace h.local_pp_name ``(eq.to_iff %%h)\n         | `(\u00ac (_ \u2227 _))  := replace h.local_pp_name ``(not_and_distrib'.mp %%h) <|>\n                            replace h.local_pp_name ``(not_and_distrib.mp %%h)\n         | `(\u00ac (_ \u2228 _))  := replace h.local_pp_name ``(not_or_distrib.mp %%h)\n         | `(\u00ac \u00ac _)      := replace h.local_pp_name ``(of_not_not %%h)\n         | `(\u00ac (_ \u2192 (_ : Prop))) := replace h.local_pp_name ``(not_imp.mp %%h)\n         | `(\u00ac (_ \u2194 _)) := replace h.local_pp_name ``(not_iff.mp %%h)\n         | `(_ \u2194 _) := replace h.local_pp_name ``(iff_iff_and_or_not_and_not.mp %%h) <|>\n                       replace h.local_pp_name ``(iff_iff_and_or_not_and_not.mp (%%h).symm) <|>\n                       () <$ tactic.cases h\n         | `(_ \u2192 _)     := replace h.local_pp_name ``(not_or_of_imp %%h)\n         | _ := failed\n         end\n\nmeta def tauto_state := ref $ expr_map (option (expr \u00d7 expr))\n\nmeta def modify_ref {\u03b1 : Type} (r : ref \u03b1) (f : \u03b1 \u2192 \u03b1) :=\nread_ref r >>= write_ref r \u2218 f\n\nmeta def add_refl (r : tauto_state) (e : expr) : tactic (expr \u00d7 expr) :=\ndo m \u2190 read_ref r,\n   p \u2190 mk_mapp `rfl [none,e],\n   write_ref r $ m.insert e none,\n   return (e,p)\n\nmeta def add_symm_proof (r : tauto_state) (e : expr) : tactic (expr \u00d7 expr) :=\ndo env \u2190 get_env,\n   let rel := e.get_app_fn.const_name,\n   some symm \u2190 pure $ environment.symm_for env rel\n     | add_refl r e,\n   (do e' \u2190 mk_meta_var `(Prop),\n       iff_t \u2190 to_expr ``(%%e = %%e'),\n       (_,p) \u2190 solve_aux iff_t\n         (applyc `iff.to_eq ; () <$ split ; applyc symm),\n       e' \u2190 instantiate_mvars e',\n       m \u2190 read_ref r,\n       write_ref r $ (m.insert e (e',p)).insert e' none,\n       return (e',p) )\n   <|> add_refl r e\n\nmeta def add_edge (r : tauto_state) (x y p : expr) : tactic unit :=\nmodify_ref r $ \u03bb m, m.insert x (y,p)\n\nmeta def root (r : tauto_state) : expr \u2192 tactic (expr \u00d7 expr) | e :=\ndo m \u2190 read_ref r,\n   let record_e : tactic (expr \u00d7 expr) :=\n       match e with\n       | v@(expr.mvar _ _ _) :=\n         (do (e,p) \u2190 get_assignment v >>= root,\n             add_edge r v e p,\n             return (e,p)) <|>\n         add_refl r e\n       | _ := add_refl r e\n       end,\n   some e' \u2190 pure $ m.find e | record_e,\n   match e' with\n   | (some (e',p')) :=\n     do (e'',p'') \u2190 root e',\n        p'' \u2190 mk_app `eq.trans [p',p''],\n        add_edge r e e'' p'',\n        pure (e'',p'')\n   | none := prod.mk e <$> mk_mapp `rfl [none,some e]\n   end\n\nmeta def symm_eq (r : tauto_state) : expr \u2192 expr \u2192 tactic expr | a b :=\ndo m \u2190 read_ref r,\n   (a',pa) \u2190 root r a,\n   (b',pb) \u2190 root r b,\n   (unify a' b' >> add_refl r a' *> mk_mapp `rfl [none,a]) <|>\n    do p \u2190 match (a', b') with\n           | (`(\u00ac %%a\u2080), `(\u00ac %%b\u2080)) :=\n             do p  \u2190 symm_eq a\u2080 b\u2080,\n                p' \u2190 mk_app `congr_arg [`(not),p],\n                add_edge r a' b' p',\n                return p'\n           | (`(%%a\u2080 \u2227 %%a\u2081), `(%%b\u2080 \u2227 %%b\u2081)) :=\n             do p\u2080 \u2190 symm_eq a\u2080 b\u2080,\n                p\u2081 \u2190 symm_eq a\u2081 b\u2081,\n                p' \u2190 to_expr ``(congr (congr_arg and %%p\u2080) %%p\u2081),\n                add_edge r a' b' p',\n                return p'\n           | (`(%%a\u2080 \u2228 %%a\u2081), `(%%b\u2080 \u2228 %%b\u2081)) :=\n             do p\u2080 \u2190 symm_eq a\u2080 b\u2080,\n                p\u2081 \u2190 symm_eq a\u2081 b\u2081,\n                p' \u2190 to_expr ``(congr (congr_arg or %%p\u2080) %%p\u2081),\n                add_edge r a' b' p',\n                return p'\n           | (`(%%a\u2080 \u2194 %%a\u2081), `(%%b\u2080 \u2194 %%b\u2081)) :=\n             (do p\u2080 \u2190 symm_eq a\u2080 b\u2080,\n                 p\u2081 \u2190 symm_eq a\u2081 b\u2081,\n                 p' \u2190 to_expr ``(congr (congr_arg iff %%p\u2080) %%p\u2081),\n                 add_edge r a' b' p',\n                 return p') <|>\n             do p\u2080 \u2190 symm_eq a\u2080 b\u2081,\n                p\u2081 \u2190 symm_eq a\u2081 b\u2080,\n                p' \u2190 to_expr ``(eq.trans (congr (congr_arg iff %%p\u2080) %%p\u2081)\n                                         (iff.to_eq iff.comm ) ),\n                add_edge r a' b' p',\n                return p'\n           | (`(%%a\u2080 \u2192 %%a\u2081), `(%%b\u2080 \u2192 %%b\u2081)) :=\n             if \u00ac a\u2081.has_var \u2227 \u00ac b\u2081.has_var then\n             do p\u2080 \u2190 symm_eq a\u2080 b\u2080,\n                p\u2081 \u2190 symm_eq a\u2081 b\u2081,\n                p' \u2190 mk_app `congr_arg [`(implies),p\u2080,p\u2081],\n                add_edge r a' b' p',\n                return p'\n             else unify a' b' >> add_refl r a' *> mk_mapp `rfl [none,a]\n           | (_, _) :=\n             (do guard $ a'.get_app_fn.is_constant \u2227\n                         a'.get_app_fn.const_name = b'.get_app_fn.const_name,\n                 (a'',pa') \u2190 add_symm_proof r a',\n                 guard $ a'' =\u2090 b',\n                 pure pa' )\n           end,\n           p' \u2190 mk_eq_trans pa p,\n           add_edge r a' b' p',\n           mk_eq_symm pb >>= mk_eq_trans p'\n\nmeta def find_eq_type (r : tauto_state) : expr \u2192 list expr \u2192 tactic (expr \u00d7 expr)\n| e []         := failed\n| e (H :: Hs) :=\n  do t  \u2190 infer_type H,\n     t' \u2190 infer_type e,\n     (prod.mk H <$> symm_eq r e t) <|> find_eq_type e Hs\n\nprivate meta def contra_p_not_p (r : tauto_state) : list expr \u2192 list expr \u2192 tactic unit\n| []         Hs := failed\n| (H1 :: Rs) Hs :=\n  do t \u2190 (extract_opt_auto_param <$> infer_type H1) >>= whnf,\n     (do a   \u2190 match_not t,\n         (H2,p)  \u2190 find_eq_type r a Hs,\n         H2 \u2190 to_expr ``( (%%p).mpr %%H2 ),\n         tgt \u2190 target,\n         pr  \u2190 mk_app `absurd [tgt, H2, H1],\n         tactic.exact pr)\n     <|> contra_p_not_p Rs Hs\n\nmeta def contradiction_with (r : tauto_state) : tactic unit :=\ncontradiction <|>\ndo tactic.try intro1,\n   ctx \u2190 local_context,\n   contra_p_not_p r ctx ctx\n\nmeta def contradiction_symm :=\nusing_new_ref (native.rb_map.mk _ _) contradiction_with\n\nmeta def assumption_with (r : tauto_state) : tactic unit :=\ndo { ctx \u2190 local_context,\n     t   \u2190 target,\n     (H,p) \u2190 find_eq_type r t ctx,\n     mk_eq_mpr p H >>= tactic.exact }\n<|> fail \"assumption tactic failed\"\n\nmeta def assumption_symm :=\nusing_new_ref (native.rb_map.mk _ _) assumption_with\n\nmeta def tautology (c : bool := ff) : tactic unit :=\ndo when c classical,\n   using_new_ref (expr_map.mk _) $ \u03bb r,\n   do try (contradiction_with r);\n      try (assumption_with r);\n      repeat (do\n        gs \u2190 get_goals,\n        repeat (() <$ tactic.intro1);\n        distrib_not;\n        casesm (some ()) [``(_ \u2227 _),``(_ \u2228 _),``(Exists _),``(false)];\n        try (contradiction_with r);\n        try (target >>= match_or >> refine ``( or_iff_not_imp_left.mpr _));\n        try (target >>= match_or >> refine ``( or_iff_not_imp_right.mpr _));\n        repeat (() <$ tactic.intro1);\n        constructor_matching (some ()) [``(_ \u2227 _),``(_ \u2194 _),``(true)];\n        try (assumption_with r),\n        gs' \u2190 get_goals,\n        guard (gs \u2260 gs') ) ;\n      repeat\n      (reflexivity <|> solve_by_elim <|>\n       constructor_matching none [``(_ \u2227 _),``(_ \u2194 _),``(Exists _),``(true)] ) ;\n      done\n\nopen interactive lean.parser\n\nnamespace interactive\nlocal postfix `?`:9001 := optional\n\n\/--\n`tautology` breaks down assumptions of the form `_ \u2227 _`, `_ \u2228 _`, `_ \u2194 _` and `\u2203 _, _`\nand splits a goal of the form `_ \u2227 _`, `_ \u2194 _` or `\u2203 _, _` until it can be discharged\nusing `reflexivity` or `solve_by_elim`.\nThis is a finishing tactic: it either closes the goal or raises an error.\nThe variant `tautology!` uses the law of excluded middle.\n-\/\nmeta def tautology (c : parse $ (tk \"!\")?) := tactic.tautology c.is_some\n\n-- Now define a shorter name for the tactic `tautology`.\n\n\/--\n`tauto` breaks down assumptions of the form `_ \u2227 _`, `_ \u2228 _`, `_ \u2194 _` and `\u2203 _, _`\nand splits a goal of the form `_ \u2227 _`, `_ \u2194 _` or `\u2203 _, _` until it can be discharged\nusing `reflexivity` or `solve_by_elim`.\nThis is a finishing tactic: it either closes the goal or raises an error.\nThe variant `tauto!` uses the law of excluded middle.\n-\/\nmeta def tauto (c : parse $ (tk \"!\")?) := tautology c\n\nadd_hint_tactic \"tauto\"\n\nend interactive\nend tactic\n","avg_line_length":36.2425531915,"max_line_length":95,"alphanum_fraction":0.5271809323}
-{"size":12014,"ext":"lean","lang":"Lean","max_stars_count":130.0,"content":"\/-\nCopyright (c) 2015 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Robert Y. Lewis\n\nBasic facts about the positive natural numbers.\n\nDeveloped primarily for use in the construction of \u211d. For the most part, the only theorems here\nare those needed for that construction.\n-\/\nimport data.rat.order data.nat\nopen nat rat subtype eq.ops\n\ndefinition pnat := { n : \u2115 | n > 0 }\n\nnamespace pnat\n\nprotected definition prio := num.pred nat.prio\n\nnotation `\u2115+` := pnat\n\ndefinition pos (n : \u2115) (H : n > 0) : \u2115+ := tag n H\n\ndefinition nat_of_pnat (p : \u2115+) : \u2115 := elt_of p\nreserve postfix `~`:std.prec.max_plus\nlocal postfix ~ := nat_of_pnat\n\ntheorem pnat_pos (p : \u2115+) : p~ > 0 := has_property p\n\nprotected definition add (p q : \u2115+) : \u2115+ :=\ntag (p~ + q~) (add_pos (pnat_pos p) (pnat_pos q))\n\nprotected definition mul (p q : \u2115+) : \u2115+ :=\ntag (p~ * q~) (mul_pos (pnat_pos p) (pnat_pos q))\n\nprotected definition le (p q : \u2115+) := p~ \u2264 q~\n\nprotected definition lt (p q : \u2115+) := p~ < q~\n\ndefinition pnat_has_add [instance] [priority pnat.prio] : has_add pnat :=\nhas_add.mk pnat.add\n\ndefinition pnat_has_mul [instance] [priority pnat.prio] : has_mul pnat :=\nhas_mul.mk pnat.mul\n\ndefinition pnat_has_le [instance] [priority pnat.prio] : has_le pnat :=\nhas_le.mk pnat.le\n\ndefinition pnat_has_lt [instance] [priority pnat.prio] : has_lt pnat :=\nhas_lt.mk pnat.lt\n\ndefinition pnat_has_one [instance] [priority pnat.prio] : has_one pnat :=\nhas_one.mk (pos (1:nat) dec_trivial)\n\nprotected lemma mul_def (p q : \u2115+) : p * q = tag (p~ * q~) (mul_pos (pnat_pos p) (pnat_pos q)) :=\nrfl\n\nprotected lemma le_def (p q : \u2115+) : (p \u2264 q) = (p~ \u2264 q~) :=\nrfl\n\nprotected lemma lt_def (p q : \u2115+) : (p < q) = (p~ < q~) :=\nrfl\n\nprotected theorem pnat.eq {p q : \u2115+} : p~ = q~ \u2192 p = q :=\nsubtype.eq\n\nprotected definition decidable_lt : decidable_rel pnat.lt :=\n\u03bba b, nat.decidable_lt a~ b~\n\nprotected theorem le_refl (a : \u2115+) : a \u2264 a :=\nbegin rewrite pnat.le_def end\n\nprotected theorem le_trans {p q r : \u2115+} : p \u2264 q \u2192 q \u2264 r \u2192 p \u2264 r :=\nbegin rewrite *pnat.le_def, apply nat.le_trans end\n\nprotected theorem le_antisymm {n m : \u2115+} : n \u2264 m \u2192 m \u2264 n \u2192 n = m :=\nbegin rewrite +pnat.le_def, intros, apply (pnat.eq (nat.le_antisymm a a_1)) end\n\nprotected theorem le_iff_lt_or_eq (m n : \u2115+) : m \u2264 n \u2194 m < n \u2228 m = n :=\nbegin\n  rewrite [pnat.lt_def, pnat.le_def], apply iff.intro,\n  { intro, apply or.elim (nat.lt_or_eq_of_le a),\n    intro, apply or.intro_left, assumption,\n    intro, apply or.intro_right, apply pnat.eq, assumption },\n  { intro, apply or.elim a, apply nat.le_of_lt, intro, rewrite a_1 }\nend\n\nprotected theorem le_total (m n : \u2115+) : m \u2264 n \u2228 n \u2264 m :=\nbegin rewrite pnat.le_def, apply nat.le_total end\n\nprotected theorem lt_irrefl (a : \u2115+) : \u00ac a < a :=\nbegin rewrite pnat.lt_def, apply nat.lt_irrefl end\n\nprotected definition decidable_linear_order [trans_instance] : decidable_linear_order pnat :=\n\u2983 decidable_linear_order,\n  le              := pnat.le,\n  le_refl         := by apply pnat.le_refl,\n  le_trans        := by apply @pnat.le_trans,\n  le_antisymm     := by apply @pnat.le_antisymm,\n  lt              := pnat.lt,\n  lt_irrefl       := by apply pnat.lt_irrefl,\n  le_iff_lt_or_eq := by apply pnat.le_iff_lt_or_eq,\n  decidable_lt    := pnat.decidable_lt,\n  le_total        := by apply pnat.le_total \u2984\n\nnotation 2 := (tag 2 dec_trivial : \u2115+)\nnotation 3 := (tag 3 dec_trivial : \u2115+)\ndefinition pone : \u2115+ := tag 1 dec_trivial\n\ndefinition rat_of_pnat [reducible] (n : \u2115+) : \u211a :=\nn~\n\ntheorem pnat.to_rat_of_nat (n : \u2115+) : rat_of_pnat n = of_nat n~ :=\nrfl\n\n-- these will come in rat\n\ntheorem rat_of_nat_nonneg (n : \u2115) : 0 \u2264 of_nat n :=\ntrivial\n\ntheorem rat_of_pnat_ge_one (n : \u2115+) : rat_of_pnat n \u2265 1 :=\nof_nat_le_of_nat_of_le (pnat_pos n)\n\ntheorem rat_of_pnat_is_pos (n : \u2115+) : rat_of_pnat n > 0 :=\nof_nat_lt_of_nat_of_lt (pnat_pos n)\n\ntheorem of_nat_le_of_nat_of_le {m n : \u2115} (H : m \u2264 n) : of_nat m \u2264 of_nat n :=\nof_nat_le_of_nat_of_le H\n\ntheorem of_nat_lt_of_nat_of_lt {m n : \u2115} (H : m < n) : of_nat m < of_nat n :=\nof_nat_lt_of_nat_of_lt H\n\ntheorem rat_of_pnat_le_of_pnat_le {m n : \u2115+} (H : m \u2264 n) : rat_of_pnat m \u2264 rat_of_pnat n :=\nbegin rewrite pnat.le_def at H, exact of_nat_le_of_nat_of_le H end\n\ntheorem rat_of_pnat_lt_of_pnat_lt {m n : \u2115+} (H : m < n) : rat_of_pnat m < rat_of_pnat n :=\nbegin rewrite pnat.lt_def at H, exact of_nat_lt_of_nat_of_lt H end\n\ntheorem pnat_le_of_rat_of_pnat_le {m n : \u2115+} (H : rat_of_pnat m \u2264 rat_of_pnat n) : m \u2264 n :=\nbegin rewrite pnat.le_def, exact le_of_of_nat_le_of_nat H end\n\ndefinition inv (n : \u2115+) : \u211a :=\n(1 : \u211a) \/ rat_of_pnat n\n\nlocal postfix `\u207b\u00b9` := inv\n\nprotected theorem inv_pos (n : \u2115+) : n\u207b\u00b9 > 0 := one_div_pos_of_pos !rat_of_pnat_is_pos\n\ntheorem inv_le_one (n : \u2115+) : n\u207b\u00b9 \u2264 (1 : \u211a) :=\nbegin\n  unfold inv,\n  change 1 \/ rat_of_pnat n \u2264 1 \/ 1,\n  apply one_div_le_one_div_of_le,\n  apply zero_lt_one,\n  apply rat_of_pnat_ge_one\nend\n\ntheorem inv_lt_one_of_gt {n : \u2115+} (H : n~ > 1) : n\u207b\u00b9 < (1 : \u211a) :=\nbegin\n  unfold inv,\n  change 1 \/ rat_of_pnat n < 1 \/ 1,\n  apply one_div_lt_one_div_of_lt,\n  apply zero_lt_one,\n  rewrite pnat.to_rat_of_nat,\n  apply (of_nat_lt_of_nat_of_lt H)\nend\n\ntheorem pone_inv : pone\u207b\u00b9 = 1 := rfl\n\ntheorem add_invs_nonneg (m n : \u2115+) : 0 \u2264 m\u207b\u00b9 + n\u207b\u00b9 :=\nbegin\n  apply le_of_lt,\n  apply add_pos,\n  repeat apply pnat.inv_pos\nend\n\nprotected theorem one_mul (n : \u2115+) : pone * n = n :=\nbegin\n  apply pnat.eq,\n  unfold pone,\n  rewrite [pnat.mul_def, \u2191nat_of_pnat, one_mul]\nend\n\ntheorem pone_le (n : \u2115+) : pone \u2264 n :=\nbegin rewrite pnat.le_def, exact succ_le_of_lt (pnat_pos n) end\n\ntheorem pnat_to_rat_mul (a b : \u2115+) : rat_of_pnat (a * b) = rat_of_pnat a * rat_of_pnat b := rfl\n\ntheorem mul_lt_mul_left {a b c : \u2115+} (H : a < b) : a * c < b * c :=\nbegin rewrite [pnat.lt_def at *], exact mul_lt_mul_of_pos_right H !pnat_pos end\n\ntheorem one_lt_two : pone < 2 :=\n!nat.le_refl\n\ntheorem inv_two_mul_lt_inv (n : \u2115+) : (2 * n)\u207b\u00b9 < n\u207b\u00b9 :=\nbegin\n  rewrite \u2191inv,\n  apply one_div_lt_one_div_of_lt,\n  apply rat_of_pnat_is_pos,\n  have H : n~ < (2 * n)~, begin\n    rewrite -(pnat.one_mul n) at {1},\n    rewrite -pnat.lt_def,\n    apply mul_lt_mul_left,\n    apply one_lt_two\n  end,\n  apply of_nat_lt_of_nat_of_lt,\n  apply H\nend\n\ntheorem inv_two_mul_le_inv (n : \u2115+) : (2 * n)\u207b\u00b9 \u2264 n\u207b\u00b9 := rat.le_of_lt !inv_two_mul_lt_inv\n\ntheorem inv_ge_of_le {p q : \u2115+} (H : p \u2264 q) : q\u207b\u00b9 \u2264 p\u207b\u00b9 :=\none_div_le_one_div_of_le !rat_of_pnat_is_pos (rat_of_pnat_le_of_pnat_le H)\n\ntheorem inv_gt_of_lt {p q : \u2115+} (H : p < q) : q\u207b\u00b9 < p\u207b\u00b9 :=\none_div_lt_one_div_of_lt !rat_of_pnat_is_pos (rat_of_pnat_lt_of_pnat_lt H)\n\ntheorem ge_of_inv_le {p q : \u2115+} (H : p\u207b\u00b9 \u2264 q\u207b\u00b9) : q \u2264 p :=\npnat_le_of_rat_of_pnat_le (le_of_one_div_le_one_div !rat_of_pnat_is_pos H)\n\ntheorem two_mul (p : \u2115+) : rat_of_pnat (2 * p) = (1 + 1) * rat_of_pnat p :=\nby rewrite pnat_to_rat_mul\n\nprotected theorem add_halves (p : \u2115+) : (2 * p)\u207b\u00b9 + (2 * p)\u207b\u00b9 = p\u207b\u00b9 :=\nbegin\n  rewrite [\u2191inv, -(add_halves (1 \/ (rat_of_pnat p))), div_div_eq_div_mul],\n  have H : rat_of_pnat (2 * p) = rat_of_pnat p * (1 + 1), by rewrite [rat.mul_comm, two_mul],\n  rewrite *H\nend\n\ntheorem add_halves_double (m n : \u2115+) :\n        m\u207b\u00b9 + n\u207b\u00b9 = ((2 * m)\u207b\u00b9 + (2 * n)\u207b\u00b9) + ((2 * m)\u207b\u00b9 + (2 * n)\u207b\u00b9) :=\nhave hsimp : \u2200 a b : \u211a, (a + a) + (b + b) = (a + b) + (a + b),\n  by intros; rewrite [rat.add_assoc, -(rat.add_assoc a b b), {_+b}rat.add_comm, -*rat.add_assoc],\nby rewrite [-pnat.add_halves m, -pnat.add_halves n, hsimp]\n\nprotected theorem inv_mul_eq_mul_inv {p q : \u2115+} : (p * q)\u207b\u00b9 = p\u207b\u00b9 * q\u207b\u00b9 :=\nbegin rewrite [\u2191inv, pnat_to_rat_mul, one_div_mul_one_div] end\n\nprotected theorem inv_mul_le_inv (p q : \u2115+) : (p * q)\u207b\u00b9 \u2264 q\u207b\u00b9 :=\nbegin\n  rewrite [pnat.inv_mul_eq_mul_inv, -{q\u207b\u00b9}rat.one_mul at {2}],\n  apply mul_le_mul,\n  apply inv_le_one,\n  apply le.refl,\n  apply le_of_lt,\n  apply pnat.inv_pos,\n  apply rat.le_of_lt rat.zero_lt_one\nend\n\ntheorem pnat_mul_le_mul_left' (a b c : \u2115+) : a \u2264 b \u2192 c * a \u2264 c * b :=\nbegin\n  rewrite +pnat.le_def, intro H,\n  apply mul_le_mul_of_nonneg_left H,\n  apply le_of_lt,\n  apply pnat_pos\nend\n\nprotected theorem mul_assoc (a b c : \u2115+) : a * b * c = a * (b * c) :=\npnat.eq !mul.assoc\n\nprotected theorem mul_comm (a b : \u2115+) : a * b = b * a :=\npnat.eq !mul.comm\n\nprotected theorem add_assoc (a b c : \u2115+) : a + b + c = a + (b + c) :=\npnat.eq !add.assoc\n\nprotected theorem mul_le_mul_left (p q : \u2115+) : q \u2264 p * q :=\nbegin\n  rewrite [-pnat.one_mul q at {1}, pnat.mul_comm, pnat.mul_comm p],\n  apply pnat_mul_le_mul_left',\n  apply pone_le\nend\n\nprotected theorem mul_le_mul_right (p q : \u2115+) : p \u2264 p * q :=\nby rewrite pnat.mul_comm; apply pnat.mul_le_mul_left\n\nprotected theorem inv_cancel_left (p : \u2115+) : rat_of_pnat p * p\u207b\u00b9 = (1 : \u211a) :=\nmul_one_div_cancel (ne.symm (ne_of_lt !rat_of_pnat_is_pos))\n\nprotected theorem inv_cancel_right (p : \u2115+) : p\u207b\u00b9 * rat_of_pnat p = (1 : \u211a) :=\nby rewrite rat.mul_comm; apply pnat.inv_cancel_left\n\ntheorem lt_add_left (p q : \u2115+) : p < p + q :=\nbegin\n  have H : p~ < p~ + q~, begin\n    rewrite -(nat.add_zero (p~)) at {1},\n    apply nat.add_lt_add_left,\n    apply pnat_pos\n  end,\n  apply H\nend\n\ntheorem inv_add_lt_left (p q : \u2115+) : (p + q)\u207b\u00b9 < p\u207b\u00b9 :=\nby apply inv_gt_of_lt; apply lt_add_left\n\ntheorem div_le_pnat (q : \u211a) (n : \u2115+) (H : q \u2265 n\u207b\u00b9) : 1 \/ q \u2264 rat_of_pnat n :=\nbegin\n  apply div_le_of_le_mul,\n  apply lt_of_lt_of_le,\n  apply pnat.inv_pos,\n  rotate 1,\n  apply H,\n  apply le_mul_of_div_le,\n  apply rat_of_pnat_is_pos,\n  apply H\nend\n\ntheorem pnat_cancel' (n m : \u2115+) : (n * n * m)\u207b\u00b9 * (rat_of_pnat n * rat_of_pnat n) = m\u207b\u00b9 :=\nhave hsimp : \u2200 a b c : \u211a, (a * a * (b * b * c)) = (a * b) * (a * b) * c,\n  begin\n    intro a b c,\n    rewrite[-*rat.mul_assoc],\n    exact (!mul.right_comm \u25b8 rfl),\n  end,\nby rewrite [rat.mul_comm, *pnat.inv_mul_eq_mul_inv, hsimp, *pnat.inv_cancel_left, *rat.one_mul]\n\ndefinition pceil (a : \u211a) : \u2115+ := tag (ubound a) !ubound_pos\n\ntheorem pceil_helper {a : \u211a} {n : \u2115+} (H : pceil a \u2264 n) (Ha : a > 0) : n\u207b\u00b9 \u2264 1 \/ a :=\nle.trans (inv_ge_of_le H) (one_div_le_one_div_of_le Ha (ubound_ge a))\n\ntheorem inv_pceil_div (a b : \u211a) (Ha : a > 0) (Hb : b > 0) : (pceil (a \/ b))\u207b\u00b9 \u2264 b \/ a :=\nhave (pceil (a \/ b))\u207b\u00b9 \u2264 1 \/ (1 \/ (b \/ a)),\n  begin\n    apply one_div_le_one_div_of_le,\n    show 0 < 1 \/ (b \/ a), from\n      one_div_pos_of_pos (div_pos_of_pos_of_pos Hb Ha),\n    show 1 \/ (b \/ a) \u2264 rat_of_pnat (pceil (a \/ b)),\n    begin\n      rewrite div_div_eq_mul_div,\n      rewrite one_mul,\n      apply ubound_ge\n    end\n  end,\nbegin\n  rewrite one_div_one_div at this,\n  exact this\nend\n\ntheorem sep_by_inv {a b : \u211a} : a > b \u2192 \u2203 N : \u2115+, a > (b + N\u207b\u00b9 + N\u207b\u00b9) :=\nbegin\n  change b < a \u2192 \u2203 N : \u2115+, (b + N\u207b\u00b9 + N\u207b\u00b9) < a,\n  intro H,\n  apply exists.elim (exists_add_lt_and_pos_of_lt H),\n  intro c Hc,\n  existsi (pceil ((1 + 1 + 1) \/ c)),\n  apply lt.trans,\n  rotate 1,\n  apply and.left Hc,\n  rewrite rat.add_assoc,\n  apply rat.add_lt_add_left,\n  rewrite -(add_halves c) at {3},\n  apply add_lt_add,\n  repeat (apply lt_of_le_of_lt;\n    apply inv_pceil_div;\n    apply dec_trivial;\n    apply and.right Hc;\n    apply div_lt_div_of_pos_of_lt_of_pos;\n    apply two_pos;\n    exact dec_trivial;\n    apply and.right Hc)\nend\n\ntheorem nonneg_of_ge_neg_invs (a : \u211a) : (\u2200 n : \u2115+, -n\u207b\u00b9 \u2264 a) \u2192 0 \u2264 a :=\nbegin\n  intro H,\n  apply le_of_not_gt,\n  suppose a < 0,\n  have H2 : 0 < -a, from neg_pos_of_neg this,\n  (not_lt_of_ge !H) (iff.mp !lt_neg_iff_lt_neg (calc\n    (pceil (of_num 2 \/ -a))\u207b\u00b9 \u2264 -a \/ of_num 2\n        : !inv_pceil_div dec_trivial H2\n                           ... < -a \/ 1\n        : div_lt_div_of_pos_of_lt_of_pos dec_trivial dec_trivial H2\n                           ... = -a : !div_one))\nend\n\ntheorem pnat_bound {\u03b5 : \u211a} (H\u03b5 : \u03b5 > 0) : \u2203 p : \u2115+, p\u207b\u00b9 \u2264 \u03b5 :=\nbegin\n  existsi (pceil (1 \/ \u03b5)),\n  rewrite -(one_div_one_div \u03b5) at {2},\n  apply pceil_helper,\n  apply pnat.le_refl,\n  apply one_div_pos_of_pos H\u03b5\nend\n\ntheorem p_add_fractions (n : \u2115+) : (2 * n)\u207b\u00b9 + (2 * 3 * n)\u207b\u00b9 + (3 * n)\u207b\u00b9 = n\u207b\u00b9 :=\nhave T : 2\u207b\u00b9 + 2\u207b\u00b9 * 3\u207b\u00b9 + 3\u207b\u00b9 = 1, from dec_trivial,\nby rewrite[*pnat.inv_mul_eq_mul_inv,-*right_distrib,T,rat.one_mul]\n\ntheorem rat_power_two_le (k : \u2115+) : rat_of_pnat k \u2264 2^k~ :=\n!binary_nat_bound\n\nend pnat\n","avg_line_length":30.3383838384,"max_line_length":97,"alphanum_fraction":0.652072582}
-{"size":13084,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2019 Neil Strickland.  All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Neil Strickland\n\nThis file sets up a version of the Euclidean algorithm that works\nonly with natural numbers.  Given a, b > 0, it computes the unique\nsystem (w, x, y, z, d) such that the following identities hold:\n\n w * z = x * y + 1\n a = (w + x) d\n b = (y + z) d\n\nThese equations force w, z, d > 0.  They also imply that\nthe integers a' = w + x = a \/ d and b' = y + z = b \/ d are coprime,\nand that d is the gcd of a and b.\n\nThis story is closely related to the structure of SL\u2082(\u2115) (as a\nfree monoid on two generators) and the theory of continued fractions.\n-\/\nimport tactic.ring\nimport tactic.abel\n\nnamespace pnat\nopen nat pnat\n\n\/-- A term of xgcd_type is a system of six naturals.  They should\n be thought of as representing the matrix\n [[w, x], [y, z]] = [[wp + 1, x], [y, zp + 1]]\n together with the vector [a, b] = [ap + 1, bp + 1].\n-\/\n@[derive inhabited]\nstructure xgcd_type :=\n(wp x y zp ap bp : \u2115)\n\nnamespace xgcd_type\n\nvariable (u : xgcd_type)\n\ninstance : has_sizeof xgcd_type := \u27e8\u03bb u, u.bp\u27e9\n\n\/-- The has_repr instance converts terms to strings in a way that\n reflects the matrix\/vector interpretation as above. -\/\ninstance : has_repr xgcd_type :=\n\u27e8\u03bb u, \"[[[\" ++ (repr (u.wp + 1)) ++ \", \" ++ (repr u.x) ++\n      \"], [\" ++ (repr u.y) ++ \", \" ++ (repr (u.zp + 1)) ++ \"]], [\" ++\n      (repr (u.ap + 1)) ++ \", \" ++ (repr (u.bp + 1)) ++ \"]]\"\u27e9\n\ndef mk' (w : \u2115+) (x : \u2115) (y : \u2115) (z : \u2115+) (a : \u2115+) (b : \u2115+) : xgcd_type :=\nmk w.val.pred x y z.val.pred a.val.pred b.val.pred\n\ndef w : \u2115+ := succ_pnat u.wp\ndef z : \u2115+ := succ_pnat u.zp\ndef a : \u2115+ := succ_pnat u.ap\ndef b : \u2115+ := succ_pnat u.bp\ndef r : \u2115 := (u.ap + 1) % (u.bp + 1)\ndef q : \u2115 := (u.ap + 1) \/ (u.bp + 1)\ndef qp : \u2115 := u.q - 1\n\n\/-- The map v gives the product of the matrix\n [[w, x], [y, z]] = [[wp + 1, x], [y, zp + 1]]\n and the vector [a, b] = [ap + 1, bp + 1].  The map\n vp gives [sp, tp] such that v = [sp + 1, tp + 1].\n-\/\ndef vp : \u2115 \u00d7 \u2115 :=\n\u27e8 u.wp + u.x + u.ap + u.wp * u.ap + u.x * u.bp,\n  u.y + u.zp + u.bp + u.y * u.ap + u.zp * u.bp \u27e9\n\ndef v : \u2115 \u00d7 \u2115 := \u27e8u.w * u.a + u.x * u.b, u.y * u.a + u.z * u.b\u27e9\ndef succ\u2082 (t : \u2115 \u00d7 \u2115) : \u2115 \u00d7 \u2115 := \u27e8t.1.succ, t.2.succ\u27e9\n\ntheorem v_eq_succ_vp : u.v = succ\u2082 u.vp :=\nby { ext; dsimp [v, vp, w, z, a, b, succ\u2082];\n     repeat { rw [nat.succ_eq_add_one] }; ring }\n\n\/-- is_special holds if the matrix has determinant one. -\/\ndef is_special : Prop := u.wp + u.zp + u.wp * u.zp = u.x * u.y\ndef is_special' : Prop := u.w * u.z = succ_pnat (u.x * u.y)\n\ntheorem is_special_iff : u.is_special \u2194 u.is_special' :=\nbegin\n  dsimp [is_special, is_special'],\n  split; intro h,\n  { apply eq, dsimp [w, z, succ_pnat], rw [\u2190 h],\n    repeat { rw [nat.succ_eq_add_one] }, ring },\n  { apply nat.succ_inj,\n    replace h := congr_arg (coe : \u2115+ \u2192 \u2115) h,\n    rw [mul_coe, w, z] at h,\n    repeat { rw [succ_pnat_coe, nat.succ_eq_add_one] at h },\n    repeat { rw [nat.succ_eq_add_one] }, rw [\u2190 h], ring }\nend\n\n\/-- is_reduced holds if the two entries in the vector are the\n same.  The reduction algorithm will produce a system with this\n property, whose product vector is the same as for the original\n system. -\/\ndef is_reduced : Prop := u.ap = u.bp\ndef is_reduced' : Prop := u.a = u.b\n\ntheorem is_reduced_iff : u.is_reduced \u2194 u.is_reduced' :=\n\u27e8 congr_arg succ_pnat, succ_pnat_inj \u27e9\n\ndef flip : xgcd_type :=\n{ wp := u.zp, x := u.y, y := u.x, zp := u.wp, ap := u.bp, bp := u.ap }\n\n@[simp] theorem flip_w : (flip u).w = u.z := rfl\n@[simp] theorem flip_x : (flip u).x = u.y := rfl\n@[simp] theorem flip_y : (flip u).y = u.x := rfl\n@[simp] theorem flip_z : (flip u).z = u.w := rfl\n@[simp] theorem flip_a : (flip u).a = u.b := rfl\n@[simp] theorem flip_b : (flip u).b = u.a := rfl\n\ntheorem flip_is_reduced : (flip u).is_reduced \u2194 u.is_reduced :=\nby { dsimp [is_reduced, flip], split; intro h; exact h.symm }\n\ntheorem flip_is_special : (flip u).is_special \u2194 u.is_special :=\nby { dsimp [is_special, flip], rw[mul_comm u.x, mul_comm u.zp, add_comm u.zp] }\n\ntheorem flip_v : (flip u).v = (u.v).swap :=\nby { dsimp [v], ext, { simp only [], ring }, { simp only [], ring } }\n\n\/-- Properties of division with remainder for a \/ b.  -\/\ntheorem rq_eq : u.r + (u.bp + 1) * u.q = u.ap + 1 :=\nnat.mod_add_div (u.ap + 1) (u.bp + 1)\n\ntheorem qp_eq (hr : u.r = 0) : u.q = u.qp + 1 :=\nbegin\n  by_cases hq : u.q = 0,\n  { let h := u.rq_eq, rw [hr, hq, mul_zero, add_zero] at h, cases h },\n  { exact (nat.succ_pred_eq_of_pos (nat.pos_of_ne_zero hq)).symm }\nend\n\n\/-- The following function provides the starting point for\n our algorithm.  We will apply an iterative reduction process\n to it, which will produce a system satisfying is_reduced.\n The gcd can be read off from this final system.\n-\/\ndef start (a b : \u2115+) : xgcd_type := \u27e80, 0, 0, 0, a - 1, b - 1\u27e9\n\ntheorem start_is_special (a b : \u2115+) : (start a b).is_special :=\nby { dsimp [start, is_special], refl }\n\ntheorem start_v (a b : \u2115+) : (start a b).v = \u27e8a, b\u27e9 :=\nbegin\n  dsimp [start, v, xgcd_type.a, xgcd_type.b, w, z],\n  rw [one_mul, one_mul, zero_mul, zero_mul, zero_add, add_zero],\n  rw [\u2190 nat.pred_eq_sub_one, \u2190 nat.pred_eq_sub_one],\n  rw [nat.succ_pred_eq_of_pos a.pos, nat.succ_pred_eq_of_pos b.pos]\nend\n\ndef finish : xgcd_type :=\nxgcd_type.mk u.wp ((u.wp + 1) * u.qp + u.x) u.y (u.y * u.qp + u.zp) u.bp u.bp\n\ntheorem finish_is_reduced : u.finish.is_reduced :=\nby { dsimp [is_reduced], refl }\n\ntheorem finish_is_special (hs : u.is_special) : u.finish.is_special :=\nbegin\n  dsimp [is_special, finish] at hs \u22a2,\n  rw [add_mul _ _ u.y, add_comm _ (u.x * u.y), \u2190 hs],\n  ring\nend\n\ntheorem finish_v (hr : u.r = 0) : u.finish.v = u.v :=\nbegin\n  let ha : u.r + u.b * u.q = u.a := u.rq_eq,\n  rw [hr, zero_add] at ha,\n  ext,\n  { change (u.wp + 1) * u.b + ((u.wp + 1) * u.qp + u.x) * u.b = u.w * u.a + u.x * u.b,\n    have : u.wp + 1 = u.w := rfl, rw [this, \u2190 ha, u.qp_eq hr], ring },\n  { change u.y * u.b + (u.y * u.qp + u.z) * u.b = u.y * u.a + u.z * u.b,\n    rw [\u2190 ha, u.qp_eq hr], ring }\nend\n\n\/-- This is the main reduction step, which is used when u.r \u2260 0, or\n equivalently b does not divide a. -\/\ndef step : xgcd_type :=\nxgcd_type.mk (u.y * u.q + u.zp) u.y ((u.wp + 1) * u.q + u.x) u.wp u.bp (u.r - 1)\n\n\/-- We will apply the above step recursively.  The following result\n is used to ensure that the process terminates. -\/\ntheorem step_wf (hr : u.r \u2260 0) : sizeof u.step < sizeof u :=\nbegin\n  change u.r - 1 < u.bp,\n  have h\u2080 : (u.r - 1) + 1 = u.r := nat.succ_pred_eq_of_pos (nat.pos_of_ne_zero hr),\n  have h\u2081 : u.r < u.bp + 1 := nat.mod_lt (u.ap + 1) u.bp.succ_pos,\n  rw[\u2190 h\u2080] at h\u2081,\n  exact lt_of_succ_lt_succ h\u2081,\nend\n\ntheorem step_is_special (hs : u.is_special) : u.step.is_special :=\nbegin\n  dsimp [is_special, step] at hs \u22a2,\n  rw [mul_add, mul_comm u.y u.x, \u2190 hs],\n  ring\nend\n\n\/-- The reduction step does not change the product vector. -\/\ntheorem step_v (hr : u.r \u2260 0) : u.step.v = (u.v).swap :=\nbegin\n  let ha : u.r + u.b * u.q = u.a := u.rq_eq,\n  let hr : (u.r - 1) + 1 = u.r :=\n    (add_comm _ 1).trans (nat.add_sub_of_le (nat.pos_of_ne_zero hr)),\n  ext,\n  { change ((u.y * u.q + u.z) * u.b + u.y * (u.r - 1 + 1) : \u2115) = u.y * u.a + u.z * u.b,\n    rw [\u2190 ha, hr], ring },\n  { change ((u.w * u.q + u.x) * u.b + u.w * (u.r - 1 + 1) : \u2115) = u.w * u.a + u.x * u.b,\n    rw [\u2190 ha, hr], ring }\nend\n\n\/-- We can now define the full reduction function, which applies\n step as long as possible, and then applies finish. Note that the\n \"have\" statement puts a fact in the local context, and the\n equation compiler uses this fact to help construct the full\n definition in terms of well-founded recursion.  The same fact\n needs to be introduced in all the inductive proofs of properties\n given below. -\/\ndef reduce : xgcd_type \u2192 xgcd_type\n| u := dite (u.r = 0)\n    (\u03bb h, u.finish)\n    (\u03bb h, have sizeof u.step < sizeof u, from u.step_wf h,\n     flip (reduce u.step))\n\ntheorem reduce_a {u : xgcd_type} (h : u.r = 0) :\nu.reduce = u.finish := by { rw [reduce], simp only [], rw [if_pos h] }\n\ntheorem reduce_b {u : xgcd_type} (h : u.r \u2260 0) :\nu.reduce = u.step.reduce.flip := by { rw [reduce], simp only [], rw [if_neg h, step] }\n\ntheorem reduce_reduced : \u2200 (u : xgcd_type), u.reduce.is_reduced\n| u := dite (u.r = 0) (\u03bb h, by { rw [reduce_a h], exact u.finish_is_reduced })\n    (\u03bb h,  have sizeof u.step < sizeof u, from u.step_wf h,\n     by { rw [reduce_b h, flip_is_reduced], apply reduce_reduced })\n\ntheorem reduce_reduced' (u : xgcd_type) : u.reduce.is_reduced' :=\n(is_reduced_iff _).mp u.reduce_reduced\n\ntheorem reduce_special : \u2200 (u : xgcd_type), u.is_special \u2192 u.reduce.is_special\n| u := dite (u.r = 0)\n    (\u03bb h hs, by { rw [reduce_a h], exact u.finish_is_special hs })\n    (\u03bb h hs, have sizeof u.step < sizeof u, from u.step_wf h,\n     by { rw [reduce_b h],\n          let u' := u.step.reduce,\n          have : u'.is_special := reduce_special u.step (u.step_is_special hs),\n          exact (flip_is_special _).mpr (reduce_special _ (u.step_is_special hs)) })\n\ntheorem reduce_special' (u : xgcd_type) (hs : u.is_special) : u.reduce.is_special' :=\n(is_special_iff _).mp (u.reduce_special hs)\n\ntheorem reduce_v : \u2200 (u : xgcd_type), u.reduce.v = u.v\n| u := dite (u.r = 0)\n (\u03bb h, by {rw[reduce_a h, finish_v u h]})\n (\u03bb h, have sizeof u.step < sizeof u, from u.step_wf h,\n       by { rw[reduce_b h, flip_v, reduce_v (step u), step_v u h, prod.swap_swap] })\n\nend xgcd_type\n\nsection gcd\n\nvariables (a b : \u2115+)\n\ndef xgcd: xgcd_type := (xgcd_type.start a b).reduce\n\ndef gcd_d : \u2115+ := (xgcd a b).a\ndef gcd_w : \u2115+ := (xgcd a b).w\ndef gcd_x : \u2115  := (xgcd a b).x\ndef gcd_y : \u2115  := (xgcd a b).y\ndef gcd_z : \u2115+ := (xgcd a b).z\n\ndef gcd_a' : \u2115+ := succ_pnat ((xgcd a b).wp + (xgcd a b).x)\ndef gcd_b' : \u2115+ := succ_pnat ((xgcd a b).y + (xgcd a b).zp)\n\ntheorem gcd_a'_coe : ((gcd_a' a b) : \u2115) = (gcd_w a b) + (gcd_x a b) :=\nby { dsimp [gcd_a', gcd_w, xgcd_type.w],\n     rw [nat.succ_eq_add_one, nat.succ_eq_add_one], ring }\n\ntheorem gcd_b'_coe : ((gcd_b' a b) : \u2115) = (gcd_y a b) + (gcd_z a b) :=\nby { dsimp [gcd_b', gcd_z, xgcd_type.z],\n     rw [nat.succ_eq_add_one, nat.succ_eq_add_one], ring }\n\ntheorem gcd_props :\n let d := gcd_d a b,\n  w := gcd_w a b, x := gcd_x a b, y := gcd_y a b, z := gcd_z a b,\n  a' := gcd_a' a b, b' := gcd_b' a b in\n (w * z = succ_pnat (x * y) \u2227\n  (a = a' * d) \u2227 (b = b' * d) \u2227\n  z * a' = succ_pnat (x * b') \u2227 w * b' = succ_pnat (y * a') \u2227\n  (z * a : \u2115) = x * b + d \u2227 (w * b : \u2115) = y * a + d\n ) :=\nbegin\n  intros,\n  let u := (xgcd_type.start a b),\n  let ur := u.reduce,\n  have ha : d = ur.a := rfl,\n  have hb : d = ur.b := u.reduce_reduced',\n  have ha' : (a' : \u2115) = w + x := gcd_a'_coe a b,\n  have hb' : (b' : \u2115) = y + z := gcd_b'_coe a b,\n  have hdet : w * z = succ_pnat (x * y) := u.reduce_special' rfl,\n  split, exact hdet,\n  have hdet' : ((w * z) : \u2115) = x * y + 1 :=\n    by { rw [\u2190 mul_coe, hdet, succ_pnat_coe] },\n  have huv : u.v = \u27e8a, b\u27e9 := (xgcd_type.start_v a b),\n  let hv : prod.mk (w * d + x * ur.b : \u2115) (y * d + z * ur.b : \u2115) = \u27e8a, b\u27e9 :=\n   u.reduce_v.trans (xgcd_type.start_v a b),\n  rw [\u2190 hb, \u2190 add_mul, \u2190 add_mul, \u2190 ha', \u2190 hb'] at hv,\n  have ha'' : (a : \u2115) = a' * d := (congr_arg prod.fst hv).symm,\n  have hb'' : (b : \u2115) = b' * d := (congr_arg prod.snd hv).symm,\n  split, exact eq ha'', split, exact eq hb'',\n  have hza' : (z * a' : \u2115) = x * b' + 1,\n  by { rw [ha', hb', mul_add, mul_add, mul_comm (z : \u2115), hdet'], ring },\n  have hwb' : (w * b' : \u2115) = y * a' + 1,\n  by { rw [ha', hb', mul_add, mul_add, hdet'], ring },\n  split,\n  { apply eq, rw [succ_pnat_coe, nat.succ_eq_add_one, mul_coe, hza'] },\n  split,\n  { apply eq, rw [succ_pnat_coe, nat.succ_eq_add_one, mul_coe, hwb'] },\n  rw [ha'', hb''], repeat { rw [\u2190 mul_assoc] }, rw [hza', hwb'],\n  split; ring,\nend\n\ntheorem gcd_eq : gcd_d a b = gcd a b :=\nbegin\n  rcases gcd_props a b with \u27e8h\u2080, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085, h\u2086\u27e9,\n  apply dvd_antisymm,\n  { apply dvd_gcd,\n    exact dvd_intro (gcd_a' a b) (h\u2081.trans (mul_comm _ _)).symm,\n    exact dvd_intro (gcd_b' a b) (h\u2082.trans (mul_comm _ _)).symm},\n  { have h\u2087 := calc\n     ((gcd a b) : \u2115) \u2223 a : nat.gcd_dvd_left a b\n      ... \u2223 (gcd_z a b) * a : dvd_mul_left _ _,\n    have h\u2088 := calc\n     ((gcd a b) : \u2115) \u2223 b : nat.gcd_dvd_right a b\n      ... \u2223 (gcd_x a b) * b : dvd_mul_left _ _,\n   rw[h\u2085] at h\u2087,\n   exact (nat.dvd_add_iff_right h\u2088).mpr h\u2087}\nend\n\ntheorem gcd_det_eq :\n  (gcd_w a b) * (gcd_z a b) = succ_pnat ((gcd_x a b) * (gcd_y a b)) :=\n(gcd_props a b).1\n\ntheorem gcd_a_eq : a = (gcd_a' a b) * (gcd a b) :=\n(gcd_eq a b) \u25b8 (gcd_props a b).2.1\n\ntheorem gcd_b_eq : b = (gcd_b' a b) * (gcd a b) :=\n(gcd_eq a b) \u25b8 (gcd_props a b).2.2.1\n\ntheorem gcd_rel_left' :\n  (gcd_z a b) * (gcd_a' a b) = succ_pnat ((gcd_x a b) * (gcd_b' a b)) :=\n(gcd_props a b).2.2.2.1\n\ntheorem gcd_rel_right' :\n  (gcd_w a b) * (gcd_b' a b) = succ_pnat ((gcd_y a b) * (gcd_a' a b)) :=\n(gcd_props a b).2.2.2.2.1\n\ntheorem gcd_rel_left :\n  ((gcd_z a b) * a : \u2115) = (gcd_x a b) * b + (gcd a b) :=\n(gcd_eq a b) \u25b8 (gcd_props a b).2.2.2.2.2.1\n\ntheorem gcd_rel_right :\n  ((gcd_w a b) * b : \u2115) = (gcd_y a b) * a + (gcd a b) :=\n(gcd_eq a b) \u25b8 (gcd_props a b).2.2.2.2.2.2\n\nend gcd\nend pnat\n","avg_line_length":35.7486338798,"max_line_length":87,"alphanum_fraction":0.592097218}
-{"size":1730,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Sebastian Ullrich\n-\/\nimport Lean.Parser.Module\nnamespace Lean.Elab\n\ndef headerToImports (header : Syntax) : List Import :=\n  let imports := if header[0].isNone then [{ module := `Init : Import }] else []\n  imports ++ header[1].getArgs.toList.map fun stx =>\n    -- `stx` is of the form `(Module.import \"import\" \"runtime\"? id)\n    let runtime := !stx[1].isNone\n    let id      := stx[2].getId\n    { module := id, runtimeOnly := runtime }\n\ndef processHeader (header : Syntax) (opts : Options) (messages : MessageLog) (inputCtx : Parser.InputContext) (trustLevel : UInt32 := 0)\n    : IO (Environment \u00d7 MessageLog) := do\n  try\n    let env \u2190 importModules (headerToImports header) opts trustLevel\n    pure (env, messages)\n  catch e =>\n    let env \u2190 mkEmptyEnvironment\n    let spos := header.getPos.getD 0\n    let pos  := inputCtx.fileMap.toPosition spos\n    pure (env, messages.add { fileName := inputCtx.fileName, data := toString e, pos := pos })\n\ndef parseImports (input : String) (fileName : Option String := none) : IO (List Import \u00d7 Position \u00d7 MessageLog) := do\n  let fileName := fileName.getD \"<input>\"\n  let inputCtx := Parser.mkInputContext input fileName\n  let (header, parserState, messages) \u2190 Parser.parseHeader inputCtx\n  pure (headerToImports header, inputCtx.fileMap.toPosition parserState.pos, messages)\n\n@[export lean_print_imports]\ndef printImports (input : String) (fileName : Option String) : IO Unit := do\n  let (deps, pos, log) \u2190 parseImports input fileName\n  for dep in deps do\n    let fname \u2190 findOLean dep.module\n    IO.println fname\n\nend Lean.Elab\n","avg_line_length":41.1904761905,"max_line_length":136,"alphanum_fraction":0.7028901734}
-{"size":14318,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-\/\n\nimport order.filter.extr topology.continuous_on\n\n\/-! # Local extrema of functions on topological spaces\n\n## Main definitions\n\nThis file defines special versions of `is_*_filter f a l`, `*=min\/max\/extr`,\nfrom `order\/filter\/extr` for two kinds of filters: `nhds_within` and `nhds`.\nThese versions are called `is_local_*_on` and `is_local_*`, respectively.\n\n## Main statements\n\nMany lemmas in this file restate those from `order\/filter\/extr`, and you can find\na detailed documentation there. These convenience lemmas are provided only to make the dot notation\nreturn propositions of expected types, not just `is_*_filter`.\n\nHere is the list of statements specific to these two types of filters:\n\n* `is_local_*.on`, `is_local_*_on.on_subset`: restrict to a subset;\n* `is_local_*_on.inter` : intersect the set with another one;\n* `is_*_on.localize` : a global extremum is a local extremum too.\n* `is_[local_]*_on.is_local_*` : if we have `is_local_*_on f s a` and `s \u2208 \ud835\udcdd a`,\n  then we have `is_local_* f a`.\n\n-\/\n\nuniverses u v w x\n\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type x} [topological_space \u03b1]\n\nopen set filter\nopen_locale topological_space\n\nsection preorder\n\nvariables [preorder \u03b2] [preorder \u03b3] (f : \u03b1 \u2192 \u03b2) (s : set \u03b1) (a : \u03b1)\n\n\/-- `is_local_min_on f s a` means that `f a \u2264 f x` for all `x \u2208 s` in some neighborhood of `a`. -\/\ndef is_local_min_on := is_min_filter f (nhds_within a s) a\n\n\/-- `is_local_max_on f s a` means that `f x \u2264 f a` for all `x \u2208 s` in some neighborhood of `a`. -\/\ndef is_local_max_on := is_max_filter f (nhds_within a s) a\n\n\/-- `is_local_extr_on f s a` means `is_local_min_on f s a \u2228 is_local_max_on f s a`. -\/\ndef is_local_extr_on := is_extr_filter f (nhds_within a s) a\n\n\/-- `is_local_min f a` means that `f a \u2264 f x` for all `x` in some neighborhood of `a`. -\/\ndef is_local_min := is_min_filter f (\ud835\udcdd a) a\n\n\/-- `is_local_max f a` means that `f x \u2264 f a` for all `x \u2208 s` in some neighborhood of `a`. -\/\ndef is_local_max := is_max_filter f (\ud835\udcdd a) a\n\n\/-- `is_local_extr_on f s a` means `is_local_min_on f s a \u2228 is_local_max_on f s a`. -\/\ndef is_local_extr := is_extr_filter f (\ud835\udcdd a) a\n\nvariables {f s a}\n\nlemma is_local_extr_on.elim {p : Prop} :\n  is_local_extr_on f s a \u2192 (is_local_min_on f s a \u2192 p) \u2192 (is_local_max_on f s a \u2192 p) \u2192 p :=\nor.elim\n\nlemma is_local_extr.elim {p : Prop} :\n  is_local_extr f a \u2192 (is_local_min f a \u2192 p) \u2192 (is_local_max f a \u2192 p) \u2192 p :=\nor.elim\n\n\/-! ### Restriction to (sub)sets -\/\n\nlemma is_local_min.on (h : is_local_min f a) (s) : is_local_min_on f s a :=\nh.filter_inf _\n\nlemma is_local_max.on (h : is_local_max f a) (s) : is_local_max_on f s a :=\nh.filter_inf _\n\nlemma is_local_extr.on (h : is_local_extr f a) (s) : is_local_extr_on f s a :=\nh.filter_inf _\n\nlemma is_local_min_on.on_subset {t : set \u03b1} (hf : is_local_min_on f t a) (h : s \u2286 t) :\n  is_local_min_on f s a :=\nhf.filter_mono $ nhds_within_mono a h\n\nlemma is_local_max_on.on_subset {t : set \u03b1} (hf : is_local_max_on f t a) (h : s \u2286 t) :\n  is_local_max_on f s a :=\nhf.filter_mono $ nhds_within_mono a h\n\nlemma is_local_extr_on.on_subset {t : set \u03b1} (hf : is_local_extr_on f t a) (h : s \u2286 t) :\n  is_local_extr_on f s a :=\nhf.filter_mono $ nhds_within_mono a h\n\nlemma is_local_min_on.inter (hf : is_local_min_on f s a) (t) : is_local_min_on f (s \u2229 t) a :=\nhf.on_subset (inter_subset_left s t)\n\nlemma is_local_max_on.inter (hf : is_local_max_on f s a) (t) : is_local_max_on f (s \u2229 t) a :=\nhf.on_subset (inter_subset_left s t)\n\nlemma is_local_extr_on.inter (hf : is_local_extr_on f s a) (t) : is_local_extr_on f (s \u2229 t) a :=\nhf.on_subset (inter_subset_left s t)\n\nlemma is_min_on.localize (hf : is_min_on f s a) : is_local_min_on f s a :=\nhf.filter_mono $ inf_le_right\n\nlemma is_max_on.localize (hf : is_max_on f s a) : is_local_max_on f s a :=\nhf.filter_mono $ inf_le_right\n\nlemma is_extr_on.localize (hf : is_extr_on f s a) : is_local_extr_on f s a :=\nhf.filter_mono $ inf_le_right\n\nlemma is_local_min_on.is_local_min (hf : is_local_min_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_min f a :=\nhave \ud835\udcdd a \u2264 principal s, from le_principal_iff.2 hs,\nhf.filter_mono $ le_inf (le_refl _) this\n\nlemma is_local_max_on.is_local_max (hf : is_local_max_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_max f a :=\nhave \ud835\udcdd a \u2264 principal s, from le_principal_iff.2 hs,\nhf.filter_mono $ le_inf (le_refl _) this\n\nlemma is_local_extr_on.is_local_extr (hf : is_local_extr_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_extr f a :=\nhf.elim (\u03bb hf, (hf.is_local_min hs).is_extr) (\u03bb hf, (hf.is_local_max hs).is_extr)\n\nlemma is_min_on.is_local_min (hf : is_min_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_min f a :=\nhf.localize.is_local_min hs\n\nlemma is_max_on.is_local_max (hf : is_max_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_max f a :=\nhf.localize.is_local_max hs\n\nlemma is_extr_on.is_local_extr (hf : is_extr_on f s a) (hs : s \u2208 \ud835\udcdd a) : is_local_extr f a :=\nhf.localize.is_local_extr hs\n\n\/-! ### Constant -\/\n\nlemma is_local_min_on_const {b : \u03b2} : is_local_min_on (\u03bb _, b) s a := is_min_filter_const\nlemma is_local_max_on_const {b : \u03b2} : is_local_max_on (\u03bb _, b) s a := is_max_filter_const\nlemma is_local_extr_on_const {b : \u03b2} : is_local_extr_on (\u03bb _, b) s a := is_extr_filter_const\n\nlemma is_local_min_const {b : \u03b2} : is_local_min (\u03bb _, b) a := is_min_filter_const\nlemma is_local_max_const {b : \u03b2} : is_local_max (\u03bb _, b) a := is_max_filter_const\nlemma is_local_extr_const {b : \u03b2} : is_local_extr (\u03bb _, b) a := is_extr_filter_const\n\n\/-! ### Composition with (anti)monotone functions -\/\n\nlemma is_local_min.comp_mono (hf : is_local_min f a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_min (g \u2218 f) a :=\nhf.comp_mono hg\n\nlemma is_local_max.comp_mono (hf : is_local_max f a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_max (g \u2218 f) a :=\nhf.comp_mono hg\n\nlemma is_local_extr.comp_mono (hf : is_local_extr f a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_extr (g \u2218 f) a :=\nhf.comp_mono hg\n\nlemma is_local_min.comp_antimono (hf : is_local_min f a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_max (g \u2218 f) a :=\nhf.comp_antimono hg\n\nlemma is_local_max.comp_antimono (hf : is_local_max f a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_min (g \u2218 f) a :=\nhf.comp_antimono hg\n\nlemma is_local_extr.comp_antimono (hf : is_local_extr f a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_extr (g \u2218 f) a :=\nhf.comp_antimono hg\n\nlemma is_local_min_on.comp_mono (hf : is_local_min_on f s a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_min_on (g \u2218 f) s a :=\nhf.comp_mono hg\n\nlemma is_local_max_on.comp_mono (hf : is_local_max_on f s a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_max_on (g \u2218 f) s a :=\nhf.comp_mono hg\n\nlemma is_local_extr_on.comp_mono (hf : is_local_extr_on f s a) {g : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n  is_local_extr_on (g \u2218 f) s a :=\nhf.comp_mono hg\n\nlemma is_local_min_on.comp_antimono (hf : is_local_min_on f s a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_max_on (g \u2218 f) s a :=\nhf.comp_antimono hg\n\nlemma is_local_max_on.comp_antimono (hf : is_local_max_on f s a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_min_on (g \u2218 f) s a :=\nhf.comp_antimono hg\n\nlemma is_local_extr_on.comp_antimono (hf : is_local_extr_on f s a) {g : \u03b2 \u2192 \u03b3}\n  (hg : \u2200 \u2983x y\u2984, x \u2264 y \u2192 g y \u2264 g x) :\n  is_local_extr_on (g \u2218 f) s a :=\nhf.comp_antimono hg\n\nlemma is_local_min.bicomp_mono [preorder \u03b4] {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (hop : ((\u2264) \u21d2 (\u2264) \u21d2 (\u2264)) op op)\n  (hf : is_local_min f a) {g : \u03b1 \u2192 \u03b3} (hg : is_local_min g a) :\n  is_local_min (\u03bb x, op (f x) (g x)) a :=\nhf.bicomp_mono hop hg\n\nlemma is_local_max.bicomp_mono [preorder \u03b4] {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (hop : ((\u2264) \u21d2 (\u2264) \u21d2 (\u2264)) op op)\n  (hf : is_local_max f a) {g : \u03b1 \u2192 \u03b3} (hg : is_local_max g a) :\n  is_local_max (\u03bb x, op (f x) (g x)) a :=\nhf.bicomp_mono hop hg\n\n-- No `extr` version because we need `hf` and `hg` to be of the same kind\n\nlemma is_local_min_on.bicomp_mono [preorder \u03b4] {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (hop : ((\u2264) \u21d2 (\u2264) \u21d2 (\u2264)) op op)\n  (hf : is_local_min_on f s a) {g : \u03b1 \u2192 \u03b3} (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, op (f x) (g x)) s a :=\nhf.bicomp_mono hop hg\n\nlemma is_local_max_on.bicomp_mono [preorder \u03b4] {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (hop : ((\u2264) \u21d2 (\u2264) \u21d2 (\u2264)) op op)\n  (hf : is_local_max_on f s a) {g : \u03b1 \u2192 \u03b3} (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, op (f x) (g x)) s a :=\nhf.bicomp_mono hop hg\n\n\/-! ### Composition with `continuous_at` -\/\n\nlemma is_local_min.comp_continuous [topological_space \u03b4] {g : \u03b4 \u2192 \u03b1} {b : \u03b4}\n  (hf : is_local_min f (g b)) (hg : continuous_at g b) :\n  is_local_min (f \u2218 g) b :=\nhg hf\n\nlemma is_local_max.comp_continuous [topological_space \u03b4] {g : \u03b4 \u2192 \u03b1} {b : \u03b4}\n  (hf : is_local_max f (g b)) (hg : continuous_at g b) :\n  is_local_max (f \u2218 g) b :=\nhg hf\n\nlemma is_local_extr.comp_continuous [topological_space \u03b4] {g : \u03b4 \u2192 \u03b1} {b : \u03b4}\n  (hf : is_local_extr f (g b)) (hg : continuous_at g b) :\n  is_local_extr (f \u2218 g) b :=\nhf.comp_tendsto hg\n\nlemma is_local_min.comp_continuous_on [topological_space \u03b4] {s : set \u03b4} {g : \u03b4 \u2192 \u03b1} {b : \u03b4}\n  (hf : is_local_min f (g b)) (hg : continuous_on g s) (hb : b \u2208 s) :\n  is_local_min_on (f \u2218 g) s b :=\nhf.comp_tendsto (hg b hb)\n\nlemma is_local_max.comp_continuous_on [topological_space \u03b4] {s : set \u03b4} {g : \u03b4 \u2192 \u03b1} {b : \u03b4}\n  (hf : is_local_max f (g b)) (hg : continuous_on g s) (hb : b \u2208 s) :\n  is_local_max_on (f \u2218 g) s b :=\nhf.comp_tendsto (hg b hb)\n\nlemma is_local_extr.comp_continuous_on [topological_space \u03b4] {s : set \u03b4} (g : \u03b4 \u2192 \u03b1) {b : \u03b4}\n  (hf : is_local_extr f (g b)) (hg : continuous_on g s) (hb : b \u2208 s) :\n  is_local_extr_on (f \u2218 g) s b :=\nhf.elim (\u03bb hf, (hf.comp_continuous_on hg hb).is_extr)\n  (\u03bb hf, (is_local_max.comp_continuous_on hf hg hb).is_extr)\n\nend preorder\n\n\/-! ### Pointwise addition -\/\nsection ordered_add_comm_monoid\n\nvariables [ordered_add_comm_monoid \u03b2] {f g : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} {l : filter \u03b1}\n\nlemma is_local_min.add (hf : is_local_min f a) (hg : is_local_min g a) :\n  is_local_min (\u03bb x, f x + g x) a :=\nhf.add hg\n\nlemma is_local_max.add (hf : is_local_max f a) (hg : is_local_max g a) :\n  is_local_max (\u03bb x, f x + g x) a :=\nhf.add hg\n\nlemma is_local_min_on.add (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, f x + g x) s a :=\nhf.add hg\n\nlemma is_local_max_on.add (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, f x + g x) s a :=\nhf.add hg\n\nend ordered_add_comm_monoid\n\n\/-! ### Pointwise negation and subtraction -\/\n\nsection ordered_add_comm_group\n\nvariables [ordered_add_comm_group \u03b2] {f g : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} {l : filter \u03b1}\n\nlemma is_local_min.neg (hf : is_local_min f a) : is_local_max (\u03bb x, -f x) a :=\nhf.neg\n\nlemma is_local_max.neg (hf : is_local_max f a) : is_local_min (\u03bb x, -f x) a :=\nhf.neg\n\nlemma is_local_extr.neg (hf : is_local_extr f a) : is_local_extr (\u03bb x, -f x) a :=\nhf.neg\n\nlemma is_local_min_on.neg (hf : is_local_min_on f s a) : is_local_max_on (\u03bb x, -f x) s a :=\nhf.neg\n\nlemma is_local_max_on.neg (hf : is_local_max_on f s a) : is_local_min_on (\u03bb x, -f x) s a :=\nhf.neg\n\nlemma is_local_extr_on.neg (hf : is_local_extr_on f s a) : is_local_extr_on (\u03bb x, -f x) s a :=\nhf.neg\n\nlemma is_local_min.sub (hf : is_local_min f a) (hg : is_local_max g a) :\n  is_local_min (\u03bb x, f x - g x) a :=\nhf.sub hg\n\nlemma is_local_max.sub (hf : is_local_max f a) (hg : is_local_min g a) :\n  is_local_max (\u03bb x, f x - g x) a :=\nhf.sub hg\n\nlemma is_local_min_on.sub (hf : is_local_min_on f s a) (hg : is_local_max_on g s a) :\n  is_local_min_on (\u03bb x, f x - g x) s a :=\nhf.sub hg\n\nlemma is_local_max_on.sub (hf : is_local_max_on f s a) (hg : is_local_min_on g s a) :\n  is_local_max_on (\u03bb x, f x - g x) s a :=\nhf.sub hg\n\nend ordered_add_comm_group\n\n\n\/-! ### Pointwise `sup`\/`inf` -\/\n\nsection semilattice_sup\n\nvariables [semilattice_sup \u03b2] {f g : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} {l : filter \u03b1}\n\nlemma is_local_min.sup (hf : is_local_min f a) (hg : is_local_min g a) :\n  is_local_min (\u03bb x, f x \u2294 g x) a :=\nhf.sup hg\n\nlemma is_local_max.sup (hf : is_local_max f a) (hg : is_local_max g a) :\n  is_local_max (\u03bb x, f x \u2294 g x) a :=\nhf.sup hg\n\nlemma is_local_min_on.sup (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, f x \u2294 g x) s a :=\nhf.sup hg\n\nlemma is_local_max_on.sup (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, f x \u2294 g x) s a :=\nhf.sup hg\n\nend semilattice_sup\n\nsection semilattice_inf\n\nvariables [semilattice_inf \u03b2] {f g : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} {l : filter \u03b1}\n\nlemma is_local_min.inf (hf : is_local_min f a) (hg : is_local_min g a) :\n  is_local_min (\u03bb x, f x \u2293 g x) a :=\nhf.inf hg\n\nlemma is_local_max.inf (hf : is_local_max f a) (hg : is_local_max g a) :\n  is_local_max (\u03bb x, f x \u2293 g x) a :=\nhf.inf hg\n\nlemma is_local_min_on.inf (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, f x \u2293 g x) s a :=\nhf.inf hg\n\nlemma is_local_max_on.inf (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, f x \u2293 g x) s a :=\nhf.inf hg\n\nend semilattice_inf\n\n\/-! ### Pointwise `min`\/`max` -\/\n\nsection decidable_linear_order\n\nvariables [decidable_linear_order \u03b2] {f g : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} {l : filter \u03b1}\n\nlemma is_local_min.min (hf : is_local_min f a) (hg : is_local_min g a) :\n  is_local_min (\u03bb x, min (f x) (g x)) a :=\nhf.min hg\n\nlemma is_local_max.min (hf : is_local_max f a) (hg : is_local_max g a) :\n  is_local_max (\u03bb x, min (f x) (g x)) a :=\nhf.min hg\n\nlemma is_local_min_on.min (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, min (f x) (g x)) s a :=\nhf.min hg\n\nlemma is_local_max_on.min (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, min (f x) (g x)) s a :=\nhf.min hg\n\nlemma is_local_min.max (hf : is_local_min f a) (hg : is_local_min g a) :\n  is_local_min (\u03bb x, max (f x) (g x)) a :=\nhf.max hg\n\nlemma is_local_max.max (hf : is_local_max f a) (hg : is_local_max g a) :\n  is_local_max (\u03bb x, max (f x) (g x)) a :=\nhf.max hg\n\nlemma is_local_min_on.max (hf : is_local_min_on f s a) (hg : is_local_min_on g s a) :\n  is_local_min_on (\u03bb x, max (f x) (g x)) s a :=\nhf.max hg\n\nlemma is_local_max_on.max (hf : is_local_max_on f s a) (hg : is_local_max_on g s a) :\n  is_local_max_on (\u03bb x, max (f x) (g x)) s a :=\nhf.max hg\n\nend decidable_linear_order\n","avg_line_length":35.1793611794,"max_line_length":104,"alphanum_fraction":0.6679005448}
-{"size":3356,"ext":"lean","lang":"Lean","max_stars_count":2.0,"content":"\/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-\/\nimport data.nat.pow\n\n\/-!\n# Natural number logarithm\n\nThis file defines `log b n`, the logarithm of `n` with base `b`, to be the largest `k` such that\n`b ^ k \u2264 n`.\n\n-\/\nnamespace nat\n\n\/-- `log b n`, is the logarithm of natural number `n` in base `b`. It returns the largest `k : \u2115`\nsuch that `b^k \u2264 n`, so if `b^k = n`, it returns exactly `k`. -\/\n@[pp_nodot] def log (b : \u2115) : \u2115 \u2192 \u2115\n| n :=\n  if h : b \u2264 n \u2227 1 < b then\n    have n \/ b < n,\n      from div_lt_self\n        (nat.lt_of_lt_of_le (lt_trans zero_lt_one h.2) h.1) h.2,\n    log (n \/ b) + 1\n  else 0\n\nlemma log_eq_zero {b n : \u2115} (hnb : n < b \u2228 b \u2264 1) : log b n = 0 :=\nbegin\n  rw [or_iff_not_and_not, not_lt, not_le] at hnb,\n  rw [log, \u2190ite_not, if_pos hnb],\nend\n\nlemma log_eq_zero_of_lt {b n : \u2115} (hn : n < b) : log b n = 0 :=\nlog_eq_zero $ or.inl hn\n\nlemma log_eq_zero_of_le {b n : \u2115} (hb : b \u2264 1) : log b n = 0 :=\nlog_eq_zero $ or.inr hb\n\nlemma log_zero_eq_zero {b : \u2115} : log b 0 = 0 :=\nby { rw log, cases b; refl }\n\nlemma log_one_eq_zero {b : \u2115} : log b 1 = 0 :=\nif h : b \u2264 1 then\n  log_eq_zero_of_le h\nelse\n  log_eq_zero_of_lt (not_le.mp h)\n\nlemma log_b_zero_eq_zero {n : \u2115} : log 0 n = 0 :=\nlog_eq_zero_of_le zero_le_one\n\nlemma log_b_one_eq_zero {n : \u2115} : log 1 n = 0 :=\nlog_eq_zero_of_le rfl.ge\n\nlemma pow_le_iff_le_log (x y : \u2115) {b} (hb : 1 < b) (hy : 1 \u2264 y) :\n  b^x \u2264 y \u2194 x \u2264 log b y :=\nbegin\n  induction y using nat.strong_induction_on with y ih\n    generalizing x,\n  rw [log], split_ifs,\n  { have h'' : 0 < b := lt_of_le_of_lt (zero_le _) hb,\n    cases h with h\u2080 h\u2081,\n    rw [\u2190 nat.sub_le_right_iff_le_add,\u2190 ih (y \/ b),\n          le_div_iff_mul_le _ _ h'',\u2190 pow_succ'],\n    { cases x; simp [h\u2080,hy] },\n    { apply div_lt_self; assumption },\n    { rwa [le_div_iff_mul_le _ _ h'',one_mul], } },\n  { replace h := lt_of_not_ge (not_and'.1 h hb),\n    split; intros h',\n    { have := lt_of_le_of_lt h' h,\n      apply le_of_succ_le_succ,\n      change x < 1, rw [\u2190 pow_lt_iff_lt_right hb,pow_one],\n      exact this },\n    { replace h' := le_antisymm h' (zero_le _),\n      rw [h',pow_zero], exact hy} },\nend\n\nlemma log_pow (b x : \u2115) (hb : 1 < b) : log b (b ^ x) = x :=\neq_of_forall_le_iff $ \u03bb z,\nby { rwa [\u2190 pow_le_iff_le_log _ _ hb,pow_le_iff_le_right],\n     rw \u2190 pow_zero b, apply pow_le_pow_of_le_right,\n     apply lt_of_le_of_lt (zero_le _) hb, apply zero_le }\n\nlemma pow_succ_log_gt_self (b x : \u2115) (hb : 1 < b) (hy : 1 \u2264 x) :\n  x < b ^ succ (log b x) :=\nbegin\n  apply lt_of_not_ge,\n  rw [(\u2265),pow_le_iff_le_log _ _ hb hy],\n  apply not_le_of_lt, apply lt_succ_self,\nend\n\nlemma pow_log_le_self (b x : \u2115) (hb : 1 < b) (hx : 1 \u2264 x) : b ^ log b x \u2264 x :=\nby rw [pow_le_iff_le_log _ _ hb hx]\n\nlemma log_le_log_of_le {b n m : \u2115} (h : n \u2264 m) : log b n \u2264 log b m :=\nbegin\n  cases le_or_lt b 1 with hb hb,\n  { rw log_eq_zero_of_le hb, exact zero_le _ },\n  { cases eq_zero_or_pos n with hn hn,\n    { rw [hn, log_zero_eq_zero], exact zero_le _ },\n    { rw \u2190pow_le_iff_le_log _ _ hb (lt_of_lt_of_le hn h),\n      exact (pow_log_le_self b n hb hn).trans h } }\nend\n\nlemma log_le_log_succ {b n : \u2115} : log b n \u2264 log b n.succ :=\nlog_le_log_of_le $ le_succ n\n\nlemma log_mono {b : \u2115} : monotone (\u03bb n : \u2115, log b n) :=\nmonotone_of_monotone_nat $ \u03bb n, log_le_log_succ\n\nend nat\n","avg_line_length":29.9642857143,"max_line_length":97,"alphanum_fraction":0.6257449344}
-{"size":17311,"ext":"hlean","lang":"Lean","max_stars_count":130.0,"content":"\/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-\/\nimport types.bool tools.helper_tactics\nopen bool eq eq.ops decidable helper_tactics\n\nnamespace pos_num\n  theorem succ_not_is_one (a : pos_num) : is_one (succ a) = ff :=\n  pos_num.rec_on a rfl (take n iH, rfl) (take n iH, rfl)\n\n  theorem succ_one                : succ one = bit0 one\n  theorem succ_bit1 (a : pos_num) : succ (bit1 a) = bit0 (succ a)\n  theorem succ_bit0 (a : pos_num) : succ (bit0 a) = bit1 a\n\n  theorem ne_of_bit0_ne_bit0 {a b : pos_num} (H\u2081 : bit0 a \u2260 bit0 b) : a \u2260 b :=\n  suppose a = b,\n  absurd rfl (this \u25b8 H\u2081)\n\n  theorem ne_of_bit1_ne_bit1 {a b : pos_num} (H\u2081 : bit1 a \u2260 bit1 b) : a \u2260 b :=\n  suppose a = b,\n  absurd rfl (this \u25b8 H\u2081)\n\n  theorem pred_succ : \u03a0 (a : pos_num), pred (succ a) = a\n  | one      := rfl\n  | (bit0 a) := by rewrite succ_bit0\n  | (bit1 a) :=\n      calc\n        pred (succ (bit1 a)) = cond (is_one (succ a)) one (bit1 (pred (succ a))) : rfl\n                       ...   = cond ff one (bit1 (pred (succ a)))                : succ_not_is_one\n                       ...   = bit1 (pred (succ a))                              : rfl\n                       ...   = bit1 a                                            : pred_succ a\n\n  section\n  variables (a b : pos_num)\n\n  theorem one_add_one     : one + one = bit0 one\n  theorem one_add_bit0    : one + (bit0 a) = bit1 a\n  theorem one_add_bit1    : one + (bit1 a) = succ (bit1 a)\n  theorem bit0_add_one    : (bit0 a) + one = bit1 a\n  theorem bit1_add_one    : (bit1 a) + one = succ (bit1 a)\n  theorem bit0_add_bit0   : (bit0 a) + (bit0 b) = bit0 (a + b)\n  theorem bit0_add_bit1   : (bit0 a) + (bit1 b) = bit1 (a + b)\n  theorem bit1_add_bit0   : (bit1 a) + (bit0 b) = bit1 (a + b)\n  theorem bit1_add_bit1   : (bit1 a) + (bit1 b) = succ (bit1 (a + b))\n  theorem one_mul         : one * a = a\n  end\n\n  theorem mul_one         : \u03a0 a, a * one = a\n  | one      := rfl\n  | (bit1 n) :=\n      calc bit1 n * one = bit0 (n * one) + one : rfl\n                   ...  = bit0 n + one         : mul_one n\n                   ...  = bit1 n               : bit0_add_one\n  | (bit0 n) :=\n      calc bit0 n * one = bit0 (n * one)       : rfl\n                    ... = bit0 n               : mul_one n\n\n  theorem decidable_eq [instance] : \u03a0 (a b : pos_num), decidable (a = b)\n  | one      one      := inl rfl\n  | one      (bit0 b) := inr (by contradiction)\n  | one      (bit1 b) := inr (by contradiction)\n  | (bit0 a) one      := inr (by contradiction)\n  | (bit0 a) (bit0 b) :=\n    match decidable_eq a b with\n    | inl H\u2081  := inl (by rewrite H\u2081)\n    | inr H\u2081  := inr (by intro H; injection H; contradiction)\n    end\n  | (bit0 a) (bit1 b) := inr (by contradiction)\n  | (bit1 a) one      := inr (by contradiction)\n  | (bit1 a) (bit0 b) := inr (by contradiction)\n  | (bit1 a) (bit1 b) :=\n    match decidable_eq a b with\n    | inl H\u2081  := inl (by rewrite H\u2081)\n    | inr H\u2081  := inr (by intro H; injection H; contradiction)\n    end\n\n  local notation a < b         := (lt a b = tt)\n  local notation a ` \u226e `:50 b:50 := (lt a b = ff)\n\n  theorem lt_one_right_eq_ff : \u03a0 a : pos_num, a \u226e one\n  | one      := rfl\n  | (bit0 a) := rfl\n  | (bit1 a) := rfl\n\n  theorem lt_one_succ_eq_tt : \u03a0 a : pos_num, one < succ a\n  | one      := rfl\n  | (bit0 a) := rfl\n  | (bit1 a) := rfl\n\n  theorem lt_of_lt_bit0_bit0 {a b : pos_num} (H : bit0 a < bit0 b) : a < b := H\n  theorem lt_of_lt_bit0_bit1 {a b : pos_num} (H : bit1 a < bit0 b) : a < b := H\n  theorem lt_of_lt_bit1_bit1 {a b : pos_num} (H : bit1 a < bit1 b) : a < b := H\n  theorem lt_of_lt_bit1_bit0 {a b : pos_num} (H : bit0 a < bit1 b) : a < succ b := H\n\n  theorem lt_bit0_bit0_eq_lt (a b : pos_num) : lt (bit0 a) (bit0 b) = lt a b :=\n  rfl\n\n  theorem lt_bit1_bit1_eq_lt (a b : pos_num) : lt (bit1 a) (bit1 b) = lt a b :=\n  rfl\n\n  theorem lt_bit1_bit0_eq_lt (a b : pos_num) : lt (bit1 a) (bit0 b) = lt a b :=\n  rfl\n\n  theorem lt_bit0_bit1_eq_lt_succ (a b : pos_num) : lt (bit0 a) (bit1 b) = lt a (succ b) :=\n  rfl\n\n  theorem lt_irrefl : \u03a0 (a : pos_num), a \u226e a\n  | one      := rfl\n  | (bit0 a) :=\n    begin\n      rewrite lt_bit0_bit0_eq_lt, apply lt_irrefl\n    end\n  | (bit1 a) :=\n    begin\n      rewrite lt_bit1_bit1_eq_lt, apply lt_irrefl\n    end\n\n  theorem ne_of_lt_eq_tt : \u03a0 {a b : pos_num}, a < b \u2192 a = b \u2192 empty\n  | one      \u231eone\u231f      H\u2081 (eq.refl one)      := absurd H\u2081 ff_ne_tt\n  | (bit0 a) \u231e(bit0 a)\u231f H\u2081 (eq.refl (bit0 a)) :=\n    begin\n      rewrite lt_bit0_bit0_eq_lt at H\u2081,\n      apply ne_of_lt_eq_tt H\u2081 (eq.refl a)\n    end\n  | (bit1 a) \u231e(bit1 a)\u231f H\u2081 (eq.refl (bit1 a)) :=\n    begin\n      rewrite lt_bit1_bit1_eq_lt at H\u2081,\n      apply ne_of_lt_eq_tt H\u2081 (eq.refl a)\n    end\n\n  theorem lt_base : \u03a0 a : pos_num, a < succ a\n  | one      := rfl\n  | (bit0 a) :=\n    begin\n      rewrite [succ_bit0, lt_bit0_bit1_eq_lt_succ],\n      apply lt_base\n    end\n  | (bit1 a) :=\n    begin\n      rewrite [succ_bit1, lt_bit1_bit0_eq_lt],\n      apply lt_base\n    end\n\n  theorem lt_step : \u03a0 {a b : pos_num}, a < b \u2192 a < succ b\n  | one      one      H := rfl\n  | one      (bit0 b) H := rfl\n  | one      (bit1 b) H := rfl\n  | (bit0 a) one      H := absurd H ff_ne_tt\n  | (bit0 a) (bit0 b) H :=\n    begin\n      rewrite [succ_bit0, lt_bit0_bit1_eq_lt_succ, lt_bit0_bit0_eq_lt at H],\n      apply lt_step H\n    end\n  | (bit0 a) (bit1 b) H :=\n    begin\n      rewrite [succ_bit1, lt_bit0_bit0_eq_lt, lt_bit0_bit1_eq_lt_succ at H],\n      exact H\n    end\n  | (bit1 a) one      H := absurd H ff_ne_tt\n  | (bit1 a) (bit0 b) H :=\n    begin\n      rewrite [succ_bit0, lt_bit1_bit1_eq_lt, lt_bit1_bit0_eq_lt at H],\n      exact H\n    end\n  | (bit1 a) (bit1 b) H :=\n    begin\n      rewrite [succ_bit1, lt_bit1_bit0_eq_lt, lt_bit1_bit1_eq_lt at H],\n      apply lt_step H\n    end\n\n  theorem lt_of_lt_succ_succ : \u03a0 {a b : pos_num}, succ a < succ b \u2192 a < b\n  | one      one      H := absurd H ff_ne_tt\n  | one      (bit0 b) H := rfl\n  | one      (bit1 b) H := rfl\n  | (bit0 a) one      H :=\n    begin\n      rewrite [succ_bit0 at H, succ_one at H, lt_bit1_bit0_eq_lt at H],\n      apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H\n    end\n  | (bit0 a) (bit0 b) H := by exact H\n  | (bit0 a) (bit1 b) H := by exact H\n  | (bit1 a) one      H :=\n    begin\n      rewrite [succ_bit1 at H, succ_one at H, lt_bit0_bit0_eq_lt at H],\n      apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff (succ a)) H\n    end\n  | (bit1 a) (bit0 b) H :=\n    begin\n      rewrite [succ_bit1 at H, succ_bit0 at H, lt_bit0_bit1_eq_lt_succ at H],\n      rewrite lt_bit1_bit0_eq_lt,\n      apply lt_of_lt_succ_succ H\n    end\n  | (bit1 a) (bit1 b) H :=\n    begin\n      rewrite [lt_bit1_bit1_eq_lt, *succ_bit1 at H, lt_bit0_bit0_eq_lt at H],\n      apply lt_of_lt_succ_succ H\n    end\n\n  theorem lt_succ_succ : \u03a0 {a b : pos_num}, a < b \u2192 succ a < succ b\n  | one      one      H := absurd H ff_ne_tt\n  | one      (bit0 b) H :=\n    begin\n      rewrite [succ_bit0, succ_one, lt_bit0_bit1_eq_lt_succ],\n      apply lt_one_succ_eq_tt\n    end\n  | one      (bit1 b) H :=\n    begin\n      rewrite [succ_one, succ_bit1, lt_bit0_bit0_eq_lt],\n      apply lt_one_succ_eq_tt\n    end\n  | (bit0 a) one      H := absurd H ff_ne_tt\n  | (bit0 a) (bit0 b) H := by exact H\n  | (bit0 a) (bit1 b) H := by exact H\n  | (bit1 a) one      H := absurd H ff_ne_tt\n  | (bit1 a) (bit0 b) H :=\n    begin\n      rewrite [succ_bit1, succ_bit0, lt_bit0_bit1_eq_lt_succ, lt_bit1_bit0_eq_lt at H],\n      apply lt_succ_succ H\n    end\n  | (bit1 a) (bit1 b) H :=\n    begin\n      rewrite [lt_bit1_bit1_eq_lt at H, *succ_bit1, lt_bit0_bit0_eq_lt],\n      apply lt_succ_succ H\n    end\n\n  theorem lt_of_lt_succ : \u03a0 {a b : pos_num}, succ a < b \u2192 a < b\n  | one      one      H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H\n  | one      (bit0 b) H := rfl\n  | one      (bit1 b) H := rfl\n  | (bit0 a) one      H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H\n  | (bit0 a) (bit0 b) H := by exact H\n  | (bit0 a) (bit1 b) H :=\n    begin\n      rewrite [succ_bit0 at H, lt_bit1_bit1_eq_lt at H, lt_bit0_bit1_eq_lt_succ],\n      apply lt_step H\n    end\n  | (bit1 a) one      H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H\n  | (bit1 a) (bit0 b) H :=\n    begin\n      rewrite [lt_bit1_bit0_eq_lt, succ_bit1 at H, lt_bit0_bit0_eq_lt at H],\n      apply lt_of_lt_succ H\n    end\n  | (bit1 a) (bit1 b) H :=\n    begin\n      rewrite [succ_bit1 at H, lt_bit0_bit1_eq_lt_succ at H, lt_bit1_bit1_eq_lt],\n      apply lt_of_lt_succ_succ H\n    end\n\n  theorem lt_of_lt_succ_of_ne : \u03a0 {a b : pos_num}, a < succ b \u2192 a \u2260 b \u2192 a < b\n  | one      one      H\u2081 H\u2082 := absurd rfl H\u2082\n  | one      (bit0 b) H\u2081 H\u2082 := rfl\n  | one      (bit1 b) H\u2081 H\u2082 := rfl\n  | (bit0 a) one      H\u2081 H\u2082 :=\n  begin\n    rewrite [succ_one at H\u2081, lt_bit0_bit0_eq_lt at H\u2081],\n    apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2081\n  end\n  | (bit0 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit0_bit0_eq_lt, succ_bit0 at H\u2081, lt_bit0_bit1_eq_lt_succ at H\u2081],\n      apply lt_of_lt_succ_of_ne H\u2081 (ne_of_bit0_ne_bit0 H\u2082)\n    end\n  | (bit0 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [succ_bit1 at H\u2081, lt_bit0_bit0_eq_lt at H\u2081, lt_bit0_bit1_eq_lt_succ],\n      exact H\u2081\n    end\n  | (bit1 a) one      H\u2081 H\u2082 :=\n    begin\n      rewrite [succ_one at H\u2081, lt_bit1_bit0_eq_lt at H\u2081],\n      apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2081\n    end\n  | (bit1 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [succ_bit0 at H\u2081, lt_bit1_bit1_eq_lt at H\u2081, lt_bit1_bit0_eq_lt],\n      exact H\u2081\n    end\n  | (bit1 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [succ_bit1 at H\u2081, lt_bit1_bit0_eq_lt at H\u2081, lt_bit1_bit1_eq_lt],\n      apply lt_of_lt_succ_of_ne H\u2081 (ne_of_bit1_ne_bit1 H\u2082)\n    end\n\n  theorem lt_trans : \u03a0 {a b c : pos_num}, a < b \u2192 b < c \u2192 a < c\n  | one      b        (bit0 c) H\u2081 H\u2082 := rfl\n  | one      b        (bit1 c) H\u2081 H\u2082 := rfl\n  | a        (bit0 b) one      H\u2081 H\u2082 := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2082\n  | a        (bit1 b) one      H\u2081 H\u2082 := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2082\n  | (bit0 a) (bit0 b) (bit0 c) H\u2081 H\u2082 :=\n    begin\n      rewrite lt_bit0_bit0_eq_lt at *, apply lt_trans H\u2081 H\u2082\n    end\n  | (bit0 a) (bit0 b) (bit1 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit0_bit1_eq_lt_succ at *, lt_bit0_bit0_eq_lt at H\u2081],\n      apply lt_trans H\u2081 H\u2082\n    end\n  | (bit0 a) (bit1 b) (bit0 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit0_bit1_eq_lt_succ at H\u2081, lt_bit1_bit0_eq_lt at H\u2082, lt_bit0_bit0_eq_lt],\n      apply @by_cases (a = b),\n      begin\n         intro H, rewrite -H at H\u2082, exact H\u2082\n      end,\n      begin\n        intro H,\n        apply lt_trans (lt_of_lt_succ_of_ne H\u2081 H) H\u2082\n      end\n    end\n  | (bit0 a) (bit1 b) (bit1 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit0_bit1_eq_lt_succ at *, lt_bit1_bit1_eq_lt at H\u2082],\n      apply lt_trans H\u2081 (lt_succ_succ H\u2082)\n    end\n  | (bit1 a) (bit0 b) (bit0 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit0_bit0_eq_lt at H\u2082, lt_bit1_bit0_eq_lt at *],\n      apply lt_trans H\u2081 H\u2082\n    end\n  | (bit1 a) (bit0 b) (bit1 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit1_bit0_eq_lt at H\u2081, lt_bit0_bit1_eq_lt_succ at H\u2082, lt_bit1_bit1_eq_lt],\n      apply @by_cases (b = c),\n      begin\n        intro H, rewrite H at H\u2081, exact H\u2081\n      end,\n      begin\n        intro H,\n        apply lt_trans H\u2081 (lt_of_lt_succ_of_ne H\u2082 H)\n      end\n    end\n  | (bit1 a) (bit1 b) (bit0 c) H\u2081 H\u2082 :=\n    begin\n      rewrite [lt_bit1_bit1_eq_lt at H\u2081, lt_bit1_bit0_eq_lt at H\u2082, lt_bit1_bit0_eq_lt],\n      apply lt_trans H\u2081 H\u2082\n    end\n  | (bit1 a) (bit1 b) (bit1 c) H\u2081 H\u2082 :=\n    begin\n      rewrite lt_bit1_bit1_eq_lt at *,\n      apply lt_trans H\u2081 H\u2082\n    end\n\n  theorem lt_antisymm : \u03a0 {a b : pos_num}, a < b \u2192 b \u226e a\n  | one      one      H := rfl\n  | one      (bit0 b) H := rfl\n  | one      (bit1 b) H := rfl\n  | (bit0 a) one      H := absurd H ff_ne_tt\n  | (bit0 a) (bit0 b) H :=\n    begin\n      rewrite lt_bit0_bit0_eq_lt at *,\n      apply lt_antisymm H\n    end\n  | (bit0 a) (bit1 b) H :=\n    begin\n      rewrite lt_bit1_bit0_eq_lt,\n      rewrite lt_bit0_bit1_eq_lt_succ at H,\n      have H\u2081 : succ b \u226e a, from lt_antisymm H,\n      apply eq_ff_of_ne_tt,\n        intro H\u2082,\n        apply @by_cases (succ b = a),\n        show succ b = a \u2192 empty,\n        begin\n          intro Hp,\n          rewrite -Hp at H,\n          apply absurd_of_eq_ff_of_eq_tt (lt_irrefl (succ b)) H\n        end,\n        show succ b \u2260 a \u2192 empty,\n        begin\n          intro Hn,\n          have H\u2083 : succ b < succ a, from lt_succ_succ H\u2082,\n          have H\u2084 : succ b < a, from lt_of_lt_succ_of_ne H\u2083 Hn,\n          apply absurd_of_eq_ff_of_eq_tt H\u2081 H\u2084\n        end,\n    end\n  | (bit1 a) one      H := absurd H ff_ne_tt\n  | (bit1 a) (bit0 b) H :=\n    begin\n      rewrite lt_bit0_bit1_eq_lt_succ,\n      rewrite lt_bit1_bit0_eq_lt at H,\n      have H\u2081 : lt b a = ff, from lt_antisymm H,\n      apply eq_ff_of_ne_tt,\n        intro H\u2082,\n        apply @by_cases (b = a),\n        show b = a \u2192 empty,\n        begin\n          intro Hp,\n          rewrite -Hp at H,\n          apply absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H\n        end,\n        show b \u2260 a \u2192 empty,\n        begin\n          intro Hn,\n          have H\u2083 : b < a, from lt_of_lt_succ_of_ne H\u2082 Hn,\n          apply absurd_of_eq_ff_of_eq_tt H\u2081 H\u2083\n        end,\n      end\n  | (bit1 a) (bit1 b) H :=\n    begin\n      rewrite lt_bit1_bit1_eq_lt at *,\n      apply lt_antisymm H\n    end\n\n  local notation a \u2264 b := (le a b = tt)\n\n  theorem le_refl : \u03a0 a : pos_num, a \u2264 a :=\n  lt_base\n\n  theorem le_eq_lt_succ {a b : pos_num} : le a b = lt a (succ b) :=\n  rfl\n\n  theorem not_lt_of_le : \u03a0 {a b : pos_num}, a \u2264 b \u2192 b < a \u2192 empty\n  | one      one      H\u2081 H\u2082 := absurd H\u2082 ff_ne_tt\n  | one      (bit0 b) H\u2081 H\u2082 := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2082\n  | one      (bit1 b) H\u2081 H\u2082 := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2082\n  | (bit0 a) one      H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_one at H\u2081, lt_bit0_bit0_eq_lt at H\u2081],\n      apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2081\n    end\n  | (bit0 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_bit0 at H\u2081, lt_bit0_bit1_eq_lt_succ at H\u2081],\n      rewrite [lt_bit0_bit0_eq_lt at H\u2082],\n      apply not_lt_of_le H\u2081 H\u2082\n    end\n  | (bit0 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_bit1 at H\u2081, lt_bit0_bit0_eq_lt at H\u2081],\n      rewrite [lt_bit1_bit0_eq_lt at H\u2082],\n      apply not_lt_of_le H\u2081 H\u2082\n    end\n  | (bit1 a) one      H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_one at H\u2081, lt_bit1_bit0_eq_lt at H\u2081],\n      apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H\u2081\n    end\n  | (bit1 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_bit0 at H\u2081, lt_bit1_bit1_eq_lt at H\u2081],\n      rewrite lt_bit0_bit1_eq_lt_succ at H\u2082,\n      have H\u2083 : a < succ b, from lt_step H\u2081,\n      apply @by_cases (b = a),\n      begin\n        intro Hba, rewrite -Hba at H\u2081,\n        apply absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H\u2081\n      end,\n      begin\n        intro Hnba,\n        have H\u2084 : b < a, from lt_of_lt_succ_of_ne H\u2082 Hnba,\n        apply not_lt_of_le H\u2083 H\u2084\n      end\n    end\n  | (bit1 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at H\u2081, succ_bit1 at H\u2081, lt_bit1_bit0_eq_lt at H\u2081],\n      rewrite [lt_bit1_bit1_eq_lt at H\u2082],\n      apply not_lt_of_le H\u2081 H\u2082\n    end\n\n  theorem le_antisymm : \u03a0 {a b : pos_num}, a \u2264 b \u2192 b \u2264 a \u2192 a = b\n  | one      one      H\u2081 H\u2082 := rfl\n  | one      (bit0 b) H\u2081 H\u2082 :=\n    by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H\u2082\n  | one      (bit1 b) H\u2081 H\u2082 :=\n    by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H\u2082\n  | (bit0 a) one      H\u2081 H\u2082 :=\n    by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H\u2081\n  | (bit0 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at *, succ_bit0 at *, lt_bit0_bit1_eq_lt_succ at *],\n      have H : a = b, from le_antisymm H\u2081 H\u2082,\n      rewrite H\n    end\n  | (bit0 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at *, succ_bit1 at H\u2081, succ_bit0 at H\u2082],\n      rewrite [lt_bit0_bit0_eq_lt at H\u2081, lt_bit1_bit1_eq_lt at H\u2082],\n      apply empty.rec _ (not_lt_of_le H\u2081 H\u2082)\n    end\n  | (bit1 a) one      H\u2081 H\u2082 :=\n    by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H\u2081\n  | (bit1 a) (bit0 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at *, succ_bit0 at H\u2081, succ_bit1 at H\u2082],\n      rewrite [lt_bit1_bit1_eq_lt at H\u2081, lt_bit0_bit0_eq_lt at H\u2082],\n      apply empty.rec _ (not_lt_of_le H\u2082 H\u2081)\n    end\n  | (bit1 a) (bit1 b) H\u2081 H\u2082 :=\n    begin\n      rewrite [le_eq_lt_succ at *, succ_bit1 at *, lt_bit1_bit0_eq_lt at *],\n      have H : a = b, from le_antisymm H\u2081 H\u2082,\n      rewrite H\n    end\n\n  theorem le_trans {a b c : pos_num} : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c :=\n  begin\n    intro H\u2081 H\u2082,\n    rewrite [le_eq_lt_succ at *],\n    apply @by_cases (a = b),\n    begin\n      intro Hab, rewrite Hab, exact H\u2082\n    end,\n    begin\n      intro Hnab,\n      have Haltb : a < b, from lt_of_lt_succ_of_ne H\u2081 Hnab,\n      apply lt_trans Haltb H\u2082\n    end,\n  end\n\nend pos_num\n\nnamespace num\n  open pos_num\n\n  theorem decidable_eq [instance] : \u03a0 (a b : num), decidable (a = b)\n  | zero zero       := inl rfl\n  | zero (pos b)    := inr (by contradiction)\n  | (pos a) zero    := inr (by contradiction)\n  | (pos a) (pos b) :=\n    if H : a = b then inl (by rewrite H) else inr (suppose pos a = pos b, begin injection this, contradiction end)\nend num\n","avg_line_length":33.036259542,"max_line_length":114,"alphanum_fraction":0.5882964589}
-{"size":4032,"ext":"lean","lang":"Lean","max_stars_count":1131.0,"content":"\/-\nCopyright (c) 2020 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-\/\nimport tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\nimport tactic.show_term\nimport topology.basic\n\/-!\n# Tactics for topology\n\nCurrently we have one domain-specific tactic for topology: `continuity`.\n\n-\/\n\n\/-!\n### `continuity` tactic\n\nAutomatically solve goals of the form `continuous f`.\n\nMark lemmas with `@[continuity]` to add them to the set of lemmas\nused by `continuity`.\n-\/\n\n\/-- User attribute used to mark tactics used by `continuity`. -\/\n@[user_attribute]\nmeta def continuity : user_attribute :=\n{ name := `continuity,\n  descr := \"lemmas usable to prove continuity\" }\n\n-- Mark some continuity lemmas already defined in `topology.basic`\nattribute [continuity]\n  continuous_id\n  continuous_const\n\n-- As we will be using `apply_rules` with `md := semireducible`,\n-- we need another version of `continuous_id`.\n@[continuity] lemma continuous_id' {\u03b1 : Type*} [topological_space \u03b1] : continuous (\u03bb a : \u03b1, a) :=\ncontinuous_id\n\nnamespace tactic\n\n\/--\nTactic to apply `continuous.comp` when appropriate.\n\nApplying `continuous.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove continuous is actually\n  constant, and that constant is a function application `f z`, then\n  continuous.comp would produce new goals `continuous f`, `continuous\n  (\u03bb _, z)`, which is silly. We avoid this by failing if we could\n  apply continuous_const.\n\n* continuous.comp will always succeed on `continuous (\u03bb x, f x)` and\n  produce new goals `continuous (\u03bb x, x)`, `continuous f`. We detect\n  this by failing if a new goal can be closed by applying\n  continuous_id.\n-\/\nmeta def apply_continuous.comp : tactic unit :=\n`[fail_if_success { exact continuous_const };\n  refine continuous.comp _ _;\n  fail_if_success { exact continuous_id }]\n\n\/-- List of tactics used by `continuity` internally. -\/\nmeta def continuity_tactics (md : transparency := reducible) : list (tactic string) :=\n[\n  intros1               >>= \u03bb ns, pure (\"intros \" ++ (\" \".intercalate (ns.map (\u03bb e, e.to_string)))),\n  apply_rules [``(continuity)] 50 { md := md }\n                        >> pure \"apply_rules continuity\",\n  apply_continuous.comp >> pure \"refine continuous.comp _ _\"\n]\n\nnamespace interactive\nsetup_tactic_parser\n\n\/--\nSolve goals of the form `continuous f`. `continuity?` reports back the proof term it found.\n-\/\nmeta def continuity\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md              := if bang.is_some then semireducible else reducible,\n    continuity_core := tactic.tidy { tactics := continuity_tactics md, ..cfg },\n    trace_fn        := if trace.is_some then show_term else id in\ntrace_fn continuity_core\n\n\/-- Version of `continuity` for use with auto_param. -\/\nmeta def continuity' : tactic unit := continuity none none {}\n\n\/--\n`continuity` solves goals of the form `continuous f` by applying lemmas tagged with the\n`continuity` user attribute.\n\n```\nexample {X Y : Type*} [topological_space X] [topological_space Y]\n  (f\u2081 f\u2082 : X \u2192 Y) (hf\u2081 : continuous f\u2081) (hf\u2082 : continuous f\u2082)\n  (g : Y \u2192 \u211d) (hg : continuous g) : continuous (\u03bb x, (max (g (f\u2081 x)) (g (f\u2082 x))) + 1) :=\nby continuity\n```\nwill discharge the goal, generating a proof term like\n`((continuous.comp hg hf\u2081).max (continuous.comp hg hf\u2082)).add continuous_const`\n\nYou can also use `continuity!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`continuity?` reports back the proof term it found.\n-\/\nadd_tactic_doc\n{ name := \"continuity \/ continuity'\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.continuity, `tactic.interactive.continuity'],\n  tags := [\"lemma application\"]\n}\n\nend interactive\n\nend tactic\n","avg_line_length":33.0491803279,"max_line_length":100,"alphanum_fraction":0.712797619}
-{"size":1943,"ext":"lean","lang":"Lean","max_stars_count":17.0,"content":"import linear_algebra\n-- 1.\n-- (ii) Let V be the vector space of all 2\u00d72 matrices over R. Which of the following functions T : V \u2212\u2192 V are linear transformations?\n-- (a) T(A)=A2 \u2200A\u2208V\udbff\udc03\n-- (b) T(A)= \u22121 1 A \u2200 A\u2208V\n-- Find a linear transformation T : R2 \u2212\u2192 R3 which sends (1,0) to (1,1,0) and (1, 1) to (1, 0, \u22121).\n-- (i) Which of the following functions T : R3 \u2212\u2192 R2 are linear transformations? (a) T(x1,x2,x3)=(x1 +x2 \u2212x3,2x1 +x2)\n-- (b) T (x1, x2, x3) = (0,\n-- (c) T(x1,x2,x3)=(x1x2,x3)\n-- (iii) (a)\n-- (b) Find two different linear transformations R3 \u2212\u2192 R2 which send (1, 1, 0) to (1, 1)\n-- *2.\n-- and (0, 1, 1) to (0, 1).\n-- (i) Give an example of a linear transformation T : R2 \u2212\u2192 R3 such that T(v) = (1,0,0) for exactly one vector v \u2208 R2.\n-- (ii) Give an example of a linear transformation T : R2 \u2212\u2192 R3 such that T(v) = (1,0,0) for no vector v \u2208 R2.\n-- (iii) Give an example of a linear transformation T : R2 \u2212\u2192 R3 such that T(v) = (1,0,0) for infinitely many vectors v \u2208 R2.\n-- (iv) Show that there is no linear transformation T : R2 \u2212\u2192 R3 such that T (v) = (1, 0, 0) for exactly two vectors v \u2208 R2.\n-- 3.\n-- (ii) Let V be the vector space of polynomials of degree at most 2 over R. Define T : V \u2212\u2192 V by\n-- T(ax2 +bx+c)=(a+b+c)x2 +(c\u2212a)x+(a+3b+5c). Find bases of Ker(T) and Im(T).\n-- (i) Define T : R3 \u2212\u2192 R3 by T(x1,x2,x3) = (x1 \u2212 x2,x2 \u2212 x3,x3 \u2212 x1). Find bases of Ker(T) and Im(T). For which values of k is the vector (1,3,k) in Ker(T) or Im(T)?\n-- (iii) Let V be as in part (ii), and define S : V \u2212\u2192 V by\n-- S(p(x)) = p(1 + x) \u2212 p(x) for p(x) \u2208 V.\n-- (So for example, S(x2) = (x+1)2\u2212x2 = 2x+1.) Show that S is a linear transformation, and find bases of Ker(S) and Im(S).\n-- 4.\n-- (b) Prove that T is surjective if and only if Ker(T ) = {0}.\n-- (ii) Find an example of a linear transformation T : R2 \u2212\u2192 R2 such that Ker(T ) = Im(T ).\n-- (iii) Prove that there does not exist a linear transformation T : R3 \u2212\u2192 R3 such that Ker(T ) = Im(T ).","avg_line_length":69.3928571429,"max_line_length":166,"alphanum_fraction":0.6031909418}
-{"size":12468,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import Mathlib.Data.Nat.Basic\nimport Mathlib.Init.Data.Nat.Lemmas\nimport Mathlib.Algebra.Group.Defs\nimport Mathlib.Algebra.GroupWithZero.Defs\nimport Mathlib.Algebra.Ring.Basic\n\ninstance : Subsingleton (Fin 0) where\n  allEq := fun.\n\ninstance : Subsingleton (Fin 1) where\n  allEq := fun \u27e80, _\u27e9 \u27e80, _\u27e9 => rfl\n\n\/-- If you actually have an element of `Fin n`, then the `n` is always positive -\/\nlemma Fin.size_positive : \u2200 (x : Fin n), 0 < n\n| \u27e8x, h\u27e9 =>\n  match Nat.eq_or_lt_of_le (Nat.zero_le x) with\n  | Or.inl h_eq => h_eq \u25b8 h\n  | Or.inr h_lt => Nat.lt_trans h_lt h\n\nlemma Fin.ext {a b : Fin n} : a.val = b.val \u2192 a = b := by\n  cases a; cases b; intro h; cases h; rfl\n\nlemma Fin.ext_iff {a b : Fin n} : a = b \u2194 a.val = b.val :=\n  \u27e8congrArg _, ext\u27e9\n\nlemma Fin.size_positive' [Nonempty (Fin n)] : 0 < n :=\n  \u2039Nonempty (Fin n)\u203a.elim fun i => Fin.size_positive i\n\nlemma zero_lt_of_lt {a : Nat} : \u2200 {x : Nat}, x < a -> 0 < a\n| 0, h   => h\n| x+1, h => Nat.lt_trans (Nat.zero_lt_succ x) h\n\nlemma Fin.val_eq_of_lt {n a : Nat} (h : a < n) : (Fin.ofNat' a (zero_lt_of_lt h)).val = a := by\n  simp only [Fin.ofNat', Nat.mod_eq_of_lt h]\n\n@[simp] lemma Fin.one_val : (1 : Fin (n + 2)).val = 1 := by\n  simp only [OfNat.ofNat, Fin.ofNat]\n  rw [Nat.mod_eq_of_lt]\n  exact Nat.succ_lt_succ (Nat.zero_lt_succ _)\n\nlemma Fin.modn_def : \u2200 (a : Fin n) (m : Nat),\n  a % m = Fin.mk ((a.val % m) % n) (Nat.mod_lt (a.val % m) (a.size_positive))\n| \u27e8a, pa\u27e9, m => rfl\n\nlemma Fin.mod_def : \u2200 (a m : Fin n),\n  a % m = Fin.mk ((a.val % m.val) % n) (Nat.mod_lt (a.val % m.val) (a.size_positive))\n| \u27e8a, pa\u27e9, \u27e8m, pm\u27e9 => rfl\n\nlemma Fin.add_def : \u2200 (a b : Fin n),\n  a + b = (Fin.mk ((a.val + b.val) % n) (Nat.mod_lt _ (a.size_positive)))\n| \u27e8a, pa\u27e9, \u27e8b, pb\u27e9 => rfl\n\nlemma Fin.mul_def : \u2200 (a b : Fin n),\n  a * b = (Fin.mk ((a.val * b.val) % n) (Nat.mod_lt _ (a.size_positive)))\n| \u27e8a, pa\u27e9, \u27e8b, pb\u27e9 => rfl\n\nlemma Fin.sub_def : \u2200 (a b : Fin n),\n  a - b = (Fin.mk ((a + (n - b)) % n) (Nat.mod_lt _ (a.size_positive)))\n| \u27e8a, pa\u27e9, \u27e8b, pb\u27e9 => rfl\n\n@[simp] lemma Fin.mod_eq (a : Fin n) : a % n = a := by\n  apply Fin.eq_of_val_eq\n  simp only [Fin.modn_def, Nat.mod_mod]\n  exact Nat.mod_eq_of_lt a.isLt\n\n@[simp] lemma Fin.mod_eq_val (a : Fin n) : a.val % n = a.val := by\n  simp only [Fin.modn_def, Nat.mod_mod]\n  exact Nat.mod_eq_of_lt a.isLt\n\ntheorem Fin.mod_eq_of_lt {a b : Fin n} (h : a < b) : a % b = a := by\n  apply Fin.eq_of_val_eq\n  simp only [Fin.mod_def]\n  rw [Nat.mod_eq_of_lt h, Nat.mod_eq_of_lt a.isLt]\n\n\/- The basic structures on `Fin` are predicated on `Fin n` being nonempty.\nThe Nonempty bound is there so that we can implement `Zero` in a way that satisfies\nthe requirements of the relevant typeclasses (for example, AddMonoid). If we were to\nuse `Fin n+1` for the `Zero` implementation, we would be shutting out some irreducible\ndefinitions (notably USize.size) that are known to be inhabited, but not defined in terms\nof `Nat.succ`. Since there's a blanket implementation of `\u2200 n, Inhabited (Fin n+1)` in\nthe prelude, this hopefully won't be a significant impediment. -\/\nsection\n\nvariable {n : Nat} [Nonempty (Fin n)]\n\ninstance : OfNat (Fin n) a where\n  ofNat := Fin.ofNat' a Fin.size_positive'\n\n@[simp] lemma Fin.ofNat'_zero : (Fin.ofNat' 0 h : Fin n) = 0 := rfl\n@[simp] lemma Fin.ofNat'_one : (Fin.ofNat' 1 h : Fin n) = 1 := rfl\n\nlemma Fin.ofNat'_succ : {n : Nat} \u2192 [Nonempty (Fin n)] \u2192\n    (Fin.ofNat' i.succ Fin.size_positive' : Fin n) = (Fin.ofNat' i Fin.size_positive' : Fin n) + 1\n  | n + 2, h => ext (by simp [Fin.ofNat', Fin.add_def])\n  | 1, h => Subsingleton.allEq _ _\n  | 0, h => Subsingleton.allEq _ _\n\ninstance : AddCommSemigroup (Fin n) where\n  add_assoc _ _ _ := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.add_def, Nat.mod_add_mod, Nat.add_mod_mod, Nat.add_assoc]\n  add_comm _ _ := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.add_def, Nat.add_comm]\n\ninstance : CommSemigroup (Fin n) where\n  mul_assoc a b c := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.mul_def]\n    generalize lhs : ((a.val * b.val) % n * c.val) % n = l\n    generalize rhs : a.val * (b.val * c.val % n) % n = r\n    rw [\u2190 Nat.mod_eq_of_lt c.isLt, (Nat.mul_mod (a.val * b.val) c.val n).symm] at lhs\n    rw [\u2190 Nat.mod_eq_of_lt a.isLt, (Nat.mul_mod a.val (b.val * c.val) n).symm,\n        \u2190 Nat.mul_assoc] at rhs\n    rw [\u2190 lhs, \u2190 rhs]\n  mul_comm (a b : Fin n) : a * b = b * a := by\n    apply Fin.eq_of_val_eq; simp only [Fin.mul_def, Nat.mul_comm]\n\n@[simp] lemma Fin.zero_def : (0 : Fin n).val = (0 : Nat) :=\n  show (Fin.ofNat' 0 Fin.size_positive').val = (0 : Nat) by simp only [Fin.ofNat', Nat.zero_mod]\n\ntheorem Fin.mod_lt : \u2200 (i : Fin n) {m : Fin n}, (0 : Fin n) < m \u2192 (i % m) < m\n| \u27e8a, aLt\u27e9, \u27e8m, mLt\u27e9, hp =>  by\n    have zero_lt : (0 : Nat) < m := Fin.zero_def \u25b8 hp\n    have a_mod_lt : a % m < m := Nat.mod_lt _ zero_lt\n    simp only [Fin.mod_def, LT.lt]\n    rw [(Nat.mod_eq_of_lt (Nat.lt_trans a_mod_lt mLt) : a % m % n = a % m)]\n    exact Nat.mod_lt _ zero_lt\n\n\/- Aux lemma that makes nsmul_succ' easier -\/\nprotected lemma Fin.nsmuls_eq (x : Nat) : \u2200 (a : Fin n),\n  ((Fin.ofNat' x Fin.size_positive') * a) = Fin.ofNat' (x * a.val) Fin.size_positive'\n| \u27e8a, isLt\u27e9 => by\n  apply Fin.eq_of_val_eq\n  simp only [Fin.ofNat', Fin.mul_def]\n  generalize hy : x * a % n = y\n  rw [\u2190 Nat.mod_eq_of_lt isLt, \u2190 Nat.mul_mod, hy]\n\n@[simp] lemma Fin.one_def : (1 : Fin n).val = (1 % n : Nat) :=\n  show (Fin.ofNat' 1 Fin.size_positive').val = 1 % n by simp [Fin.ofNat']\n\ndef Fin.addOverflows? (a b : Fin n) : Bool := n <= a.val + b.val\n\ndef Fin.mulOverflows? (a b : Fin n) : Bool := n <= a.val * b.val\n\ndef Fin.subUnderflows? (a b : Fin n) : Bool := a.val < b.val\n\ndef Fin.overflowingAdd (a b : Fin n) : (Bool \u00d7 Fin n) := (n <= a.val + b.val, a + b)\n\ndef Fin.overflowingMul (a b : Fin n) : (Bool \u00d7 Fin n) := (n <= a.val * b.val, a * b)\n\ndef Fin.underflowingSub (a b : Fin n) : (Bool \u00d7 Fin n) := (a.val < b.val, a - b)\n\ndef Fin.checkedAdd (a b : Fin n) : Option (Fin n) :=\n  match Fin.overflowingAdd a b with\n  | (true, _) => none\n  | (false, sum) => some (sum)\n\ndef Fin.checkedMul (a b : Fin n) : Option (Fin n) :=\n  match Fin.overflowingMul a b with\n  | (true, _) => none\n  | (false, prod) => some (prod)\n\ndef Fin.checkedSub (a b : Fin n) : Option (Fin n) :=\n  match Fin.underflowingSub a b with\n  | (true, _) => none\n  | (false, diff) => some (diff)\n\nlemma Fin.checked_add_spec (a b : Fin n) :\n  (Fin.checkedAdd a b).isSome = true \u2194 a.val + b.val < n := by\n  by_cases n <= a.val + b.val <;>\n    simp_all [checkedAdd, Option.isSome, overflowingAdd, decide_eq_true, decide_eq_false]\n\nlemma Fin.checked_mul_spec (a b : Fin n) :\n  (Fin.checkedMul a b).isSome = true \u2194 a.val * b.val < n := by\n  simp only [checkedMul, overflowingMul, Option.isSome]\n  refine Iff.intro ?mp ?mpr <;> intro h\n  case mp =>\n    by_cases hx : n <= a.val * b.val\n    case pos => simp only [(decide_eq_true_iff (n <= a.val * b.val)).mpr hx] at h\n    case neg => exact Nat.lt_of_not_le hx\n  case mpr => simp only [decide_eq_false (Nat.not_le_of_lt h : \u00acn <= a.val * b.val)]\n\nlemma Fin.checked_sub_spec (a b : Fin n) :\n  (Fin.checkedSub a b).isSome = true \u2194 b.val <= a.val := by\n  simp only [checkedSub, underflowingSub, Option.isSome]\n  refine Iff.intro ?mp ?mpr <;> intro h\n  case mp =>\n    by_cases hx : a.val < b.val\n    case pos => simp only [(decide_eq_true_iff (a.val < b.val)).mpr hx] at h\n    case neg => exact Nat.le_of_not_lt hx\n  case mpr => simp only [decide_eq_false (Nat.not_lt_of_le h : \u00aca.val < b.val)]\n\ninstance : AddCommMonoid (Fin n) where\n  __ := inferInstanceAs (AddCommSemigroup (Fin n))\n\n  add_zero (a : Fin n) : a + 0 = a := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.add_def, Fin.zero_def, Nat.add_zero]\n    exact Nat.mod_eq_of_lt a.isLt\n  zero_add (a : Fin n) : 0 + a = a := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.add_def, Fin.zero_def, Nat.zero_add]\n    exact Nat.mod_eq_of_lt a.isLt\n\n  nsmul := fun x a => (Fin.ofNat' x a.size_positive) * a\n  nsmul_zero' := fun _ => by\n    apply Fin.eq_of_val_eq\n    simp [Fin.mul_def, Fin.ofNat', Fin.zero_def, Nat.zero_mul, Nat.zero_mod]\n  nsmul_succ' := fun x a => by\n    simp only [Fin.nsmuls_eq]\n    simp [Fin.ofNat', Fin.add_def]\n    exact congrArg (fun x => x % n) (Nat.add_comm (x * a.val) (a.val) \u25b8 Nat.succ_mul x a.val)\n\ninstance : AddMonoidWithOne (Fin n) where\n  __ := inferInstanceAs (AddCommMonoid (Fin n))\n  natCast n := Fin.ofNat' n Fin.size_positive'\n  natCast_zero := rfl\n  natCast_succ _ := Fin.ofNat'_succ\n\nprivate theorem Fin.mul_one (a : Fin n) : a * 1 = a := by\n  apply Fin.eq_of_val_eq\n  simp only [Fin.mul_def, Fin.one_def]\n  cases n with\n  | zero => exact (False.elim a.elim0)\n  | succ n =>\n    match Nat.lt_or_eq_of_le (Nat.mod_le 1 n.succ) with\n    | Or.inl h_lt =>\n      have h_eq : 1 % n.succ = 0 := Nat.eq_zero_of_le_zero (Nat.le_of_lt_succ h_lt)\n      have hnz : n = 0 := Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ (Nat.le_of_mod_lt h_lt))\n      have haz : a.val = 0 := Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ (hnz \u25b8 a.isLt))\n      rw [h_eq, haz]\n      simp only [Nat.zero_mul, Nat.zero_mod]\n    | Or.inr h_eq => simp only [h_eq, Nat.mul_one, Nat.mod_eq_of_lt (a.isLt)]\n\ninstance : MonoidWithZero (Fin n) where\n  __ := inferInstanceAs (CommSemigroup (Fin n))\n  mul_one := Fin.mul_one\n  one_mul _ := by rw [mul_comm, Fin.mul_one]\n  npow_zero' _ := rfl\n  npow_succ' _ _ := rfl\n  zero_mul x := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.mul_def, Fin.zero_def, Nat.zero_mul, Nat.zero_mod]\n  mul_zero x := by\n    apply Fin.eq_of_val_eq\n    simp only [Fin.mul_def, Fin.zero_def, Nat.mul_zero, Nat.zero_mod]\n\nprivate theorem Fin.mul_add (a b c : Fin n) : a * (b + c) = a * b + a * c := by\n    apply Fin.eq_of_val_eq\n    simp [Fin.mul_def, Fin.add_def]\n    generalize lhs : a.val * ((b.val + c.val) % n) % n = l\n    rw [(Nat.mod_eq_of_lt a.isLt).symm, \u2190 Nat.mul_mod] at lhs\n    rw [\u2190 lhs, left_distrib]\n\ninstance : CommSemiring (Fin n) where\n  __ := inferInstanceAs (MonoidWithZero (Fin n))\n  __ := inferInstanceAs (CommSemigroup (Fin n))\n  __ := inferInstanceAs (AddCommMonoid (Fin n))\n  __ := inferInstanceAs (AddMonoidWithOne (Fin n))\n  left_distrib := Fin.mul_add\n  right_distrib a b c := (by rw [mul_comm, Fin.mul_add, mul_comm c, mul_comm c])\n\ninstance : Neg (Fin n) where\n  neg a := \u27e8(n - a) % n, Nat.mod_lt _ (lt_of_le_of_lt (Nat.zero_le _) a.isLt)\u27e9\n\nlemma Fin.neg_def : (-a : Fin n) = \u27e8(n - a) % n, Nat.mod_lt _ (lt_of_le_of_lt (Nat.zero_le _) a.isLt)\u27e9 := rfl\n\nprotected def Fin.ofInt'' : Int \u2192 Fin n\n  | Int.ofNat a => Fin.ofNat' a Fin.size_positive'\n  | Int.negSucc a => -(Fin.ofNat' a.succ Fin.size_positive')\n\nprivate theorem Fin.sub_eq_add_neg : \u2200 (a b : Fin n), a - b = a + -b := by\n  simp [Fin.add_def, Fin.sub_def, Neg.neg]\n\nprivate theorem Fin.add_left_neg (a : Fin n) : -a + a = 0 := by\n    rw [add_comm, \u2190 Fin.sub_eq_add_neg]\n    apply Fin.eq_of_val_eq\n    simp [Fin.sub_def, (Nat.add_sub_cancel' (Nat.le_of_lt a.isLt)), Nat.mod_self]\n\ndef Fin.ofInt' : \u2124 \u2192 Fin n\n  | (i : \u2115) => i\n  | (Int.negSucc i) => -\u2191(i + 1 : \u2115)\n\ninstance : AddGroupWithOne (Fin n) where\n  __ := inferInstanceAs (AddMonoidWithOne (Fin n))\n  gsmul_zero' := by simp [gsmul_rec, nsmul_rec]\n  gsmul_succ' := by simp [gsmul_rec, nsmul_rec, -Int.ofNat_eq_cast]\n  gsmul_neg' := by simp [gsmul_rec, nsmul_rec, -Int.ofNat_eq_cast]\n  sub_eq_add_neg := Fin.sub_eq_add_neg\n  add_left_neg := Fin.add_left_neg\n  intCast := Fin.ofInt'\n  intCast_ofNat _ := rfl\n  intCast_negSucc _ := rfl\n\ninstance : CommRing (Fin n) where\n  __ := inferInstanceAs (AddGroupWithOne (Fin n))\n  __ := inferInstanceAs (CommSemiring (Fin n))\n\nlemma Fin.gt_wf : WellFounded (fun a b : Fin n => b < a) :=\n  Subrelation.wf (@fun a i h => \u27e8h, i.2\u27e9) (invImage (fun i => i.1) (Nat.upRel n)).wf\n\n\/-- A well-ordered relation for \"upwards\" induction on `Fin n`. -\/\ndef Fin.upRel (n : \u2115) : WellFoundedRelation (Fin n) := \u27e8_, gt_wf\u27e9\n\nlemma Fin.le_refl (f : Fin n) : f \u2264 f := Nat.le_refl _\nlemma Fin.le_trans (a b c : Fin n) : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c := Nat.le_trans\nlemma Fin.lt_iff_le_not_le (a b : Fin n) : a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a := Nat.lt_iff_le_not_le\nlemma Fin.le_antisymm (a b : Fin n) : a \u2264 b \u2192 b \u2264 a \u2192 a = b := by\n  intro h1 h2\n  apply Fin.eq_of_val_eq\n  exact Nat.le_antisymm h1 h2\n\nlemma Fin.le_total (a b : Fin n) : a \u2264 b \u2228 b \u2264 a := Nat.le_total _ _\n\ninstance : LinearOrder (Fin n) where\n  le_refl := Fin.le_refl\n  le_trans := Fin.le_trans\n  lt_iff_le_not_le := Fin.lt_iff_le_not_le\n  le_antisymm := Fin.le_antisymm\n  le_total := Fin.le_total\n  decidable_le := inferInstance\n\nend\n","avg_line_length":38.3630769231,"max_line_length":109,"alphanum_fraction":0.635707411}
-{"size":24285,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-\/\n\nimport field_theory.finite.basic\nimport data.zmod.basic\nimport data.nat.parity\n\n\/-!\n# Quadratic reciprocity.\n\nThis file contains results about quadratic residues modulo a prime number.\n\nThe main results are the law of quadratic reciprocity, `quadratic_reciprocity`, as well as the\ninterpretations in terms of existence of square roots depending on the congruence mod 4,\n`exists_sq_eq_prime_iff_of_mod_four_eq_one`, and\n`exists_sq_eq_prime_iff_of_mod_four_eq_three`.\n\nAlso proven are conditions for `-1` and `2` to be a square modulo a prime,\n`exists_sq_eq_neg_one_iff_mod_four_ne_three` and\n`exists_sq_eq_two_iff`\n\n## Implementation notes\n\nThe proof of quadratic reciprocity implemented uses Gauss' lemma and Eisenstein's lemma\n-\/\n\nopen function finset nat finite_field zmod\nopen_locale big_operators nat\n\nnamespace zmod\n\nvariables (p q : \u2115) [fact p.prime] [fact q.prime]\n\n\/-- Euler's Criterion: A unit `x` of `zmod p` is a square if and only if `x ^ (p \/ 2) = 1`. -\/\nlemma euler_criterion_units (x : units (zmod p)) :\n  (\u2203 y : units (zmod p), y ^ 2 = x) \u2194 x ^ (p \/ 2) = 1 :=\nbegin\n  cases nat.prime.eq_two_or_odd (fact.out p.prime) with hp2 hp_odd,\n  { substI p, refine iff_of_true \u27e81, _\u27e9 _; apply subsingleton.elim },\n  obtain \u27e8g, hg\u27e9 := is_cyclic.exists_generator (units (zmod p)),\n  obtain \u27e8n, hn\u27e9 : x \u2208 submonoid.powers g, { rw mem_powers_iff_mem_gpowers, apply hg },\n  split,\n  { rintro \u27e8y, rfl\u27e9, rw [\u2190 pow_mul, two_mul_odd_div_two hp_odd, units_pow_card_sub_one_eq_one], },\n  { subst x, assume h,\n    have key : 2 * (p \/ 2) \u2223 n * (p \/ 2),\n    { rw [\u2190 pow_mul] at h,\n      rw [two_mul_odd_div_two hp_odd, \u2190 card_units, \u2190 order_of_eq_card_of_forall_mem_gpowers hg],\n      apply order_of_dvd_of_pow_eq_one h },\n    have : 0 < p \/ 2 := nat.div_pos (fact.out (1 < p)) dec_trivial,\n    obtain \u27e8m, rfl\u27e9 := dvd_of_mul_dvd_mul_right this key,\n    refine \u27e8g ^ m, _\u27e9,\n    rw [mul_comm, pow_mul], },\nend\n\n\/-- Euler's Criterion: a nonzero `a : zmod p` is a square if and only if `x ^ (p \/ 2) = 1`. -\/\nlemma euler_criterion {a : zmod p} (ha : a \u2260 0) :\n  (\u2203 y : zmod p, y ^ 2 = a) \u2194 a ^ (p \/ 2) = 1 :=\nbegin\n  apply (iff_congr _ (by simp [units.ext_iff])).mp (euler_criterion_units p (units.mk0 a ha)),\n  simp only [units.ext_iff, sq, units.coe_mk0, units.coe_mul],\n  split, { rintro \u27e8y, hy\u27e9, exact \u27e8y, hy\u27e9 },\n  { rintro \u27e8y, rfl\u27e9,\n    have hy : y \u2260 0, { rintro rfl, simpa [zero_pow] using ha, },\n    refine \u27e8units.mk0 y hy, _\u27e9, simp, }\nend\n\nlemma exists_sq_eq_neg_one_iff_mod_four_ne_three :\n  (\u2203 y : zmod p, y ^ 2 = -1) \u2194 p % 4 \u2260 3 :=\nbegin\n  cases nat.prime.eq_two_or_odd (fact.out p.prime) with hp2 hp_odd,\n  { substI p, exact dec_trivial },\n  haveI := fact.mk hp_odd,\n  have neg_one_ne_zero : (-1 : zmod p) \u2260 0, from mt neg_eq_zero.1 one_ne_zero,\n  rw [euler_criterion p neg_one_ne_zero, neg_one_pow_eq_pow_mod_two],\n  cases mod_two_eq_zero_or_one (p \/ 2) with p_half_even p_half_odd,\n  { rw [p_half_even, pow_zero, eq_self_iff_true, true_iff],\n    contrapose! p_half_even with hp,\n    rw [\u2190 nat.mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp],\n    exact dec_trivial },\n  { rw [p_half_odd, pow_one,\n        iff_false_intro (ne_neg_self p one_ne_zero).symm, false_iff, not_not],\n    rw [\u2190 nat.mod_mul_right_div_self, show 2 * 2 = 4, from rfl] at p_half_odd,\n    rw [\u2190 nat.mod_mul_left_mod _ 2, show 2 * 2 = 4, from rfl] at hp_odd,\n    have hp : p % 4 < 4, from nat.mod_lt _ dec_trivial,\n    revert hp hp_odd p_half_odd,\n    generalize : p % 4 = k, dec_trivial! }\nend\n\nlemma pow_div_two_eq_neg_one_or_one {a : zmod p} (ha : a \u2260 0) :\n  a ^ (p \/ 2) = 1 \u2228 a ^ (p \/ 2) = -1 :=\nbegin\n  cases nat.prime.eq_two_or_odd (fact.out p.prime) with hp2 hp_odd,\n  { substI p, revert a ha, exact dec_trivial },\n  rw [\u2190 mul_self_eq_one_iff, \u2190 pow_add, \u2190 two_mul, two_mul_odd_div_two hp_odd],\n  exact pow_card_sub_one_eq_one ha\nend\n\n\/-- **Wilson's Lemma**: the product of `1`, ..., `p-1` is `-1` modulo `p`. -\/\n@[simp] lemma wilsons_lemma : ((p - 1)! : zmod p) = -1 :=\nbegin\n  refine\n  calc ((p - 1)! : zmod p) = (\u220f x in Ico 1 (succ (p - 1)), x) :\n    by rw [\u2190 finset.prod_Ico_id_eq_factorial, prod_nat_cast]\n                               ... = (\u220f x : units (zmod p), x) : _\n                               ... = -1 : by simp_rw [\u2190 units.coe_hom_apply,\n    \u2190 (units.coe_hom (zmod p)).map_prod, prod_univ_units_id_eq_neg_one, units.coe_hom_apply,\n    units.coe_neg, units.coe_one],\n  have hp : 0 < p := (fact.out p.prime).pos,\n  symmetry,\n  refine prod_bij (\u03bb a _, (a : zmod p).val) _ _ _ _,\n  { intros a ha,\n    rw [Ico.mem, \u2190 nat.succ_sub hp, nat.succ_sub_one],\n    split,\n    { apply nat.pos_of_ne_zero, rw \u2190 @val_zero p,\n      assume h, apply units.ne_zero a (val_injective p h) },\n    { exact val_lt _ } },\n  { intros a ha, simp only [cast_id, nat_cast_val], },\n  { intros _ _ _ _ h, rw units.ext_iff, exact val_injective p h },\n  { intros b hb,\n    rw [Ico.mem, nat.succ_le_iff, \u2190 succ_sub hp, succ_sub_one, pos_iff_ne_zero] at hb,\n    refine \u27e8units.mk0 b _, finset.mem_univ _, _\u27e9,\n    { assume h, apply hb.1, apply_fun val at h,\n      simpa only [val_cast_of_lt hb.right, val_zero] using h },\n    { simp only [val_cast_of_lt hb.right, units.coe_mk0], } }\nend\n\n@[simp] lemma prod_Ico_one_prime : (\u220f x in Ico 1 p, (x : zmod p)) = -1 :=\nbegin\n  conv in (Ico 1 p) { rw [\u2190 succ_sub_one p, succ_sub (fact.out p.prime).pos] },\n  rw [\u2190 prod_nat_cast, finset.prod_Ico_id_eq_factorial, wilsons_lemma]\nend\n\nend zmod\n\n\/-- The image of the map sending a non zero natural number `x \u2264 p \/ 2` to the absolute value\n  of the element of interger in the interval `(-p\/2, p\/2]` congruent to `a * x` mod p is the set\n  of non zero natural numbers `x` such that `x \u2264 p \/ 2` -\/\nlemma Ico_map_val_min_abs_nat_abs_eq_Ico_map_id\n  (p : \u2115) [hp : fact p.prime] (a : zmod p) (hap : a \u2260 0) :\n  (Ico 1 (p \/ 2).succ).1.map (\u03bb x, (a * x).val_min_abs.nat_abs) =\n  (Ico 1 (p \/ 2).succ).1.map (\u03bb a, a) :=\nbegin\n  have he : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 x \u2260 0 \u2227 x \u2264 p \/ 2,\n    by simp [nat.lt_succ_iff, nat.succ_le_iff, pos_iff_ne_zero] {contextual := tt},\n  have hep : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 x < p,\n    from \u03bb x hx, lt_of_le_of_lt (he hx).2 (nat.div_lt_self hp.1.pos dec_trivial),\n  have hpe : \u2200 {x}, x \u2208 Ico 1 (p \/ 2).succ \u2192 \u00ac p \u2223 x,\n    from \u03bb x hx hpx, not_lt_of_ge (le_of_dvd (nat.pos_of_ne_zero (he hx).1) hpx) (hep hx),\n  have hmem : \u2200 (x : \u2115) (hx : x \u2208 Ico 1 (p \/ 2).succ),\n    (a * x : zmod p).val_min_abs.nat_abs \u2208 Ico 1 (p \/ 2).succ,\n  { assume x hx,\n    simp [hap, char_p.cast_eq_zero_iff (zmod p) p, hpe hx, lt_succ_iff, succ_le_iff,\n        pos_iff_ne_zero, nat_abs_val_min_abs_le _], },\n  have hsurj : \u2200 (b : \u2115) (hb : b \u2208 Ico 1 (p \/ 2).succ),\n    \u2203 x \u2208 Ico 1 (p \/ 2).succ, b = (a * x : zmod p).val_min_abs.nat_abs,\n  { assume b hb,\n    refine \u27e8(b \/ a : zmod p).val_min_abs.nat_abs, Ico.mem.mpr \u27e8_, _\u27e9, _\u27e9,\n    { apply nat.pos_of_ne_zero,\n      simp only [div_eq_mul_inv, hap, char_p.cast_eq_zero_iff (zmod p) p, hpe hb, not_false_iff,\n        val_min_abs_eq_zero, inv_eq_zero, int.nat_abs_eq_zero, ne.def, mul_eq_zero, or_self] },\n      { apply lt_succ_of_le, apply nat_abs_val_min_abs_le },\n      { rw nat_cast_nat_abs_val_min_abs,\n        split_ifs,\n        { erw [mul_div_cancel' _ hap, val_min_abs_def_pos, val_cast_of_lt (hep hb),\n            if_pos (le_of_lt_succ (Ico.mem.1 hb).2), int.nat_abs_of_nat], },\n        { erw [mul_neg_eq_neg_mul_symm, mul_div_cancel' _ hap, nat_abs_val_min_abs_neg,\n            val_min_abs_def_pos, val_cast_of_lt (hep hb), if_pos (le_of_lt_succ (Ico.mem.1 hb).2),\n            int.nat_abs_of_nat] } } },\n  exact multiset.map_eq_map_of_bij_of_nodup _ _ (finset.nodup _) (finset.nodup _)\n    (\u03bb x _, (a * x : zmod p).val_min_abs.nat_abs) hmem (\u03bb _ _, rfl)\n    (inj_on_of_surj_on_of_card_le _ hmem hsurj (le_refl _)) hsurj\nend\n\nprivate lemma gauss_lemma_aux\u2081 (p : \u2115) [fact p.prime] [fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  (a^(p \/ 2) * (p \/ 2)! : zmod p) =\n  (-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card * (p \/ 2)! :=\ncalc (a ^ (p \/ 2) * (p \/ 2)! : zmod p) =\n    (\u220f x in Ico 1 (p \/ 2).succ, a * x) :\n  by rw [prod_mul_distrib, \u2190 prod_nat_cast, \u2190 prod_nat_cast, prod_Ico_id_eq_factorial,\n      prod_const, Ico.card, succ_sub_one]; simp\n... = (\u220f x in Ico 1 (p \/ 2).succ, (a * x : zmod p).val) : by simp\n... = (\u220f x in Ico 1 (p \/ 2).succ,\n    (if (a * x : zmod p).val \u2264 p \/ 2 then 1 else -1) *\n      (a * x : zmod p).val_min_abs.nat_abs) :\n  prod_congr rfl $ \u03bb _ _, begin\n    simp only [nat_cast_nat_abs_val_min_abs],\n    split_ifs; simp\n  end\n... = (-1)^((Ico 1 (p \/ 2).succ).filter\n      (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card *\n    (\u220f x in Ico 1 (p \/ 2).succ, (a * x : zmod p).val_min_abs.nat_abs) :\n  have (\u220f x in Ico 1 (p \/ 2).succ,\n        if (a * x : zmod p).val \u2264 p \/ 2 then (1 : zmod p) else -1) =\n      (\u220f x in (Ico 1 (p \/ 2).succ).filter\n        (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2), -1),\n    from prod_bij_ne_one (\u03bb x _ _, x)\n      (\u03bb x, by split_ifs; simp * at * {contextual := tt})\n      (\u03bb _ _ _ _ _ _, id)\n      (\u03bb b h _, \u27e8b, by simp [-not_le, *] at *\u27e9)\n      (by intros; split_ifs at *; simp * at *),\n  by rw [prod_mul_distrib, this]; simp\n... = (-1)^((Ico 1 (p \/ 2).succ).filter\n      (\u03bb x : \u2115, \u00ac(a * x : zmod p).val \u2264 p \/ 2)).card * (p \/ 2)! :\n  by rw [\u2190 prod_nat_cast, finset.prod_eq_multiset_prod,\n      Ico_map_val_min_abs_nat_abs_eq_Ico_map_id p a hap,\n      \u2190 finset.prod_eq_multiset_prod, prod_Ico_id_eq_factorial]\n\nprivate lemma gauss_lemma_aux\u2082 (p : \u2115) [hp : fact p.prime] [fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  (a^(p \/ 2) : zmod p) = (-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card :=\n(mul_left_inj'\n    (show ((p \/ 2)! : zmod p) \u2260 0,\n      by rw [ne.def, char_p.cast_eq_zero_iff (zmod p) p, hp.1.dvd_factorial, not_le];\n          exact nat.div_lt_self hp.1.pos dec_trivial)).1 $\n  by simpa using gauss_lemma_aux\u2081 p hap\n\nprivate lemma eisenstein_lemma_aux\u2081 (p : \u2115) [fact p.prime] [hp2 : fact (p % 2 = 1)]\n  {a : \u2115} (hap : (a : zmod p) \u2260 0) :\n  ((\u2211 x in Ico 1 (p \/ 2).succ, a * x : \u2115) : zmod 2) =\n    ((Ico 1 (p \/ 2).succ).filter\n      ((\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val))).card +\n      \u2211 x in Ico 1 (p \/ 2).succ, x\n    + (\u2211 x in Ico 1 (p \/ 2).succ, (a * x) \/ p : \u2115) :=\nhave hp2 : (p : zmod 2) = (1 : \u2115), from (eq_iff_modeq_nat _).2 hp2.1,\ncalc ((\u2211 x in Ico 1 (p \/ 2).succ, a * x : \u2115) : zmod 2)\n    = ((\u2211 x in Ico 1 (p \/ 2).succ, ((a * x) % p + p * ((a * x) \/ p)) : \u2115) : zmod 2) :\n  by simp only [mod_add_div]\n... = (\u2211 x in Ico 1 (p \/ 2).succ, ((a * x : \u2115) : zmod p).val : \u2115) +\n    (\u2211 x in Ico 1 (p \/ 2).succ, (a * x) \/ p : \u2115) :\n  by simp only [val_nat_cast];\n    simp [sum_add_distrib, mul_sum.symm, nat.cast_add, nat.cast_mul, nat.cast_sum, hp2]\n... = _ : congr_arg2 (+)\n  (calc ((\u2211 x in Ico 1 (p \/ 2).succ, ((a * x : \u2115) : zmod p).val : \u2115) : zmod 2)\n      = \u2211 x in Ico 1 (p \/ 2).succ,\n          ((((a * x : zmod p).val_min_abs +\n            (if (a * x : zmod p).val \u2264 p \/ 2 then 0 else p)) : \u2124) : zmod 2) :\n        by simp only [(val_eq_ite_val_min_abs _).symm]; simp [nat.cast_sum]\n  ... = ((Ico 1 (p \/ 2).succ).filter\n        (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card +\n      ((\u2211 x in Ico 1 (p \/ 2).succ, (a * x : zmod p).val_min_abs.nat_abs) : \u2115) :\n    by { simp [ite_cast, add_comm, sum_add_distrib, finset.sum_ite, hp2, nat.cast_sum], }\n  ... = _ : by rw [finset.sum_eq_multiset_sum,\n      Ico_map_val_min_abs_nat_abs_eq_Ico_map_id p a hap,\n      \u2190 finset.sum_eq_multiset_sum];\n    simp [nat.cast_sum]) rfl\n\nprivate lemma eisenstein_lemma_aux\u2082 (p : \u2115) [fact p.prime] [fact (p % 2 = 1)]\n  {a : \u2115} (ha2 : a % 2 = 1) (hap : (a : zmod p) \u2260 0) :\n  ((Ico 1 (p \/ 2).succ).filter\n    ((\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val))).card\n  \u2261 \u2211 x in Ico 1 (p \/ 2).succ, (x * a) \/ p [MOD 2] :=\nhave ha2 : (a : zmod 2) = (1 : \u2115), from (eq_iff_modeq_nat _).2 ha2,\n(eq_iff_modeq_nat 2).1 $ sub_eq_zero.1 $\n  by simpa [add_left_comm, sub_eq_add_neg, finset.mul_sum.symm, mul_comm, ha2, nat.cast_sum,\n            add_neg_eq_iff_eq_add.symm, neg_eq_self_mod_two, add_assoc]\n    using eq.symm (eisenstein_lemma_aux\u2081 p hap)\n\nlemma div_eq_filter_card {a b c : \u2115} (hb0 : 0 < b) (hc : a \/ b \u2264 c) : a \/ b =\n  ((Ico 1 c.succ).filter (\u03bb x, x * b \u2264 a)).card :=\ncalc a \/ b = (Ico 1 (a \/ b).succ).card : by simp\n... = ((Ico 1 c.succ).filter (\u03bb x, x * b \u2264 a)).card :\n  congr_arg _ $ finset.ext $ \u03bb x,\n    have x * b \u2264 a \u2192 x \u2264 c,\n      from \u03bb h, le_trans (by rwa [le_div_iff_mul_le _ _ hb0]) hc,\n    by simp [lt_succ_iff, le_div_iff_mul_le _ _ hb0]; tauto\n\n\/-- The given sum is the number of integer points in the triangle formed by the diagonal of the\n  rectangle `(0, p\/2) \u00d7 (0, q\/2)`  -\/\nprivate lemma sum_Ico_eq_card_lt {p q : \u2115} :\n  \u2211 a in Ico 1 (p \/ 2).succ, (a * q) \/ p =\n  (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n  (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q)).card :=\nif hp0 : p = 0 then by simp [hp0, finset.ext_iff]\nelse\n  calc \u2211 a in Ico 1 (p \/ 2).succ, (a * q) \/ p =\n    \u2211 a in Ico 1 (p \/ 2).succ,\n      ((Ico 1 (q \/ 2).succ).filter (\u03bb x, x * p \u2264 a * q)).card :\n    finset.sum_congr rfl $ \u03bb x hx,\n      div_eq_filter_card (nat.pos_of_ne_zero hp0)\n        (calc x * q \/ p \u2264 (p \/ 2) * q \/ p :\n            nat.div_le_div_right (mul_le_mul_of_nonneg_right\n              (le_of_lt_succ $ by finish)\n              (nat.zero_le _))\n          ... \u2264 _ : nat.div_mul_div_le_div _ _ _)\n  ... = _ : by rw [\u2190 card_sigma];\n    exact card_congr (\u03bb a _, \u27e8a.1, a.2\u27e9)\n      (by simp only [mem_filter, mem_sigma, and_self, forall_true_iff, mem_product]\n        {contextual := tt})\n      (\u03bb \u27e8_, _\u27e9 \u27e8_, _\u27e9, by simp only [prod.mk.inj_iff, eq_self_iff_true, and_self, heq_iff_eq,\n        forall_true_iff] {contextual := tt})\n      (\u03bb \u27e8b\u2081, b\u2082\u27e9 h, \u27e8\u27e8b\u2081, b\u2082\u27e9,\n        by revert h; simp only [mem_filter, eq_self_iff_true, exists_prop_of_true, mem_sigma,\n          and_self, forall_true_iff, mem_product] {contextual := tt}\u27e9)\n\n\/-- Each of the sums in this lemma is the cardinality of the set integer points in each of the\n  two triangles formed by the diagonal of the rectangle `(0, p\/2) \u00d7 (0, q\/2)`. Adding them\n  gives the number of points in the rectangle. -\/\nprivate lemma sum_mul_div_add_sum_mul_div_eq_mul (p q : \u2115) [hp : fact p.prime]\n  (hq0 : (q : zmod p) \u2260 0) :\n  \u2211 a in Ico 1 (p \/ 2).succ, (a * q) \/ p +\n  \u2211 a in Ico 1 (q \/ 2).succ, (a * p) \/ q =\n  (p \/ 2) * (q \/ 2) :=\nbegin\n  have hswap : (((Ico 1 (q \/ 2).succ).product (Ico 1 (p \/ 2).succ)).filter\n    (\u03bb x : \u2115 \u00d7 \u2115, x.2 * q \u2264 x.1 * p)).card =\n  (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n    (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p)).card :=\n  card_congr (\u03bb x _, prod.swap x)\n    (\u03bb \u27e8_, _\u27e9, by simp only [mem_filter, and_self, prod.swap_prod_mk, forall_true_iff, mem_product]\n      {contextual := tt})\n    (\u03bb \u27e8_, _\u27e9 \u27e8_, _\u27e9, by simp only [prod.mk.inj_iff, eq_self_iff_true, and_self, prod.swap_prod_mk,\n      forall_true_iff] {contextual := tt})\n    (\u03bb \u27e8x\u2081, x\u2082\u27e9 h, \u27e8\u27e8x\u2082, x\u2081\u27e9, by revert h; simp only [mem_filter, eq_self_iff_true, and_self,\n      exists_prop_of_true, prod.swap_prod_mk, forall_true_iff, mem_product] {contextual := tt}\u27e9),\n  have hdisj : disjoint\n    (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q))\n    (((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p)),\n  { apply disjoint_filter.2 (\u03bb x hx hpq hqp, _),\n    have hxp : x.1 < p, from lt_of_le_of_lt\n      (show x.1 \u2264 p \/ 2, by simp only [*, lt_succ_iff, Ico.mem, mem_product] at *; tauto)\n      (nat.div_lt_self hp.1.pos dec_trivial),\n    have : (x.1 : zmod p) = 0,\n      { simpa [hq0] using congr_arg (coe : \u2115 \u2192 zmod p) (le_antisymm hpq hqp) },\n    apply_fun zmod.val at this,\n    rw [val_cast_of_lt hxp, val_zero] at this,\n    simpa only [this, nonpos_iff_eq_zero, Ico.mem, one_ne_zero, false_and, mem_product] using hx },\n  have hunion : ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.2 * p \u2264 x.1 * q) \u222a\n    ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)).filter\n      (\u03bb x : \u2115 \u00d7 \u2115, x.1 * q \u2264 x.2 * p) =\n    ((Ico 1 (p \/ 2).succ).product (Ico 1 (q \/ 2).succ)),\n  from finset.ext (\u03bb x, by have := le_total (x.2 * p) (x.1 * q);\n    simp only [mem_union, mem_filter, Ico.mem, mem_product]; tauto),\n  rw [sum_Ico_eq_card_lt, sum_Ico_eq_card_lt, hswap, \u2190 card_disjoint_union hdisj, hunion,\n    card_product],\n  simp only [Ico.card, nat.sub_zero, succ_sub_succ_eq_sub]\nend\n\nvariables (p q : \u2115) [fact p.prime] [fact q.prime]\n\nnamespace zmod\n\n\/-- The Legendre symbol of `a` and `p` is an integer defined as\n\n* `0` if `a` is `0` modulo `p`;\n* `1` if `a ^ (p \/ 2)` is `1` modulo `p`\n   (by `euler_criterion` this is equivalent to \u201c`a` is a square modulo `p`\u201d);\n* `-1` otherwise.\n\n-\/\ndef legendre_sym (a p : \u2115) : \u2124 :=\nif      (a : zmod p) = 0           then  0\nelse if (a : zmod p) ^ (p \/ 2) = 1 then  1\n                                   else -1\n\nlemma legendre_sym_eq_pow (a p : \u2115) [hp : fact p.prime] :\n  (legendre_sym a p : zmod p) = (a ^ (p \/ 2)) :=\nbegin\n  rw legendre_sym,\n  by_cases ha : (a : zmod p) = 0,\n  { simp only [if_pos, ha, zero_pow (nat.div_pos (hp.1.two_le) (succ_pos 1)), int.cast_zero] },\n  cases hp.1.eq_two_or_odd with hp2 hp_odd,\n  { substI p,\n    generalize : (a : (zmod 2)) = b, revert b, dec_trivial, },\n  { haveI := fact.mk hp_odd,\n    rw if_neg ha,\n    have : (-1 : zmod p) \u2260 1, from (ne_neg_self p one_ne_zero).symm,\n    cases pow_div_two_eq_neg_one_or_one p ha with h h,\n    { rw [if_pos h, h, int.cast_one], },\n    { rw [h, if_neg this, int.cast_neg, int.cast_one], } }\nend\n\nlemma legendre_sym_eq_one_or_neg_one (a p : \u2115) (ha : (a : zmod p) \u2260 0) :\n  legendre_sym a p = -1 \u2228 legendre_sym a p = 1 :=\nby unfold legendre_sym; split_ifs; simp only [*, eq_self_iff_true, or_true, true_or] at *\n\nlemma legendre_sym_eq_zero_iff (a p : \u2115) :\n  legendre_sym a p = 0 \u2194 (a : zmod p) = 0 :=\nbegin\n  split,\n  { classical, contrapose,\n    assume ha, cases legendre_sym_eq_one_or_neg_one a p ha with h h,\n    all_goals { rw h, norm_num } },\n  { assume ha, rw [legendre_sym, if_pos ha] }\nend\n\n\/-- Gauss' lemma. The legendre symbol can be computed by considering the number of naturals less\n  than `p\/2` such that `(a * x) % p > p \/ 2` -\/\nlemma gauss_lemma {a : \u2115} [fact (p % 2 = 1)] (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = (-1) ^ ((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card :=\nhave (legendre_sym a p : zmod p) = (((-1)^((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card : \u2124) : zmod p),\n  by rw [legendre_sym_eq_pow, gauss_lemma_aux\u2082 p ha0]; simp,\nbegin\n  cases legendre_sym_eq_one_or_neg_one a p ha0;\n  cases neg_one_pow_eq_or \u2124 ((Ico 1 (p \/ 2).succ).filter\n    (\u03bb x : \u2115, p \/ 2 < (a * x : zmod p).val)).card;\n  simp [*, ne_neg_self p one_ne_zero, (ne_neg_self p one_ne_zero).symm] at *\nend\n\nlemma legendre_sym_eq_one_iff {a : \u2115} (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = 1 \u2194 (\u2203 b : zmod p, b ^ 2 = a) :=\nbegin\n  rw [euler_criterion p ha0, legendre_sym, if_neg ha0],\n  split_ifs,\n  { simp only [h, eq_self_iff_true] },\n  finish -- this is quite slow. I'm actually surprised that it can close the goal at all!\nend\n\nlemma eisenstein_lemma [fact (p % 2 = 1)] {a : \u2115} (ha1 : a % 2 = 1) (ha0 : (a : zmod p) \u2260 0) :\n  legendre_sym a p = (-1)^\u2211 x in Ico 1 (p \/ 2).succ, (x * a) \/ p :=\nby rw [neg_one_pow_eq_pow_mod_two, gauss_lemma p ha0, neg_one_pow_eq_pow_mod_two,\n    show _ = _, from eisenstein_lemma_aux\u2082 p ha1 ha0]\n\n\/-- **Quadratic reciprocity theorem** -\/\ntheorem quadratic_reciprocity [hp1 : fact (p % 2 = 1)] [hq1 : fact (q % 2 = 1)] (hpq : p \u2260 q) :\n  legendre_sym p q * legendre_sym q p = (-1) ^ ((p \/ 2) * (q \/ 2)) :=\nhave hpq0 : (p : zmod q) \u2260 0, from prime_ne_zero q p hpq.symm,\nhave hqp0 : (q : zmod p) \u2260 0, from prime_ne_zero p q hpq,\nby rw [eisenstein_lemma q hp1.1 hpq0, eisenstein_lemma p hq1.1 hqp0,\n  \u2190 pow_add, sum_mul_div_add_sum_mul_div_eq_mul q p hpq0, mul_comm]\n\n-- move this\nlocal attribute [instance]\nlemma fact_prime_two : fact (nat.prime 2) := \u27e8nat.prime_two\u27e9\n\nlemma legendre_sym_two [hp1 : fact (p % 2 = 1)] : legendre_sym 2 p = (-1) ^ (p \/ 4 + p \/ 2) :=\nhave hp2 : p \u2260 2, from mt (congr_arg (% 2)) (by simpa using hp1.1),\nhave hp22 : p \/ 2 \/ 2 = _ := div_eq_filter_card (show 0 < 2, from dec_trivial)\n  (nat.div_le_self (p \/ 2) 2),\nhave hcard : (Ico 1 (p \/ 2).succ).card = p \/ 2, by simp,\nhave hx2 : \u2200 x \u2208 Ico 1 (p \/ 2).succ, (2 * x : zmod p).val = 2 * x,\n  from \u03bb x hx, have h2xp : 2 * x < p,\n      from calc 2 * x \u2264 2 * (p \/ 2) : mul_le_mul_of_nonneg_left\n        (le_of_lt_succ $ by finish) dec_trivial\n      ... < _ : by conv_rhs {rw [\u2190 div_add_mod p 2, hp1.1]}; exact lt_succ_self _,\n    by rw [\u2190 nat.cast_two, \u2190 nat.cast_mul, val_cast_of_lt h2xp],\nhave hdisj : disjoint\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, p \/ 2 < ((2 : \u2115) * x : zmod p).val))\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, x * 2 \u2264 p \/ 2)),\n  from disjoint_filter.2 (\u03bb x hx, by simp [hx2 _ hx, mul_comm]),\nhave hunion :\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, p \/ 2 < ((2 : \u2115) * x : zmod p).val)) \u222a\n    ((Ico 1 (p \/ 2).succ).filter (\u03bb x, x * 2 \u2264 p \/ 2)) =\n    Ico 1 (p \/ 2).succ,\n  begin\n    rw [filter_union_right],\n    conv_rhs {rw [\u2190 @filter_true _ (Ico 1 (p \/ 2).succ)]},\n    exact filter_congr (\u03bb x hx, by simp [hx2 _ hx, lt_or_le, mul_comm])\n  end,\nbegin\n  rw [gauss_lemma p (prime_ne_zero p 2 hp2),\n    neg_one_pow_eq_pow_mod_two, @neg_one_pow_eq_pow_mod_two _ _ (p \/ 4 + p \/ 2)],\n  refine congr_arg2 _ rfl ((eq_iff_modeq_nat 2).1 _),\n  rw [show 4 = 2 * 2, from rfl, \u2190 nat.div_div_eq_div_mul, hp22, nat.cast_add,\n    \u2190 sub_eq_iff_eq_add', sub_eq_add_neg, neg_eq_self_mod_two,\n    \u2190 nat.cast_add, \u2190 card_disjoint_union hdisj, hunion, hcard]\nend\n\nlemma exists_sq_eq_two_iff [hp1 : fact (p % 2 = 1)] :\n  (\u2203 a : zmod p, a ^ 2 = 2) \u2194 p % 8 = 1 \u2228 p % 8 = 7 :=\nhave hp2 : ((2 : \u2115) : zmod p) \u2260 0,\n  from prime_ne_zero p 2 (\u03bb h, by simpa [h] using hp1.1),\nhave hpm4 : p % 4 = p % 8 % 4, from (nat.mod_mul_left_mod p 2 4).symm,\nhave hpm2 : p % 2 = p % 8 % 2, from (nat.mod_mul_left_mod p 4 2).symm,\nbegin\n  rw [show (2 : zmod p) = (2 : \u2115), by simp, \u2190 legendre_sym_eq_one_iff p hp2,\n    legendre_sym_two p, neg_one_pow_eq_one_iff_even (show (-1 : \u2124) \u2260 1, from dec_trivial),\n    even_add, even_div, even_div],\n  have := nat.mod_lt p (show 0 < 8, from dec_trivial),\n  resetI, rw fact_iff at hp1,\n  revert this hp1,\n  erw [hpm4, hpm2],\n  generalize hm : p % 8 = m, unfreezingI {clear_dependent p},\n  dec_trivial!,\nend\n\nlemma exists_sq_eq_prime_iff_of_mod_four_eq_one (hp1 : p % 4 = 1) [hq1 : fact (q % 2 = 1)] :\n  (\u2203 a : zmod p, a ^ 2 = q) \u2194 \u2203 b : zmod q, b ^ 2 = p :=\nif hpq : p = q then by substI hpq else\nhave h1 : ((p \/ 2) * (q \/ 2)) % 2 = 0,\n  from (dvd_iff_mod_eq_zero _ _).1\n    (dvd_mul_of_dvd_left ((dvd_iff_mod_eq_zero _ _).2 $\n    by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp1]; refl) _),\nbegin\n  haveI hp_odd : fact (p % 2 = 1) := \u27e8odd_of_mod_four_eq_one hp1\u27e9,\n  have hpq0 : (p : zmod q) \u2260 0 := prime_ne_zero q p (ne.symm hpq),\n  have hqp0 : (q : zmod p) \u2260 0 := prime_ne_zero p q hpq,\n  have := quadratic_reciprocity p q hpq,\n  rw [neg_one_pow_eq_pow_mod_two, h1, legendre_sym, legendre_sym,\n    if_neg hqp0, if_neg hpq0] at this,\n  rw [euler_criterion q hpq0, euler_criterion p hqp0],\n  split_ifs at this; simp *; contradiction,\nend\n\nlemma exists_sq_eq_prime_iff_of_mod_four_eq_three (hp3 : p % 4 = 3)\n  (hq3 : q % 4 = 3) (hpq : p \u2260 q) : (\u2203 a : zmod p, a ^ 2 = q) \u2194 \u00ac\u2203 b : zmod q, b ^ 2 = p :=\nhave h1 : ((p \/ 2) * (q \/ 2)) % 2 = 1,\n  from nat.odd_mul_odd\n    (by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hp3]; refl)\n    (by rw [\u2190 mod_mul_right_div_self, show 2 * 2 = 4, from rfl, hq3]; refl),\nbegin\n  haveI hp_odd : fact (p % 2 = 1) := \u27e8odd_of_mod_four_eq_three hp3\u27e9,\n  haveI hq_odd : fact (q % 2 = 1) := \u27e8odd_of_mod_four_eq_three hq3\u27e9,\n  have hpq0 : (p : zmod q) \u2260 0 := prime_ne_zero q p (ne.symm hpq),\n  have hqp0 : (q : zmod p) \u2260 0 := prime_ne_zero p q hpq,\n  have := quadratic_reciprocity p q hpq,\n  rw [neg_one_pow_eq_pow_mod_two, h1, legendre_sym, legendre_sym,\n    if_neg hpq0, if_neg hqp0] at this,\n  rw [euler_criterion q hpq0, euler_criterion p hqp0],\n  split_ifs at this; simp *; contradiction\nend\n\nend zmod\n","avg_line_length":46.3454198473,"max_line_length":99,"alphanum_fraction":0.6069178505}
-{"size":16604,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Aur\u00e9lien Saue\n-\/\nimport Lean.Elab.Tactic.Basic\nimport Mathlib.Algebra.Ring.Basic\nimport Mathlib.Tactic.NormNum\n\n\/-!\n# `ring`\n\nEvaluate expressions in the language of commutative (semi)rings.\nBased on <http:\/\/www.cs.ru.nl\/~freek\/courses\/tt-2014\/read\/10.1.1.61.3041.pdf> .\n-\/\n\nopen Lean Parser.Tactic Elab Command Elab.Tactic Meta\n\nnamespace Tactic\nnamespace Ring\n\n\/-- This cache contains data required by the `ring` tactic during execution. -\/\nstructure Cache :=\n  \u03b1 : Expr\n  univ : Level\n  cs : Expr\n\nstructure State :=\n  atoms : Array Expr := #[]\n  numAtoms : Nat     := 0\n\n\/-- The monad that `ring` works in. This is a reader monad containing a cache and\nthe list of atoms-up-to-defeq encountered thus far, used for atom sorting. -\/\nabbrev RingM := ReaderT Cache $ StateRefT State MetaM\n\ndef RingM.run (ty : Expr) (m : RingM \u03b1) : MetaM \u03b1 := do\n  let Level.succ u _ \u2190 getLevel ty | throwError \"fail\"\n  let inst \u2190 synthInstance (mkApp (mkConst ``CommSemiring [u]) ty)\n  (m {\u03b1 := ty, univ := u, cs := inst }).run' {}\n\ndef mkAppCS (f : Name) (args : Array Expr) : RingM Expr := do\n  let c \u2190 read\n  pure $ mkAppN (mkConst f [c.univ]) (#[c.\u03b1, c.cs] ++ args)\n\n\/-- Get the index corresponding to an atomic expression, if it has already been encountered, or\nput it in the list of atoms and return the new index, otherwise. -\/\ndef addAtom (e : Expr) : RingM Nat := do\n  let c \u2190 get\n  for i in [:c.numAtoms] do\n    if \u2190 isDefEq e c.atoms[i] then\n      return i\n  modify \u03bb c => { c with atoms := c.atoms.push e, numAtoms := c.numAtoms + 1}\n  return c.numAtoms\n\n\/-- The normal form that `ring` uses is mediated by the function `horner a x n b := a * x ^ n + b`.\nThe reason we use a definition rather than the (more readable) expression on the right is because\nthis expression contains a number of typeclass arguments in different positions, while `horner`\ncontains only one `CommSemiring` instance at the top level. See also `HornerExpr` for a\ndescription of normal form. -\/\n@[reducible] def horner {\u03b1} [CommSemiring \u03b1] (a x : \u03b1) (n : \u2115) (b : \u03b1) := a * (x ^ n) + b\n\n\/-- Every expression in the language of commutative semirings can be viewed as a sum of monomials,\nwhere each monomial is a product of powers of atoms. We fix a global order on atoms (up to\ndefinitional equality), and then separate the terms according to their smallest atom. So the top\nlevel expression is `a * x^n + b` where `x` is the smallest atom and `n > 0` is a numeral, and\n`n` is maximal (so `a` contains at least one monomial not containing an `x`), and `b` contains no\nmonomials with an `x` (hence all atoms in `b` are larger than `x`).\n\nIf there is no `x` satisfying these constraints, then the expression must be a numeral. Even though\nwe are working over rings, we allow rational constants when these can be interpreted in the ring,\nso we can solve problems like `x \/ 3 = 1 \/ 3 * x` even though these are not technically in the\nlanguage of rings.\n\nThese constraints ensure that there is a unique normal form for each ring expression, and so the\nalgorithm is simply to calculate the normal form of each side and compare for equality.\n\nTo allow us to efficiently pattern match on normal forms, we maintain this inductive type that\nholds a normalized expression together with its structure. All the `Expr`s in this type could be\nremoved without loss of information, and conversely the `horner_expr` structure and the `\u2115` and\n`\u211a` values can be recovered from the top level `Expr`, but we keep both in order to keep proof\nproducing normalization functions efficient. -\/\ninductive HornerExpr : Type\n| const (e : Expr) (coeff : \u2115) : HornerExpr --TODO : coeff in \u211a\n| xadd (e : Expr) (a : HornerExpr) (x : Expr \u00d7 \u2115) (n : Expr \u00d7 \u2115) (b : HornerExpr) : HornerExpr\n\ninstance : Inhabited HornerExpr := \u27e8HornerExpr.const (mkRawNatLit 0) 0\u27e9\n\nnamespace HornerExpr\n\n\/-- Get the expression corresponding to a `HornerExpr`. This can be calculated recursively from\nthe structure, but we cache the exprs in all subterms so that this function can be computed in\nconstant time. -\/\ndef e : HornerExpr \u2192 Expr\n| (HornerExpr.const e _) => e\n| (HornerExpr.xadd e _ _ _ _) => e\n\ninstance : Coe HornerExpr Expr := \u27e8e\u27e9\n\n\/-- Is this expr the constant `0`? -\/\ndef isZero : HornerExpr \u2192 Bool\n| (HornerExpr.const _ c) => c = 0\n| _ => false\n\n\/-- Construct a `xadd` node -\/\ndef xadd' (a : HornerExpr) (x : Expr \u00d7 \u2115) (n : Expr \u00d7 \u2115) (b : HornerExpr) : RingM HornerExpr := do\n  pure $ xadd (\u2190 mkAppCS ``horner #[a, x.1, n.1, b]) a x n b\n\n\/-- Reflexivity conversion for a `HornerExpr`. -\/\ndef reflConv (e : HornerExpr) : RingM (HornerExpr \u00d7 Expr) := do pure (e, \u2190 mkEqRefl e)\n\n\/-- Pretty printer for `horner_expr`. -\/\ndef pp : HornerExpr \u2192 MetaM Format\n| (const e c) => pure $ toString c\n| (xadd e a x (_, n) b) => do\n  let pa \u2190 a.pp\n  let pb \u2190 b.pp\n  let px \u2190 PrettyPrinter.ppExpr x.1\n  return \"(\" ++ pa ++ \") * (\" ++ px ++ \")^\" ++ toString n ++ \" + \" ++ pb\n\nend HornerExpr\n\nopen HornerExpr\n\ntheorem zero_horner {\u03b1} [CommSemiring \u03b1] (x n b) :\n  @horner \u03b1 _ 0 x n b = b :=\nby simp [horner]\n\ntheorem horner_horner {\u03b1} [CommSemiring \u03b1] (a\u2081 x n\u2081 n\u2082 b n')\n  (h : n\u2081 + n\u2082 = n') :\n  @horner \u03b1 _ (horner a\u2081 x n\u2081 0) x n\u2082 b = horner a\u2081 x n' b :=\nby simp [h.symm, horner, pow_add, mul_assoc]\n\n\/-- Evaluate `horner a n x b` where `a` and `b` are already in normal form. -\/\ndef evalHorner : HornerExpr \u2192 Expr \u00d7 \u2115 \u2192 Expr \u00d7 \u2115 \u2192 HornerExpr \u2192 RingM (HornerExpr \u00d7 Expr)\n| ha@(const a coeff), x, n, b => do\n  if coeff = 0 then\n    return (b, \u2190 mkAppCS ``zero_horner #[x.1, n.1, b])\n  else (\u2190 xadd' ha x n b).reflConv\n| ha@(xadd a a\u2081 x\u2081 n\u2081 b\u2081), x, n, b => do\n  if x\u2081.2 = x.2 \u2227 b\u2081.e.numeral? = some 0 then do\n    let n' := mkRawNatLit (n\u2081.2 + n.2)\n    let h \u2190 mkEqRefl n'\n    return (\u2190 xadd' a\u2081 x (n', n\u2081.2 + n.2) b,\n      \u2190 mkAppCS ``horner_horner #[a\u2081, x.1, n\u2081.1, n.1, b, n', h])\n  else (\u2190 xadd' ha x n b).reflConv\n\n\ntheorem const_add_horner {\u03b1} [CommSemiring \u03b1] (k a x n b b') (h : k + b = b') :\n  k + @horner \u03b1 _ a x n b = horner a x n b' :=\nby\n  simp [horner, h.symm, add_comm k, add_assoc]\n\ntheorem horner_add_const {\u03b1} [CommSemiring \u03b1] (a x n b k b') (h : b + k = b') :\n  @horner \u03b1 _ a x n b + k = horner a x n b' :=\nby\n  simp [horner, h.symm ,add_left_comm k, add_assoc]\n\ntheorem horner_add_horner_lt {\u03b1} [CommSemiring \u03b1] (a\u2081 x n\u2081 b\u2081 a\u2082 n\u2082 b\u2082 k a' b')\n  (h\u2081 : n\u2081 + k = n\u2082) (h\u2082 : (a\u2081 + horner a\u2082 x k 0 : \u03b1) = a') (h\u2083 : b\u2081 + b\u2082 = b') :\n  @horner \u03b1 _ a\u2081 x n\u2081 b\u2081 + horner a\u2082 x n\u2082 b\u2082 = horner a' x n\u2081 b' :=\nby\n  rw [\u2190 h\u2081, \u2190 h\u2082, \u2190 h\u2083]\n  simp [horner, add_assoc, add_mul, pow_add, mul_assoc, add_comm n\u2081 k, add_left_comm b\u2081]\n\ntheorem horner_add_horner_gt {\u03b1} [CommSemiring \u03b1] (a\u2081 x n\u2081 b\u2081 a\u2082 n\u2082 b\u2082 k a' b')\n  (h\u2081 : n\u2082 + k = n\u2081) (h\u2082 : (horner a\u2081 x k 0 + a\u2082 : \u03b1) = a') (h\u2083 : b\u2081 + b\u2082 = b') :\n  @horner \u03b1 _ a\u2081 x n\u2081 b\u2081 + horner a\u2082 x n\u2082 b\u2082 = horner a' x n\u2082 b' :=\nby\n  rw [\u2190 h\u2081, \u2190 h\u2082, \u2190 h\u2083]\n  simp [horner, add_assoc, mul_assoc, add_mul, add_comm n\u2082 k, pow_add, add_left_comm b\u2081]\n\ntheorem horner_add_horner_eq {\u03b1} [CommSemiring \u03b1] (a\u2081 x n b\u2081 a\u2082 b\u2082 a' b' t)\n  (h\u2081 : a\u2081 + a\u2082 = a') (h\u2082 : b\u2081 + b\u2082 = b') (h\u2083 : horner a' x n b' = t) :\n  @horner \u03b1 _ a\u2081 x n b\u2081 + horner a\u2082 x n b\u2082 = t :=\nby\n  rw [\u2190 h\u2083, \u2190 h\u2081, \u2190 h\u2082]\n  simp only [horner, add_assoc, mul_assoc, add_mul, zero_mul, zero_add, pow_add, add_left_comm b\u2081]\n\npartial def evalAdd : HornerExpr \u2192 HornerExpr \u2192 RingM (HornerExpr \u00d7 Expr)\n| (const e\u2081 c\u2081), (const e\u2082 c\u2082) => do\n  let (e', p) \u2190 NormNum.eval $ \u2190 mkAdd e\u2081 e\u2082\n  pure (const e' (c\u2081 + c\u2082), p)\n| he\u2081@(const e\u2081 c\u2081), he\u2082@(xadd e\u2082 a x n b) => do\n\n  if c\u2081 = 0 then\n    let p \u2190 mkAppM ``zero_add #[e\u2082]\n    return (he\u2082, p)\n  else\n    let (b', h) \u2190 evalAdd he\u2081 b\n    return (\u2190 xadd' a x n b',\n      \u2190 mkAppCS ``const_add_horner #[e\u2081, a, x.1, n.1, b, b', h])\n| he\u2081@(xadd e\u2081 a x n b), he\u2082@(const e\u2082 c\u2082) => do\n  if c\u2082 = 0 then\n    let p \u2190 mkAppM ``add_zero #[e\u2081]\n    return (he\u2081, p)\n  else\n    let (b', h) \u2190 evalAdd b he\u2082\n    return (\u2190 xadd' a x n b',\n      \u2190 mkAppCS ``horner_add_const #[a, x.1, n.1, b, e\u2082, b', h])\n| he\u2081@(xadd e\u2081 a\u2081 x\u2081 n\u2081 b\u2081), he\u2082@(xadd e\u2082 a\u2082 x\u2082 n\u2082 b\u2082) => do\n  let c \u2190 get\n  if x\u2081.2 < x\u2082.2 then\n    let (b', h) \u2190 evalAdd b\u2081 he\u2082\n    return (\u2190 xadd' a\u2081 x\u2081 n\u2081 b',\n      \u2190 mkAppCS ``horner_add_const #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, e\u2082, b', h])\n  else if x\u2081.2 \u2260 x\u2082.2 then\n    let (b', h) \u2190 evalAdd he\u2081 b\u2082\n    return (\u2190 xadd' a\u2082 x\u2082 n\u2082 b',\n      \u2190 mkAppCS ``const_add_horner #[e\u2081, a\u2082, x\u2082.1, n\u2082.1, b\u2082, b', h])\n  else if n\u2081.2 < n\u2082.2 then do\n    let k := n\u2082.2 - n\u2081.2\n    let ek := mkRawNatLit k\n    let h\u2081 \u2190 mkEqRefl n\u2082.1\n    let c \u2190 read\n    let \u03b10 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]\n    let (a', h\u2082) \u2190 evalAdd a\u2081 (\u2190 xadd' a\u2082 x\u2081 (ek, k) (const \u03b10 0))\n    let (b', h\u2083) \u2190 evalAdd b\u2081 b\u2082\n    return (\u2190 xadd' a' x\u2081 n\u2081 b',\n      \u2190 mkAppCS ``horner_add_horner_lt #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, a\u2082, n\u2082.1, b\u2082, ek, a', b', h\u2081, h\u2082, h\u2083])\n  else if n\u2081.2 \u2260 n\u2082.2 then do\n    let k := n\u2081.2 - n\u2082.2\n    let ek := mkRawNatLit k\n    let h\u2081 \u2190 mkEqRefl n\u2081.1\n    let \u03b10 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]\n    let (a', h\u2082) \u2190 evalAdd (\u2190 xadd' a\u2081 x\u2081 (ek, k) (const \u03b10 0)) a\u2082\n    let (b', h\u2083) \u2190 evalAdd b\u2081 b\u2082\n    return (\u2190 xadd' a' x\u2081 n\u2082 b',\n      \u2190 mkAppCS ``horner_add_horner_gt #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, a\u2082, n\u2082.1, b\u2082, ek, a', b', h\u2081, h\u2082, h\u2083])\n  else do\n    let (a', h\u2081) \u2190 evalAdd a\u2081 a\u2082\n    let (b', h\u2082) \u2190 evalAdd b\u2081 b\u2082\n    let (t, h\u2083) \u2190 evalHorner a' x\u2081 n\u2081 b'\n    return (t, \u2190 mkAppCS ``horner_add_horner_eq\n      #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, a\u2082, b\u2082, a', b', t, h\u2081, h\u2082, h\u2083])\n\n\ntheorem horner_const_mul {\u03b1} [CommSemiring \u03b1] (c a x n b a' b')\n  (h\u2081 : c * a = a') (h\u2082 : c * b = b') :\n  c * @horner \u03b1 _ a x n b = horner a' x n b' :=\nby simp [h\u2082.symm, h\u2081.symm, horner, mul_add, mul_assoc]\n\ntheorem horner_mul_const {\u03b1} [CommSemiring \u03b1] (a x n b c a' b')\n  (h\u2081 : a * c = a') (h\u2082 : b * c = b') :\n  @horner \u03b1 _ a x n b * c = horner a' x n b' :=\nby simp [horner, \u2190 h\u2081, \u2190 h\u2082, add_mul, mul_assoc, mul_comm c]\n\n\/-- Evaluate `k * a` where `k` is a rational numeral and `a` is in normal form. -\/\ndef evalConstMul (k : Expr \u00d7 \u2115) : HornerExpr \u2192 RingM (HornerExpr \u00d7 Expr)\n| (const e coeff) => do\n  let (e', p) \u2190 NormNum.eval $ \u2190 mkMul k.1 e\n  return (const e' (k.2 * coeff), p)\n| (xadd e a x n b) => do\n  let (a', h\u2081) \u2190 evalConstMul k a\n  let (b', h\u2082) \u2190 evalConstMul k b\n  return (\u2190 xadd' a' x n b',\n    \u2190 mkAppCS ``horner_const_mul #[k.1, a, x.1, n.1, b, a', b', h\u2081, h\u2082])\n\n\ntheorem horner_mul_horner_zero {\u03b1} [CommSemiring \u03b1] (a\u2081 x n\u2081 b\u2081 a\u2082 n\u2082 aa t)\n  (h\u2081 : @horner \u03b1 _ a\u2081 x n\u2081 b\u2081 * a\u2082 = aa)\n  (h\u2082 : horner aa x n\u2082 0 = t) :\n  horner a\u2081 x n\u2081 b\u2081 * horner a\u2082 x n\u2082 0 = t :=\nby\n  rw [\u2190 h\u2082, \u2190 h\u2081]\n  simp [horner, add_mul, mul_assoc]\n\ntheorem horner_mul_horner {\u03b1} [CommSemiring \u03b1]\n  (a\u2081 x n\u2081 b\u2081 a\u2082 n\u2082 b\u2082 aa haa ab bb t)\n  (h\u2081 : @horner \u03b1 _ a\u2081 x n\u2081 b\u2081 * a\u2082 = aa)\n  (h\u2082 : horner aa x n\u2082 0 = haa)\n  (h\u2083 : a\u2081 * b\u2082 = ab) (h\u2084 : b\u2081 * b\u2082 = bb)\n  (H : haa + horner ab x n\u2081 bb = t) :\n  horner a\u2081 x n\u2081 b\u2081 * horner a\u2082 x n\u2082 b\u2082 = t :=\nby\n  rw [\u2190 H, \u2190 h\u2082, \u2190 h\u2081, \u2190 h\u2083, \u2190 h\u2084]\n  simp [horner, add_mul, mul_add, mul_assoc, mul_comm b\u2082]\n\n\/-- Evaluate `a * b` where `a` and `b` are in normal form. -\/\npartial def evalMul : HornerExpr \u2192 HornerExpr \u2192 RingM (HornerExpr \u00d7 Expr)\n| (const e\u2081 c\u2081), (const e\u2082 c\u2082) => do\n  let (e', p) \u2190 NormNum.eval $ \u2190 mkMul e\u2081 e\u2082\n  return (const e' (c\u2081 * c\u2082), p)\n| (const e\u2081 c\u2081), e\u2082 =>\n  if c\u2081 = 0 then do\n    let \u03b10 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]\n    let p \u2190 mkAppM ``zero_mul #[e\u2082]\n    return (const \u03b10 0, p)\n  else if c\u2081 = 1 then do\n    let p \u2190  mkAppM ``one_mul #[e\u2082]\n    return (e\u2082, p)\n  else evalConstMul (e\u2081, c\u2081) e\u2082\n| e\u2081, he\u2082@(const e\u2082 c\u2082) => do\n  let p\u2081 \u2190 mkAppM ``mul_comm #[e\u2081, e\u2082]\n  let (e', p\u2082) \u2190 evalMul he\u2082 e\u2081\n  let p \u2190 mkEqTrans p\u2081 p\u2082\n  return (e', p)\n| he\u2081@(xadd e\u2081 a\u2081 x\u2081 n\u2081 b\u2081), he\u2082@(xadd e\u2082 a\u2082 x\u2082 n\u2082 b\u2082) => do\n  if x\u2081.2 < x\u2082.2 then do\n    let (a', h\u2081) \u2190 evalMul a\u2081 he\u2082\n    let (b', h\u2082) \u2190 evalMul b\u2081 he\u2082\n    return (\u2190 xadd' a' x\u2081 n\u2081 b',\n      \u2190 mkAppCS ``horner_mul_const #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, e\u2082, a', b', h\u2081, h\u2082])\n  else if x\u2081.2 \u2260 x\u2082.2 then do\n    let (a', h\u2081) \u2190 evalMul he\u2081 a\u2082\n    let (b', h\u2082) \u2190 evalMul he\u2081 b\u2082\n    return (\u2190 xadd' a' x\u2082 n\u2082 b',\n      \u2190 mkAppCS ``horner_const_mul #[e\u2081, a\u2082, x\u2082.1, n\u2082.1, b\u2082, a', b', h\u2081, h\u2082])\n  else do\n    let (aa, h\u2081) \u2190 evalMul he\u2081 a\u2082\n    let \u03b10 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]\n    let (haa, h\u2082) \u2190 evalHorner aa x\u2081 n\u2082 (const \u03b10 0)\n    if b\u2082.isZero then\n      return (haa, \u2190 mkAppCS ``horner_mul_horner_zero\n        #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, a\u2082, n\u2082.1, aa, haa, h\u2081, h\u2082])\n    else do\n      let (ab, h\u2083) \u2190 evalMul a\u2081 b\u2082\n      let (bb, h\u2084) \u2190 evalMul b\u2081 b\u2082\n      let (t, H) \u2190 evalAdd haa (\u2190 xadd' ab x\u2081 n\u2081 bb)\n      return (t, \u2190 mkAppCS ``horner_mul_horner\n        #[a\u2081, x\u2081.1, n\u2081.1, b\u2081, a\u2082, n\u2082.1, b\u2082, aa, haa, ab, bb, t, h\u2081, h\u2082, h\u2083, h\u2084, H])\n\n\ntheorem horner_pow {\u03b1} [CommSemiring \u03b1] (a x : \u03b1) (n m n' : \u2115) (a') (h\u2081 : n * m = n') (h\u2082 : a ^ m = (a' : \u03b1)) :\n  @horner \u03b1 _ a x n 0 ^ m = horner a' x n' 0 :=\nby\n  simp [h\u2081.symm, h\u2082.symm, horner,  mul_pow a, pow_mul]\n\ntheorem pow_succ_eq {\u03b1} [CommSemiring \u03b1] (a : \u03b1) (n : \u2115) (b c)\n  (h\u2081 : a ^ n = b) (h\u2082 : b * a = c) : a ^ (n + 1) = c :=\nby rw [\u2190 h\u2082, \u2190 h\u2081, pow_succ]\n\n\/-- Evaluate `a ^ n` where `a` is in normal form and `n` is a natural numeral. -\/\npartial def evalPow : HornerExpr \u2192 Expr \u00d7 \u2115 \u2192 RingM (HornerExpr \u00d7 Expr)\n| e, (_, 0) => do\n  let \u03b11 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 1, none]\n  let p \u2190 mkAppM ``pow_zero #[e]\n  pure (const \u03b11 1, p)\n| e, (_, 1) => do\n  let p \u2190 mkAppM ``pow_one #[e]\n  pure (e, p)\n| const e coeff, (e\u2082, m) => do\n  let (e', p) \u2190 NormNum.eval $ \u2190 mkAppM ``HPow.hPow #[e, e\u2082]\n  pure (const e' (coeff ^ m), p)\n| he@(xadd e a x n b), m =>\n  match b.e.numeral? with\n  | some 0 => do\n    let n' := mkRawNatLit (n.2 * m.2)\n    let h\u2081 \u2190 mkEqRefl n'\n    let (a', h\u2082) \u2190 evalPow a m\n    let \u03b10 \u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]\n    pure (\u2190 xadd' a' x (n', n.2 * m.2) (const \u03b10 0),\n      \u2190 mkAppCS ``horner_pow #[a, x.1, n.1, m.1, n', a', h\u2081, h\u2082])\n  | _ => do\n    let e\u2082 := mkRawNatLit (m.2 - 1)\n    let (tl, hl) \u2190 evalPow he (e\u2082, m.2-1)\n    let (t, p\u2082) \u2190 evalMul tl he\n    pure (t, \u2190 mkAppCS ``pow_succ_eq #[e, e\u2082, tl, t, hl, p\u2082])\n\n\ntheorem horner_atom {\u03b1} [CommSemiring \u03b1] (x : \u03b1) : x = horner 1 x 1 0 := by\n  simp [horner]\n\n\/-- Evaluate `a` where `a` is an atom. -\/\ndef evalAtom (e : Expr) : RingM (HornerExpr \u00d7 Expr) := do\n  let i \u2190 addAtom e\n  let zero := const (\u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 0, none]) 0\n  let one := const (\u2190 mkAppOptM ``OfNat.ofNat #[(\u2190 read).\u03b1, mkRawNatLit 1, none]) 1\n  pure (\u2190 xadd' one (e,i) (mkRawNatLit 1,1) zero, \u2190 mkAppCS ``horner_atom #[e])\n\ntheorem subst_into_add {\u03b1} [Add \u03b1] (l r tl tr t)\n  (prl : (l : \u03b1) = tl) (prr : r = tr) (prt : tl + tr = t) : l + r = t :=\nby rw [prl, prr, prt]\n\ntheorem subst_into_mul {\u03b1} [Mul \u03b1] (l r tl tr t)\n  (prl : (l : \u03b1) = tl) (prr : r = tr) (prt : tl * tr = t) : l * r = t :=\nby rw [prl, prr, prt]\n\ntheorem subst_into_pow {\u03b1} [Monoid \u03b1] (l r tl tr t)\n  (prl : (l : \u03b1) = tl) (prr : (r : \u2115) = tr) (prt : tl ^ tr = t) : l ^ r = t :=\nby rw [prl, prr, prt]\n\npartial def eval (e : Expr) : RingM (HornerExpr \u00d7 Expr) :=\n  match e.getAppFnArgs with\n  | (``HAdd.hAdd, #[_,_,_,_,e\u2081,e\u2082]) => do\n    let (e\u2081', p\u2081) \u2190 eval e\u2081\n    let (e\u2082', p\u2082) \u2190 eval e\u2082\n    let (e', p') \u2190 evalAdd e\u2081' e\u2082'\n    let p \u2190 mkAppM ``subst_into_add #[e\u2081, e\u2082, e\u2081', e\u2082', e', p\u2081, p\u2082, p']\n    pure (e', p)\n  | (``HMul.hMul, #[_,_,_,_,e\u2081,e\u2082]) => do\n    let (e\u2081', p\u2081) \u2190 eval e\u2081\n    let (e\u2082', p\u2082) \u2190 eval e\u2082\n    let (e', p') \u2190 evalMul e\u2081' e\u2082'\n    let p \u2190 mkAppM ``subst_into_mul #[e\u2081, e\u2082, e\u2081', e\u2082', e', p\u2081, p\u2082, p']\n    return (e', p)\n  | (``HPow.hPow, #[_,_,_,P,e\u2081,e\u2082]) => do\n    -- let (e\u2082', p\u2082) \u2190 lift $ norm_num.derive e\u2082 <|> refl_conv e\u2082,\n    let (e\u2082', p\u2082) := (e\u2082, \u2190 mkEqRefl e\u2082)\n    match e\u2082'.numeral?, P.getAppFn with\n    | some k, Expr.const ``Monoid.HPow _ _ => do\n      let (e\u2081', p\u2081) \u2190 eval e\u2081\n      let (e', p') \u2190 evalPow e\u2081' (e\u2082, k)\n      let p \u2190 mkAppM ``subst_into_pow #[e\u2081, e\u2082, e\u2081', e\u2082', e', p\u2081, p\u2082, p']\n      return (e', p)\n    | _, _ => evalAtom e\n  | _ =>\n    match e.numeral? with\n    | some n => (const e n).reflConv\n    | _ => evalAtom e\n\nelab \"ring\" : tactic => liftMetaMAtMain fun g => do\n  match (\u2190 instantiateMVars (\u2190 getMVarDecl g).type).getAppFnArgs with\n  | (`Eq, #[ty, e\u2081, e\u2082]) =>\n    let ((e\u2081', p\u2081), (e\u2082', p\u2082)) \u2190 RingM.run ty $ do pure (\u2190 eval e\u2081, \u2190 eval e\u2082)\n    if \u2190 isDefEq e\u2081' e\u2082' then\n      assignExprMVar g (\u2190 mkEqTrans p\u2081 (\u2190 mkEqSymm p\u2082))\n    else\n      throwError \"ring failed, ring expressions not equal: \\n{\u2190 e\u2081'.pp}\\n  !=\\n{\u2190 e\u2082'.pp}\"\n  | _ => throwError \"ring failed: not an equality\"\n","avg_line_length":39.5333333333,"max_line_length":111,"alphanum_fraction":0.5988316068}
-{"size":93620,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2019 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, S\u00e9bastien Gou\u00ebzel, Fr\u00e9d\u00e9ric Dupuis, Heather Macbeth\n-\/\n\nimport linear_algebra.bilinear_form\nimport linear_algebra.sesquilinear_form\nimport topology.metric_space.pi_Lp\nimport data.complex.is_R_or_C\n\n\/-!\n# Inner Product Space\n\nThis file defines inner product spaces and proves its basic properties.\n\nAn inner product space is a vector space endowed with an inner product. It generalizes the notion of\ndot product in `\u211d^n` and provides the means of defining the length of a vector and the angle between\ntwo vectors. In particular vectors `x` and `y` are orthogonal if their inner product equals zero.\nWe define both the real and complex cases at the same time using the `is_R_or_C` typeclass.\n\n## Main results\n\n- We define the class `inner_product_space \ud835\udd5c E` extending `normed_space \ud835\udd5c E` with a number of basic\n  properties, most notably the Cauchy-Schwarz inequality. Here `\ud835\udd5c` is understood to be either `\u211d`\n  or `\u2102`, through the `is_R_or_C` typeclass.\n- We show that if `f i` is an inner product space for each `i`, then so is `\u03a0 i, f i`\n- We define `euclidean_space \ud835\udd5c n` to be `n \u2192 \ud835\udd5c` for any `fintype n`, and show that\n  this an inner product space.\n- Existence of orthogonal projection onto nonempty complete subspace:\n  Let `u` be a point in an inner product space, and let `K` be a nonempty complete subspace.\n  Then there exists a unique `v` in `K` that minimizes the distance `\u2225u - v\u2225` to `u`.\n  The point `v` is usually called the orthogonal projection of `u` onto `K`.\n\n## Notation\n\nWe globally denote the real and complex inner products by `\u27ea\u00b7, \u00b7\u27eb_\u211d` and `\u27ea\u00b7, \u00b7\u27eb_\u2102` respectively.\nWe also provide two notation namespaces: `real_inner_product_space`, `complex_inner_product_space`,\nwhich respectively introduce the plain notation `\u27ea\u00b7, \u00b7\u27eb` for the the real and complex inner product.\n\nThe orthogonal complement of a submodule `K` is denoted by `K\u15ee`.\n\n## Implementation notes\n\nWe choose the convention that inner products are conjugate linear in the first argument and linear\nin the second.\n\n## TODO\n\n- Fix the section on the existence of minimizers and orthogonal projections to make sure that it\n  also applies in the complex case.\n\n## Tags\n\ninner product space, norm\n\n## References\n*  [Cl\u00e9ment & Martin, *The Lax-Milgram Theorem. A detailed proof to be formalized in Coq*]\n*  [Cl\u00e9ment & Martin, *A Coq formal proof of the Lax\u2013Milgram theorem*]\n\nThe Coq code is available at the following address: <http:\/\/www.lri.fr\/~sboldo\/elfic\/index.html>\n-\/\n\nnoncomputable theory\n\nopen is_R_or_C real filter\nopen_locale big_operators classical topological_space\n\nvariables {\ud835\udd5c E F : Type*} [is_R_or_C \ud835\udd5c]\n\n\/-- Syntactic typeclass for types endowed with an inner product -\/\nclass has_inner (\ud835\udd5c E : Type*) := (inner : E \u2192 E \u2192 \ud835\udd5c)\n\nexport has_inner (inner)\n\nnotation `\u27ea`x`, `y`\u27eb_\u211d` := @inner \u211d _ _ x y\nnotation `\u27ea`x`, `y`\u27eb_\u2102` := @inner \u2102 _ _ x y\n\nsection notations\n\nlocalized \"notation `\u27ea`x`, `y`\u27eb` := @inner \u211d _ _ x y\" in real_inner_product_space\nlocalized \"notation `\u27ea`x`, `y`\u27eb` := @inner \u2102 _ _ x y\" in complex_inner_product_space\n\nend notations\n\n\/--\nAn inner product space is a vector space with an additional operation called inner product.\nThe norm could be derived from the inner product, instead we require the existence of a norm and\nthe fact that `\u2225x\u2225^2 = re \u27eax, x\u27eb` to be able to put instances on `\ud835\udd42` or product\nspaces.\n\nTo construct a norm from an inner product, see `inner_product_space.of_core`.\n-\/\nclass inner_product_space (\ud835\udd5c : Type*) (E : Type*) [is_R_or_C \ud835\udd5c]\n  extends normed_group E, normed_space \ud835\udd5c E, has_inner \ud835\udd5c E :=\n(norm_sq_eq_inner : \u2200 (x : E), \u2225x\u2225^2 = re (inner x x))\n(conj_sym  : \u2200 x y, conj (inner y x) = inner x y)\n(nonneg_im : \u2200 x, im (inner x x) = 0)\n(add_left  : \u2200 x y z, inner (x + y) z = inner x z + inner y z)\n(smul_left : \u2200 x y r, inner (r \u2022 x) y = (conj r) * inner x y)\n\nattribute [nolint dangerous_instance] inner_product_space.to_normed_group\n-- note [is_R_or_C instance]\n\n\/-!\n### Constructing a normed space structure from an inner product\n\nIn the definition of an inner product space, we require the existence of a norm, which is equal\n(but maybe not defeq) to the square root of the scalar product. This makes it possible to put\nan inner product space structure on spaces with a preexisting norm (for instance `\u211d`), with good\nproperties. However, sometimes, one would like to define the norm starting only from a well-behaved\nscalar product. This is what we implement in this paragraph, starting from a structure\n`inner_product_space.core` stating that we have a nice scalar product.\n\nOur goal here is not to develop a whole theory with all the supporting API, as this will be done\nbelow for `inner_product_space`. Instead, we implement the bare minimum to go as directly as\npossible to the construction of the norm and the proof of the triangular inequality.\n\nWarning: Do not use this `core` structure if the space you are interested in already has a norm\ninstance defined on it, otherwise this will create a second non-defeq norm instance!\n-\/\n\n\/-- A structure requiring that a scalar product is positive definite and symmetric, from which one\ncan construct an `inner_product_space` instance in `inner_product_space.of_core`. -\/\n@[nolint has_inhabited_instance]\nstructure inner_product_space.core\n  (\ud835\udd5c : Type*) (F : Type*)\n  [is_R_or_C \ud835\udd5c] [add_comm_group F] [semimodule \ud835\udd5c F] :=\n(inner     : F \u2192 F \u2192 \ud835\udd5c)\n(conj_sym  : \u2200 x y, conj (inner y x) = inner x y)\n(nonneg_im : \u2200 x, im (inner x x) = 0)\n(nonneg_re : \u2200 x, 0 \u2264 re (inner x x))\n(definite  : \u2200 x, inner x x = 0 \u2192 x = 0)\n(add_left  : \u2200 x y z, inner (x + y) z = inner x z + inner y z)\n(smul_left : \u2200 x y r, inner (r \u2022 x) y = (conj r) * inner x y)\n\n\/- We set `inner_product_space.core` to be a class as we will use it as such in the construction\nof the normed space structure that it produces. However, all the instances we will use will be\nlocal to this proof. -\/\nattribute [class] inner_product_space.core\n\nnamespace inner_product_space.of_core\n\nvariables [add_comm_group F] [semimodule \ud835\udd5c F] [c : inner_product_space.core \ud835\udd5c F]\ninclude c\n\nlocal notation `\u27ea`x`, `y`\u27eb` := @inner \ud835\udd5c F _ x y\nlocal notation `norm_sqK` := @is_R_or_C.norm_sq \ud835\udd5c _\nlocal notation `reK` := @is_R_or_C.re \ud835\udd5c _\nlocal notation `absK` := @is_R_or_C.abs \ud835\udd5c _\nlocal notation `ext_iff` := @is_R_or_C.ext_iff \ud835\udd5c _\nlocal postfix `\u2020`:90 := @is_R_or_C.conj \ud835\udd5c _\n\n\/-- Inner product defined by the `inner_product_space.core` structure. -\/\ndef to_has_inner : has_inner \ud835\udd5c F := { inner := c.inner }\nlocal attribute [instance] to_has_inner\n\n\/-- The norm squared function for `inner_product_space.core` structure. -\/\ndef norm_sq (x : F) := reK \u27eax, x\u27eb\n\nlocal notation `norm_sqF` := @norm_sq \ud835\udd5c F _ _ _ _\n\nlemma inner_conj_sym (x y : F) : \u27eay, x\u27eb\u2020 = \u27eax, y\u27eb := c.conj_sym x y\n\nlemma inner_self_nonneg {x : F} : 0 \u2264 re \u27eax, x\u27eb := c.nonneg_re _\n\nlemma inner_self_nonneg_im {x : F} : im \u27eax, x\u27eb = 0 := c.nonneg_im _\n\nlemma inner_self_im_zero {x : F} : im \u27eax, x\u27eb = 0 := c.nonneg_im _\n\nlemma inner_add_left {x y z : F} : \u27eax + y, z\u27eb = \u27eax, z\u27eb + \u27eay, z\u27eb :=\nc.add_left _ _ _\n\nlemma inner_add_right {x y z : F} : \u27eax, y + z\u27eb = \u27eax, y\u27eb + \u27eax, z\u27eb :=\nby rw [\u2190inner_conj_sym, inner_add_left, ring_hom.map_add]; simp only [inner_conj_sym]\n\nlemma inner_norm_sq_eq_inner_self (x : F) : (norm_sqF x : \ud835\udd5c) = \u27eax, x\u27eb :=\nbegin\n  rw ext_iff,\n  exact \u27e8by simp only [of_real_re]; refl, by simp only [inner_self_nonneg_im, of_real_im]\u27e9\nend\n\nlemma inner_re_symm {x y : F} : re \u27eax, y\u27eb = re \u27eay, x\u27eb :=\nby rw [\u2190inner_conj_sym, conj_re]\n\nlemma inner_im_symm {x y : F} : im \u27eax, y\u27eb = -im \u27eay, x\u27eb :=\nby rw [\u2190inner_conj_sym, conj_im]\n\nlemma inner_smul_left {x y : F} {r : \ud835\udd5c} : \u27ear \u2022 x, y\u27eb = r\u2020 * \u27eax, y\u27eb :=\nc.smul_left _ _ _\n\nlemma inner_smul_right {x y : F} {r : \ud835\udd5c} : \u27eax, r \u2022 y\u27eb = r * \u27eax, y\u27eb :=\nby rw [\u2190inner_conj_sym, inner_smul_left]; simp only [conj_conj, inner_conj_sym, ring_hom.map_mul]\n\nlemma inner_zero_left {x : F} : \u27ea0, x\u27eb = 0 :=\nby rw [\u2190zero_smul \ud835\udd5c (0 : F), inner_smul_left]; simp only [zero_mul, ring_hom.map_zero]\n\nlemma inner_zero_right {x : F} : \u27eax, 0\u27eb = 0 :=\nby rw [\u2190inner_conj_sym, inner_zero_left]; simp only [ring_hom.map_zero]\n\nlemma inner_self_eq_zero {x : F} : \u27eax, x\u27eb = 0 \u2194 x = 0 :=\niff.intro (c.definite _) (by { rintro rfl, exact inner_zero_left })\n\nlemma inner_self_re_to_K {x : F} : (re \u27eax, x\u27eb : \ud835\udd5c) = \u27eax, x\u27eb :=\nby norm_num [ext_iff, inner_self_nonneg_im]\n\nlemma inner_abs_conj_sym {x y : F} : abs \u27eax, y\u27eb = abs \u27eay, x\u27eb :=\n  by rw [\u2190inner_conj_sym, abs_conj]\n\nlemma inner_neg_left {x y : F} : \u27ea-x, y\u27eb = -\u27eax, y\u27eb :=\nby { rw [\u2190 neg_one_smul \ud835\udd5c x, inner_smul_left], simp }\n\nlemma inner_neg_right {x y : F} : \u27eax, -y\u27eb = -\u27eax, y\u27eb :=\nby rw [\u2190inner_conj_sym, inner_neg_left]; simp only [ring_hom.map_neg, inner_conj_sym]\n\nlemma inner_sub_left {x y z : F} : \u27eax - y, z\u27eb = \u27eax, z\u27eb - \u27eay, z\u27eb :=\nby { simp [sub_eq_add_neg, inner_add_left, inner_neg_left] }\n\nlemma inner_sub_right {x y z : F} : \u27eax, y - z\u27eb = \u27eax, y\u27eb - \u27eax, z\u27eb :=\nby { simp [sub_eq_add_neg, inner_add_right, inner_neg_right] }\n\nlemma inner_mul_conj_re_abs {x y : F} : re (\u27eax, y\u27eb * \u27eay, x\u27eb) = abs (\u27eax, y\u27eb * \u27eay, x\u27eb) :=\nby { rw[\u2190inner_conj_sym, mul_comm], exact re_eq_abs_of_mul_conj (inner y x), }\n\n\/-- Expand `inner (x + y) (x + y)` -\/\nlemma inner_add_add_self {x y : F} : \u27eax + y, x + y\u27eb = \u27eax, x\u27eb + \u27eax, y\u27eb + \u27eay, x\u27eb + \u27eay, y\u27eb :=\nby simp only [inner_add_left, inner_add_right]; ring\n\n\/- Expand `inner (x - y) (x - y)` -\/\nlemma inner_sub_sub_self {x y : F} : \u27eax - y, x - y\u27eb = \u27eax, x\u27eb - \u27eax, y\u27eb - \u27eay, x\u27eb + \u27eay, y\u27eb :=\nby simp only [inner_sub_left, inner_sub_right]; ring\n\n\/--\nCauchy\u2013Schwarz inequality. This proof follows \"Proof 2\" on Wikipedia.\nWe need this for the `core` structure to prove the triangle inequality below when\nshowing the core is a normed group.\n-\/\nlemma inner_mul_inner_self_le (x y : F) : abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \u2264 re \u27eax, x\u27eb * re \u27eay, y\u27eb :=\nbegin\n  by_cases hy : y = 0,\n  { rw [hy], simp only [is_R_or_C.abs_zero, inner_zero_left, mul_zero, add_monoid_hom.map_zero] },\n  { change y \u2260 0 at hy,\n    have hy' : \u27eay, y\u27eb \u2260 0 := \u03bb h, by rw [inner_self_eq_zero] at h; exact hy h,\n    set T := \u27eay, x\u27eb \/ \u27eay, y\u27eb with hT,\n    have h\u2081 : re \u27eay, x\u27eb = re \u27eax, y\u27eb := inner_re_symm,\n    have h\u2082 : im \u27eay, x\u27eb = -im \u27eax, y\u27eb := inner_im_symm,\n    have h\u2083 : \u27eay, x\u27eb * \u27eax, y\u27eb * \u27eay, y\u27eb \/ (\u27eay, y\u27eb * \u27eay, y\u27eb) = \u27eay, x\u27eb * \u27eax, y\u27eb \/ \u27eay, y\u27eb,\n    { rw [mul_div_assoc],\n      have : \u27eay, y\u27eb \/ (\u27eay, y\u27eb * \u27eay, y\u27eb) = 1 \/ \u27eay, y\u27eb :=\n        by rw [div_mul_eq_div_mul_one_div, div_self hy', one_mul],\n      rw [this, div_eq_mul_inv, one_mul, \u2190div_eq_mul_inv] },\n    have h\u2084 : \u27eay, y\u27eb = re \u27eay, y\u27eb := by simp only [inner_self_re_to_K],\n    have h\u2085 : re \u27eay, y\u27eb > 0,\n    { refine lt_of_le_of_ne inner_self_nonneg _,\n      intro H,\n      apply hy',\n      rw ext_iff,\n      exact \u27e8by simp [H],by simp [inner_self_nonneg_im]\u27e9 },\n    have h\u2086 : re \u27eay, y\u27eb \u2260 0 := ne_of_gt h\u2085,\n    have hmain := calc\n      0   \u2264 re \u27eax - T \u2022 y, x - T \u2022 y\u27eb\n                  : inner_self_nonneg\n      ... = re \u27eax, x\u27eb - re \u27eaT \u2022 y, x\u27eb - re \u27eax, T \u2022 y\u27eb + re \u27eaT \u2022 y, T \u2022 y\u27eb\n                  : by simp [inner_sub_sub_self, inner_smul_left, inner_smul_right, h\u2081, h\u2082]\n      ... = re \u27eax, x\u27eb - re (T\u2020 * \u27eay, x\u27eb) - re (T * \u27eax, y\u27eb) + re (T * T\u2020 * \u27eay, y\u27eb)\n                  : by simp [inner_smul_left, inner_smul_right, mul_assoc]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb \/ \u27eay, y\u27eb * \u27eay, x\u27eb)\n                  : by field_simp [-mul_re, inner_conj_sym, hT, conj_div, h\u2081, h\u2083]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb \/ \u27eay, y\u27eb)\n                  : by rw [div_mul_eq_mul_div_comm, \u2190mul_div_assoc]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb \/ re \u27eay, y\u27eb)\n                  : by conv_lhs { rw [h\u2084] }\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb) \/ re \u27eay, y\u27eb\n                  : by rw [div_re_of_real]\n      ... = re \u27eax, x\u27eb - abs (\u27eax, y\u27eb * \u27eay, x\u27eb) \/ re \u27eay, y\u27eb\n                  : by rw [inner_mul_conj_re_abs]\n      ... = re \u27eax, x\u27eb - abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \/ re \u27eay, y\u27eb\n                  : by rw is_R_or_C.abs_mul,\n    have hmain' : abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \/ re \u27eay, y\u27eb \u2264 re \u27eax, x\u27eb := by linarith,\n    have := (mul_le_mul_right h\u2085).mpr hmain',\n    rwa [div_mul_cancel (abs \u27eax, y\u27eb * abs \u27eay, x\u27eb) h\u2086] at this }\nend\n\n\/-- Norm constructed from a `inner_product_space.core` structure, defined to be the square root\nof the scalar product. -\/\ndef to_has_norm : has_norm F :=\n{ norm := \u03bb x, sqrt (re \u27eax, x\u27eb) }\n\nlocal attribute [instance] to_has_norm\n\nlemma norm_eq_sqrt_inner (x : F) : \u2225x\u2225 = sqrt (re \u27eax, x\u27eb) := rfl\n\nlemma inner_self_eq_norm_square (x : F) : re \u27eax, x\u27eb = \u2225x\u2225 * \u2225x\u2225 :=\nby rw[norm_eq_sqrt_inner, \u2190sqrt_mul inner_self_nonneg (re \u27eax, x\u27eb),\n  sqrt_mul_self inner_self_nonneg]\n\nlemma sqrt_norm_sq_eq_norm {x : F} : sqrt (norm_sqF x) = \u2225x\u2225 := rfl\n\n\/-- Cauchy\u2013Schwarz inequality with norm -\/\nlemma abs_inner_le_norm (x y : F) : abs \u27eax, y\u27eb \u2264 \u2225x\u2225 * \u2225y\u2225 :=\nnonneg_le_nonneg_of_squares_le (mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _))\nbegin\n  have H : \u2225x\u2225 * \u2225y\u2225 * (\u2225x\u2225 * \u2225y\u2225) = re \u27eay, y\u27eb * re \u27eax, x\u27eb,\n  { simp only [inner_self_eq_norm_square], ring, },\n  rw H,\n  conv\n  begin\n    to_lhs, congr, rw[inner_abs_conj_sym],\n  end,\n  exact inner_mul_inner_self_le y x,\nend\n\n\/-- Normed group structure constructed from an `inner_product_space.core` structure -\/\ndef to_normed_group : normed_group F :=\nnormed_group.of_core F\n{ norm_eq_zero_iff := assume x,\n  begin\n    split,\n    { intro H,\n      change sqrt (re \u27eax, x\u27eb) = 0 at H,\n      rw [sqrt_eq_zero inner_self_nonneg] at H,\n      apply (inner_self_eq_zero : \u27eax, x\u27eb = 0 \u2194 x = 0).mp,\n      rw ext_iff,\n      exact \u27e8by simp [H], by simp [inner_self_im_zero]\u27e9 },\n    { rintro rfl,\n      change sqrt (re \u27ea0, 0\u27eb) = 0,\n      simp only [sqrt_zero, inner_zero_right, add_monoid_hom.map_zero] }\n  end,\n  triangle := assume x y,\n  begin\n    have h\u2081 : abs \u27eax, y\u27eb \u2264 \u2225x\u2225 * \u2225y\u2225 := abs_inner_le_norm _ _,\n    have h\u2082 : re \u27eax, y\u27eb \u2264 abs \u27eax, y\u27eb := re_le_abs _,\n    have h\u2083 : re \u27eax, y\u27eb \u2264 \u2225x\u2225 * \u2225y\u2225 := by linarith,\n    have h\u2084 : re \u27eay, x\u27eb \u2264 \u2225x\u2225 * \u2225y\u2225 := by rwa [\u2190inner_conj_sym, conj_re],\n    have : \u2225x + y\u2225 * \u2225x + y\u2225 \u2264 (\u2225x\u2225 + \u2225y\u2225) * (\u2225x\u2225 + \u2225y\u2225),\n    { simp [\u2190inner_self_eq_norm_square, inner_add_add_self, add_mul, mul_add, mul_comm],\n      linarith },\n    exact nonneg_le_nonneg_of_squares_le (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this\n  end,\n  norm_neg := \u03bb x, by simp only [norm, inner_neg_left, neg_neg, inner_neg_right] }\n\nlocal attribute [instance] to_normed_group\n\n\/-- Normed space structure constructed from a `inner_product_space.core` structure -\/\ndef to_normed_space : normed_space \ud835\udd5c F :=\n{ norm_smul_le := assume r x,\n  begin\n    rw [norm_eq_sqrt_inner, inner_smul_left, inner_smul_right, \u2190mul_assoc],\n    rw [conj_mul_eq_norm_sq_left, of_real_mul_re, sqrt_mul, \u2190inner_norm_sq_eq_inner_self,\n        of_real_re],\n    { simp [sqrt_norm_sq_eq_norm, is_R_or_C.sqrt_norm_sq_eq_norm] },\n    { exact norm_sq_nonneg r }\n  end }\n\nend inner_product_space.of_core\n\n\/-- Given a `inner_product_space.core` structure on a space, one can use it to turn\nthe space into an inner product space, constructing the norm out of the inner product -\/\ndef inner_product_space.of_core [add_comm_group F] [semimodule \ud835\udd5c F]\n  (c : inner_product_space.core \ud835\udd5c F) : inner_product_space \ud835\udd5c F :=\nbegin\n  letI : normed_group F := @inner_product_space.of_core.to_normed_group \ud835\udd5c F _ _ _ c,\n  letI : normed_space \ud835\udd5c F := @inner_product_space.of_core.to_normed_space \ud835\udd5c F _ _ _ c,\n  exact { norm_sq_eq_inner := \u03bb x,\n    begin\n      have h\u2081 : \u2225x\u2225^2 = (sqrt (re (c.inner x x))) ^ 2 := rfl,\n      have h\u2082 : 0 \u2264 re (c.inner x x) := inner_product_space.of_core.inner_self_nonneg,\n      simp [h\u2081, sqr_sqrt, h\u2082],\n    end,\n    ..c }\nend\n\n\/-! ### Properties of inner product spaces -\/\n\nvariables [inner_product_space \ud835\udd5c E] [inner_product_space \u211d F]\nlocal notation `\u27ea`x`, `y`\u27eb` := @inner \ud835\udd5c _ _ x y\nlocal notation `IK` := @is_R_or_C.I \ud835\udd5c _\nlocal notation `absR` := _root_.abs\nlocal notation `absK` := @is_R_or_C.abs \ud835\udd5c _\nlocal postfix `\u2020`:90 := @is_R_or_C.conj \ud835\udd5c _\nlocal postfix `\u22c6`:90 := complex.conj\n\nexport inner_product_space (norm_sq_eq_inner)\n\nsection basic_properties\n\nlemma inner_conj_sym (x y : E) : \u27eay, x\u27eb\u2020 = \u27eax, y\u27eb := inner_product_space.conj_sym _ _\nlemma real_inner_comm (x y : F) : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := inner_conj_sym x y\n\nlemma inner_eq_zero_sym {x y : E} : \u27eax, y\u27eb = 0 \u2194 \u27eay, x\u27eb = 0 :=\n\u27e8\u03bb h, by simp [\u2190inner_conj_sym, h], \u03bb h, by simp [\u2190inner_conj_sym, h]\u27e9\n\nlemma inner_self_nonneg_im {x : E} : im \u27eax, x\u27eb = 0 := inner_product_space.nonneg_im _\n\nlemma inner_self_im_zero {x : E} : im \u27eax, x\u27eb = 0 := inner_product_space.nonneg_im _\n\nlemma inner_add_left {x y z : E} : \u27eax + y, z\u27eb = \u27eax, z\u27eb + \u27eay, z\u27eb :=\ninner_product_space.add_left _ _ _\n\nlemma inner_add_right {x y z : E} : \u27eax, y + z\u27eb = \u27eax, y\u27eb + \u27eax, z\u27eb :=\nbegin\n  rw [\u2190inner_conj_sym, inner_add_left, ring_hom.map_add],\n  conv_rhs { rw \u2190inner_conj_sym, conv { congr, skip, rw \u2190inner_conj_sym } }\nend\n\nlemma inner_re_symm {x y : E} : re \u27eax, y\u27eb = re \u27eay, x\u27eb :=\nby rw [\u2190inner_conj_sym, conj_re]\n\nlemma inner_im_symm {x y : E} : im \u27eax, y\u27eb = -im \u27eay, x\u27eb :=\nby rw [\u2190inner_conj_sym, conj_im]\n\nlemma inner_smul_left {x y : E} {r : \ud835\udd5c} : \u27ear \u2022 x, y\u27eb = r\u2020 * \u27eax, y\u27eb :=\ninner_product_space.smul_left _ _ _\nlemma real_inner_smul_left {x y : F} {r : \u211d} : \u27ear \u2022 x, y\u27eb_\u211d = r * \u27eax, y\u27eb_\u211d := inner_smul_left\n\nlemma inner_smul_real_left {x y : E} {r : \u211d} : \u27ea(r : \ud835\udd5c) \u2022 x, y\u27eb = r \u2022 \u27eax, y\u27eb :=\nby { rw [inner_smul_left, conj_of_real, algebra.smul_def], refl }\n\nlemma inner_smul_right {x y : E} {r : \ud835\udd5c} : \u27eax, r \u2022 y\u27eb = r * \u27eax, y\u27eb :=\nby rw [\u2190inner_conj_sym, inner_smul_left, ring_hom.map_mul, conj_conj, inner_conj_sym]\nlemma real_inner_smul_right {x y : F} {r : \u211d} : \u27eax, r \u2022 y\u27eb_\u211d = r * \u27eax, y\u27eb_\u211d := inner_smul_right\n\nlemma inner_smul_real_right {x y : E} {r : \u211d} : \u27eax, (r : \ud835\udd5c) \u2022 y\u27eb = r \u2022 \u27eax, y\u27eb :=\nby { rw [inner_smul_right, algebra.smul_def], refl }\n\n\/-- The inner product as a sesquilinear form. -\/\ndef sesq_form_of_inner : sesq_form \ud835\udd5c E (conj_to_ring_equiv \ud835\udd5c) :=\n{ sesq := \u03bb x y, \u27eay, x\u27eb,    -- Note that sesquilinear forms are linear in the first argument\n  sesq_add_left := \u03bb x y z, inner_add_right,\n  sesq_add_right := \u03bb x y z, inner_add_left,\n  sesq_smul_left := \u03bb r x y, inner_smul_right,\n  sesq_smul_right := \u03bb r x y, inner_smul_left }\n\n\/-- The real inner product as a bilinear form. -\/\ndef bilin_form_of_real_inner : bilin_form \u211d F :=\n{ bilin := inner,\n  bilin_add_left := \u03bb x y z, inner_add_left,\n  bilin_smul_left := \u03bb a x y, inner_smul_left,\n  bilin_add_right := \u03bb x y z, inner_add_right,\n  bilin_smul_right := \u03bb a x y, inner_smul_right }\n\n\/-- An inner product with a sum on the left. -\/\nlemma sum_inner {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 E) (x : E) :\n  \u27ea\u2211 i in s, f i, x\u27eb = \u2211 i in s, \u27eaf i, x\u27eb :=\nsesq_form.map_sum_right (sesq_form_of_inner) _ _ _\n\n\/-- An inner product with a sum on the right. -\/\nlemma inner_sum {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 E) (x : E) :\n  \u27eax, \u2211 i in s, f i\u27eb = \u2211 i in s, \u27eax, f i\u27eb :=\nsesq_form.map_sum_left (sesq_form_of_inner) _ _ _\n\n@[simp] lemma inner_zero_left {x : E} : \u27ea0, x\u27eb = 0 :=\nby rw [\u2190 zero_smul \ud835\udd5c (0:E), inner_smul_left, ring_hom.map_zero, zero_mul]\n\nlemma inner_re_zero_left {x : E} : re \u27ea0, x\u27eb = 0 :=\nby simp only [inner_zero_left, add_monoid_hom.map_zero]\n\n@[simp] lemma inner_zero_right {x : E} : \u27eax, 0\u27eb = 0 :=\nby rw [\u2190inner_conj_sym, inner_zero_left, ring_hom.map_zero]\n\nlemma inner_re_zero_right {x : E} : re \u27eax, 0\u27eb = 0 :=\nby simp only [inner_zero_right, add_monoid_hom.map_zero]\n\nlemma inner_self_nonneg {x : E} : 0 \u2264 re \u27eax, x\u27eb :=\nby rw [\u2190norm_sq_eq_inner]; exact pow_nonneg (norm_nonneg x) 2\nlemma real_inner_self_nonneg {x : F} : 0 \u2264 \u27eax, x\u27eb_\u211d := @inner_self_nonneg \u211d F _ _ x\n\n@[simp] lemma inner_self_eq_zero {x : E} : \u27eax, x\u27eb = 0 \u2194 x = 0 :=\nbegin\n  split,\n  { intro h,\n    have h\u2081 : re \u27eax, x\u27eb = 0 := by rw is_R_or_C.ext_iff at h; simp [h.1],\n    rw [\u2190norm_sq_eq_inner x] at h\u2081,\n    rw [\u2190norm_eq_zero],\n    exact pow_eq_zero h\u2081 },\n  { rintro rfl,\n    exact inner_zero_left }\nend\n\n@[simp] lemma inner_self_nonpos {x : E} : re \u27eax, x\u27eb \u2264 0 \u2194 x = 0 :=\nbegin\n  split,\n  { intro h,\n    rw \u2190inner_self_eq_zero,\n    have H\u2081 : re \u27eax, x\u27eb \u2265 0, exact inner_self_nonneg,\n    have H\u2082 : re \u27eax, x\u27eb = 0, exact le_antisymm h H\u2081,\n    rw is_R_or_C.ext_iff,\n    exact \u27e8by simp [H\u2082], by simp [inner_self_nonneg_im]\u27e9 },\n  { rintro rfl,\n    simp only [inner_zero_left, add_monoid_hom.map_zero] }\nend\n\nlemma real_inner_self_nonpos {x : F} : \u27eax, x\u27eb_\u211d \u2264 0 \u2194 x = 0 :=\nby { have h := @inner_self_nonpos \u211d F _ _ x, simpa using h }\n\n@[simp] lemma inner_self_re_to_K {x : E} : (re \u27eax, x\u27eb : \ud835\udd5c) = \u27eax, x\u27eb :=\nby rw is_R_or_C.ext_iff; exact \u27e8by simp, by simp [inner_self_nonneg_im]\u27e9\n\nlemma inner_self_re_abs {x : E} : re \u27eax, x\u27eb = abs \u27eax, x\u27eb :=\nbegin\n  have H : \u27eax, x\u27eb = (re \u27eax, x\u27eb : \ud835\udd5c) + im \u27eax, x\u27eb * I,\n  { rw re_add_im, },\n  rw [H, is_add_hom.map_add re ((re \u27eax, x\u27eb) : \ud835\udd5c) (((im \u27eax, x\u27eb) : \ud835\udd5c) * I)],\n  rw [mul_re, I_re, mul_zero, I_im, zero_sub, tactic.ring.add_neg_eq_sub],\n  rw [of_real_re, of_real_im, sub_zero, inner_self_nonneg_im],\n  simp only [abs_of_real, add_zero, of_real_zero, zero_mul],\n  exact (_root_.abs_of_nonneg inner_self_nonneg).symm,\nend\n\nlemma inner_self_abs_to_K {x : E} : (absK \u27eax, x\u27eb : \ud835\udd5c) = \u27eax, x\u27eb :=\nby { rw[\u2190inner_self_re_abs], exact inner_self_re_to_K }\n\nlemma real_inner_self_abs {x : F} : absR \u27eax, x\u27eb_\u211d = \u27eax, x\u27eb_\u211d :=\nby { have h := @inner_self_abs_to_K \u211d F _ _ x, simpa using h }\n\nlemma inner_abs_conj_sym {x y : E} : abs \u27eax, y\u27eb = abs \u27eay, x\u27eb :=\nby rw [\u2190inner_conj_sym, abs_conj]\n\n@[simp] lemma inner_neg_left {x y : E} : \u27ea-x, y\u27eb = -\u27eax, y\u27eb :=\nby { rw [\u2190 neg_one_smul \ud835\udd5c x, inner_smul_left], simp }\n\n@[simp] lemma inner_neg_right {x y : E} : \u27eax, -y\u27eb = -\u27eax, y\u27eb :=\nby rw [\u2190inner_conj_sym, inner_neg_left]; simp only [ring_hom.map_neg, inner_conj_sym]\n\nlemma inner_neg_neg {x y : E} : \u27ea-x, -y\u27eb = \u27eax, y\u27eb := by simp\n\n@[simp] lemma inner_self_conj {x : E} : \u27eax, x\u27eb\u2020 = \u27eax, x\u27eb :=\nby rw [is_R_or_C.ext_iff]; exact \u27e8by rw [conj_re], by rw [conj_im, inner_self_im_zero, neg_zero]\u27e9\n\nlemma inner_sub_left {x y z : E} : \u27eax - y, z\u27eb = \u27eax, z\u27eb - \u27eay, z\u27eb :=\nby { simp [sub_eq_add_neg, inner_add_left] }\n\nlemma inner_sub_right {x y z : E} : \u27eax, y - z\u27eb = \u27eax, y\u27eb - \u27eax, z\u27eb :=\nby { simp [sub_eq_add_neg, inner_add_right] }\n\nlemma inner_mul_conj_re_abs {x y : E} : re (\u27eax, y\u27eb * \u27eay, x\u27eb) = abs (\u27eax, y\u27eb * \u27eay, x\u27eb) :=\nby { rw[\u2190inner_conj_sym, mul_comm], exact re_eq_abs_of_mul_conj (inner y x), }\n\n\/-- Expand `\u27eax + y, x + y\u27eb` -\/\nlemma inner_add_add_self {x y : E} : \u27eax + y, x + y\u27eb = \u27eax, x\u27eb + \u27eax, y\u27eb + \u27eay, x\u27eb + \u27eay, y\u27eb :=\nby simp only [inner_add_left, inner_add_right]; ring\n\n\/-- Expand `\u27eax + y, x + y\u27eb_\u211d` -\/\nlemma real_inner_add_add_self {x y : F} : \u27eax + y, x + y\u27eb_\u211d = \u27eax, x\u27eb_\u211d + 2 * \u27eax, y\u27eb_\u211d + \u27eay, y\u27eb_\u211d :=\nbegin\n  have : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190inner_conj_sym]; refl,\n  simp [inner_add_add_self, this],\n  ring,\nend\n\n\/- Expand `\u27eax - y, x - y\u27eb` -\/\nlemma inner_sub_sub_self {x y : E} : \u27eax - y, x - y\u27eb = \u27eax, x\u27eb - \u27eax, y\u27eb - \u27eay, x\u27eb + \u27eay, y\u27eb :=\nby simp only [inner_sub_left, inner_sub_right]; ring\n\n\/-- Expand `\u27eax - y, x - y\u27eb_\u211d` -\/\nlemma real_inner_sub_sub_self {x y : F} : \u27eax - y, x - y\u27eb_\u211d = \u27eax, x\u27eb_\u211d - 2 * \u27eax, y\u27eb_\u211d + \u27eay, y\u27eb_\u211d :=\nbegin\n  have : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190inner_conj_sym]; refl,\n  simp [inner_sub_sub_self, this],\n  ring,\nend\n\n\/-- Parallelogram law -\/\nlemma parallelogram_law {x y : E} :\n  \u27eax + y, x + y\u27eb + \u27eax - y, x - y\u27eb = 2 * (\u27eax, x\u27eb + \u27eay, y\u27eb) :=\nby simp [inner_add_add_self, inner_sub_sub_self, two_mul, sub_eq_add_neg, add_comm, add_left_comm]\n\n\/-- Cauchy\u2013Schwarz inequality. This proof follows \"Proof 2\" on Wikipedia. -\/\nlemma inner_mul_inner_self_le (x y : E) : abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \u2264 re \u27eax, x\u27eb * re \u27eay, y\u27eb :=\nbegin\n  by_cases hy : y = 0,\n  { rw [hy], simp only [is_R_or_C.abs_zero, inner_zero_left, mul_zero, add_monoid_hom.map_zero] },\n  { change y \u2260 0 at hy,\n    have hy' : \u27eay, y\u27eb \u2260 0 := \u03bb h, by rw [inner_self_eq_zero] at h; exact hy h,\n    set T := \u27eay, x\u27eb \/ \u27eay, y\u27eb with hT,\n    have h\u2081 : re \u27eay, x\u27eb = re \u27eax, y\u27eb := inner_re_symm,\n    have h\u2082 : im \u27eay, x\u27eb = -im \u27eax, y\u27eb := inner_im_symm,\n    have h\u2083 : \u27eay, x\u27eb * \u27eax, y\u27eb * \u27eay, y\u27eb \/ (\u27eay, y\u27eb * \u27eay, y\u27eb) = \u27eay, x\u27eb * \u27eax, y\u27eb \/ \u27eay, y\u27eb,\n    { rw [mul_div_assoc],\n      have : \u27eay, y\u27eb \/ (\u27eay, y\u27eb * \u27eay, y\u27eb) = 1 \/ \u27eay, y\u27eb :=\n        by rw [div_mul_eq_div_mul_one_div, div_self hy', one_mul],\n      rw [this, div_eq_mul_inv, one_mul, \u2190div_eq_mul_inv] },\n    have h\u2084 : \u27eay, y\u27eb = re \u27eay, y\u27eb := by simp,\n    have h\u2085 : re \u27eay, y\u27eb > 0,\n    { refine lt_of_le_of_ne inner_self_nonneg _,\n      intro H,\n      apply hy',\n      rw is_R_or_C.ext_iff,\n      exact \u27e8by simp [H],by simp [inner_self_nonneg_im]\u27e9 },\n    have h\u2086 : re \u27eay, y\u27eb \u2260 0 := ne_of_gt h\u2085,\n    have hmain := calc\n      0   \u2264 re \u27eax - T \u2022 y, x - T \u2022 y\u27eb\n                  : inner_self_nonneg\n      ... = re \u27eax, x\u27eb - re \u27eaT \u2022 y, x\u27eb - re \u27eax, T \u2022 y\u27eb + re \u27eaT \u2022 y, T \u2022 y\u27eb\n                  : by simp [inner_sub_sub_self, inner_smul_left, inner_smul_right, h\u2081, h\u2082]\n      ... = re \u27eax, x\u27eb - re (T\u2020 * \u27eay, x\u27eb) - re (T * \u27eax, y\u27eb) + re (T * T\u2020 * \u27eay, y\u27eb)\n                  : by simp [inner_smul_left, inner_smul_right, mul_assoc]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb \/ \u27eay, y\u27eb * \u27eay, x\u27eb)\n                  : by field_simp [-mul_re, hT, conj_div, h\u2081, h\u2083, inner_conj_sym]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb \/ \u27eay, y\u27eb)\n                  : by rw [div_mul_eq_mul_div_comm, \u2190mul_div_assoc]\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb \/ re \u27eay, y\u27eb)\n                  : by conv_lhs { rw [h\u2084] }\n      ... = re \u27eax, x\u27eb - re (\u27eax, y\u27eb * \u27eay, x\u27eb) \/ re \u27eay, y\u27eb\n                  : by rw [div_re_of_real]\n      ... = re \u27eax, x\u27eb - abs (\u27eax, y\u27eb * \u27eay, x\u27eb) \/ re \u27eay, y\u27eb\n                  : by rw [inner_mul_conj_re_abs]\n      ... = re \u27eax, x\u27eb - abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \/ re \u27eay, y\u27eb\n                  : by rw is_R_or_C.abs_mul,\n    have hmain' : abs \u27eax, y\u27eb * abs \u27eay, x\u27eb \/ re \u27eay, y\u27eb \u2264 re \u27eax, x\u27eb := by linarith,\n    have := (mul_le_mul_right h\u2085).mpr hmain',\n    rwa [div_mul_cancel (abs \u27eax, y\u27eb * abs \u27eay, x\u27eb) h\u2086] at this }\nend\n\n\/-- Cauchy\u2013Schwarz inequality for real inner products. -\/\nlemma real_inner_mul_inner_self_le (x y : F) : \u27eax, y\u27eb_\u211d * \u27eax, y\u27eb_\u211d \u2264 \u27eax, x\u27eb_\u211d * \u27eay, y\u27eb_\u211d :=\nbegin\n  have h\u2081 : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190inner_conj_sym]; refl,\n  have h\u2082 := @inner_mul_inner_self_le \u211d F _ _ x y,\n  dsimp at h\u2082,\n  have h\u2083 := abs_mul_abs_self \u27eax, y\u27eb_\u211d,\n  rw [h\u2081] at h\u2082,\n  simpa [h\u2083] using h\u2082,\nend\n\n\/-- A family of vectors is linearly independent if they are nonzero\nand orthogonal. -\/\nlemma linear_independent_of_ne_zero_of_inner_eq_zero {\u03b9 : Type*} {v : \u03b9 \u2192 E}\n  (hz : \u2200 i, v i \u2260 0) (ho : \u2200 i j, i \u2260 j \u2192 \u27eav i, v j\u27eb = 0) : linear_independent \ud835\udd5c v :=\nbegin\n  rw linear_independent_iff',\n  intros s g hg i hi,\n  have h' : g i * inner (v i) (v i) = inner (v i) (\u2211 j in s, g j \u2022 v j),\n  { rw inner_sum,\n    symmetry,\n    convert finset.sum_eq_single i _ _,\n    { rw inner_smul_right },\n    { intros j hj hji,\n      rw [inner_smul_right, ho i j hji.symm, mul_zero] },\n    { exact \u03bb h, false.elim (h hi) } },\n  simpa [hg, hz] using h'\nend\n\nend basic_properties\n\nsection norm\n\nlemma norm_eq_sqrt_inner (x : E) : \u2225x\u2225 = sqrt (re \u27eax, x\u27eb) :=\nbegin\n  have h\u2081 : \u2225x\u2225^2 = re \u27eax, x\u27eb := norm_sq_eq_inner x,\n  have h\u2082 := congr_arg sqrt h\u2081,\n  simpa using h\u2082,\nend\n\nlemma norm_eq_sqrt_real_inner (x : F) : \u2225x\u2225 = sqrt \u27eax, x\u27eb_\u211d :=\nby { have h := @norm_eq_sqrt_inner \u211d F _ _ x, simpa using h }\n\nlemma inner_self_eq_norm_square (x : E) : re \u27eax, x\u27eb = \u2225x\u2225 * \u2225x\u2225 :=\nby rw[norm_eq_sqrt_inner, \u2190sqrt_mul inner_self_nonneg (re \u27eax, x\u27eb),\n  sqrt_mul_self inner_self_nonneg]\n\nlemma real_inner_self_eq_norm_square (x : F) : \u27eax, x\u27eb_\u211d = \u2225x\u2225 * \u2225x\u2225 :=\nby { have h := @inner_self_eq_norm_square \u211d F _ _ x, simpa using h }\n\n\n\/-- Expand the square -\/\nlemma norm_add_pow_two {x y : E} : \u2225x + y\u2225^2 = \u2225x\u2225^2 + 2 * (re \u27eax, y\u27eb) + \u2225y\u2225^2 :=\nbegin\n  repeat {rw [pow_two, \u2190inner_self_eq_norm_square]},\n  rw[inner_add_add_self, two_mul],\n  simp only [add_assoc, add_left_inj, add_right_inj, add_monoid_hom.map_add],\n  rw [\u2190inner_conj_sym, conj_re],\nend\n\n\/-- Expand the square -\/\nlemma norm_add_pow_two_real {x y : F} : \u2225x + y\u2225^2 = \u2225x\u2225^2 + 2 * \u27eax, y\u27eb_\u211d + \u2225y\u2225^2 :=\nby { have h := @norm_add_pow_two \u211d F _ _, simpa using h }\n\n\/-- Expand the square -\/\nlemma norm_add_mul_self {x y : E} : \u2225x + y\u2225 * \u2225x + y\u2225 = \u2225x\u2225 * \u2225x\u2225 + 2 * (re \u27eax, y\u27eb) + \u2225y\u2225 * \u2225y\u2225 :=\nby { repeat {rw [\u2190 pow_two]}, exact norm_add_pow_two }\n\n\/-- Expand the square -\/\nlemma norm_add_mul_self_real {x y : F} : \u2225x + y\u2225 * \u2225x + y\u2225 = \u2225x\u2225 * \u2225x\u2225 + 2 * \u27eax, y\u27eb_\u211d + \u2225y\u2225 * \u2225y\u2225 :=\nby { have h := @norm_add_mul_self \u211d F _ _, simpa using h }\n\n\/-- Expand the square -\/\nlemma norm_sub_pow_two {x y : E} : \u2225x - y\u2225^2 = \u2225x\u2225^2 - 2 * (re \u27eax, y\u27eb) + \u2225y\u2225^2 :=\nbegin\n  repeat {rw [pow_two, \u2190inner_self_eq_norm_square]},\n  rw[inner_sub_sub_self],\n  calc\n    re (\u27eax, x\u27eb - \u27eax, y\u27eb - \u27eay, x\u27eb + \u27eay, y\u27eb)\n        = re \u27eax, x\u27eb - re \u27eax, y\u27eb - re \u27eay, x\u27eb + re \u27eay, y\u27eb  : by simp\n    ... = -re \u27eay, x\u27eb - re \u27eax, y\u27eb + re \u27eax, x\u27eb + re \u27eay, y\u27eb  : by ring\n    ... = -re (\u27eax, y\u27eb\u2020) - re \u27eax, y\u27eb + re \u27eax, x\u27eb + re \u27eay, y\u27eb : by rw[inner_conj_sym]\n    ... = -re \u27eax, y\u27eb - re \u27eax, y\u27eb + re \u27eax, x\u27eb + re \u27eay, y\u27eb : by rw[conj_re]\n    ... = re \u27eax, x\u27eb - 2*re \u27eax, y\u27eb + re \u27eay, y\u27eb : by ring\nend\n\n\/-- Expand the square -\/\nlemma norm_sub_pow_two_real {x y : F} : \u2225x - y\u2225^2 = \u2225x\u2225^2 - 2 * \u27eax, y\u27eb_\u211d + \u2225y\u2225^2 :=\nby { have h := @norm_sub_pow_two \u211d F _ _, simpa using h }\n\n\/-- Expand the square -\/\nlemma norm_sub_mul_self {x y : E} : \u2225x - y\u2225 * \u2225x - y\u2225 = \u2225x\u2225 * \u2225x\u2225 - 2 * re \u27eax, y\u27eb + \u2225y\u2225 * \u2225y\u2225 :=\nby { repeat {rw [\u2190 pow_two]}, exact norm_sub_pow_two }\n\n\/-- Expand the square -\/\nlemma norm_sub_mul_self_real {x y : F} : \u2225x - y\u2225 * \u2225x - y\u2225 = \u2225x\u2225 * \u2225x\u2225 - 2 * \u27eax, y\u27eb_\u211d + \u2225y\u2225 * \u2225y\u2225 :=\nby { have h := @norm_sub_mul_self \u211d F _ _, simpa using h }\n\n\/-- Cauchy\u2013Schwarz inequality with norm -\/\nlemma abs_inner_le_norm (x y : E) : abs \u27eax, y\u27eb \u2264 \u2225x\u2225 * \u2225y\u2225 :=\nnonneg_le_nonneg_of_squares_le (mul_nonneg (norm_nonneg _) (norm_nonneg _))\nbegin\n  have : \u2225x\u2225 * \u2225y\u2225 * (\u2225x\u2225 * \u2225y\u2225) = (re \u27eax, x\u27eb) * (re \u27eay, y\u27eb),\n    simp only [inner_self_eq_norm_square], ring,\n  rw this,\n  conv_lhs { congr, skip, rw [inner_abs_conj_sym] },\n  exact inner_mul_inner_self_le _ _\nend\n\n\/-- Cauchy\u2013Schwarz inequality with norm -\/\nlemma abs_real_inner_le_norm (x y : F) : absR \u27eax, y\u27eb_\u211d \u2264 \u2225x\u2225 * \u2225y\u2225 :=\nby { have h := @abs_inner_le_norm \u211d F _ _ x y, simpa using h }\n\n\/-- Cauchy\u2013Schwarz inequality with norm -\/\nlemma real_inner_le_norm (x y : F) : \u27eax, y\u27eb_\u211d \u2264 \u2225x\u2225 * \u2225y\u2225 :=\nle_trans (le_abs_self _) (abs_real_inner_le_norm _ _)\n\ninclude \ud835\udd5c\nlemma parallelogram_law_with_norm {x y : E} :\n  \u2225x + y\u2225 * \u2225x + y\u2225 + \u2225x - y\u2225 * \u2225x - y\u2225 = 2 * (\u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225) :=\nbegin\n  simp only [(inner_self_eq_norm_square _).symm],\n  rw[\u2190add_monoid_hom.map_add, parallelogram_law, two_mul, two_mul],\n  simp only [add_monoid_hom.map_add],\nend\nomit \ud835\udd5c\n\nlemma parallelogram_law_with_norm_real {x y : F} :\n  \u2225x + y\u2225 * \u2225x + y\u2225 + \u2225x - y\u2225 * \u2225x - y\u2225 = 2 * (\u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225) :=\nby { have h := @parallelogram_law_with_norm \u211d F _ _ x y, simpa using h }\n\n\/-- Polarization identity: The real inner product, in terms of the norm. -\/\nlemma real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : F) :\n  \u27eax, y\u27eb_\u211d = (\u2225x + y\u2225 * \u2225x + y\u2225 - \u2225x\u2225 * \u2225x\u2225 - \u2225y\u2225 * \u2225y\u2225) \/ 2 :=\nby rw norm_add_mul_self; ring\n\n\/-- Polarization identity: The real inner product, in terms of the norm. -\/\nlemma real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : F) :\n  \u27eax, y\u27eb_\u211d = (\u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 - \u2225x - y\u2225 * \u2225x - y\u2225) \/ 2 :=\nby rw norm_sub_mul_self; ring\n\n\/-- Pythagorean theorem, if-and-only-if vector inner product form. -\/\nlemma norm_add_square_eq_norm_square_add_norm_square_iff_real_inner_eq_zero (x y : F) :\n  \u2225x + y\u2225 * \u2225x + y\u2225 = \u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 \u2194 \u27eax, y\u27eb_\u211d = 0 :=\nbegin\n  rw [norm_add_mul_self, add_right_cancel_iff, add_right_eq_self, mul_eq_zero],\n  norm_num\nend\n\n\/-- Pythagorean theorem, vector inner product form. -\/\nlemma norm_add_square_eq_norm_square_add_norm_square_of_inner_eq_zero (x y : E) (h : \u27eax, y\u27eb = 0) :\n  \u2225x + y\u2225 * \u2225x + y\u2225 = \u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 :=\nbegin\n  rw [norm_add_mul_self, add_right_cancel_iff, add_right_eq_self, mul_eq_zero],\n  apply or.inr,\n  simp only [h, zero_re'],\nend\n\n\/-- Pythagorean theorem, vector inner product form. -\/\nlemma norm_add_square_eq_norm_square_add_norm_square_real {x y : F} (h : \u27eax, y\u27eb_\u211d = 0) :\n  \u2225x + y\u2225 * \u2225x + y\u2225 = \u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 :=\n(norm_add_square_eq_norm_square_add_norm_square_iff_real_inner_eq_zero x y).2 h\n\n\/-- Pythagorean theorem, subtracting vectors, if-and-only-if vector\ninner product form. -\/\nlemma norm_sub_square_eq_norm_square_add_norm_square_iff_real_inner_eq_zero (x y : F) :\n  \u2225x - y\u2225 * \u2225x - y\u2225 = \u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 \u2194 \u27eax, y\u27eb_\u211d = 0 :=\nbegin\n  rw [norm_sub_mul_self, add_right_cancel_iff, sub_eq_add_neg, add_right_eq_self, neg_eq_zero,\n      mul_eq_zero],\n  norm_num\nend\n\n\/-- Pythagorean theorem, subtracting vectors, vector inner product\nform. -\/\nlemma norm_sub_square_eq_norm_square_add_norm_square_real {x y : F} (h : \u27eax, y\u27eb_\u211d = 0) :\n  \u2225x - y\u2225 * \u2225x - y\u2225 = \u2225x\u2225 * \u2225x\u2225 + \u2225y\u2225 * \u2225y\u2225 :=\n(norm_sub_square_eq_norm_square_add_norm_square_iff_real_inner_eq_zero x y).2 h\n\n\/-- The sum and difference of two vectors are orthogonal if and only\nif they have the same norm. -\/\nlemma real_inner_add_sub_eq_zero_iff (x y : F) : \u27eax + y, x - y\u27eb_\u211d = 0 \u2194 \u2225x\u2225 = \u2225y\u2225 :=\nbegin\n  conv_rhs { rw \u2190mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _) },\n  simp only [\u2190inner_self_eq_norm_square, inner_add_left, inner_sub_right,\n            real_inner_comm y x, sub_eq_zero, re_to_real],\n  split,\n  { intro h,\n    rw [add_comm] at h,\n    linarith },\n  { intro h,\n    linarith }\nend\n\n\/-- The real inner product of two vectors, divided by the product of their\nnorms, has absolute value at most 1. -\/\nlemma abs_real_inner_div_norm_mul_norm_le_one (x y : F) : absR (\u27eax, y\u27eb_\u211d \/ (\u2225x\u2225 * \u2225y\u2225)) \u2264 1 :=\nbegin\n  rw _root_.abs_div,\n  by_cases h : 0 = absR (\u2225x\u2225 * \u2225y\u2225),\n  { rw [\u2190h, div_zero],\n    norm_num },\n  { change 0 \u2260 absR (\u2225x\u2225 * \u2225y\u2225) at h,\n    rw div_le_iff' (lt_of_le_of_ne (ge_iff_le.mp (_root_.abs_nonneg (\u2225x\u2225 * \u2225y\u2225))) h),\n    convert abs_real_inner_le_norm x y using 1,\n    rw [_root_.abs_mul, _root_.abs_of_nonneg (norm_nonneg x), _root_.abs_of_nonneg (norm_nonneg y),\n        mul_one] }\nend\n\n\/-- The inner product of a vector with a multiple of itself. -\/\nlemma real_inner_smul_self_left (x : F) (r : \u211d) : \u27ear \u2022 x, x\u27eb_\u211d = r * (\u2225x\u2225 * \u2225x\u2225) :=\nby rw [real_inner_smul_left, \u2190real_inner_self_eq_norm_square]\n\n\/-- The inner product of a vector with a multiple of itself. -\/\nlemma real_inner_smul_self_right (x : F) (r : \u211d) : \u27eax, r \u2022 x\u27eb_\u211d = r * (\u2225x\u2225 * \u2225x\u2225) :=\nby rw [inner_smul_right, \u2190real_inner_self_eq_norm_square]\n\n\/-- The inner product of a nonzero vector with a nonzero multiple of\nitself, divided by the product of their norms, has absolute value\n1. -\/\nlemma abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul\n  {x : E} {r : \ud835\udd5c} (hx : x \u2260 0) (hr : r \u2260 0) : abs \u27eax, r \u2022 x\u27eb \/ (\u2225x\u2225 * \u2225r \u2022 x\u2225) = 1 :=\nbegin\n  have hx' : \u2225x\u2225 \u2260 0 := by simp [norm_eq_zero, hx],\n  have hr' : abs r \u2260 0 := by simp [is_R_or_C.abs_eq_zero, hr],\n  rw [inner_smul_right, is_R_or_C.abs_mul, \u2190inner_self_re_abs, inner_self_eq_norm_square,\n      norm_smul],\n  rw [is_R_or_C.norm_eq_abs, \u2190mul_assoc, \u2190div_div_eq_div_mul, mul_div_cancel _ hx',\n     \u2190div_div_eq_div_mul, mul_comm, mul_div_cancel _ hr', div_self hx'],\nend\n\n\/-- The inner product of a nonzero vector with a nonzero multiple of\nitself, divided by the product of their norms, has absolute value\n1. -\/\nlemma abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul\n  {x : F} {r : \u211d} (hx : x \u2260 0) (hr : r \u2260 0) : absR \u27eax, r \u2022 x\u27eb_\u211d \/ (\u2225x\u2225 * \u2225r \u2022 x\u2225) = 1 :=\nbegin\n  rw \u2190 abs_to_real,\n  exact abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr\nend\n\n\/-- The inner product of a nonzero vector with a positive multiple of\nitself, divided by the product of their norms, has value 1. -\/\nlemma real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul\n  {x : F} {r : \u211d} (hx : x \u2260 0) (hr : 0 < r) : \u27eax, r \u2022 x\u27eb_\u211d \/ (\u2225x\u2225 * \u2225r \u2022 x\u2225) = 1 :=\nbegin\n  rw [real_inner_smul_self_right, norm_smul, real.norm_eq_abs, \u2190mul_assoc \u2225x\u2225, mul_comm _ (absR r),\n      mul_assoc, _root_.abs_of_nonneg (le_of_lt hr), div_self],\n  exact mul_ne_zero (ne_of_gt hr)\n    (\u03bb h, hx (norm_eq_zero.1 (eq_zero_of_mul_self_eq_zero h)))\nend\n\n\/-- The inner product of a nonzero vector with a negative multiple of\nitself, divided by the product of their norms, has value -1. -\/\nlemma real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul\n  {x : F} {r : \u211d} (hx : x \u2260 0) (hr : r < 0) : \u27eax, r \u2022 x\u27eb_\u211d \/ (\u2225x\u2225 * \u2225r \u2022 x\u2225) = -1 :=\nbegin\n  rw [real_inner_smul_self_right, norm_smul, real.norm_eq_abs, \u2190mul_assoc \u2225x\u2225, mul_comm _ (absR r),\n      mul_assoc, abs_of_neg hr, \u2190neg_mul_eq_neg_mul, div_neg_eq_neg_div, div_self],\n  exact mul_ne_zero (ne_of_lt hr)\n    (\u03bb h, hx (norm_eq_zero.1 (eq_zero_of_mul_self_eq_zero h)))\nend\n\n\/-- The inner product of two vectors, divided by the product of their\nnorms, has absolute value 1 if and only if they are nonzero and one is\na multiple of the other. One form of equality case for Cauchy-Schwarz. -\/\nlemma abs_inner_div_norm_mul_norm_eq_one_iff (x y : E) :\n  abs (\u27eax, y\u27eb \/ (\u2225x\u2225 * \u2225y\u2225)) = 1 \u2194 (x \u2260 0 \u2227 \u2203 (r : \ud835\udd5c), r \u2260 0 \u2227 y = r \u2022 x) :=\nbegin\n  split,\n  { intro h,\n    have hx0 : x \u2260 0,\n    { intro hx0,\n      rw [hx0, inner_zero_left, zero_div] at h,\n      norm_num at h,\n      exact h },\n    refine and.intro hx0 _,\n    set r := \u27eax, y\u27eb \/ (\u2225x\u2225 * \u2225x\u2225) with hr,\n    use r,\n    set t := y - r \u2022 x with ht,\n    have ht0 : \u27eax, t\u27eb = 0,\n    { rw [ht, inner_sub_right, inner_smul_right, hr],\n      norm_cast,\n      rw [\u2190inner_self_eq_norm_square, inner_self_re_to_K,\n          div_mul_cancel _ (\u03bb h, hx0 (inner_self_eq_zero.1 h)), sub_self] },\n    replace h : \u2225r \u2022 x\u2225 \/ \u2225t + r \u2022 x\u2225 = 1,\n    { rw [\u2190sub_add_cancel y (r \u2022 x), \u2190ht, inner_add_right, ht0, zero_add, inner_smul_right,\n        is_R_or_C.abs_div, is_R_or_C.abs_mul, \u2190inner_self_re_abs,\n        inner_self_eq_norm_square] at h,\n      norm_cast at h,\n      rwa [_root_.abs_mul, abs_norm_eq_norm, abs_norm_eq_norm, \u2190mul_assoc, mul_comm,\n        mul_div_mul_left _ _ (\u03bb h, hx0 (norm_eq_zero.1 h)), \u2190is_R_or_C.norm_eq_abs,\n        \u2190norm_smul] at h },\n    have hr0 : r \u2260 0,\n    { intro hr0,\n      rw [hr0, zero_smul, norm_zero, zero_div] at h,\n      norm_num at h },\n    refine and.intro hr0 _,\n    have h2 : \u2225r \u2022 x\u2225 ^ 2 = \u2225t + r \u2022 x\u2225 ^ 2,\n    { rw [eq_of_div_eq_one h] },\n    replace h2 : \u27ear \u2022 x, r \u2022 x\u27eb = \u27eat, t\u27eb + \u27eat, r \u2022 x\u27eb + \u27ear \u2022 x, t\u27eb + \u27ear \u2022 x, r \u2022 x\u27eb,\n    { rw [pow_two, pow_two, \u2190inner_self_eq_norm_square, \u2190inner_self_eq_norm_square ] at h2,\n      have h2' := congr_arg (\u03bb z : \u211d, (z : \ud835\udd5c)) h2,\n      simp_rw [inner_self_re_to_K, inner_add_add_self] at h2',\n      exact h2' },\n    conv at h2 in \u27ear \u2022 x, t\u27eb { rw [inner_smul_left, ht0, mul_zero] },\n    symmetry' at h2,\n    have h\u2081 : \u27eat, r \u2022 x\u27eb = 0 := by { rw [inner_smul_right, \u2190inner_conj_sym, ht0], simp },\n    rw [add_zero, h\u2081, add_left_eq_self, add_zero, inner_self_eq_zero] at h2,\n    rw h2 at ht,\n    exact eq_of_sub_eq_zero ht.symm },\n  { intro h,\n    rcases h with \u27e8hx, \u27e8r, \u27e8hr, hy\u27e9\u27e9\u27e9,\n    rw [hy, is_R_or_C.abs_div],\n    norm_cast,\n    rw [_root_.abs_mul, abs_norm_eq_norm, abs_norm_eq_norm],\n    exact abs_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr }\nend\n\n\/-- The inner product of two vectors, divided by the product of their\nnorms, has absolute value 1 if and only if they are nonzero and one is\na multiple of the other. One form of equality case for Cauchy-Schwarz. -\/\nlemma abs_real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :\n  absR (\u27eax, y\u27eb_\u211d \/ (\u2225x\u2225 * \u2225y\u2225)) = 1 \u2194 (x \u2260 0 \u2227 \u2203 (r : \u211d), r \u2260 0 \u2227 y = r \u2022 x) :=\nbegin\n  have := @abs_inner_div_norm_mul_norm_eq_one_iff \u211d F _ _ x y,\n  simpa [coe_real_eq_id] using this,\nend\n\n\/--\nIf the inner product of two vectors is equal to the product of their norms, then the two vectors\nare multiples of each other. One form of the equality case for Cauchy-Schwarz.\nCompare `inner_eq_norm_mul_iff`, which takes the stronger hypothesis `\u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225`. -\/\nlemma abs_inner_eq_norm_iff (x y : E) (hx0 : x \u2260 0) (hy0 : y \u2260 0):\n  abs \u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225 \u2194 \u2203 (r : \ud835\udd5c), r \u2260 0 \u2227 y = r \u2022 x :=\nbegin\n  have hx0' : \u2225x\u2225 \u2260 0 := by simp [norm_eq_zero, hx0],\n  have hy0' : \u2225y\u2225 \u2260 0 := by simp [norm_eq_zero, hy0],\n  have hxy0 : \u2225x\u2225 * \u2225y\u2225 \u2260 0 := by simp [hx0', hy0'],\n  have h\u2081 : abs \u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225 \u2194 abs (\u27eax, y\u27eb \/ (\u2225x\u2225 * \u2225y\u2225)) = 1,\n  { refine \u27e8_ ,_\u27e9,\n    { intro h,\n      norm_cast,\n      rw [is_R_or_C.abs_div, h, abs_of_real, _root_.abs_mul, abs_norm_eq_norm, abs_norm_eq_norm],\n      exact div_self hxy0 },\n    { intro h,\n      norm_cast at h,\n      rwa [is_R_or_C.abs_div, abs_of_real, _root_.abs_mul, abs_norm_eq_norm, abs_norm_eq_norm,\n          div_eq_one_iff_eq hxy0] at h } },\n  rw [h\u2081, abs_inner_div_norm_mul_norm_eq_one_iff x y],\n  have : x \u2260 0 := \u03bb h, (hx0' $ norm_eq_zero.mpr h),\n  simp [this]\nend\n\n\/-- The inner product of two vectors, divided by the product of their\nnorms, has value 1 if and only if they are nonzero and one is\na positive multiple of the other. -\/\nlemma real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :\n  \u27eax, y\u27eb_\u211d \/ (\u2225x\u2225 * \u2225y\u2225) = 1 \u2194 (x \u2260 0 \u2227 \u2203 (r : \u211d), 0 < r \u2227 y = r \u2022 x) :=\nbegin\n  split,\n  { intro h,\n    have ha := h,\n    apply_fun absR at ha,\n    norm_num at ha,\n    rcases (abs_real_inner_div_norm_mul_norm_eq_one_iff x y).1 ha with \u27e8hx, \u27e8r, \u27e8hr, hy\u27e9\u27e9\u27e9,\n    use [hx, r],\n    refine and.intro _ hy,\n    by_contradiction hrneg,\n    rw hy at h,\n    rw real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul hx\n      (lt_of_le_of_ne (le_of_not_lt hrneg) hr) at h,\n    norm_num at h },\n  { intro h,\n    rcases h with \u27e8hx, \u27e8r, \u27e8hr, hy\u27e9\u27e9\u27e9,\n    rw hy,\n    exact real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul hx hr }\nend\n\n\/-- The inner product of two vectors, divided by the product of their\nnorms, has value -1 if and only if they are nonzero and one is\na negative multiple of the other. -\/\nlemma real_inner_div_norm_mul_norm_eq_neg_one_iff (x y : F) :\n  \u27eax, y\u27eb_\u211d \/ (\u2225x\u2225 * \u2225y\u2225) = -1 \u2194 (x \u2260 0 \u2227 \u2203 (r : \u211d), r < 0 \u2227 y = r \u2022 x) :=\nbegin\n  split,\n  { intro h,\n    have ha := h,\n    apply_fun absR at ha,\n    norm_num at ha,\n    rcases (abs_real_inner_div_norm_mul_norm_eq_one_iff x y).1 ha with \u27e8hx, \u27e8r, \u27e8hr, hy\u27e9\u27e9\u27e9,\n    use [hx, r],\n    refine and.intro _ hy,\n    by_contradiction hrpos,\n    rw hy at h,\n    rw real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul hx\n      (lt_of_le_of_ne (le_of_not_lt hrpos) hr.symm) at h,\n    norm_num at h },\n  { intro h,\n    rcases h with \u27e8hx, \u27e8r, \u27e8hr, hy\u27e9\u27e9\u27e9,\n    rw hy,\n    exact real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul hx hr }\nend\n\n\/-- If the inner product of two vectors is equal to the product of their norms (i.e.,\n`\u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225`), then the two vectors are nonnegative real multiples of each other. One form\nof the equality case for Cauchy-Schwarz.\nCompare `abs_inner_eq_norm_iff`, which takes the weaker hypothesis `abs \u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225`. -\/\nlemma inner_eq_norm_mul_iff {x y : E} :\n  \u27eax, y\u27eb = (\u2225x\u2225 : \ud835\udd5c) * \u2225y\u2225 \u2194 (\u2225y\u2225 : \ud835\udd5c) \u2022 x = (\u2225x\u2225 : \ud835\udd5c) \u2022 y :=\nbegin\n  by_cases h : (x = 0 \u2228 y = 0), -- WLOG `x` and `y` are nonzero\n  { cases h; simp [h] },\n  calc \u27eax, y\u27eb = (\u2225x\u2225 : \ud835\udd5c) * \u2225y\u2225 \u2194 \u2225x\u2225 * \u2225y\u2225 = re \u27eax, y\u27eb :\n  begin\n    norm_cast,\n    split,\n    { intros h',\n      simp [h'] },\n    { have cauchy_schwarz := abs_inner_le_norm x y,\n      intros h',\n      rw h' at \u22a2 cauchy_schwarz,\n      rwa re_eq_self_of_le }\n  end\n  ... \u2194 2 * \u2225x\u2225 * \u2225y\u2225 * (\u2225x\u2225 * \u2225y\u2225 - re \u27eax, y\u27eb) = 0 :\n    by simp [h, show (2:\u211d) \u2260 0, by norm_num, sub_eq_zero]\n  ... \u2194 \u2225(\u2225y\u2225:\ud835\udd5c) \u2022 x - (\u2225x\u2225:\ud835\udd5c) \u2022 y\u2225 * \u2225(\u2225y\u2225:\ud835\udd5c) \u2022 x - (\u2225x\u2225:\ud835\udd5c) \u2022 y\u2225 = 0 :\n  begin\n    simp only [norm_sub_mul_self, inner_smul_left, inner_smul_right, norm_smul, conj_of_real,\n      is_R_or_C.norm_eq_abs, abs_of_real, of_real_im, of_real_re, mul_re, abs_norm_eq_norm],\n    refine eq.congr _ rfl,\n    ring\n  end\n  ... \u2194 (\u2225y\u2225 : \ud835\udd5c) \u2022 x = (\u2225x\u2225 : \ud835\udd5c) \u2022 y : by simp [norm_sub_eq_zero_iff]\nend\n\n\/-- If the inner product of two vectors is equal to the product of their norms (i.e.,\n`\u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225`), then the two vectors are nonnegative real multiples of each other. One form\nof the equality case for Cauchy-Schwarz.\nCompare `abs_inner_eq_norm_iff`, which takes the weaker hypothesis `abs \u27eax, y\u27eb = \u2225x\u2225 * \u2225y\u2225`. -\/\nlemma inner_eq_norm_mul_iff_real {x y : F} : \u27eax, y\u27eb_\u211d = \u2225x\u2225 * \u2225y\u2225 \u2194 \u2225y\u2225 \u2022 x = \u2225x\u2225 \u2022 y :=\ninner_eq_norm_mul_iff\n\n\/-- If the inner product of two unit vectors is `1`, then the two vectors are equal. One form of\nthe equality case for Cauchy-Schwarz. -\/\nlemma inner_eq_norm_mul_iff_of_norm_one {x y : E} (hx : \u2225x\u2225 = 1) (hy : \u2225y\u2225 = 1) :\n  \u27eax, y\u27eb = 1 \u2194 x = y :=\nby { convert inner_eq_norm_mul_iff using 2; simp [hx, hy] }\n\nlemma inner_lt_norm_mul_iff_real {x y : F} :\n  \u27eax, y\u27eb_\u211d < \u2225x\u2225 * \u2225y\u2225 \u2194 \u2225y\u2225 \u2022 x \u2260 \u2225x\u2225 \u2022 y :=\ncalc \u27eax, y\u27eb_\u211d < \u2225x\u2225 * \u2225y\u2225\n    \u2194 \u27eax, y\u27eb_\u211d \u2260 \u2225x\u2225 * \u2225y\u2225 : \u27e8ne_of_lt, lt_of_le_of_ne (real_inner_le_norm _ _)\u27e9\n... \u2194 \u2225y\u2225 \u2022 x \u2260 \u2225x\u2225 \u2022 y : not_congr inner_eq_norm_mul_iff_real\n\n\/-- If the inner product of two unit vectors is strictly less than `1`, then the two vectors are\ndistinct. One form of the equality case for Cauchy-Schwarz. -\/\nlemma inner_lt_one_iff_real_of_norm_one {x y : F} (hx : \u2225x\u2225 = 1) (hy : \u2225y\u2225 = 1) :\n  \u27eax, y\u27eb_\u211d < 1 \u2194 x \u2260 y :=\nby { convert inner_lt_norm_mul_iff_real; simp [hx, hy] }\n\n\/-- The inner product of two weighted sums, where the weights in each\nsum add to 0, in terms of the norms of pairwise differences. -\/\nlemma inner_sum_smul_sum_smul_of_sum_eq_zero {\u03b9\u2081 : Type*} {s\u2081 : finset \u03b9\u2081} {w\u2081 : \u03b9\u2081 \u2192 \u211d}\n    (v\u2081 : \u03b9\u2081 \u2192 F) (h\u2081 : \u2211 i in s\u2081, w\u2081 i = 0) {\u03b9\u2082 : Type*} {s\u2082 : finset \u03b9\u2082} {w\u2082 : \u03b9\u2082 \u2192 \u211d}\n    (v\u2082 : \u03b9\u2082 \u2192 F) (h\u2082 : \u2211 i in s\u2082, w\u2082 i = 0) :\n  \u27ea(\u2211 i\u2081 in s\u2081, w\u2081 i\u2081 \u2022 v\u2081 i\u2081), (\u2211 i\u2082 in s\u2082, w\u2082 i\u2082 \u2022 v\u2082 i\u2082)\u27eb_\u211d =\n    (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (\u2225v\u2081 i\u2081 - v\u2082 i\u2082\u2225 * \u2225v\u2081 i\u2081 - v\u2082 i\u2082\u2225)) \/ 2 :=\nby simp_rw [sum_inner, inner_sum, real_inner_smul_left, real_inner_smul_right,\n            real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two,\n            \u2190div_sub_div_same, \u2190div_add_div_same, mul_sub_left_distrib, left_distrib,\n            finset.sum_sub_distrib, finset.sum_add_distrib, \u2190finset.mul_sum, \u2190finset.sum_mul,\n            h\u2081, h\u2082, zero_mul, mul_zero, finset.sum_const_zero, zero_add, zero_sub, finset.mul_sum,\n            neg_div, finset.sum_div, mul_div_assoc, mul_assoc]\n\n\/-- The inner product with a fixed left element, as a continuous linear map.  This can be upgraded\nto a continuous map which is jointly conjugate-linear in the left argument and linear in the right\nargument, once (TODO) conjugate-linear maps have been defined. -\/\ndef inner_right (v : E) : E \u2192L[\ud835\udd5c] \ud835\udd5c :=\nlinear_map.mk_continuous\n  { to_fun := \u03bb w, \u27eav, w\u27eb,\n    map_add' := \u03bb x y, inner_add_right,\n    map_smul' := \u03bb c x, inner_smul_right }\n  \u2225v\u2225\n  (by simpa [is_R_or_C.norm_eq_abs] using abs_inner_le_norm v)\n\n@[simp] lemma inner_right_coe (v : E) : (inner_right v : E \u2192 \ud835\udd5c) = \u03bb w, \u27eav, w\u27eb := rfl\n\n@[simp] lemma inner_right_apply (v w : E) : inner_right v w = \u27eav, w\u27eb := rfl\n\nend norm\n\n\/-! ### Inner product space structure on product spaces -\/\n\n\/-\n If `\u03b9` is a finite type and each space `f i`, `i : \u03b9`, is an inner product space,\nthen `\u03a0 i, f i` is an inner product space as well. Since `\u03a0 i, f i` is endowed with the sup norm,\nwe use instead `pi_Lp 2 one_le_two f` for the product space, which is endowed with the `L^2` norm.\n-\/\ninstance pi_Lp.inner_product_space {\u03b9 : Type*} [fintype \u03b9] (f : \u03b9 \u2192 Type*)\n  [\u03a0 i, inner_product_space \ud835\udd5c (f i)] : inner_product_space \ud835\udd5c (pi_Lp 2 one_le_two f) :=\n{ inner := \u03bb x y, \u2211 i, inner (x i) (y i),\n  norm_sq_eq_inner :=\n  begin\n    intro x,\n    have h\u2081 : \u2211 (i : \u03b9), \u2225x i\u2225 ^ (2 : \u2115) = \u2211 (i : \u03b9), \u2225x i\u2225 ^ (2 : \u211d),\n    { apply finset.sum_congr rfl,\n      intros j hj,\n      simp [\u2190rpow_nat_cast] },\n    have h\u2082 : 0 \u2264 \u2211 (i : \u03b9), \u2225x i\u2225 ^ (2 : \u211d),\n    { rw [\u2190h\u2081],\n      exact finset.sum_nonneg (\u03bb j (hj : j \u2208 finset.univ), pow_nonneg (norm_nonneg (x j)) 2) },\n    simp [norm, add_monoid_hom.map_sum, \u2190norm_sq_eq_inner],\n    rw [\u2190rpow_nat_cast ((\u2211 (i : \u03b9), \u2225x i\u2225 ^ (2 : \u211d)) ^ (2 : \u211d)\u207b\u00b9) 2],\n    rw [\u2190rpow_mul h\u2082],\n    norm_num [h\u2081],\n  end,\n  conj_sym :=\n  begin\n    intros x y,\n    unfold inner,\n    rw [\u2190finset.sum_hom finset.univ conj],\n    apply finset.sum_congr rfl,\n    rintros z -,\n    apply inner_conj_sym,\n    apply_instance\n  end,\n  nonneg_im :=\n  begin\n    intro x,\n    unfold inner,\n    rw[\u2190finset.sum_hom finset.univ im],\n    apply finset.sum_eq_zero,\n    rintros z -,\n    exact inner_self_nonneg_im,\n    apply_instance\n  end,\n  add_left := \u03bb x y z,\n    show \u2211 i, inner (x i + y i) (z i) = \u2211 i, inner (x i) (z i) + \u2211 i, inner (y i) (z i),\n    by simp only [inner_add_left, finset.sum_add_distrib],\n  smul_left := \u03bb x y r,\n    show \u2211 (i : \u03b9), inner (r \u2022 x i) (y i) = (conj r) * \u2211 i, inner (x i) (y i),\n    by simp only [finset.mul_sum, inner_smul_left]\n}\n\n\/-- A field `\ud835\udd5c` satisfying `is_R_or_C` is itself a `\ud835\udd5c`-inner product space. -\/\ninstance is_R_or_C.inner_product_space : inner_product_space \ud835\udd5c \ud835\udd5c :=\n{ inner := (\u03bb x y, (conj x) * y),\n  norm_sq_eq_inner := \u03bb x,\n    by { unfold inner, rw [mul_comm, mul_conj, of_real_re, norm_sq_eq_def'] },\n  conj_sym := \u03bb x y, by simp [mul_comm],\n  nonneg_im := \u03bb x, by rw [mul_im, conj_re, conj_im]; ring,\n  add_left := \u03bb x y z, by simp [inner, add_mul],\n  smul_left := \u03bb x y z, by simp [inner, mul_assoc] }\n\n\/-- The standard real\/complex Euclidean space, functions on a finite type. For an `n`-dimensional\nspace use `euclidean_space \ud835\udd5c (fin n)`. -\/\n@[reducible, nolint unused_arguments]\ndef euclidean_space (\ud835\udd5c : Type*) [is_R_or_C \ud835\udd5c]\n  (n : Type*) [fintype n] : Type* := pi_Lp 2 one_le_two (\u03bb (i : n), \ud835\udd5c)\n\nsection is_R_or_C_to_real\n\nvariables {G : Type*}\n\nvariables (\ud835\udd5c E)\ninclude \ud835\udd5c\n\n\/-- A general inner product implies a real inner product. This is not registered as an instance\nsince it creates problems with the case `\ud835\udd5c = \u211d`. -\/\ndef has_inner.is_R_or_C_to_real : has_inner \u211d E :=\n{ inner := \u03bb x y, re \u27eax, y\u27eb }\n\n\/-- A general inner product space structure implies a real inner product structure. This is not\nregistered as an instance since it creates problems with the case `\ud835\udd5c = \u211d`, but in can be used in a\nproof to obtain a real inner product space structure from a given `\ud835\udd5c`-inner product space\nstructure. -\/\ndef inner_product_space.is_R_or_C_to_real : inner_product_space \u211d E :=\n{ norm_sq_eq_inner := norm_sq_eq_inner,\n  conj_sym := \u03bb x y, inner_re_symm,\n  nonneg_im := \u03bb x, rfl,\n  add_left := \u03bb x y z, by {\n    change re \u27eax + y, z\u27eb = re \u27eax, z\u27eb + re \u27eay, z\u27eb,\n    simp [inner_add_left] },\n  smul_left := \u03bb x y r, by {\n    change re \u27ea(r : \ud835\udd5c) \u2022 x, y\u27eb = r * re \u27eax, y\u27eb,\n    simp [inner_smul_left] },\n  ..has_inner.is_R_or_C_to_real \ud835\udd5c E,\n  ..normed_space.restrict_scalars \u211d \ud835\udd5c E }\n\nvariable {E}\n\nlemma real_inner_eq_re_inner (x y : E) :\n  @has_inner.inner \u211d E (has_inner.is_R_or_C_to_real \ud835\udd5c E) x y = re \u27eax, y\u27eb := rfl\n\nomit \ud835\udd5c\n\n\/-- A complex inner product implies a real inner product -\/\ninstance inner_product_space.complex_to_real [inner_product_space \u2102 G] : inner_product_space \u211d G :=\ninner_product_space.is_R_or_C_to_real \u2102 G\n\nend is_R_or_C_to_real\n\nsection deriv\n\n\/-!\n### Derivative of the inner product\n\nIn this section we prove that the inner product and square of the norm in an inner space are\ninfinitely `\u211d`-smooth. In order to state these results, we need a `normed_space \u211d E`\ninstance. Though we can deduce this structure from `inner_product_space \ud835\udd5c E`, this instance may be\nnot definitionally equal to some other \u201cnatural\u201d instance. So, we assume `[normed_space \u211d E]` and\n`[is_scalar_tower \u211d \ud835\udd5c E]`. In both interesting cases `\ud835\udd5c = \u211d` and `\ud835\udd5c = \u2102` we have these instances.\n\n-\/\n\nvariables [normed_space \u211d E] [is_scalar_tower \u211d \ud835\udd5c E]\n\nlemma is_bounded_bilinear_map_inner : is_bounded_bilinear_map \u211d (\u03bb p : E \u00d7 E, \u27eap.1, p.2\u27eb) :=\n{ add_left := \u03bb _ _ _, inner_add_left,\n  smul_left := \u03bb r x y,\n    by simp only [\u2190 algebra_map_smul \ud835\udd5c r x, algebra_map_eq_of_real, inner_smul_real_left],\n  add_right := \u03bb _ _ _, inner_add_right,\n  smul_right := \u03bb r x y,\n    by simp only [\u2190 algebra_map_smul \ud835\udd5c r y, algebra_map_eq_of_real, inner_smul_real_right],\n  bound := \u27e81, zero_lt_one, \u03bb x y,\n    by { rw [one_mul, is_R_or_C.norm_eq_abs], exact abs_inner_le_norm x y, }\u27e9 }\n\n\/-- Derivative of the inner product. -\/\ndef fderiv_inner_clm (p : E \u00d7 E) : E \u00d7 E \u2192L[\u211d] \ud835\udd5c := is_bounded_bilinear_map_inner.deriv p\n\n@[simp] lemma fderiv_inner_clm_apply (p x : E \u00d7 E) :\n  fderiv_inner_clm  p x = \u27eap.1, x.2\u27eb + \u27eax.1, p.2\u27eb := rfl\n\nlemma times_cont_diff_inner {n} : times_cont_diff \u211d n (\u03bb p : E \u00d7 E, \u27eap.1, p.2\u27eb) :=\nis_bounded_bilinear_map_inner.times_cont_diff\n\nlemma times_cont_diff_at_inner {p : E \u00d7 E} {n} :\n  times_cont_diff_at \u211d n (\u03bb p : E \u00d7 E, \u27eap.1, p.2\u27eb) p :=\ntimes_cont_diff_inner.times_cont_diff_at\n\nlemma differentiable_inner : differentiable \u211d (\u03bb p : E \u00d7 E, \u27eap.1, p.2\u27eb) :=\nis_bounded_bilinear_map_inner.differentiable_at\n\nvariables {G : Type*} [normed_group G] [normed_space \u211d G]\n  {f g : G \u2192 E} {f' g' : G \u2192L[\u211d] E} {s : set G} {x : G} {n : with_top \u2115}\n\ninclude \ud835\udd5c\n\nlemma times_cont_diff_within_at.inner (hf : times_cont_diff_within_at \u211d n f s x)\n  (hg : times_cont_diff_within_at \u211d n g s x) :\n  times_cont_diff_within_at \u211d n (\u03bb x, \u27eaf x, g x\u27eb) s x :=\ntimes_cont_diff_at_inner.comp_times_cont_diff_within_at x (hf.prod hg)\n\nlemma times_cont_diff_at.inner (hf : times_cont_diff_at \u211d n f x)\n  (hg : times_cont_diff_at \u211d n g x) :\n  times_cont_diff_at \u211d n (\u03bb x, \u27eaf x, g x\u27eb) x :=\nhf.inner hg\n\nlemma times_cont_diff_on.inner (hf : times_cont_diff_on \u211d n f s) (hg : times_cont_diff_on \u211d n g s) :\n  times_cont_diff_on \u211d n (\u03bb x, \u27eaf x, g x\u27eb) s :=\n\u03bb x hx, (hf x hx).inner (hg x hx)\n\nlemma times_cont_diff.inner (hf : times_cont_diff \u211d n f) (hg : times_cont_diff \u211d n g) :\n  times_cont_diff \u211d n (\u03bb x, \u27eaf x, g x\u27eb) :=\ntimes_cont_diff_inner.comp (hf.prod hg)\n\nlemma has_fderiv_within_at.inner (hf : has_fderiv_within_at f f' s x)\n  (hg : has_fderiv_within_at g g' s x) :\n  has_fderiv_within_at (\u03bb t, \u27eaf t, g t\u27eb) ((fderiv_inner_clm (f x, g x)).comp $ f'.prod g') s x :=\n(is_bounded_bilinear_map_inner.has_fderiv_at (f x, g x)).comp_has_fderiv_within_at x (hf.prod hg)\n\nlemma has_fderiv_at.inner (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :\n  has_fderiv_at (\u03bb t, \u27eaf t, g t\u27eb) ((fderiv_inner_clm (f x, g x)).comp $ f'.prod g') x :=\n(is_bounded_bilinear_map_inner.has_fderiv_at (f x, g x)).comp x (hf.prod hg)\n\nlemma has_deriv_within_at.inner {f g : \u211d \u2192 E} {f' g' : E} {s : set \u211d} {x : \u211d}\n  (hf : has_deriv_within_at f f' s x) (hg : has_deriv_within_at g g' s x) :\n  has_deriv_within_at (\u03bb t, \u27eaf t, g t\u27eb) (\u27eaf x, g'\u27eb + \u27eaf', g x\u27eb) s x :=\nby simpa using (hf.has_fderiv_within_at.inner hg.has_fderiv_within_at).has_deriv_within_at\n\nlemma has_deriv_at.inner {f g : \u211d \u2192 E} {f' g' : E} {x : \u211d} :\n  has_deriv_at f f' x \u2192  has_deriv_at g g' x \u2192\n  has_deriv_at (\u03bb t, \u27eaf t, g t\u27eb) (\u27eaf x, g'\u27eb + \u27eaf', g x\u27eb) x :=\nby simpa only [\u2190 has_deriv_within_at_univ] using has_deriv_within_at.inner\n\nlemma differentiable_within_at.inner (hf : differentiable_within_at \u211d f s x)\n  (hg : differentiable_within_at \u211d g s x) :\n  differentiable_within_at \u211d (\u03bb x, \u27eaf x, g x\u27eb) s x :=\n((differentiable_inner _).has_fderiv_at.comp_has_fderiv_within_at x\n  (hf.prod hg).has_fderiv_within_at).differentiable_within_at\n\nlemma differentiable_at.inner (hf : differentiable_at \u211d f x) (hg : differentiable_at \u211d g x) :\n  differentiable_at \u211d (\u03bb x, \u27eaf x, g x\u27eb) x :=\n(differentiable_inner _).comp x (hf.prod hg)\n\nlemma differentiable_on.inner (hf : differentiable_on \u211d f s) (hg : differentiable_on \u211d g s) :\n  differentiable_on \u211d (\u03bb x, \u27eaf x, g x\u27eb) s :=\n\u03bb x hx, (hf x hx).inner (hg x hx)\n\nlemma differentiable.inner (hf : differentiable \u211d f) (hg : differentiable \u211d g) :\n  differentiable \u211d (\u03bb x, \u27eaf x, g x\u27eb) :=\n\u03bb x, (hf x).inner (hg x)\n\nlemma fderiv_inner_apply (hf : differentiable_at \u211d f x) (hg : differentiable_at \u211d g x) (y : G) :\n  fderiv \u211d (\u03bb t, \u27eaf t, g t\u27eb) x y = \u27eaf x, fderiv \u211d g x y\u27eb + \u27eafderiv \u211d f x y, g x\u27eb :=\nby { rw [(hf.has_fderiv_at.inner hg.has_fderiv_at).fderiv], refl }\n\nlemma deriv_inner_apply {f g : \u211d \u2192 E} {x : \u211d} (hf : differentiable_at \u211d f x)\n  (hg : differentiable_at \u211d g x) :\n  deriv (\u03bb t, \u27eaf t, g t\u27eb) x = \u27eaf x, deriv g x\u27eb + \u27eaderiv f x, g x\u27eb :=\n(hf.has_deriv_at.inner hg.has_deriv_at).deriv\n\nlemma times_cont_diff_norm_square : times_cont_diff \u211d n (\u03bb x : E, \u2225x\u2225 ^ 2) :=\nbegin\n  simp only [pow_two, \u2190 inner_self_eq_norm_square],\n  exact (re_clm : \ud835\udd5c \u2192L[\u211d] \u211d).times_cont_diff.comp (times_cont_diff_id.inner times_cont_diff_id)\nend\n\nlemma times_cont_diff.norm_square (hf : times_cont_diff \u211d n f) :\n  times_cont_diff \u211d n (\u03bb x, \u2225f x\u2225 ^ 2) :=\ntimes_cont_diff_norm_square.comp hf\n\nlemma times_cont_diff_within_at.norm_square (hf : times_cont_diff_within_at \u211d n f s x) :\n  times_cont_diff_within_at \u211d n (\u03bb y, \u2225f y\u2225 ^ 2) s x :=\ntimes_cont_diff_norm_square.times_cont_diff_at.comp_times_cont_diff_within_at x hf\n\nlemma times_cont_diff_at.norm_square (hf : times_cont_diff_at \u211d n f x) :\n  times_cont_diff_at \u211d n (\u03bb y, \u2225f y\u2225 ^ 2) x :=\nhf.norm_square\n\nlemma times_cont_diff_on.norm_square (hf : times_cont_diff_on \u211d n f s) :\n  times_cont_diff_on \u211d n (\u03bb y, \u2225f y\u2225 ^ 2) s :=\n(\u03bb x hx, (hf x hx).norm_square)\n\nlemma differentiable_at.norm_square (hf : differentiable_at \u211d f x) :\n  differentiable_at \u211d (\u03bb y, \u2225f y\u2225 ^ 2) x :=\n(times_cont_diff_norm_square.differentiable le_rfl).differentiable_at.comp x hf\n\nlemma differentiable.norm_square (hf : differentiable \u211d f) : differentiable \u211d (\u03bb y, \u2225f y\u2225 ^ 2) :=\n\u03bb x, (hf x).norm_square\n\nlemma differentiable_within_at.norm_square (hf : differentiable_within_at \u211d f s x) :\n  differentiable_within_at \u211d (\u03bb y, \u2225f y\u2225 ^ 2) s x :=\n(times_cont_diff_norm_square.differentiable le_rfl).differentiable_at.comp_differentiable_within_at\n  x hf\n\nlemma differentiable_on.norm_square (hf : differentiable_on \u211d f s) :\n  differentiable_on \u211d (\u03bb y, \u2225f y\u2225 ^ 2) s :=\n\u03bb x hx, (hf x hx).norm_square\n\nend deriv\n\nsection continuous\n\n\/-!\n### Continuity and measurability of the inner product\n\nSince the inner product is `\u211d`-smooth, it is continuous. We do not need a `[normed_space \u211d E]`\nstructure to *state* this fact and its corollaries, so we introduce them in the proof instead.\n-\/\n\nlemma continuous_inner : continuous (\u03bb p : E \u00d7 E, \u27eap.1, p.2\u27eb) :=\nbegin\n  letI : inner_product_space \u211d E := inner_product_space.is_R_or_C_to_real \ud835\udd5c E,\n  letI : is_scalar_tower \u211d \ud835\udd5c E := restrict_scalars.is_scalar_tower _ _ _,\n  exact differentiable_inner.continuous\nend\n\nvariables {\u03b1 : Type*}\n\nlemma filter.tendsto.inner {f g : \u03b1 \u2192 E} {l : filter \u03b1} {x y : E} (hf : tendsto f l (\ud835\udcdd x))\n  (hg : tendsto g l (\ud835\udcdd y)) :\n  tendsto (\u03bb t, \u27eaf t, g t\u27eb) l (\ud835\udcdd \u27eax, y\u27eb) :=\n(continuous_inner.tendsto _).comp (hf.prod_mk_nhds hg)\n\nlemma measurable.inner [measurable_space \u03b1] [measurable_space E] [opens_measurable_space E]\n  [topological_space.second_countable_topology E] [measurable_space \ud835\udd5c] [borel_space \ud835\udd5c]\n  {f g : \u03b1 \u2192 E} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb t, \u27eaf t, g t\u27eb) :=\ncontinuous.measurable2 continuous_inner hf hg\n\nvariables [topological_space \u03b1] {f g : \u03b1 \u2192 E} {x : \u03b1} {s : set \u03b1}\n\ninclude \ud835\udd5c\n\nlemma continuous_within_at.inner (hf : continuous_within_at f s x)\n  (hg : continuous_within_at g s x) :\n  continuous_within_at (\u03bb t, \u27eaf t, g t\u27eb) s x :=\nhf.inner hg\n\nlemma continuous_at.inner (hf : continuous_at f x) (hg : continuous_at g x) :\n  continuous_at (\u03bb t, \u27eaf t, g t\u27eb) x :=\nhf.inner hg\n\nlemma continuous_on.inner (hf : continuous_on f s) (hg : continuous_on g s) :\n  continuous_on (\u03bb t, \u27eaf t, g t\u27eb) s :=\n\u03bb x hx, (hf x hx).inner (hg x hx)\n\nlemma continuous.inner (hf : continuous f) (hg : continuous g) : continuous (\u03bb t, \u27eaf t, g t\u27eb) :=\ncontinuous_iff_continuous_at.2 $ \u03bb x, hf.continuous_at.inner hg.continuous_at\n\nend continuous\n\nsection pi_Lp\nlocal attribute [reducible] pi_Lp\nvariables {\u03b9 : Type*} [fintype \u03b9]\n\ninstance : finite_dimensional \ud835\udd5c (euclidean_space \ud835\udd5c \u03b9) := by apply_instance\n\n@[simp] lemma findim_euclidean_space :\n  finite_dimensional.findim \ud835\udd5c (euclidean_space \ud835\udd5c \u03b9) = fintype.card \u03b9 := by simp\n\nlemma findim_euclidean_space_fin {n : \u2115} :\n  finite_dimensional.findim \ud835\udd5c (euclidean_space \ud835\udd5c (fin n)) = n := by simp\n\nend pi_Lp\n\n\n\/-! ### Orthogonal projection in inner product spaces -\/\n\nsection orthogonal\n\nopen filter\n\n\/--\nExistence of minimizers\nLet `u` be a point in a real inner product space, and let `K` be a nonempty complete convex subset.\nThen there exists a (unique) `v` in `K` that minimizes the distance `\u2225u - v\u2225` to `u`.\n -\/\n-- FIXME this monolithic proof causes a deterministic timeout with `-T50000`\n-- It should be broken in a sequence of more manageable pieces,\n-- perhaps with individual statements for the three steps below.\ntheorem exists_norm_eq_infi_of_complete_convex {K : set F} (ne : K.nonempty) (h\u2081 : is_complete K)\n  (h\u2082 : convex K) : \u2200 u : F, \u2203 v \u2208 K, \u2225u - v\u2225 = \u2a05 w : K, \u2225u - w\u2225 := assume u,\nbegin\n  let \u03b4 := \u2a05 w : K, \u2225u - w\u2225,\n  letI : nonempty K := ne.to_subtype,\n  have zero_le_\u03b4 : 0 \u2264 \u03b4 := le_cinfi (\u03bb _, norm_nonneg _),\n  have \u03b4_le : \u2200 w : K, \u03b4 \u2264 \u2225u - w\u2225,\n    from cinfi_le \u27e80, set.forall_range_iff.2 $ \u03bb _, norm_nonneg _\u27e9,\n  have \u03b4_le' : \u2200 w \u2208 K, \u03b4 \u2264 \u2225u - w\u2225 := assume w hw, \u03b4_le \u27e8w, hw\u27e9,\n  -- Step 1: since `\u03b4` is the infimum, can find a sequence `w : \u2115 \u2192 K` in `K`\n  -- such that `\u2225u - w n\u2225 < \u03b4 + 1 \/ (n + 1)` (which implies `\u2225u - w n\u2225 --> \u03b4`);\n  -- maybe this should be a separate lemma\n  have exists_seq : \u2203 w : \u2115 \u2192 K, \u2200 n, \u2225u - w n\u2225 < \u03b4 + 1 \/ (n + 1),\n  { have h\u03b4 : \u2200n:\u2115, \u03b4 < \u03b4 + 1 \/ (n + 1), from\n      \u03bb n, lt_add_of_le_of_pos (le_refl _) nat.one_div_pos_of_nat,\n    have h := \u03bb n, exists_lt_of_cinfi_lt (h\u03b4 n),\n    let w : \u2115 \u2192 K := \u03bb n, classical.some (h n),\n    exact \u27e8w, \u03bb n, classical.some_spec (h n)\u27e9 },\n  rcases exists_seq with \u27e8w, hw\u27e9,\n  have norm_tendsto : tendsto (\u03bb n, \u2225u - w n\u2225) at_top (nhds \u03b4),\n  { have h : tendsto (\u03bb n:\u2115, \u03b4) at_top (nhds \u03b4) := tendsto_const_nhds,\n    have h' : tendsto (\u03bb n:\u2115, \u03b4 + 1 \/ (n + 1)) at_top (nhds \u03b4),\n    { convert h.add tendsto_one_div_add_at_top_nhds_0_nat, simp only [add_zero] },\n    exact tendsto_of_tendsto_of_tendsto_of_le_of_le h h'\n      (\u03bb x, \u03b4_le _) (\u03bb x, le_of_lt (hw _)) },\n  -- Step 2: Prove that the sequence `w : \u2115 \u2192 K` is a Cauchy sequence\n  have seq_is_cauchy : cauchy_seq (\u03bb n, ((w n):F)),\n  { rw cauchy_seq_iff_le_tendsto_0, -- splits into three goals\n    let b := \u03bb n:\u2115, (8 * \u03b4 * (1\/(n+1)) + 4 * (1\/(n+1)) * (1\/(n+1))),\n    use (\u03bbn, sqrt (b n)),\n    split,\n    -- first goal :  `\u2200 (n : \u2115), 0 \u2264 sqrt (b n)`\n    assume n, exact sqrt_nonneg _,\n    split,\n    -- second goal : `\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)`\n    assume p q N hp hq,\n    let wp := ((w p):F), let wq := ((w q):F),\n    let a := u - wq, let b := u - wp,\n    let half := 1 \/ (2:\u211d), let div := 1 \/ ((N:\u211d) + 1),\n    have : 4 * \u2225u - half \u2022 (wq + wp)\u2225 * \u2225u - half \u2022 (wq + wp)\u2225 + \u2225wp - wq\u2225 * \u2225wp - wq\u2225 =\n      2 * (\u2225a\u2225 * \u2225a\u2225 + \u2225b\u2225 * \u2225b\u2225) :=\n    calc\n      4 * \u2225u - half\u2022(wq + wp)\u2225 * \u2225u - half\u2022(wq + wp)\u2225 + \u2225wp - wq\u2225 * \u2225wp - wq\u2225\n          = (2*\u2225u - half\u2022(wq + wp)\u2225) * (2 * \u2225u - half\u2022(wq + wp)\u2225) + \u2225wp-wq\u2225*\u2225wp-wq\u2225 : by ring\n      ... = (absR ((2:\u211d)) * \u2225u - half\u2022(wq + wp)\u2225) * (absR ((2:\u211d)) * \u2225u - half\u2022(wq+wp)\u2225) +\n            \u2225wp-wq\u2225*\u2225wp-wq\u2225 :\n      by { rw _root_.abs_of_nonneg, exact zero_le_two }\n      ... = \u2225(2:\u211d) \u2022 (u - half \u2022 (wq + wp))\u2225 * \u2225(2:\u211d) \u2022 (u - half \u2022 (wq + wp))\u2225 +\n            \u2225wp-wq\u2225 * \u2225wp-wq\u2225 :\n      by simp [norm_smul]\n      ... = \u2225a + b\u2225 * \u2225a + b\u2225 + \u2225a - b\u2225 * \u2225a - b\u2225 :\n      begin\n        rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : \u211d) \u2260 0),\n            \u2190 one_add_one_eq_two, add_smul],\n        simp only [one_smul],\n        have eq\u2081 : wp - wq = a - b, from (sub_sub_sub_cancel_left _ _ _).symm,\n        have eq\u2082 : u + u - (wq + wp) = a + b, show u + u - (wq + wp) = (u - wq) + (u - wp), abel,\n        rw [eq\u2081, eq\u2082],\n      end\n      ... = 2 * (\u2225a\u2225 * \u2225a\u2225 + \u2225b\u2225 * \u2225b\u2225) : parallelogram_law_with_norm,\n    have eq : \u03b4 \u2264 \u2225u - half \u2022 (wq + wp)\u2225,\n    { rw smul_add,\n      apply \u03b4_le', apply h\u2082,\n        repeat {exact subtype.mem _},\n        repeat {exact le_of_lt one_half_pos},\n        exact add_halves 1 },\n    have eq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2225u - half \u2022 (wq + wp)\u2225 * \u2225u - half \u2022 (wq + wp)\u2225,\n    {  mono, mono, norm_num, apply mul_nonneg, norm_num, exact norm_nonneg _ },\n    have eq\u2082 : \u2225a\u2225 * \u2225a\u2225 \u2264 (\u03b4 + div) * (\u03b4 + div) :=\n      mul_self_le_mul_self (norm_nonneg _)\n        (le_trans (le_of_lt $ hw q) (add_le_add_left (nat.one_div_le_one_div hq) _)),\n    have eq\u2082' : \u2225b\u2225 * \u2225b\u2225 \u2264 (\u03b4 + div) * (\u03b4 + div) :=\n      mul_self_le_mul_self (norm_nonneg _)\n        (le_trans (le_of_lt $ hw p) (add_le_add_left (nat.one_div_le_one_div hp) _)),\n    rw dist_eq_norm,\n    apply nonneg_le_nonneg_of_squares_le, { exact sqrt_nonneg _ },\n    rw mul_self_sqrt,\n    exact calc\n      \u2225wp - wq\u2225 * \u2225wp - wq\u2225 = 2 * (\u2225a\u2225*\u2225a\u2225 + \u2225b\u2225*\u2225b\u2225) -\n        4 * \u2225u - half \u2022 (wq+wp)\u2225 * \u2225u - half \u2022 (wq+wp)\u2225 : by { rw \u2190 this, simp }\n      ... \u2264 2 * (\u2225a\u2225 * \u2225a\u2225 + \u2225b\u2225 * \u2225b\u2225) - 4 * \u03b4 * \u03b4 : sub_le_sub_left eq\u2081 _\n      ... \u2264 2 * ((\u03b4 + div) * (\u03b4 + div) + (\u03b4 + div) * (\u03b4 + div)) - 4 * \u03b4 * \u03b4 :\n        sub_le_sub_right (mul_le_mul_of_nonneg_left (add_le_add eq\u2082 eq\u2082') (by norm_num)) _\n      ... = 8 * \u03b4 * div + 4 * div * div : by ring,\n    exact add_nonneg\n      (mul_nonneg (mul_nonneg (by norm_num) zero_le_\u03b4) (le_of_lt nat.one_div_pos_of_nat))\n      (mul_nonneg (mul_nonneg (by norm_num) nat.one_div_pos_of_nat.le) nat.one_div_pos_of_nat.le),\n    -- third goal : `tendsto (\u03bb (n : \u2115), sqrt (b n)) at_top (\ud835\udcdd 0)`\n    apply tendsto.comp,\n    { convert continuous_sqrt.continuous_at, exact sqrt_zero.symm },\n    have eq\u2081 : tendsto (\u03bb (n : \u2115), 8 * \u03b4 * (1 \/ (n + 1))) at_top (nhds (0:\u211d)),\n    { convert (@tendsto_const_nhds _ _ _ (8 * \u03b4) _).mul tendsto_one_div_add_at_top_nhds_0_nat,\n      simp only [mul_zero] },\n    have : tendsto (\u03bb (n : \u2115), (4:\u211d) * (1 \/ (n + 1))) at_top (nhds (0:\u211d)),\n    { convert (@tendsto_const_nhds _ _ _ (4:\u211d) _).mul tendsto_one_div_add_at_top_nhds_0_nat,\n      simp only [mul_zero] },\n    have eq\u2082 : tendsto (\u03bb (n : \u2115), (4:\u211d) * (1 \/ (n + 1)) * (1 \/ (n + 1))) at_top (nhds (0:\u211d)),\n    { convert this.mul tendsto_one_div_add_at_top_nhds_0_nat,\n      simp only [mul_zero] },\n    convert eq\u2081.add eq\u2082, simp only [add_zero] },\n  -- Step 3: By completeness of `K`, let `w : \u2115 \u2192 K` converge to some `v : K`.\n  -- Prove that it satisfies all requirements.\n  rcases cauchy_seq_tendsto_of_is_complete h\u2081 (\u03bb n, _) seq_is_cauchy with \u27e8v, hv, w_tendsto\u27e9,\n  use v, use hv,\n  have h_cont : continuous (\u03bb v, \u2225u - v\u2225) :=\n    continuous.comp continuous_norm (continuous.sub continuous_const continuous_id),\n  have : tendsto (\u03bb n, \u2225u - w n\u2225) at_top (nhds \u2225u - v\u2225),\n    convert (tendsto.comp h_cont.continuous_at w_tendsto),\n  exact tendsto_nhds_unique this norm_tendsto,\n  exact subtype.mem _\nend\n\n\/-- Characterization of minimizers for the projection on a convex set in a real inner product\nspace. -\/\ntheorem norm_eq_infi_iff_real_inner_le_zero {K : set F} (h : convex K) {u : F} {v : F}\n  (hv : v \u2208 K) : \u2225u - v\u2225 = (\u2a05 w : K, \u2225u - w\u2225) \u2194 \u2200 w \u2208 K, \u27eau - v, w - v\u27eb_\u211d \u2264 0 :=\niff.intro\nbegin\n  assume eq w hw,\n  let \u03b4 := \u2a05 w : K, \u2225u - w\u2225, let p := \u27eau - v, w - v\u27eb_\u211d, let q := \u2225w - v\u2225^2,\n  letI : nonempty K := \u27e8\u27e8v, hv\u27e9\u27e9,\n  have zero_le_\u03b4 : 0 \u2264 \u03b4,\n    apply le_cinfi, intro, exact norm_nonneg _,\n  have \u03b4_le : \u2200 w : K, \u03b4 \u2264 \u2225u - w\u2225,\n    assume w, apply cinfi_le, use (0:\u211d), rintros _ \u27e8_, rfl\u27e9, exact norm_nonneg _,\n  have \u03b4_le' : \u2200 w \u2208 K, \u03b4 \u2264 \u2225u - w\u2225 := assume w hw, \u03b4_le \u27e8w, hw\u27e9,\n  have : \u2200\u03b8:\u211d, 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q,\n    assume \u03b8 h\u03b8\u2081 h\u03b8\u2082,\n    have : \u2225u - v\u2225^2 \u2264 \u2225u - v\u2225^2 - 2 * \u03b8 * \u27eau - v, w - v\u27eb_\u211d + \u03b8*\u03b8*\u2225w - v\u2225^2 :=\n    calc\n      \u2225u - v\u2225^2 \u2264 \u2225u - (\u03b8\u2022w + (1-\u03b8)\u2022v)\u2225^2 :\n      begin\n        simp only [pow_two], apply mul_self_le_mul_self (norm_nonneg _),\n        rw [eq], apply \u03b4_le',\n        apply h hw hv,\n        exacts [le_of_lt h\u03b8\u2081, sub_nonneg.2 h\u03b8\u2082, add_sub_cancel'_right _ _],\n      end\n      ... = \u2225(u - v) - \u03b8 \u2022 (w - v)\u2225^2 :\n      begin\n        have : u - (\u03b8\u2022w + (1-\u03b8)\u2022v) = (u - v) - \u03b8 \u2022 (w - v),\n        { rw [smul_sub, sub_smul, one_smul],\n          simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev] },\n        rw this\n      end\n      ... = \u2225u - v\u2225^2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8*\u03b8*\u2225w - v\u2225^2 :\n      begin\n        rw [norm_sub_pow_two, inner_smul_right, norm_smul],\n        simp only [pow_two],\n        show \u2225u-v\u2225*\u2225u-v\u2225-2*(\u03b8*inner(u-v)(w-v))+absR (\u03b8)*\u2225w-v\u2225*(absR (\u03b8)*\u2225w-v\u2225)=\n                \u2225u-v\u2225*\u2225u-v\u2225-2*\u03b8*inner(u-v)(w-v)+\u03b8*\u03b8*(\u2225w-v\u2225*\u2225w-v\u2225),\n        rw abs_of_pos h\u03b8\u2081, ring\n      end,\n    have eq\u2081 : \u2225u-v\u2225^2-2*\u03b8*inner(u-v)(w-v)+\u03b8*\u03b8*\u2225w-v\u2225^2=\u2225u-v\u2225^2+(\u03b8*\u03b8*\u2225w-v\u2225^2-2*\u03b8*inner(u-v)(w-v)),\n      by abel,\n    rw [eq\u2081, le_add_iff_nonneg_right] at this,\n    have eq\u2082 : \u03b8*\u03b8*\u2225w-v\u2225^2-2*\u03b8*inner(u-v)(w-v)=\u03b8*(\u03b8*\u2225w-v\u2225^2-2*inner(u-v)(w-v)), ring,\n    rw eq\u2082 at this,\n    have := le_of_sub_nonneg (nonneg_of_mul_nonneg_left this h\u03b8\u2081),\n    exact this,\n  by_cases hq : q = 0,\n  { rw hq at this,\n    have : p \u2264 0,\n      have := this (1:\u211d) (by norm_num) (by norm_num),\n      linarith,\n    exact this },\n  { have q_pos : 0 < q,\n      apply lt_of_le_of_ne, exact pow_two_nonneg _, intro h, exact hq h.symm,\n    by_contradiction hp, rw not_le at hp,\n    let \u03b8 := min (1:\u211d) (p \/ q),\n    have eq\u2081 : \u03b8*q \u2264 p := calc\n      \u03b8*q \u2264 (p\/q) * q : mul_le_mul_of_nonneg_right (min_le_right _ _) (pow_two_nonneg _)\n      ... = p : div_mul_cancel _ hq,\n    have : 2 * p \u2264 p := calc\n      2 * p \u2264 \u03b8*q : by { refine this \u03b8 (lt_min (by norm_num) (div_pos hp q_pos)) (by norm_num) }\n      ... \u2264 p : eq\u2081,\n    linarith }\nend\nbegin\n  assume h,\n  letI : nonempty K := \u27e8\u27e8v, hv\u27e9\u27e9,\n  apply le_antisymm,\n  { apply le_cinfi, assume w,\n    apply nonneg_le_nonneg_of_squares_le (norm_nonneg _),\n    have := h w w.2,\n    exact calc\n      \u2225u - v\u2225 * \u2225u - v\u2225 \u2264 \u2225u - v\u2225 * \u2225u - v\u2225 - 2 * inner (u - v) ((w:F) - v) : by linarith\n      ... \u2264 \u2225u - v\u2225^2 - 2 * inner (u - v) ((w:F) - v) + \u2225(w:F) - v\u2225^2 :\n        by { rw pow_two, refine le_add_of_nonneg_right _, exact pow_two_nonneg _ }\n      ... = \u2225(u - v) - (w - v)\u2225^2 : norm_sub_pow_two.symm\n      ... = \u2225u - w\u2225 * \u2225u - w\u2225 :\n        by { have : (u - v) - (w - v) = u - w, abel, rw [this, pow_two] } },\n  { show (\u2a05 (w : K), \u2225u - w\u2225) \u2264 (\u03bbw:K, \u2225u - w\u2225) \u27e8v, hv\u27e9,\n      apply cinfi_le, use 0, rintros y \u27e8z, rfl\u27e9, exact norm_nonneg _ }\nend\n\nvariables (K : submodule \ud835\udd5c E)\n\n\/--\nExistence of projections on complete subspaces.\nLet `u` be a point in an inner product space, and let `K` be a nonempty complete subspace.\nThen there exists a (unique) `v` in `K` that minimizes the distance `\u2225u - v\u2225` to `u`.\nThis point `v` is usually called the orthogonal projection of `u` onto `K`.\n-\/\ntheorem exists_norm_eq_infi_of_complete_subspace\n  (h : is_complete (\u2191K : set E)) : \u2200 u : E, \u2203 v \u2208 K, \u2225u - v\u2225 = \u2a05 w : (K : set E), \u2225u - w\u2225 :=\nbegin\n  letI : inner_product_space \u211d E := inner_product_space.is_R_or_C_to_real \ud835\udd5c E,\n  letI : module \u211d E := restrict_scalars.semimodule \u211d \ud835\udd5c E,\n  letI : is_scalar_tower \u211d \ud835\udd5c E := restrict_scalars.is_scalar_tower _ _ _,\n  let K' : submodule \u211d E := submodule.restrict_scalars \u211d K,\n  exact exists_norm_eq_infi_of_complete_convex \u27e80, K'.zero_mem\u27e9 h K'.convex\nend\n\n\/--\nCharacterization of minimizers in the projection on a subspace, in the real case.\nLet `u` be a point in a real inner product space, and let `K` be a nonempty subspace.\nThen point `v` minimizes the distance `\u2225u - v\u2225` over points in `K` if and only if\nfor all `w \u2208 K`, `\u27eau - v, w\u27eb = 0` (i.e., `u - v` is orthogonal to the subspace `K`).\nThis is superceded by `norm_eq_infi_iff_inner_eq_zero` that gives the same conclusion over\nany `is_R_or_C` field.\n-\/\ntheorem norm_eq_infi_iff_real_inner_eq_zero (K : submodule \u211d F) {u : F} {v : F}\n  (hv : v \u2208 K) : \u2225u - v\u2225 = (\u2a05 w : (\u2191K : set F), \u2225u - w\u2225) \u2194 \u2200 w \u2208 K, \u27eau - v, w\u27eb_\u211d = 0 :=\niff.intro\nbegin\n  assume h,\n  have h : \u2200 w \u2208 K, \u27eau - v, w - v\u27eb_\u211d \u2264 0,\n  { rwa [norm_eq_infi_iff_real_inner_le_zero] at h, exacts [K.convex, hv] },\n  assume w hw,\n  have le : \u27eau - v, w\u27eb_\u211d \u2264 0,\n    let w' := w + v,\n    have : w' \u2208 K := submodule.add_mem _ hw hv,\n    have h\u2081 := h w' this,\n    have h\u2082 : w' - v = w, simp only [add_neg_cancel_right, sub_eq_add_neg],\n    rw h\u2082 at h\u2081, exact h\u2081,\n  have ge : \u27eau - v, w\u27eb_\u211d \u2265 0,\n    let w'' := -w + v,\n    have : w'' \u2208 K := submodule.add_mem _ (submodule.neg_mem _ hw) hv,\n    have h\u2081 := h w'' this,\n    have h\u2082 : w'' - v = -w, simp only [neg_inj, add_neg_cancel_right, sub_eq_add_neg],\n    rw [h\u2082, inner_neg_right] at h\u2081,\n    linarith,\n    exact le_antisymm le ge\nend\nbegin\n  assume h,\n  have : \u2200 w \u2208 K, \u27eau - v, w - v\u27eb_\u211d \u2264 0,\n    assume w hw,\n    let w' := w - v,\n    have : w' \u2208 K := submodule.sub_mem _ hw hv,\n    have h\u2081 := h w' this,\n    exact le_of_eq h\u2081,\n  rwa norm_eq_infi_iff_real_inner_le_zero,\n  exacts [submodule.convex _, hv]\nend\n\n\/--\nCharacterization of minimizers in the projection on a subspace.\nLet `u` be a point in an inner product space, and let `K` be a nonempty subspace.\nThen point `v` minimizes the distance `\u2225u - v\u2225` over points in `K` if and only if\nfor all `w \u2208 K`, `\u27eau - v, w\u27eb = 0` (i.e., `u - v` is orthogonal to the subspace `K`)\n-\/\ntheorem norm_eq_infi_iff_inner_eq_zero {u : E} {v : E}\n  (hv : v \u2208 K) : \u2225u - v\u2225 = (\u2a05 w : (\u2191K : set E), \u2225u - w\u2225) \u2194 \u2200 w \u2208 K, \u27eau - v, w\u27eb = 0 :=\nbegin\n  letI : inner_product_space \u211d E := inner_product_space.is_R_or_C_to_real \ud835\udd5c E,\n  letI : module \u211d E := restrict_scalars.semimodule \u211d \ud835\udd5c E,\n  letI : is_scalar_tower \u211d \ud835\udd5c E := restrict_scalars.is_scalar_tower _ _ _,\n  let K' : submodule \u211d E := K.restrict_scalars \u211d,\n  split,\n  { assume H,\n    have A : \u2200 w \u2208 K, re \u27eau - v, w\u27eb = 0 := (norm_eq_infi_iff_real_inner_eq_zero K' hv).1 H,\n    assume w hw,\n    apply ext,\n    { simp [A w hw] },\n    { symmetry, calc\n      im (0 : \ud835\udd5c) = 0 : im.map_zero\n      ... = re \u27eau - v, (-I) \u2022 w\u27eb : (A _ (K.smul_mem (-I) hw)).symm\n      ... = re ((-I) * \u27eau - v, w\u27eb) : by rw inner_smul_right\n      ... = im \u27eau - v, w\u27eb : by simp } },\n  { assume H,\n    have : \u2200 w \u2208 K', \u27eau - v, w\u27eb_\u211d = 0,\n    { assume w hw,\n      rw [real_inner_eq_re_inner, H w hw],\n      exact zero_re' },\n    exact (norm_eq_infi_iff_real_inner_eq_zero K' hv).2 this }\nend\n\nsection orthogonal_projection\nvariables [complete_space K]\n\n\/-- The orthogonal projection onto a complete subspace, as an\nunbundled function.  This definition is only intended for use in\nsetting up the bundled version `orthogonal_projection` and should not\nbe used once that is defined. -\/\ndef orthogonal_projection_fn (v : E) :=\n(exists_norm_eq_infi_of_complete_subspace K (complete_space_coe_iff_is_complete.mp \u2039_\u203a) v).some\n\nvariables {K}\n\n\/-- The unbundled orthogonal projection is in the given subspace.\nThis lemma is only intended for use in setting up the bundled version\nand should not be used once that is defined. -\/\nlemma orthogonal_projection_fn_mem (v : E) : orthogonal_projection_fn K v \u2208 K :=\n(exists_norm_eq_infi_of_complete_subspace K\n  (complete_space_coe_iff_is_complete.mp \u2039_\u203a) v).some_spec.some\n\n\/-- The characterization of the unbundled orthogonal projection.  This\nlemma is only intended for use in setting up the bundled version\nand should not be used once that is defined. -\/\nlemma orthogonal_projection_fn_inner_eq_zero (v : E) :\n  \u2200 w \u2208 K, \u27eav - orthogonal_projection_fn K v, w\u27eb = 0 :=\nbegin\n  rw \u2190norm_eq_infi_iff_inner_eq_zero K (orthogonal_projection_fn_mem v),\n  exact (exists_norm_eq_infi_of_complete_subspace K\n    (complete_space_coe_iff_is_complete.mp \u2039_\u203a) v).some_spec.some_spec\nend\n\n\/-- The unbundled orthogonal projection is the unique point in `K`\nwith the orthogonality property.  This lemma is only intended for use\nin setting up the bundled version and should not be used once that is\ndefined. -\/\nlemma eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero\n  {u v : E} (hvm : v \u2208 K) (hvo : \u2200 w \u2208 K, \u27eau - v, w\u27eb = 0) :\n  orthogonal_projection_fn K u = v :=\nbegin\n  rw [\u2190sub_eq_zero, \u2190inner_self_eq_zero],\n  have hvs : orthogonal_projection_fn K u - v \u2208 K :=\n    submodule.sub_mem K (orthogonal_projection_fn_mem u) hvm,\n  have huo : \u27eau - orthogonal_projection_fn K u, orthogonal_projection_fn K u - v\u27eb = 0 :=\n    orthogonal_projection_fn_inner_eq_zero u _ hvs,\n  have huv : \u27eau - v, orthogonal_projection_fn K u - v\u27eb = 0 := hvo _ hvs,\n  have houv : \u27ea(u - v) - (u - orthogonal_projection_fn K u), orthogonal_projection_fn K u - v\u27eb = 0,\n  { rw [inner_sub_left, huo, huv, sub_zero] },\n  rwa sub_sub_sub_cancel_left at houv\nend\n\nvariables (K)\n\nlemma orthogonal_projection_fn_norm_sq (v : E) :\n  \u2225v\u2225 * \u2225v\u2225 = \u2225v - (orthogonal_projection_fn K v)\u2225 * \u2225v - (orthogonal_projection_fn K v)\u2225\n            + \u2225orthogonal_projection_fn K v\u2225 * \u2225orthogonal_projection_fn K v\u2225 :=\nbegin\n  set p := orthogonal_projection_fn K v,\n  have h' : \u27eav - p, p\u27eb = 0,\n  { exact orthogonal_projection_fn_inner_eq_zero _ _ (orthogonal_projection_fn_mem v) },\n  convert norm_add_square_eq_norm_square_add_norm_square_of_inner_eq_zero (v - p) p h' using 2;\n  simp,\nend\n\n\/-- The orthogonal projection onto a complete subspace. -\/\ndef orthogonal_projection : E \u2192L[\ud835\udd5c] K :=\nlinear_map.mk_continuous\n  { to_fun := \u03bb v, \u27e8orthogonal_projection_fn K v, orthogonal_projection_fn_mem v\u27e9,\n    map_add' := \u03bb x y, begin\n      have hm : orthogonal_projection_fn K x + orthogonal_projection_fn K y \u2208 K :=\n        submodule.add_mem K (orthogonal_projection_fn_mem x) (orthogonal_projection_fn_mem y),\n      have ho :\n        \u2200 w \u2208 K, \u27eax + y - (orthogonal_projection_fn K x + orthogonal_projection_fn K y), w\u27eb = 0,\n      { intros w hw,\n        rw [add_sub_comm, inner_add_left, orthogonal_projection_fn_inner_eq_zero _ w hw,\n            orthogonal_projection_fn_inner_eq_zero _ w hw, add_zero] },\n      ext,\n      simp [eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hm ho]\n    end,\n    map_smul' := \u03bb c x, begin\n      have hm : c \u2022 orthogonal_projection_fn K x \u2208 K :=\n        submodule.smul_mem K _ (orthogonal_projection_fn_mem x),\n      have ho : \u2200 w \u2208 K, \u27eac \u2022 x - c \u2022 orthogonal_projection_fn K x, w\u27eb = 0,\n      { intros w hw,\n        rw [\u2190smul_sub, inner_smul_left, orthogonal_projection_fn_inner_eq_zero _ w hw, mul_zero] },\n      ext,\n      simp [eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hm ho]\n    end }\n  1\n  (\u03bb x, begin\n    simp only [one_mul, linear_map.coe_mk],\n    refine le_of_pow_le_pow 2 (norm_nonneg _) (by norm_num) _,\n    change \u2225orthogonal_projection_fn K x\u2225 ^ 2 \u2264 \u2225x\u2225 ^ 2,\n    nlinarith [orthogonal_projection_fn_norm_sq K x]\n  end)\n\nvariables {K}\n\n@[simp]\nlemma orthogonal_projection_fn_eq (v : E) :\n  orthogonal_projection_fn K v = (orthogonal_projection K v : E) :=\nrfl\n\n\/-- The characterization of the orthogonal projection.  -\/\n@[simp]\nlemma orthogonal_projection_inner_eq_zero (v : E) :\n  \u2200 w \u2208 K, \u27eav - orthogonal_projection K v, w\u27eb = 0 :=\northogonal_projection_fn_inner_eq_zero v\n\n\/-- The orthogonal projection is the unique point in `K` with the\northogonality property. -\/\nlemma eq_orthogonal_projection_of_mem_of_inner_eq_zero\n  {u v : E} (hvm : v \u2208 K) (hvo : \u2200 w \u2208 K, \u27eau - v, w\u27eb = 0) :\n  (orthogonal_projection K u : E) = v :=\neq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hvm hvo\n\n\/-- The orthogonal projections onto equal subspaces are coerced back to the same point in `E`. -\/\nlemma eq_orthogonal_projection_of_eq_submodule\n  {K' : submodule \ud835\udd5c E} [complete_space K'] (h : K = K') (u : E) :\n  (orthogonal_projection K u : E) = (orthogonal_projection K' u : E) :=\nbegin\n  change orthogonal_projection_fn K u = orthogonal_projection_fn K' u,\n  congr,\n  exact h\nend\n\n\/-- The orthogonal projection sends elements of `K` to themselves. -\/\n@[simp] lemma orthogonal_projection_mem_subspace_eq_self (v : K) : orthogonal_projection K v = v :=\nby { ext, apply eq_orthogonal_projection_of_mem_of_inner_eq_zero; simp }\n\nlocal attribute [instance] finite_dimensional_bot\n\n\/-- The orthogonal projection onto the trivial submodule is the zero map. -\/\n@[simp] lemma orthogonal_projection_bot : orthogonal_projection (\u22a5 : submodule \ud835\udd5c E) = 0 :=\nbegin\n  ext u,\n  apply eq_orthogonal_projection_of_mem_of_inner_eq_zero,\n  { simp },\n  { intros w hw,\n    simp [(submodule.mem_bot \ud835\udd5c).mp hw] }\nend\n\nvariables (K)\n\n\/-- The orthogonal projection has norm `\u2264 1`. -\/\nlemma orthogonal_projection_norm_le : \u2225orthogonal_projection K\u2225 \u2264 1 :=\nlinear_map.mk_continuous_norm_le _ (by norm_num) _\n\nvariables (\ud835\udd5c)\n\nlemma smul_orthogonal_projection_singleton {v : E} (w : E) :\n  (\u2225v\u2225 ^ 2 : \ud835\udd5c) \u2022 (orthogonal_projection (\ud835\udd5c \u2219 v) w : E) = \u27eav, w\u27eb \u2022 v :=\nbegin\n  suffices : \u2191(orthogonal_projection (\ud835\udd5c \u2219 v) ((\u2225v\u2225 ^ 2 : \ud835\udd5c) \u2022 w)) = \u27eav, w\u27eb \u2022 v,\n  { simpa using this },\n  apply eq_orthogonal_projection_of_mem_of_inner_eq_zero,\n  { rw submodule.mem_span_singleton,\n    use \u27eav, w\u27eb },\n  { intros x hx,\n    obtain \u27e8c, rfl\u27e9 := submodule.mem_span_singleton.mp hx,\n    have hv : \u2191\u2225v\u2225 ^ 2 = \u27eav, v\u27eb := by { norm_cast, simp [norm_sq_eq_inner] },\n    simp [inner_sub_left, inner_smul_left, inner_smul_right, is_R_or_C.conj_div, mul_comm, hv,\n      inner_product_space.conj_sym, hv] }\nend\n\n\/-- Formula for orthogonal projection onto a single vector. -\/\nlemma orthogonal_projection_singleton {v : E} (w : E) :\n  (orthogonal_projection (\ud835\udd5c \u2219 v) w : E) = (\u27eav, w\u27eb \/ \u2225v\u2225 ^ 2) \u2022 v :=\nbegin\n  by_cases hv : v = 0,\n  { rw [hv, eq_orthogonal_projection_of_eq_submodule submodule.span_zero_singleton],\n    { simp },\n    { apply_instance } },\n  have hv' : \u2225v\u2225 \u2260 0 := ne_of_gt (norm_pos_iff.mpr hv),\n  have key : ((\u2225v\u2225 ^ 2 : \ud835\udd5c)\u207b\u00b9 * \u2225v\u2225 ^ 2) \u2022 \u2191(orthogonal_projection (\ud835\udd5c \u2219 v) w)\n              = ((\u2225v\u2225 ^ 2 : \ud835\udd5c)\u207b\u00b9 * \u27eav, w\u27eb) \u2022 v,\n  { simp [mul_smul, smul_orthogonal_projection_singleton \ud835\udd5c w] },\n  convert key;\n  field_simp [hv']\nend\n\n\/-- Formula for orthogonal projection onto a single unit vector. -\/\nlemma orthogonal_projection_unit_singleton {v : E} (hv : \u2225v\u2225 = 1) (w : E) :\n  (orthogonal_projection (\ud835\udd5c \u2219 v) w : E) = \u27eav, w\u27eb \u2022 v :=\nby { rw \u2190 smul_orthogonal_projection_singleton \ud835\udd5c w, simp [hv] }\n\nend orthogonal_projection\n\n\/-- The subspace of vectors orthogonal to a given subspace. -\/\ndef submodule.orthogonal : submodule \ud835\udd5c E :=\n{ carrier := {v | \u2200 u \u2208 K, \u27eau, v\u27eb = 0},\n  zero_mem' := \u03bb _ _, inner_zero_right,\n  add_mem' := \u03bb x y hx hy u hu, by rw [inner_add_right, hx u hu, hy u hu, add_zero],\n  smul_mem' := \u03bb c x hx u hu, by rw [inner_smul_right, hx u hu, mul_zero] }\n\nnotation K`\u15ee`:1200 := submodule.orthogonal K\n\n\/-- When a vector is in `K\u15ee`. -\/\nlemma submodule.mem_orthogonal (v : E) : v \u2208 K\u15ee \u2194 \u2200 u \u2208 K, \u27eau, v\u27eb = 0 := iff.rfl\n\n\/-- When a vector is in `K\u15ee`, with the inner product the\nother way round. -\/\nlemma submodule.mem_orthogonal' (v : E) : v \u2208 K\u15ee \u2194 \u2200 u \u2208 K, \u27eav, u\u27eb = 0 :=\nby simp_rw [submodule.mem_orthogonal, inner_eq_zero_sym]\n\nvariables {K}\n\n\/-- A vector in `K` is orthogonal to one in `K\u15ee`. -\/\nlemma submodule.inner_right_of_mem_orthogonal {u v : E} (hu : u \u2208 K) (hv : v \u2208 K\u15ee) : \u27eau, v\u27eb = 0 :=\n(K.mem_orthogonal v).1 hv u hu\n\n\/-- A vector in `K\u15ee` is orthogonal to one in `K`. -\/\nlemma submodule.inner_left_of_mem_orthogonal {u v : E} (hu : u \u2208 K) (hv : v \u2208 K\u15ee) : \u27eav, u\u27eb = 0 :=\nby rw [inner_eq_zero_sym]; exact submodule.inner_right_of_mem_orthogonal hu hv\n\n\/-- A vector in `(\ud835\udd5c \u2219 u)\u15ee` is orthogonal to `u`. -\/\nlemma inner_right_of_mem_orthogonal_singleton (u : E) {v : E} (hv : v \u2208 (\ud835\udd5c \u2219 u)\u15ee) : \u27eau, v\u27eb = 0 :=\nsubmodule.inner_right_of_mem_orthogonal (submodule.mem_span_singleton_self u) hv\n\n\/-- A vector in `(\ud835\udd5c \u2219 u)\u15ee` is orthogonal to `u`. -\/\nlemma inner_left_of_mem_orthogonal_singleton (u : E) {v : E} (hv : v \u2208 (\ud835\udd5c \u2219 u)\u15ee) : \u27eav, u\u27eb = 0 :=\nsubmodule.inner_left_of_mem_orthogonal (submodule.mem_span_singleton_self u) hv\n\nvariables (K)\n\n\/-- `K` and `K\u15ee` have trivial intersection. -\/\nlemma submodule.orthogonal_disjoint : disjoint K K\u15ee :=\nbegin\n  simp_rw [submodule.disjoint_def, submodule.mem_orthogonal],\n  exact \u03bb x hx ho, inner_self_eq_zero.1 (ho x hx)\nend\n\n\/-- `K\u15ee` can be characterized as the intersection of the kernels of the operations of\ninner product with each of the elements of `K`. -\/\nlemma orthogonal_eq_inter : K\u15ee = \u2a05 v : K, (inner_right (v:E)).ker :=\nbegin\n  apply le_antisymm,\n  { rw le_infi_iff,\n    rintros \u27e8v, hv\u27e9 w hw,\n    simpa using hw _ hv },\n  { intros v hv w hw,\n    simp only [submodule.mem_infi] at hv,\n    exact hv \u27e8w, hw\u27e9 }\nend\n\n\/-- The orthogonal complement of any submodule `K` is closed. -\/\nlemma submodule.is_closed_orthogonal : is_closed (K\u15ee : set E) :=\nbegin\n  rw orthogonal_eq_inter K,\n  convert is_closed_Inter (\u03bb v : K, (inner_right (v:E)).is_closed_ker),\n  simp\nend\n\n\/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -\/\ninstance [complete_space E] : complete_space K\u15ee := K.is_closed_orthogonal.complete_space_coe\n\nvariables (\ud835\udd5c E)\n\n\/-- `submodule.orthogonal` gives a `galois_connection` between\n`submodule \ud835\udd5c E` and its `order_dual`. -\/\nlemma submodule.orthogonal_gc :\n  @galois_connection (submodule \ud835\udd5c E) (order_dual $ submodule \ud835\udd5c E) _ _\n    submodule.orthogonal submodule.orthogonal :=\n\u03bb K\u2081 K\u2082, \u27e8\u03bb h v hv u hu, submodule.inner_left_of_mem_orthogonal hv (h hu),\n          \u03bb h v hv u hu, submodule.inner_left_of_mem_orthogonal hv (h hu)\u27e9\n\nvariables {\ud835\udd5c E}\n\n\/-- `submodule.orthogonal` reverses the `\u2264` ordering of two\nsubspaces. -\/\nlemma submodule.orthogonal_le {K\u2081 K\u2082 : submodule \ud835\udd5c E} (h : K\u2081 \u2264 K\u2082) : K\u2082\u15ee \u2264 K\u2081\u15ee :=\n(submodule.orthogonal_gc \ud835\udd5c E).monotone_l h\n\n\/-- `K` is contained in `K\u15ee\u15ee`. -\/\nlemma submodule.le_orthogonal_orthogonal : K \u2264 K\u15ee\u15ee := (submodule.orthogonal_gc \ud835\udd5c E).le_u_l _\n\n\/-- The inf of two orthogonal subspaces equals the subspace orthogonal\nto the sup. -\/\nlemma submodule.inf_orthogonal (K\u2081 K\u2082 : submodule \ud835\udd5c E) : K\u2081\u15ee \u2293 K\u2082\u15ee = (K\u2081 \u2294 K\u2082)\u15ee :=\n(submodule.orthogonal_gc \ud835\udd5c E).l_sup.symm\n\n\/-- The inf of an indexed family of orthogonal subspaces equals the\nsubspace orthogonal to the sup. -\/\nlemma submodule.infi_orthogonal {\u03b9 : Type*} (K : \u03b9 \u2192 submodule \ud835\udd5c E) : (\u2a05 i, (K i)\u15ee) = (supr K)\u15ee :=\n(submodule.orthogonal_gc \ud835\udd5c E).l_supr.symm\n\n\/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -\/\nlemma submodule.Inf_orthogonal (s : set $ submodule \ud835\udd5c E) : (\u2a05 K \u2208 s, K\u15ee) = (Sup s)\u15ee :=\n(submodule.orthogonal_gc \ud835\udd5c E).l_Sup.symm\n\n\/-- If `K\u2081` is complete and contained in `K\u2082`, `K\u2081` and `K\u2081\u15ee \u2293 K\u2082` span `K\u2082`. -\/\nlemma submodule.sup_orthogonal_inf_of_is_complete {K\u2081 K\u2082 : submodule \ud835\udd5c E} (h : K\u2081 \u2264 K\u2082)\n  (hc : is_complete (K\u2081 : set E)) : K\u2081 \u2294 (K\u2081\u15ee \u2293 K\u2082) = K\u2082 :=\nbegin\n  ext x,\n  rw submodule.mem_sup,\n  rcases exists_norm_eq_infi_of_complete_subspace K\u2081 hc x with \u27e8v, hv, hvm\u27e9,\n  rw norm_eq_infi_iff_inner_eq_zero K\u2081 hv at hvm,\n  split,\n  { rintro \u27e8y, hy, z, hz, rfl\u27e9,\n    exact K\u2082.add_mem (h hy) hz.2 },\n  { exact \u03bb hx, \u27e8v, hv, x - v, \u27e8(K\u2081.mem_orthogonal' _).2 hvm, K\u2082.sub_mem hx (h hv)\u27e9,\n                 add_sub_cancel'_right _ _\u27e9 }\nend\n\nvariables {K}\n\n\/-- If `K` is complete, `K` and `K\u15ee` span the whole space. -\/\nlemma submodule.sup_orthogonal_of_is_complete (h : is_complete (K : set E)) : K \u2294 K\u15ee = \u22a4 :=\nbegin\n  convert submodule.sup_orthogonal_inf_of_is_complete (le_top : K \u2264 \u22a4) h,\n  simp\nend\n\n\/-- If `K` is complete, `K` and `K\u15ee` span the whole space. Version using `complete_space`. -\/\nlemma submodule.sup_orthogonal_of_complete_space [complete_space K] : K \u2294 K\u15ee = \u22a4 :=\nsubmodule.sup_orthogonal_of_is_complete (complete_space_coe_iff_is_complete.mp \u2039_\u203a)\n\nvariables (K)\n\n\/-- If `K` is complete, any `v` in `E` can be expressed as a sum of elements of `K` and `K\u15ee`. -\/\nlemma submodule.exists_sum_mem_mem_orthogonal [complete_space K] (v : E) :\n  \u2203 (y \u2208 K) (z \u2208 K\u15ee), v = y + z :=\nbegin\n  have h_mem : v \u2208 K \u2294 K\u15ee := by simp [submodule.sup_orthogonal_of_complete_space],\n  obtain \u27e8y, hy, z, hz, hyz\u27e9 := submodule.mem_sup.mp h_mem,\n  exact \u27e8y, hy, z, hz, hyz.symm\u27e9\nend\n\n\/-- If `K` is complete, then the orthogonal complement of its orthogonal complement is itself. -\/\n@[simp] lemma submodule.orthogonal_orthogonal [complete_space K] : K\u15ee\u15ee = K :=\nbegin\n  ext v,\n  split,\n  { obtain \u27e8y, hy, z, hz, rfl\u27e9 := K.exists_sum_mem_mem_orthogonal v,\n    intros hv,\n    have hz' : z = 0,\n    { have hyz : \u27eaz, y\u27eb = 0 := by simp [hz y hy, inner_eq_zero_sym],\n      simpa [inner_add_right, hyz] using hv z hz },\n    simp [hy, hz'] },\n  { intros hv w hw,\n    rw inner_eq_zero_sym,\n    exact hw v hv }\nend\n\nvariables {K}\n\n\/-- If `K` is complete, `K` and `K\u15ee` are complements of each other. -\/\nlemma submodule.is_compl_orthogonal_of_is_complete (h : is_complete (K : set E)) : is_compl K K\u15ee :=\n\u27e8K.orthogonal_disjoint, le_of_eq (submodule.sup_orthogonal_of_is_complete h).symm\u27e9\n\n@[simp] lemma submodule.top_orthogonal_eq_bot : (\u22a4 : submodule \ud835\udd5c E)\u15ee = \u22a5 :=\nbegin\n  ext,\n  rw [submodule.mem_bot, submodule.mem_orthogonal],\n  exact \u27e8\u03bb h, inner_self_eq_zero.mp (h x submodule.mem_top), by { rintro rfl, simp }\u27e9\nend\n\n@[simp] lemma submodule.bot_orthogonal_eq_top : (\u22a5 : submodule \ud835\udd5c E)\u15ee = \u22a4 :=\nbegin\n  rw [\u2190 submodule.top_orthogonal_eq_bot, eq_top_iff],\n  exact submodule.le_orthogonal_orthogonal \u22a4\nend\n\n@[simp] lemma submodule.orthogonal_eq_bot_iff (hK : is_complete (K : set E)) :\n  K\u15ee = \u22a5 \u2194 K = \u22a4 :=\nbegin\n  refine \u27e8_, by { rintro rfl, exact submodule.top_orthogonal_eq_bot }\u27e9,\n  intro h,\n  have : K \u2294 K\u15ee = \u22a4 := submodule.sup_orthogonal_of_is_complete hK,\n  rwa [h, sup_comm, bot_sup_eq] at this,\nend\n\n@[simp] lemma submodule.orthogonal_eq_top_iff : K\u15ee = \u22a4 \u2194 K = \u22a5 :=\nbegin\n  refine \u27e8_, by { rintro rfl, exact submodule.bot_orthogonal_eq_top }\u27e9,\n  intro h,\n  have : K \u2293 K\u15ee = \u22a5 := K.orthogonal_disjoint.eq_bot,\n  rwa [h, inf_comm, top_inf_eq] at this\nend\n\n\/-- A point in `K` with the orthogonality property (here characterized in terms of `K\u15ee`) must be the\northogonal projection. -\/\nlemma eq_orthogonal_projection_of_mem_orthogonal\n  [complete_space K] {u v : E} (hv : v \u2208 K) (hvo : u - v \u2208 K\u15ee) :\n  (orthogonal_projection K u : E) = v :=\neq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hv (\u03bb w, inner_eq_zero_sym.mp \u2218 (hvo w))\n\n\/-- A point in `K` with the orthogonality property (here characterized in terms of `K\u15ee`) must be the\northogonal projection. -\/\nlemma eq_orthogonal_projection_of_mem_orthogonal'\n  [complete_space K] {u v z : E} (hv : v \u2208 K) (hz : z \u2208 K\u15ee) (hu : u = v + z) :\n  (orthogonal_projection K u : E) = v :=\neq_orthogonal_projection_of_mem_orthogonal hv (by simpa [hu])\n\n\/-- The orthogonal projection onto `K` of an element of `K\u15ee` is zero. -\/\nlemma orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero\n  [complete_space K] {v : E} (hv : v \u2208 K\u15ee) :\n  orthogonal_projection K v = 0 :=\nby { ext, convert eq_orthogonal_projection_of_mem_orthogonal _ _; simp [hv] }\n\n\/-- The orthogonal projection onto `K\u15ee` of an element of `K` is zero. -\/\nlemma orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero\n  [complete_space E] {v : E} (hv : v \u2208 K) :\n  orthogonal_projection K\u15ee v = 0 :=\northogonal_projection_mem_subspace_orthogonal_complement_eq_zero (K.le_orthogonal_orthogonal hv)\n\n\/-- The orthogonal projection onto `(\ud835\udd5c \u2219 v)\u15ee` of `v` is zero. -\/\nlemma orthogonal_projection_orthogonal_complement_singleton_eq_zero [complete_space E] (v : E) :\n  orthogonal_projection (\ud835\udd5c \u2219 v)\u15ee v = 0 :=\northogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero\n  (submodule.mem_span_singleton_self v)\n\nvariables (K)\n\n\/-- In a complete space `E`, a vector splits as the sum of its orthogonal projections onto a\ncomplete submodule `K` and onto the orthogonal complement of `K`.-\/\nlemma eq_sum_orthogonal_projection_self_orthogonal_complement\n  [complete_space E] [complete_space K] (w : E) :\n  w = (orthogonal_projection K w : E) + (orthogonal_projection K\u15ee w : E) :=\nbegin\n  obtain \u27e8y, hy, z, hz, hwyz\u27e9 := K.exists_sum_mem_mem_orthogonal w,\n  convert hwyz,\n  { exact eq_orthogonal_projection_of_mem_orthogonal' hy hz hwyz },\n  { rw add_comm at hwyz,\n    refine eq_orthogonal_projection_of_mem_orthogonal' hz _ hwyz,\n    simp [hy] }\nend\n\n\/-- In a complete space `E`, the projection maps onto a complete subspace `K` and its orthogonal\ncomplement sum to the identity. -\/\nlemma id_eq_sum_orthogonal_projection_self_orthogonal_complement\n  [complete_space E] [complete_space K] :\n  continuous_linear_map.id \ud835\udd5c E\n  = K.subtype_continuous.comp (orthogonal_projection K)\n  + K\u15ee.subtype_continuous.comp (orthogonal_projection K\u15ee) :=\nby { ext w, exact eq_sum_orthogonal_projection_self_orthogonal_complement K w }\n\nopen finite_dimensional\n\n\/-- Given a finite-dimensional subspace `K\u2082`, and a subspace `K\u2081`\ncontainined in it, the dimensions of `K\u2081` and the intersection of its\northogonal subspace with `K\u2082` add to that of `K\u2082`. -\/\nlemma submodule.findim_add_inf_findim_orthogonal {K\u2081 K\u2082 : submodule \ud835\udd5c E}\n  [finite_dimensional \ud835\udd5c K\u2082] (h : K\u2081 \u2264 K\u2082) :\n  findim \ud835\udd5c K\u2081 + findim \ud835\udd5c (K\u2081\u15ee \u2293 K\u2082 : submodule \ud835\udd5c E) = findim \ud835\udd5c K\u2082 :=\nbegin\n  haveI := submodule.finite_dimensional_of_le h,\n  have hd := submodule.dim_sup_add_dim_inf_eq K\u2081 (K\u2081\u15ee \u2293 K\u2082),\n  rw [\u2190inf_assoc, (submodule.orthogonal_disjoint K\u2081).eq_bot, bot_inf_eq, findim_bot,\n      submodule.sup_orthogonal_inf_of_is_complete h\n        (submodule.complete_of_finite_dimensional _)] at hd,\n  rw add_zero at hd,\n  exact hd.symm\nend\n\n\/-- Given a finite-dimensional subspace `K\u2082`, and a subspace `K\u2081`\ncontainined in it, the dimensions of `K\u2081` and the intersection of its\northogonal subspace with `K\u2082` add to that of `K\u2082`. -\/\nlemma submodule.findim_add_inf_findim_orthogonal' {K\u2081 K\u2082 : submodule \ud835\udd5c E}\n  [finite_dimensional \ud835\udd5c K\u2082] (h : K\u2081 \u2264 K\u2082) {n : \u2115} (h_dim : findim \ud835\udd5c K\u2081 + n = findim \ud835\udd5c K\u2082) :\n  findim \ud835\udd5c (K\u2081\u15ee \u2293 K\u2082 : submodule \ud835\udd5c E) = n :=\nby { rw \u2190 add_right_inj (findim \ud835\udd5c K\u2081), simp [submodule.findim_add_inf_findim_orthogonal h, h_dim] }\n\n\/-- Given a finite-dimensional space `E` and subspace `K`, the dimensions of `K` and `K\u15ee` add to\nthat of `E`. -\/\nlemma submodule.findim_add_findim_orthogonal [finite_dimensional \ud835\udd5c E] {K : submodule \ud835\udd5c E} :\n  findim \ud835\udd5c K + findim \ud835\udd5c K\u15ee = findim \ud835\udd5c E :=\nbegin\n  convert submodule.findim_add_inf_findim_orthogonal (le_top : K \u2264 \u22a4) using 1,\n  { rw inf_top_eq },\n  { simp }\nend\n\n\/-- Given a finite-dimensional space `E` and subspace `K`, the dimensions of `K` and `K\u15ee` add to\nthat of `E`. -\/\nlemma submodule.findim_add_findim_orthogonal' [finite_dimensional \ud835\udd5c E] {K : submodule \ud835\udd5c E} {n : \u2115}\n  (h_dim : findim \ud835\udd5c K + n = findim \ud835\udd5c E) :\n  findim \ud835\udd5c K\u15ee = n :=\nby { rw \u2190 add_right_inj (findim \ud835\udd5c K), simp [submodule.findim_add_findim_orthogonal, h_dim] }\n\n\/-- In a finite-dimensional inner product space, the dimension of the orthogonal complement of the\nspan of a nonzero vector is one less than the dimension of the space. -\/\nlemma findim_orthogonal_span_singleton [finite_dimensional \ud835\udd5c E] {v : E} (hv : v \u2260 0) :\n  findim \ud835\udd5c (\ud835\udd5c \u2219 v)\u15ee = findim \ud835\udd5c E - 1 :=\nbegin\n  haveI : nontrivial E := \u27e8\u27e8v, 0, hv\u27e9\u27e9,\n  apply submodule.findim_add_findim_orthogonal',\n  simp only [findim_span_singleton hv, findim_euclidean_space, fintype.card_fin],\n  exact nat.add_sub_cancel' (nat.succ_le_iff.mpr findim_pos)\nend\nend orthogonal\n","avg_line_length":42.0575022462,"max_line_length":100,"alphanum_fraction":0.6450544755}
-{"size":45103,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-\/\nimport data.finset.fold\nimport data.finset.option\nimport data.multiset.lattice\nimport order.order_dual\nimport order.complete_lattice\n\n\/-!\n# Lattice operations on finsets\n-\/\n\nvariables {\u03b1 \u03b2 \u03b3 : Type*}\n\nnamespace finset\nopen multiset order_dual\n\n\/-! ### sup -\/\nsection sup\nvariables [semilattice_sup_bot \u03b1]\n\n\/-- Supremum of a finite set: `sup {a, b, c} f = f a \u2294 f b \u2294 f c` -\/\ndef sup (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) : \u03b1 := s.fold (\u2294) \u22a5 f\n\nvariables {s s\u2081 s\u2082 : finset \u03b2} {f g : \u03b2 \u2192 \u03b1}\n\nlemma sup_def : s.sup f = (s.1.map f).sup := rfl\n\n@[simp] lemma sup_empty : (\u2205 : finset \u03b2).sup f = \u22a5 :=\nfold_empty\n\n@[simp] lemma sup_cons {b : \u03b2} (h : b \u2209 s) : (cons b s h).sup f = f b \u2294 s.sup f :=\nfold_cons h\n\n@[simp] lemma sup_insert [decidable_eq \u03b2] {b : \u03b2} : (insert b s : finset \u03b2).sup f = f b \u2294 s.sup f :=\nfold_insert_idem\n\nlemma sup_image [decidable_eq \u03b2] (s : finset \u03b3) (f : \u03b3 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  (s.image f).sup g = s.sup (g \u2218 f) :=\nfold_image_idem\n\n@[simp] lemma sup_map (s : finset \u03b3) (f : \u03b3 \u21aa \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  (s.map f).sup g = s.sup (g \u2218 f) :=\nfold_map\n\n@[simp] lemma sup_singleton {b : \u03b2} : ({b} : finset \u03b2).sup f = f b :=\nsup_singleton\n\nlemma sup_union [decidable_eq \u03b2] : (s\u2081 \u222a s\u2082).sup f = s\u2081.sup f \u2294 s\u2082.sup f :=\nfinset.induction_on s\u2081 (by rw [empty_union, sup_empty, bot_sup_eq]) $ \u03bb a s has ih,\nby rw [insert_union, sup_insert, sup_insert, ih, sup_assoc]\n\nlemma sup_sup : s.sup (f \u2294 g) = s.sup f \u2294 s.sup g :=\nbegin\n  refine finset.cons_induction_on s _ (\u03bb b t _ h, _),\n  { rw [sup_empty, sup_empty, sup_empty, bot_sup_eq] },\n  { rw [sup_cons, sup_cons, sup_cons, h],\n    exact sup_sup_sup_comm _ _ _ _ }\nend\n\ntheorem sup_congr {f g : \u03b2 \u2192 \u03b1} (hs : s\u2081 = s\u2082) (hfg : \u2200a\u2208s\u2082, f a = g a) : s\u2081.sup f = s\u2082.sup g :=\nby subst hs; exact finset.fold_congr hfg\n\n@[simp] lemma sup_le_iff {a : \u03b1} : s.sup f \u2264 a \u2194 (\u2200b \u2208 s, f b \u2264 a) :=\nbegin\n  apply iff.trans multiset.sup_le,\n  simp only [multiset.mem_map, and_imp, exists_imp_distrib],\n  exact \u27e8\u03bb k b hb, k _ _ hb rfl, \u03bb k a' b hb h, h \u25b8 k _ hb\u27e9,\nend\n\n@[simp] lemma sup_bUnion [decidable_eq \u03b2] (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b2) :\n  (s.bUnion t).sup f = s.sup (\u03bb x, (t x).sup f) :=\neq_of_forall_ge_iff $ \u03bb c, by simp [@forall_swap _ \u03b2]\n\nlemma sup_const {s : finset \u03b2} (h : s.nonempty) (c : \u03b1) : s.sup (\u03bb _, c) = c :=\neq_of_forall_ge_iff $ \u03bb b, sup_le_iff.trans h.forall_const\n\nlemma sup_le {a : \u03b1} : (\u2200b \u2208 s, f b \u2264 a) \u2192 s.sup f \u2264 a :=\nsup_le_iff.2\n\nlemma le_sup {b : \u03b2} (hb : b \u2208 s) : f b \u2264 s.sup f :=\nsup_le_iff.1 (le_refl _) _ hb\n\nlemma sup_mono_fun {g : \u03b2 \u2192 \u03b1} (h : \u2200b\u2208s, f b \u2264 g b) : s.sup f \u2264 s.sup g :=\nsup_le (\u03bb b hb, le_trans (h b hb) (le_sup hb))\n\nlemma sup_mono (h : s\u2081 \u2286 s\u2082) : s\u2081.sup f \u2264 s\u2082.sup f :=\nsup_le $ assume b hb, le_sup (h hb)\n\n@[simp] lemma sup_lt_iff [is_total \u03b1 (\u2264)] {a : \u03b1} (ha : \u22a5 < a) : s.sup f < a \u2194 (\u2200 b \u2208 s, f b < a) :=\n\u27e8(\u03bb hs b hb, lt_of_le_of_lt (le_sup hb) hs), finset.cons_induction_on s (\u03bb _, ha)\n  (\u03bb c t hc, by simpa only [sup_cons, sup_lt_iff, mem_cons, forall_eq_or_imp] using and.imp_right)\u27e9\n\n@[simp] lemma le_sup_iff [is_total \u03b1 (\u2264)] {a : \u03b1} (ha : \u22a5 < a) : a \u2264 s.sup f \u2194 \u2203 b \u2208 s, a \u2264 f b :=\n\u27e8finset.cons_induction_on s (\u03bb h, absurd h (not_le_of_lt ha))\n  (\u03bb c t hc ih, by simpa using @or.rec _ _ (\u2203 b, (b = c \u2228 b \u2208 t) \u2227 a \u2264 f b)\n    (\u03bb h, \u27e8c, or.inl rfl, h\u27e9) (\u03bb h, let \u27e8b, hb, hle\u27e9 := ih h in \u27e8b, or.inr hb, hle\u27e9)),\n(\u03bb \u27e8b, hb, hle\u27e9, trans hle (le_sup hb))\u27e9\n\n@[simp] lemma lt_sup_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : a < s.sup f \u2194 \u2203 b \u2208 s, a < f b :=\n\u27e8finset.cons_induction_on s (\u03bb h, absurd h not_lt_bot)\n  (\u03bb c t hc ih, by simpa using @or.rec _ _ (\u2203 b, (b = c \u2228 b \u2208 t) \u2227 a < f b)\n    (\u03bb h, \u27e8c, or.inl rfl, h\u27e9) (\u03bb h, let \u27e8b, hb, hlt\u27e9 := ih h in \u27e8b, or.inr hb, hlt\u27e9)),\n(\u03bb \u27e8b, hb, hlt\u27e9, lt_of_lt_of_le hlt (le_sup hb))\u27e9\n\n@[simp] lemma sup_attach (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) : s.attach.sup (\u03bb x, f x) = s.sup f :=\n(s.attach.sup_map (function.embedding.subtype _) f).symm.trans $ congr_arg _ attach_map_val\n\n@[simp] lemma sup_erase_bot [decidable_eq \u03b1] (s : finset \u03b1) : (s.erase \u22a5).sup id = s.sup id :=\nbegin\n  refine (sup_mono (s.erase_subset _)).antisymm (finset.sup_le_iff.2 $ \u03bb a ha, _),\n  obtain rfl | ha' := eq_or_ne a \u22a5,\n  { exact bot_le },\n  { exact le_sup (mem_erase.2 \u27e8ha', ha\u27e9) }\nend\n\nlemma sup_sdiff_right {\u03b1 \u03b2 : Type*} [generalized_boolean_algebra \u03b1] (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1)\n  (a : \u03b1) :\n  s.sup (\u03bb b, f b \\ a) = s.sup f \\ a :=\nbegin\n  refine finset.cons_induction_on s _ (\u03bb b t _ h, _),\n  { rw [sup_empty, sup_empty, bot_sdiff] },\n  { rw [sup_cons, sup_cons, h, sup_sdiff] }\nend\n\nlemma comp_sup_eq_sup_comp [semilattice_sup_bot \u03b3] {s : finset \u03b2}\n  {f : \u03b2 \u2192 \u03b1} (g : \u03b1 \u2192 \u03b3) (g_sup : \u2200 x y, g (x \u2294 y) = g x \u2294 g y) (bot : g \u22a5 = \u22a5) :\n  g (s.sup f) = s.sup (g \u2218 f) :=\nfinset.cons_induction_on s bot (\u03bb c t hc ih, by rw [sup_cons, sup_cons, g_sup, ih])\n\nlemma comp_sup_eq_sup_comp_of_is_total [is_total \u03b1 (\u2264)] {\u03b3 : Type} [semilattice_sup_bot \u03b3]\n  (g : \u03b1 \u2192 \u03b3) (mono_g : monotone g) (bot : g \u22a5 = \u22a5) : g (s.sup f) = s.sup (g \u2218 f) :=\ncomp_sup_eq_sup_comp g mono_g.map_sup bot\n\n\/-- Computing `sup` in a subtype (closed under `sup`) is the same as computing it in `\u03b1`. -\/\nlemma sup_coe {P : \u03b1 \u2192 Prop}\n  {Pbot : P \u22a5} {Psup : \u2200{{x y}}, P x \u2192 P y \u2192 P (x \u2294 y)}\n  (t : finset \u03b2) (f : \u03b2 \u2192 {x : \u03b1 \/\/ P x}) :\n  (@sup _ _ (subtype.semilattice_sup_bot Pbot Psup) t f : \u03b1) = t.sup (\u03bb x, f x) :=\nby { rw [comp_sup_eq_sup_comp coe]; intros; refl }\n\n@[simp] lemma sup_to_finset {\u03b1 \u03b2} [decidable_eq \u03b2]\n  (s : finset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) :\n  (s.sup f).to_finset = s.sup (\u03bb x, (f x).to_finset) :=\ncomp_sup_eq_sup_comp multiset.to_finset to_finset_union rfl\n\ntheorem subset_range_sup_succ (s : finset \u2115) : s \u2286 range (s.sup id).succ :=\n\u03bb n hn, mem_range.2 $ nat.lt_succ_of_le $ le_sup hn\n\ntheorem exists_nat_subset_range (s : finset \u2115) : \u2203n : \u2115, s \u2286 range n :=\n\u27e8_, s.subset_range_sup_succ\u27e9\n\nlemma sup_induction {p : \u03b1 \u2192 Prop} (hb : p \u22a5) (hp : \u2200 (a\u2081 a\u2082 : \u03b1), p a\u2081 \u2192 p a\u2082 \u2192 p (a\u2081 \u2294 a\u2082))\n  (hs : \u2200 b \u2208 s, p (f b)) : p (s.sup f) :=\nbegin\n  induction s using finset.cons_induction with c s hc ih,\n  { exact hb, },\n  { rw sup_cons,\n    apply hp,\n    { exact hs c (mem_cons.2 (or.inl rfl)), },\n    { exact ih (\u03bb b h, hs b (mem_cons.2 (or.inr h))), }, },\nend\n\nlemma sup_le_of_le_directed {\u03b1 : Type*} [semilattice_sup_bot \u03b1] (s : set \u03b1)\n  (hs : s.nonempty) (hdir : directed_on (\u2264) s) (t : finset \u03b1) :\n  (\u2200 x \u2208 t, \u2203 y \u2208 s, x \u2264 y) \u2192 \u2203 x, x \u2208 s \u2227 t.sup id \u2264 x :=\nbegin\n  classical,\n  apply finset.induction_on t,\n  { simpa only [forall_prop_of_true, and_true, forall_prop_of_false, bot_le, not_false_iff,\n      sup_empty, forall_true_iff, not_mem_empty], },\n  { intros a r har ih h,\n    have incs : \u2191r \u2286 \u2191(insert a r), by { rw finset.coe_subset, apply finset.subset_insert, },\n    -- x \u2208 s is above the sup of r\n    obtain \u27e8x, \u27e8hxs, hsx_sup\u27e9\u27e9 := ih (\u03bb x hx, h x $ incs hx),\n    -- y \u2208 s is above a\n    obtain \u27e8y, hys, hay\u27e9 := h a (finset.mem_insert_self a r),\n    -- z \u2208 s is above x and y\n    obtain \u27e8z, hzs, \u27e8hxz, hyz\u27e9\u27e9 := hdir x hxs y hys,\n    use [z, hzs],\n    rw [sup_insert, id.def, _root_.sup_le_iff],\n    exact \u27e8le_trans hay hyz, le_trans hsx_sup hxz\u27e9, },\nend\n\n-- If we acquire sublattices\n-- the hypotheses should be reformulated as `s : subsemilattice_sup_bot`\nlemma sup_mem\n  (s : set \u03b1) (w\u2081 : \u22a5 \u2208 s) (w\u2082 : \u2200 x y \u2208 s, x \u2294 y \u2208 s)\n  {\u03b9 : Type*} (t : finset \u03b9) (p : \u03b9 \u2192 \u03b1) (h : \u2200 i \u2208 t, p i \u2208 s) :\n  t.sup p \u2208 s :=\n@sup_induction _ _ _ _ _ (\u2208 s) w\u2081 w\u2082 h\n\nend sup\n\nlemma disjoint_sup_right [distrib_lattice_bot \u03b1] {a : \u03b1} {s : finset \u03b2} {f : \u03b2 \u2192 \u03b1} :\n  disjoint a (s.sup f) \u2194 \u2200 i \u2208 s, disjoint a (f i) :=\n\u27e8\u03bb h i hi, h.mono_right (le_sup hi), sup_induction disjoint_bot_right (\u03bb b c, disjoint.sup_right)\u27e9\n\nlemma disjoint_sup_left [distrib_lattice_bot \u03b1] {a : \u03b1} {s : finset \u03b2} {f : \u03b2 \u2192 \u03b1} :\n  disjoint (s.sup f) a \u2194 \u2200 i \u2208 s, disjoint (f i) a :=\nby { simp_rw @disjoint.comm _ _ _ a, exact disjoint_sup_right }\n\nlemma sup_eq_supr [complete_lattice \u03b2] (s : finset \u03b1) (f : \u03b1 \u2192 \u03b2) : s.sup f = (\u2a06a\u2208s, f a) :=\nle_antisymm\n  (finset.sup_le $ assume a ha, le_supr_of_le a $ le_supr _ ha)\n  (supr_le $ assume a, supr_le $ assume ha, le_sup ha)\n\nlemma sup_id_eq_Sup [complete_lattice \u03b1] (s : finset \u03b1) : s.sup id = Sup s :=\nby simp [Sup_eq_supr, sup_eq_supr]\n\nlemma sup_eq_Sup_image [complete_lattice \u03b2] (s : finset \u03b1) (f : \u03b1 \u2192 \u03b2) : s.sup f = Sup (f '' s) :=\nbegin\n  classical,\n  rw [\u2190finset.coe_image, \u2190sup_id_eq_Sup, sup_image, function.comp.left_id],\nend\n\n\/-! ### inf -\/\nsection inf\nvariables [semilattice_inf_top \u03b1]\n\n\/-- Infimum of a finite set: `inf {a, b, c} f = f a \u2293 f b \u2293 f c` -\/\ndef inf (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) : \u03b1 := s.fold (\u2293) \u22a4 f\n\nvariables {s s\u2081 s\u2082 : finset \u03b2} {f g : \u03b2 \u2192 \u03b1}\n\nlemma inf_def : s.inf f = (s.1.map f).inf := rfl\n\n@[simp] lemma inf_empty : (\u2205 : finset \u03b2).inf f = \u22a4 :=\nfold_empty\n\n@[simp] lemma inf_cons {b : \u03b2} (h : b \u2209 s) : (cons b s h).inf f = f b \u2293 s.inf f :=\n@sup_cons (order_dual \u03b1) _ _ _ _ _ h\n\n@[simp] lemma inf_insert [decidable_eq \u03b2] {b : \u03b2} : (insert b s : finset \u03b2).inf f = f b \u2293 s.inf f :=\nfold_insert_idem\n\nlemma inf_image [decidable_eq \u03b2] (s : finset \u03b3) (f : \u03b3 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  (s.image f).inf g = s.inf (g \u2218 f) :=\nfold_image_idem\n\n@[simp] lemma inf_map (s : finset \u03b3) (f : \u03b3 \u21aa \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  (s.map f).inf g = s.inf (g \u2218 f) :=\nfold_map\n\n@[simp] lemma inf_singleton {b : \u03b2} : ({b} : finset \u03b2).inf f = f b :=\ninf_singleton\n\nlemma inf_union [decidable_eq \u03b2] : (s\u2081 \u222a s\u2082).inf f = s\u2081.inf f \u2293 s\u2082.inf f :=\n@sup_union (order_dual \u03b1) _ _ _ _ _ _\n\nlemma inf_inf : s.inf (f \u2293 g) = s.inf f \u2293 s.inf g :=\n@sup_sup (order_dual \u03b1) _ _ _ _ _\n\ntheorem inf_congr {f g : \u03b2 \u2192 \u03b1} (hs : s\u2081 = s\u2082) (hfg : \u2200a\u2208s\u2082, f a = g a) : s\u2081.inf f = s\u2082.inf g :=\nby subst hs; exact finset.fold_congr hfg\n\n@[simp] lemma inf_bUnion [decidable_eq \u03b2] (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b2) :\n  (s.bUnion t).inf f = s.inf (\u03bb x, (t x).inf f) :=\n@sup_bUnion (order_dual \u03b1) _ _ _ _ _ _ _\n\nlemma inf_const {s : finset \u03b2} (h : s.nonempty) (c : \u03b1) : s.inf (\u03bb _, c) = c :=\n@sup_const (order_dual \u03b1) _ _ _ h _\n\nlemma le_inf_iff {a : \u03b1} : a \u2264 s.inf f \u2194 \u2200 b \u2208 s, a \u2264 f b :=\n@sup_le_iff (order_dual \u03b1) _ _ _ _ _\n\nlemma inf_le {b : \u03b2} (hb : b \u2208 s) : s.inf f \u2264 f b :=\nle_inf_iff.1 (le_refl _) _ hb\n\nlemma le_inf {a : \u03b1} : (\u2200b \u2208 s, a \u2264 f b) \u2192 a \u2264 s.inf f :=\nle_inf_iff.2\n\nlemma inf_mono_fun {g : \u03b2 \u2192 \u03b1} (h : \u2200b\u2208s, f b \u2264 g b) : s.inf f \u2264 s.inf g :=\nle_inf (\u03bb b hb, le_trans (inf_le hb) (h b hb))\n\nlemma inf_mono (h : s\u2081 \u2286 s\u2082) : s\u2082.inf f \u2264 s\u2081.inf f :=\nle_inf $ assume b hb, inf_le (h hb)\n\n@[simp] lemma lt_inf_iff [is_total \u03b1 (\u2264)] {a : \u03b1} (ha : a < \u22a4) : a < s.inf f \u2194 (\u2200 b \u2208 s, a < f b) :=\n@sup_lt_iff (order_dual \u03b1) _ _ _ _ _ _ ha\n\n@[simp] lemma inf_le_iff [is_total \u03b1 (\u2264)] {a : \u03b1} (ha : a < \u22a4) : s.inf f \u2264 a \u2194 (\u2203 b \u2208 s, f b \u2264 a) :=\n@le_sup_iff (order_dual \u03b1) _ _ _ _ _ _ ha\n\n@[simp] lemma inf_lt_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : s.inf f < a \u2194 (\u2203 b \u2208 s, f b < a) :=\n@lt_sup_iff (order_dual \u03b1) _ _ _ _ _ _\n\nlemma inf_attach (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) : s.attach.inf (\u03bb x, f x) = s.inf f :=\n@sup_attach (order_dual \u03b1) _ _ _ _\n\n@[simp] lemma inf_erase_top [decidable_eq \u03b1] (s : finset \u03b1) : (s.erase \u22a4).inf id = s.inf id :=\n@sup_erase_bot (order_dual \u03b1) _ _ _\n\nlemma sup_sdiff_left {\u03b1 \u03b2 : Type*} [boolean_algebra \u03b1] (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) (a : \u03b1) :\n  s.sup (\u03bb b, a \\ f b) = a \\ s.inf f :=\nbegin\n  refine finset.cons_induction_on s _ (\u03bb b t _ h, _),\n  { rw [sup_empty, inf_empty, sdiff_top] },\n  { rw [sup_cons, inf_cons, h, sdiff_inf] }\nend\n\nlemma inf_sdiff_left {\u03b1 \u03b2 : Type*} [boolean_algebra \u03b1] {s : finset \u03b2} (hs : s.nonempty) (f : \u03b2 \u2192 \u03b1)\n  (a : \u03b1) :\n  s.inf (\u03bb b, a \\ f b) = a \\ s.sup f :=\nbegin\n  refine hs.cons_induction (\u03bb b, _) (\u03bb b t _ h, _),\n  { rw [sup_singleton, inf_singleton] },\n  { rw [sup_cons, inf_cons, h, sdiff_sup] }\nend\n\nlemma inf_sdiff_right {\u03b1 \u03b2 : Type*} [boolean_algebra \u03b1] {s : finset \u03b2} (hs : s.nonempty) (f : \u03b2 \u2192 \u03b1)\n  (a : \u03b1) :\n  s.inf (\u03bb b, f b \\ a) = s.inf f \\ a :=\nbegin\n  refine hs.cons_induction (\u03bb b, _) (\u03bb b t _ h, _),\n  { rw [inf_singleton, inf_singleton] },\n  { rw [inf_cons, inf_cons, h, inf_sdiff] }\nend\n\nlemma comp_inf_eq_inf_comp [semilattice_inf_top \u03b3] {s : finset \u03b2}\n  {f : \u03b2 \u2192 \u03b1} (g : \u03b1 \u2192 \u03b3) (g_inf : \u2200 x y, g (x \u2293 y) = g x \u2293 g y) (top : g \u22a4 = \u22a4) :\n  g (s.inf f) = s.inf (g \u2218 f) :=\n@comp_sup_eq_sup_comp (order_dual \u03b1) _ (order_dual \u03b3) _ _ _ _ _ g_inf top\n\nlemma comp_inf_eq_inf_comp_of_is_total [h : is_total \u03b1 (\u2264)] {\u03b3 : Type} [semilattice_inf_top \u03b3]\n  (g : \u03b1 \u2192 \u03b3) (mono_g : monotone g) (top : g \u22a4 = \u22a4) : g (s.inf f) = s.inf (g \u2218 f) :=\ncomp_inf_eq_inf_comp g mono_g.map_inf top\n\n\/-- Computing `inf` in a subtype (closed under `inf`) is the same as computing it in `\u03b1`. -\/\nlemma inf_coe {P : \u03b1 \u2192 Prop}\n  {Ptop : P \u22a4} {Pinf : \u2200{{x y}}, P x \u2192 P y \u2192 P (x \u2293 y)}\n  (t : finset \u03b2) (f : \u03b2 \u2192 {x : \u03b1 \/\/ P x}) :\n  (@inf _ _ (subtype.semilattice_inf_top Ptop Pinf) t f : \u03b1) = t.inf (\u03bb x, f x) :=\n@sup_coe (order_dual \u03b1) _ _ _ Ptop Pinf t f\n\nlemma inf_induction {p : \u03b1 \u2192 Prop} (ht : p \u22a4) (hp : \u2200 (a\u2081 a\u2082 : \u03b1), p a\u2081 \u2192 p a\u2082 \u2192 p (a\u2081 \u2293 a\u2082))\n  (hs : \u2200 b \u2208 s, p (f b)) : p (s.inf f) :=\n@sup_induction (order_dual \u03b1) _ _ _ _ _ ht hp hs\n\nlemma inf_mem\n  (s : set \u03b1) (w\u2081 : \u22a4 \u2208 s) (w\u2082 : \u2200 x y \u2208 s, x \u2293 y \u2208 s)\n  {\u03b9 : Type*} (t : finset \u03b9) (p : \u03b9 \u2192 \u03b1) (h : \u2200 i \u2208 t, p i \u2208 s) :\n  t.inf p \u2208 s :=\n@inf_induction _ _ _ _ _ (\u2208 s) w\u2081 w\u2082 h\n\nend inf\n\nlemma inf_eq_infi [complete_lattice \u03b2] (s : finset \u03b1) (f : \u03b1 \u2192 \u03b2) : s.inf f = (\u2a05a\u2208s, f a) :=\n@sup_eq_supr _ (order_dual \u03b2) _ _ _\n\nlemma inf_id_eq_Inf [complete_lattice \u03b1] (s : finset \u03b1) : s.inf id = Inf s :=\n@sup_id_eq_Sup (order_dual \u03b1) _ _\n\nlemma inf_eq_Inf_image [complete_lattice \u03b2] (s : finset \u03b1) (f : \u03b1 \u2192 \u03b2) : s.inf f = Inf (f '' s) :=\n@sup_eq_Sup_image _ (order_dual \u03b2) _ _ _\n\nsection sup'\nvariables [semilattice_sup \u03b1]\n\nlemma sup_of_mem {s : finset \u03b2} (f : \u03b2 \u2192 \u03b1) {b : \u03b2} (h : b \u2208 s) :\n  \u2203 (a : \u03b1), s.sup (coe \u2218 f : \u03b2 \u2192 with_bot \u03b1) = \u2191a :=\nExists.imp (\u03bb a, Exists.fst) (@le_sup (with_bot \u03b1) _ _ _ _ _ h (f b) rfl)\n\n\/-- Given nonempty finset `s` then `s.sup' H f` is the supremum of its image under `f` in (possibly\nunbounded) join-semilattice `\u03b1`, where `H` is a proof of nonemptiness. If `\u03b1` has a bottom element\nyou may instead use `finset.sup` which does not require `s` nonempty. -\/\ndef sup' (s : finset \u03b2) (H : s.nonempty) (f : \u03b2 \u2192 \u03b1) : \u03b1 :=\noption.get $ let \u27e8b, hb\u27e9 := H in option.is_some_iff_exists.2 (sup_of_mem f hb)\n\nvariables {s : finset \u03b2} (H : s.nonempty) (f : \u03b2 \u2192 \u03b1)\n\n@[simp] lemma coe_sup' : ((s.sup' H f : \u03b1) : with_bot \u03b1) = s.sup (coe \u2218 f) :=\nby rw [sup', \u2190with_bot.some_eq_coe, option.some_get]\n\n@[simp] lemma sup'_cons {b : \u03b2} {hb : b \u2209 s} {h : (cons b s hb).nonempty} :\n  (cons b s hb).sup' h f = f b \u2294 s.sup' H f :=\nby { rw \u2190with_bot.coe_eq_coe, simp only [coe_sup', sup_cons, with_bot.coe_sup], }\n\n@[simp] lemma sup'_insert [decidable_eq \u03b2] {b : \u03b2} {h : (insert b s).nonempty} :\n  (insert b s).sup' h f = f b \u2294 s.sup' H f :=\nby { rw \u2190with_bot.coe_eq_coe, simp only [coe_sup', sup_insert, with_bot.coe_sup], }\n\n@[simp] lemma sup'_singleton {b : \u03b2} {h : ({b} : finset \u03b2).nonempty} :\n  ({b} : finset \u03b2).sup' h f = f b := rfl\n\nlemma sup'_le {a : \u03b1} (hs : \u2200 b \u2208 s, f b \u2264 a) : s.sup' H f \u2264 a :=\nby { rw [\u2190with_bot.coe_le_coe, coe_sup'], exact sup_le (\u03bb b h, with_bot.coe_le_coe.2 $ hs b h), }\n\nlemma le_sup' {b : \u03b2} (h : b \u2208 s) : f b \u2264 s.sup' \u27e8b, h\u27e9 f :=\nby { rw [\u2190with_bot.coe_le_coe, coe_sup'], exact le_sup h, }\n\n@[simp] lemma sup'_const (a : \u03b1) : s.sup' H (\u03bb b, a) = a :=\nbegin\n  apply le_antisymm,\n  { apply sup'_le, intros, apply le_refl, },\n  { apply le_sup' (\u03bb b, a) H.some_spec, }\nend\n\n@[simp] lemma sup'_le_iff {a : \u03b1} : s.sup' H f \u2264 a \u2194 \u2200 b \u2208 s, f b \u2264 a :=\niff.intro (\u03bb h b hb, trans (le_sup' f hb) h) (sup'_le H f)\n\n@[simp] lemma sup'_lt_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : s.sup' H f < a \u2194 (\u2200 b \u2208 s, f b < a) :=\nbegin\n  rw [\u2190with_bot.coe_lt_coe, coe_sup', sup_lt_iff (with_bot.bot_lt_coe a)],\n  exact ball_congr (\u03bb b hb, with_bot.coe_lt_coe),\nend\n\n@[simp] lemma le_sup'_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : a \u2264 s.sup' H f \u2194 (\u2203 b \u2208 s, a \u2264 f b) :=\nbegin\n  rw [\u2190with_bot.coe_le_coe, coe_sup', le_sup_iff (with_bot.bot_lt_coe a)],\n  exact bex_congr (\u03bb b hb, with_bot.coe_le_coe),\nend\n\n@[simp] lemma lt_sup'_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : a < s.sup' H f \u2194 (\u2203 b \u2208 s, a < f b) :=\nbegin\n  rw [\u2190with_bot.coe_lt_coe, coe_sup', lt_sup_iff],\n  exact bex_congr (\u03bb b hb, with_bot.coe_lt_coe),\nend\n\nlemma sup'_bUnion [decidable_eq \u03b2] {s : finset \u03b3} (Hs : s.nonempty) {t : \u03b3 \u2192 finset \u03b2}\n  (Ht : \u2200 b, (t b).nonempty) :\n  (s.bUnion t).sup' (Hs.bUnion (\u03bb b _, Ht b)) f = s.sup' Hs (\u03bb b, (t b).sup' (Ht b) f) :=\neq_of_forall_ge_iff $ \u03bb c, by simp [@forall_swap _ \u03b2]\n\nlemma comp_sup'_eq_sup'_comp [semilattice_sup \u03b3] {s : finset \u03b2} (H : s.nonempty)\n  {f : \u03b2 \u2192 \u03b1} (g : \u03b1 \u2192 \u03b3) (g_sup : \u2200 x y, g (x \u2294 y) = g x \u2294 g y) :\n  g (s.sup' H f) = s.sup' H (g \u2218 f) :=\nbegin\n  rw [\u2190with_bot.coe_eq_coe, coe_sup'],\n  let g' : with_bot \u03b1 \u2192 with_bot \u03b3 := with_bot.rec_bot_coe \u22a5 (\u03bb x, \u2191(g x)),\n  show g' \u2191(s.sup' H f) = s.sup (\u03bb a, g' \u2191(f a)),\n  rw coe_sup',\n  refine comp_sup_eq_sup_comp g' _ rfl,\n  intros f\u2081 f\u2082,\n  cases f\u2081,\n  { rw [with_bot.none_eq_bot, bot_sup_eq], exact bot_sup_eq.symm, },\n  { cases f\u2082, refl,\n    exact congr_arg coe (g_sup f\u2081 f\u2082), },\nend\n\nlemma sup'_induction {p : \u03b1 \u2192 Prop} (hp : \u2200 (a\u2081 a\u2082 : \u03b1), p a\u2081 \u2192 p a\u2082 \u2192 p (a\u2081 \u2294 a\u2082))\n  (hs : \u2200 b \u2208 s, p (f b)) : p (s.sup' H f) :=\nbegin\n  show @with_bot.rec_bot_coe \u03b1 (\u03bb _, Prop) true p \u2191(s.sup' H f),\n  rw coe_sup',\n  refine sup_induction trivial _ hs,\n  intros a\u2081 a\u2082 h\u2081 h\u2082,\n  cases a\u2081,\n  { rw [with_bot.none_eq_bot, bot_sup_eq], exact h\u2082, },\n  { cases a\u2082, exact h\u2081, exact hp a\u2081 a\u2082 h\u2081 h\u2082, },\nend\n\nlemma exists_mem_eq_sup' [is_total \u03b1 (\u2264)] : \u2203 b, b \u2208 s \u2227 s.sup' H f = f b :=\nbegin\n  induction s using finset.cons_induction with c s hc ih,\n  { exact false.elim (not_nonempty_empty H), },\n  { rcases s.eq_empty_or_nonempty with rfl | hs,\n    { exact \u27e8c, mem_singleton_self c, rfl\u27e9, },\n    { rcases ih hs with \u27e8b, hb, h'\u27e9,\n      rw [sup'_cons hs, h'],\n      cases total_of (\u2264) (f b) (f c) with h h,\n      { exact \u27e8c, mem_cons.2 (or.inl rfl), sup_eq_left.2 h\u27e9, },\n      { exact \u27e8b, mem_cons.2 (or.inr hb), sup_eq_right.2 h\u27e9, }, }, },\nend\n\nlemma sup'_mem\n  (s : set \u03b1) (w : \u2200 x y \u2208 s, x \u2294 y \u2208 s)\n  {\u03b9 : Type*} (t : finset \u03b9) (H : t.nonempty) (p : \u03b9 \u2192 \u03b1) (h : \u2200 i \u2208 t, p i \u2208 s) :\n  t.sup' H p \u2208 s :=\nsup'_induction H p w h\n\n@[congr] lemma sup'_congr {t : finset \u03b2} {f g : \u03b2 \u2192 \u03b1} (h\u2081 : s = t) (h\u2082 : \u2200 x \u2208 s, f x = g x) :\n  s.sup' H f = t.sup' (h\u2081 \u25b8 H) g :=\nbegin\n  subst s,\n  refine eq_of_forall_ge_iff (\u03bb c, _),\n  simp only [sup'_le_iff, h\u2082] { contextual := tt }\nend\n\nend sup'\n\nsection inf'\nvariables [semilattice_inf \u03b1]\n\nlemma inf_of_mem {s : finset \u03b2} (f : \u03b2 \u2192 \u03b1) {b : \u03b2} (h : b \u2208 s) :\n  \u2203 (a : \u03b1), s.inf (coe \u2218 f : \u03b2 \u2192 with_top \u03b1) = \u2191a :=\n@sup_of_mem (order_dual \u03b1) _ _ _ f _ h\n\n\/-- Given nonempty finset `s` then `s.inf' H f` is the infimum of its image under `f` in (possibly\nunbounded) meet-semilattice `\u03b1`, where `H` is a proof of nonemptiness. If `\u03b1` has a top element you\nmay instead use `finset.inf` which does not require `s` nonempty. -\/\ndef inf' (s : finset \u03b2) (H : s.nonempty) (f : \u03b2 \u2192 \u03b1) : \u03b1 :=\n@sup' (order_dual \u03b1) _ _ s H f\n\nvariables {s : finset \u03b2} (H : s.nonempty) (f : \u03b2 \u2192 \u03b1)\n\n@[simp] lemma coe_inf' : ((s.inf' H f : \u03b1) : with_top \u03b1) = s.inf (coe \u2218 f) :=\n@coe_sup' (order_dual \u03b1) _ _ _ H f\n\n@[simp] lemma inf'_cons {b : \u03b2} {hb : b \u2209 s} {h : (cons b s hb).nonempty} :\n  (cons b s hb).inf' h f = f b \u2293 s.inf' H f :=\n@sup'_cons (order_dual \u03b1) _ _ _ H f _ _ _\n\n@[simp] lemma inf'_insert [decidable_eq \u03b2] {b : \u03b2} {h : (insert b s).nonempty} :\n  (insert b s).inf' h f = f b \u2293 s.inf' H f :=\n@sup'_insert (order_dual \u03b1) _ _ _ H f _ _ _\n\n@[simp] lemma inf'_singleton {b : \u03b2} {h : ({b} : finset \u03b2).nonempty} :\n  ({b} : finset \u03b2).inf' h f = f b := rfl\n\nlemma le_inf' {a : \u03b1} (hs : \u2200 b \u2208 s, a \u2264 f b) : a \u2264 s.inf' H f :=\n@sup'_le (order_dual \u03b1) _ _ _ H f _ hs\n\nlemma inf'_le {b : \u03b2} (h : b \u2208 s) : s.inf' \u27e8b, h\u27e9 f \u2264 f b :=\n@le_sup' (order_dual \u03b1) _ _ _ f _ h\n\n@[simp] lemma inf'_const (a : \u03b1) : s.inf' H (\u03bb b, a) = a :=\n@sup'_const (order_dual \u03b1) _ _ _ _ _\n\n@[simp] lemma le_inf'_iff {a : \u03b1} : a \u2264 s.inf' H f \u2194 \u2200 b \u2208 s, a \u2264 f b :=\n@sup'_le_iff (order_dual \u03b1) _ _ _ H f _\n\n@[simp] lemma lt_inf'_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : a < s.inf' H f \u2194 (\u2200 b \u2208 s, a < f b) :=\n@sup'_lt_iff (order_dual \u03b1) _ _ _ H f _ _\n\n@[simp] lemma inf'_le_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : s.inf' H f \u2264 a \u2194 (\u2203 b \u2208 s, f b \u2264 a) :=\n@le_sup'_iff (order_dual \u03b1) _ _ _ H f _ _\n\n@[simp] lemma inf'_lt_iff [is_total \u03b1 (\u2264)] {a : \u03b1} : s.inf' H f < a \u2194 (\u2203 b \u2208 s, f b < a) :=\n@lt_sup'_iff (order_dual \u03b1) _ _ _ H f _ _\n\nlemma inf'_bUnion [decidable_eq \u03b2] {s : finset \u03b3} (Hs : s.nonempty) {t : \u03b3 \u2192 finset \u03b2}\n  (Ht : \u2200 b, (t b).nonempty) :\n  (s.bUnion t).inf' (Hs.bUnion (\u03bb b _, Ht b)) f = s.inf' Hs (\u03bb b, (t b).inf' (Ht b) f) :=\n@sup'_bUnion (order_dual \u03b1) _ _ _ _ _ _ Hs _ Ht\n\nlemma comp_inf'_eq_inf'_comp [semilattice_inf \u03b3] {s : finset \u03b2} (H : s.nonempty)\n  {f : \u03b2 \u2192 \u03b1} (g : \u03b1 \u2192 \u03b3) (g_inf : \u2200 x y, g (x \u2293 y) = g x \u2293 g y) :\n  g (s.inf' H f) = s.inf' H (g \u2218 f) :=\n@comp_sup'_eq_sup'_comp (order_dual \u03b1) _ (order_dual \u03b3) _ _ _ H f g g_inf\n\nlemma inf'_induction {p : \u03b1 \u2192 Prop} (hp : \u2200 (a\u2081 a\u2082 : \u03b1), p a\u2081 \u2192 p a\u2082 \u2192 p (a\u2081 \u2293 a\u2082))\n  (hs : \u2200 b \u2208 s, p (f b)) : p (s.inf' H f) :=\n@sup'_induction (order_dual \u03b1) _ _ _ H f _ hp hs\n\nlemma exists_mem_eq_inf' [is_total \u03b1 (\u2264)] : \u2203 b, b \u2208 s \u2227 s.inf' H f = f b :=\n@exists_mem_eq_sup' (order_dual \u03b1) _ _ _ H f _\n\nlemma inf'_mem (s : set \u03b1) (w : \u2200 x y \u2208 s, x \u2293 y \u2208 s)\n  {\u03b9 : Type*} (t : finset \u03b9) (H : t.nonempty) (p : \u03b9 \u2192 \u03b1) (h : \u2200 i \u2208 t, p i \u2208 s) :\n  t.inf' H p \u2208 s :=\ninf'_induction H p w h\n\n@[congr] lemma inf'_congr {t : finset \u03b2} {f g : \u03b2 \u2192 \u03b1} (h\u2081 : s = t) (h\u2082 : \u2200 x \u2208 s, f x = g x) :\n  s.inf' H f = t.inf' (h\u2081 \u25b8 H) g :=\n@sup'_congr (order_dual \u03b1) _ _ _ H _ _ _ h\u2081 h\u2082\n\nend inf'\n\nsection sup\nvariable [semilattice_sup_bot \u03b1]\n\nlemma sup'_eq_sup {s : finset \u03b2} (H : s.nonempty) (f : \u03b2 \u2192 \u03b1) : s.sup' H f = s.sup f :=\nle_antisymm (sup'_le H f (\u03bb b, le_sup)) (sup_le (\u03bb b, le_sup' f))\n\nlemma sup_closed_of_sup_closed {s : set \u03b1} (t : finset \u03b1) (htne : t.nonempty) (h_subset : \u2191t \u2286 s)\n  (h : \u2200\u2983a b\u2984, a \u2208 s \u2192 b \u2208 s \u2192 a \u2294 b \u2208 s) : t.sup id \u2208 s :=\nsup'_eq_sup htne id \u25b8 sup'_induction _ _ h h_subset\n\nlemma exists_mem_eq_sup [is_total \u03b1 (\u2264)] (s : finset \u03b2) (h : s.nonempty) (f : \u03b2 \u2192 \u03b1) :\n  \u2203 b, b \u2208 s \u2227 s.sup f = f b :=\nsup'_eq_sup h f \u25b8 exists_mem_eq_sup' h f\n\nend sup\n\nsection inf\nvariable [semilattice_inf_top \u03b1]\n\nlemma inf'_eq_inf {s : finset \u03b2} (H : s.nonempty) (f : \u03b2 \u2192 \u03b1) : s.inf' H f = s.inf f :=\n@sup'_eq_sup (order_dual \u03b1) _ _ _ H f\n\nlemma inf_closed_of_inf_closed {s : set \u03b1} (t : finset \u03b1) (htne : t.nonempty) (h_subset : \u2191t \u2286 s)\n  (h : \u2200\u2983a b\u2984, a \u2208 s \u2192 b \u2208 s \u2192 a \u2293 b \u2208 s) : t.inf id \u2208 s :=\n@sup_closed_of_sup_closed (order_dual \u03b1) _ _ t htne h_subset h\n\nlemma exists_mem_eq_inf [is_total \u03b1 (\u2264)] (s : finset \u03b2) (h : s.nonempty) (f : \u03b2 \u2192 \u03b1) :\n  \u2203 a, a \u2208 s \u2227 s.inf f = f a :=\n@exists_mem_eq_sup (order_dual \u03b1) _ _ _ _ h f\n\nend inf\n\nsection sup\nvariables {C : \u03b2 \u2192 Type*} [\u03a0 (b : \u03b2), semilattice_sup_bot (C b)]\n\n@[simp]\nprotected lemma sup_apply (s : finset \u03b1) (f : \u03b1 \u2192 \u03a0 (b : \u03b2), C b) (b : \u03b2) :\n  s.sup f b = s.sup (\u03bb a, f a b) :=\ncomp_sup_eq_sup_comp (\u03bb x : \u03a0 b : \u03b2, C b, x b) (\u03bb i j, rfl) rfl\n\nend sup\n\nsection inf\nvariables {C : \u03b2 \u2192 Type*} [\u03a0 (b : \u03b2), semilattice_inf_top (C b)]\n\n@[simp]\nprotected lemma inf_apply (s : finset \u03b1) (f : \u03b1 \u2192 \u03a0 (b : \u03b2), C b) (b : \u03b2) :\n  s.inf f b = s.inf (\u03bb a, f a b) :=\n@finset.sup_apply _ _ (\u03bb b, order_dual (C b)) _ s f b\n\nend inf\n\nsection sup'\nvariables {C : \u03b2 \u2192 Type*} [\u03a0 (b : \u03b2), semilattice_sup (C b)]\n\n@[simp]\nprotected lemma sup'_apply {s : finset \u03b1} (H : s.nonempty) (f : \u03b1 \u2192 \u03a0 (b : \u03b2), C b) (b : \u03b2) :\n  s.sup' H f b = s.sup' H (\u03bb a, f a b) :=\ncomp_sup'_eq_sup'_comp H (\u03bb x : \u03a0 b : \u03b2, C b, x b) (\u03bb i j, rfl)\n\nend sup'\n\nsection inf'\nvariables {C : \u03b2 \u2192 Type*} [\u03a0 (b : \u03b2), semilattice_inf (C b)]\n\n@[simp]\nprotected lemma inf'_apply {s : finset \u03b1} (H : s.nonempty) (f : \u03b1 \u2192 \u03a0 (b : \u03b2), C b) (b : \u03b2) :\n  s.inf' H f b = s.inf' H (\u03bb a, f a b) :=\n@finset.sup'_apply _ _ (\u03bb b, order_dual (C b)) _ _ H f b\n\nend inf'\n\n\/-! ### max and min of finite sets -\/\nsection max_min\nvariables [linear_order \u03b1]\n\n\/-- Let `s` be a finset in a linear order. Then `s.max` is the maximum of `s` if `s` is not empty,\nand `none` otherwise. It belongs to `option \u03b1`. If you want to get an element of `\u03b1`, see\n`s.max'`. -\/\nprotected def max : finset \u03b1 \u2192 option \u03b1 :=\nfold (option.lift_or_get max) none some\n\ntheorem max_eq_sup_with_bot (s : finset \u03b1) :\n  s.max = @sup (with_bot \u03b1) \u03b1 _ s some := rfl\n\n@[simp] theorem max_empty : (\u2205 : finset \u03b1).max = none := rfl\n\n@[simp] theorem max_insert {a : \u03b1} {s : finset \u03b1} :\n  (insert a s).max = option.lift_or_get max (some a) s.max := fold_insert_idem\n\n@[simp] theorem max_singleton {a : \u03b1} : finset.max {a} = some a :=\nby { rw [\u2190 insert_emptyc_eq], exact max_insert }\n\ntheorem max_of_mem {s : finset \u03b1} {a : \u03b1} (h : a \u2208 s) : \u2203 b, b \u2208 s.max :=\n(@le_sup (with_bot \u03b1) _ _ _ _ _ h _ rfl).imp $ \u03bb b, Exists.fst\n\ntheorem max_of_nonempty {s : finset \u03b1} (h : s.nonempty) : \u2203 a, a \u2208 s.max :=\nlet \u27e8a, ha\u27e9 := h in max_of_mem ha\n\ntheorem max_eq_none {s : finset \u03b1} : s.max = none \u2194 s = \u2205 :=\n\u27e8\u03bb h, s.eq_empty_or_nonempty.elim id\n  (\u03bb H, let \u27e8a, ha\u27e9 := max_of_nonempty H in by rw h at ha; cases ha),\n  \u03bb h, h.symm \u25b8 max_empty\u27e9\n\ntheorem mem_of_max {s : finset \u03b1} : \u2200 {a : \u03b1}, a \u2208 s.max \u2192 a \u2208 s :=\nfinset.induction_on s (\u03bb _ H, by cases H)\n  (\u03bb b s _ (ih : \u2200 {a}, a \u2208 s.max \u2192 a \u2208 s) a (h : a \u2208 (insert b s).max),\n  begin\n    by_cases p : b = a,\n    { induction p, exact mem_insert_self b s },\n    { cases option.lift_or_get_choice max_choice (some b) s.max with q q;\n      rw [max_insert, q] at h,\n      { cases h, cases p rfl },\n      { exact mem_insert_of_mem (ih h) } }\n  end)\n\ntheorem le_max_of_mem {s : finset \u03b1} {a b : \u03b1} (h\u2081 : a \u2208 s) (h\u2082 : b \u2208 s.max) : a \u2264 b :=\nby rcases @le_sup (with_bot \u03b1) _ _ _ _ _ h\u2081 _ rfl with \u27e8b', hb, ab\u27e9;\n   cases h\u2082.symm.trans hb; assumption\n\n\/-- Let `s` be a finset in a linear order. Then `s.min` is the minimum of `s` if `s` is not empty,\nand `none` otherwise. It belongs to `option \u03b1`. If you want to get an element of `\u03b1`, see\n`s.min'`. -\/\nprotected def min : finset \u03b1 \u2192 option \u03b1 :=\nfold (option.lift_or_get min) none some\n\ntheorem min_eq_inf_with_top (s : finset \u03b1) :\n  s.min = @inf (with_top \u03b1) \u03b1 _ s some := rfl\n\n@[simp] theorem min_empty : (\u2205 : finset \u03b1).min = none := rfl\n\n@[simp] theorem min_insert {a : \u03b1} {s : finset \u03b1} :\n  (insert a s).min = option.lift_or_get min (some a) s.min :=\nfold_insert_idem\n\n@[simp] theorem min_singleton {a : \u03b1} : finset.min {a} = some a :=\nby { rw \u2190 insert_emptyc_eq, exact min_insert }\n\ntheorem min_of_mem {s : finset \u03b1} {a : \u03b1} (h : a \u2208 s) : \u2203 b, b \u2208 s.min :=\n(@inf_le (with_top \u03b1) _ _ _ _ _ h _ rfl).imp $ \u03bb b, Exists.fst\n\ntheorem min_of_nonempty {s : finset \u03b1} (h : s.nonempty) : \u2203 a, a \u2208 s.min :=\nlet \u27e8a, ha\u27e9 := h in min_of_mem ha\n\ntheorem min_eq_none {s : finset \u03b1} : s.min = none \u2194 s = \u2205 :=\n\u27e8\u03bb h, s.eq_empty_or_nonempty.elim id\n  (\u03bb H, let \u27e8a, ha\u27e9 := min_of_nonempty H in by rw h at ha; cases ha),\n  \u03bb h, h.symm \u25b8 min_empty\u27e9\n\ntheorem mem_of_min {s : finset \u03b1} : \u2200 {a : \u03b1}, a \u2208 s.min \u2192 a \u2208 s :=\n@mem_of_max (order_dual \u03b1) _ s\n\ntheorem min_le_of_mem {s : finset \u03b1} {a b : \u03b1} (h\u2081 : b \u2208 s) (h\u2082 : a \u2208 s.min) : a \u2264 b :=\nby rcases @inf_le (with_top \u03b1) _ _ _ _ _ h\u2081 _ rfl with \u27e8b', hb, ab\u27e9;\n   cases h\u2082.symm.trans hb; assumption\n\n\/-- Given a nonempty finset `s` in a linear order `\u03b1 `, then `s.min' h` is its minimum, as an\nelement of `\u03b1`, where `h` is a proof of nonemptiness. Without this assumption, use instead `s.min`,\ntaking values in `option \u03b1`. -\/\ndef min' (s : finset \u03b1) (H : s.nonempty) : \u03b1 :=\n@option.get _ s.min $\n  let \u27e8k, hk\u27e9 := H in\n  let \u27e8b, hb\u27e9 := min_of_mem hk in by simp at hb; simp [hb]\n\n\/-- Given a nonempty finset `s` in a linear order `\u03b1 `, then `s.max' h` is its maximum, as an\nelement of `\u03b1`, where `h` is a proof of nonemptiness. Without this assumption, use instead `s.max`,\ntaking values in `option \u03b1`. -\/\ndef max' (s : finset \u03b1) (H : s.nonempty) : \u03b1 :=\n@option.get _ s.max $\n  let \u27e8k, hk\u27e9 := H in\n  let \u27e8b, hb\u27e9 := max_of_mem hk in by simp at hb; simp [hb]\n\nvariables (s : finset \u03b1) (H : s.nonempty)\n\ntheorem min'_mem : s.min' H \u2208 s := mem_of_min $ by simp [min']\n\ntheorem min'_le (x) (H2 : x \u2208 s) : s.min' \u27e8x, H2\u27e9 \u2264 x := min_le_of_mem H2 $ option.get_mem _\n\ntheorem le_min' (x) (H2 : \u2200 y \u2208 s, x \u2264 y) : x \u2264 s.min' H := H2 _ $ min'_mem _ _\n\ntheorem is_least_min' : is_least \u2191s (s.min' H) := \u27e8min'_mem _ _, min'_le _\u27e9\n\n@[simp] lemma le_min'_iff {x} : x \u2264 s.min' H \u2194 \u2200 y \u2208 s, x \u2264 y :=\nle_is_glb_iff (is_least_min' s H).is_glb\n\n\/-- `{a}.min' _` is `a`. -\/\n@[simp] lemma min'_singleton (a : \u03b1) :\n  ({a} : finset \u03b1).min' (singleton_nonempty _) = a :=\nby simp [min']\n\ntheorem max'_mem : s.max' H \u2208 s := mem_of_max $ by simp [max']\n\ntheorem le_max' (x) (H2 : x \u2208 s) : x \u2264 s.max' \u27e8x, H2\u27e9 := le_max_of_mem H2 $ option.get_mem _\n\ntheorem max'_le (x) (H2 : \u2200 y \u2208 s, y \u2264 x) : s.max' H \u2264 x := H2 _ $ max'_mem _ _\n\ntheorem is_greatest_max' : is_greatest \u2191s (s.max' H) := \u27e8max'_mem _ _, le_max' _\u27e9\n\n@[simp] lemma max'_le_iff {x} : s.max' H \u2264 x \u2194 \u2200 y \u2208 s, y \u2264 x :=\nis_lub_le_iff (is_greatest_max' s H).is_lub\n\n@[simp] lemma max'_lt_iff {x} : s.max' H < x \u2194 \u2200 y \u2208 s, y < x :=\n\u27e8\u03bb Hlt y hy, (s.le_max' y hy).trans_lt Hlt, \u03bb H, H _ $ s.max'_mem _\u27e9\n\n@[simp] lemma lt_min'_iff {x} : x < s.min' H \u2194 \u2200 y \u2208 s, x < y :=\n@max'_lt_iff (order_dual \u03b1) _ _ H _\n\nlemma max'_eq_sup' : s.max' H = s.sup' H id :=\neq_of_forall_ge_iff $ \u03bb a, (max'_le_iff _ _).trans (sup'_le_iff _ _).symm\n\nlemma min'_eq_inf' : s.min' H = s.inf' H id :=\n@max'_eq_sup' (order_dual \u03b1) _ s H\n\n\/-- `{a}.max' _` is `a`. -\/\n@[simp] lemma max'_singleton (a : \u03b1) :\n  ({a} : finset \u03b1).max' (singleton_nonempty _) = a :=\nby simp [max']\n\ntheorem min'_lt_max' {i j} (H1 : i \u2208 s) (H2 : j \u2208 s) (H3 : i \u2260 j) :\n  s.min' \u27e8i, H1\u27e9 < s.max' \u27e8i, H1\u27e9 :=\nis_glb_lt_is_lub_of_ne (s.is_least_min' _).is_glb (s.is_greatest_max' _).is_lub H1 H2 H3\n\n\/--\nIf there's more than 1 element, the min' is less than the max'. An alternate version of\n`min'_lt_max'` which is sometimes more convenient.\n-\/\nlemma min'_lt_max'_of_card (h\u2082 : 1 < card s) :\n  s.min' (finset.card_pos.mp $ lt_trans zero_lt_one h\u2082) <\n  s.max' (finset.card_pos.mp $ lt_trans zero_lt_one h\u2082) :=\nbegin\n  rcases one_lt_card.1 h\u2082 with \u27e8a, ha, b, hb, hab\u27e9,\n  exact s.min'_lt_max' ha hb hab\nend\n\nlemma max'_eq_of_dual_min' {s : finset \u03b1} (hs : s.nonempty) :\n  max' s hs = of_dual (min' (image to_dual s) (nonempty.image hs to_dual)) :=\nbegin\n  rw [of_dual, to_dual, equiv.coe_fn_mk, equiv.coe_fn_symm_mk, id.def],\n  simp_rw (@image_id (order_dual \u03b1) (s : finset (order_dual \u03b1))),\n  refl,\nend\n\nlemma min'_eq_of_dual_max' {s : finset \u03b1} (hs : s.nonempty) :\n  min' s hs = of_dual (max' (image to_dual s) (nonempty.image hs to_dual)) :=\nbegin\n  rw [of_dual, to_dual, equiv.coe_fn_mk, equiv.coe_fn_symm_mk, id.def],\n  simp_rw (@image_id (order_dual \u03b1) (s : finset (order_dual \u03b1))),\n  refl,\nend\n\n@[simp] lemma of_dual_max_eq_min_of_dual {a b : \u03b1} :\n  of_dual (max a b) = min (of_dual a) (of_dual b) := rfl\n\n@[simp] lemma of_dual_min_eq_max_of_dual {a b : \u03b1} :\n  of_dual (min a b) = max (of_dual a) (of_dual b) := rfl\n\nlemma max'_subset {s t : finset \u03b1} (H : s.nonempty) (hst : s \u2286 t) :\n  s.max' H \u2264 t.max' (H.mono hst) :=\nle_max' _ _ (hst (s.max'_mem H))\n\nlemma min'_subset {s t : finset \u03b1} (H : s.nonempty) (hst : s \u2286 t) :\n  t.min' (H.mono hst) \u2264 s.min' H :=\nmin'_le _ _ (hst (s.min'_mem H))\n\nlemma max'_insert (a : \u03b1) (s : finset \u03b1) (H : s.nonempty) :\n  (insert a s).max' (s.insert_nonempty a) = max (s.max' H) a :=\n(is_greatest_max' _ _).unique $\n  by { rw [coe_insert, max_comm], exact (is_greatest_max' _ _).insert _ }\n\nlemma min'_insert (a : \u03b1) (s : finset \u03b1) (H : s.nonempty) :\n  (insert a s).min' (s.insert_nonempty a) = min (s.min' H) a :=\n(is_least_min' _ _).unique $\n  by { rw [coe_insert, min_comm], exact (is_least_min' _ _).insert _ }\n\nlemma lt_max'_of_mem_erase_max' [decidable_eq \u03b1] {a : \u03b1} (ha : a \u2208 s.erase (s.max' H)) :\n  a < s.max' H :=\nlt_of_le_of_ne (le_max' _ _ (mem_of_mem_erase ha)) $ ne_of_mem_of_not_mem ha $ not_mem_erase _ _\n\nlemma min'_lt_of_mem_erase_min' [decidable_eq \u03b1] {a : \u03b1} (ha : a \u2208 s.erase (s.min' H)) :\n  s.min' H < a :=\n@lt_max'_of_mem_erase_max' (order_dual \u03b1) _ s H _ a ha\n\n@[simp] lemma max'_image [linear_order \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : monotone f) (s : finset \u03b1) (h : (s.image f).nonempty) :\n  (s.image f).max' h = f (s.max' ((nonempty.image_iff f).mp h)) :=\nbegin\n  refine le_antisymm (max'_le _ _ _ (\u03bb y hy, _))\n    (le_max' _ _ (mem_image.mpr \u27e8_, max'_mem _ _, rfl\u27e9)),\n  obtain \u27e8x, hx, rfl\u27e9 := mem_image.mp hy,\n  exact hf (le_max' _ _ hx)\nend\n\n@[simp] lemma min'_image [linear_order \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : monotone f) (s : finset \u03b1) (h : (s.image f).nonempty) :\n  (s.image f).min' h = f (s.min' ((nonempty.image_iff f).mp h)) :=\nbegin\n  refine le_antisymm (min'_le _ _ (mem_image.mpr \u27e8_, min'_mem _ _, rfl\u27e9))\n    (le_min' _ _ _ (\u03bb y hy, _)),\n  obtain \u27e8x, hx, rfl\u27e9 := mem_image.mp hy,\n  exact hf (min'_le _ _ hx)\nend\n\n\/-- Induction principle for `finset`s in a linearly ordered type: a predicate is true on all\n`s : finset \u03b1` provided that:\n\n* it is true on the empty `finset`,\n* for every `s : finset \u03b1` and an element `a` strictly greater than all elements of `s`, `p s`\n  implies `p (insert a s)`. -\/\n@[elab_as_eliminator]\nlemma induction_on_max [decidable_eq \u03b1] {p : finset \u03b1 \u2192 Prop} (s : finset \u03b1) (h0 : p \u2205)\n  (step : \u2200 a s, (\u2200 x \u2208 s, x < a) \u2192 p s \u2192 p (insert a s)) : p s :=\nbegin\n  induction s using finset.strong_induction_on with s ihs,\n  rcases s.eq_empty_or_nonempty with rfl|hne,\n  { exact h0 },\n  { have H : s.max' hne \u2208 s, from max'_mem s hne,\n    rw \u2190 insert_erase H,\n    exact step _ _ (\u03bb x, s.lt_max'_of_mem_erase_max' hne) (ihs _ $ erase_ssubset H) }\nend\n\n\/-- Induction principle for `finset`s in a linearly ordered type: a predicate is true on all\n`s : finset \u03b1` provided that:\n\n* it is true on the empty `finset`,\n* for every `s : finset \u03b1` and an element `a` strictly less than all elements of `s`, `p s`\n  implies `p (insert a s)`. -\/\n@[elab_as_eliminator]\nlemma induction_on_min [decidable_eq \u03b1] {p : finset \u03b1 \u2192 Prop} (s : finset \u03b1) (h0 : p \u2205)\n  (step : \u2200 a s, (\u2200 x \u2208 s, a < x) \u2192 p s \u2192 p (insert a s)) : p s :=\n@induction_on_max (order_dual \u03b1) _ _ _ s h0 step\n\nend max_min\n\nsection exists_max_min\n\nvariables [linear_order \u03b1]\nlemma exists_max_image (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) (h : s.nonempty) :\n  \u2203 x \u2208 s, \u2200 x' \u2208 s, f x' \u2264 f x :=\nbegin\n  cases max_of_nonempty (h.image f) with y hy,\n  rcases mem_image.mp (mem_of_max hy) with \u27e8x, hx, rfl\u27e9,\n  exact \u27e8x, hx, \u03bb x' hx', le_max_of_mem (mem_image_of_mem f hx') hy\u27e9,\nend\n\nlemma exists_min_image (s : finset \u03b2) (f : \u03b2 \u2192 \u03b1) (h : s.nonempty) :\n  \u2203 x \u2208 s, \u2200 x' \u2208 s, f x \u2264 f x' :=\n@exists_max_image (order_dual \u03b1) \u03b2 _ s f h\n\nend exists_max_min\nend finset\n\nnamespace multiset\n\nlemma count_sup [decidable_eq \u03b2] (s : finset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) (b : \u03b2) :\n  count b (s.sup f) = s.sup (\u03bba, count b (f a)) :=\nbegin\n  letI := classical.dec_eq \u03b1,\n  refine s.induction _ _,\n  { exact count_zero _ },\n  { assume i s his ih,\n    rw [finset.sup_insert, sup_eq_union, count_union, finset.sup_insert, ih],\n    refl }\nend\n\nlemma mem_sup {\u03b1 \u03b2} [decidable_eq \u03b2] {s : finset \u03b1} {f : \u03b1 \u2192 multiset \u03b2}\n  {x : \u03b2} : x \u2208 s.sup f \u2194 \u2203 v \u2208 s, x \u2208 f v :=\nbegin\n  classical,\n  apply s.induction_on,\n  { simp },\n  { intros a s has hxs,\n    rw [finset.sup_insert, multiset.sup_eq_union, multiset.mem_union],\n    split,\n    { intro hxi,\n      cases hxi with hf hf,\n      { refine \u27e8a, _, hf\u27e9,\n        simp only [true_or, eq_self_iff_true, finset.mem_insert] },\n      { rcases hxs.mp hf with \u27e8v, hv, hfv\u27e9,\n        refine \u27e8v, _, hfv\u27e9,\n        simp only [hv, or_true, finset.mem_insert] } },\n    { rintros \u27e8v, hv, hfv\u27e9,\n      rw [finset.mem_insert] at hv,\n      rcases hv with rfl | hv,\n      { exact or.inl hfv },\n      { refine or.inr (hxs.mpr \u27e8v, hv, hfv\u27e9) } } },\nend\n\nend multiset\n\nnamespace finset\n\nlemma mem_sup {\u03b1 \u03b2} [decidable_eq \u03b2] {s : finset \u03b1} {f : \u03b1 \u2192 finset \u03b2}\n  {x : \u03b2} : x \u2208 s.sup f \u2194 \u2203 v \u2208 s, x \u2208 f v :=\nbegin\n  change _ \u2194 \u2203 v \u2208 s, x \u2208 (f v).val,\n  rw [\u2190multiset.mem_sup, \u2190multiset.mem_to_finset, sup_to_finset],\n  simp_rw [val_to_finset],\nend\n\nlemma sup_eq_bUnion {\u03b1 \u03b2} [decidable_eq \u03b2] (s : finset \u03b1) (t : \u03b1 \u2192 finset \u03b2) :\n  s.sup t = s.bUnion t :=\nby { ext, rw [mem_sup, mem_bUnion], }\n\n@[simp] lemma sup_singleton' [decidable_eq \u03b1] (s : finset \u03b1) : s.sup singleton = s :=\nbegin\n  refine (finset.sup_le $ \u03bb a, _).antisymm (\u03bb a ha, mem_sup.2 \u27e8a, ha, mem_singleton_self a\u27e9),\n  exact singleton_subset_iff.2,\nend\n\nend finset\n\nsection lattice\nvariables {\u03b9 : Type*} {\u03b9' : Sort*} [complete_lattice \u03b1]\n\n\/-- Supremum of `s i`, `i : \u03b9`, is equal to the supremum over `t : finset \u03b9` of suprema\n`\u2a06 i \u2208 t, s i`. This version assumes `\u03b9` is a `Type*`. See `supr_eq_supr_finset'` for a version\nthat works for `\u03b9 : Sort*`. -\/\nlemma supr_eq_supr_finset (s : \u03b9 \u2192 \u03b1) :\n  (\u2a06i, s i) = (\u2a06t:finset \u03b9, \u2a06i\u2208t, s i) :=\nbegin\n  classical,\n  exact le_antisymm\n    (supr_le $ assume b, le_supr_of_le {b} $ le_supr_of_le b $ le_supr_of_le\n      (by simp) $ le_refl _)\n    (supr_le $ assume t, supr_le $ assume b, supr_le $ assume hb, le_supr _ _)\nend\n\n\/-- Supremum of `s i`, `i : \u03b9`, is equal to the supremum over `t : finset \u03b9` of suprema\n`\u2a06 i \u2208 t, s i`. This version works for `\u03b9 : Sort*`. See `supr_eq_supr_finset` for a version\nthat assumes `\u03b9 : Type*` but has no `plift`s. -\/\nlemma supr_eq_supr_finset' (s : \u03b9' \u2192 \u03b1) :\n  (\u2a06i, s i) = (\u2a06t:finset (plift \u03b9'), \u2a06i\u2208t, s (plift.down i)) :=\nby rw [\u2190 supr_eq_supr_finset, \u2190 equiv.plift.surjective.supr_comp]; refl\n\n\/-- Infimum of `s i`, `i : \u03b9`, is equal to the infimum over `t : finset \u03b9` of infima\n`\u2a05 i \u2208 t, s i`. This version assumes `\u03b9` is a `Type*`. See `infi_eq_infi_finset'` for a version\nthat works for `\u03b9 : Sort*`. -\/\nlemma infi_eq_infi_finset (s : \u03b9 \u2192 \u03b1) :\n  (\u2a05i, s i) = (\u2a05t:finset \u03b9, \u2a05i\u2208t, s i) :=\n@supr_eq_supr_finset (order_dual \u03b1) _ _ _\n\n\/-- Infimum of `s i`, `i : \u03b9`, is equal to the infimum over `t : finset \u03b9` of infima\n`\u2a05 i \u2208 t, s i`. This version works for `\u03b9 : Sort*`. See `infi_eq_infi_finset` for a version\nthat assumes `\u03b9 : Type*` but has no `plift`s. -\/\nlemma infi_eq_infi_finset' (s : \u03b9' \u2192 \u03b1) :\n  (\u2a05i, s i) = (\u2a05t:finset (plift \u03b9'), \u2a05i\u2208t, s (plift.down i)) :=\n@supr_eq_supr_finset' (order_dual \u03b1) _ _ _\n\nend lattice\n\nnamespace set\nvariables {\u03b9 : Type*} {\u03b9' : Sort*}\n\n\/-- Union of an indexed family of sets `s : \u03b9 \u2192 set \u03b1` is equal to the union of the unions\nof finite subfamilies. This version assumes `\u03b9 : Type*`. See also `Union_eq_Union_finset'` for\na version that works for `\u03b9 : Sort*`. -\/\nlemma Union_eq_Union_finset (s : \u03b9 \u2192 set \u03b1) :\n  (\u22c3i, s i) = (\u22c3t:finset \u03b9, \u22c3i\u2208t, s i) :=\nsupr_eq_supr_finset s\n\n\/-- Union of an indexed family of sets `s : \u03b9 \u2192 set \u03b1` is equal to the union of the unions\nof finite subfamilies. This version works for `\u03b9 : Sort*`. See also `Union_eq_Union_finset` for\na version that assumes `\u03b9 : Type*` but avoids `plift`s in the right hand side. -\/\nlemma Union_eq_Union_finset' (s : \u03b9' \u2192 set \u03b1) :\n  (\u22c3i, s i) = (\u22c3t:finset (plift \u03b9'), \u22c3i\u2208t, s (plift.down i)) :=\nsupr_eq_supr_finset' s\n\n\/-- Intersection of an indexed family of sets `s : \u03b9 \u2192 set \u03b1` is equal to the intersection of the\nintersections of finite subfamilies. This version assumes `\u03b9 : Type*`. See also\n`Inter_eq_Inter_finset'` for a version that works for `\u03b9 : Sort*`. -\/\nlemma Inter_eq_Inter_finset (s : \u03b9 \u2192 set \u03b1) :\n  (\u22c2i, s i) = (\u22c2t:finset \u03b9, \u22c2i\u2208t, s i) :=\ninfi_eq_infi_finset s\n\n\/-- Intersection of an indexed family of sets `s : \u03b9 \u2192 set \u03b1` is equal to the intersection of the\nintersections of finite subfamilies. This version works for `\u03b9 : Sort*`. See also\n`Inter_eq_Inter_finset` for a version that assumes `\u03b9 : Type*` but avoids `plift`s in the right\nhand side. -\/\nlemma Inter_eq_Inter_finset' (s : \u03b9' \u2192 set \u03b1) :\n  (\u22c2i, s i) = (\u22c2t:finset (plift \u03b9'), \u22c2i\u2208t, s (plift.down i)) :=\ninfi_eq_infi_finset' s\n\nend set\n\nnamespace finset\n\nopen function\n\n\/-! ### Interaction with big lattice\/set operations -\/\n\nsection lattice\n\nlemma supr_coe [has_Sup \u03b2] (f : \u03b1 \u2192 \u03b2) (s : finset \u03b1) :\n  (\u2a06 x \u2208 (\u2191s : set \u03b1), f x) = \u2a06 x \u2208 s, f x :=\nrfl\n\nlemma infi_coe [has_Inf \u03b2] (f : \u03b1 \u2192 \u03b2) (s : finset \u03b1) :\n  (\u2a05 x \u2208 (\u2191s : set \u03b1), f x) = \u2a05 x \u2208 s, f x :=\nrfl\n\nvariables [complete_lattice \u03b2]\n\ntheorem supr_singleton (a : \u03b1) (s : \u03b1 \u2192 \u03b2) : (\u2a06 x \u2208 ({a} : finset \u03b1), s x) = s a :=\nby simp\n\ntheorem infi_singleton (a : \u03b1) (s : \u03b1 \u2192 \u03b2) : (\u2a05 x \u2208 ({a} : finset \u03b1), s x) = s a :=\nby simp\n\nlemma supr_option_to_finset (o : option \u03b1) (f : \u03b1 \u2192 \u03b2) :\n  (\u2a06 x \u2208 o.to_finset, f x) = \u2a06 x \u2208 o, f x :=\nby simp\n\nlemma infi_option_to_finset (o : option \u03b1) (f : \u03b1 \u2192 \u03b2) :\n  (\u2a05 x \u2208 o.to_finset, f x) = \u2a05 x \u2208 o, f x :=\n@supr_option_to_finset _ (order_dual \u03b2) _ _ _\n\nvariables [decidable_eq \u03b1]\n\ntheorem supr_union {f : \u03b1 \u2192 \u03b2} {s t : finset \u03b1} :\n  (\u2a06 x \u2208 s \u222a t, f x) = (\u2a06x\u2208s, f x) \u2294 (\u2a06x\u2208t, f x) :=\nby simp [supr_or, supr_sup_eq]\n\ntheorem infi_union {f : \u03b1 \u2192 \u03b2} {s t : finset \u03b1} :\n  (\u2a05 x \u2208 s \u222a t, f x) = (\u2a05 x \u2208 s, f x) \u2293 (\u2a05 x \u2208 t, f x) :=\n@supr_union \u03b1 (order_dual \u03b2) _ _ _ _ _\n\nlemma supr_insert (a : \u03b1) (s : finset \u03b1) (t : \u03b1 \u2192 \u03b2) :\n  (\u2a06 x \u2208 insert a s, t x) = t a \u2294 (\u2a06 x \u2208 s, t x) :=\nby { rw insert_eq, simp only [supr_union, finset.supr_singleton] }\n\nlemma infi_insert (a : \u03b1) (s : finset \u03b1) (t : \u03b1 \u2192 \u03b2) :\n  (\u2a05 x \u2208 insert a s, t x) = t a \u2293 (\u2a05 x \u2208 s, t x) :=\n@supr_insert \u03b1 (order_dual \u03b2)  _ _ _ _ _\n\nlemma supr_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 \u03b2} {s : finset \u03b3} :\n  (\u2a06 x \u2208 s.image f, g x) = (\u2a06 y \u2208 s, g (f y)) :=\nby rw [\u2190 supr_coe, coe_image, supr_image, supr_coe]\n\nlemma sup_finset_image {\u03b2 \u03b3 : Type*} [semilattice_sup_bot \u03b2]\n  (f : \u03b3 \u2192 \u03b1) (g : \u03b1 \u2192 \u03b2) (s : finset \u03b3) :\n  (s.image f).sup g = s.sup (g \u2218 f) :=\nbegin\n  classical,\n  induction s using finset.induction_on with a s' ha ih; simp *\nend\n\nlemma infi_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 \u03b2} {s : finset \u03b3} :\n  (\u2a05 x \u2208 s.image f, g x) = (\u2a05 y \u2208 s, g (f y)) :=\nby rw [\u2190 infi_coe, coe_image, infi_image, infi_coe]\n\nlemma supr_insert_update {x : \u03b1} {t : finset \u03b1} (f : \u03b1 \u2192 \u03b2) {s : \u03b2} (hx : x \u2209 t) :\n  (\u2a06 (i \u2208 insert x t), function.update f x s i) = (s \u2294 \u2a06 (i \u2208 t), f i) :=\nbegin\n  simp only [finset.supr_insert, update_same],\n  rcongr i hi, apply update_noteq, rintro rfl, exact hx hi\nend\n\nlemma infi_insert_update {x : \u03b1} {t : finset \u03b1} (f : \u03b1 \u2192 \u03b2) {s : \u03b2} (hx : x \u2209 t) :\n  (\u2a05 (i \u2208 insert x t), update f x s i) = (s \u2293 \u2a05 (i \u2208 t), f i) :=\n@supr_insert_update \u03b1 (order_dual \u03b2) _ _ _ _ f _ hx\n\nlemma supr_bUnion (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n  (\u2a06 y \u2208 s.bUnion t, f y) = \u2a06 (x \u2208 s) (y \u2208 t x), f y :=\nby simp [@supr_comm _ \u03b1, supr_and]\n\nlemma infi_bUnion (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n  (\u2a05 y \u2208 s.bUnion t, f y) = \u2a05 (x \u2208 s) (y \u2208 t x), f y :=\n@supr_bUnion _ (order_dual \u03b2) _ _ _ _ _ _\n\nend lattice\n\ntheorem set_bUnion_coe (s : finset \u03b1) (t : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 x \u2208 (\u2191s : set \u03b1), t x) = \u22c3 x \u2208 s, t x :=\nrfl\n\ntheorem set_bInter_coe (s : finset \u03b1) (t : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 x \u2208 (\u2191s : set \u03b1), t x) = \u22c2 x \u2208 s, t x :=\nrfl\n\ntheorem set_bUnion_singleton (a : \u03b1) (s : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 x \u2208 ({a} : finset \u03b1), s x) = s a :=\nsupr_singleton a s\n\ntheorem set_bInter_singleton (a : \u03b1) (s : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 x \u2208 ({a} : finset \u03b1), s x) = s a :=\ninfi_singleton a s\n\n@[simp] lemma set_bUnion_preimage_singleton (f : \u03b1 \u2192 \u03b2) (s : finset \u03b2) :\n  (\u22c3 y \u2208 s, f \u207b\u00b9' {y}) = f \u207b\u00b9' s :=\nset.bUnion_preimage_singleton f s\n\nlemma set_bUnion_option_to_finset (o : option \u03b1) (f : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 x \u2208 o.to_finset, f x) = \u22c3 x \u2208 o, f x :=\nsupr_option_to_finset o f\n\nlemma set_bInter_option_to_finset (o : option \u03b1) (f : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 x \u2208 o.to_finset, f x) = \u22c2 x \u2208 o, f x :=\ninfi_option_to_finset o f\n\nlemma subset_set_bUnion_of_mem {s : finset \u03b1} {f : \u03b1 \u2192 set \u03b2} {x : \u03b1} (h : x \u2208 s) :\n  f x \u2286 \u22c3 (y \u2208 s), f y :=\nshow f x \u2264 (\u2a06 y \u2208 s, f y), from le_supr_of_le x $ le_supr _ h\n\nvariables [decidable_eq \u03b1]\n\nlemma set_bUnion_union (s t : finset \u03b1) (u : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 x \u2208 s \u222a t, u x) = (\u22c3 x \u2208 s, u x) \u222a (\u22c3 x \u2208 t, u x) :=\nsupr_union\n\nlemma set_bInter_inter (s t : finset \u03b1) (u : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 x \u2208 s \u222a t, u x) = (\u22c2 x \u2208 s, u x) \u2229 (\u22c2 x \u2208 t, u x) :=\ninfi_union\n\nlemma set_bUnion_insert (a : \u03b1) (s : finset \u03b1) (t : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 x \u2208 insert a s, t x) = t a \u222a (\u22c3 x \u2208 s, t x) :=\nsupr_insert a s t\n\nlemma set_bInter_insert (a : \u03b1) (s : finset \u03b1) (t : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 x \u2208 insert a s, t x) = t a \u2229 (\u22c2 x \u2208 s, t x) :=\ninfi_insert a s t\n\nlemma set_bUnion_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 set \u03b2} {s : finset \u03b3} :\n  (\u22c3x \u2208 s.image f, g x) = (\u22c3y \u2208 s, g (f y)) :=\nsupr_finset_image\n\nlemma set_bInter_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 set \u03b2} {s : finset \u03b3} :\n  (\u22c2 x \u2208 s.image f, g x) = (\u22c2 y \u2208 s, g (f y)) :=\ninfi_finset_image\n\nlemma set_bUnion_insert_update {x : \u03b1} {t : finset \u03b1} (f : \u03b1 \u2192 set \u03b2) {s : set \u03b2} (hx : x \u2209 t) :\n  (\u22c3 (i \u2208 insert x t), @update _ _ _ f x s i) = (s \u222a \u22c3 (i \u2208 t), f i) :=\nsupr_insert_update f hx\n\nlemma set_bInter_insert_update {x : \u03b1} {t : finset \u03b1} (f : \u03b1 \u2192 set \u03b2) {s : set \u03b2} (hx : x \u2209 t) :\n  (\u22c2 (i \u2208 insert x t), @update _ _ _ f x s i) = (s \u2229 \u22c2 (i \u2208 t), f i) :=\ninfi_insert_update f hx\n\nlemma set_bUnion_bUnion (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b1) (f : \u03b1 \u2192 set \u03b2) :\n  (\u22c3 y \u2208 s.bUnion t, f y) = \u22c3 (x \u2208 s) (y \u2208 t x), f y :=\nsupr_bUnion s t f\n\nlemma set_bInter_bUnion (s : finset \u03b3) (t : \u03b3 \u2192 finset \u03b1) (f : \u03b1 \u2192 set \u03b2) :\n  (\u22c2 y \u2208 s.bUnion t, f y) = \u22c2 (x \u2208 s) (y \u2208 t x), f y :=\ninfi_bUnion s t f\n\nend finset\n","avg_line_length":37.306038048,"max_line_length":100,"alphanum_fraction":0.5936190497}
-{"size":3485,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import algebra.algebra.basic tactic.ring\n\/-\nMia LaRusso mnl6sc 12\/08\/21\nRead, understand (collaborating if necessary) the material\nin chapter 17 of Jeremy Avigad's *Logic and Proof.* It's here:\nhttps:\/\/leanprover.github.io\/logic_and_proof\/the_natural_numbers_and_induction.html\n-\/\n\n\/-\nThe following problems are listed in the Problems section of \nthe chapter. \n\n#1. Solve probem #1, first sentence only.\nWrite the principle of complete induction using the notation of symbolic logic:\n\u2200 P, (P(0) \u2227 (\u2200 n', P(n') \u2192 P(n'+1)) \u2192 (\u2200 n, P n))\n\n#2. Solve at least one of #2 and #3. Give detailed but informal proofs. \n\nDetailed Informal Proof #2: \nWe first define a function that returns the sum\nof all natural numbers from 0 to n (the argument). \nLets call this function sum_n. \n\nUsing this function, sum_n, we can state that for\na given value n, sum_n n = n * (n + 1) * (1 + (2 * n)) \/ 6. \nor 6 * sum_n n = n * (n + 1) * (1 + (2 * n)). This is \nour proposition. \n\nNext we can define a function, P, that takes a value\nn, of type nat, and reduced to the proposition: \n6 * (sum_n n) = n * (n + 1) * (1 + (2 * n)).\n\nUsing induction on n, we will be proving that every\nvalue of type n has property P: (forall n, P n). \nWe can use this to conclude that for every nat, n, \n6 * (sum_n n) = n * (n + 1) * (1 + (2 * n)).\n\nBy using proof by induction on n, we have the base\ncase: proof of (P 0) which provides that sum_n 0 = 0 * (1) * (1).\n\nThe second case assumes an arbitrary n, lets call it n\u2019\nand proves that (P n\u2019) holds: sum_n n\u2019 = n\u2019 * (n\u2019 + 1) * (1 + (2 * n\u2019)) \/ 6.\n\nGiven these cases we have shown that if P holds \nfor n\u2019 then it holds for (n\u2019+1). With both cases \n(base & second) proved, we can finally apply induction \nto get a proof of forall n P n. (every value of type \nn has property P). QED.\n-\/\n\n\/-\nTo test out of the final exam ...\n***\n#1: Give a formal proof for #2 or #3.\n#2: Formal or detailed informal proofs 10-13\n#3 (Extra Credit): #5 or #9\n\n\n-\/\n\n--Exam Exemption--\n-- #1. Foraml proof of #2:\ndef sum_n : \u2115 \u2192 \u2115 \n| (nat.zero)    := nat.zero                \n| (nat.succ n')  := (sum_n n') + (n'.succ * n'.succ) \n\ndef P : \u2115 \u2192 Prop := \n  \u03bb n, 6 * sum_n n = n * (n + 1) * (1 + (2 * n))\n\ndef conjecture_p := \u2200 n, P n \n\ntheorem sum_n_opt : conjecture_p := \nbegin\n    unfold conjecture_p,\n    assume n,\n    unfold P,\n    induction n with n' ih_n',\n    apply rfl,\n    simp [sum_n],\n    rw mul_add,\n    rw ih_n',\n    rw nat.succ_eq_add_one,\n    ring,\nend\n\n-- #2 10-13:\n-- 10\ndef mult_one : \u2115 \u2192 \u2115 \u2192 \u2115 \n| q (nat.zero) := nat.zero\n| q (nat.succ n') := q * n' + q --(define multiplication: m * succ(n) = m * n + n)\n\ndef Q : \u2115 \u2192 Prop :=\n  \u03bb n, mult_one 1 n = n\n\ndef conjecture_q := \u2200 n, Q n \n\ntheorem mult_one_opt : conjecture_q :=\nbegin\n    unfold conjecture_q,\n    assume n,\n    unfold Q,\n    induction n,\n    apply rfl,\n    simp[mult_one],\nend\n-- 11\nexample : \u2200 (m n k : \u2115), m * (n + k) = m * n + m * k :=\nbegin\n    assume m n k,\n    induction m,\n    induction n,\n    induction k,\n    apply rfl,\n    rw mul_add,\n    rw nat.succ_eq_add_one,\n    ring,\n    ring,\nend\n-- 12\nexample : \u2200 (m n k: \u2115), m * (n * k) = (m * n) * k :=\nbegin\n    assume m n k,\n    induction m,\n    induction n,\n    induction k,\n    apply rfl,\n    exact k_ih,\n    rw nat.succ_eq_add_one,\n    ring,\n    ring,\nend\n-- 13\nexample : \u2200 (m n k: \u2115), m * n  = n * m :=\nbegin\n    assume m n k,\n    induction m,\n    induction n,\n    induction k,\n    apply rfl,\n    exact k_ih,\n    exact n_ih,\n    rw nat.succ_eq_add_one,\n    ring,\nend","avg_line_length":24.2013888889,"max_line_length":83,"alphanum_fraction":0.6086083214}
-{"size":19924,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau.\n-\/\n\nimport algebra.polynomial.big_operators\nimport field_theory.minimal_polynomial\nimport field_theory.splitting_field\n\n\/-!\n\n# Separable polynomials\n\nWe define a polynomial to be separable if it is coprime with its derivative. We prove basic\nproperties about separable polynomials here.\n\n## Main definitions\n\n* `polynomial.separable f`: a polynomial `f` is separable iff it is coprime with its derivative.\n* `polynomial.expand R p f`: expand the polynomial `f` with coefficients in a\n  commutative semiring `R` by a factor of p, so `expand R p (\u2211 a\u2099 x\u207f)` is `\u2211 a\u2099 x\u207f\u1d56`.\n* `polynomial.contract p f`: the opposite of `expand`, so it sends `\u2211 a\u2099 x\u207f\u1d56` to `\u2211 a\u2099 x\u207f`.\n\n-\/\n\nuniverses u v w\nopen_locale classical big_operators\nopen finset\n\nnamespace polynomial\n\nsection comm_semiring\n\nvariables {R : Type u} [comm_semiring R] {S : Type v} [comm_semiring S]\n\n\/-- A polynomial is separable iff it is coprime with its derivative. -\/\ndef separable (f : polynomial R) : Prop :=\nis_coprime f f.derivative\n\nlemma separable_def (f : polynomial R) :\n  f.separable \u2194 is_coprime f f.derivative :=\niff.rfl\n\nlemma separable_def' (f : polynomial R) :\n  f.separable \u2194 \u2203 a b : polynomial R, a * f + b * f.derivative = 1 :=\niff.rfl\n\nlemma separable_one : (1 : polynomial R).separable :=\nis_coprime_one_left\n\nlemma separable_X_add_C (a : R) : (X + C a).separable :=\nby { rw [separable_def, derivative_add, derivative_X, derivative_C, add_zero],\n  exact is_coprime_one_right }\n\nlemma separable_X : (X : polynomial R).separable :=\nby { rw [separable_def, derivative_X], exact is_coprime_one_right }\n\nlemma separable_C (r : R) : (C r).separable \u2194 is_unit r :=\nby rw [separable_def, derivative_C, is_coprime_zero_right, is_unit_C]\n\nlemma separable.of_mul_left {f g : polynomial R} (h : (f * g).separable) : f.separable :=\nbegin\n  have := h.of_mul_left_left, rw derivative_mul at this,\n  exact is_coprime.of_mul_right_left (is_coprime.of_add_mul_left_right this)\nend\n\nlemma separable.of_mul_right {f g : polynomial R} (h : (f * g).separable) : g.separable :=\nby { rw mul_comm at h, exact h.of_mul_left }\n\nlemma separable.of_dvd {f g : polynomial R} (hf : f.separable) (hfg : g \u2223 f) : g.separable :=\nby { rcases hfg with \u27e8f', rfl\u27e9, exact separable.of_mul_left hf }\n\nlemma separable.is_coprime {f g : polynomial R} (h : (f * g).separable) : is_coprime f g :=\nbegin\n  have := h.of_mul_left_left, rw derivative_mul at this,\n  exact is_coprime.of_mul_right_right (is_coprime.of_add_mul_left_right this)\nend\n\ntheorem separable.of_pow' {f : polynomial R} :\n  \u2200 {n : \u2115} (h : (f ^ n).separable), is_unit f \u2228 (f.separable \u2227 n = 1) \u2228 n = 0\n| 0     := \u03bb h, or.inr $ or.inr rfl\n| 1     := \u03bb h, or.inr $ or.inl \u27e8pow_one f \u25b8 h, rfl\u27e9\n| (n+2) := \u03bb h, or.inl $ is_coprime_self.1 h.is_coprime.of_mul_right_left\n\ntheorem separable.of_pow {f : polynomial R} (hf : \u00acis_unit f) {n : \u2115} (hn : n \u2260 0)\n  (hfs : (f ^ n).separable) : f.separable \u2227 n = 1 :=\n(hfs.of_pow'.resolve_left hf).resolve_right hn\n\ntheorem separable.map {p : polynomial R} (h : p.separable) {f : R \u2192+* S} : (p.map f).separable :=\nlet \u27e8a, b, H\u27e9 := h in \u27e8a.map f, b.map f,\nby rw [derivative_map, \u2190 map_mul, \u2190 map_mul, \u2190 map_add, H, map_one]\u27e9\n\nvariables (R) (p q : \u2115)\n\n\/-- Expand the polynomial by a factor of p, so `\u2211 a\u2099 x\u207f` becomes `\u2211 a\u2099 x\u207f\u1d56`. -\/\nnoncomputable def expand : polynomial R \u2192\u2090[R] polynomial R :=\n{ commutes' := \u03bb r, eval\u2082_C _ _,\n  .. (eval\u2082_ring_hom C (X ^ p) : polynomial R \u2192+* polynomial R) }\n\nlemma coe_expand : (expand R p : polynomial R \u2192 polynomial R) = eval\u2082 C (X ^ p) := rfl\n\nvariables {R}\n\nlemma expand_eq_sum {f : polynomial R} :\n  expand R p f = f.sum (\u03bb e a, C a * (X ^ p) ^ e) :=\nby { dsimp [expand, eval\u2082], refl, }\n\n@[simp] lemma expand_C (r : R) : expand R p (C r) = C r := eval\u2082_C _ _\n@[simp] lemma expand_X : expand R p X = X ^ p := eval\u2082_X _ _\n@[simp] lemma expand_monomial (r : R) : expand R p (monomial q r) = monomial (q * p) r :=\nby simp_rw [monomial_eq_smul_X, alg_hom.map_smul, alg_hom.map_pow, expand_X, mul_comm, pow_mul]\n\ntheorem expand_expand (f : polynomial R) : expand R p (expand R q f) = expand R (p * q) f :=\npolynomial.induction_on f (\u03bb r, by simp_rw expand_C)\n  (\u03bb f g ihf ihg, by simp_rw [alg_hom.map_add, ihf, ihg])\n  (\u03bb n r ih, by simp_rw [alg_hom.map_mul, expand_C, alg_hom.map_pow, expand_X,\n    alg_hom.map_pow, expand_X, pow_mul])\n\ntheorem expand_mul (f : polynomial R) : expand R (p * q) f = expand R p (expand R q f) :=\n(expand_expand p q f).symm\n\n@[simp] theorem expand_one (f : polynomial R) : expand R 1 f = f :=\npolynomial.induction_on f\n  (\u03bb r, by rw expand_C)\n  (\u03bb f g ihf ihg, by rw [alg_hom.map_add, ihf, ihg])\n  (\u03bb n r ih, by rw [alg_hom.map_mul, expand_C, alg_hom.map_pow, expand_X, pow_one])\n\ntheorem expand_pow (f : polynomial R) : expand R (p ^ q) f = (expand R p ^[q] f) :=\nnat.rec_on q (by rw [pow_zero, expand_one, function.iterate_zero, id]) $ \u03bb n ih,\nby rw [function.iterate_succ_apply', pow_succ, expand_mul, ih]\n\ntheorem derivative_expand (f : polynomial R) :\n  (expand R p f).derivative = expand R p f.derivative * (p * X ^ (p - 1)) :=\nby rw [coe_expand, derivative_eval\u2082_C, derivative_pow, derivative_X, mul_one]\n\ntheorem coeff_expand {p : \u2115} (hp : 0 < p) (f : polynomial R) (n : \u2115) :\n  (expand R p f).coeff n = if p \u2223 n then f.coeff (n \/ p) else 0 :=\nbegin\n  simp only [expand_eq_sum],\n  change (show \u2115 \u2192\u2080 R, from (f.sum (\u03bb e a, C a * (X ^ p) ^ e) : polynomial R)) n = _,\n  simp_rw [finsupp.sum_apply, finsupp.sum, \u2190 pow_mul, C_mul', \u2190 monomial_eq_smul_X,\n    monomial, finsupp.single_apply],\n  split_ifs with h,\n  { rw [finset.sum_eq_single (n\/p), nat.mul_div_cancel' h, if_pos rfl], refl,\n    { intros b hb1 hb2, rw if_neg, intro hb3, apply hb2, rw [\u2190 hb3, nat.mul_div_cancel_left b hp] },\n    { intro hn, rw finsupp.not_mem_support_iff.1 hn, split_ifs; refl } },\n  { rw finset.sum_eq_zero, intros k hk, rw if_neg, exact \u03bb hkn, h \u27e8k, hkn.symm\u27e9, },\nend\n\n@[simp] theorem coeff_expand_mul {p : \u2115} (hp : 0 < p) (f : polynomial R) (n : \u2115) :\n  (expand R p f).coeff (n * p) = f.coeff n :=\nby rw [coeff_expand hp, if_pos (dvd_mul_left _ _), nat.mul_div_cancel _ hp]\n\n@[simp] theorem coeff_expand_mul' {p : \u2115} (hp : 0 < p) (f : polynomial R) (n : \u2115) :\n  (expand R p f).coeff (p * n) = f.coeff n :=\nby rw [mul_comm, coeff_expand_mul hp]\n\ntheorem expand_eq_map_domain (p : \u2115) (f : polynomial R) :\n  expand R p f = f.map_domain (*p) :=\nfinsupp.induction f (by { simp only [expand_eq_sum], refl }) $ \u03bb n r f hf hr ih,\nby rw [finsupp.map_domain_add, finsupp.map_domain_single, alg_hom.map_add, \u2190 monomial,\n  expand_monomial, \u2190 monomial, ih]\n\ntheorem expand_inj {p : \u2115} (hp : 0 < p) {f g : polynomial R} :\n  expand R p f = expand R p g \u2194 f = g :=\n\u27e8\u03bb H, ext $ \u03bb n, by rw [\u2190 coeff_expand_mul hp, H, coeff_expand_mul hp], congr_arg _\u27e9\n\ntheorem expand_eq_zero {p : \u2115} (hp : 0 < p) {f : polynomial R} : expand R p f = 0 \u2194 f = 0 :=\nby rw [\u2190 (expand R p).map_zero, expand_inj hp, alg_hom.map_zero]\n\ntheorem expand_eq_C {p : \u2115} (hp : 0 < p) {f : polynomial R} {r : R} :\n  expand R p f = C r \u2194 f = C r :=\nby rw [\u2190 expand_C, expand_inj hp, expand_C]\n\ntheorem nat_degree_expand (p : \u2115) (f : polynomial R) :\n  (expand R p f).nat_degree = f.nat_degree * p :=\nbegin\n  cases p.eq_zero_or_pos with hp hp,\n  { rw [hp, coe_expand, pow_zero, mul_zero, \u2190 C_1, eval\u2082_hom, nat_degree_C] },\n  by_cases hf : f = 0,\n  { rw [hf, alg_hom.map_zero, nat_degree_zero, zero_mul] },\n  have hf1 : expand R p f \u2260 0 := mt (expand_eq_zero hp).1 hf,\n  rw [\u2190 with_bot.coe_eq_coe, \u2190 degree_eq_nat_degree hf1],\n  refine le_antisymm ((degree_le_iff_coeff_zero _ _).2 $ \u03bb n hn, _) _,\n  { rw coeff_expand hp, split_ifs with hpn,\n    { rw coeff_eq_zero_of_nat_degree_lt, contrapose! hn,\n      rw [with_bot.coe_le_coe, \u2190 nat.div_mul_cancel hpn], exact nat.mul_le_mul_right p hn },\n    { refl } },\n  { refine le_degree_of_ne_zero _,\n    rw [coeff_expand_mul hp, \u2190 leading_coeff], exact mt leading_coeff_eq_zero.1 hf }\nend\n\ntheorem map_expand {p : \u2115} (hp : 0 < p) {f : R \u2192+* S} {q : polynomial R} :\n  map f (expand R p q) = expand S p (map f q) :=\nby { ext, rw [coeff_map, coeff_expand hp, coeff_expand hp], split_ifs; simp, }\n\nend comm_semiring\n\nsection comm_ring\n\nvariables {R : Type u} [comm_ring R]\n\nlemma separable_X_sub_C {x : R} : separable (X - C x) :=\nby simpa only [C_neg] using separable_X_add_C (-x)\n\nlemma separable.mul {f g : polynomial R} (hf : f.separable) (hg : g.separable)\n  (h : is_coprime f g) : (f * g).separable :=\nby { rw [separable_def, derivative_mul], exact ((hf.mul_right h).add_mul_left_right _).mul_left\n  ((h.symm.mul_right hg).mul_add_right_right _) }\n\nlemma separable_prod' {\u03b9 : Sort*} {f : \u03b9 \u2192 polynomial R} {s : finset \u03b9} :\n  (\u2200x\u2208s, \u2200y\u2208s, x \u2260 y \u2192 is_coprime (f x) (f y)) \u2192 (\u2200x\u2208s, (f x).separable) \u2192 (\u220f x in s, f x).separable :=\nfinset.induction_on s (\u03bb _ _, separable_one) $ \u03bb a s has ih h1 h2, begin\n  simp_rw [finset.forall_mem_insert, forall_and_distrib] at h1 h2, rw prod_insert has,\n  exact h2.1.mul (ih h1.2.2 h2.2) (is_coprime.prod_right $ \u03bb i his, h1.1.2 i his $\n    ne.symm $ ne_of_mem_of_not_mem his has)\nend\n\nlemma separable_prod {\u03b9 : Sort*} [fintype \u03b9] {f : \u03b9 \u2192 polynomial R}\n  (h1 : pairwise (is_coprime on f)) (h2 : \u2200 x, (f x).separable) : (\u220f x, f x).separable :=\nseparable_prod' (\u03bb x hx y hy hxy, h1 x y hxy) (\u03bb x hx, h2 x)\n\nlemma separable.inj_of_prod_X_sub_C [nontrivial R] {\u03b9 : Sort*} {f : \u03b9 \u2192 R} {s : finset \u03b9}\n  (hfs : (\u220f i in s, (X - C (f i))).separable)\n  {x y : \u03b9} (hx : x \u2208 s) (hy : y \u2208 s) (hfxy : f x = f y) : x = y :=\nbegin\n  by_contra hxy,\n  rw [\u2190 insert_erase hx, prod_insert (not_mem_erase _ _),\n      \u2190 insert_erase (mem_erase_of_ne_of_mem (ne.symm hxy) hy),\n      prod_insert (not_mem_erase _ _), \u2190 mul_assoc, hfxy, \u2190 pow_two] at hfs,\n  cases (hfs.of_mul_left.of_pow (by exact not_is_unit_X_sub_C) two_ne_zero).2\nend\n\nlemma separable.injective_of_prod_X_sub_C [nontrivial R] {\u03b9 : Sort*} [fintype \u03b9] {f : \u03b9 \u2192 R}\n  (hfs : (\u220f i, (X - C (f i))).separable) : function.injective f :=\n\u03bb x y hfxy, hfs.inj_of_prod_X_sub_C (mem_univ _) (mem_univ _) hfxy\n\nlemma is_unit_of_self_mul_dvd_separable {p q : polynomial R}\n  (hp : p.separable) (hq : q * q \u2223 p) : is_unit q :=\nbegin\n  obtain \u27e8p, rfl\u27e9 := hq,\n  apply is_coprime_self.mp,\n  have : is_coprime (q * (q * p)) (q * (q.derivative * p + q.derivative * p + q * p.derivative)),\n  { simp only [\u2190 mul_assoc, mul_add],\n    convert hp,\n    rw [derivative_mul, derivative_mul],\n    ring },\n  exact is_coprime.of_mul_right_left (is_coprime.of_mul_left_left this)\nend\n\nend comm_ring\n\nsection integral_domain\n\nvariables (R : Type u) [integral_domain R]\n\ntheorem is_local_ring_hom_expand {p : \u2115} (hp : 0 < p) :\n  is_local_ring_hom (\u2191(expand R p) : polynomial R \u2192+* polynomial R) :=\nbegin\n  refine \u27e8\u03bb f hf1, _\u27e9, rw \u2190 coe_fn_coe_base at hf1,\n  have hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_is_unit hf1),\n  rw [coeff_expand hp, if_pos (dvd_zero _), p.zero_div] at hf2,\n  rw [hf2, is_unit_C] at hf1, rw expand_eq_C hp at hf2, rwa [hf2, is_unit_C]\nend\n\nend integral_domain\n\nsection field\n\nvariables {F : Type u} [field F] {K : Type v} [field K]\n\ntheorem separable_iff_derivative_ne_zero {f : polynomial F} (hf : irreducible f) :\n  f.separable \u2194 f.derivative \u2260 0 :=\n\u27e8\u03bb h1 h2, hf.1 $ is_coprime_zero_right.1 $ h2 \u25b8 h1,\n\u03bb h, is_coprime_of_dvd (mt and.right h) $ \u03bb g hg1 hg2 \u27e8p, hg3\u27e9 hg4,\nlet \u27e8u, hu\u27e9 := (hf.2 _ _ hg3).resolve_left hg1 in\nhave f \u2223 f.derivative, by { conv_lhs { rw [hg3, \u2190 hu] }, rwa units.mul_right_dvd },\nnot_lt_of_le (nat_degree_le_of_dvd this h) $ nat_degree_derivative_lt h\u27e9\n\ntheorem separable_map (f : F \u2192+* K) {p : polynomial F} : (p.map f).separable \u2194 p.separable :=\nby simp_rw [separable_def, derivative_map, is_coprime_map]\n\nsection char_p\n\nvariables (p : \u2115) [hp : fact p.prime]\ninclude hp\n\n\/-- The opposite of `expand`: sends `\u2211 a\u2099 x\u207f\u1d56` to `\u2211 a\u2099 x\u207f`. -\/\nnoncomputable def contract (f : polynomial F) : polynomial F :=\n\u27e8f.support.preimage (*p) $ \u03bb _ _ _ _, (nat.mul_left_inj hp.pos).1,\n\u03bb n, f.coeff (n * p),\n\u03bb n, by { rw [finset.mem_preimage, finsupp.mem_support_iff], refl }\u27e9\n\ntheorem coeff_contract (f : polynomial F) (n : \u2115) : (contract p f).coeff n = f.coeff (n * p) := rfl\n\ntheorem of_irreducible_expand {f : polynomial F} (hf : irreducible (expand F p f)) :\n  irreducible f :=\n@@of_irreducible_map _ _ _ (is_local_ring_hom_expand F hp.pos) hf\n\ntheorem of_irreducible_expand_pow {f : polynomial F} {n : \u2115} :\n  irreducible (expand F (p ^ n) f) \u2192 irreducible f :=\nnat.rec_on n (\u03bb hf, by rwa [pow_zero, expand_one] at hf) $ \u03bb n ih hf,\nih $ of_irreducible_expand p $ by rwa [expand_expand]\n\nvariables [HF : char_p F p]\ninclude HF\n\ntheorem expand_char (f : polynomial F) :\n  map (frobenius F p) (expand F p f) = f ^ p :=\nbegin\n  refine f.induction_on' (\u03bb a b ha hb, _) (\u03bb n a, _),\n  { rw [alg_hom.map_add, map_add, ha, hb, add_pow_char], },\n  { rw [expand_monomial, map_monomial, single_eq_C_mul_X, single_eq_C_mul_X,\n        mul_pow, \u2190 C.map_pow, frobenius_def],\n    ring_exp }\nend\n\ntheorem map_expand_pow_char (f : polynomial F) (n : \u2115) :\n   map ((frobenius F p) ^ n) (expand F (p ^ n) f) = f ^ (p ^ n) :=\nbegin\n  induction n, {simp [ring_hom.one_def]},\n  symmetry,\n  rw [pow_succ', pow_mul, \u2190 n_ih, \u2190 expand_char, pow_succ, ring_hom.mul_def, \u2190 map_map, mul_comm,\n      expand_mul, \u2190 map_expand (nat.prime.pos hp)],\nend\n\ntheorem expand_contract {f : polynomial F} (hf : f.derivative = 0) :\n  expand F p (contract p f) = f :=\nbegin\n  ext n, rw [coeff_expand hp.pos, coeff_contract], split_ifs with h,\n  { rw nat.div_mul_cancel h },\n  { cases n, { exact absurd (dvd_zero p) h },\n    have := coeff_derivative f n, rw [hf, coeff_zero, zero_eq_mul] at this, cases this, { rw this },\n    rw [\u2190 nat.cast_succ, char_p.cast_eq_zero_iff F p] at this,\n    exact absurd this h }\nend\n\ntheorem separable_or {f : polynomial F} (hf : irreducible f) : f.separable \u2228\n  \u00acf.separable \u2227 \u2203 g : polynomial F, irreducible g \u2227 expand F p g = f :=\nif H : f.derivative = 0 then or.inr\n  \u27e8by rw [separable_iff_derivative_ne_zero hf, not_not, H],\n  contract p f,\n  by haveI := is_local_ring_hom_expand F hp.pos; exact\n    of_irreducible_map \u2191(expand F p) (by rwa \u2190 expand_contract p H at hf),\n  expand_contract p H\u27e9\nelse or.inl $ (separable_iff_derivative_ne_zero hf).2 H\n\ntheorem exists_separable_of_irreducible {f : polynomial F} (hf : irreducible f) (hf0 : f \u2260 0) :\n  \u2203 (n : \u2115) (g : polynomial F), g.separable \u2227 expand F (p ^ n) g = f :=\nbegin\n  generalize hn : f.nat_degree = N, unfreezingI { revert f },\n  apply nat.strong_induction_on N, intros N ih f hf hf0 hn,\n  rcases separable_or p hf with h | \u27e8h1, g, hg, hgf\u27e9,\n  { refine \u27e80, f, h, _\u27e9, rw [pow_zero, expand_one] },\n  { cases N with N,\n    { rw [nat_degree_eq_zero_iff_degree_le_zero, degree_le_zero_iff] at hn,\n      rw [hn, separable_C, is_unit_iff_ne_zero, not_not] at h1,\n      rw [h1, C_0] at hn, exact absurd hn hf0 },\n    have hg1 : g.nat_degree * p = N.succ,\n    { rwa [\u2190 nat_degree_expand, hgf] },\n    have hg2 : g.nat_degree \u2260 0,\n    { intro this, rw [this, zero_mul] at hg1, cases hg1 },\n    have hg3 : g.nat_degree < N.succ,\n    { rw [\u2190 mul_one g.nat_degree, \u2190 hg1],\n      exact nat.mul_lt_mul_of_pos_left hp.one_lt (nat.pos_of_ne_zero hg2) },\n    have hg4 : g \u2260 0,\n    { rintro rfl, exact hg2 nat_degree_zero },\n    rcases ih _ hg3 hg hg4 rfl with \u27e8n, g, hg5, rfl\u27e9, refine \u27e8n+1, g, hg5, _\u27e9,\n    rw [\u2190 hgf, expand_expand, pow_succ] }\nend\n\ntheorem is_unit_or_eq_zero_of_separable_expand {f : polynomial F} (n : \u2115)\n  (hf : (expand F (p ^ n) f).separable) : is_unit f \u2228 n = 0 :=\nbegin\n  rw or_iff_not_imp_right, intro hn,\n  have hf2 : (expand F (p ^ n) f).derivative = 0,\n  { by rw [derivative_expand, nat.cast_pow, char_p.cast_eq_zero,\n      zero_pow (nat.pos_of_ne_zero hn), zero_mul, mul_zero] },\n  rw [separable_def, hf2, is_coprime_zero_right, is_unit_iff] at hf, rcases hf with \u27e8r, hr, hrf\u27e9,\n  rw [eq_comm, expand_eq_C (pow_pos hp.pos _)] at hrf,\n  rwa [hrf, is_unit_C]\nend\n\ntheorem unique_separable_of_irreducible {f : polynomial F} (hf : irreducible f) (hf0 : f \u2260 0)\n  (n\u2081 : \u2115) (g\u2081 : polynomial F) (hg\u2081 : g\u2081.separable) (hgf\u2081 : expand F (p ^ n\u2081) g\u2081 = f)\n  (n\u2082 : \u2115) (g\u2082 : polynomial F) (hg\u2082 : g\u2082.separable) (hgf\u2082 : expand F (p ^ n\u2082) g\u2082 = f) :\n  n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082 :=\nbegin\n  revert g\u2081 g\u2082, wlog hn : n\u2081 \u2264 n\u2082 := le_total n\u2081 n\u2082 using [n\u2081 n\u2082, n\u2082 n\u2081] tactic.skip,\n  unfreezingI { intros, rw le_iff_exists_add at hn, rcases hn with \u27e8k, rfl\u27e9,\n    rw [\u2190 hgf\u2081, pow_add, expand_mul, expand_inj (pow_pos hp.pos n\u2081)] at hgf\u2082, subst hgf\u2082,\n    subst hgf\u2081,\n    rcases is_unit_or_eq_zero_of_separable_expand p k hg\u2081 with h | rfl,\n    { rw is_unit_iff at h, rcases h with \u27e8r, hr, rfl\u27e9,\n      simp_rw expand_C at hf, exact absurd (is_unit_C.2 hr) hf.1 },\n    { rw [add_zero, pow_zero, expand_one], split; refl } },\n  exact \u03bb g\u2081 g\u2082 hg\u2081 hgf\u2081 hg\u2082 hgf\u2082, let \u27e8hn, hg\u27e9 := this g\u2082 g\u2081 hg\u2082 hgf\u2082 hg\u2081 hgf\u2081 in \u27e8hn.symm, hg.symm\u27e9\nend\n\nend char_p\n\nlemma separable_prod_X_sub_C_iff' {\u03b9 : Sort*} {f : \u03b9 \u2192 F} {s : finset \u03b9} :\n  (\u220f i in s, (X - C (f i))).separable \u2194 (\u2200 (x \u2208 s) (y \u2208 s), f x = f y \u2192 x = y) :=\n\u27e8\u03bb hfs x hx y hy hfxy, hfs.inj_of_prod_X_sub_C hx hy hfxy,\n\u03bb H, by { rw \u2190 prod_attach, exact separable_prod' (\u03bb x hx y hy hxy,\n    @pairwise_coprime_X_sub _ _ { x \/\/ x \u2208 s } (\u03bb x, f x)\n      (\u03bb x y hxy, subtype.eq $ H x.1 x.2 y.1 y.2 hxy) _ _ hxy)\n  (\u03bb _ _, separable_X_sub_C) }\u27e9\n\nlemma separable_prod_X_sub_C_iff {\u03b9 : Sort*} [fintype \u03b9] {f : \u03b9 \u2192 F} :\n  (\u220f i, (X - C (f i))).separable \u2194 function.injective f :=\nseparable_prod_X_sub_C_iff'.trans $ by simp_rw [mem_univ, true_implies_iff]\n\nsection splits\n\nopen_locale big_operators\n\nvariables {i : F \u2192+* K}\n\nlemma not_unit_X_sub_C (a : F) : \u00ac is_unit (X - C a) :=\n\u03bb h, have one_eq_zero : (1 : with_bot \u2115) = 0, by simpa using degree_eq_zero_of_is_unit h,\none_ne_zero (option.some_injective _ one_eq_zero)\n\nlemma nodup_of_separable_prod {s : multiset F}\n  (hs : separable (multiset.map (\u03bb a, X - C a) s).prod) : s.nodup :=\nbegin\n  rw multiset.nodup_iff_ne_cons_cons,\n  rintros a t rfl,\n  refine not_unit_X_sub_C a (is_unit_of_self_mul_dvd_separable hs _),\n  simpa only [multiset.map_cons, multiset.prod_cons] using mul_dvd_mul_left _ (dvd_mul_right _ _)\nend\n\nlemma multiplicity_le_one_of_seperable {p q : polynomial F} (hq : \u00ac is_unit q)\n  (hsep : separable p) : multiplicity q p \u2264 1 :=\nbegin\n  contrapose! hq,\n  apply is_unit_of_self_mul_dvd_separable hsep,\n  rw \u2190 pow_two,\n  apply multiplicity.pow_dvd_of_le_multiplicity,\n  exact_mod_cast (enat.add_one_le_of_lt hq)\nend\n\nlemma root_multiplicity_le_one_of_seperable {p : polynomial F} (hp : p \u2260 0)\n  (hsep : separable p) (x : F) : root_multiplicity x p \u2264 1 :=\nbegin\n  rw [root_multiplicity_eq_multiplicity, dif_neg hp, \u2190 enat.coe_le_coe, enat.coe_get],\n  exact multiplicity_le_one_of_seperable (not_unit_X_sub_C _) hsep\nend\n\nlemma count_roots_le_one {p : polynomial F} (hsep : separable p) (x : F) :\n  p.roots.count x \u2264 1 :=\nbegin\n  by_cases hp : p = 0,\n  { simp [hp] },\n  rw count_roots hp,\n  exact root_multiplicity_le_one_of_seperable hp hsep x\nend\n\nlemma nodup_roots {p : polynomial F} (hsep : separable p) :\n  p.roots.nodup :=\nmultiset.nodup_iff_count_le_one.mpr (count_roots_le_one hsep)\n\nend splits\n\nend field\n\nend polynomial\n\nopen polynomial\n\ntheorem irreducible.separable {F : Type u} [field F] [char_zero F] {f : polynomial F}\n  (hf : irreducible f) (hf0 : f \u2260 0) : f.separable :=\nbegin\n  rw [separable_iff_derivative_ne_zero hf, ne, \u2190 degree_eq_bot, degree_derivative_eq], rintro \u27e8\u27e9,\n  rw [nat.pos_iff_ne_zero, ne, nat_degree_eq_zero_iff_degree_le_zero, degree_le_zero_iff],\n  refine \u03bb hf1, hf.1 _, rw [hf1, is_unit_C, is_unit_iff_ne_zero],\n  intro hf2, rw [hf2, C_0] at hf1, exact absurd hf1 hf0\nend\n\n\/-- Typeclass for separable field extension: `K` is a separable field extension of `F` iff\nthe minimal polynomial of every `x : K` is separable. -\/\n@[class] def is_separable (F K : Sort*) [field F] [field K] [algebra F K] : Prop :=\n\u2200 x : K, \u2203 H : is_integral F x, (minimal_polynomial H).separable\n","avg_line_length":41.0804123711,"max_line_length":103,"alphanum_fraction":0.6754667737}
-{"size":480,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"mutual def even, odd\nwith even : nat \u2192 bool\n| 0     := tt\n| (a+1) := odd a\nwith odd : nat \u2192 bool\n| 0     := ff\n| (a+1) := even a\nusing_well_founded {rel_tac := \u03bb f eqns, tactic.trace f >> tactic.trace eqns >> tactic.apply_instance}\n\nexample (a : nat) : even (a + 1) = odd a :=\nby simp [even]\n\nexample (a : nat) : odd (a + 1) = even a :=\nby simp [odd]\n\nlemma even_eq_not_odd : \u2200 a, even a = bnot (odd a) :=\nbegin\n  intro a, induction a,\n  simp [even, odd],\n  simph [even, odd]\nend\n","avg_line_length":21.8181818182,"max_line_length":102,"alphanum_fraction":0.5916666667}
-{"size":125,"ext":"lean","lang":"Lean","max_stars_count":2232.0,"content":"import system.io\n\nopen io\ndef tst : io unit :=\n   put_str \"hello\\n\"\n>> put_str \"world\\n\"\n>> put_str \"from Lean\\n\"\n\n#eval tst\n","avg_line_length":12.5,"max_line_length":24,"alphanum_fraction":0.648}
-{"size":15046,"ext":"lean","lang":"Lean","max_stars_count":2.0,"content":"\/-\nCopyright (c) 2020 Joseph Myers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Myers\n-\/\nimport linear_algebra.affine_space.independent\nimport linear_algebra.finite_dimensional\n\n\/-!\n# Finite-dimensional subspaces of affine spaces.\n\nThis file provides a few results relating to finite-dimensional\nsubspaces of affine spaces.\n\n## Main definitions\n\n* `collinear` defines collinear sets of points as those that span a\n  subspace of dimension at most 1.\n\n-\/\n\nnoncomputable theory\nopen_locale big_operators classical affine\n\nsection affine_space'\n\nvariables (k : Type*) {V : Type*} {P : Type*} [field k] [add_comm_group V] [module k V]\n          [affine_space V P]\nvariables {\u03b9 : Type*}\ninclude V\n\nopen affine_subspace finite_dimensional module\n\n\/-- The `vector_span` of a finite set is finite-dimensional. -\/\nlemma finite_dimensional_vector_span_of_finite {s : set P} (h : set.finite s) :\n  finite_dimensional k (vector_span k s) :=\nspan_of_finite k $ h.vsub h\n\n\/-- The `vector_span` of a family indexed by a `fintype` is\nfinite-dimensional. -\/\ninstance finite_dimensional_vector_span_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P) :\n  finite_dimensional k (vector_span k (set.range p)) :=\nfinite_dimensional_vector_span_of_finite k (set.finite_range _)\n\n\/-- The `vector_span` of a subset of a family indexed by a `fintype`\nis finite-dimensional. -\/\ninstance finite_dimensional_vector_span_image_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P)\n  (s : set \u03b9) : finite_dimensional k (vector_span k (p '' s)) :=\nfinite_dimensional_vector_span_of_finite k ((set.finite.of_fintype _).image _)\n\n\/-- The direction of the affine span of a finite set is\nfinite-dimensional. -\/\nlemma finite_dimensional_direction_affine_span_of_finite {s : set P} (h : set.finite s) :\n  finite_dimensional k (affine_span k s).direction :=\n(direction_affine_span k s).symm \u25b8 finite_dimensional_vector_span_of_finite k h\n\n\/-- The direction of the affine span of a family indexed by a\n`fintype` is finite-dimensional. -\/\ninstance finite_dimensional_direction_affine_span_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P) :\n  finite_dimensional k (affine_span k (set.range p)).direction :=\nfinite_dimensional_direction_affine_span_of_finite k (set.finite_range _)\n\n\/-- The direction of the affine span of a subset of a family indexed\nby a `fintype` is finite-dimensional. -\/\ninstance finite_dimensional_direction_affine_span_image_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P)\n  (s : set \u03b9) : finite_dimensional k (affine_span k (p '' s)).direction :=\nfinite_dimensional_direction_affine_span_of_finite k ((set.finite.of_fintype _).image _)\n\nvariables {k}\n\n\/-- The `vector_span` of a finite subset of an affinely independent\nfamily has dimension one less than its cardinality. -\/\nlemma finrank_vector_span_image_finset_of_affine_independent {p : \u03b9 \u2192 P}\n  (hi : affine_independent k p) {s : finset \u03b9} {n : \u2115} (hc : finset.card s = n + 1) :\n  finrank k (vector_span k (s.image p : set P)) = n :=\nbegin\n  have hi' := affine_independent_of_subset_affine_independent\n    (affine_independent_set_of_affine_independent hi) (set.image_subset_range p \u2191s),\n  have hc' : (s.image p).card = n + 1,\n  { rwa [s.card_image_of_injective (injective_of_affine_independent hi)] },\n  have hn : (s.image p).nonempty,\n  { simp [hc', \u2190finset.card_pos] },\n  rcases hn with \u27e8p\u2081, hp\u2081\u27e9,\n  have hp\u2081' : p\u2081 \u2208 p '' s := by simpa using hp\u2081,\n  rw [affine_independent_set_iff_linear_independent_vsub k hp\u2081', \u2190 finset.coe_singleton,\n      \u2190 finset.coe_image, \u2190 finset.coe_sdiff, finset.sdiff_singleton_eq_erase,\n      \u2190 finset.coe_image] at hi',\n  have hc : (finset.image (\u03bb (p : P), p -\u1d65 p\u2081) ((finset.image p s).erase p\u2081)).card = n,\n  { rw [finset.card_image_of_injective _ (vsub_left_injective _),\n        finset.card_erase_of_mem hp\u2081],\n    exact nat.pred_eq_of_eq_succ hc' },\n  rwa [vector_span_eq_span_vsub_finset_right_ne k hp\u2081, finrank_span_finset_eq_card, hc]\nend\n\n\/-- The `vector_span` of a finite affinely independent family has\ndimension one less than its cardinality. -\/\nlemma finrank_vector_span_of_affine_independent [fintype \u03b9] {p : \u03b9 \u2192 P}\n  (hi : affine_independent k p) {n : \u2115} (hc : fintype.card \u03b9 = n + 1) :\n  finrank k (vector_span k (set.range p)) = n :=\nbegin\n  rw \u2190 finset.card_univ at hc,\n  rw [\u2190 set.image_univ, \u2190 finset.coe_univ, \u2190 finset.coe_image],\n  exact finrank_vector_span_image_finset_of_affine_independent hi hc\nend\n\n\/-- If the `vector_span` of a finite subset of an affinely independent\nfamily lies in a submodule with dimension one less than its\ncardinality, it equals that submodule. -\/\nlemma vector_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {s : finset \u03b9} {sm : submodule k V}\n  [finite_dimensional k sm] (hle : vector_span k (s.image p : set P) \u2264 sm)\n  (hc : finset.card s = finrank k sm + 1) : vector_span k (s.image p : set P) = sm :=\neq_of_le_of_finrank_eq hle $ finrank_vector_span_image_finset_of_affine_independent hi hc\n\n\/-- If the `vector_span` of a finite affinely independent\nfamily lies in a submodule with dimension one less than its\ncardinality, it equals that submodule. -\/\nlemma vector_span_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one [fintype \u03b9]\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {sm : submodule k V} [finite_dimensional k sm]\n  (hle : vector_span k (set.range p) \u2264 sm) (hc : fintype.card \u03b9 = finrank k sm + 1) :\n  vector_span k (set.range p) = sm :=\neq_of_le_of_finrank_eq hle $ finrank_vector_span_of_affine_independent hi hc\n\n\/-- If the `affine_span` of a finite subset of an affinely independent\nfamily lies in an affine subspace whose direction has dimension one\nless than its cardinality, it equals that subspace. -\/\nlemma affine_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {s : finset \u03b9} {sp : affine_subspace k P}\n  [finite_dimensional k sp.direction] (hle : affine_span k (s.image p : set P) \u2264 sp)\n  (hc : finset.card s = finrank k sp.direction + 1) : affine_span k (s.image p : set P) = sp :=\nbegin\n  have hn : (s.image p).nonempty,\n  { rw [finset.nonempty.image_iff, \u2190 finset.card_pos, hc], apply nat.succ_pos },\n  refine eq_of_direction_eq_of_nonempty_of_le _ ((affine_span_nonempty k _).2 hn) hle,\n  have hd := direction_le hle,\n  rw direction_affine_span at \u22a2 hd,\n  exact vector_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one hi hd hc\nend\n\n\/-- If the `affine_span` of a finite affinely independent family lies\nin an affine subspace whose direction has dimension one less than its\ncardinality, it equals that subspace. -\/\nlemma affine_span_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one [fintype \u03b9]\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {sp : affine_subspace k P}\n  [finite_dimensional k sp.direction] (hle : affine_span k (set.range p) \u2264 sp)\n  (hc : fintype.card \u03b9 = finrank k sp.direction + 1) : affine_span k (set.range p) = sp :=\nbegin\n  rw \u2190finset.card_univ at hc,\n  rw [\u2190set.image_univ, \u2190finset.coe_univ, \u2190 finset.coe_image] at \u22a2 hle,\n  exact affine_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one hi hle hc\nend\n\n\/-- The `vector_span` of a finite affinely independent family whose\ncardinality is one more than that of the finite-dimensional space is\n`\u22a4`. -\/\nlemma vector_span_eq_top_of_affine_independent_of_card_eq_finrank_add_one [finite_dimensional k V]\n  [fintype \u03b9] {p : \u03b9 \u2192 P} (hi : affine_independent k p) (hc : fintype.card \u03b9 = finrank k V + 1) :\n  vector_span k (set.range p) = \u22a4 :=\neq_top_of_finrank_eq $ finrank_vector_span_of_affine_independent hi hc\n\n\/-- The `affine_span` of a finite affinely independent family whose\ncardinality is one more than that of the finite-dimensional space is\n`\u22a4`. -\/\nlemma affine_span_eq_top_of_affine_independent_of_card_eq_finrank_add_one [finite_dimensional k V]\n  [fintype \u03b9] {p : \u03b9 \u2192 P} (hi : affine_independent k p) (hc : fintype.card \u03b9 = finrank k V + 1) :\n  affine_span k (set.range p) = \u22a4 :=\nbegin\n  rw [\u2190finrank_top, \u2190direction_top k V P] at hc,\n  exact affine_span_eq_of_le_of_affine_independent_of_card_eq_finrank_add_one hi le_top hc\nend\n\nvariables (k)\n\n\/-- The `vector_span` of `n + 1` points in an indexed family has\ndimension at most `n`. -\/\nlemma finrank_vector_span_image_finset_le (p : \u03b9 \u2192 P) (s : finset \u03b9) {n : \u2115}\n  (hc : finset.card s = n + 1) : finrank k (vector_span k (s.image p : set P)) \u2264 n :=\nbegin\n  have hn : (s.image p).nonempty,\n  { rw [finset.nonempty.image_iff, \u2190 finset.card_pos, hc], apply nat.succ_pos },\n  rcases hn with \u27e8p\u2081, hp\u2081\u27e9,\n  rw [vector_span_eq_span_vsub_finset_right_ne k hp\u2081],\n  refine le_trans (finrank_span_finset_le_card (((s.image p).erase p\u2081).image (\u03bb p, p -\u1d65 p\u2081))) _,\n  rw [finset.card_image_of_injective _ (vsub_left_injective p\u2081), finset.card_erase_of_mem hp\u2081,\n      nat.pred_le_iff, nat.succ_eq_add_one, \u2190 hc],\n  apply finset.card_image_le\nend\n\n\/-- The `vector_span` of an indexed family of `n + 1` points has\ndimension at most `n`. -\/\nlemma finrank_vector_span_range_le [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) : finrank k (vector_span k (set.range p)) \u2264 n :=\nbegin\n  rw [\u2190set.image_univ, \u2190finset.coe_univ, \u2190 finset.coe_image],\n  rw \u2190finset.card_univ at hc,\n  exact finrank_vector_span_image_finset_le _ _ _ hc\nend\n\n\/-- `n + 1` points are affinely independent if and only if their\n`vector_span` has dimension `n`. -\/\nlemma affine_independent_iff_finrank_vector_span_eq [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) :\n  affine_independent k p \u2194 finrank k (vector_span k (set.range p)) = n :=\nbegin\n  have hn : nonempty \u03b9, by simp [\u2190fintype.card_pos_iff, hc],\n  cases hn with i\u2081,\n  rw [affine_independent_iff_linear_independent_vsub _ _ i\u2081,\n      linear_independent_iff_card_eq_finrank_span, eq_comm,\n      vector_span_range_eq_span_range_vsub_right_ne k p i\u2081],\n  congr',\n  rw \u2190finset.card_univ at hc,\n  rw fintype.subtype_card,\n  simp [finset.filter_ne', finset.card_erase_of_mem, hc]\nend\n\n\/-- `n + 1` points are affinely independent if and only if their\n`vector_span` has dimension at least `n`. -\/\nlemma affine_independent_iff_le_finrank_vector_span [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) :\n  affine_independent k p \u2194 n \u2264 finrank k (vector_span k (set.range p)) :=\nbegin\n  rw affine_independent_iff_finrank_vector_span_eq k p hc,\n  split,\n  { rintro rfl,\n    refl },\n  { exact \u03bb hle, le_antisymm (finrank_vector_span_range_le k p hc) hle }\nend\n\n\/-- `n + 2` points are affinely independent if and only if their\n`vector_span` does not have dimension at most `n`. -\/\nlemma affine_independent_iff_not_finrank_vector_span_le [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 2) :\n  affine_independent k p \u2194 \u00ac finrank k (vector_span k (set.range p)) \u2264 n :=\nby rw [affine_independent_iff_le_finrank_vector_span k p hc, \u2190nat.lt_iff_add_one_le, lt_iff_not_ge]\n\n\/-- `n + 2` points have a `vector_span` with dimension at most `n` if\nand only if they are not affinely independent. -\/\nlemma finrank_vector_span_le_iff_not_affine_independent [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 2) :\n  finrank k (vector_span k (set.range p)) \u2264 n \u2194 \u00ac affine_independent k p :=\n(not_iff_comm.1 (affine_independent_iff_not_finrank_vector_span_le k p hc).symm).symm\n\n\/-- A set of points is collinear if their `vector_span` has dimension\nat most `1`. -\/\ndef collinear (s : set P) : Prop := module.rank k (vector_span k s) \u2264 1\n\n\/-- The definition of `collinear`. -\/\nlemma collinear_iff_dim_le_one (s : set P) : collinear k s \u2194 module.rank k (vector_span k s) \u2264 1 :=\niff.rfl\n\n\/-- A set of points, whose `vector_span` is finite-dimensional, is\ncollinear if and only if their `vector_span` has dimension at most\n`1`. -\/\nlemma collinear_iff_finrank_le_one (s : set P) [finite_dimensional k (vector_span k s)] :\n  collinear k s \u2194 finrank k (vector_span k s) \u2264 1 :=\nbegin\n  have h := collinear_iff_dim_le_one k s,\n  rw \u2190finrank_eq_dim at h,\n  exact_mod_cast h\nend\n\nvariables (P)\n\n\/-- The empty set is collinear. -\/\nlemma collinear_empty : collinear k (\u2205 : set P) :=\nbegin\n  rw [collinear_iff_dim_le_one, vector_span_empty],\n  simp\nend\n\nvariables {P}\n\n\/-- A single point is collinear. -\/\nlemma collinear_singleton (p : P) : collinear k ({p} : set P) :=\nbegin\n  rw [collinear_iff_dim_le_one, vector_span_singleton],\n  simp\nend\n\n\/-- Given a point `p\u2080` in a set of points, that set is collinear if and\nonly if the points can all be expressed as multiples of the same\nvector, added to `p\u2080`. -\/\nlemma collinear_iff_of_mem {s : set P} {p\u2080 : P} (h : p\u2080 \u2208 s) :\n  collinear k s \u2194 \u2203 v : V, \u2200 p \u2208 s, \u2203 r : k, p = r \u2022 v +\u1d65 p\u2080 :=\nbegin\n  simp_rw [collinear_iff_dim_le_one, dim_submodule_le_one_iff', submodule.le_span_singleton_iff],\n  split,\n  { rintro \u27e8v\u2080, hv\u27e9,\n    use v\u2080,\n    intros p hp,\n    obtain \u27e8r, hr\u27e9 := hv (p -\u1d65 p\u2080) (vsub_mem_vector_span k hp h),\n    use r,\n    rw eq_vadd_iff_vsub_eq,\n    exact hr.symm },\n  { rintro \u27e8v, hp\u2080v\u27e9,\n    use v,\n    intros w hw,\n    have hs : vector_span k s \u2264 k \u2219 v,\n    { rw [vector_span_eq_span_vsub_set_right k h, submodule.span_le, set.subset_def],\n      intros x hx,\n      rw [set_like.mem_coe, submodule.mem_span_singleton],\n      rw set.mem_image at hx,\n      rcases hx with \u27e8p, hp, rfl\u27e9,\n      rcases hp\u2080v p hp with \u27e8r, rfl\u27e9,\n      use r,\n      simp },\n    have hw' := set_like.le_def.1 hs hw,\n    rwa submodule.mem_span_singleton at hw' }\nend\n\n\/-- A set of points is collinear if and only if they can all be\nexpressed as multiples of the same vector, added to the same base\npoint. -\/\nlemma collinear_iff_exists_forall_eq_smul_vadd (s : set P) :\n  collinear k s \u2194 \u2203 (p\u2080 : P) (v : V), \u2200 p \u2208 s, \u2203 r : k, p = r \u2022 v +\u1d65 p\u2080 :=\nbegin\n  rcases set.eq_empty_or_nonempty s with rfl | \u27e8\u27e8p\u2081, hp\u2081\u27e9\u27e9,\n  { simp [collinear_empty] },\n  { rw collinear_iff_of_mem k hp\u2081,\n    split,\n    { exact \u03bb h, \u27e8p\u2081, h\u27e9 },\n    { rintros \u27e8p, v, hv\u27e9,\n      use v,\n      intros p\u2082 hp\u2082,\n      rcases hv p\u2082 hp\u2082 with \u27e8r, rfl\u27e9,\n      rcases hv p\u2081 hp\u2081 with \u27e8r\u2081, rfl\u27e9,\n      use r - r\u2081,\n      simp [vadd_vadd, \u2190add_smul] } }\nend\n\n\/-- Two points are collinear. -\/\nlemma collinear_insert_singleton (p\u2081 p\u2082 : P) : collinear k ({p\u2081, p\u2082} : set P) :=\nbegin\n  rw collinear_iff_exists_forall_eq_smul_vadd,\n  use [p\u2081, p\u2082 -\u1d65 p\u2081],\n  intros p hp,\n  rw [set.mem_insert_iff, set.mem_singleton_iff] at hp,\n  cases hp,\n  { use 0,\n    simp [hp] },\n  { use 1,\n    simp [hp] }\nend\n\n\/-- Three points are affinely independent if and only if they are not\ncollinear. -\/\nlemma affine_independent_iff_not_collinear (p : fin 3 \u2192 P) :\n  affine_independent k p \u2194 \u00ac collinear k (set.range p) :=\nby rw [collinear_iff_finrank_le_one,\n       affine_independent_iff_not_finrank_vector_span_le k p (fintype.card_fin 3)]\n\n\/-- Three points are collinear if and only if they are not affinely\nindependent. -\/\nlemma collinear_iff_not_affine_independent (p : fin 3 \u2192 P) :\n  collinear k (set.range p) \u2194 \u00ac affine_independent k p :=\nby rw [collinear_iff_finrank_le_one,\n       finrank_vector_span_le_iff_not_affine_independent k p (fintype.card_fin 3)]\n\nend affine_space'\n","avg_line_length":41.7944444444,"max_line_length":100,"alphanum_fraction":0.7239133324}
-{"size":90,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"inductive t | one | two\n\nexample (h : False) : t.one = t.two := by\n  simp\n  contradiction\n","avg_line_length":15.0,"max_line_length":41,"alphanum_fraction":0.6222222222}
-{"size":124907,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nMultisets.\n-\/\nimport logic.function order.boolean_algebra\n  data.equiv.basic data.list.basic data.list.perm data.list.sort data.quot data.string\n  algebra.order_functions algebra.group_power algebra.ordered_group\n  category.traversable.lemmas tactic.interactive\n  category.traversable.instances category.basic\n\nopen list subtype nat lattice\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n\nlocal infix ` \u2022 ` := add_monoid.smul\n\n\/-- `multiset \u03b1` is the quotient of `list \u03b1` by list permutation. The result\n  is a type of finite sets with duplicates allowed.  -\/\ndef {u} multiset (\u03b1 : Type u) : Type u :=\nquotient (list.is_setoid \u03b1)\n\nnamespace multiset\n\ninstance : has_coe (list \u03b1) (multiset \u03b1) := \u27e8quot.mk _\u27e9\n\n@[simp] theorem quot_mk_to_coe (l : list \u03b1) : @eq (multiset \u03b1) \u27e6l\u27e7 l := rfl\n\n@[simp] theorem quot_mk_to_coe' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk (\u2248) l) l := rfl\n\n@[simp] theorem quot_mk_to_coe'' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk setoid.r l) l := rfl\n\n@[simp] theorem coe_eq_coe {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) = l\u2082 \u2194 l\u2081 ~ l\u2082 := quotient.eq\n\ninstance has_decidable_eq [decidable_eq \u03b1] : decidable_eq (multiset \u03b1)\n| s\u2081 s\u2082 := quotient.rec_on_subsingleton\u2082 s\u2081 s\u2082 $ \u03bb l\u2081 l\u2082,\n  decidable_of_iff' _ quotient.eq\n\n\/- empty multiset -\/\n\n\/-- `0 : multiset \u03b1` is the empty set -\/\nprotected def zero : multiset \u03b1 := @nil \u03b1\n\ninstance : has_zero (multiset \u03b1)   := \u27e8multiset.zero\u27e9\ninstance : has_emptyc (multiset \u03b1) := \u27e80\u27e9\ninstance : inhabited (multiset \u03b1)  := \u27e80\u27e9\n\n@[simp] theorem coe_nil_eq_zero : (@nil \u03b1 : multiset \u03b1) = 0 := rfl\n@[simp] theorem empty_eq_zero : (\u2205 : multiset \u03b1) = 0 := rfl\n\ntheorem coe_eq_zero (l : list \u03b1) : (l : multiset \u03b1) = 0 \u2194 l = [] :=\niff.trans coe_eq_coe perm_nil\n\n\/- cons -\/\n\n\/-- `cons a s` is the multiset which contains `s` plus one more\n  instance of `a`. -\/\ndef cons (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (a :: l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound ((perm_cons a).2 p))\n\nnotation a :: b := cons a b\n\ninstance : has_insert \u03b1 (multiset \u03b1) := \u27e8cons\u27e9\n\n@[simp] theorem insert_eq_cons (a : \u03b1) (s : multiset \u03b1) :\n  insert a s = a::s := rfl\n\n@[simp] theorem cons_coe (a : \u03b1) (l : list \u03b1) :\n  (a::l : multiset \u03b1) = (a::l : list \u03b1) := rfl\n\ntheorem singleton_coe (a : \u03b1) : (a::0 : multiset \u03b1) = ([a] : list \u03b1) := rfl\n\n@[simp] theorem cons_inj_left {a b : \u03b1} (s : multiset \u03b1) :\n  a::s = b::s \u2194 a = b :=\n\u27e8quot.induction_on s $ \u03bb l e,\n  have [a] ++ l ~ [b] ++ l, from quotient.exact e,\n  eq_singleton_of_perm $ (perm_app_right_iff _).1 this, congr_arg _\u27e9\n\n@[simp] theorem cons_inj_right (a : \u03b1) : \u2200{s t : multiset \u03b1}, a::s = a::t \u2194 s = t :=\nby rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9; simp [perm_cons]\n\n@[recursor 5] protected theorem induction {p : multiset \u03b1 \u2192 Prop}\n  (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : \u2200s, p s :=\nby rintros \u27e8l\u27e9; induction l with _ _ ih; [exact h\u2081, exact h\u2082 ih]\n\n@[elab_as_eliminator] protected theorem induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : p s :=\nmultiset.induction h\u2081 h\u2082 s\n\ntheorem cons_swap (a b : \u03b1) (s : multiset \u03b1) : a :: b :: s = b :: a :: s :=\nquot.induction_on s $ \u03bb l, quotient.sound $ perm.swap _ _ _\n\nsection rec\nvariables {C : multiset \u03b1 \u2192 Sort*}\n\n\/-- Dependent recursor on multisets.\n\nTODO: should be @[recursor 6], but then the definition of `multiset.pi` failes with a stack\noverflow in `whnf`.\n-\/\nprotected def rec\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b))\n  (m : multiset \u03b1) : C m :=\nquotient.hrec_on m (@list.rec \u03b1 (\u03bbl, C \u27e6l\u27e7) C_0 (\u03bba l b, C_cons a \u27e6l\u27e7 b)) $\n  assume l l' h,\n  list.rec_heq_of_perm h\n    (assume a l l' b b' hl, have \u27e6l\u27e7 = \u27e6l'\u27e7, from quot.sound hl, by cc)\n    (assume a a' l, C_cons_heq a a' \u27e6l\u27e7)\n\n@[elab_as_eliminator]\nprotected def rec_on (m : multiset \u03b1)\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)) :\n  C m :=\nmultiset.rec C_0 C_cons C_cons_heq m\n\nvariables {C_0 : C 0} {C_cons : \u03a0a m, C m \u2192 C (a::m)}\n  {C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)}\n\n@[simp] lemma rec_on_0 : @multiset.rec_on \u03b1 C (0:multiset \u03b1) C_0 C_cons C_cons_heq = C_0 :=\nrfl\n\n@[simp] lemma rec_on_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).rec_on C_0 C_cons C_cons_heq = C_cons a m (m.rec_on C_0 C_cons C_cons_heq) :=\nquotient.induction_on m $ assume l, rfl\n\nend rec\n\nsection mem\n\n\/-- `a \u2208 s` means that `a` has nonzero multiplicity in `s`. -\/\ndef mem (a : \u03b1) (s : multiset \u03b1) : Prop :=\nquot.lift_on s (\u03bb l, a \u2208 l) (\u03bb l\u2081 l\u2082 (e : l\u2081 ~ l\u2082), propext $ mem_of_perm e)\n\ninstance : has_mem \u03b1 (multiset \u03b1) := \u27e8mem\u27e9\n\n@[simp] lemma mem_coe {a : \u03b1} {l : list \u03b1} : a \u2208 (l : multiset \u03b1) \u2194 a \u2208 l := iff.rfl\n\ninstance decidable_mem [decidable_eq \u03b1] (a : \u03b1) (s : multiset \u03b1) : decidable (a \u2208 s) :=\nquot.rec_on_subsingleton s $ list.decidable_mem a\n\n@[simp] theorem mem_cons {a b : \u03b1} {s : multiset \u03b1} : a \u2208 b :: s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, iff.rfl\n\nlemma mem_cons_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 b :: s :=\nmem_cons.2 $ or.inr h\n\n@[simp] theorem mem_cons_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 a :: s :=\nmem_cons.2 (or.inl rfl)\n\ntheorem exists_cons_of_mem {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 \u2203 t, s = a :: t :=\nquot.induction_on s $ \u03bb l (h : a \u2208 l),\nlet \u27e8l\u2081, l\u2082, e\u27e9 := mem_split h in\ne.symm \u25b8 \u27e8(l\u2081++l\u2082 : list \u03b1), quot.sound perm_middle\u27e9\n\n@[simp] theorem not_mem_zero (a : \u03b1) : a \u2209 (0 : multiset \u03b1) := id\n\ntheorem eq_zero_of_forall_not_mem {s : multiset \u03b1} : (\u2200x, x \u2209 s) \u2192 s = 0 :=\nquot.induction_on s $ \u03bb l H, by rw eq_nil_iff_forall_not_mem.mpr H; refl\n\ntheorem exists_mem_of_ne_zero {s : multiset \u03b1} : s \u2260 0 \u2192 \u2203 a : \u03b1, a \u2208 s :=\nquot.induction_on s $ assume l hl,\n  match l, hl with\n  | [] := assume h, false.elim $ h rfl\n  | (a :: l) := assume _, \u27e8a, by simp\u27e9\n  end\n\n@[simp] lemma zero_ne_cons {a : \u03b1} {m : multiset \u03b1} : 0 \u2260 a :: m :=\nassume h, have a \u2208 (0:multiset \u03b1), from h.symm \u25b8 mem_cons_self _ _, not_mem_zero _ this\n\n@[simp] lemma cons_ne_zero {a : \u03b1} {m : multiset \u03b1} : a :: m \u2260 0 := zero_ne_cons.symm\n\nlemma cons_eq_cons {a b : \u03b1} {as bs : multiset \u03b1} :\n  a :: as = b :: bs \u2194 ((a = b \u2227 as = bs) \u2228 (a \u2260 b \u2227 \u2203cs, as = b :: cs \u2227 bs = a :: cs)) :=\nbegin\n  haveI : decidable_eq \u03b1 := classical.dec_eq \u03b1,\n  split,\n  { assume eq,\n    by_cases a = b,\n    { subst h, simp * at * },\n    { have : a \u2208 b :: bs, from eq \u25b8 mem_cons_self _ _,\n      have : a \u2208 bs, by simpa [h],\n      rcases exists_cons_of_mem this with \u27e8cs, hcs\u27e9,\n      simp [h, hcs],\n      have : a :: as = b :: a :: cs, by simp [eq, hcs],\n      have : a :: as = a :: b :: cs, by rwa [cons_swap],\n      simpa using this } },\n  { assume h,\n    rcases h with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n    { simp * },\n    { simp [*, cons_swap a b] } }\nend\n\nend mem\n\n\/- subset -\/\nsection subset\n\n\/-- `s \u2286 t` is the lift of the list subset relation. It means that any\n  element with nonzero multiplicity in `s` has nonzero multiplicity in `t`,\n  but it does not imply that the multiplicity of `a` in `s` is less or equal than in `t`;\n  see `s \u2264 t` for this relation. -\/\nprotected def subset (s t : multiset \u03b1) : Prop := \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 a \u2208 t\n\ninstance : has_subset (multiset \u03b1) := \u27e8multiset.subset\u27e9\n\n@[simp] theorem coe_subset {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2286 l\u2082 \u2194 l\u2081 \u2286 l\u2082 := iff.rfl\n\n@[simp] theorem subset.refl (s : multiset \u03b1) : s \u2286 s := \u03bb a h, h\n\ntheorem subset.trans {s t u : multiset \u03b1} : s \u2286 t \u2192 t \u2286 u \u2192 s \u2286 u :=\n\u03bb h\u2081 h\u2082 a m, h\u2082 (h\u2081 m)\n\ntheorem subset_iff {s t : multiset \u03b1} : s \u2286 t \u2194 (\u2200\u2983x\u2984, x \u2208 s \u2192 x \u2208 t) := iff.rfl\n\ntheorem mem_of_subset {s t : multiset \u03b1} {a : \u03b1} (h : s \u2286 t) : a \u2208 s \u2192 a \u2208 t := @h _\n\n@[simp] theorem zero_subset (s : multiset \u03b1) : 0 \u2286 s :=\n\u03bb a, (not_mem_nil a).elim\n\n@[simp] theorem cons_subset {a : \u03b1} {s t : multiset \u03b1} : (a :: s) \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t :=\nby simp [subset_iff, or_imp_distrib, forall_and_distrib]\n\ntheorem eq_zero_of_subset_zero {s : multiset \u03b1} (h : s \u2286 0) : s = 0 :=\neq_zero_of_forall_not_mem h\n\ntheorem subset_zero {s : multiset \u03b1} : s \u2286 0 \u2194 s = 0 :=\n\u27e8eq_zero_of_subset_zero, \u03bb xeq, xeq.symm \u25b8 subset.refl 0\u27e9\n\nend subset\n\n\/- multiset order -\/\n\n\/-- `s \u2264 t` means that `s` is a sublist of `t` (up to permutation).\n  Equivalently, `s \u2264 t` means that `count a s \u2264 count a t` for all `a`. -\/\nprotected def le (s t : multiset \u03b1) : Prop :=\nquotient.lift_on\u2082 s t (<+~) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  propext (p\u2082.subperm_left.trans p\u2081.subperm_right)\n\ninstance : partial_order (multiset \u03b1) :=\n{ le          := multiset.le,\n  le_refl     := by rintros \u27e8l\u27e9; exact subperm.refl _,\n  le_trans    := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 \u27e8l\u2083\u27e9; exact @subperm.trans _ _ _ _,\n  le_antisymm := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 h\u2081 h\u2082; exact quot.sound (subperm.antisymm h\u2081 h\u2082) }\n\ntheorem subset_of_le {s t : multiset \u03b1} : s \u2264 t \u2192 s \u2286 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subset_of_subperm\n\ntheorem mem_of_le {s t : multiset \u03b1} {a : \u03b1} (h : s \u2264 t) : a \u2208 s \u2192 a \u2208 t :=\nmem_of_subset (subset_of_le h)\n\n@[simp] theorem coe_le {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2264 l\u2082 \u2194 l\u2081 <+~ l\u2082 := iff.rfl\n\n@[elab_as_eliminator] theorem le_induction_on {C : multiset \u03b1 \u2192 multiset \u03b1 \u2192 Prop}\n  {s t : multiset \u03b1} (h : s \u2264 t)\n  (H : \u2200 {l\u2081 l\u2082 : list \u03b1}, l\u2081 <+ l\u2082 \u2192 C l\u2081 l\u2082) : C s t :=\nquotient.induction_on\u2082 s t (\u03bb l\u2081 l\u2082 \u27e8l, p, s\u27e9,\n  (show \u27e6l\u27e7 = \u27e6l\u2081\u27e7, from quot.sound p) \u25b8 H s) h\n\ntheorem zero_le (s : multiset \u03b1) : 0 \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ nil_sublist l\n\ntheorem le_zero {s : multiset \u03b1} : s \u2264 0 \u2194 s = 0 :=\n\u27e8\u03bb h, le_antisymm h (zero_le _), le_of_eq\u27e9\n\ntheorem lt_cons_self (s : multiset \u03b1) (a : \u03b1) : s < a :: s :=\nquot.induction_on s $ \u03bb l,\nsuffices l <+~ a :: l \u2227 (\u00acl ~ a :: l),\n  by simpa [lt_iff_le_and_ne],\n\u27e8subperm_of_sublist (sublist_cons _ _),\n \u03bb p, ne_of_lt (lt_succ_self (length l)) (perm_length p)\u27e9\n\n\ntheorem le_cons_self (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s :=\nle_of_lt $ lt_cons_self _ _\n\ntheorem cons_le_cons_iff (a : \u03b1) {s t : multiset \u03b1} : a :: s \u2264 a :: t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_cons a\n\ntheorem cons_le_cons (a : \u03b1) {s t : multiset \u03b1} : s \u2264 t \u2192 a :: s \u2264 a :: t :=\n(cons_le_cons_iff a).2\n\ntheorem le_cons_of_not_mem {a : \u03b1} {s t : multiset \u03b1} (m : a \u2209 s) : s \u2264 a :: t \u2194 s \u2264 t :=\nbegin\n  refine \u27e8_, \u03bb h, le_trans h $ le_cons_self _ _\u27e9,\n  suffices : \u2200 {t'} (_ : s \u2264 t') (_ : a \u2208 t'), a :: s \u2264 t',\n  { exact \u03bb h, (cons_le_cons_iff a).1 (this h (mem_cons_self _ _)) },\n  introv h, revert m, refine le_induction_on h _,\n  introv s m\u2081 m\u2082,\n  rcases mem_split m\u2082 with \u27e8r\u2081, r\u2082, rfl\u27e9,\n  exact perm_middle.subperm_left.2 ((subperm_cons _).2 $ subperm_of_sublist $\n    (sublist_or_mem_of_sublist s).resolve_right m\u2081)\nend\n\n\/- cardinality -\/\n\n\/-- The cardinality of a multiset is the sum of the multiplicities\n  of all its elements, or simply the length of the underlying list. -\/\ndef card (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s length $ \u03bb l\u2081 l\u2082, perm_length\n\n@[simp] theorem coe_card (l : list \u03b1) : card (l : multiset \u03b1) = length l := rfl\n\n@[simp] theorem card_zero : @card \u03b1 0 = 0 := rfl\n\n@[simp] theorem card_cons (a : \u03b1) (s : multiset \u03b1) : card (a :: s) = card s + 1 :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem card_singleton (a : \u03b1) : card (a::0) = 1 := by simp\n\ntheorem card_le_of_le {s t : multiset \u03b1} (h : s \u2264 t) : card s \u2264 card t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, length_le_of_sublist\n\ntheorem eq_of_le_of_card_le {s t : multiset \u03b1} (h : s \u2264 t) : card t \u2264 card s \u2192 s = t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 s h\u2082, congr_arg coe $ eq_of_sublist_of_length_le s h\u2082\n\ntheorem card_lt_of_lt {s t : multiset \u03b1} (h : s < t) : card s < card t :=\nlt_of_not_ge $ \u03bb h\u2082, ne_of_lt h $ eq_of_le_of_card_le (le_of_lt h) h\u2082\n\ntheorem lt_iff_cons_le {s t : multiset \u03b1} : s < t \u2194 \u2203 a, a :: s \u2264 t :=\n\u27e8quotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\n  subperm.exists_of_length_lt (le_of_lt h) (card_lt_of_lt h),\n\u03bb \u27e8a, h\u27e9, lt_of_lt_of_le (lt_cons_self _ _) h\u27e9\n\n@[simp] theorem card_eq_zero {s : multiset \u03b1} : card s = 0 \u2194 s = 0 :=\n\u27e8\u03bb h, (eq_of_le_of_card_le (zero_le _) (le_of_eq h)).symm, \u03bb e, by simp [e]\u27e9\n\ntheorem card_pos {s : multiset \u03b1} : 0 < card s \u2194 s \u2260 0 :=\npos_iff_ne_zero.trans $ not_congr card_eq_zero\n\ntheorem card_pos_iff_exists_mem {s : multiset \u03b1} : 0 < card s \u2194 \u2203 a, a \u2208 s :=\nquot.induction_on s $ \u03bb l, length_pos_iff_exists_mem\n\n@[elab_as_eliminator] def strong_induction_on {p : multiset \u03b1 \u2192 Sort*} :\n  \u2200 (s : multiset \u03b1), (\u2200 s, (\u2200t < s, p t) \u2192 p s) \u2192 p s\n| s := \u03bb ih, ih s $ \u03bb t h,\n  have card t < card s, from card_lt_of_lt h,\n  strong_induction_on t ih\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf card\u27e9]}\n\ntheorem strong_induction_eq {p : multiset \u03b1 \u2192 Sort*}\n  (s : multiset \u03b1) (H) : @strong_induction_on _ p s H =\n    H s (\u03bb t h, @strong_induction_on _ p t H) :=\nby rw [strong_induction_on]\n\n@[elab_as_eliminator] lemma case_strong_induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2080 : p 0) (h\u2081 : \u2200 a s, (\u2200t \u2264 s, p t) \u2192 p (a :: s)) : p s :=\nmultiset.strong_induction_on s $ assume s,\nmultiset.induction_on s (\u03bb _, h\u2080) $ \u03bb a s _ ih, h\u2081 _ _ $\n\u03bb t h, ih _ $ lt_of_le_of_lt h $ lt_cons_self _ _\n\n\/- singleton -\/\n@[simp] theorem singleton_eq_singleton (a : \u03b1) : singleton a = a::0 := rfl\n\n@[simp] theorem mem_singleton {a b : \u03b1} : b \u2208 a::0 \u2194 b = a := by simp\n\ntheorem mem_singleton_self (a : \u03b1) : a \u2208 (a::0 : multiset \u03b1) := mem_cons_self _ _\n\ntheorem singleton_inj {a b : \u03b1} : a::0 = b::0 \u2194 a = b := cons_inj_left _\n\n@[simp] theorem singleton_ne_zero (a : \u03b1) : a::0 \u2260 0 :=\nne_of_gt (lt_cons_self _ _)\n\n@[simp] theorem singleton_le {a : \u03b1} {s : multiset \u03b1} : a::0 \u2264 s \u2194 a \u2208 s :=\n\u27e8\u03bb h, mem_of_le h (mem_singleton_self _),\n \u03bb h, let \u27e8t, e\u27e9 := exists_cons_of_mem h in e.symm \u25b8 cons_le_cons _ (zero_le _)\u27e9\n\ntheorem card_eq_one {s : multiset \u03b1} : card s = 1 \u2194 \u2203 a, s = a::0 :=\n\u27e8quot.induction_on s $ \u03bb l h,\n  (list.length_eq_one.1 h).imp $ \u03bb a, congr_arg coe,\n \u03bb \u27e8a, e\u27e9, e.symm \u25b8 rfl\u27e9\n\n\/- add -\/\n\n\/-- The sum of two multisets is the lift of the list append operation.\n  This adds the multiplicities of each element,\n  i.e. `count a (s + t) = count a s + count a t`. -\/\nprotected def add (s\u2081 s\u2082 : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s\u2081 s\u2082 (\u03bb l\u2081 l\u2082, ((l\u2081 ++ l\u2082 : list \u03b1) : multiset \u03b1)) $\n  \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082, quot.sound $ perm_app p\u2081 p\u2082\n\ninstance : has_add (multiset \u03b1) := \u27e8multiset.add\u27e9\n\n@[simp] theorem coe_add (s t : list \u03b1) : (s + t : multiset \u03b1) = (s ++ t : list \u03b1) := rfl\n\nprotected theorem add_comm (s t : multiset \u03b1) : s + t = t + s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quot.sound perm_app_comm\n\nprotected theorem zero_add (s : multiset \u03b1) : 0 + s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\ntheorem singleton_add (a : \u03b1) (s : multiset \u03b1) : \u2191[a] + s = a::s := rfl\n\nprotected theorem add_le_add_left (s) {t u : multiset \u03b1} : s + t \u2264 s + u \u2194 t \u2264 u :=\nquotient.induction_on\u2083 s t u $ \u03bb l\u2081 l\u2082 l\u2083, subperm_app_left _\n\nprotected theorem add_left_cancel (s) {t u : multiset \u03b1} (h : s + t = s + u) : t = u :=\nle_antisymm ((multiset.add_le_add_left _).1 (le_of_eq h))\n  ((multiset.add_le_add_left _).1 (le_of_eq h.symm))\n\ninstance : ordered_cancel_comm_monoid (multiset \u03b1) :=\n{ zero                  := 0,\n  add                   := (+),\n  add_comm              := multiset.add_comm,\n  add_assoc             := \u03bb s\u2081 s\u2082 s\u2083, quotient.induction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb l\u2081 l\u2082 l\u2083,\n    congr_arg coe $ append_assoc l\u2081 l\u2082 l\u2083,\n  zero_add              := multiset.zero_add,\n  add_zero              := \u03bb s, by rw [multiset.add_comm, multiset.zero_add],\n  add_left_cancel       := multiset.add_left_cancel,\n  add_right_cancel      := \u03bb s\u2081 s\u2082 s\u2083 h, multiset.add_left_cancel s\u2082 $\n    by simpa [multiset.add_comm] using h,\n  add_le_add_left       := \u03bb s\u2081 s\u2082 h s\u2083, (multiset.add_le_add_left _).2 h,\n  le_of_add_le_add_left := \u03bb s\u2081 s\u2082 s\u2083, (multiset.add_le_add_left _).1,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem cons_add (a : \u03b1) (s t : multiset \u03b1) : a :: s + t = a :: (s + t) :=\nby rw [\u2190 singleton_add, \u2190 singleton_add, add_assoc]\n\n@[simp] theorem add_cons (a : \u03b1) (s t : multiset \u03b1) : s + a :: t = a :: (s + t) :=\nby rw [add_comm, cons_add, add_comm]\n\ntheorem le_add_right (s t : multiset \u03b1) : s \u2264 s + t :=\nby simpa using add_le_add_left (zero_le t) s\n\ntheorem le_add_left (s t : multiset \u03b1) : s \u2264 t + s :=\nby simpa using add_le_add_right (zero_le t) s\n\n@[simp] theorem card_add (s t : multiset \u03b1) : card (s + t) = card s + card t :=\nquotient.induction_on\u2082 s t length_append\n\nlemma card_smul (s : multiset \u03b1) (n : \u2115) :\n  (n \u2022 s).card = n * s.card :=\nby induction n; simp [succ_smul, *, nat.succ_mul]\n\n@[simp] theorem mem_add {a : \u03b1} {s t : multiset \u03b1} : a \u2208 s + t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, mem_append\n\ntheorem le_iff_exists_add {s t : multiset \u03b1} : s \u2264 t \u2194 \u2203 u, t = s + u :=\n\u27e8\u03bb h, le_induction_on h $ \u03bb l\u2081 l\u2082 s,\n  let \u27e8l, p\u27e9 := exists_perm_append_of_sublist s in \u27e8l, quot.sound p\u27e9,\n\u03bb\u27e8u, e\u27e9, e.symm \u25b8 le_add_right s u\u27e9\n\ninstance : canonically_ordered_monoid (multiset \u03b1) :=\n{ lt_of_add_lt_add_left := @lt_of_add_lt_add_left _ _,\n  le_iff_exists_add     := @le_iff_exists_add _,\n  bot                   := 0,\n  bot_le                := multiset.zero_le,\n  ..multiset.ordered_cancel_comm_monoid }\n\n\/- repeat -\/\n\n\/-- `repeat a n` is the multiset containing only `a` with multiplicity `n`. -\/\ndef repeat (a : \u03b1) (n : \u2115) : multiset \u03b1 := repeat a n\n\n@[simp] lemma repeat_zero (a : \u03b1) : repeat a 0 = 0 := rfl\n\n@[simp] lemma repeat_succ (a : \u03b1) (n) : repeat a (n+1) = a :: repeat a n := by simp [repeat]\n\n@[simp] lemma repeat_one (a : \u03b1) : repeat a 1 = a :: 0 := by simp\n\n@[simp] lemma card_repeat : \u2200 (a : \u03b1) n, card (repeat a n) = n := length_repeat\n\ntheorem eq_of_mem_repeat {a b : \u03b1} {n} : b \u2208 repeat a n \u2192 b = a := eq_of_mem_repeat\n\ntheorem eq_repeat' {a : \u03b1} {s : multiset \u03b1} : s = repeat a s.card \u2194 \u2200 b \u2208 s, b = a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  (perm_repeat.1 $ (quotient.exact h).symm).symm, congr_arg coe\u27e9 eq_repeat'\n\ntheorem eq_repeat_of_mem {a : \u03b1} {s : multiset \u03b1} : (\u2200 b \u2208 s, b = a) \u2192 s = repeat a s.card :=\neq_repeat'.2\n\ntheorem eq_repeat {a : \u03b1} {n} {s : multiset \u03b1} : s = repeat a n \u2194 card s = n \u2227 \u2200 b \u2208 s, b = a :=\n\u27e8\u03bb h, h.symm \u25b8 \u27e8card_repeat _ _, \u03bb b, eq_of_mem_repeat\u27e9,\n \u03bb \u27e8e, al\u27e9, e \u25b8 eq_repeat_of_mem al\u27e9\n\ntheorem repeat_subset_singleton : \u2200 (a : \u03b1) n, repeat a n \u2286 a::0 := repeat_subset_singleton\n\ntheorem repeat_le_coe {a : \u03b1} {n} {l : list \u03b1} : repeat a n \u2264 l \u2194 list.repeat a n <+ l :=\n\u27e8\u03bb \u27e8l', p, s\u27e9, (perm_repeat.1 p.symm).symm \u25b8 s, subperm_of_sublist\u27e9\n\n\/- range -\/\n\n\/-- `range n` is the multiset lifted from the list `range n`,\n  that is, the set `{0, 1, ..., n-1}`. -\/\ndef range (n : \u2115) : multiset \u2115 := range n\n\n@[simp] theorem range_zero : range 0 = 0 := rfl\n\n@[simp] theorem range_succ (n : \u2115) : range (succ n) = n :: range n :=\nby rw [range, range_concat, \u2190 coe_add, add_comm]; refl\n\n@[simp] theorem card_range (n : \u2115) : card (range n) = n := length_range _\n\ntheorem range_subset {m n : \u2115} : range m \u2286 range n \u2194 m \u2264 n := range_subset\n\n@[simp] theorem mem_range {m n : \u2115} : m \u2208 range n \u2194 m < n := mem_range\n\n@[simp] theorem not_mem_range_self {n : \u2115} : n \u2209 range n := not_mem_range_self\n\n\/- erase -\/\nsection erase\nvariables [decidable_eq \u03b1] {s t : multiset \u03b1} {a b : \u03b1}\n\n\/-- `erase s a` is the multiset that subtracts 1 from the\n  multiplicity of `a`. -\/\ndef erase (s : multiset \u03b1) (a : \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase a : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (erase_perm_erase a p))\n\n@[simp] theorem coe_erase (l : list \u03b1) (a : \u03b1) :\n  erase (l : multiset \u03b1) a = l.erase a := rfl\n\n@[simp] theorem erase_zero (a : \u03b1) : (0 : multiset \u03b1).erase a = 0 := rfl\n\n@[simp] theorem erase_cons_head (a : \u03b1) (s : multiset \u03b1) : (a :: s).erase a = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_head a l\n\n@[simp] theorem erase_cons_tail {a b : \u03b1} (s : multiset \u03b1) (h : b \u2260 a) : (b::s).erase a = b :: s.erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_tail l h\n\n@[simp] theorem erase_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 s.erase a = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ erase_of_not_mem h\n\n@[simp] theorem cons_erase {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 a :: s.erase a = s :=\nquot.induction_on s $ \u03bb l h, quot.sound (perm_erase h).symm\n\ntheorem le_cons_erase (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s.erase a :=\nif h : a \u2208 s then le_of_eq (cons_erase h).symm\nelse by rw erase_of_not_mem h; apply le_cons_self\n\n@[simp] theorem card_erase_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 card (s.erase a) = pred (card s) :=\nquot.induction_on s $ \u03bb l, length_erase_of_mem\n\ntheorem erase_add_left_pos {a : \u03b1} {s : multiset \u03b1} (t) : a \u2208 s \u2192 (s + t).erase a = s.erase a + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_left l\u2082 h\n\ntheorem erase_add_right_pos {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2208 t) : (s + t).erase a = s + t.erase a :=\nby rw [add_comm, erase_add_left_pos s h, add_comm]\n\ntheorem erase_add_right_neg {a : \u03b1} {s : multiset \u03b1} (t) : a \u2209 s \u2192 (s + t).erase a = s + t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_right l\u2082 h\n\ntheorem erase_add_left_neg {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2209 t) : (s + t).erase a = s.erase a + t :=\nby rw [add_comm, erase_add_right_neg s h, add_comm]\n\ntheorem erase_le (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist (erase_sublist a l)\n\n@[simp] theorem erase_lt {a : \u03b1} {s : multiset \u03b1} : s.erase a < s \u2194 a \u2208 s :=\n\u27e8\u03bb h, not_imp_comm.1 erase_of_not_mem (ne_of_lt h),\n \u03bb h, by simpa [h] using lt_cons_self (s.erase a) a\u27e9\n\ntheorem erase_subset (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2286 s :=\nsubset_of_le (erase_le a s)\n\ntheorem mem_erase_of_ne {a b : \u03b1} {s : multiset \u03b1} (ab : a \u2260 b) : a \u2208 s.erase b \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, list.mem_erase_of_ne ab\n\ntheorem mem_of_mem_erase {a b : \u03b1} {s : multiset \u03b1} : a \u2208 s.erase b \u2192 a \u2208 s :=\nmem_of_subset (erase_subset _ _)\n\ntheorem erase_comm (s : multiset \u03b1) (a b : \u03b1) : (s.erase a).erase b = (s.erase b).erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ l.erase_comm a b\n\ntheorem erase_le_erase {s t : multiset \u03b1} (a : \u03b1) (h : s \u2264 t) : s.erase a \u2264 t.erase a :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist (erase_sublist_erase _ h)\n\ntheorem erase_le_iff_le_cons {s t : multiset \u03b1} {a : \u03b1} : s.erase a \u2264 t \u2194 s \u2264 a :: t :=\n\u27e8\u03bb h, le_trans (le_cons_erase _ _) (cons_le_cons _ h),\n \u03bb h, if m : a \u2208 s\n  then by rw \u2190 cons_erase m at h; exact (cons_le_cons_iff _).1 h\n  else le_trans (erase_le _ _) ((le_cons_of_not_mem m).1 h)\u27e9\n\nend erase\n\n@[simp] theorem coe_reverse (l : list \u03b1) : (reverse l : multiset \u03b1) = l :=\nquot.sound $ reverse_perm _\n\n\/- map -\/\n\n\/-- `map f s` is the lift of the list `map` operation. The multiplicity\n  of `b` in `map f s` is the number of `a \u2208 s` (counting multiplicity)\n  such that `f a = b`. -\/\ndef map (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l : list \u03b1, (l.map f : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (perm_map f p))\n\n@[simp] theorem coe_map (f : \u03b1 \u2192 \u03b2) (l : list \u03b1) : map f \u2191l = l.map f := rfl\n\n@[simp] theorem map_zero (f : \u03b1 \u2192 \u03b2) : map f 0 = 0 := rfl\n\n@[simp] theorem map_cons (f : \u03b1 \u2192 \u03b2) (a s) : map f (a::s) = f a :: map f s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] lemma map_singleton (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : ({a} : multiset \u03b1).map f = {f a} := rfl\n\n@[simp] theorem map_add (f : \u03b1 \u2192 \u03b2) (s t) : map f (s + t) = map f s + map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ map_append _ _ _\n\ninstance (f : \u03b1 \u2192 \u03b2) : is_add_monoid_hom (map f) :=\n{ map_add := map_add _, map_zero := map_zero _ }\n\n@[simp] theorem mem_map {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : multiset \u03b1} :\n  b \u2208 map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = b :=\nquot.induction_on s $ \u03bb l, mem_map\n\n@[simp] theorem card_map (f : \u03b1 \u2192 \u03b2) (s) : card (map f s) = card s :=\nquot.induction_on s $ \u03bb l, length_map _ _\n\ntheorem mem_map_of_mem (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : f a \u2208 map f s :=\nmem_map.2 \u27e8_, h, rfl\u27e9\n\n@[simp] theorem mem_map_of_inj {f : \u03b1 \u2192 \u03b2} (H : function.injective f) {a : \u03b1} {s : multiset \u03b1} :\n  f a \u2208 map f s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_map_of_inj H\n\n@[simp] theorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : map g (map f s) = map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ list.map_map _ _ _\n\n@[simp] theorem map_id (s : multiset \u03b1) : map id s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_id _\n\n@[simp] lemma map_id' (s : multiset \u03b1) : map (\u03bbx, x) s = s := map_id s\n\n@[simp] theorem map_const (s : multiset \u03b1) (b : \u03b2) : map (function.const \u03b1 b) s = repeat b s.card :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_const _ _\n\n@[congr] theorem map_congr {f g : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200 x \u2208 s, f x = g x) \u2192 map f s = map g s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_congr H\n\nlemma map_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : map f m == map f' m :=\nbegin subst h, simp at hf, simp [map_congr hf] end\n\ntheorem eq_of_mem_map_const {b\u2081 b\u2082 : \u03b2} {l : list \u03b1} (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) l) : b\u2081 = b\u2082 :=\neq_of_mem_repeat $ by rwa map_const at h\n\n@[simp] theorem map_le_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (h : s \u2264 t) : map f s \u2264 map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ map_sublist_map f h\n\n@[simp] theorem map_subset_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (H : s \u2286 t) : map f s \u2286 map f t :=\n\u03bb b m, let \u27e8a, h, e\u27e9 := mem_map.1 m in mem_map.2 \u27e8a, H h, e\u27e9\n\n\/- fold -\/\n\n\/-- `foldl f H b s` is the lift of the list operation `foldl f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is right-commutative,\n  that is, `f (f b a\u2081) a\u2082 = f (f b a\u2082) a\u2081`. -\/\ndef foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldl f b l)\n  (\u03bb l\u2081 l\u2082 p, foldl_eq_of_perm H p b)\n\n@[simp] theorem foldl_zero (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b) : foldl f H b 0 = b := rfl\n\n@[simp] theorem foldl_cons (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b a s) : foldl f H b (a :: s) = foldl f H (f b a) s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldl_add (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b s t) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldl_append _ _ _ _\n\n\/-- `foldr f H b s` is the lift of the list operation `foldr f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is left-commutative,\n  that is, `f a\u2081 (f a\u2082 b) = f a\u2082 (f a\u2081 b)`. -\/\ndef foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldr f b l)\n  (\u03bb l\u2081 l\u2082 p, foldr_eq_of_perm H p b)\n\n@[simp] theorem foldr_zero (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b) : foldr f H b 0 = b := rfl\n\n@[simp] theorem foldr_cons (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b a s) : foldr f H b (a :: s) = f a (foldr f H b s) :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldr_add (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b s t) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldr_append _ _ _ _\n\n@[simp] theorem coe_foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldr f b := rfl\n\n@[simp] theorem coe_foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldl f H b l = l.foldl f b := rfl\n\ntheorem coe_foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldl (\u03bb x y, f y x) b :=\n(congr_arg (foldr f H b) (coe_reverse l)).symm.trans $ foldr_reverse _ _ _\n\ntheorem foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldr f H b s = foldl (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\nquot.induction_on s $ \u03bb l, coe_foldr_swap _ _ _ _\n\ntheorem foldl_swap (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldl f H b s = foldr (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\n(foldr_swap _ _ _ _).symm\n\n\/-- Product of a multiset given a commutative monoid structure on `\u03b1`.\n  `prod {a, b, c} = a * b * c` -\/\ndef prod [comm_monoid \u03b1] : multiset \u03b1 \u2192 \u03b1 :=\nfoldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1\nattribute [to_additive multiset.sum._proof_1] prod._proof_1\nattribute [to_additive multiset.sum] prod\n\n@[to_additive multiset.sum_eq_foldr]\ntheorem prod_eq_foldr [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1 s := rfl\n\n@[to_additive multiset.sum_eq_foldl]\ntheorem prod_eq_foldl [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldl (*) (\u03bb x y z, by simp [mul_right_comm]) 1 s :=\n(foldr_swap _ _ _ _).trans (by simp [mul_comm])\n\n@[simp, to_additive multiset.coe_sum]\ntheorem coe_prod [comm_monoid \u03b1] (l : list \u03b1) : prod \u2191l = l.prod :=\nprod_eq_foldl _\n\n@[simp, to_additive multiset.sum_zero]\ntheorem prod_zero [comm_monoid \u03b1] : @prod \u03b1 _ 0 = 1 := rfl\n\n@[simp, to_additive multiset.sum_cons]\ntheorem prod_cons [comm_monoid \u03b1] (a : \u03b1) (s) : prod (a :: s) = a * prod s :=\nfoldr_cons _ _ _ _ _\n\n@[to_additive multiset.sum_singleton]\ntheorem prod_singleton [comm_monoid \u03b1] (a : \u03b1) : prod (a :: 0) = a := by simp\n\n@[simp, to_additive multiset.sum_add]\ntheorem prod_add [comm_monoid \u03b1] (s t : multiset \u03b1) : prod (s + t) = prod s * prod t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, by simp\n\ninstance sum.is_add_monoid_hom [add_comm_monoid \u03b1] : is_add_monoid_hom (sum : multiset \u03b1 \u2192 \u03b1) :=\n{ map_add := sum_add, map_zero := sum_zero }\n\nlemma prod_smul {\u03b1 : Type*} [comm_monoid \u03b1] (m : multiset \u03b1) :\n  \u2200n, (add_monoid.smul n m).prod = m.prod ^ n\n| 0       := rfl\n| (n + 1) :=\n  by rw [add_monoid.add_smul, add_monoid.one_smul, _root_.pow_add, _root_.pow_one, prod_add, prod_smul n]\n\n@[simp] theorem prod_repeat [comm_monoid \u03b1] (a : \u03b1) (n : \u2115) : prod (multiset.repeat a n) = a ^ n :=\nby simp [repeat, list.prod_repeat]\n@[simp] theorem sum_repeat [add_comm_monoid \u03b1] : \u2200 (a : \u03b1) (n : \u2115), sum (multiset.repeat a n) = n \u2022 a :=\n@prod_repeat (multiplicative \u03b1) _\nattribute [to_additive multiset.sum_repeat] prod_repeat\n\n@[simp] lemma prod_map_one [comm_monoid \u03b3] {m : multiset \u03b1} :\n  prod (m.map (\u03bba, (1 : \u03b3))) = (1 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\n@[simp] lemma sum_map_zero [add_comm_monoid \u03b3] {m : multiset \u03b1} :\n  sum (m.map (\u03bba, (0 : \u03b3))) = (0 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\nattribute [to_additive multiset.sum_map_zero] prod_map_one\n\n@[simp, to_additive multiset.sum_map_add]\nlemma prod_map_mul [comm_monoid \u03b3] {m : multiset \u03b1} {f g : \u03b1 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, f a * g a) = prod (m.map f) * prod (m.map g) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih]; cc)\n\nlemma prod_map_prod_map [comm_monoid \u03b3] (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, prod $ n.map $ \u03bbb, f a b) = prod (n.map $ \u03bbb, prod $ m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih])\n\nlemma sum_map_sum_map [add_comm_monoid \u03b3] : \u2200 (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3},\n  sum (m.map $ \u03bba, sum $ n.map $ \u03bbb, f a b) = sum (n.map $ \u03bbb, sum $ m.map $ \u03bba, f a b) :=\n@prod_map_prod_map _ _ (multiplicative \u03b3) _\nattribute [to_additive multiset.sum_map_sum_map] prod_map_prod_map\n\nlemma sum_map_mul_left [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, b * f a)) = b * sum (s.map f) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, mul_add])\n\nlemma sum_map_mul_right [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, f a * b)) = sum (s.map f) * b :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, add_mul])\n\nlemma prod_hom [comm_monoid \u03b1] [comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_monoid_hom f] (s : multiset \u03b1) :\n  (s.map f).prod = f s.prod :=\nmultiset.induction_on s (by simp [is_monoid_hom.map_one f])\n  (by simp [is_monoid_hom.map_mul f] {contextual := tt})\n\nlemma dvd_prod [comm_semiring \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 a \u2223 s.prod :=\nquotient.induction_on s (\u03bb l a h, by simpa using list.dvd_prod h) a\n\nlemma sum_hom [add_comm_monoid \u03b1] [add_comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_add_monoid_hom f] (s : multiset \u03b1) :\n  (s.map f).sum = f s.sum :=\nmultiset.induction_on s (by simp [is_add_monoid_hom.map_zero f])\n  (by simp [is_add_monoid_hom.map_add f] {contextual := tt})\nattribute [to_additive multiset.sum_hom] multiset.prod_hom\n\nlemma le_sum_of_subadditive [add_comm_monoid \u03b1] [ordered_comm_monoid \u03b2]\n  (f : \u03b1 \u2192 \u03b2) (h_zero : f 0 = 0) (h_add : \u2200x y, f (x + y) \u2264 f x + f y) (s : multiset \u03b1) :\n  f s.sum \u2264 (s.map f).sum :=\nmultiset.induction_on s (le_of_eq h_zero) $\n  assume a s ih, by rw [sum_cons, map_cons, sum_cons];\n    from le_trans (h_add a s.sum) (add_le_add_left' ih)\n\nlemma abs_sum_le_sum_abs [discrete_linear_ordered_field \u03b1] {s : multiset \u03b1} :\n  abs s.sum \u2264 (s.map abs).sum :=\nle_sum_of_subadditive _ abs_zero abs_add s\n\n\/- join -\/\n\n\/-- `join S`, where `S` is a multiset of multisets, is the lift of the list join\n  operation, that is, the union of all the sets.\n\n     join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2} -\/\ndef join : multiset (multiset \u03b1) \u2192 multiset \u03b1 := sum\n\ntheorem coe_join : \u2200 L : list (list \u03b1),\n  join (L.map (@coe _ (multiset \u03b1) _) : multiset (multiset \u03b1)) = L.join\n| []       := rfl\n| (l :: L) := congr_arg (\u03bb s : multiset \u03b1, \u2191l + s) (coe_join L)\n\n@[simp] theorem join_zero : @join \u03b1 0 = 0 := rfl\n\n@[simp] theorem join_cons (s S) : @join \u03b1 (s :: S) = s + join S :=\nsum_cons _ _\n\n@[simp] theorem join_add (S T) : @join \u03b1 (S + T) = join S + join T :=\nsum_add _ _\n\n@[simp] theorem mem_join {a S} : a \u2208 @join \u03b1 S \u2194 \u2203 s \u2208 S, a \u2208 s :=\nmultiset.induction_on S (by simp) $\n  by simp [or_and_distrib_right, exists_or_distrib] {contextual := tt}\n\n@[simp] theorem card_join (S) : card (@join \u03b1 S) = sum (map card S) :=\nmultiset.induction_on S (by simp) (by simp)\n\n\/- bind -\/\n\n\/-- `bind s f` is the monad bind operation, defined as `join (map f s)`.\n  It is the union of `f a` as `a` ranges over `s`. -\/\ndef bind (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : multiset \u03b2 :=\njoin (map f s)\n\n@[simp] theorem coe_bind (l : list \u03b1) (f : \u03b1 \u2192 list \u03b2) :\n  @bind \u03b1 \u03b2 l (\u03bb a, f a) = l.bind f :=\nby rw [list.bind, \u2190 coe_join, list.map_map]; refl\n\n@[simp] theorem zero_bind (f : \u03b1 \u2192 multiset \u03b2) : bind 0 f = 0 := rfl\n\n@[simp] theorem cons_bind (a s) (f : \u03b1 \u2192 multiset \u03b2) : bind (a::s) f = f a + bind s f :=\nby simp [bind]\n\n@[simp] theorem add_bind (s t) (f : \u03b1 \u2192 multiset \u03b2) : bind (s + t) f = bind s f + bind t f :=\nby simp [bind]\n\n@[simp] theorem bind_zero (s : multiset \u03b1) : bind s (\u03bba, 0 : \u03b1 \u2192 multiset \u03b2) = 0 :=\nby simp [bind, -map_const, join]\n\n@[simp] theorem bind_add (s : multiset \u03b1) (f g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a + g a) = bind s f + bind s g :=\nby simp [bind, join]\n\n@[simp] theorem bind_cons (s : multiset \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a :: g a) = map f s + bind s g :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\n@[simp] theorem mem_bind {b s} {f : \u03b1 \u2192 multiset \u03b2} : b \u2208 bind s f \u2194 \u2203 a \u2208 s, b \u2208 f a :=\nby simp [bind]; simp [-exists_and_distrib_right, exists_and_distrib_right.symm];\n   rw exists_swap; simp [and_assoc]\n\n@[simp] theorem card_bind (s) (f : \u03b1 \u2192 multiset \u03b2) : card (bind s f) = sum (map (card \u2218 f) s) :=\nby simp [bind]\n\nlemma bind_congr {f g : \u03b1 \u2192 multiset \u03b2} {m : multiset \u03b1} : (\u2200a\u2208m, f a = g a) \u2192 bind m f = bind m g :=\nby simp [bind] {contextual := tt}\n\nlemma bind_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {f' : \u03b1 \u2192 multiset \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : bind m f == bind m f' :=\nbegin subst h, simp at hf, simp [bind_congr hf] end\n\nlemma map_bind (m : multiset \u03b1) (n : \u03b1 \u2192 multiset \u03b2) (f : \u03b2 \u2192 \u03b3) :\n  map f (bind m n) = bind m (\u03bba, map f (n a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map (m : multiset \u03b1) (n : \u03b2 \u2192 multiset \u03b3) (f : \u03b1 \u2192 \u03b2) :\n  bind (map f m) n = bind m (\u03bba, n (f a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_assoc {s : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b2 \u2192 multiset \u03b3} :\n  (s.bind f).bind g = s.bind (\u03bba, (f a).bind g) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma bind_bind (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 multiset \u03b3} :\n  (bind m $ \u03bba, bind n $ \u03bbb, f a b) = (bind n $ \u03bbb, bind m $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map_comm (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  (bind m $ \u03bba, n.map $ \u03bbb, f a b) = (bind n $ \u03bbb, m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\n@[simp, to_additive multiset.sum_bind]\nlemma prod_bind [comm_monoid \u03b2] (s : multiset \u03b1) (t : \u03b1 \u2192 multiset \u03b2) :\n  prod (bind s t) = prod (s.map $ \u03bba, prod (t a)) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, cons_bind])\n\n\/- product -\/\n\n\/-- The multiplicity of `(a, b)` in `product s t` is\n  the product of the multiplicity of `a` in `s` and `b` in `t`. -\/\ndef product (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u00d7 \u03b2) :=\ns.bind $ \u03bb a, t.map $ prod.mk a\n\n@[simp] theorem coe_product (l\u2081 : list \u03b1) (l\u2082 : list \u03b2) :\n  @product \u03b1 \u03b2 l\u2081 l\u2082 = l\u2081.product l\u2082 :=\nby rw [product, list.product, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_product (t) : @product \u03b1 \u03b2 0 t = 0 := rfl\n\n@[simp] theorem cons_product (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b2) :\n  product (a :: s) t = map (prod.mk a) t + product s t :=\nby simp [product]\n\n@[simp] theorem product_singleton (a : \u03b1) (b : \u03b2) : product (a::0) (b::0) = (a,b)::0 := rfl\n\n@[simp] theorem add_product (s t : multiset \u03b1) (u : multiset \u03b2) :\n  product (s + t) u = product s u + product t u :=\nby simp [product]\n\n@[simp] theorem product_add (s : multiset \u03b1) : \u2200 t u : multiset \u03b2,\n  product s (t + u) = product s t + product s u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_product, IH]; simp\n\n@[simp] theorem mem_product {s t} : \u2200 {p : \u03b1 \u00d7 \u03b2}, p \u2208 @product \u03b1 \u03b2 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t\n| (a, b) := by simp [product, and.left_comm]\n\n@[simp] theorem card_product (s : multiset \u03b1) (t : multiset \u03b2) : card (product s t) = card s * card t :=\nby simp [product, repeat, (\u2218), mul_comm]\n\n\/- sigma -\/\nsection\nvariable {\u03c3 : \u03b1 \u2192 Type*}\n\n\/-- `sigma s t` is the dependent version of `product`. It is the sum of\n  `(a, b)` as `a` ranges over `s` and `b` ranges over `t a`. -\/\nprotected def sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) : multiset (\u03a3 a, \u03c3 a) :=\ns.bind $ \u03bb a, (t a).map $ sigma.mk a\n\n@[simp] theorem coe_sigma (l\u2081 : list \u03b1) (l\u2082 : \u03a0 a, list (\u03c3 a)) :\n  @multiset.sigma \u03b1 \u03c3 l\u2081 (\u03bb a, l\u2082 a) = l\u2081.sigma l\u2082 :=\nby rw [multiset.sigma, list.sigma, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_sigma (t) : @multiset.sigma \u03b1 \u03c3 0 t = 0 := rfl\n\n@[simp] theorem cons_sigma (a : \u03b1) (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  (a :: s).sigma t = map (sigma.mk a) (t a) + s.sigma t :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_singleton (a : \u03b1) (b : \u03b1 \u2192 \u03b2) :\n  (a::0).sigma (\u03bb a, b a::0) = \u27e8a, b a\u27e9::0 := rfl\n\n@[simp] theorem add_sigma (s t : multiset \u03b1) (u : \u03a0 a, multiset (\u03c3 a)) :\n  (s + t).sigma u = s.sigma u + t.sigma u :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_add (s : multiset \u03b1) : \u2200 t u : \u03a0 a, multiset (\u03c3 a),\n  s.sigma (\u03bb a, t a + u a) = s.sigma t + s.sigma u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_sigma, IH]; simp\n\n@[simp] theorem mem_sigma {s t} : \u2200 {p : \u03a3 a, \u03c3 a},\n  p \u2208 @multiset.sigma \u03b1 \u03c3 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t p.1\n| \u27e8a, b\u27e9 := by simp [multiset.sigma, and_assoc, and.left_comm]\n\n@[simp] theorem card_sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  card (s.sigma t) = sum (map (\u03bb a, card (t a)) s) :=\nby simp [multiset.sigma, (\u2218)]\n\nend\n\n\/- map for partial functions -\/\n\n\/-- Lift of the list `pmap` operation. Map a partial function `f` over a multiset\n  `s` whose elements are all in the domain of `f`. -\/\ndef pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (s : multiset \u03b1) : (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2 :=\nquot.rec_on s (\u03bb l H, \u2191(pmap f l H)) $ \u03bb l\u2081 l\u2082 (pp : l\u2081 ~ l\u2082),\nfunext $ \u03bb (H\u2082 : \u2200 a \u2208 l\u2082, p a),\nhave H\u2081 : \u2200 a \u2208 l\u2081, p a, from \u03bb a h, H\u2082 a ((mem_of_perm pp).1 h),\nhave \u2200 {s\u2082 e H}, @eq.rec (multiset \u03b1) l\u2081\n  (\u03bb s, (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2) (\u03bb _, \u2191(pmap f l\u2081 H\u2081))\n  s\u2082 e H = \u2191(pmap f l\u2081 H\u2081), by intros s\u2082 e _; subst e,\nthis.trans $ quot.sound $ perm_pmap f pp\n\n@[simp] theorem coe_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (l : list \u03b1) (H : \u2200 a \u2208 l, p a) : pmap f l H = l.pmap f H := rfl\n\n@[simp] lemma pmap_zero {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (h : \u2200a\u2208(0:multiset \u03b1), p a) :\n  pmap f 0 h = 0 := rfl\n\n@[simp] lemma pmap_cons {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (a : \u03b1) (m : multiset \u03b1) :\n  \u2200(h : \u2200b\u2208a::m, p b), pmap f (a :: m) h =\n    f a (h a (mem_cons_self a m)) :: pmap f m (\u03bba ha, h a $ mem_cons_of_mem ha) :=\nquotient.induction_on m $ assume l h, rfl\n\n\/-- \"Attach\" a proof that `a \u2208 s` to each element `a` in `s` to produce\n  a multiset on `{x \/\/ x \u2208 s}`. -\/\ndef attach (s : multiset \u03b1) : multiset {x \/\/ x \u2208 s} := pmap subtype.mk s (\u03bb a, id)\n\n@[simp] theorem coe_attach (l : list \u03b1) :\n @eq (multiset {x \/\/ x \u2208 l}) (@attach \u03b1 l) l.attach := rfl\n\ntheorem pmap_eq_map (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  \u2200 H, @pmap _ _ p (\u03bb a _, f a) s H = map f s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map p f l H\n\ntheorem pmap_congr {p q : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {g : \u03a0 a, q a \u2192 \u03b2}\n  (s : multiset \u03b1) {H\u2081 H\u2082} (h : \u2200 a h\u2081 h\u2082, f a h\u2081 = g a h\u2082) :\n  pmap f s H\u2081 = pmap g s H\u2082 :=\nquot.induction_on s (\u03bb l H\u2081 H\u2082, congr_arg coe $ pmap_congr l h) H\u2081 H\u2082\n\ntheorem map_pmap {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3) (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, map g (pmap f s H) = pmap (\u03bb a h, g (f a h)) s H :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_pmap g f l H\n\ntheorem pmap_eq_map_attach {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, pmap f s H = s.attach.map (\u03bb x, f x.1 (H _ x.2)) :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map_attach f l H\n\ntheorem attach_map_val (s : multiset \u03b1) : s.attach.map subtype.val = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ attach_map_val l\n\n@[simp] theorem mem_attach (s : multiset \u03b1) : \u2200 x, x \u2208 s.attach :=\nquot.induction_on s $ \u03bb l, mem_attach _\n\n@[simp] theorem mem_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2}\n  {s H b} : b \u2208 pmap f s H \u2194 \u2203 a (h : a \u2208 s), f a (H a h) = b :=\nquot.induction_on s (\u03bb l H, mem_pmap) H\n\n@[simp] theorem card_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s H) : card (pmap f s H) = card s :=\nquot.induction_on s (\u03bb l H, length_pmap) H\n\n@[simp] theorem card_attach {m : multiset \u03b1} : card (attach m) = card m := card_pmap _ _ _\n\n@[simp] lemma attach_zero : (0 : multiset \u03b1).attach = 0 := rfl\n\nlemma attach_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).attach = \u27e8a, mem_cons_self a m\u27e9 :: (m.attach.map $ \u03bbp, \u27e8p.1, mem_cons_of_mem p.2\u27e9) :=\nquotient.induction_on m $ assume l, congr_arg coe $ congr_arg (list.cons _) $\n  by rw [list.map_pmap]; exact list.pmap_congr _ (assume a' h\u2081 h\u2082, subtype.eq rfl)\n\nsection decidable_pi_exists\nvariables {m : multiset \u03b1}\n\nprotected def decidable_forall_multiset {p : \u03b1 \u2192 Prop} [hp : \u2200a, decidable (p a)] :\n  decidable (\u2200a\u2208m, p a) :=\nquotient.rec_on_subsingleton m (\u03bbl, decidable_of_iff (\u2200a\u2208l, p a) $ by simp)\n\ninstance decidable_dforall_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2200a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_forall_multiset {a \/\/ a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (assume h a ha, h \u27e8a, ha\u27e9 (mem_attach _ _)) (assume h \u27e8a, ha\u27e9 _, h _ _))\n\n\/-- decidable equality for functions whose domain is bounded by multisets -\/\ninstance decidable_eq_pi_multiset {\u03b2 : \u03b1 \u2192 Type*} [h : \u2200a, decidable_eq (\u03b2 a)] :\n  decidable_eq (\u03a0a\u2208m, \u03b2 a) :=\nassume f g, decidable_of_iff (\u2200a (h : a \u2208 m), f a h = g a h) (by simp [function.funext_iff])\n\ndef decidable_exists_multiset {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  decidable (\u2203 x \u2208 m, p x) :=\nquotient.rec_on_subsingleton m list.decidable_exists_mem\n\ninstance decidable_dexists_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2203a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_exists_multiset {a \/\/ a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (\u03bb \u27e8\u27e8a, ha\u2081\u27e9, _, ha\u2082\u27e9, \u27e8a, ha\u2081, ha\u2082\u27e9)\n    (\u03bb \u27e8a, ha\u2081, ha\u2082\u27e9, \u27e8\u27e8a, ha\u2081\u27e9, mem_attach _ _, ha\u2082\u27e9))\n\nend decidable_pi_exists\n\n\/- subtraction -\/\nsection\nvariables [decidable_eq \u03b1] {s t u : multiset \u03b1} {a b : \u03b1}\n\n\/-- `s - t` is the multiset such that\n  `count a (s - t) = count a s - count a t` for all `a`. -\/\nprotected def sub (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.diff l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_diff_right w\u2081 p\u2082 \u25b8 perm_diff_left _ p\u2081\n\ninstance : has_sub (multiset \u03b1) := \u27e8multiset.sub\u27e9\n\n@[simp] theorem coe_sub (s t : list \u03b1) : (s - t : multiset \u03b1) = (s.diff t : list \u03b1) := rfl\n\ntheorem sub_eq_fold_erase (s t : multiset \u03b1) : s - t = foldl erase erase_comm s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nshow \u2191(l\u2081.diff l\u2082) = foldl erase erase_comm \u2191l\u2081 \u2191l\u2082,\nby rw diff_eq_foldl l\u2081 l\u2082; exact foldl_hom _ _ _ _ (\u03bb x y, rfl) _\n\n@[simp] theorem sub_zero (s : multiset \u03b1) : s - 0 = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem sub_cons (a : \u03b1) (s t : multiset \u03b1) : s - a::t = s.erase a - t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ diff_cons _ _ _\n\ntheorem add_sub_of_le (h : s \u2264 t) : s + (t - s) = t :=\nbegin\n  revert t,\n  refine multiset.induction_on s (by simp) (\u03bb a s IH t h, _),\n  have := cons_erase (mem_of_le h (mem_cons_self _ _)),\n  rw [cons_add, sub_cons, IH, this],\n  exact (cons_le_cons_iff a).1 (this.symm \u25b8 h)\nend\n\ntheorem sub_add' : s - (t + u) = s - t - u :=\nquotient.induction_on\u2083 s t u $\n\u03bb l\u2081 l\u2082 l\u2083, congr_arg coe $ diff_append _ _ _\n\ntheorem sub_add_cancel (h : t \u2264 s) : s - t + t = s :=\nby rw [add_comm, add_sub_of_le h]\n\n@[simp] theorem add_sub_cancel_left (s : multiset \u03b1) : \u2200 t, s + t - s = t :=\nmultiset.induction_on s (by simp)\n  (\u03bb a s IH t, by rw [cons_add, sub_cons, erase_cons_head, IH])\n\n@[simp] theorem add_sub_cancel (s t : multiset \u03b1) : s + t - t = s :=\nby rw [add_comm, add_sub_cancel_left]\n\ntheorem sub_le_sub_right (h : s \u2264 t) (u) : s - u \u2264 t - u :=\nby revert s t h; exact\nmultiset.induction_on u (by simp {contextual := tt})\n  (\u03bb a u IH s t h, by simp [IH, erase_le_erase a h])\n\ntheorem sub_le_sub_left (h : s \u2264 t) : \u2200 u, u - t \u2264 u - s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, begin\n  induction h with l\u2081 l\u2082 a s IH l\u2081 l\u2082 a s IH; intro u,\n  { refl },\n  { rw [\u2190 cons_coe, sub_cons],\n    exact le_trans (sub_le_sub_right (erase_le _ _) _) (IH u) },\n  { rw [\u2190 cons_coe, sub_cons, \u2190 cons_coe, sub_cons],\n    exact IH _ }\nend\n\ntheorem sub_le_iff_le_add : s - t \u2264 u \u2194 s \u2264 u + t :=\nby revert s; exact\nmultiset.induction_on t (by simp)\n  (\u03bb a t IH s, by simp [IH, erase_le_iff_le_cons])\n\ntheorem le_sub_add (s t : multiset \u03b1) : s \u2264 s - t + t :=\nsub_le_iff_le_add.1 (le_refl _)\n\ntheorem sub_le_self (s t : multiset \u03b1) : s - t \u2264 s :=\nsub_le_iff_le_add.2 (le_add_right _ _)\n\n@[simp] theorem card_sub {s t : multiset \u03b1} (h : t \u2264 s) : card (s - t) = card s - card t :=\n(nat.sub_eq_of_eq_add $ by rw [add_comm, \u2190 card_add, sub_add_cancel h]).symm\n\n\/- union -\/\n\n\/-- `s \u222a t` is the lattice join operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u222a t` is the maximum\n  of the multiplicities in `s` and `t`. -\/\ndef union (s t : multiset \u03b1) : multiset \u03b1 := s - t + t\n\ninstance : has_union (multiset \u03b1) := \u27e8union\u27e9\n\ntheorem union_def (s t : multiset \u03b1) : s \u222a t = s - t + t := rfl\n\ntheorem le_union_left (s t : multiset \u03b1) : s \u2264 s \u222a t := le_sub_add _ _\n\ntheorem le_union_right (s t : multiset \u03b1) : t \u2264 s \u222a t := le_add_left _ _\n\ntheorem eq_union_left : t \u2264 s \u2192 s \u222a t = s := sub_add_cancel\n\ntheorem union_le_union_right (h : s \u2264 t) (u) : s \u222a u \u2264 t \u222a u :=\nadd_le_add_right (sub_le_sub_right h _) u\n\ntheorem union_le (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u :=\nby rw \u2190 eq_union_left h\u2082; exact union_le_union_right h\u2081 t\n\n@[simp] theorem mem_union : a \u2208 s \u222a t \u2194 a \u2208 s \u2228 a \u2208 t :=\n\u27e8\u03bb h, (mem_add.1 h).imp_left (mem_of_le $ sub_le_self _ _),\n or.rec (mem_of_le $ le_union_left _ _) (mem_of_le $ le_union_right _ _)\u27e9\n\n@[simp] theorem map_union [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : function.injective f) {s t : multiset \u03b1} :\n  map f (s \u222a t) = map f s \u222a map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\ncongr_arg coe (by rw [list.map_append f, list.map_diff finj])\n\n\/- inter -\/\n\n\/-- `s \u2229 t` is the lattice meet operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u2229 t` is the minimum\n  of the multiplicities in `s` and `t`. -\/\ndef inter (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.bag_inter l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_bag_inter_right w\u2081 p\u2082 \u25b8 perm_bag_inter_left _ p\u2081\n\ninstance : has_inter (multiset \u03b1) := \u27e8inter\u27e9\n\n@[simp] theorem inter_zero (s : multiset \u03b1) : s \u2229 0 = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.bag_inter_nil\n\n@[simp] theorem zero_inter (s : multiset \u03b1) : 0 \u2229 s = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.nil_bag_inter\n\n@[simp] theorem cons_inter_of_pos {a} (s : multiset \u03b1) {t} :\n  a \u2208 t \u2192 (a :: s) \u2229 t = a :: s \u2229 t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_pos _ h\n\n@[simp] theorem cons_inter_of_neg {a} (s : multiset \u03b1) {t} :\n  a \u2209 t \u2192 (a :: s) \u2229 t = s \u2229 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_neg _ h\n\ntheorem inter_le_left (s t : multiset \u03b1) : s \u2229 t \u2264 s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ bag_inter_sublist_left _ _\n\ntheorem inter_le_right (s : multiset \u03b1) : \u2200 t, s \u2229 t \u2264 t :=\nmultiset.induction_on s (\u03bb t, (zero_inter t).symm \u25b8 zero_le _) $\n\u03bb a s IH t, if h : a \u2208 t\n  then by simpa [h] using cons_le_cons a (IH (t.erase a))\n  else by simp [h, IH]\n\ntheorem le_inter (h\u2081 : s \u2264 t) (h\u2082 : s \u2264 u) : s \u2264 t \u2229 u :=\nbegin\n  revert s u, refine multiset.induction_on t _ (\u03bb a t IH, _); intros,\n  { simp [h\u2081] },\n  by_cases a \u2208 u,\n  { rw [cons_inter_of_pos _ h, \u2190 erase_le_iff_le_cons],\n    exact IH (erase_le_iff_le_cons.2 h\u2081) (erase_le_erase _ h\u2082) },\n  { rw cons_inter_of_neg _ h,\n    exact IH ((le_cons_of_not_mem $ mt (mem_of_le h\u2082) h).1 h\u2081) h\u2082 }\nend\n\n@[simp] theorem mem_inter : a \u2208 s \u2229 t \u2194 a \u2208 s \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (inter_le_left _ _) h, mem_of_le (inter_le_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, by rw [\u2190 cons_erase h\u2081, cons_inter_of_pos _ h\u2082]; apply mem_cons_self\u27e9\n\ninstance : lattice (multiset \u03b1) :=\n{ sup          := (\u222a),\n  sup_le       := @union_le _ _,\n  le_sup_left  := le_union_left,\n  le_sup_right := le_union_right,\n  inf          := (\u2229),\n  le_inf       := @le_inter _ _,\n  inf_le_left  := inter_le_left,\n  inf_le_right := inter_le_right,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem sup_eq_union (s t : multiset \u03b1) : s \u2294 t = s \u222a t := rfl\n@[simp] theorem inf_eq_inter (s t : multiset \u03b1) : s \u2293 t = s \u2229 t := rfl\n\n@[simp] theorem le_inter_iff : s \u2264 t \u2229 u \u2194 s \u2264 t \u2227 s \u2264 u := le_inf_iff\n@[simp] theorem union_le_iff : s \u222a t \u2264 u \u2194 s \u2264 u \u2227 t \u2264 u := sup_le_iff\n\ninstance : semilattice_inf_bot (multiset \u03b1) :=\n{ bot := 0, bot_le := zero_le, ..multiset.lattice.lattice }\n\ntheorem union_comm (s t : multiset \u03b1) : s \u222a t = t \u222a s := sup_comm\ntheorem inter_comm (s t : multiset \u03b1) : s \u2229 t = t \u2229 s := inf_comm\n\ntheorem eq_union_right (h : s \u2264 t) : s \u222a t = t :=\nby rw [union_comm, eq_union_left h]\n\ntheorem union_le_union_left (h : s \u2264 t) (u) : u \u222a s \u2264 u \u222a t :=\nsup_le_sup_left h _\n\ntheorem union_le_add (s t : multiset \u03b1) : s \u222a t \u2264 s + t :=\nunion_le (le_add_right _ _) (le_add_left _ _)\n\ntheorem union_add_distrib (s t u : multiset \u03b1) : (s \u222a t) + u = (s + u) \u222a (t + u) :=\nby simpa [(\u222a), union, eq_comm] using show s + u - (t + u) = s - t,\nby rw [add_comm t, sub_add', add_sub_cancel]\n\ntheorem add_union_distrib (s t u : multiset \u03b1) : s + (t \u222a u) = (s + t) \u222a (s + u) :=\nby rw [add_comm, union_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_union_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u222a t) = (a :: s) \u222a (a :: t) :=\nby simpa using add_union_distrib (a::0) s t\n\ntheorem inter_add_distrib (s t u : multiset \u03b1) : (s \u2229 t) + u = (s + u) \u2229 (t + u) :=\nbegin\n  by_contra h,\n  cases lt_iff_cons_le.1 (lt_of_le_of_ne (le_inter\n    (add_le_add_right (inter_le_left s t) u)\n    (add_le_add_right (inter_le_right s t) u)) h) with a hl,\n  rw \u2190 cons_add at hl,\n  exact not_le_of_lt (lt_cons_self (s \u2229 t) a) (le_inter\n    (le_of_add_le_add_right (le_trans hl (inter_le_left _ _)))\n    (le_of_add_le_add_right (le_trans hl (inter_le_right _ _))))\nend\n\ntheorem add_inter_distrib (s t u : multiset \u03b1) : s + (t \u2229 u) = (s + t) \u2229 (s + u) :=\nby rw [add_comm, inter_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_inter_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u2229 t) = (a :: s) \u2229 (a :: t) :=\nby simp\n\ntheorem union_add_inter (s t : multiset \u03b1) : s \u222a t + s \u2229 t = s + t :=\nbegin\n  apply le_antisymm,\n  { rw union_add_distrib,\n    refine union_le (add_le_add_left (inter_le_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (inter_le_left _ _) _ },\n  { rw [add_comm, add_inter_distrib],\n    refine le_inter (add_le_add_right (le_union_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (le_union_left _ _) _ }\nend\n\ntheorem sub_add_inter (s t : multiset \u03b1) : s - t + s \u2229 t = s :=\nbegin\n  rw [inter_comm],\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  by_cases a \u2208 s,\n  { rw [cons_inter_of_pos _ h, sub_cons, add_cons, IH, cons_erase h] },\n  { rw [cons_inter_of_neg _ h, sub_cons, erase_of_not_mem h, IH] }\nend\n\ntheorem sub_inter (s t : multiset \u03b1) : s - (s \u2229 t) = s - t :=\nadd_right_cancel $\nby rw [sub_add_inter s t, sub_add_cancel (inter_le_left _ _)]\n\nend\n\n\n\/- filter -\/\nsection\nvariables {p : \u03b1 \u2192 Prop} [decidable_pred p]\n\n\/-- `filter p s` returns the elements in `s` (with the same multiplicities)\n  which satisfy `p`, and removes the rest. -\/\ndef filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p] (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (filter p l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $ perm_filter p h)\n\n@[simp] theorem coe_filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p]\n  (l : list \u03b1) : filter p (\u2191l) = l.filter p := rfl\n\n@[simp] theorem filter_zero (p : \u03b1 \u2192 Prop) [h : decidable_pred p] : filter p 0 = 0 := rfl\n\n@[simp] theorem filter_cons_of_pos {a : \u03b1} (s) : p a \u2192 filter p (a::s) = a :: filter p s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_cons_of_pos l h\n\n@[simp] theorem filter_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 filter p (a::s) = filter p s :=\nquot.induction_on s $ \u03bb l h, @congr_arg _ _ _ _ coe $ filter_cons_of_neg l h\n\nlemma filter_congr {p q : \u03b1 \u2192 Prop} [decidable_pred p] [decidable_pred q]\n  {s : multiset \u03b1} : (\u2200 x \u2208 s, p x \u2194 q x) \u2192 filter p s = filter q s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_congr h\n\n@[simp] theorem filter_add (s t : multiset \u03b1) :\n  filter p (s + t) = filter p s + filter p t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ filter_append _ _\n\n@[simp] theorem filter_le (s : multiset \u03b1) : filter p s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ filter_sublist _\n\n@[simp] theorem filter_subset (s : multiset \u03b1) : filter p s \u2286 s :=\nsubset_of_le $ filter_le _\n\n@[simp] theorem mem_filter {a : \u03b1} {s} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a :=\nquot.induction_on s $ \u03bb l, mem_filter\n\ntheorem of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : p a :=\n(mem_filter.1 h).2\n\ntheorem mem_of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : a \u2208 s :=\n(mem_filter.1 h).1\n\ntheorem mem_filter_of_mem {a : \u03b1} {l} (m : a \u2208 l) (h : p a) : a \u2208 filter p l :=\nmem_filter.2 \u27e8m, h\u27e9\n\ntheorem filter_eq_self {s} : filter p s = s \u2194 \u2200 a \u2208 s, p a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_of_sublist_of_length_eq (filter_sublist _) (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_self\n\ntheorem filter_eq_nil {s} : filter p s = 0 \u2194 \u2200 a \u2208 s, \u00acp a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_nil_of_length_eq_zero (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_nil\n\ntheorem filter_le_filter {s t} (h : s \u2264 t) : filter p s \u2264 filter p t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ filter_sublist_filter h\n\ntheorem le_filter {s t} : s \u2264 filter p t \u2194 s \u2264 t \u2227 \u2200 a \u2208 s, p a :=\n\u27e8\u03bb h, \u27e8le_trans h (filter_le _), \u03bb a m, of_mem_filter (mem_of_le h m)\u27e9,\n \u03bb \u27e8h, al\u27e9, filter_eq_self.2 al \u25b8 filter_le_filter h\u27e9\n\n@[simp] theorem filter_sub [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s - t) = filter p s - filter p t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  rw [sub_cons, IH],\n  by_cases p a,\n  { rw [filter_cons_of_pos _ h, sub_cons], congr,\n    by_cases m : a \u2208 s,\n    { rw [\u2190 cons_inj_right a, \u2190 filter_cons_of_pos _ h,\n          cons_erase (mem_filter_of_mem m h), cons_erase m] },\n    { rw [erase_of_not_mem m, erase_of_not_mem (mt mem_of_mem_filter m)] } },\n  { rw [filter_cons_of_neg _ h],\n    by_cases m : a \u2208 s,\n    { rw [(by rw filter_cons_of_neg _ h : filter p (erase s a) = filter p (a :: erase s a)),\n          cons_erase m] },\n    { rw [erase_of_not_mem m] } }\nend\n\n@[simp] theorem filter_union [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u222a t) = filter p s \u222a filter p t :=\nby simp [(\u222a), union]\n\n@[simp] theorem filter_inter [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u2229 t) = filter p s \u2229 filter p t :=\nle_antisymm (le_inter\n    (filter_le_filter $ inter_le_left _ _)\n    (filter_le_filter $ inter_le_right _ _)) $ le_filter.2\n\u27e8inf_le_inf (filter_le _) (filter_le _),\n  \u03bb a h, of_mem_filter (mem_of_le (inter_le_left _ _) h)\u27e9\n\n@[simp] theorem filter_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p (filter q s) = filter (\u03bb a, p a \u2227 q a) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter l\n\ntheorem filter_add_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p s + filter q s = filter (\u03bb a, p a \u2228 q a) s + filter (\u03bb a, p a \u2227 q a) s :=\nmultiset.induction_on s rfl $ \u03bb a s IH,\nby by_cases p a; by_cases q a; simp *\n\ntheorem filter_add_not (s : multiset \u03b1) :\n  filter p s + filter (\u03bb a, \u00ac p a) s = s :=\nby rw [filter_add_filter, filter_eq_self.2, filter_eq_nil.2]; simp [decidable.em]\n\n\/- filter_map -\/\n\n\/-- `filter_map f s` is a combination filter\/map operation on `s`.\n  The function `f : \u03b1 \u2192 option \u03b2` is applied to each element of `s`;\n  if `f a` is `some b` then `b` is added to the result, otherwise\n  `a` is removed from the resulting multiset. -\/\ndef filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l, (filter_map f l : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $perm_filter_map f h)\n\n@[simp] theorem coe_filter_map (f : \u03b1 \u2192 option \u03b2) (l : list \u03b1) : filter_map f l = l.filter_map f := rfl\n\n@[simp] theorem filter_map_zero (f : \u03b1 \u2192 option \u03b2) : filter_map f 0 = 0 := rfl\n\n@[simp] theorem filter_map_cons_none {f : \u03b1 \u2192 option \u03b2} (a : \u03b1) (s : multiset \u03b1) (h : f a = none) :\n  filter_map f (a :: s) = filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_none a l h\n\n@[simp] theorem filter_map_cons_some (f : \u03b1 \u2192 option \u03b2)\n  (a : \u03b1) (s : multiset \u03b1) {b : \u03b2} (h : f a = some b) :\n  filter_map f (a :: s) = b :: filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_some f a l h\n\ntheorem filter_map_eq_map (f : \u03b1 \u2192 \u03b2) : filter_map (some \u2218 f) = map f :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_map f) l\n\ntheorem filter_map_eq_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  filter_map (option.guard p) = filter p :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_filter p) l\n\ntheorem filter_map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (filter_map f s) = filter_map (\u03bb x, (f x).bind g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter_map f g l\n\ntheorem map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b3) (s : multiset \u03b1) :\n  map g (filter_map f s) = filter_map (\u03bb x, (f x).map g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map f g l\n\ntheorem filter_map_map (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (map f s) = filter_map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_map f g l\n\ntheorem filter_filter_map (f : \u03b1 \u2192 option \u03b2) (p : \u03b2 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n  filter p (filter_map f s) = filter_map (\u03bb x, (f x).filter p) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter_map f p l\n\ntheorem filter_map_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) :\n  filter_map f (filter p s) = filter_map (\u03bb x, if p x then f x else none) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter p f l\n\n@[simp] theorem filter_map_some (s : multiset \u03b1) : filter_map some s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_some l\n\n@[simp] theorem mem_filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) {b : \u03b2} :\n  b \u2208 filter_map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = some b :=\nquot.induction_on s $ \u03bb l, mem_filter_map f l\n\ntheorem map_filter_map_of_inv (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b1)\n  (H : \u2200 x : \u03b1, (f x).map g = some x) (s : multiset \u03b1) :\n  map g (filter_map f s) = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map_of_inv f g H l\n\ntheorem filter_map_le_filter_map (f : \u03b1 \u2192 option \u03b2) {s t : multiset \u03b1}\n  (h : s \u2264 t) : filter_map f s \u2264 filter_map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h,\nsubperm_of_sublist $ filter_map_sublist_filter_map _ h\n\n\/- powerset -\/\n\ndef powerset_aux (l : list \u03b1) : list (multiset \u03b1) :=\n0 :: sublists_aux l (\u03bb x y, x :: y)\n\ntheorem powerset_aux_eq_map_coe {l : list \u03b1} :\n  powerset_aux l = (sublists l).map coe :=\nby simp [powerset_aux, sublists];\n   rw [\u2190 show @sublists_aux\u2081 \u03b1 (multiset \u03b1) l (\u03bb x, [\u2191x]) =\n              sublists_aux l (\u03bb x, list.cons \u2191x),\n         from sublists_aux\u2081_eq_sublists_aux _ _,\n       sublists_aux_cons_eq_sublists_aux\u2081,\n       \u2190 bind_ret_eq_map, sublists_aux\u2081_bind]; refl\n\n@[simp] theorem mem_powerset_aux {l : list \u03b1} {s} :\n  s \u2208 powerset_aux l \u2194 s \u2264 \u2191l :=\nquotient.induction_on s $\nby simp [powerset_aux_eq_map_coe, subperm, and.comm]\n\ndef powerset_aux' (l : list \u03b1) : list (multiset \u03b1) := (sublists' l).map coe\n\ntheorem powerset_aux_perm_powerset_aux' {l : list \u03b1} :\n  powerset_aux l ~ powerset_aux' l :=\nby rw powerset_aux_eq_map_coe; exact\nperm_map _ (sublists_perm_sublists' _)\n\n@[simp] theorem powerset_aux'_nil : powerset_aux' (@nil \u03b1) = [0] := rfl\n\n@[simp] theorem powerset_aux'_cons (a : \u03b1) (l : list \u03b1) :\n  powerset_aux' (a::l) = powerset_aux' l ++ list.map (cons a) (powerset_aux' l) :=\nby simp [powerset_aux']; refl\n\ntheorem powerset_aux'_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux' l\u2081 ~ powerset_aux' l\u2082 :=\nbegin\n  induction p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 p\u2082 IH\u2081 IH\u2082, {simp},\n  { simp, exact perm_app IH (perm_map _ IH) },\n  { simp, apply perm_app_right,\n    rw [\u2190 append_assoc, \u2190 append_assoc,\n        (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)],\n    exact perm_app_left _ perm_app_comm },\n  { exact IH\u2081.trans IH\u2082 }\nend\n\ntheorem powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux l\u2081 ~ powerset_aux l\u2082 :=\npowerset_aux_perm_powerset_aux'.trans $\n(powerset_aux'_perm p).trans powerset_aux_perm_powerset_aux'.symm\n\ndef powerset (s : multiset \u03b1) : multiset (multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (powerset_aux l : multiset (multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (powerset_aux_perm h))\n\ntheorem powerset_coe (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists l).map coe : list (multiset \u03b1)) :=\ncongr_arg coe powerset_aux_eq_map_coe\n\n@[simp] theorem powerset_coe' (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists' l).map coe : list (multiset \u03b1)) :=\nquot.sound powerset_aux_perm_powerset_aux'\n\n@[simp] theorem powerset_zero : @powerset \u03b1 0 = 0::0 := rfl\n\n@[simp] theorem powerset_cons (a : \u03b1) (s) :\n  powerset (a::s) = powerset s + map (cons a) (powerset s) :=\nquotient.induction_on s $ \u03bb l, by simp; refl\n\n@[simp] theorem mem_powerset {s t : multiset \u03b1} :\n  s \u2208 powerset t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ by simp [subperm, and.comm]\n\ntheorem map_single_le_powerset (s : multiset \u03b1) :\n  s.map (\u03bb a, a::0) \u2264 powerset s :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [powerset_coe],\n  show l.map (coe \u2218 list.ret) <+~ (sublists l).map coe,\n  rw \u2190 list.map_map,\n  exact subperm_of_sublist\n    (map_sublist_map _ (map_ret_sublist_sublists _))\nend\n\n@[simp] theorem card_powerset (s : multiset \u03b1) :\n  card (powerset s) = 2 ^ card s :=\nquotient.induction_on s $ by simp\n\n\/- diagonal -\/\n\ntheorem revzip_powerset_aux {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux_eq_map_coe, \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists _ _ _ h)\nend\n\ntheorem revzip_powerset_aux' {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux' l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux', \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists' _ _ _ h)\nend\n\ntheorem revzip_powerset_aux_lemma [decidable_eq \u03b1] (l : list \u03b1)\n  {l' : list (multiset \u03b1)} (H : \u2200 \u2983s t\u2984, (s, t) \u2208 revzip l' \u2192 s + t = \u2191l) :\n  revzip l' = l'.map (\u03bb x, (x, \u2191l - x)) :=\nbegin\n  have : forall\u2082 (\u03bb (p : multiset \u03b1 \u00d7 multiset \u03b1) (s : multiset \u03b1), p = (s, \u2191l - s))\n    (revzip l') ((revzip l').map prod.fst),\n  { rw forall\u2082_map_right_iff,\n    apply forall\u2082_same, rintro \u27e8s, t\u27e9 h,\n    dsimp, rw [\u2190 H h, add_sub_cancel_left] },\n  rw [\u2190 forall\u2082_eq_eq_eq, forall\u2082_map_right_iff], simpa\nend\n\ntheorem revzip_powerset_aux_perm_aux' {l : list \u03b1} :\n  revzip (powerset_aux l) ~ revzip (powerset_aux' l) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  rw [revzip_powerset_aux_lemma l revzip_powerset_aux,\n      revzip_powerset_aux_lemma l revzip_powerset_aux'],\n  exact perm_map _ powerset_aux_perm_powerset_aux',\nend\n\ntheorem revzip_powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  revzip (powerset_aux l\u2081) ~ revzip (powerset_aux l\u2082) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  simp [\u03bb l:list \u03b1, revzip_powerset_aux_lemma l revzip_powerset_aux, coe_eq_coe.2 p],\n  exact perm_map _ (powerset_aux_perm p)\nend\n\ndef diagonal (s : multiset \u03b1) : multiset (multiset \u03b1 \u00d7 multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (revzip (powerset_aux l) : multiset (multiset \u03b1 \u00d7 multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (revzip_powerset_aux_perm h))\n\ntheorem diagonal_coe (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux l) := rfl\n\n@[simp] theorem diagonal_coe' (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux' l) :=\nquot.sound revzip_powerset_aux_perm_aux'\n\n@[simp] theorem mem_diagonal {s\u2081 s\u2082 t : multiset \u03b1} :\n  (s\u2081, s\u2082) \u2208 diagonal t \u2194 s\u2081 + s\u2082 = t :=\nquotient.induction_on t $ \u03bb l, begin\n  simp [diagonal_coe], refine \u27e8\u03bb h, revzip_powerset_aux h, \u03bb h, _\u27e9,\n  haveI := classical.dec_eq \u03b1,\n  simp [revzip_powerset_aux_lemma l revzip_powerset_aux, h.symm],\n  exact \u27e8_, le_add_right _ _, rfl, add_sub_cancel_left _ _\u27e9\nend\n\n@[simp] theorem diagonal_map_fst (s : multiset \u03b1) :\n  (diagonal s).map prod.fst = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_map_snd (s : multiset \u03b1) :\n  (diagonal s).map prod.snd = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_zero : @diagonal \u03b1 0 = (0, 0)::0 := rfl\n\n@[simp] theorem diagonal_cons (a : \u03b1) (s) : diagonal (a::s) =\n  map (prod.map id (cons a)) (diagonal s) +\n  map (prod.map (cons a) id) (diagonal s) :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [revzip, reverse_append],\n  rw [\u2190 zip_map, \u2190 zip_map, zip_append, (_ : _++_=_)],\n  {congr; simp}, {simp}\nend\n\n@[simp] theorem card_diagonal (s : multiset \u03b1) :\n  card (diagonal s) = 2 ^ card s :=\nby have := card_powerset s;\n   rwa [\u2190 diagonal_map_fst, card_map] at this\n\nlemma prod_map_add [comm_semiring \u03b2] {s : multiset \u03b1} {f g : \u03b1 \u2192 \u03b2} :\n  prod (s.map (\u03bba, f a + g a)) = sum ((diagonal s).map (\u03bbp, (p.1.map f).prod * (p.2.map g).prod)) :=\nbegin\n  refine s.induction_on _ _,\n  { simp },\n  { assume a s ih, simp [ih, add_mul, mul_comm, mul_left_comm, mul_assoc, sum_map_mul_left.symm] },\nend\n\n\/- powerset_len -\/\n\ndef powerset_len_aux (n : \u2115) (l : list \u03b1) : list (multiset \u03b1) :=\nsublists_len_aux n l coe []\n\ntheorem powerset_len_aux_eq_map_coe {n} {l : list \u03b1} :\n  powerset_len_aux n l = (sublists_len n l).map coe :=\nby rw [powerset_len_aux, sublists_len_aux_eq, append_nil]\n\n@[simp] theorem mem_powerset_len_aux {n} {l : list \u03b1} {s} :\n  s \u2208 powerset_len_aux n l \u2194 s \u2264 \u2191l \u2227 card s = n :=\nquotient.induction_on s $\nby simp [powerset_len_aux_eq_map_coe, subperm]; exact\n  \u03bb l\u2081, \u27e8\u03bb \u27e8l\u2082, \u27e8s, e\u27e9, p\u27e9, \u27e8\u27e8_, p, s\u27e9, (perm_length p.symm).trans e\u27e9,\n    \u03bb \u27e8\u27e8l\u2082, p, s\u27e9, e\u27e9, \u27e8_, \u27e8s, (perm_length p).trans e\u27e9, p\u27e9\u27e9\n\n@[simp] theorem powerset_len_aux_zero (l : list \u03b1) :\n  powerset_len_aux 0 l = [0] :=\nby simp [powerset_len_aux_eq_map_coe]\n\n@[simp] theorem powerset_len_aux_nil (n : \u2115) :\n  powerset_len_aux (n+1) (@nil \u03b1) = [] := rfl\n\n@[simp] theorem powerset_len_aux_cons (n : \u2115) (a : \u03b1) (l : list \u03b1) :\n  powerset_len_aux (n+1) (a::l) =\n  powerset_len_aux (n+1) l ++ list.map (cons a) (powerset_len_aux n l) :=\nby simp [powerset_len_aux_eq_map_coe]; refl\n\ntheorem powerset_len_aux_perm {n} {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_len_aux n l\u2081 ~ powerset_len_aux n l\u2082 :=\nbegin\n  induction n with n IHn generalizing l\u2081 l\u2082, {simp},\n  induction p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 p\u2082 IH\u2081 IH\u2082, {refl},\n  { simp, exact perm_app IH (perm_map _ (IHn p)) },\n  { simp, apply perm_app_right,\n    cases n, {simp, apply perm.swap},\n    simp,\n    rw [\u2190 append_assoc, \u2190 append_assoc,\n        (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)],\n    exact perm_app_left _ perm_app_comm },\n  { exact IH\u2081.trans IH\u2082 }\nend\n\ndef powerset_len (n : \u2115) (s : multiset \u03b1) : multiset (multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (powerset_len_aux n l : multiset (multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (powerset_len_aux_perm h))\n\ntheorem powerset_len_coe' (n) (l : list \u03b1) :\n  @powerset_len \u03b1 n l = powerset_len_aux n l := rfl\n\ntheorem powerset_len_coe (n) (l : list \u03b1) :\n  @powerset_len \u03b1 n l = ((sublists_len n l).map coe : list (multiset \u03b1)) :=\ncongr_arg coe powerset_len_aux_eq_map_coe\n\n@[simp] theorem powerset_len_zero_left (s : multiset \u03b1) :\n  powerset_len 0 s = 0::0 :=\nquotient.induction_on s $ \u03bb l, by simp [powerset_len_coe']; refl\n\n@[simp] theorem powerset_len_zero_right (n : \u2115) :\n  @powerset_len \u03b1 (n + 1) 0 = 0 := rfl\n\n@[simp] theorem powerset_len_cons (n : \u2115) (a : \u03b1) (s) :\n  powerset_len (n + 1) (a::s) =\n  powerset_len (n + 1) s + map (cons a) (powerset_len n s) :=\nquotient.induction_on s $ \u03bb l, by simp [powerset_len_coe']; refl\n\n@[simp] theorem mem_powerset_len {n : \u2115} {s t : multiset \u03b1} :\n  s \u2208 powerset_len n t \u2194 s \u2264 t \u2227 card s = n :=\nquotient.induction_on t $ \u03bb l, by simp [powerset_len_coe']\n\n@[simp] theorem card_powerset_len (n : \u2115) (s : multiset \u03b1) :\n  card (powerset_len n s) = nat.choose (card s) n :=\nquotient.induction_on s $ by simp [powerset_len_coe]\n\ntheorem powerset_len_le_powerset (n : \u2115) (s : multiset \u03b1) :\n  powerset_len n s \u2264 powerset s :=\nquotient.induction_on s $ \u03bb l, by simp [powerset_len_coe]; exact\n  subperm_of_sublist (map_sublist_map _ (sublists_len_sublist_sublists' _ _))\n\ntheorem powerset_len_mono (n : \u2115) {s t : multiset \u03b1} (h : s \u2264 t) :\n  powerset_len n s \u2264 powerset_len n t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, by simp [powerset_len_coe]; exact\n  subperm_of_sublist (map_sublist_map _ (sublists_len_sublist_of_sublist _ h))\n\n\/- countp -\/\n\n\/-- `countp p s` counts the number of elements of `s` (with multiplicity) that\n  satisfy `p`. -\/\ndef countp (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s (countp p) (\u03bb l\u2081 l\u2082, perm_countp p)\n\n@[simp] theorem coe_countp (l : list \u03b1) : countp p l = l.countp p := rfl\n\n@[simp] theorem countp_zero (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p 0 = 0 := rfl\n\n@[simp] theorem countp_cons_of_pos {a : \u03b1} (s) : p a \u2192 countp p (a::s) = countp p s + 1 :=\nquot.induction_on s countp_cons_of_pos\n\n@[simp] theorem countp_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 countp p (a::s) = countp p s :=\nquot.induction_on s countp_cons_of_neg\n\ntheorem countp_eq_card_filter (s) : countp p s = card (filter p s) :=\nquot.induction_on s $ \u03bb l, countp_eq_length_filter _\n\n@[simp] theorem countp_add (s t) : countp p (s + t) = countp p s + countp p t :=\nby simp [countp_eq_card_filter]\n\ninstance countp.is_add_monoid_hom : is_add_monoid_hom (countp p : multiset \u03b1 \u2192 \u2115) :=\n{ map_add := countp_add, map_zero := countp_zero _ }\n\ntheorem countp_pos {s} : 0 < countp p s \u2194 \u2203 a \u2208 s, p a :=\nby simp [countp_eq_card_filter, card_pos_iff_exists_mem]\n\n@[simp] theorem countp_sub [decidable_eq \u03b1] {s t : multiset \u03b1} (h : t \u2264 s) :\n  countp p (s - t) = countp p s - countp p t :=\nby simp [countp_eq_card_filter, h, filter_le_filter]\n\ntheorem countp_pos_of_mem {s a} (h : a \u2208 s) (pa : p a) : 0 < countp p s :=\ncountp_pos.2 \u27e8_, h, pa\u27e9\n\ntheorem countp_le_of_le {s t} (h : s \u2264 t) : countp p s \u2264 countp p t :=\nby simpa [countp_eq_card_filter] using card_le_of_le (filter_le_filter h)\n\n@[simp] theorem countp_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  countp p (filter q s) = countp (\u03bb a, p a \u2227 q a) s :=\nby simp [countp_eq_card_filter]\n\nend\n\n\/- count -\/\n\nsection\nvariable [decidable_eq \u03b1]\n\n\/-- `count a s` is the multiplicity of `a` in `s`. -\/\ndef count (a : \u03b1) : multiset \u03b1 \u2192 \u2115 := countp (eq a)\n\n@[simp] theorem coe_count (a : \u03b1) (l : list \u03b1) : count a (\u2191l) = l.count a := coe_countp _\n\n@[simp] theorem count_zero (a : \u03b1) : count a 0 = 0 := rfl\n\n@[simp] theorem count_cons_self (a : \u03b1) (s : multiset \u03b1) : count a (a::s) = succ (count a s) :=\ncountp_cons_of_pos _ rfl\n\n@[simp] theorem count_cons_of_ne {a b : \u03b1} (h : a \u2260 b) (s : multiset \u03b1) : count a (b::s) = count a s :=\ncountp_cons_of_neg _ h\n\ntheorem count_le_of_le (a : \u03b1) {s t} : s \u2264 t \u2192 count a s \u2264 count a t :=\ncountp_le_of_le\n\ntheorem count_le_count_cons (a b : \u03b1) (s : multiset \u03b1) : count a s \u2264 count a (b :: s) :=\ncount_le_of_le _ (le_cons_self _ _)\n\ntheorem count_singleton (a : \u03b1) : count a (a::0) = 1 :=\nby simp\n\n@[simp] theorem count_add (a : \u03b1) : \u2200 s t, count a (s + t) = count a s + count a t :=\ncountp_add\n\ninstance count.is_add_monoid_hom (a : \u03b1) : is_add_monoid_hom (count a : multiset \u03b1 \u2192 \u2115) :=\ncountp.is_add_monoid_hom\n\n@[simp] theorem count_smul (a : \u03b1) (n s) : count a (n \u2022 s) = n * count a s :=\nby induction n; simp [*, succ_smul', succ_mul]\n\ntheorem count_pos {a : \u03b1} {s : multiset \u03b1} : 0 < count a s \u2194 a \u2208 s :=\nby simp [count, countp_pos]\n\n@[simp] theorem count_eq_zero_of_not_mem {a : \u03b1} {s : multiset \u03b1} (h : a \u2209 s) : count a s = 0 :=\nby_contradiction $ \u03bb h', h $ count_pos.1 (nat.pos_of_ne_zero h')\n\ntheorem count_eq_zero {a : \u03b1} {s : multiset \u03b1} : count a s = 0 \u2194 a \u2209 s :=\niff_not_comm.1 $ count_pos.symm.trans pos_iff_ne_zero\n\n@[simp] theorem count_repeat (a : \u03b1) (n : \u2115) : count a (repeat a n) = n :=\nby simp [repeat]\n\n@[simp] theorem count_erase_self (a : \u03b1) (s : multiset \u03b1) : count a (erase s a) = pred (count a s) :=\nbegin\n  by_cases a \u2208 s,\n  { rw [(by rw cons_erase h : count a s = count a (a::erase s a)),\n        count_cons_self]; refl },\n  { rw [erase_of_not_mem h, count_eq_zero.2 h]; refl }\nend\n\n@[simp] theorem count_erase_of_ne {a b : \u03b1} (ab : a \u2260 b) (s : multiset \u03b1) : count a (erase s b) = count a s :=\nbegin\n  by_cases b \u2208 s,\n  { rw [\u2190 count_cons_of_ne ab, cons_erase h] },\n  { rw [erase_of_not_mem h] }\nend\n\n@[simp] theorem count_sub (a : \u03b1) (s t : multiset \u03b1) : count a (s - t) = count a s - count a t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb b t IH s, _),\n  rw [sub_cons, IH],\n  by_cases ab : a = b,\n  { subst b, rw [count_erase_self, count_cons_self, sub_succ, pred_sub] },\n  { rw [count_erase_of_ne ab, count_cons_of_ne ab] }\nend\n\n@[simp] theorem count_union (a : \u03b1) (s t : multiset \u03b1) : count a (s \u222a t) = max (count a s) (count a t) :=\nby simp [(\u222a), union, sub_add_eq_max, -add_comm]\n\n@[simp] theorem count_inter (a : \u03b1) (s t : multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t) :=\nbegin\n  apply @nat.add_left_cancel (count a (s - t)),\n  rw [\u2190 count_add, sub_add_inter, count_sub, sub_add_min],\nend\n\nlemma count_bind {m : multiset \u03b2} {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} :\n  count a (bind m f) = sum (m.map $ \u03bbb, count a $ f b) :=\nmultiset.induction_on m (by simp) (by simp)\n\ntheorem le_count_iff_repeat_le {a : \u03b1} {s : multiset \u03b1} {n : \u2115} : n \u2264 count a s \u2194 repeat a n \u2264 s :=\nquot.induction_on s $ \u03bb l, le_count_iff_repeat_sublist.trans repeat_le_coe.symm\n\n@[simp] theorem count_filter {p} [decidable_pred p]\n  {a} {s : multiset \u03b1} (h : p a) : count a (filter p s) = count a s :=\nquot.induction_on s $ \u03bb l, count_filter h\n\ntheorem ext {s t : multiset \u03b1} : s = t \u2194 \u2200 a, count a s = count a t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quotient.eq.trans perm_iff_count\n\n@[extensionality]\ntheorem ext' {s t : multiset \u03b1} : (\u2200 a, count a s = count a t) \u2192 s = t :=\next.2\n\n@[simp] theorem coe_inter (s t : list \u03b1) : (s \u2229 t : multiset \u03b1) = (s.bag_inter t : list \u03b1) :=\nby ext; simp\n\ntheorem le_iff_count {s t : multiset \u03b1} : s \u2264 t \u2194 \u2200 a, count a s \u2264 count a t :=\n\u27e8\u03bb h a, count_le_of_le a h, \u03bb al,\n by rw \u2190 (ext.2 (\u03bb a, by simp [max_eq_right (al a)]) : s \u222a t = t);\n    apply le_union_left\u27e9\n\ninstance : distrib_lattice (multiset \u03b1) :=\n{ le_sup_inf := \u03bb s t u, le_of_eq $ eq.symm $\n    ext.2 $ \u03bb a, by simp [max_min_distrib_left],\n  ..multiset.lattice.lattice }\n\ninstance : semilattice_sup_bot (multiset \u03b1) :=\n{ bot := 0,\n  bot_le := zero_le,\n  ..multiset.lattice.lattice }\n\nend\n\n\/- relator -\/\n\nsection rel\n\n\/-- `rel r s t` -- lift the relation `r` between two elements to a relation between `s` and `t`,\ns.t. there is a one-to-one mapping betweem elements in `s` and `t` following `r`. -\/\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : multiset \u03b1 \u2192 multiset \u03b2 \u2192 Prop\n| zero {} : rel 0 0\n| cons {a b as bs} : r a b \u2192 rel as bs \u2192 rel (a :: as) (b :: bs)\n\nrun_cmd tactic.mk_iff_of_inductive_prop `multiset.rel `multiset.rel_iff\n\nvariables {\u03b4 : Type*} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop}\n\nprivate lemma rel_flip_aux {s t} (h : rel r s t) : rel (flip r) t s :=\nrel.rec_on h rel.zero (assume _ _ _ _ h\u2080 h\u2081 ih, rel.cons h\u2080 ih)\n\nlemma rel_flip {s t} : rel (flip r) s t \u2194 rel r t s :=\n\u27e8rel_flip_aux, rel_flip_aux\u27e9\n\nlemma rel_eq_refl {s : multiset \u03b1} : rel (=) s s :=\nmultiset.induction_on s rel.zero (assume a s, rel.cons rfl)\n\nlemma rel_eq {s t : multiset \u03b1} : rel (=) s t \u2194 s = t :=\nbegin\n  split,\n  { assume h, induction h; simp * },\n  { assume h, subst h, exact rel_eq_refl }\nend\n\nlemma rel.mono {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (h : \u2200a b, r a b \u2192 p a b) (hst : rel r s t) : rel p s t :=\nbegin\n  induction hst,\n  case rel.zero { exact rel.zero },\n  case rel.cons : a b s t hab hst ih { exact ih.cons (h a b hab) }\nend\n\nlemma rel.add {s t u v} (hst : rel r s t) (huv : rel r u v) : rel r (s + u) (t + v) :=\nbegin\n  induction hst,\n  case rel.zero { simpa using huv },\n  case rel.cons : a b s t hab hst ih { simpa using ih.cons hab }\nend\n\nlemma rel_flip_eq  {s t : multiset \u03b1} : rel (\u03bba b, b = a) s t \u2194 s = t :=\nshow rel (flip (=)) s t \u2194 s = t, by rw [rel_flip, rel_eq, eq_comm]\n\n@[simp] lemma rel_zero_left {b : multiset \u03b2} : rel r 0 b \u2194 b = 0 :=\nby rw [rel_iff]; simp\n\n@[simp] lemma rel_zero_right {a : multiset \u03b1} : rel r a 0 \u2194 a = 0 :=\nby rw [rel_iff]; simp\n\nlemma rel_cons_left {a as bs} :\n  rel r (a :: as) bs \u2194 (\u2203b bs', r a b \u2227 rel r as bs' \u2227 bs = b :: bs') :=\nbegin\n  split,\n  { generalize hm : a :: as = m,\n    assume h,\n    induction h generalizing as,\n    case rel.zero { simp at hm, contradiction },\n    case rel.cons : a' b as' bs ha'b h ih {\n      rcases cons_eq_cons.1 hm with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n      { subst eq\u2081, subst eq\u2082, exact \u27e8b, bs, ha'b, h, rfl\u27e9 },\n      { rcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9,\n        exact \u27e8b', b::bs', h\u2081, eq\u2081.symm \u25b8 rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9  }\n    } },\n  { exact assume \u27e8b, bs', hab, h, eq\u27e9, eq.symm \u25b8 rel.cons hab h }\nend\n\nlemma rel_cons_right {as b bs} :\n  rel r as (b :: bs) \u2194 (\u2203a as', r a b \u2227 rel r as' bs \u2227 as = a :: as') :=\nbegin\n  rw [\u2190 rel_flip, rel_cons_left],\n  apply exists_congr, assume a,\n  apply exists_congr, assume as',\n  rw [rel_flip, flip]\nend\n\nlemma rel_add_left {as\u2080 as\u2081} :\n  \u2200{bs}, rel r (as\u2080 + as\u2081) bs \u2194 (\u2203bs\u2080 bs\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081) :=\nmultiset.induction_on as\u2080 (by simp)\n  begin\n    assume a s ih bs,\n    simp only [ih, cons_add, rel_cons_left],\n    split,\n    { assume h,\n      rcases h with \u27e8b, bs', hab, h, rfl\u27e9,\n      rcases h with \u27e8bs\u2080, bs\u2081, h\u2080, h\u2081, rfl\u27e9,\n      exact \u27e8b :: bs\u2080, bs\u2081, \u27e8b, bs\u2080, hab, h\u2080, rfl\u27e9, h\u2081, by simp\u27e9 },\n    { assume h,\n      rcases h with \u27e8bs\u2080, bs\u2081, h, h\u2081, rfl\u27e9,\n      rcases h with \u27e8b, bs, hab, h\u2080, rfl\u27e9,\n      exact \u27e8b, bs + bs\u2081, hab, \u27e8bs, bs\u2081, h\u2080, h\u2081, rfl\u27e9, by simp\u27e9 }\n  end\n\nlemma rel_add_right {as bs\u2080 bs\u2081} :\n  rel r as (bs\u2080 + bs\u2081) \u2194 (\u2203as\u2080 as\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081) :=\nby rw [\u2190 rel_flip, rel_add_left]; simp [rel_flip]\n\nlemma rel_map_left {s : multiset \u03b3} {f : \u03b3 \u2192 \u03b1} :\n  \u2200{t}, rel r (s.map f) t \u2194 rel (\u03bba b, r (f a) b) s t :=\nmultiset.induction_on s (by simp) (by simp [rel_cons_left] {contextual := tt})\n\nlemma rel_map_right {s : multiset \u03b1} {t : multiset \u03b3} {f : \u03b3 \u2192 \u03b2} :\n  rel r s (t.map f) \u2194 rel (\u03bba b, r a (f b)) s t :=\nby rw [\u2190 rel_flip, rel_map_left, \u2190 rel_flip]; refl\n\nlemma rel_join {s t} (h : rel (rel r) s t) : rel r s.join t.join :=\nbegin\n  induction h,\n  case rel.zero { simp },\n  case rel.cons : a b s t hab hst ih { simpa using hab.add ih }\nend\n\nlemma rel_map {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} (h : (r \u21d2 p) f g) (hst : rel r s t) :\n  rel p (s.map f) (t.map g) :=\nby rw [rel_map_left, rel_map_right]; exact hst.mono (assume a b, h)\n\nlemma rel_bind {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 multiset \u03b3} {g : \u03b2 \u2192 multiset \u03b4}\n  (h : (r \u21d2 rel p) f g) (hst : rel r s t) :\n  rel p (s.bind f) (t.bind g) :=\nby apply rel_join; apply rel_map; assumption\n\nlemma card_eq_card_of_rel {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) :\n  card s = card t :=\nby induction h; simp [*]\n\nlemma exists_mem_of_rel_of_mem {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) :\n  \u2200 {a : \u03b1} (ha : a \u2208 s), \u2203 b \u2208 t, r a b :=\nbegin\n  induction h with x y s t hxy hst ih,\n  { simp },\n  { assume a ha,\n    cases mem_cons.1 ha with ha ha,\n    { exact \u27e8y, mem_cons_self _ _, ha.symm \u25b8 hxy\u27e9 },\n    { rcases ih ha with \u27e8b, hbt, hab\u27e9,\n      exact \u27e8b, mem_cons.2 (or.inr hbt), hab\u27e9 } }\nend\n\nend rel\n\nsection map\n\ntheorem map_eq_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) {s t : multiset \u03b1} :\n  s.map f = t.map f \u2194 s = t :=\nby rw [\u2190 rel_eq, \u2190 rel_eq, rel_map_left, rel_map_right]; simp [hf.eq_iff]\n\ntheorem injective_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  function.injective (multiset.map f) :=\nassume x y, (map_eq_map hf).1\n\nend map\n\nsection quot\n\ntheorem map_mk_eq_map_mk_of_rel {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s t : multiset \u03b1} (hst : s.rel r t) :\n s.map (quot.mk r) = t.map (quot.mk r) :=\nrel.rec_on hst rfl $ assume a b s t hab hst ih, by simp [ih, quot.sound hab]\n\ntheorem exists_multiset_eq_map_quot_mk {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : multiset (quot r)) :\n  \u2203t:multiset \u03b1, s = t.map (quot.mk r) :=\nmultiset.induction_on s \u27e80, rfl\u27e9 $\n  assume a s \u27e8t, ht\u27e9, quot.induction_on a $ assume a, ht.symm \u25b8 \u27e8a::t, (map_cons _ _ _).symm\u27e9\n\ntheorem induction_on_multiset_quot\n  {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {p : multiset (quot r) \u2192 Prop} (s : multiset (quot r)) :\n  (\u2200s:multiset \u03b1, p (s.map (quot.mk r))) \u2192 p s :=\nmatch s, exists_multiset_eq_map_quot_mk s with _, \u27e8t, rfl\u27e9 := assume h, h _ end\n\nend quot\n\n\/- disjoint -\/\n\n\/-- `disjoint s t` means that `s` and `t` have no elements in common. -\/\ndef disjoint (s t : multiset \u03b1) : Prop := \u2200 \u2983a\u2984, a \u2208 s \u2192 a \u2208 t \u2192 false\n\n@[simp] theorem coe_disjoint (l\u2081 l\u2082 : list \u03b1) : @disjoint \u03b1 l\u2081 l\u2082 \u2194 l\u2081.disjoint l\u2082 := iff.rfl\n\ntheorem disjoint.symm {s t : multiset \u03b1} (d : disjoint s t) : disjoint t s\n| a i\u2082 i\u2081 := d i\u2081 i\u2082\n\n@[simp] theorem disjoint_comm {s t : multiset \u03b1} : disjoint s t \u2194 disjoint t s :=\n\u27e8disjoint.symm, disjoint.symm\u27e9\n\ntheorem disjoint_left {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 s \u2192 a \u2209 t := iff.rfl\n\ntheorem disjoint_right {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 t \u2192 a \u2209 s :=\ndisjoint_comm\n\ntheorem disjoint_iff_ne {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 a \u2208 s, \u2200 b \u2208 t, a \u2260 b :=\nby simp [disjoint_left, imp_not_comm]\n\ntheorem disjoint_of_subset_left {s t u : multiset \u03b1} (h : s \u2286 u) (d : disjoint u t) : disjoint s t\n| x m\u2081 := d (h m\u2081)\n\ntheorem disjoint_of_subset_right {s t u : multiset \u03b1} (h : t \u2286 u) (d : disjoint s u) : disjoint s t\n| x m m\u2081 := d m (h m\u2081)\n\ntheorem disjoint_of_le_left {s t u : multiset \u03b1} (h : s \u2264 u) : disjoint u t \u2192 disjoint s t :=\ndisjoint_of_subset_left (subset_of_le h)\n\ntheorem disjoint_of_le_right {s t u : multiset \u03b1} (h : t \u2264 u) : disjoint s u \u2192 disjoint s t :=\ndisjoint_of_subset_right (subset_of_le h)\n\n@[simp] theorem zero_disjoint (l : multiset \u03b1) : disjoint 0 l\n| a := (not_mem_nil a).elim\n\n@[simp] theorem singleton_disjoint {l : multiset \u03b1} {a : \u03b1} : disjoint (a::0) l \u2194 a \u2209 l :=\nby simp [disjoint]; refl\n\n@[simp] theorem disjoint_singleton {l : multiset \u03b1} {a : \u03b1} : disjoint l (a::0) \u2194 a \u2209 l :=\nby rw disjoint_comm; simp\n\n@[simp] theorem disjoint_add_left {s t u : multiset \u03b1} :\n  disjoint (s + t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_add_right {s t u : multiset \u03b1} :\n  disjoint s (t + u) \u2194 disjoint s t \u2227 disjoint s u :=\ndisjoint_comm.trans $ by simp [disjoint_append_left]\n\n@[simp] theorem disjoint_cons_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (a::s) t \u2194 a \u2209 t \u2227 disjoint s t :=\n(@disjoint_add_left _ (a::0) s t).trans $ by simp\n\n@[simp] theorem disjoint_cons_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (a::t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp [disjoint_cons_left]\n\ntheorem inter_eq_zero_iff_disjoint [decidable_eq \u03b1] {s t : multiset \u03b1} : s \u2229 t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\n@[simp] theorem disjoint_union_left [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint (s \u222a t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_union_right [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint s (t \u222a u) \u2194 disjoint s t \u2227 disjoint s u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\nlemma disjoint_map_map {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {s : multiset \u03b1} {t : multiset \u03b2} :\n  disjoint (s.map f) (t.map g) \u2194 (\u2200a\u2208s, \u2200b\u2208t, f a \u2260 g b) :=\nbegin\n  simp [disjoint],\n  split,\n  from assume h a ha b hb eq, h _ ha rfl _ hb eq.symm,\n  from assume h c a ha eq\u2081 b hb eq\u2082, h _ ha _ hb (eq\u2082.symm \u25b8 eq\u2081)\nend\n\n\/-- `pairwise r m` states that there exists a list of the elements s.t. `r` holds pairwise on this list. -\/\ndef pairwise (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (m : multiset \u03b1) : Prop :=\n\u2203l:list \u03b1, m = l \u2227 l.pairwise r\n\nlemma pairwise_coe_iff_pairwise {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (hr : symmetric r) {l : list \u03b1} :\n  multiset.pairwise r l \u2194 l.pairwise r :=\niff.intro\n  (assume \u27e8l', eq, h\u27e9, (list.perm_pairwise hr (quotient.exact eq)).2 h)\n  (assume h, \u27e8l, rfl, h\u27e9)\n\n\/- nodup -\/\n\n\/-- `nodup s` means that `s` has no duplicates, i.e. the multiplicity of\n  any element is at most 1. -\/\ndef nodup (s : multiset \u03b1) : Prop :=\nquot.lift_on s nodup (\u03bb s t p, propext $ perm_nodup p)\n\n@[simp] theorem coe_nodup {l : list \u03b1} : @nodup \u03b1 l \u2194 l.nodup := iff.rfl\n\n@[simp] theorem forall_mem_ne {a : \u03b1} {l : list \u03b1} : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 \u00aca = a') \u2194 a \u2209 l :=\n\u27e8\u03bb h m, h _ m rfl, \u03bb h a' m e, h (e.symm \u25b8 m)\u27e9\n\n@[simp] theorem nodup_zero : @nodup \u03b1 0 := pairwise.nil\n\n@[simp] theorem nodup_cons {a : \u03b1} {s : multiset \u03b1} : nodup (a::s) \u2194 a \u2209 s \u2227 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_cons\n\ntheorem nodup_cons_of_nodup {a : \u03b1} {s : multiset \u03b1} (m : a \u2209 s) (n : nodup s) : nodup (a::s) :=\nnodup_cons.2 \u27e8m, n\u27e9\n\ntheorem nodup_singleton : \u2200 a : \u03b1, nodup (a::0) := nodup_singleton\n\ntheorem nodup_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : nodup s :=\n(nodup_cons.1 h).2\n\ntheorem not_mem_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : a \u2209 s :=\n(nodup_cons.1 h).1\n\ntheorem nodup_of_le {s t : multiset \u03b1} (h : s \u2264 t) : nodup t \u2192 nodup s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, nodup_of_sublist\n\ntheorem not_nodup_pair : \u2200 a : \u03b1, \u00ac nodup (a::a::0) := not_nodup_pair\n\ntheorem nodup_iff_le {s : multiset \u03b1} : nodup s \u2194 \u2200 a : \u03b1, \u00ac a::a::0 \u2264 s :=\nquot.induction_on s $ \u03bb l, nodup_iff_sublist.trans $ forall_congr $ \u03bb a,\nnot_congr (@repeat_le_coe _ a 2 _).symm\n\ntheorem nodup_iff_count_le_one [decidable_eq \u03b1] {s : multiset \u03b1} : nodup s \u2194 \u2200 a, count a s \u2264 1 :=\nquot.induction_on s $ \u03bb l, nodup_iff_count_le_one\n\n@[simp] theorem count_eq_one_of_mem [decidable_eq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n  (d : nodup s) (h : a \u2208 s) : count a s = 1 :=\nle_antisymm (nodup_iff_count_le_one.1 d a) (count_pos.2 h)\n\nlemma pairwise_of_nodup {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : multiset \u03b1} :\n  (\u2200a\u2208s, \u2200b\u2208s, a \u2260 b \u2192 r a b) \u2192 nodup s \u2192 pairwise r s :=\nquotient.induction_on s $ assume l h hl, \u27e8l, rfl, hl.imp_of_mem $ assume a b ha hb, h a ha b hb\u27e9\n\nlemma forall_of_pairwise {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (H : symmetric r) {s : multiset \u03b1}\n   (hs : pairwise r s) : (\u2200a\u2208s, \u2200b\u2208s, a \u2260 b \u2192 r a b) :=\nlet \u27e8l, hl\u2081, hl\u2082\u27e9 := hs in hl\u2081.symm \u25b8 list.forall_of_pairwise H hl\u2082\n\ntheorem nodup_add {s t : multiset \u03b1} : nodup (s + t) \u2194 nodup s \u2227 nodup t \u2227 disjoint s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, nodup_append\n\ntheorem disjoint_of_nodup_add {s t : multiset \u03b1} (d : nodup (s + t)) : disjoint s t :=\n(nodup_add.1 d).2.2\n\ntheorem nodup_add_of_nodup {s t : multiset \u03b1} (d\u2081 : nodup s) (d\u2082 : nodup t) : nodup (s + t) \u2194 disjoint s t :=\nby simp [nodup_add, d\u2081, d\u2082]\n\ntheorem nodup_of_nodup_map (f : \u03b1 \u2192 \u03b2) {s : multiset \u03b1} : nodup (map f s) \u2192 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_of_nodup_map f\n\ntheorem nodup_map_on {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200x\u2208s, \u2200y\u2208s, f x = f y \u2192 x = y) \u2192\n  nodup s \u2192 nodup (map f s) :=\nquot.induction_on s $ \u03bb l, nodup_map_on\n\ntheorem nodup_map {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} (hf : function.injective f) : nodup s \u2192 nodup (map f s) :=\nnodup_map_on (\u03bb x _ y _ h, hf h)\n\ntheorem nodup_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] {s} : nodup s \u2192 nodup (filter p s) :=\nquot.induction_on s $ \u03bb l, nodup_filter p\n\n@[simp] theorem nodup_attach {s : multiset \u03b1} : nodup (attach s) \u2194 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_attach\n\ntheorem nodup_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {s : multiset \u03b1} {H}\n  (hf : \u2200 a ha b hb, f a ha = f b hb \u2192 a = b) : nodup s \u2192 nodup (pmap f s H) :=\nquot.induction_on s (\u03bb l H, nodup_pmap hf) H\n\ninstance nodup_decidable [decidable_eq \u03b1] (s : multiset \u03b1) : decidable (nodup s) :=\nquotient.rec_on_subsingleton s $ \u03bb l, l.nodup_decidable\n\ntheorem nodup_erase_eq_filter [decidable_eq \u03b1] (a : \u03b1) {s} : nodup s \u2192 s.erase a = filter (\u2260 a) s :=\nquot.induction_on s $ \u03bb l d, congr_arg coe $ nodup_erase_eq_filter a d\n\ntheorem nodup_erase_of_nodup [decidable_eq \u03b1] (a : \u03b1) {l} : nodup l \u2192 nodup (l.erase a) :=\nnodup_of_le (erase_le _ _)\n\ntheorem mem_erase_iff_of_nodup [decidable_eq \u03b1] {a b : \u03b1} {l} (d : nodup l) :\n  a \u2208 l.erase b \u2194 a \u2260 b \u2227 a \u2208 l :=\nby rw nodup_erase_eq_filter b d; simp [and_comm]\n\ntheorem mem_erase_of_nodup [decidable_eq \u03b1] {a : \u03b1} {l} (h : nodup l) : a \u2209 l.erase a :=\nby rw mem_erase_iff_of_nodup h; simp\n\ntheorem nodup_product {s : multiset \u03b1} {t : multiset \u03b2} : nodup s \u2192 nodup t \u2192 nodup (product s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, by simp [nodup_product d\u2081 d\u2082]\n\ntheorem nodup_sigma {\u03c3 : \u03b1 \u2192 Type*} {s : multiset \u03b1} {t : \u03a0 a, multiset (\u03c3 a)} :\n  nodup s \u2192 (\u2200 a, nodup (t a)) \u2192 nodup (s.sigma t) :=\nquot.induction_on s $ assume l\u2081,\nbegin\n  choose f hf using assume a, quotient.exists_rep (t a),\n  rw show t = \u03bb a, f a, from (eq.symm $ funext $ \u03bb a, hf a),\n  simpa using nodup_sigma\nend\n\ntheorem nodup_filter_map (f : \u03b1 \u2192 option \u03b2) {s : multiset \u03b1}\n  (H : \u2200 (a a' : \u03b1) (b : \u03b2), b \u2208 f a \u2192 b \u2208 f a' \u2192 a = a') :\n  nodup s \u2192 nodup (filter_map f s) :=\nquot.induction_on s $ \u03bb l, nodup_filter_map H\n\ntheorem nodup_range (n : \u2115) : nodup (range n) := nodup_range _\n\ntheorem nodup_inter_left [decidable_eq \u03b1] {s : multiset \u03b1} (t) : nodup s \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_left _ _\n\ntheorem nodup_inter_right [decidable_eq \u03b1] (s) {t : multiset \u03b1} : nodup t \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_right _ _\n\n@[simp] theorem nodup_union [decidable_eq \u03b1] {s t : multiset \u03b1} : nodup (s \u222a t) \u2194 nodup s \u2227 nodup t :=\n\u27e8\u03bb h, \u27e8nodup_of_le (le_union_left _ _) h, nodup_of_le (le_union_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, nodup_iff_count_le_one.2 $ \u03bb a, by rw [count_union]; exact\n   max_le (nodup_iff_count_le_one.1 h\u2081 a) (nodup_iff_count_le_one.1 h\u2082 a)\u27e9\n\n@[simp] theorem nodup_powerset {s : multiset \u03b1} : nodup (powerset s) \u2194 nodup s :=\n\u27e8\u03bb h, nodup_of_nodup_map _ (nodup_of_le (map_single_le_powerset _) h),\n  quotient.induction_on s $ \u03bb l h,\n  by simp; refine list.nodup_map_on _ (nodup_sublists'.2 h); exact\n  \u03bb x sx y sy e,\n    (perm_ext_sublist_nodup h (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1\n      (quotient.exact e)\u27e9\n\ntheorem nodup_powerset_len {n : \u2115} {s : multiset \u03b1}\n  (h : nodup s) : nodup (powerset_len n s) :=\nnodup_of_le (powerset_len_le_powerset _ _) (nodup_powerset.2 h)\n\n@[simp] lemma nodup_bind {s : multiset \u03b1} {t : \u03b1 \u2192 multiset \u03b2} :\n  nodup (bind s t) \u2194 ((\u2200a\u2208s, nodup (t a)) \u2227 (s.pairwise (\u03bba b, disjoint (t a) (t b)))) :=\nhave h\u2081 : \u2200a, \u2203l:list \u03b2, t a = l, from\n  assume a, quot.induction_on (t a) $ assume l, \u27e8l, rfl\u27e9,\nlet \u27e8t', h'\u27e9 := classical.axiom_of_choice h\u2081 in\nhave t = \u03bba, t' a, from funext h',\nhave hd : symmetric (\u03bba b, list.disjoint (t' a) (t' b)), from assume a b h, h.symm,\nquot.induction_on s $ by simp [this, list.nodup_bind, pairwise_coe_iff_pairwise hd]\n\ntheorem nodup_ext {s t : multiset \u03b1} : nodup s \u2192 nodup t \u2192 (s = t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, quotient.eq.trans $ perm_ext d\u2081 d\u2082\n\ntheorem le_iff_subset {s t : multiset \u03b1} : nodup s \u2192 (s \u2264 t \u2194 s \u2286 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d, \u27e8subset_of_le, subperm_of_subset_nodup d\u27e9\n\ntheorem range_le {m n : \u2115} : range m \u2264 range n \u2194 m \u2264 n :=\n(le_iff_subset (nodup_range _)).trans range_subset\n\ntheorem mem_sub_of_nodup [decidable_eq \u03b1] {a : \u03b1} {s t : multiset \u03b1} (d : nodup s) :\n  a \u2208 s - t \u2194 a \u2208 s \u2227 a \u2209 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (sub_le_self _ _) h, \u03bb h',\n  by refine count_eq_zero.1 _ h; rw [count_sub a s t, nat.sub_eq_zero_iff_le];\n     exact le_trans (nodup_iff_count_le_one.1 d _) (count_pos.2 h')\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, or.resolve_right (mem_add.1 $ mem_of_le (le_sub_add _ _) h\u2081) h\u2082\u27e9\n\nsection\nvariable [decidable_eq \u03b1]\n\n\/- erase_dup -\/\n\n\/-- `erase_dup s` removes duplicates from `s`, yielding a `nodup` multiset. -\/\ndef erase_dup (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase_dup : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_erase_dup_of_perm p))\n\n@[simp] theorem coe_erase_dup (l : list \u03b1) : @erase_dup \u03b1 _ l = l.erase_dup := rfl\n\n@[simp] theorem erase_dup_zero : @erase_dup \u03b1 _ 0 = 0 := rfl\n\n@[simp] theorem mem_erase_dup {a : \u03b1} {s : multiset \u03b1} : a \u2208 erase_dup s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_erase_dup\n\n@[simp] theorem erase_dup_cons_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192\n  erase_dup (a::s) = erase_dup s :=\nquot.induction_on s $ \u03bb l m, @congr_arg _ _ _ _ coe $ erase_dup_cons_of_mem m\n\n@[simp] theorem erase_dup_cons_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192\n  erase_dup (a::s) = a :: erase_dup s :=\nquot.induction_on s $ \u03bb l m, congr_arg coe $ erase_dup_cons_of_not_mem m\n\ntheorem erase_dup_le (s : multiset \u03b1) : erase_dup s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ erase_dup_sublist _\n\ntheorem erase_dup_subset (s : multiset \u03b1) : erase_dup s \u2286 s :=\nsubset_of_le $ erase_dup_le _\n\ntheorem subset_erase_dup (s : multiset \u03b1) : s \u2286 erase_dup s :=\n\u03bb a, mem_erase_dup.2\n\n@[simp] theorem erase_dup_subset' {s t : multiset \u03b1} : erase_dup s \u2286 t \u2194 s \u2286 t :=\n\u27e8subset.trans (subset_erase_dup _), subset.trans (erase_dup_subset _)\u27e9\n\n@[simp] theorem subset_erase_dup' {s t : multiset \u03b1} : s \u2286 erase_dup t \u2194 s \u2286 t :=\n\u27e8\u03bb h, subset.trans h (erase_dup_subset _), \u03bb h, subset.trans h (subset_erase_dup _)\u27e9\n\n@[simp] theorem nodup_erase_dup (s : multiset \u03b1) : nodup (erase_dup s) :=\nquot.induction_on s nodup_erase_dup\n\ntheorem erase_dup_eq_self {s : multiset \u03b1} : erase_dup s = s \u2194 nodup s :=\n\u27e8\u03bb e, e \u25b8 nodup_erase_dup s,\n quot.induction_on s $ \u03bb l h, congr_arg coe $ erase_dup_eq_self.2 h\u27e9\n\ntheorem erase_dup_eq_zero {s : multiset \u03b1} : erase_dup s = 0 \u2194 s = 0 :=\n\u27e8\u03bb h, eq_zero_of_subset_zero $ h \u25b8 subset_erase_dup _,\n \u03bb h, h.symm \u25b8 erase_dup_zero\u27e9\n\n@[simp] theorem erase_dup_singleton {a : \u03b1} : erase_dup (a :: 0) = a :: 0 :=\nerase_dup_eq_self.2 $ nodup_singleton _\n\ntheorem le_erase_dup {s t : multiset \u03b1} : s \u2264 erase_dup t \u2194 s \u2264 t \u2227 nodup s :=\n\u27e8\u03bb h, \u27e8le_trans h (erase_dup_le _), nodup_of_le h (nodup_erase_dup _)\u27e9,\n \u03bb \u27e8l, d\u27e9, (le_iff_subset d).2 $ subset.trans (subset_of_le l) (subset_erase_dup _)\u27e9\n\ntheorem erase_dup_ext {s t : multiset \u03b1} : erase_dup s = erase_dup t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t :=\nby simp [nodup_ext]\n\ntheorem erase_dup_map_erase_dup_eq [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  erase_dup (map f (erase_dup s)) = erase_dup (map f s) := by simp [erase_dup_ext]\n\n\/- finset insert -\/\n\n\/-- `ndinsert a s` is the lift of the list `insert` operation. This operation\n  does not respect multiplicities, unlike `cons`, but it is suitable as\n  an insert operation on `finset`. -\/\ndef ndinsert (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.insert a : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_insert a p))\n\n@[simp] theorem coe_ndinsert (a : \u03b1) (l : list \u03b1) : ndinsert a l = (insert a l : list \u03b1) := rfl\n\n@[simp] theorem ndinsert_zero (a : \u03b1) : ndinsert a 0 = a::0 := rfl\n\n@[simp] theorem ndinsert_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 ndinsert a s = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_mem h\n\n@[simp] theorem ndinsert_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 ndinsert a s = a :: s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_not_mem h\n\n@[simp] theorem mem_ndinsert {a b : \u03b1} {s : multiset \u03b1} : a \u2208 ndinsert b s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_insert_iff\n\n@[simp] theorem le_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : s \u2264 ndinsert a s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ sublist_of_suffix $ suffix_insert _ _\n\n@[simp] theorem mem_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 ndinsert a s :=\nmem_ndinsert.2 (or.inl rfl)\n\n@[simp] theorem mem_ndinsert_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 ndinsert b s :=\nmem_ndinsert.2 (or.inr h)\n\n@[simp] theorem length_ndinsert_of_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2208 s) :\n  card (ndinsert a s) = card s :=\nby simp [h]\n\n@[simp] theorem length_ndinsert_of_not_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2209 s) :\n  card (ndinsert a s) = card s + 1 :=\nby simp [h]\n\ntheorem erase_dup_cons {a : \u03b1} {s : multiset \u03b1} :\n  erase_dup (a::s) = ndinsert a (erase_dup s) :=\nby by_cases a \u2208 s; simp [h]\n\ntheorem nodup_ndinsert (a : \u03b1) {s : multiset \u03b1} : nodup s \u2192 nodup (ndinsert a s) :=\nquot.induction_on s $ \u03bb l, nodup_insert\n\ntheorem ndinsert_le {a : \u03b1} {s t : multiset \u03b1} : ndinsert a s \u2264 t \u2194 s \u2264 t \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8le_trans (le_ndinsert_self _ _) h, mem_of_le h (mem_ndinsert_self _ _)\u27e9,\n \u03bb \u27e8l, m\u27e9, if h : a \u2208 s then by simp [h, l] else\n   by rw [ndinsert_of_not_mem h, \u2190 cons_erase m, cons_le_cons_iff,\n          \u2190 le_cons_of_not_mem h, cons_erase m]; exact l\u27e9\n\nlemma attach_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (s.ndinsert a).attach =\n    ndinsert \u27e8a, mem_ndinsert_self a s\u27e9 (s.attach.map $ \u03bbp, \u27e8p.1, mem_ndinsert_of_mem p.2\u27e9) :=\nhave eq : \u2200h : \u2200(p : {x \/\/ x \u2208 s}), p.1 \u2208 s,\n    (\u03bb (p : {x \/\/ x \u2208 s}), \u27e8p.val, h p\u27e9 : {x \/\/ x \u2208 s} \u2192 {x \/\/ x \u2208 s}) = id, from\n  assume h, funext $ assume p, subtype.eq rfl,\nhave \u2200t (eq : s.ndinsert a = t), t.attach = ndinsert \u27e8a, eq \u25b8 mem_ndinsert_self a s\u27e9\n  (s.attach.map $ \u03bbp, \u27e8p.1, eq \u25b8 mem_ndinsert_of_mem p.2\u27e9),\nbegin\n  intros t ht,\n  by_cases a \u2208 s,\n  { rw [ndinsert_of_mem h] at ht,\n    subst ht,\n    rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)] },\n  { rw [ndinsert_of_not_mem h] at ht,\n    subst ht,\n    simp [attach_cons, h] }\nend,\nthis _ rfl\n\n@[simp] theorem disjoint_ndinsert_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (ndinsert a s) t \u2194 a \u2209 t \u2227 disjoint s t :=\niff.trans (by simp [disjoint]) disjoint_cons_left\n\n@[simp] theorem disjoint_ndinsert_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (ndinsert a t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp\n\n\/- finset union -\/\n\n\/-- `ndunion s t` is the lift of the list `union` operation. This operation\n  does not respect multiplicities, unlike `s \u222a t`, but it is suitable as\n  a union operation on `finset`. (`s \u222a t` would also work as a union operation\n  on finset, but this is more efficient.) -\/\ndef ndunion (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.union l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_union p\u2081 p\u2082\n\n@[simp] theorem coe_ndunion (l\u2081 l\u2082 : list \u03b1) : @ndunion \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u222a l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndunion (s : multiset \u03b1) : ndunion 0 s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem cons_ndunion (s t : multiset \u03b1) (a : \u03b1) : ndunion (a :: s) t = ndinsert a (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, rfl\n\n@[simp] theorem mem_ndunion {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndunion s t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.mem_union\n\ntheorem le_ndunion_right (s t : multiset \u03b1) : t \u2264 ndunion s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ sublist_of_suffix $ suffix_union_right _ _\n\ntheorem ndunion_le_add (s t : multiset \u03b1) : ndunion s t \u2264 s + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_of_sublist $ union_sublist_append _ _\n\ntheorem ndunion_le {s t u : multiset \u03b1} : ndunion s t \u2264 u \u2194 s \u2286 u \u2227 t \u2264 u :=\nmultiset.induction_on s (by simp) (by simp [ndinsert_le, and_comm, and.left_comm] {contextual := tt})\n\ntheorem subset_ndunion_left (s t : multiset \u03b1) : s \u2286 ndunion s t :=\n\u03bb a h, mem_ndunion.2 $ or.inl h\n\ntheorem le_ndunion_left {s} (t : multiset \u03b1) (d : nodup s) : s \u2264 ndunion s t :=\n(le_iff_subset d).2 $ subset_ndunion_left _ _\n\ntheorem ndunion_le_union (s t : multiset \u03b1) : ndunion s t \u2264 s \u222a t :=\nndunion_le.2 \u27e8subset_of_le (le_union_left _ _), le_union_right _ _\u27e9\n\ntheorem nodup_ndunion (s : multiset \u03b1) {t : multiset \u03b1} : nodup t \u2192 nodup (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.nodup_union _\n\n@[simp] theorem ndunion_eq_union {s t : multiset \u03b1} (d : nodup s) : ndunion s t = s \u222a t :=\nle_antisymm (ndunion_le_union _ _) $ union_le (le_ndunion_left _ d) (le_ndunion_right _ _)\n\ntheorem erase_dup_add (s t : multiset \u03b1) : erase_dup (s + t) = ndunion s (erase_dup t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ erase_dup_append _ _\n\n\/- finset inter -\/\n\n\/-- `ndinter s t` is the lift of the list `\u2229` operation. This operation\n  does not respect multiplicities, unlike `s \u2229 t`, but it is suitable as\n  an intersection operation on `finset`. (`s \u2229 t` would also work as a union operation\n  on finset, but this is more efficient.) -\/\ndef ndinter (s t : multiset \u03b1) : multiset \u03b1 := filter (\u2208 t) s\n\n@[simp] theorem coe_ndinter (l\u2081 l\u2082 : list \u03b1) : @ndinter \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u2229 l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndinter (s : multiset \u03b1) : ndinter 0 s = 0 := rfl\n\n@[simp] theorem cons_ndinter_of_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2208 t) :\n  ndinter (a::s) t = a :: (ndinter s t) := by simp [ndinter, h]\n\n@[simp] theorem ndinter_cons_of_not_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2209 t) :\n  ndinter (a::s) t = ndinter s t := by simp [ndinter, h]\n\n@[simp] theorem mem_ndinter {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndinter s t \u2194 a \u2208 s \u2227 a \u2208 t :=\nmem_filter\n\ntheorem nodup_ndinter {s : multiset \u03b1} (t : multiset \u03b1) : nodup s \u2192 nodup (ndinter s t) :=\nnodup_filter _\n\ntheorem le_ndinter {s t u : multiset \u03b1} : s \u2264 ndinter t u \u2194 s \u2264 t \u2227 s \u2286 u :=\nby simp [ndinter, le_filter, subset_iff]\n\ntheorem ndinter_le_left (s t : multiset \u03b1) : ndinter s t \u2264 s :=\n(le_ndinter.1 (le_refl _)).1\n\ntheorem ndinter_subset_right (s t : multiset \u03b1) : ndinter s t \u2286 t :=\n(le_ndinter.1 (le_refl _)).2\n\ntheorem ndinter_le_right {s} (t : multiset \u03b1) (d : nodup s) : ndinter s t \u2264 t :=\n(le_iff_subset $ nodup_ndinter _ d).2 (ndinter_subset_right _ _)\n\ntheorem inter_le_ndinter (s t : multiset \u03b1) : s \u2229 t \u2264 ndinter s t :=\nle_ndinter.2 \u27e8inter_le_left _ _, subset_of_le $ inter_le_right _ _\u27e9\n\n@[simp] theorem ndinter_eq_inter {s t : multiset \u03b1} (d : nodup s) : ndinter s t = s \u2229 t :=\nle_antisymm (le_inter (ndinter_le_left _ _) (ndinter_le_right _ d)) (inter_le_ndinter _ _)\n\ntheorem ndinter_eq_zero_iff_disjoint {s t : multiset \u03b1} : ndinter s t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\nend\n\n\/- fold -\/\nsection fold\nvariables (op : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [hc : is_commutative \u03b1 op] [ha : is_associative \u03b1 op]\nlocal notation a * b := op a b\ninclude hc ha\n\n\/-- `fold op b s` folds a commutative associative operation `op` over\n  the multiset `s`. -\/\ndef fold : \u03b1 \u2192 multiset \u03b1 \u2192 \u03b1 := foldr op (left_comm _ hc.comm ha.assoc)\n\ntheorem fold_eq_foldr (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldr op (left_comm _ hc.comm ha.assoc) b s := rfl\n\n@[simp] theorem coe_fold_r (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldr op b := rfl\n\ntheorem coe_fold_l (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldl op b :=\n(coe_foldr_swap op _ b l).trans $ by simp [hc.comm]\n\ntheorem fold_eq_foldl (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldl op (right_comm _ hc.comm ha.assoc) b s :=\nquot.induction_on s $ \u03bb l, coe_fold_l _ _ _\n\n@[simp] theorem fold_zero (b : \u03b1) : (0 : multiset \u03b1).fold op b = b := rfl\n\n@[simp] theorem fold_cons_left : \u2200 (b a : \u03b1) (s : multiset \u03b1),\n  (a :: s).fold op b = a * s.fold op b := foldr_cons _ _\n\ntheorem fold_cons_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op b * a :=\nby simp [hc.comm]\n\ntheorem fold_cons'_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (b * a) :=\nby rw [fold_eq_foldl, foldl_cons, \u2190 fold_eq_foldl]\n\ntheorem fold_cons'_left (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (a * b) :=\nby rw [fold_cons'_right, hc.comm]\n\ntheorem fold_add (b\u2081 b\u2082 : \u03b1) (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).fold op (b\u2081 * b\u2082) = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nmultiset.induction_on s\u2082\n  (by rw [add_zero, fold_zero, \u2190 fold_cons'_right, \u2190 fold_cons_right op])\n  (by simp {contextual := tt}; cc)\n\ntheorem fold_singleton (b a : \u03b1) : (a::0 : multiset \u03b1).fold op b = a * b := by simp\n\ntheorem fold_distrib {f g : \u03b2 \u2192 \u03b1} (u\u2081 u\u2082 : \u03b1) (s : multiset \u03b2) :\n  (s.map (\u03bbx, f x * g x)).fold op (u\u2081 * u\u2082) = (s.map f).fold op u\u2081 * (s.map g).fold op u\u2082 :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt}; cc)\n\ntheorem fold_hom {op' : \u03b2 \u2192 \u03b2 \u2192 \u03b2} [is_commutative \u03b2 op'] [is_associative \u03b2 op']\n  {m : \u03b1 \u2192 \u03b2} (hm : \u2200x y, m (op x y) = op' (m x) (m y)) (b : \u03b1) (s : multiset \u03b1) :\n  (s.map m).fold op' (m b) = m (s.fold op b) :=\nmultiset.induction_on s (by simp) (by simp [hm] {contextual := tt})\n\ntheorem fold_union_inter [decidable_eq \u03b1] (s\u2081 s\u2082 : multiset \u03b1) (b\u2081 b\u2082 : \u03b1) :\n  (s\u2081 \u222a s\u2082).fold op b\u2081 * (s\u2081 \u2229 s\u2082).fold op b\u2082 = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nby rw [\u2190 fold_add op, union_add_inter, fold_add op]\n\n@[simp] theorem fold_erase_dup_idem [decidable_eq \u03b1] [hi : is_idempotent \u03b1 op] (s : multiset \u03b1) (b : \u03b1) :\n  (erase_dup s).fold op b = s.fold op b :=\nmultiset.induction_on s (by simp) $ \u03bb a s IH, begin\n  by_cases a \u2208 s; simp [IH, h],\n  show fold op b s = op a (fold op b s),\n  rw [\u2190 cons_erase h, fold_cons_left, \u2190 ha.assoc, hi.idempotent],\nend\n\nend fold\n\ntheorem le_smul_erase_dup [decidable_eq \u03b1] (s : multiset \u03b1) :\n  \u2203 n : \u2115, s \u2264 n \u2022 erase_dup s :=\n\u27e8(s.map (\u03bb a, count a s)).fold max 0, le_iff_count.2 $ \u03bb a, begin\n  rw count_smul, by_cases a \u2208 s,\n  { refine le_trans _ (mul_le_mul_left _ $ count_pos.2 $ mem_erase_dup.2 h),\n    have : count a s \u2264 fold max 0 (map (\u03bb a, count a s) (a :: erase s a));\n    [simp [le_max_left], simpa [cons_erase h]] },\n  { simp [count_eq_zero.2 h, nat.zero_le] }\nend\u27e9\n\nsection sup\nvariables [semilattice_sup_bot \u03b1]\n\n\/-- Supremum of a multiset: `sup {a, b, c} = a \u2294 b \u2294 c` -\/\ndef sup (s : multiset \u03b1) : \u03b1 := s.fold (\u2294) \u22a5\n\n@[simp] lemma sup_zero : (0 : multiset \u03b1).sup = \u22a5 :=\nfold_zero _ _\n\n@[simp] lemma sup_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).sup = a \u2294 s.sup :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma sup_singleton {a : \u03b1} : (a::0).sup = a := by simp\n\n@[simp] lemma sup_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\neq.trans (by simp [sup]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma sup_erase_dup (s : multiset \u03b1) : (erase_dup s).sup = s.sup :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma sup_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).sup = a \u2294 s.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_cons]; simp\n\nlemma sup_le {s : multiset \u03b1} {a : \u03b1} : s.sup \u2264 a \u2194 (\u2200b \u2208 s, b \u2264 a) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma le_sup {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : a \u2264 s.sup :=\nsup_le.1 (le_refl _) _ h\n\nlemma sup_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2081.sup \u2264 s\u2082.sup :=\nsup_le.2 $ assume b hb, le_sup (h hb)\n\nend sup\n\nsection inf\nvariables [semilattice_inf_top \u03b1]\n\n\/-- Infimum of a multiset: `inf {a, b, c} = a \u2293 b \u2293 c` -\/\ndef inf (s : multiset \u03b1) : \u03b1 := s.fold (\u2293) \u22a4\n\n@[simp] lemma inf_zero : (0 : multiset \u03b1).inf = \u22a4 :=\nfold_zero _ _\n\n@[simp] lemma inf_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).inf = a \u2293 s.inf :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma inf_singleton {a : \u03b1} : (a::0).inf = a := by simp\n\n@[simp] lemma inf_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\neq.trans (by simp [inf]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma inf_erase_dup (s : multiset \u03b1) : (erase_dup s).inf = s.inf :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma inf_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).inf = a \u2293 s.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_cons]; simp\n\nlemma le_inf {s : multiset \u03b1} {a : \u03b1} : a \u2264 s.inf \u2194 (\u2200b \u2208 s, a \u2264 b) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma inf_le {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : s.inf \u2264 a :=\nle_inf.1 (le_refl _) _ h\n\nlemma inf_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2082.inf \u2264 s\u2081.inf :=\nle_inf.2 $ assume b hb, inf_le (h hb)\n\nend inf\n\nsection sort\nvariables (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [decidable_rel r]\n  [is_trans \u03b1 r] [is_antisymm \u03b1 r] [is_total \u03b1 r]\n\n\/-- `sort s` constructs a sorted list from the multiset `s`.\n  (Uses merge sort algorithm.) -\/\ndef sort (s : multiset \u03b1) : list \u03b1 :=\nquot.lift_on s (merge_sort r) $ \u03bb a b h,\neq_of_sorted_of_perm\n  ((perm_merge_sort _ _).trans $ h.trans (perm_merge_sort _ _).symm)\n  (sorted_merge_sort r _)\n  (sorted_merge_sort r _)\n\n@[simp] theorem coe_sort (l : list \u03b1) : sort r l = merge_sort r l := rfl\n\n@[simp] theorem sort_sorted (s : multiset \u03b1) : sorted r (sort r s) :=\nquot.induction_on s $ \u03bb l, sorted_merge_sort r _\n\n@[simp] theorem sort_eq (s : multiset \u03b1) : \u2191(sort r s) = s :=\nquot.induction_on s $ \u03bb l, quot.sound $ perm_merge_sort _ _\n\n@[simp] theorem mem_sort {s : multiset \u03b1} {a : \u03b1} : a \u2208 sort r s \u2194 a \u2208 s :=\nby rw [\u2190 mem_coe, sort_eq]\n\nend sort\n\ninstance [has_repr \u03b1] : has_repr (multiset \u03b1) :=\n\u27e8\u03bb s, \"{\" ++ string.intercalate \", \" ((s.map repr).sort (\u2264)) ++ \"}\"\u27e9\n\nsection sections\n\ndef sections (s : multiset (multiset \u03b1)) : multiset (multiset \u03b1) :=\nmultiset.rec_on s {0} (\u03bbs _ c, s.bind $ \u03bba, c.map ((::) a))\n  (assume a\u2080 a\u2081 s pi, by simp [map_bind, bind_bind a\u2080 a\u2081, cons_swap])\n\n@[simp] lemma sections_zero : sections (0 : multiset (multiset \u03b1)) = 0::0 :=\nrfl\n\n@[simp] lemma sections_cons (s : multiset (multiset \u03b1)) (m : multiset \u03b1) :\n  sections (m :: s) = m.bind (\u03bba, (sections s).map ((::) a)) :=\nrec_on_cons m s\n\nlemma coe_sections : \u2200(l : list (list \u03b1)),\n  sections ((l.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1)) =\n    ((l.sections.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1))\n| [] := rfl\n| (a :: l) :=\n  begin\n    simp,\n    rw [\u2190 cons_coe, sections_cons, bind_map_comm, coe_sections l],\n    simp [list.sections, (\u2218), list.bind]\n  end\n\n@[simp] lemma sections_add (s t : multiset (multiset \u03b1)) :\n  sections (s + t) = (sections s).bind (\u03bbm, (sections t).map ((+) m)) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, bind_assoc, map_bind, bind_map, -add_comm])\n\nlemma mem_sections {s : multiset (multiset \u03b1)} :\n  \u2200{a}, a \u2208 sections s \u2194 s.rel (\u03bbs a, a \u2208 s) a :=\nmultiset.induction_on s (by simp)\n  (assume a s ih a',\n    by simp [ih, rel_cons_left, -exists_and_distrib_left, exists_and_distrib_left.symm, eq_comm])\n\nlemma card_sections {s : multiset (multiset \u03b1)} : card (sections s) = prod (s.map card) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma prod_map_sum [comm_semiring \u03b1] {s : multiset (multiset \u03b1)} :\n  prod (s.map sum) = sum ((sections s).map prod) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, map_bind, sum_map_mul_left, sum_map_mul_right])\n\nend sections\n\nsection pi\nvariables [decidable_eq \u03b1] {\u03b4 : \u03b1 \u2192 Type*}\nopen function\n\ndef pi.cons (m : multiset \u03b1) (a : \u03b1) (b : \u03b4 a) (f : \u03a0a\u2208m, \u03b4 a) : \u03a0a'\u2208a::m, \u03b4 a' :=\n\u03bba' ha', if h : a' = a then eq.rec b h.symm else f a' $ (mem_cons.1 ha').resolve_left h\n\ndef pi.empty (\u03b4 : \u03b1 \u2192 Type*) : (\u03a0a\u2208(0:multiset \u03b1), \u03b4 a) .\n\nlemma pi.cons_same {m : multiset \u03b1} {a : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2208 a :: m) :\n  pi.cons m a b f a h = b :=\ndif_pos rfl\n\nlemma pi.cons_ne {m : multiset \u03b1} {a a' : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h' : a' \u2208 a :: m) (h : a' \u2260 a) :\n  pi.cons m a b f a' h' = f a' ((mem_cons.1 h').resolve_left h) :=\ndif_neg h\n\nlemma pi.cons_swap {a a' : \u03b1} {b : \u03b4 a} {b' : \u03b4 a'} {m : multiset \u03b1} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2260 a') :\n  pi.cons (a' :: m) a b (pi.cons m a' b' f) == pi.cons (a :: m) a' b' (pi.cons m a b f) :=\nbegin\n  apply hfunext, { refl }, intros a'' _ h, subst h,\n  apply hfunext, { rw [cons_swap] }, intros ha\u2081 ha\u2082 h,\n  by_cases h\u2081 : a'' = a; by_cases h\u2082 : a'' = a';\n    simp [*, pi.cons_same, pi.cons_ne] at *,\n  { subst h\u2081, rw [pi.cons_same, pi.cons_same] },\n  { subst h\u2082, rw [pi.cons_same, pi.cons_same] }\nend\n\n\/-- `pi m t` constructs the Cartesian product over `t` indexed by `m`. -\/\ndef pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) : multiset (\u03a0a\u2208m, \u03b4 a) :=\nm.rec_on {pi.empty \u03b4} (\u03bba m (p : multiset (\u03a0a\u2208m, \u03b4 a)), (t a).bind $ \u03bbb, p.map $ pi.cons m a b)\nbegin\n  intros a a' m n,\n  by_cases eq : a = a',\n  { subst eq },\n  { simp [map_bind, bind_bind (t a') (t a)],\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b hb,\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b' hb',\n    apply map_hcongr, { rw [cons_swap a a'] },\n    intros f hf,\n    exact pi.cons_swap eq }\nend\n\n@[simp] lemma pi_zero (t : \u03a0a, multiset (\u03b4 a)) : pi 0 t = pi.empty \u03b4 :: 0 := rfl\n\n@[simp] lemma pi_cons (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) (a : \u03b1) :\n  pi (a :: m) t = ((t a).bind $ \u03bbb, (pi m t).map $ pi.cons m a b) :=\nrec_on_cons a m\n\nlemma injective_pi_cons {a : \u03b1} {b : \u03b4 a} {s : multiset \u03b1} (hs : a \u2209 s) :\n  function.injective (pi.cons s a b) :=\nassume f\u2081 f\u2082 eq, funext $ assume a', funext $ assume h',\nhave ne : a \u2260 a', from assume h, hs $ h.symm \u25b8 h',\nhave a' \u2208 a :: s, from mem_cons_of_mem h',\ncalc f\u2081 a' h' = pi.cons s a b f\u2081 a' this : by rw [pi.cons_ne this ne.symm]\n  ... = pi.cons s a b f\u2082 a' this : by rw [eq]\n  ... = f\u2082 a' h' : by rw [pi.cons_ne this ne.symm]\n\nlemma card_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  card (pi m t) = prod (m.map $ \u03bba, card (t a)) :=\nmultiset.induction_on m (by simp) (by simp [mul_comm] {contextual := tt})\n\nlemma nodup_pi {s : multiset \u03b1} {t : \u03a0a, multiset (\u03b4 a)} :\n  nodup s \u2192 (\u2200a\u2208s, nodup (t a)) \u2192 nodup (pi s t) :=\nmultiset.induction_on s (assume _ _, nodup_singleton _)\nbegin\n  assume a s ih hs ht,\n  have has : a \u2209 s, by simp at hs; exact hs.1,\n  have hs : nodup s, by simp at hs; exact hs.2,\n  simp,\n  split,\n  { assume b hb,\n    from nodup_map (injective_pi_cons has) (ih hs $ assume a' h', ht a' $ mem_cons_of_mem h') },\n  { apply pairwise_of_nodup _ (ht a $ mem_cons_self _ _),\n    from assume b\u2081 hb\u2081 b\u2082 hb\u2082 neb, disjoint_map_map.2 (assume f hf g hg eq,\n      have pi.cons s a b\u2081 f a (mem_cons_self _ _) = pi.cons s a b\u2082 g a (mem_cons_self _ _),\n        by rw [eq],\n      neb $ show b\u2081 = b\u2082, by rwa [pi.cons_same, pi.cons_same] at this) }\nend\n\nlemma mem_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  \u2200f:\u03a0a\u2208m, \u03b4 a, (f \u2208 pi m t) \u2194 (\u2200a (h : a \u2208 m), f a h \u2208 t a) :=\nbegin\n  refine multiset.induction_on m (\u03bb f, _) (\u03bb a m ih f, _),\n  { simpa using show f = pi.empty \u03b4, by funext a ha; exact ha.elim },\n  simp, split,\n  { rintro \u27e8b, hb, f', hf', rfl\u27e9 a' ha',\n    rw [ih] at hf',\n    by_cases a' = a,\n    { subst h, rwa [pi.cons_same] },\n    { rw [pi.cons_ne _ h], apply hf' } },\n  { intro hf,\n    refine \u27e8_, hf a (mem_cons_self a _), \u03bba ha, f a (mem_cons_of_mem ha),\n      (ih _).2 (\u03bb a' h', hf _ _), _\u27e9,\n    funext a' h',\n    by_cases a' = a,\n    { subst h, rw [pi.cons_same] },\n    { rw [pi.cons_ne _ h] } }\nend\n\nend pi\nend multiset\n\nnamespace multiset\n\ninstance : functor multiset :=\n{ map := @map }\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u_1 \u2192 Type u_1} [applicative F] [is_comm_applicative F]\nvariables {\u03b1' \u03b2' : Type u_1} (f : \u03b1' \u2192 F \u03b2')\n\ndef traverse : multiset \u03b1' \u2192 F (multiset \u03b2') :=\nquotient.lift (functor.map coe \u2218 traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.skip {\n    have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2081) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2082),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap {\n    have : (\u03bba b (l:list \u03b2'), (\u2191(a :: b :: l) : multiset \u03b2')) <$> f p_y <*> f p_x =\n      (\u03bba b l, \u2191(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(\u2218), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\ninstance : monad multiset :=\n{ pure := \u03bb \u03b1 x, x::0,\n  bind := @bind,\n  .. multiset.functor }\n\ninstance : is_lawful_monad multiset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, multiset.induction_on s rfl $ \u03bb a s ih,\n    by rw [bind_cons, map_cons, bind_zero, add_zero],\n  pure_bind := \u03bb \u03b1 \u03b2 x f, by simp only [cons_bind, zero_bind, add_zero],\n  bind_assoc := @bind_assoc }\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_beta {\u03b1 \u03b2 : Type*} (x : list \u03b1) (f : list \u03b1 \u2192 \u03b2)\n  (h : \u2200 a b : list \u03b1, a \u2248 b \u2192 f a = f b) :\n  quotient.lift f h (x : multiset \u03b1) = f x :=\nquotient.lift_beta _ _ _\n\n@[simp]\nlemma map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 coe = (coe \u2218 functor.map h : list \u03b1 \u2192 multiset \u03b2) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {\u03b1 : Type*} (x : multiset \u03b1) :\n  traverse id.mk x = x :=\nquotient.induction_on x\n(by { intro, rw [traverse,quotient.lift_beta,function.comp],\n      simp, congr })\n\nlemma comp_traverse {G H : Type* \u2192 Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : multiset \u03b1) :\n  traverse (comp.mk \u2218 functor.map h \u2218 g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(\u2218)] with functor_norm)\n\nlemma map_traverse {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3)\n               (x : multiset \u03b1) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [comp_map,map_traverse])\n\nlemma traverse_map {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3)\n               (x : multiset \u03b1) :\n  traverse h (map g x) =\n  traverse (h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [\u2190 traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* \u2192 Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 G \u03b2) (x : multiset \u03b1) :\n  eta (traverse f x) = traverse (@eta _ \u2218 f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nsection choose\nvariables (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n\ndef choose_x : \u03a0 hp : (\u2203! a, a \u2208 l \u2227 p a), { a \/\/ a \u2208 l \u2227 p a } :=\nquotient.rec_on l (\u03bb l' ex_unique, list.choose_x p l' (exists_of_exists_unique ex_unique)) begin\n  intros,\n  funext hp,\n  suffices all_equal : \u2200 x y : { t \/\/ t \u2208 b \u2227 p t }, x = y,\n  { apply all_equal },\n  { rintros \u27e8x, px\u27e9 \u27e8y, py\u27e9,\n    rcases hp with \u27e8z, \u27e8z_mem_l, pz\u27e9, z_unique\u27e9,\n    congr,\n    calc x = z : z_unique x px\n    ...    = y : (z_unique y py).symm }\nend\n\ndef choose (hp : \u2203! a, a \u2208 l \u2227 p a) : \u03b1 := choose_x p l hp\n\nlemma choose_spec (hp : \u2203! a, a \u2208 l \u2227 p a) : choose p l hp \u2208 l \u2227 p (choose p l hp) :=\n(choose_x p l hp).property\n\nlemma choose_mem (hp : \u2203! a, a \u2208 l \u2227 p a) : choose p l hp \u2208 l := (choose_spec _ _ _).1\n\nlemma choose_property (hp : \u2203! a, a \u2208 l \u2227 p a) : p (choose p l hp) := (choose_spec _ _ _).2\n\nend choose\n\n\/- Ico -\/\n\n\/-- `Ico n m` is the multiset lifted from the list `Ico n m`, e.g. the set `{n, n+1, ..., m-1}`. -\/\ndef Ico (n m : \u2115) : multiset \u2115 := Ico n m\n\nnamespace Ico\n\ntheorem map_add (n m k : \u2115) : (Ico n m).map ((+) k) = Ico (n + k) (m + k) :=\ncongr_arg coe $ list.Ico.map_add _ _ _\n\ntheorem map_sub (n m k : \u2115) (h : k \u2264 n) : (Ico n m).map (\u03bb x, x - k) = Ico (n - k) (m - k) :=\ncongr_arg coe $ list.Ico.map_sub _ _ _ h\n\ntheorem zero_bot (n : \u2115) : Ico 0 n = range n :=\ncongr_arg coe $ list.Ico.zero_bot _\n\n@[simp] theorem card (n m : \u2115) : (Ico n m).card = m - n :=\nlist.Ico.length _ _\n\ntheorem nodup (n m : \u2115) : nodup (Ico n m) := Ico.nodup _ _\n\n@[simp] theorem mem {n m l : \u2115} : l \u2208 Ico n m \u2194 n \u2264 l \u2227 l < m :=\nlist.Ico.mem\n\ntheorem eq_zero_of_le {n m : \u2115} (h : m \u2264 n) : Ico n m = 0 :=\ncongr_arg coe $ list.Ico.eq_nil_of_le h\n\n@[simp] theorem self_eq_zero {n : \u2115} : Ico n n = 0 :=\neq_zero_of_le $ le_refl n\n\n@[simp] theorem eq_zero_iff {n m : \u2115} : Ico n m = 0 \u2194 m \u2264 n :=\niff.trans (coe_eq_zero _) list.Ico.eq_empty_iff\n\nlemma add_consecutive {n m l : \u2115} (hnm : n \u2264 m) (hml : m \u2264 l) :\n  Ico n m + Ico m l = Ico n l :=\ncongr_arg coe $ list.Ico.append_consecutive hnm hml\n\n@[simp] lemma inter_consecutive (n m l : \u2115) : Ico n m \u2229 Ico m l = 0 :=\ncongr_arg coe $ list.Ico.bag_inter_consecutive n m l\n\n@[simp] theorem succ_singleton {n : \u2115} : Ico n (n+1) = {n} :=\ncongr_arg coe $ list.Ico.succ_singleton\n\ntheorem succ_top {n m : \u2115} (h : n \u2264 m) : Ico n (m + 1) = m :: Ico n m :=\nby rw [Ico, list.Ico.succ_top h, \u2190 coe_add, add_comm]; refl\n\ntheorem eq_cons {n m : \u2115} (h : n < m) : Ico n m = n :: Ico (n + 1) m :=\ncongr_arg coe $ list.Ico.eq_cons h\n\n@[simp] theorem pred_singleton {m : \u2115} (h : m > 0) : Ico (m - 1) m = {m - 1} :=\ncongr_arg coe $ list.Ico.pred_singleton h\n\n@[simp] theorem not_mem_top {n m : \u2115} : m \u2209 Ico n m :=\nlist.Ico.not_mem_top\n\nlemma filter_lt_of_top_le {n m l : \u2115} (hml : m \u2264 l) : (Ico n m).filter (\u03bb x, x < l) = Ico n m :=\ncongr_arg coe $ list.Ico.filter_lt_of_top_le hml\n\nlemma filter_lt_of_le_bot {n m l : \u2115} (hln : l \u2264 n) : (Ico n m).filter (\u03bb x, x < l) = \u2205 :=\ncongr_arg coe $ list.Ico.filter_lt_of_le_bot hln\n\nlemma filter_lt_of_ge {n m l : \u2115} (hlm : l \u2264 m) : (Ico n m).filter (\u03bb x, x < l) = Ico n l :=\ncongr_arg coe $ list.Ico.filter_lt_of_ge hlm\n\n@[simp] lemma filter_lt (n m l : \u2115) : (Ico n m).filter (\u03bb x, x < l) = Ico n (min m l) :=\ncongr_arg coe $ list.Ico.filter_lt n m l\n\nlemma filter_ge_of_le_bot {n m l : \u2115} (hln : l \u2264 n) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico n m :=\ncongr_arg coe $ list.Ico.filter_ge_of_le_bot hln\n\nlemma filter_ge_of_top_le {n m l : \u2115} (hml : m \u2264 l) : (Ico n m).filter (\u03bb x, x \u2265 l) = \u2205 :=\ncongr_arg coe $ list.Ico.filter_ge_of_top_le hml\n\nlemma filter_ge_of_ge {n m l : \u2115} (hnl : n \u2264 l) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico l m :=\ncongr_arg coe $ list.Ico.filter_ge_of_ge hnl\n\n@[simp] lemma filter_ge (n m l : \u2115) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico (max n l) m :=\ncongr_arg coe $ list.Ico.filter_ge n m l\n\nend Ico\n\nvariable (\u03b1)\n\ndef subsingleton_equiv [subsingleton \u03b1] : list \u03b1 \u2243 multiset \u03b1 :=\n{ to_fun := coe,\n  inv_fun := quot.lift id $ \u03bb (a b : list \u03b1) (h : a ~ b),\n    list.ext_le (perm_length h) $ \u03bb n h\u2081 h\u2082, subsingleton.elim _ _,\n  left_inv := \u03bb l, rfl,\n  right_inv := \u03bb m, quot.induction_on m $ \u03bb l, rfl }\n\nend multiset\n","avg_line_length":39.3284005038,"max_line_length":114,"alphanum_fraction":0.6270665375}
-{"size":71386,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-\/\nimport data.array.lemmas\nimport data.finset.pi\nimport data.finset.powerset\nimport data.sym.basic\nimport group_theory.perm.basic\nimport order.well_founded\nimport tactic.wlog\n\n\/-!\n# Finite types\n\nThis file defines a typeclass to state that a type is finite.\n\n## Main declarations\n\n* `fintype \u03b1`:  Typeclass saying that a type is finite. It takes as fields a `finset` and a proof\n  that all terms of type `\u03b1` are in it.\n* `finset.univ`: The finset of all elements of a fintype.\n* `fintype.card \u03b1`: Cardinality of a fintype. Equal to `finset.univ.card`.\n* `perms_of_finset s`: The finset of permutations of the finset `s`.\n* `fintype.trunc_equiv_fin`: A fintype `\u03b1` is computably equivalent to `fin (card \u03b1)`. The\n  `trunc`-free, noncomputable version is `fintype.equiv_fin`.\n* `fintype.trunc_equiv_of_card_eq` `fintype.equiv_of_card_eq`: Two fintypes of same cardinality are\n  equivalent. See above.\n* `fin.equiv_iff_eq`: `fin m \u2243 fin n` iff `m = n`.\n* `infinite \u03b1`: Typeclass saying that a type is infinite. Defined as `fintype \u03b1 \u2192 false`.\n* `not_fintype`: No `fintype` has an `infinite` instance.\n* `infinite.nat_embedding`: An embedding of `\u2115` into an infinite type.\n\nWe also provide the following versions of the pigeonholes principle.\n* `fintype.exists_ne_map_eq_of_card_lt` and `is_empty_of_card_lt`: Finitely many pigeons and\n  pigeonholes. Weak formulation.\n* `fintype.exists_ne_map_eq_of_infinite`: Infinitely many pigeons in finitely many pigeonholes.\n  Weak formulation.\n* `fintype.exists_infinite_fiber`: Infinitely many pigeons in finitely many pigeonholes. Strong\n  formulation.\n\nSome more pigeonhole-like statements can be found in `data.fintype.card_embedding`.\n\n## Instances\n\nAmong others, we provide `fintype` instances for\n* A `subtype` of a fintype. See `fintype.subtype`.\n* The `option` of a fintype.\n* The product of two fintypes.\n* The sum of two fintypes.\n* `Prop`.\n\nand `infinite` instances for\n* `\u2115`\n* `\u2124`\n\nalong with some machinery\n* Types which have a surjection from\/an injection to a `fintype` are themselves fintypes. See\n  `fintype.of_injective` and `fintype.of_surjective`.\n* Types which have an injection from\/a surjection to an `infinite` type are themselves `infinite`.\n  See `infinite.of_injective` and `infinite.of_surjective`.\n-\/\n\nopen_locale nat\n\nuniverses u v\n\nvariables {\u03b1 \u03b2 \u03b3 : Type*}\n\n\/-- `fintype \u03b1` means that `\u03b1` is finite, i.e. there are only\n  finitely many distinct elements of type `\u03b1`. The evidence of this\n  is a finset `elems` (a list up to permutation without duplicates),\n  together with a proof that everything of type `\u03b1` is in the list. -\/\nclass fintype (\u03b1 : Type*) :=\n(elems [] : finset \u03b1)\n(complete : \u2200 x : \u03b1, x \u2208 elems)\n\nnamespace finset\nvariable [fintype \u03b1]\n\n\/-- `univ` is the universal finite set of type `finset \u03b1` implied from\n  the assumption `fintype \u03b1`. -\/\ndef univ : finset \u03b1 := fintype.elems \u03b1\n\n@[simp] theorem mem_univ (x : \u03b1) : x \u2208 (univ : finset \u03b1) :=\nfintype.complete x\n\n@[simp] theorem mem_univ_val : \u2200 x, x \u2208 (univ : finset \u03b1).1 := mem_univ\n\n@[simp] lemma coe_univ : \u2191(univ : finset \u03b1) = (set.univ : set \u03b1) :=\nby ext; simp\n\nlemma univ_nonempty_iff : (univ : finset \u03b1).nonempty \u2194 nonempty \u03b1 :=\nby rw [\u2190 coe_nonempty, coe_univ, set.nonempty_iff_univ_nonempty]\n\nlemma univ_nonempty [nonempty \u03b1] : (univ : finset \u03b1).nonempty :=\nuniv_nonempty_iff.2 \u2039_\u203a\n\nlemma univ_eq_empty : (univ : finset \u03b1) = \u2205 \u2194 \u00acnonempty \u03b1 :=\nby rw [\u2190 univ_nonempty_iff, nonempty_iff_ne_empty, ne.def, not_not]\n\nlemma univ_eq_empty' : (univ : finset \u03b1) = \u2205 \u2194 is_empty \u03b1 :=\nuniv_eq_empty.trans (not_nonempty_iff)\n\n@[simp] theorem subset_univ (s : finset \u03b1) : s \u2286 univ := \u03bb a _, mem_univ a\n\ninstance : order_top (finset \u03b1) :=\n{ top := univ,\n  le_top := subset_univ,\n  .. finset.partial_order }\n\ninstance [decidable_eq \u03b1] : boolean_algebra (finset \u03b1) :=\n{ compl := \u03bb s, univ \\ s,\n  inf_compl_le_bot := \u03bb s x hx, by simpa using hx,\n  top_le_sup_compl := \u03bb s x hx, by simp,\n  sdiff_eq := \u03bb s t, by simp [ext_iff, compl],\n  ..finset.order_top,\n  ..finset.generalized_boolean_algebra }\n\nlemma compl_eq_univ_sdiff [decidable_eq \u03b1] (s : finset \u03b1) : s\u1d9c = univ \\ s := rfl\n\n@[simp] lemma mem_compl [decidable_eq \u03b1] {s : finset \u03b1} {x : \u03b1} : x \u2208 s\u1d9c \u2194 x \u2209 s :=\nby simp [compl_eq_univ_sdiff]\n\n@[simp, norm_cast] lemma coe_compl [decidable_eq \u03b1] (s : finset \u03b1) : \u2191(s\u1d9c) = (\u2191s : set \u03b1)\u1d9c :=\nset.ext $ \u03bb x, mem_compl\n\n@[simp] theorem union_compl [decidable_eq \u03b1] (s : finset \u03b1) : s \u222a s\u1d9c = finset.univ :=\nsup_compl_eq_top\n\n@[simp] lemma compl_filter [decidable_eq \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p]\n  [\u03a0 x, decidable (\u00acp x)] :\n  (univ.filter p)\u1d9c = univ.filter (\u03bb x, \u00acp x) :=\n(filter_not _ _).symm\n\ntheorem eq_univ_iff_forall {s : finset \u03b1} : s = univ \u2194 \u2200 x, x \u2208 s :=\nby simp [ext_iff]\n\nlemma compl_ne_univ_iff_nonempty [decidable_eq \u03b1] (s : finset \u03b1) : s\u1d9c \u2260 univ \u2194 s.nonempty :=\nby simp [eq_univ_iff_forall, finset.nonempty]\n\n@[simp] lemma univ_inter [decidable_eq \u03b1] (s : finset \u03b1) :\n  univ \u2229 s = s := ext $ \u03bb a, by simp\n\n@[simp] lemma inter_univ [decidable_eq \u03b1] (s : finset \u03b1) :\n  s \u2229 univ = s :=\nby rw [inter_comm, univ_inter]\n\n@[simp] lemma piecewise_univ [\u03a0 i : \u03b1, decidable (i \u2208 (univ : finset \u03b1))]\n  {\u03b4 : \u03b1 \u2192 Sort*} (f g : \u03a0 i, \u03b4 i) : univ.piecewise f g = f :=\nby { ext i, simp [piecewise] }\n\nlemma piecewise_compl [decidable_eq \u03b1] (s : finset \u03b1) [\u03a0 i : \u03b1, decidable (i \u2208 s)]\n  [\u03a0 i : \u03b1, decidable (i \u2208 s\u1d9c)] {\u03b4 : \u03b1 \u2192 Sort*} (f g : \u03a0 i, \u03b4 i) :\n  s\u1d9c.piecewise f g = s.piecewise g f :=\nby { ext i, simp [piecewise] }\n\nlemma univ_map_equiv_to_embedding {\u03b1 \u03b2 : Type*} [fintype \u03b1] [fintype \u03b2] (e : \u03b1 \u2243 \u03b2) :\n  univ.map e.to_embedding = univ :=\neq_univ_iff_forall.mpr (\u03bb b, mem_map.mpr \u27e8e.symm b, mem_univ _, by simp\u27e9)\n\n@[simp] lemma univ_filter_exists (f : \u03b1 \u2192 \u03b2) [fintype \u03b2]\n  [decidable_pred (\u03bb y, \u2203 x, f x = y)] [decidable_eq \u03b2] :\n  finset.univ.filter (\u03bb y, \u2203 x, f x = y) = finset.univ.image f :=\nby { ext, simp }\n\n\/-- Note this is a special case of `(finset.image_preimage f univ _).symm`. -\/\nlemma univ_filter_mem_range (f : \u03b1 \u2192 \u03b2) [fintype \u03b2]\n  [decidable_pred (\u03bb y, y \u2208 set.range f)] [decidable_eq \u03b2] :\n  finset.univ.filter (\u03bb y, y \u2208 set.range f) = finset.univ.image f :=\nuniv_filter_exists f\n\n\/-- A special case of `finset.sup_eq_supr` that omits the useless `x \u2208 univ` binder. -\/\nlemma sup_univ_eq_supr [complete_lattice \u03b2] (f : \u03b1 \u2192 \u03b2) : finset.univ.sup f = supr f :=\n(sup_eq_supr _ f).trans $ congr_arg _ $ funext $ \u03bb a, supr_pos (mem_univ _)\n\n\/-- A special case of `finset.inf_eq_infi` that omits the useless `x \u2208 univ` binder. -\/\nlemma inf_univ_eq_infi [complete_lattice \u03b2] (f : \u03b1 \u2192 \u03b2) : finset.univ.inf f = infi f :=\nsup_univ_eq_supr (by exact f : \u03b1 \u2192 order_dual \u03b2)\n\nend finset\n\nopen finset function\n\nnamespace fintype\n\ninstance decidable_pi_fintype {\u03b1} {\u03b2 : \u03b1 \u2192 Type*} [\u2200 a, decidable_eq (\u03b2 a)] [fintype \u03b1] :\n  decidable_eq (\u03a0 a, \u03b2 a) :=\n\u03bb f g, decidable_of_iff (\u2200 a \u2208 fintype.elems \u03b1, f a = g a)\n  (by simp [function.funext_iff, fintype.complete])\n\ninstance decidable_forall_fintype {p : \u03b1 \u2192 Prop} [decidable_pred p] [fintype \u03b1] :\n  decidable (\u2200 a, p a) :=\ndecidable_of_iff (\u2200 a \u2208 @univ \u03b1 _, p a) (by simp)\n\ninstance decidable_exists_fintype {p : \u03b1 \u2192 Prop} [decidable_pred p] [fintype \u03b1] :\n  decidable (\u2203 a, p a) :=\ndecidable_of_iff (\u2203 a \u2208 @univ \u03b1 _, p a) (by simp)\n\ninstance decidable_mem_range_fintype [fintype \u03b1] [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  decidable_pred (\u2208 set.range f) :=\n\u03bb x, fintype.decidable_exists_fintype\n\nsection bundled_homs\n\ninstance decidable_eq_equiv_fintype [decidable_eq \u03b2] [fintype \u03b1] :\n  decidable_eq (\u03b1 \u2243 \u03b2) :=\n\u03bb a b, decidable_of_iff (a.1 = b.1) equiv.coe_fn_injective.eq_iff\n\ninstance decidable_eq_embedding_fintype [decidable_eq \u03b2] [fintype \u03b1] :\n  decidable_eq (\u03b1 \u21aa \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) function.embedding.coe_injective.eq_iff\n\n@[to_additive]\ninstance decidable_eq_one_hom_fintype [decidable_eq \u03b2] [fintype \u03b1] [has_one \u03b1] [has_one \u03b2]:\n  decidable_eq (one_hom \u03b1 \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) (injective.eq_iff one_hom.coe_inj)\n\n@[to_additive]\ninstance decidable_eq_mul_hom_fintype [decidable_eq \u03b2] [fintype \u03b1] [has_mul \u03b1] [has_mul \u03b2]:\n  decidable_eq (mul_hom \u03b1 \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) (injective.eq_iff mul_hom.coe_inj)\n\n@[to_additive]\ninstance decidable_eq_monoid_hom_fintype [decidable_eq \u03b2] [fintype \u03b1]\n  [mul_one_class \u03b1] [mul_one_class \u03b2]:\n  decidable_eq (\u03b1 \u2192* \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) (injective.eq_iff monoid_hom.coe_inj)\n\ninstance decidable_eq_monoid_with_zero_hom_fintype [decidable_eq \u03b2] [fintype \u03b1]\n  [mul_zero_one_class \u03b1] [mul_zero_one_class \u03b2]:\n  decidable_eq (monoid_with_zero_hom \u03b1 \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) (injective.eq_iff monoid_with_zero_hom.coe_inj)\n\ninstance decidable_eq_ring_hom_fintype [decidable_eq \u03b2] [fintype \u03b1]\n  [semiring \u03b1] [semiring \u03b2]:\n  decidable_eq (\u03b1 \u2192+* \u03b2) :=\n\u03bb a b, decidable_of_iff (\u21d1a = b) (injective.eq_iff ring_hom.coe_inj)\n\nend bundled_homs\n\ninstance decidable_injective_fintype [decidable_eq \u03b1] [decidable_eq \u03b2] [fintype \u03b1] :\n  decidable_pred (injective : (\u03b1 \u2192 \u03b2) \u2192 Prop) := \u03bb x, by unfold injective; apply_instance\n\ninstance decidable_surjective_fintype [decidable_eq \u03b2] [fintype \u03b1] [fintype \u03b2] :\n  decidable_pred (surjective : (\u03b1 \u2192 \u03b2) \u2192 Prop) := \u03bb x, by unfold surjective; apply_instance\n\ninstance decidable_bijective_fintype [decidable_eq \u03b1] [decidable_eq \u03b2] [fintype \u03b1] [fintype \u03b2] :\n  decidable_pred (bijective : (\u03b1 \u2192 \u03b2) \u2192 Prop) := \u03bb x, by unfold bijective; apply_instance\n\ninstance decidable_right_inverse_fintype [decidable_eq \u03b1] [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  decidable (function.right_inverse f g) :=\nshow decidable (\u2200 x, g (f x) = x), by apply_instance\n\ninstance decidable_left_inverse_fintype [decidable_eq \u03b2] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  decidable (function.left_inverse f g) :=\nshow decidable (\u2200 x, f (g x) = x), by apply_instance\n\nlemma exists_max [fintype \u03b1] [nonempty \u03b1] {\u03b2 : Type*} [linear_order \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u2203 x\u2080 : \u03b1, \u2200 x, f x \u2264 f x\u2080 :=\nby simpa using exists_max_image univ f univ_nonempty\n\nlemma exists_min [fintype \u03b1] [nonempty \u03b1]\n  {\u03b2 : Type*} [linear_order \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u2203 x\u2080 : \u03b1, \u2200 x, f x\u2080 \u2264 f x :=\nby simpa using exists_min_image univ f univ_nonempty\n\n\/-- Construct a proof of `fintype \u03b1` from a universal multiset -\/\ndef of_multiset [decidable_eq \u03b1] (s : multiset \u03b1) (H : \u2200 x : \u03b1, x \u2208 s) :\n  fintype \u03b1 :=\n\u27e8s.to_finset, by simpa using H\u27e9\n\n\/-- Construct a proof of `fintype \u03b1` from a universal list -\/\ndef of_list [decidable_eq \u03b1] (l : list \u03b1) (H : \u2200 x : \u03b1, x \u2208 l) :\n  fintype \u03b1 :=\n\u27e8l.to_finset, by simpa using H\u27e9\n\ntheorem exists_univ_list (\u03b1) [fintype \u03b1] :\n  \u2203 l : list \u03b1, l.nodup \u2227 \u2200 x : \u03b1, x \u2208 l :=\nlet \u27e8l, e\u27e9 := quotient.exists_rep (@univ \u03b1 _).1 in\nby have := and.intro univ.2 mem_univ_val;\n   exact \u27e8_, by rwa \u2190e at this\u27e9\n\n\/-- `card \u03b1` is the number of elements in `\u03b1`, defined when `\u03b1` is a fintype. -\/\ndef card (\u03b1) [fintype \u03b1] : \u2115 := (@univ \u03b1 _).card\n\n\/-- If `l` lists all the elements of `\u03b1` without duplicates, then `\u03b1 \u2243 fin (l.length)`. -\/\ndef equiv_fin_of_forall_mem_list {\u03b1} [decidable_eq \u03b1]\n  {l : list \u03b1} (h : \u2200 x : \u03b1, x \u2208 l) (nd : l.nodup) : \u03b1 \u2243 fin (l.length) :=\n\u27e8\u03bb a, \u27e8_, list.index_of_lt_length.2 (h a)\u27e9,\n \u03bb i, l.nth_le i.1 i.2,\n \u03bb a, by simp,\n \u03bb \u27e8i, h\u27e9, fin.eq_of_veq $ list.nodup_iff_nth_le_inj.1 nd _ _\n   (list.index_of_lt_length.2 (list.nth_le_mem _ _ _)) h $ by simp\u27e9\n\n\/-- There is (computably) a bijection between `\u03b1` and `fin (card \u03b1)`.\n\nSince it is not unique, and depends on which permutation\nof the universe list is used, the bijection is wrapped in `trunc` to\npreserve computability.\n\nSee `fintype.equiv_fin` for the noncomputable version,\nand `fintype.trunc_equiv_fin_of_card_eq` and `fintype.equiv_fin_of_card_eq`\nfor an equiv `\u03b1 \u2243 fin n` given `fintype.card \u03b1 = n`.\n-\/\ndef trunc_equiv_fin (\u03b1) [decidable_eq \u03b1] [fintype \u03b1] : trunc (\u03b1 \u2243 fin (card \u03b1)) :=\nby unfold card finset.card; exact\nquot.rec_on_subsingleton (@univ \u03b1 _).1\n  (\u03bb l (h : \u2200 x : \u03b1, x \u2208 l) (nd : l.nodup), trunc.mk (equiv_fin_of_forall_mem_list h nd))\n  mem_univ_val univ.2\n\n\/-- There is a (noncomputable) bijection between `\u03b1` and `fin (card \u03b1)`.\n\nSee `fintype.trunc_equiv_fin` for the computable version,\nand `fintype.trunc_equiv_fin_of_card_eq` and `fintype.equiv_fin_of_card_eq`\nfor an equiv `\u03b1 \u2243 fin n` given `fintype.card \u03b1 = n`.\n-\/\nnoncomputable def equiv_fin (\u03b1) [fintype \u03b1] : \u03b1 \u2243 fin (card \u03b1) :=\nby { letI := classical.dec_eq \u03b1, exact (trunc_equiv_fin \u03b1).out }\n\ninstance (\u03b1 : Type*) : subsingleton (fintype \u03b1) :=\n\u27e8\u03bb \u27e8s\u2081, h\u2081\u27e9 \u27e8s\u2082, h\u2082\u27e9, by congr; simp [finset.ext_iff, h\u2081, h\u2082]\u27e9\n\n\/-- Given a predicate that can be represented by a finset, the subtype\nassociated to the predicate is a fintype. -\/\nprotected def subtype {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 x : \u03b1, x \u2208 s \u2194 p x) :\n  fintype {x \/\/ p x} :=\n\u27e8\u27e8multiset.pmap subtype.mk s.1 (\u03bb x, (H x).1),\n  multiset.nodup_pmap (\u03bb a _ b _, congr_arg subtype.val) s.2\u27e9,\n\u03bb \u27e8x, px\u27e9, multiset.mem_pmap.2 \u27e8x, (H x).2 px, rfl\u27e9\u27e9\n\ntheorem subtype_card {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 x : \u03b1, x \u2208 s \u2194 p x) :\n  @card {x \/\/ p x} (fintype.subtype s H) = s.card :=\nmultiset.card_pmap _ _ _\n\ntheorem card_of_subtype {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 x : \u03b1, x \u2208 s \u2194 p x)\n  [fintype {x \/\/ p x}] :\n  card {x \/\/ p x} = s.card :=\nby { rw \u2190 subtype_card s H, congr }\n\n\/-- Construct a fintype from a finset with the same elements. -\/\ndef of_finset {p : set \u03b1} (s : finset \u03b1) (H : \u2200 x, x \u2208 s \u2194 x \u2208 p) : fintype p :=\nfintype.subtype s H\n\n@[simp] theorem card_of_finset {p : set \u03b1} (s : finset \u03b1) (H : \u2200 x, x \u2208 s \u2194 x \u2208 p) :\n  @fintype.card p (of_finset s H) = s.card :=\nfintype.subtype_card s H\n\ntheorem card_of_finset' {p : set \u03b1} (s : finset \u03b1)\n  (H : \u2200 x, x \u2208 s \u2194 x \u2208 p) [fintype p] : fintype.card p = s.card :=\nby rw \u2190card_of_finset s H; congr\n\n\/-- If `f : \u03b1 \u2192 \u03b2` is a bijection and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -\/\ndef of_bijective [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (H : function.bijective f) : fintype \u03b2 :=\n\u27e8univ.map \u27e8f, H.1\u27e9,\n\u03bb b, let \u27e8a, e\u27e9 := H.2 b in e \u25b8 mem_map_of_mem _ (mem_univ _)\u27e9\n\n\/-- If `f : \u03b1 \u2192 \u03b2` is a surjection and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -\/\ndef of_surjective [decidable_eq \u03b2] [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (H : function.surjective f) :\n  fintype \u03b2 :=\n\u27e8univ.image f, \u03bb b, let \u27e8a, e\u27e9 := H b in e \u25b8 mem_image_of_mem _ (mem_univ _)\u27e9\n\nend fintype\n\nsection inv\n\nnamespace function\n\nvariables [fintype \u03b1] [decidable_eq \u03b2]\n\nnamespace injective\n\nvariables {f : \u03b1 \u2192 \u03b2} (hf : function.injective f)\n\n\/--\nThe inverse of an `hf : injective` function `f : \u03b1 \u2192 \u03b2`, of the type `\u21a5(set.range f) \u2192 \u03b1`.\nThis is the computable version of `function.inv_fun` that requires `fintype \u03b1` and `decidable_eq \u03b2`,\nor the function version of applying `(equiv.of_injective f hf).symm`.\nThis function should not usually be used for actual computation because for most cases,\nan explicit inverse can be stated that has better computational properties.\nThis function computes by checking all terms `a : \u03b1` to find the `f a = b`, so it is O(N) where\n`N = fintype.card \u03b1`.\n-\/\ndef inv_of_mem_range : set.range f \u2192 \u03b1 :=\n\u03bb b, finset.choose (\u03bb a, f a = b) finset.univ ((exists_unique_congr (by simp)).mp\n  (hf.exists_unique_of_mem_range b.property))\n\nlemma left_inv_of_inv_of_mem_range (b : set.range f) :\n  f (hf.inv_of_mem_range b) = b :=\n(finset.choose_spec (\u03bb a, f a = b) _ _).right\n\n@[simp] lemma right_inv_of_inv_of_mem_range (a : \u03b1) :\n  hf.inv_of_mem_range (\u27e8f a, set.mem_range_self a\u27e9) = a :=\nhf (finset.choose_spec (\u03bb a', f a' = f a) _ _).right\n\nlemma inv_fun_restrict [nonempty \u03b1] :\n  (set.range f).restrict (inv_fun f) = hf.inv_of_mem_range :=\nbegin\n  ext \u27e8b, h\u27e9,\n  apply hf,\n  simp [hf.left_inv_of_inv_of_mem_range, @inv_fun_eq _ _ _ f b (set.mem_range.mp h)]\nend\n\nlemma inv_of_mem_range_surjective : function.surjective hf.inv_of_mem_range :=\n\u03bb a, \u27e8\u27e8f a, set.mem_range_self a\u27e9, by simp\u27e9\n\nend injective\n\nnamespace embedding\nvariables (f : \u03b1 \u21aa \u03b2) (b : set.range f)\n\n\/--\nThe inverse of an embedding `f : \u03b1 \u21aa \u03b2`, of the type `\u21a5(set.range f) \u2192 \u03b1`.\nThis is the computable version of `function.inv_fun` that requires `fintype \u03b1` and `decidable_eq \u03b2`,\nor the function version of applying `(equiv.of_injective f f.injective).symm`.\nThis function should not usually be used for actual computation because for most cases,\nan explicit inverse can be stated that has better computational properties.\nThis function computes by checking all terms `a : \u03b1` to find the `f a = b`, so it is O(N) where\n`N = fintype.card \u03b1`.\n-\/\ndef inv_of_mem_range : \u03b1 :=\nf.injective.inv_of_mem_range b\n\n@[simp] lemma left_inv_of_inv_of_mem_range :\n  f (f.inv_of_mem_range b) = b :=\nf.injective.left_inv_of_inv_of_mem_range b\n\n@[simp] lemma right_inv_of_inv_of_mem_range (a : \u03b1) :\n  f.inv_of_mem_range \u27e8f a, set.mem_range_self a\u27e9 = a :=\nf.injective.right_inv_of_inv_of_mem_range a\n\nlemma inv_fun_restrict [nonempty \u03b1] :\n  (set.range f).restrict (inv_fun f) = f.inv_of_mem_range :=\nbegin\n  ext \u27e8b, h\u27e9,\n  apply f.injective,\n  simp [f.left_inv_of_inv_of_mem_range, @inv_fun_eq _ _ _ f b (set.mem_range.mp h)]\nend\n\nlemma inv_of_mem_range_surjective : function.surjective f.inv_of_mem_range :=\n\u03bb a, \u27e8\u27e8f a, set.mem_range_self a\u27e9, by simp\u27e9\n\nend embedding\n\nend function\n\nend inv\n\nnamespace fintype\n\n\/-- Given an injective function to a fintype, the domain is also a\nfintype. This is noncomputable because injectivity alone cannot be\nused to construct preimages. -\/\nnoncomputable def of_injective [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (H : function.injective f) : fintype \u03b1 :=\nby letI := classical.dec; exact\nif h\u03b1 : nonempty \u03b1 then by letI := classical.inhabited_of_nonempty h\u03b1;\n  exact of_surjective (inv_fun f) (inv_fun_surjective H)\nelse \u27e8\u2205, \u03bb x, (h\u03b1 \u27e8x\u27e9).elim\u27e9\n\n\/-- If `f : \u03b1 \u2243 \u03b2` and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -\/\ndef of_equiv (\u03b1 : Type*) [fintype \u03b1] (f : \u03b1 \u2243 \u03b2) : fintype \u03b2 := of_bijective _ f.bijective\n\ntheorem of_equiv_card [fintype \u03b1] (f : \u03b1 \u2243 \u03b2) :\n  @card \u03b2 (of_equiv \u03b1 f) = card \u03b1 :=\nmultiset.card_map _ _\n\ntheorem card_congr {\u03b1 \u03b2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2243 \u03b2) : card \u03b1 = card \u03b2 :=\nby rw \u2190 of_equiv_card f; congr\n\nsection\n\nvariables [fintype \u03b1] [fintype \u03b2]\n\n\/-- If the cardinality of `\u03b1` is `n`, there is computably a bijection between `\u03b1` and `fin n`.\n\nSee `fintype.equiv_fin_of_card_eq` for the noncomputable definition,\nand `fintype.trunc_equiv_fin` and `fintype.equiv_fin` for the bijection `\u03b1 \u2243 fin (card \u03b1)`.\n-\/\ndef trunc_equiv_fin_of_card_eq [decidable_eq \u03b1] {n : \u2115} (h : fintype.card \u03b1 = n) :\n  trunc (\u03b1 \u2243 fin n) :=\n(trunc_equiv_fin \u03b1).map (\u03bb e, e.trans (fin.cast h).to_equiv)\n\n\n\/-- If the cardinality of `\u03b1` is `n`, there is noncomputably a bijection between `\u03b1` and `fin n`.\n\nSee `fintype.trunc_equiv_fin_of_card_eq` for the computable definition,\nand `fintype.trunc_equiv_fin` and `fintype.equiv_fin` for the bijection `\u03b1 \u2243 fin (card \u03b1)`.\n-\/\nnoncomputable def equiv_fin_of_card_eq {n : \u2115} (h : fintype.card \u03b1 = n) :\n  \u03b1 \u2243 fin n :=\nby { letI := classical.dec_eq \u03b1, exact (trunc_equiv_fin_of_card_eq h).out }\n\n\/-- Two `fintype`s with the same cardinality are (computably) in bijection.\n\nSee `fintype.equiv_of_card_eq` for the noncomputable version,\nand `fintype.trunc_equiv_fin_of_card_eq` and `fintype.equiv_fin_of_card_eq` for\nthe specialization to `fin`.\n-\/\ndef trunc_equiv_of_card_eq [decidable_eq \u03b1] [decidable_eq \u03b2] (h : card \u03b1 = card \u03b2) :\n  trunc (\u03b1 \u2243 \u03b2) :=\n(trunc_equiv_fin_of_card_eq h).bind (\u03bb e, (trunc_equiv_fin \u03b2).map (\u03bb e', e.trans e'.symm))\n\n\/-- Two `fintype`s with the same cardinality are (noncomputably) in bijection.\n\nSee `fintype.trunc_equiv_of_card_eq` for the computable version,\nand `fintype.trunc_equiv_fin_of_card_eq` and `fintype.equiv_fin_of_card_eq` for\nthe specialization to `fin`.\n-\/\nnoncomputable def equiv_of_card_eq (h : card \u03b1 = card \u03b2) : \u03b1 \u2243 \u03b2 :=\nby { letI := classical.dec_eq \u03b1, letI := classical.dec_eq \u03b2,\n     exact (trunc_equiv_of_card_eq h).out }\n\nend\n\ntheorem card_eq {\u03b1 \u03b2} [F : fintype \u03b1] [G : fintype \u03b2] : card \u03b1 = card \u03b2 \u2194 nonempty (\u03b1 \u2243 \u03b2) :=\n\u27e8\u03bb h, by { haveI := classical.prop_decidable, exact (trunc_equiv_of_card_eq h).nonempty },\n \u03bb \u27e8f\u27e9, card_congr f\u27e9\n\n\/-- Any subsingleton type with a witness is a fintype (with one term). -\/\ndef of_subsingleton (a : \u03b1) [subsingleton \u03b1] : fintype \u03b1 :=\n\u27e8{a}, \u03bb b, finset.mem_singleton.2 (subsingleton.elim _ _)\u27e9\n\n@[simp] theorem univ_of_subsingleton (a : \u03b1) [subsingleton \u03b1] :\n  @univ _ (of_subsingleton a) = {a} := rfl\n\n\/-- Note: this lemma is specifically about `fintype.of_subsingleton`. For a statement about\narbitrary `fintype` instances, use either `fintype.card_le_one_iff_subsingleton` or\n`fintype.card_unique`. -\/\n@[simp] theorem card_of_subsingleton (a : \u03b1) [subsingleton \u03b1] :\n  @fintype.card _ (of_subsingleton a) = 1 := rfl\n\n@[simp] theorem card_unique [unique \u03b1] [h : fintype \u03b1] :\n  fintype.card \u03b1 = 1 :=\nsubsingleton.elim (of_subsingleton $ default \u03b1) h \u25b8 card_of_subsingleton _\n\n@[priority 100] -- see Note [lower instance priority]\ninstance of_is_empty [is_empty \u03b1] : fintype \u03b1 := \u27e8\u2205, is_empty_elim\u27e9\n\n\/-- Note: this lemma is specifically about `fintype.of_is_empty`. For a statement about\narbitrary `fintype` instances, use `fintype.univ_is_empty`. -\/\n-- no-lint since while `fintype.of_is_empty` can prove this, it isn't applicable for `dsimp`.\n@[simp, nolint simp_nf] theorem univ_of_is_empty [is_empty \u03b1] : @univ \u03b1 _ = \u2205 := rfl\n\n\/-- Note: this lemma is specifically about `fintype.of_is_empty`. For a statement about\narbitrary `fintype` instances, use `fintype.card_eq_zero_iff`. -\/\n@[simp] theorem card_of_is_empty [is_empty \u03b1] : fintype.card \u03b1 = 0 := rfl\n\nopen_locale classical\nvariables (\u03b1)\n\n\/-- Any subsingleton type is (noncomputably) a fintype (with zero or one term). -\/\n@[priority 5] -- see Note [lower instance priority]\nnoncomputable instance of_subsingleton' [subsingleton \u03b1] : fintype \u03b1 :=\nif h : nonempty \u03b1 then\n  of_subsingleton (nonempty.some h)\nelse\n  @fintype.of_is_empty _ $ not_nonempty_iff.mp h\n\nend fintype\n\nnamespace set\n\n\/-- Construct a finset enumerating a set `s`, given a `fintype` instance.  -\/\ndef to_finset (s : set \u03b1) [fintype s] : finset \u03b1 :=\n\u27e8(@finset.univ s _).1.map subtype.val,\n multiset.nodup_map (\u03bb a b, subtype.eq) finset.univ.2\u27e9\n\n@[simp] theorem mem_to_finset {s : set \u03b1} [fintype s] {a : \u03b1} : a \u2208 s.to_finset \u2194 a \u2208 s :=\nby simp [to_finset]\n\n@[simp] theorem mem_to_finset_val {s : set \u03b1} [fintype s] {a : \u03b1} : a \u2208 s.to_finset.1 \u2194 a \u2208 s :=\nmem_to_finset\n\n-- We use an arbitrary `[fintype s]` instance here,\n-- not necessarily coming from a `[fintype \u03b1]`.\n@[simp]\nlemma to_finset_card {\u03b1 : Type*} (s : set \u03b1) [fintype s] :\n  s.to_finset.card = fintype.card s :=\nmultiset.card_map subtype.val finset.univ.val\n\n@[simp] theorem coe_to_finset (s : set \u03b1) [fintype s] : (\u2191s.to_finset : set \u03b1) = s :=\nset.ext $ \u03bb _, mem_to_finset\n\n@[simp] theorem to_finset_inj {s t : set \u03b1} [fintype s] [fintype t] :\n  s.to_finset = t.to_finset \u2194 s = t :=\n\u27e8\u03bb h, by rw [\u2190s.coe_to_finset, h, t.coe_to_finset], \u03bb h, by simp [h]; congr\u27e9\n\n@[simp, mono] theorem to_finset_mono {s t : set \u03b1} [fintype s] [fintype t] :\n  s.to_finset \u2286 t.to_finset \u2194 s \u2286 t :=\nby simp [finset.subset_iff, set.subset_def]\n\n@[simp, mono] theorem to_finset_strict_mono {s t : set \u03b1} [fintype s] [fintype t] :\n  s.to_finset \u2282 t.to_finset \u2194 s \u2282 t :=\nbegin\n  rw [\u2190lt_eq_ssubset, \u2190finset.lt_iff_ssubset, lt_iff_le_and_ne, lt_iff_le_and_ne],\n  simp\nend\n\n@[simp] theorem to_finset_disjoint_iff [decidable_eq \u03b1] {s t : set \u03b1} [fintype s] [fintype t] :\n  disjoint s.to_finset t.to_finset \u2194 disjoint s t :=\n\u27e8\u03bb h x hx, h (by simpa using hx), \u03bb h x hx, h (by simpa using hx)\u27e9\n\nend set\n\nlemma finset.card_univ [fintype \u03b1] : (finset.univ : finset \u03b1).card = fintype.card \u03b1 :=\nrfl\n\nlemma finset.eq_univ_of_card [fintype \u03b1] (s : finset \u03b1) (hs : s.card = fintype.card \u03b1) :\n  s = univ :=\neq_of_subset_of_card_le (subset_univ _) $ by rw [hs, finset.card_univ]\n\nlemma finset.card_eq_iff_eq_univ [fintype \u03b1] (s : finset \u03b1) :\n  s.card = fintype.card \u03b1 \u2194 s = finset.univ :=\n\u27e8s.eq_univ_of_card, by { rintro rfl, exact finset.card_univ, }\u27e9\n\nlemma finset.card_le_univ [fintype \u03b1] (s : finset \u03b1) :\n  s.card \u2264 fintype.card \u03b1 :=\ncard_le_of_subset (subset_univ s)\n\nlemma finset.card_lt_univ_of_not_mem [fintype \u03b1] {s : finset \u03b1} {x : \u03b1} (hx : x \u2209 s) :\n  s.card < fintype.card \u03b1 :=\ncard_lt_card \u27e8subset_univ s, not_forall.2 \u27e8x, \u03bb hx', hx (hx' $ mem_univ x)\u27e9\u27e9\n\nlemma finset.card_lt_iff_ne_univ [fintype \u03b1] (s : finset \u03b1) :\n  s.card < fintype.card \u03b1 \u2194 s \u2260 finset.univ :=\ns.card_le_univ.lt_iff_ne.trans (not_iff_not_of_iff s.card_eq_iff_eq_univ)\n\nlemma finset.card_compl_lt_iff_nonempty [fintype \u03b1] [decidable_eq \u03b1] (s : finset \u03b1) :\n  s\u1d9c.card < fintype.card \u03b1 \u2194 s.nonempty :=\ns\u1d9c.card_lt_iff_ne_univ.trans s.compl_ne_univ_iff_nonempty\n\nlemma finset.card_univ_diff [decidable_eq \u03b1] [fintype \u03b1] (s : finset \u03b1) :\n  (finset.univ \\ s).card = fintype.card \u03b1 - s.card :=\nfinset.card_sdiff (subset_univ s)\n\nlemma finset.card_compl [decidable_eq \u03b1] [fintype \u03b1] (s : finset \u03b1) :\n  s\u1d9c.card = fintype.card \u03b1 - s.card :=\nfinset.card_univ_diff s\n\ninstance (n : \u2115) : fintype (fin n) :=\n\u27e8finset.fin_range n, finset.mem_fin_range\u27e9\n\nlemma fin.univ_def (n : \u2115) : (univ : finset (fin n)) = finset.fin_range n := rfl\n\n@[simp] theorem fintype.card_fin (n : \u2115) : fintype.card (fin n) = n :=\nlist.length_fin_range n\n\n@[simp] lemma finset.card_fin (n : \u2115) : finset.card (finset.univ : finset (fin n)) = n :=\nby rw [finset.card_univ, fintype.card_fin]\n\nlemma card_finset_fin_le {n : \u2115} (s : finset (fin n)) : s.card \u2264 n :=\nby simpa only [fintype.card_fin] using s.card_le_univ\n\nlemma fin.equiv_iff_eq {m n : \u2115} : nonempty (fin m \u2243 fin n) \u2194 m = n :=\n  \u27e8\u03bb \u27e8h\u27e9, by simpa using fintype.card_congr h, \u03bb h, \u27e8equiv.cast $ h \u25b8 rfl \u27e9 \u27e9\n\n\/-- Embed `fin n` into `fin (n + 1)` by prepending zero to the `univ` -\/\nlemma fin.univ_succ (n : \u2115) :\n  (univ : finset (fin (n + 1))) = insert 0 (univ.image fin.succ) :=\nbegin\n  ext m,\n  simp only [mem_univ, mem_insert, true_iff, mem_image, exists_prop],\n  exact fin.cases (or.inl rfl) (\u03bb i, or.inr \u27e8i, trivial, rfl\u27e9) m\nend\n\n\/-- Embed `fin n` into `fin (n + 1)` by appending a new `fin.last n` to the `univ` -\/\nlemma fin.univ_cast_succ (n : \u2115) :\n  (univ : finset (fin (n + 1))) = insert (fin.last n) (univ.image fin.cast_succ) :=\nbegin\n  ext m,\n  simp only [mem_univ, mem_insert, true_iff, mem_image, exists_prop, true_and],\n  by_cases h : m.val < n,\n  { right,\n    use fin.cast_lt m h,\n    rw fin.cast_succ_cast_lt },\n  { left,\n    exact fin.eq_last_of_not_lt h }\nend\n\n\/-- Embed `fin n` into `fin (n + 1)` by inserting\naround a specified pivot `p : fin (n + 1)` into the `univ` -\/\nlemma fin.univ_succ_above (n : \u2115) (p : fin (n + 1)) :\n  (univ : finset (fin (n + 1))) = insert p (univ.image (fin.succ_above p)) :=\nbegin\n  obtain hl | rfl := (fin.le_last p).lt_or_eq,\n  { ext m,\n    simp only [finset.mem_univ, finset.mem_insert, true_iff, finset.mem_image, exists_prop],\n    refine or_iff_not_imp_left.mpr (\u03bb h, _),\n    cases n,\n    { have : m = p := by simp,\n      exact absurd this h },\n    use p.cast_pred.pred_above m,\n    rw fin.pred_above,\n    split_ifs with H,\n    { simp only [fin.coe_cast_succ, true_and, fin.coe_coe_eq_self, coe_coe],\n      rw fin.lt_last_iff_coe_cast_pred at hl,\n      rw fin.succ_above_above,\n      { simp },\n      { simp only [fin.lt_iff_coe_lt_coe, fin.coe_cast_succ] at H,\n        simpa [fin.le_iff_coe_le_coe, \u2190hl] using nat.le_pred_of_lt H } },\n    { rw fin.succ_above_below,\n      { simp },\n      { simp only [fin.cast_succ_cast_pred hl, not_lt] at H,\n        simpa using lt_of_le_of_ne H h } } },\n  { rw fin.succ_above_last,\n    exact fin.univ_cast_succ n }\nend\n\n@[instance, priority 10] def unique.fintype {\u03b1 : Type*} [unique \u03b1] : fintype \u03b1 :=\nfintype.of_subsingleton (default \u03b1)\n\n\/-- Short-circuit instance to decrease search for `unique.fintype`,\nsince that relies on a subsingleton elimination for `unique`. -\/\ninstance fintype.subtype_eq (y : \u03b1) : fintype {x \/\/ x = y} :=\nfintype.subtype {y} (by simp)\n\n\/-- Short-circuit instance to decrease search for `unique.fintype`,\nsince that relies on a subsingleton elimination for `unique`. -\/\ninstance fintype.subtype_eq' (y : \u03b1) : fintype {x \/\/ y = x} :=\nfintype.subtype {y} (by simp [eq_comm])\n\n@[simp] lemma univ_unique {\u03b1 : Type*} [unique \u03b1] [f : fintype \u03b1] : @finset.univ \u03b1 _ = {default \u03b1} :=\nby rw [subsingleton.elim f (@unique.fintype \u03b1 _)]; refl\n\n@[simp] lemma univ_is_empty {\u03b1 : Type*} [is_empty \u03b1] [fintype \u03b1] : @finset.univ \u03b1 _ = \u2205 :=\nfinset.ext is_empty_elim\n\n@[simp] lemma fintype.card_subtype_eq (y : \u03b1)  :\n  fintype.card {x \/\/ x = y} = 1 :=\nbegin\n  convert fintype.card_unique,\n  exact unique.subtype_eq _\nend\n\n@[simp] lemma fintype.card_subtype_eq' (y : \u03b1) :\n  fintype.card {x \/\/ y = x} = 1 :=\nbegin\n  convert fintype.card_unique,\n  exact unique.subtype_eq' _\nend\n\n@[simp] theorem fintype.univ_empty : @univ empty _ = \u2205 := rfl\n\n@[simp] theorem fintype.card_empty : fintype.card empty = 0 := rfl\n\n@[simp] theorem fintype.univ_pempty : @univ pempty _ = \u2205 := rfl\n\n@[simp] theorem fintype.card_pempty : fintype.card pempty = 0 := rfl\n\ninstance : fintype unit := fintype.of_subsingleton ()\n\ntheorem fintype.univ_unit : @univ unit _ = {()} := rfl\n\ntheorem fintype.card_unit : fintype.card unit = 1 := rfl\n\ninstance : fintype punit := fintype.of_subsingleton punit.star\n\n@[simp] theorem fintype.univ_punit : @univ punit _ = {punit.star} := rfl\n\n@[simp] theorem fintype.card_punit : fintype.card punit = 1 := rfl\n\ninstance : fintype bool := \u27e8\u27e8tt ::\u2098 ff ::\u2098 0, by simp\u27e9, \u03bb x, by cases x; simp\u27e9\n\n@[simp] theorem fintype.univ_bool : @univ bool _ = {tt, ff} := rfl\n\ninstance units_int.fintype : fintype (units \u2124) :=\n\u27e8{1, -1}, \u03bb x, by cases int.units_eq_one_or x; simp *\u27e9\n\n@[simp] lemma units_int.univ : (finset.univ : finset (units \u2124)) = {1, -1} := rfl\n\ninstance additive.fintype : \u03a0 [fintype \u03b1], fintype (additive \u03b1) := id\n\ninstance multiplicative.fintype : \u03a0 [fintype \u03b1], fintype (multiplicative \u03b1) := id\n\n@[simp] theorem fintype.card_units_int : fintype.card (units \u2124) = 2 := rfl\n\nnoncomputable instance [monoid \u03b1] [fintype \u03b1] : fintype (units \u03b1) :=\nby classical; exact fintype.of_injective units.val units.ext\n\n@[simp] theorem fintype.card_bool : fintype.card bool = 2 := rfl\n\n\/-- Given a finset on `\u03b1`, lift it to being a finset on `option \u03b1`\nusing `option.some` and then insert `option.none`. -\/\ndef finset.insert_none (s : finset \u03b1) : finset (option \u03b1) :=\n\u27e8none ::\u2098 s.1.map some, multiset.nodup_cons.2\n  \u27e8by simp, multiset.nodup_map (\u03bb a b, option.some.inj) s.2\u27e9\u27e9\n\n@[simp] theorem finset.mem_insert_none {s : finset \u03b1} : \u2200 {o : option \u03b1},\n  o \u2208 s.insert_none \u2194 \u2200 a \u2208 o, a \u2208 s\n| none     := iff_of_true (multiset.mem_cons_self _ _) (\u03bb a h, by cases h)\n| (some a) := multiset.mem_cons.trans $ by simp; refl\n\ntheorem finset.some_mem_insert_none {s : finset \u03b1} {a : \u03b1} :\n  some a \u2208 s.insert_none \u2194 a \u2208 s := by simp\n\ninstance {\u03b1 : Type*} [fintype \u03b1] : fintype (option \u03b1) :=\n\u27e8univ.insert_none, \u03bb a, by simp\u27e9\n\n@[simp] theorem fintype.card_option {\u03b1 : Type*} [fintype \u03b1] :\n  fintype.card (option \u03b1) = fintype.card \u03b1 + 1 :=\n(multiset.card_cons _ _).trans (by rw multiset.card_map; refl)\n\ninstance {\u03b1 : Type*} (\u03b2 : \u03b1 \u2192 Type*)\n  [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] : fintype (sigma \u03b2) :=\n\u27e8univ.sigma (\u03bb _, univ), \u03bb \u27e8a, b\u27e9, by simp\u27e9\n\n@[simp] lemma finset.univ_sigma_univ {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] :\n  (univ : finset \u03b1).sigma (\u03bb a, (univ : finset (\u03b2 a))) = univ := rfl\n\ninstance (\u03b1 \u03b2 : Type*) [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u00d7 \u03b2) :=\n\u27e8univ.product univ, \u03bb \u27e8a, b\u27e9, by simp\u27e9\n\n@[simp] lemma finset.univ_product_univ {\u03b1 \u03b2 : Type*} [fintype \u03b1] [fintype \u03b2] :\n  (univ : finset \u03b1).product (univ : finset \u03b2) = univ :=\nrfl\n\n@[simp] theorem fintype.card_prod (\u03b1 \u03b2 : Type*) [fintype \u03b1] [fintype \u03b2] :\n  fintype.card (\u03b1 \u00d7 \u03b2) = fintype.card \u03b1 * fintype.card \u03b2 :=\ncard_product _ _\n\n\/-- Given that `\u03b1 \u00d7 \u03b2` is a fintype, `\u03b1` is also a fintype. -\/\ndef fintype.prod_left {\u03b1 \u03b2} [decidable_eq \u03b1] [fintype (\u03b1 \u00d7 \u03b2)] [nonempty \u03b2] : fintype \u03b1 :=\n\u27e8(fintype.elems (\u03b1 \u00d7 \u03b2)).image prod.fst,\n  \u03bb a, let \u27e8b\u27e9 := \u2039nonempty \u03b2\u203a in by simp; exact \u27e8b, fintype.complete _\u27e9\u27e9\n\n\/-- Given that `\u03b1 \u00d7 \u03b2` is a fintype, `\u03b2` is also a fintype. -\/\ndef fintype.prod_right {\u03b1 \u03b2} [decidable_eq \u03b2] [fintype (\u03b1 \u00d7 \u03b2)] [nonempty \u03b1] : fintype \u03b2 :=\n\u27e8(fintype.elems (\u03b1 \u00d7 \u03b2)).image prod.snd,\n  \u03bb b, let \u27e8a\u27e9 := \u2039nonempty \u03b1\u203a in by simp; exact \u27e8a, fintype.complete _\u27e9\u27e9\n\ninstance (\u03b1 : Type*) [fintype \u03b1] : fintype (ulift \u03b1) :=\nfintype.of_equiv _ equiv.ulift.symm\n\n@[simp] theorem fintype.card_ulift (\u03b1 : Type*) [fintype \u03b1] :\n  fintype.card (ulift \u03b1) = fintype.card \u03b1 :=\nfintype.of_equiv_card _\n\nlemma univ_sum_type {\u03b1 \u03b2 : Type*} [fintype \u03b1] [fintype \u03b2] [fintype (\u03b1 \u2295 \u03b2)] [decidable_eq (\u03b1 \u2295 \u03b2)] :\n  (univ : finset (\u03b1 \u2295 \u03b2)) = map function.embedding.inl univ \u222a map function.embedding.inr univ :=\nbegin\n  rw [eq_comm, eq_univ_iff_forall], simp only [mem_union, mem_map, exists_prop, mem_univ, true_and],\n  rintro (x|y), exacts [or.inl \u27e8x, rfl\u27e9, or.inr \u27e8y, rfl\u27e9]\nend\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u2295 \u03b2) :=\n@fintype.of_equiv _ _ (@sigma.fintype _\n    (\u03bb b, cond b (ulift \u03b1) (ulift.{(max u v) v} \u03b2)) _\n    (\u03bb b, by cases b; apply ulift.fintype))\n  ((equiv.sum_equiv_sigma_bool _ _).symm.trans\n    (equiv.sum_congr equiv.ulift equiv.ulift))\n\n\/-- Given that `\u03b1 \u2295 \u03b2` is a fintype, `\u03b1` is also a fintype. This is non-computable as it uses\nthat `sum.inl` is an injection, but there's no clear inverse if `\u03b1` is empty. -\/\nnoncomputable def fintype.sum_left {\u03b1 \u03b2} [fintype (\u03b1 \u2295 \u03b2)] : fintype \u03b1 :=\nfintype.of_injective (sum.inl : \u03b1 \u2192 \u03b1 \u2295 \u03b2) sum.inl_injective\n\n\/-- Given that `\u03b1 \u2295 \u03b2` is a fintype, `\u03b2` is also a fintype. This is non-computable as it uses\nthat `sum.inr` is an injection, but there's no clear inverse if `\u03b2` is empty. -\/\nnoncomputable def fintype.sum_right {\u03b1 \u03b2} [fintype (\u03b1 \u2295 \u03b2)] : fintype \u03b2 :=\nfintype.of_injective (sum.inr : \u03b2 \u2192 \u03b1 \u2295 \u03b2) sum.inr_injective\n\n@[simp] theorem fintype.card_sum [fintype \u03b1] [fintype \u03b2] :\n  fintype.card (\u03b1 \u2295 \u03b2) = fintype.card \u03b1 + fintype.card \u03b2 :=\nbegin\n  classical,\n  rw [\u2190finset.card_univ, univ_sum_type, finset.card_union_eq],\n  { simp [finset.card_univ] },\n  { intros x hx,\n    suffices : (\u2203 (a : \u03b1), sum.inl a = x) \u2227 \u2203 (b : \u03b2), sum.inr b = x,\n    { obtain \u27e8\u27e8a, rfl\u27e9, \u27e8b, hb\u27e9\u27e9 := this,\n      simpa using hb },\n    simpa using hx }\nend\n\nsection finset\n\n\/-! ### `fintype (s : finset \u03b1)` -\/\n\ninstance finset.fintype_coe_sort {\u03b1 : Type u} (s : finset \u03b1) : fintype s :=\n\u27e8s.attach, s.mem_attach\u27e9\n\n@[simp] lemma finset.univ_eq_attach {\u03b1 : Type u} (s : finset \u03b1) :\n  (univ : finset s) = s.attach :=\nrfl\n\nend finset\n\nnamespace fintype\nvariables [fintype \u03b1] [fintype \u03b2]\n\nlemma card_le_of_injective (f : \u03b1 \u2192 \u03b2) (hf : function.injective f) : card \u03b1 \u2264 card \u03b2 :=\nfinset.card_le_card_of_inj_on f (\u03bb _ _, finset.mem_univ _) (\u03bb _ _ _ _ h, hf h)\n\nlemma card_le_of_embedding (f : \u03b1 \u21aa \u03b2) : card \u03b1 \u2264 card \u03b2 := card_le_of_injective f f.2\n\nlemma card_lt_of_injective_of_not_mem (f : \u03b1 \u2192 \u03b2) (h : function.injective f)\n  {b : \u03b2} (w : b \u2209 set.range f) : card \u03b1 < card \u03b2 :=\ncalc card \u03b1 = (univ.map \u27e8f, h\u27e9).card : (card_map _).symm\n... < card \u03b2 : finset.card_lt_univ_of_not_mem $\n                 by rwa [\u2190 mem_coe, coe_map, coe_univ, set.image_univ]\n\nlemma card_lt_of_injective_not_surjective (f : \u03b1 \u2192 \u03b2) (h : function.injective f)\n  (h' : \u00acfunction.surjective f) : card \u03b1 < card \u03b2 :=\nlet \u27e8y, hy\u27e9 := not_forall.1 h' in card_lt_of_injective_of_not_mem f h hy\n\nlemma card_le_of_surjective (f : \u03b1 \u2192 \u03b2) (h : function.surjective f) : card \u03b2 \u2264 card \u03b1 :=\ncard_le_of_injective _ (function.injective_surj_inv h)\n\n\/--\nThe pigeonhole principle for finitely many pigeons and pigeonholes.\nThis is the `fintype` version of `finset.exists_ne_map_eq_of_card_lt_of_maps_to`.\n-\/\nlemma exists_ne_map_eq_of_card_lt (f : \u03b1 \u2192 \u03b2) (h : fintype.card \u03b2 < fintype.card \u03b1) :\n  \u2203 x y, x \u2260 y \u2227 f x = f y :=\nlet \u27e8x, _, y, _, h\u27e9 := finset.exists_ne_map_eq_of_card_lt_of_maps_to h (\u03bb x _, mem_univ (f x))\nin \u27e8x, y, h\u27e9\n\nlemma card_eq_one_iff : card \u03b1 = 1 \u2194 (\u2203 x : \u03b1, \u2200 y, y = x) :=\nby rw [\u2190card_unit, card_eq]; exact\n\u27e8\u03bb \u27e8a\u27e9, \u27e8a.symm (), \u03bb y, a.injective (subsingleton.elim _ _)\u27e9,\n  \u03bb \u27e8x, hx\u27e9, \u27e8\u27e8\u03bb _, (), \u03bb _, x, \u03bb _, (hx _).trans (hx _).symm,\n    \u03bb _, subsingleton.elim _ _\u27e9\u27e9\u27e9\n\nlemma card_eq_zero_iff : card \u03b1 = 0 \u2194 is_empty \u03b1 :=\n\u27e8\u03bb h, \u27e8\u03bb a, have e : \u03b1 \u2243 empty := classical.choice (card_eq.1 (by simp [h])), (e a).elim\u27e9,\n  \u03bb h, by { have e : \u03b1 \u2243 empty, exactI equiv.equiv_empty \u03b1, simp [card_congr e] }\u27e9\n\nlemma card_eq_one_iff_nonempty_unique : card \u03b1 = 1 \u2194 nonempty (unique \u03b1) :=\n\u27e8\u03bb h, let \u27e8d, h\u27e9 := fintype.card_eq_one_iff.mp h in \u27e8{ default := d, uniq := h}\u27e9,\n \u03bb \u27e8h\u27e9, by exactI fintype.card_unique\u27e9\n\n\/-- A `fintype` with cardinality zero is equivalent to `empty`. -\/\ndef card_eq_zero_equiv_equiv_empty : card \u03b1 = 0 \u2243 (\u03b1 \u2243 empty) :=\n(equiv.of_iff card_eq_zero_iff).trans (equiv.equiv_empty_equiv \u03b1).symm\n\nlemma card_pos_iff : 0 < card \u03b1 \u2194 nonempty \u03b1 :=\npos_iff_ne_zero.trans $ not_iff_comm.mp $ not_nonempty_iff.trans card_eq_zero_iff.symm\n\nlemma card_le_one_iff : card \u03b1 \u2264 1 \u2194 (\u2200 a b : \u03b1, a = b) :=\nlet n := card \u03b1 in\nhave hn : n = card \u03b1 := rfl,\nmatch n, hn with\n| 0     := \u03bb ha, \u27e8\u03bb h, \u03bb a, (card_eq_zero_iff.1 ha.symm).elim a, \u03bb _, ha \u25b8 nat.le_succ _\u27e9\n| 1     := \u03bb ha, \u27e8\u03bb h, \u03bb a b, let \u27e8x, hx\u27e9 := card_eq_one_iff.1 ha.symm in\n  by rw [hx a, hx b],\n    \u03bb _, ha \u25b8 le_refl _\u27e9\n| (n+2) := \u03bb ha, \u27e8\u03bb h, by rw \u2190 ha at h; exact absurd h dec_trivial,\n  (\u03bb h, card_unit \u25b8 card_le_of_injective (\u03bb _, ())\n    (\u03bb _ _ _, h _ _))\u27e9\nend\n\nlemma card_le_one_iff_subsingleton : card \u03b1 \u2264 1 \u2194 subsingleton \u03b1 :=\ncard_le_one_iff.trans subsingleton_iff.symm\n\nlemma one_lt_card_iff_nontrivial : 1 < card \u03b1 \u2194 nontrivial \u03b1 :=\nbegin\n  classical,\n  rw \u2190not_iff_not,\n  push_neg,\n  rw [not_nontrivial_iff_subsingleton, card_le_one_iff_subsingleton]\nend\n\nlemma exists_ne_of_one_lt_card (h : 1 < card \u03b1) (a : \u03b1) : \u2203 b : \u03b1, b \u2260 a :=\nby { haveI : nontrivial \u03b1 := one_lt_card_iff_nontrivial.1 h, exact exists_ne a }\n\nlemma exists_pair_of_one_lt_card (h : 1 < card \u03b1) : \u2203 (a b : \u03b1), a \u2260 b :=\nby { haveI : nontrivial \u03b1 := one_lt_card_iff_nontrivial.1 h, exact exists_pair_ne \u03b1 }\n\nlemma card_eq_one_of_forall_eq {i : \u03b1} (h : \u2200 j, j = i) : card \u03b1 = 1 :=\nfintype.card_eq_one_iff.2 \u27e8i,h\u27e9\n\nlemma injective_iff_surjective {f : \u03b1 \u2192 \u03b1} : injective f \u2194 surjective f :=\nby haveI := classical.prop_decidable; exact\nhave \u2200 {f : \u03b1 \u2192 \u03b1}, injective f \u2192 surjective f,\nfrom \u03bb f hinj x,\n  have h\u2081 : image f univ = univ := eq_of_subset_of_card_le (subset_univ _)\n    ((card_image_of_injective univ hinj).symm \u25b8 le_refl _),\n  have h\u2082 : x \u2208 image f univ := h\u2081.symm \u25b8 mem_univ _,\n  exists_of_bex (mem_image.1 h\u2082),\n\u27e8this,\n  \u03bb hsurj, has_left_inverse.injective\n    \u27e8surj_inv hsurj, left_inverse_of_surjective_of_right_inverse\n      (this (injective_surj_inv _)) (right_inverse_surj_inv _)\u27e9\u27e9\n\nlemma injective_iff_bijective {f : \u03b1 \u2192 \u03b1} : injective f \u2194 bijective f :=\nby simp [bijective, injective_iff_surjective]\n\nlemma surjective_iff_bijective {f : \u03b1 \u2192 \u03b1} : surjective f \u2194 bijective f :=\nby simp [bijective, injective_iff_surjective]\n\nlemma injective_iff_surjective_of_equiv {\u03b2 : Type*} {f : \u03b1 \u2192 \u03b2} (e : \u03b1 \u2243 \u03b2) :\n  injective f \u2194 surjective f :=\nhave injective (e.symm \u2218 f) \u2194 surjective (e.symm \u2218 f), from injective_iff_surjective,\n\u27e8\u03bb hinj, by simpa [function.comp] using\n  e.surjective.comp (this.1 (e.symm.injective.comp hinj)),\n\u03bb hsurj, by simpa [function.comp] using\n  e.injective.comp (this.2 (e.symm.surjective.comp hsurj))\u27e9\n\nlemma bijective_iff_injective_and_card (f : \u03b1 \u2192 \u03b2) :\n  bijective f \u2194 injective f \u2227 card \u03b1 = card \u03b2 :=\nbegin\n  split,\n  { intro h, exact \u27e8h.1, card_congr (equiv.of_bijective f h)\u27e9 },\n  { rintro \u27e8hf, h\u27e9,\n    refine \u27e8hf, _\u27e9,\n    rwa \u2190injective_iff_surjective_of_equiv (equiv_of_card_eq h) }\nend\n\nlemma bijective_iff_surjective_and_card (f : \u03b1 \u2192 \u03b2) :\n  bijective f \u2194 surjective f \u2227 card \u03b1 = card \u03b2 :=\nbegin\n  split,\n  { intro h, exact \u27e8h.2, card_congr (equiv.of_bijective f h)\u27e9, },\n  { rintro \u27e8hf, h\u27e9,\n    refine \u27e8_, hf\u27e9,\n    rwa injective_iff_surjective_of_equiv (equiv_of_card_eq h) }\nend\n\nlemma right_inverse_of_left_inverse_of_card_le {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n  (hfg : left_inverse f g) (hcard : card \u03b1 \u2264 card \u03b2) :\n  right_inverse f g :=\nhave hsurj : surjective f, from surjective_iff_has_right_inverse.2 \u27e8g, hfg\u27e9,\nright_inverse_of_injective_of_left_inverse\n  ((bijective_iff_surjective_and_card _).2\n    \u27e8hsurj, le_antisymm hcard (card_le_of_surjective f hsurj)\u27e9 ).1\n  hfg\n\nlemma left_inverse_of_right_inverse_of_card_le {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n  (hfg : right_inverse f g) (hcard : card \u03b2 \u2264 card \u03b1) :\n  left_inverse f g :=\nright_inverse_of_left_inverse_of_card_le hfg hcard\n\nend fintype\n\nlemma fintype.coe_image_univ [fintype \u03b1] [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  \u2191(finset.image f finset.univ) = set.range f :=\nby { ext x, simp }\n\ninstance list.subtype.fintype [decidable_eq \u03b1] (l : list \u03b1) : fintype {x \/\/ x \u2208 l} :=\nfintype.of_list l.attach l.mem_attach\n\ninstance multiset.subtype.fintype [decidable_eq \u03b1] (s : multiset \u03b1) : fintype {x \/\/ x \u2208 s} :=\nfintype.of_multiset s.attach s.mem_attach\n\ninstance finset.subtype.fintype (s : finset \u03b1) : fintype {x \/\/ x \u2208 s} :=\n\u27e8s.attach, s.mem_attach\u27e9\n\ninstance finset_coe.fintype (s : finset \u03b1) : fintype (\u2191s : set \u03b1) :=\nfinset.subtype.fintype s\n\n@[simp] lemma fintype.card_coe (s : finset \u03b1) :\n  fintype.card s = s.card := card_attach\n\nlemma finset.attach_eq_univ {s : finset \u03b1} : s.attach = finset.univ := rfl\n\ninstance plift.fintype (p : Prop) [decidable p] : fintype (plift p) :=\n\u27e8if h : p then {\u27e8h\u27e9} else \u2205, \u03bb \u27e8h\u27e9, by simp [h]\u27e9\n\ninstance Prop.fintype : fintype Prop :=\n\u27e8\u27e8true ::\u2098 false ::\u2098 0, by simp [true_ne_false]\u27e9,\n classical.cases (by simp) (by simp)\u27e9\n\ninstance subtype.fintype (p : \u03b1 \u2192 Prop) [decidable_pred p] [fintype \u03b1] : fintype {x \/\/ p x} :=\nfintype.subtype (univ.filter p) (by simp)\n\n\/-- A set on a fintype, when coerced to a type, is a fintype. -\/\ndef set_fintype {\u03b1} [fintype \u03b1] (s : set \u03b1) [decidable_pred (\u2208 s)] : fintype s :=\nsubtype.fintype (\u03bb x, x \u2208 s)\n\nlemma set_fintype_card_le_univ {\u03b1 : Type*} [fintype \u03b1] (s : set \u03b1) [fintype \u21a5s] :\n  fintype.card \u21a5s \u2264 fintype.card \u03b1 :=\nfintype.card_le_of_embedding (function.embedding.subtype s)\n\nnamespace function.embedding\n\n\/-- An embedding from a `fintype` to itself can be promoted to an equivalence. -\/\nnoncomputable def equiv_of_fintype_self_embedding {\u03b1 : Type*} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) : \u03b1 \u2243 \u03b1 :=\nequiv.of_bijective e (fintype.injective_iff_bijective.1 e.2)\n\n@[simp]\nlemma equiv_of_fintype_self_embedding_to_embedding {\u03b1 : Type*} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) :\n  e.equiv_of_fintype_self_embedding.to_embedding = e :=\nby { ext, refl, }\n\n\/-- If `\u2016\u03b2\u2016 < \u2016\u03b1\u2016` there are no embeddings `\u03b1 \u21aa \u03b2`.\nThis is a formulation of the pigeonhole principle.\n\nNote this cannot be an instance as it needs `h`. -\/\n@[simp] lemma is_empty_of_card_lt {\u03b1 \u03b2} [fintype \u03b1] [fintype \u03b2]\n  (h : fintype.card \u03b2 < fintype.card \u03b1) : is_empty (\u03b1 \u21aa \u03b2) :=\n\u27e8\u03bb f, let \u27e8x, y, ne, feq\u27e9 := fintype.exists_ne_map_eq_of_card_lt f h in ne $ f.injective feq\u27e9\n\nend function.embedding\n\n@[simp]\nlemma finset.univ_map_embedding {\u03b1 : Type*} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) :\n  univ.map e = univ :=\nby rw [\u2190e.equiv_of_fintype_self_embedding_to_embedding, univ_map_equiv_to_embedding]\n\nnamespace fintype\n\nlemma card_lt_of_surjective_not_injective [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2)\n  (h : function.surjective f) (h' : \u00acfunction.injective f) : card \u03b2 < card \u03b1 :=\ncard_lt_of_injective_not_surjective _ (function.injective_surj_inv h) $ \u03bb hg,\nhave w : function.bijective (function.surj_inv h) := \u27e8function.injective_surj_inv h, hg\u27e9,\nh' $ (injective_iff_surjective_of_equiv (equiv.of_bijective _ w).symm).mpr h\n\nvariables [decidable_eq \u03b1] [fintype \u03b1] {\u03b4 : \u03b1 \u2192 Type*}\n\n\/-- Given for all `a : \u03b1` a finset `t a` of `\u03b4 a`, then one can define the\nfinset `fintype.pi_finset t` of all functions taking values in `t a` for all `a`. This is the\nanalogue of `finset.pi` where the base finset is `univ` (but formally they are not the same, as\nthere is an additional condition `i \u2208 finset.univ` in the `finset.pi` definition). -\/\ndef pi_finset (t : \u03a0 a, finset (\u03b4 a)) : finset (\u03a0 a, \u03b4 a) :=\n(finset.univ.pi t).map \u27e8\u03bb f a, f a (mem_univ a), \u03bb _ _, by simp [function.funext_iff]\u27e9\n\n@[simp] lemma mem_pi_finset {t : \u03a0 a, finset (\u03b4 a)} {f : \u03a0 a, \u03b4 a} :\n  f \u2208 pi_finset t \u2194 (\u2200 a, f a \u2208 t a) :=\nbegin\n  split,\n  { simp only [pi_finset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ,\n               exists_imp_distrib, mem_pi],\n    rintro g hg hgf a,\n    rw \u2190 hgf,\n    exact hg a },\n  { simp only [pi_finset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi],\n    exact \u03bb hf, \u27e8\u03bb a ha, f a, hf, rfl\u27e9 }\nend\n\nlemma pi_finset_subset (t\u2081 t\u2082 : \u03a0 a, finset (\u03b4 a)) (h : \u2200 a, t\u2081 a \u2286 t\u2082 a) :\n  pi_finset t\u2081 \u2286 pi_finset t\u2082 :=\n\u03bb g hg, mem_pi_finset.2 $ \u03bb a, h a $ mem_pi_finset.1 hg a\n\nlemma pi_finset_disjoint_of_disjoint [\u2200 a, decidable_eq (\u03b4 a)]\n  (t\u2081 t\u2082 : \u03a0 a, finset (\u03b4 a)) {a : \u03b1} (h : disjoint (t\u2081 a) (t\u2082 a)) :\n  disjoint (pi_finset t\u2081) (pi_finset t\u2082) :=\ndisjoint_iff_ne.2 $ \u03bb f\u2081 hf\u2081 f\u2082 hf\u2082 eq\u2081\u2082,\ndisjoint_iff_ne.1 h (f\u2081 a) (mem_pi_finset.1 hf\u2081 a) (f\u2082 a) (mem_pi_finset.1 hf\u2082 a) (congr_fun eq\u2081\u2082 a)\n\nend fintype\n\n\/-! ### pi -\/\n\n\/-- A dependent product of fintypes, indexed by a fintype, is a fintype. -\/\ninstance pi.fintype {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*}\n  [decidable_eq \u03b1] [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] : fintype (\u03a0 a, \u03b2 a) :=\n\u27e8fintype.pi_finset (\u03bb _, univ), by simp\u27e9\n\n@[simp] lemma fintype.pi_finset_univ {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*}\n  [decidable_eq \u03b1] [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] :\n  fintype.pi_finset (\u03bb a : \u03b1, (finset.univ : finset (\u03b2 a))) = (finset.univ : finset (\u03a0 a, \u03b2 a)) :=\nrfl\n\ninstance d_array.fintype {n : \u2115} {\u03b1 : fin n \u2192 Type*}\n  [\u2200 n, fintype (\u03b1 n)] : fintype (d_array n \u03b1) :=\nfintype.of_equiv _ (equiv.d_array_equiv_fin _).symm\n\ninstance array.fintype {n : \u2115} {\u03b1 : Type*} [fintype \u03b1] : fintype (array n \u03b1) :=\nd_array.fintype\n\ninstance vector.fintype {\u03b1 : Type*} [fintype \u03b1] {n : \u2115} : fintype (vector \u03b1 n) :=\nfintype.of_equiv _ (equiv.vector_equiv_fin _ _).symm\n\ninstance quotient.fintype [fintype \u03b1] (s : setoid \u03b1)\n  [decidable_rel ((\u2248) : \u03b1 \u2192 \u03b1 \u2192 Prop)] : fintype (quotient s) :=\nfintype.of_surjective quotient.mk (\u03bb x, quotient.induction_on x (\u03bb x, \u27e8x, rfl\u27e9))\n\ninstance finset.fintype [fintype \u03b1] : fintype (finset \u03b1) :=\n\u27e8univ.powerset, \u03bb x, finset.mem_powerset.2 (finset.subset_univ _)\u27e9\n\n-- irreducible due to this conversation on Zulip:\n-- https:\/\/leanprover.zulipchat.com\/#narrow\/stream\/113488-general\/\n-- topic\/.60simp.60.20ignoring.20lemmas.3F\/near\/241824115\n@[irreducible] instance function.embedding.fintype {\u03b1 \u03b2} [fintype \u03b1] [fintype \u03b2]\n  [decidable_eq \u03b1] [decidable_eq \u03b2] : fintype (\u03b1 \u21aa \u03b2) :=\nfintype.of_equiv _ (equiv.subtype_injective_equiv_embedding \u03b1 \u03b2)\n\ninstance [decidable_eq \u03b1] [fintype \u03b1] {n : \u2115} : fintype (sym.sym' \u03b1 n) :=\nquotient.fintype _\n\ninstance [decidable_eq \u03b1] [fintype \u03b1] {n : \u2115} : fintype (sym \u03b1 n) :=\nfintype.of_equiv _ sym.sym_equiv_sym'.symm\n\n@[simp] lemma fintype.card_finset [fintype \u03b1] :\n  fintype.card (finset \u03b1) = 2 ^ (fintype.card \u03b1) :=\nfinset.card_powerset finset.univ\n\n@[simp] lemma finset.univ_filter_card_eq (\u03b1 : Type*) [fintype \u03b1] (k : \u2115) :\n  (finset.univ : finset (finset \u03b1)).filter (\u03bb s, s.card = k) = finset.univ.powerset_len k :=\nby { ext, simp [finset.mem_powerset_len] }\n\n@[simp] lemma fintype.card_finset_len [fintype \u03b1] (k : \u2115) :\n  fintype.card {s : finset \u03b1 \/\/ s.card = k} = nat.choose (fintype.card \u03b1) k :=\nby simp [fintype.subtype_card, finset.card_univ]\n\n@[simp] lemma set.to_finset_univ [fintype \u03b1] :\n  (set.univ : set \u03b1).to_finset = finset.univ :=\nby { ext, simp only [set.mem_univ, mem_univ, set.mem_to_finset] }\n\n@[simp] lemma set.to_finset_eq_empty_iff {s : set \u03b1} [fintype s] :\n  s.to_finset = \u2205 \u2194 s = \u2205 :=\nby simp [ext_iff, set.ext_iff]\n\n@[simp] lemma set.to_finset_empty :\n  (\u2205 : set \u03b1).to_finset = \u2205 :=\nset.to_finset_eq_empty_iff.mpr rfl\n\n@[simp] lemma set.to_finset_range [decidable_eq \u03b1] [fintype \u03b2] (f : \u03b2 \u2192 \u03b1) [fintype (set.range f)] :\n  (set.range f).to_finset = finset.univ.image f :=\nby simp [ext_iff]\n\ntheorem fintype.card_subtype_le [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  fintype.card {x \/\/ p x} \u2264 fintype.card \u03b1 :=\nfintype.card_le_of_embedding (function.embedding.subtype _)\n\ntheorem fintype.card_subtype_lt [fintype \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p]\n  {x : \u03b1} (hx : \u00ac p x) : fintype.card {x \/\/ p x} < fintype.card \u03b1 :=\nfintype.card_lt_of_injective_of_not_mem coe subtype.coe_injective $ by rwa subtype.range_coe_subtype\n\nlemma fintype.card_subtype [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  fintype.card {x \/\/ p x} = ((finset.univ : finset \u03b1).filter p).card :=\nbegin\n  refine fintype.card_of_subtype _ _,\n  simp\nend\n\nlemma fintype.card_subtype_or (p q : \u03b1 \u2192 Prop)\n  [fintype {x \/\/ p x}] [fintype {x \/\/ q x}] [fintype {x \/\/ p x \u2228 q x}] :\n  fintype.card {x \/\/ p x \u2228 q x} \u2264 fintype.card {x \/\/ p x} + fintype.card {x \/\/ q x} :=\nbegin\n  classical,\n  convert fintype.card_le_of_embedding (subtype_or_left_embedding p q),\n  rw fintype.card_sum\nend\n\nlemma fintype.card_subtype_or_disjoint (p q : \u03b1 \u2192 Prop) (h : disjoint p q)\n  [fintype {x \/\/ p x}] [fintype {x \/\/ q x}] [fintype {x \/\/ p x \u2228 q x}] :\n  fintype.card {x \/\/ p x \u2228 q x} = fintype.card {x \/\/ p x} + fintype.card {x \/\/ q x} :=\nbegin\n  classical,\n  convert fintype.card_congr (subtype_or_equiv p q h),\n  simp\nend\n\ntheorem fintype.card_quotient_le [fintype \u03b1] (s : setoid \u03b1) [decidable_rel ((\u2248) : \u03b1 \u2192 \u03b1 \u2192 Prop)] :\n  fintype.card (quotient s) \u2264 fintype.card \u03b1 :=\nfintype.card_le_of_surjective _ (surjective_quotient_mk _)\n\ntheorem fintype.card_quotient_lt [fintype \u03b1] {s : setoid \u03b1} [decidable_rel ((\u2248) : \u03b1 \u2192 \u03b1 \u2192 Prop)]\n  {x y : \u03b1} (h1 : x \u2260 y) (h2 : x \u2248 y) : fintype.card (quotient s) < fintype.card \u03b1 :=\nfintype.card_lt_of_surjective_not_injective _ (surjective_quotient_mk _) $ \u03bb w,\nh1 (w $ quotient.eq.mpr h2)\n\ninstance psigma.fintype {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] :\n  fintype (\u03a3' a, \u03b2 a) :=\nfintype.of_equiv _ (equiv.psigma_equiv_sigma _).symm\n\ninstance psigma.fintype_prop_left {\u03b1 : Prop} {\u03b2 : \u03b1 \u2192 Type*} [decidable \u03b1] [\u2200 a, fintype (\u03b2 a)] :\n  fintype (\u03a3' a, \u03b2 a) :=\nif h : \u03b1 then fintype.of_equiv (\u03b2 h) \u27e8\u03bb x, \u27e8h, x\u27e9, psigma.snd, \u03bb _, rfl, \u03bb \u27e8_, _\u27e9, rfl\u27e9\nelse \u27e8\u2205, \u03bb x, h x.1\u27e9\n\ninstance psigma.fintype_prop_right {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Prop} [\u2200 a, decidable (\u03b2 a)] [fintype \u03b1] :\n  fintype (\u03a3' a, \u03b2 a) :=\nfintype.of_equiv {a \/\/ \u03b2 a} \u27e8\u03bb \u27e8x, y\u27e9, \u27e8x, y\u27e9, \u03bb \u27e8x, y\u27e9, \u27e8x, y\u27e9, \u03bb \u27e8x, y\u27e9, rfl, \u03bb \u27e8x, y\u27e9, rfl\u27e9\n\ninstance psigma.fintype_prop_prop {\u03b1 : Prop} {\u03b2 : \u03b1 \u2192 Prop} [decidable \u03b1] [\u2200 a, decidable (\u03b2 a)] :\n  fintype (\u03a3' a, \u03b2 a) :=\nif h : \u2203 a, \u03b2 a then \u27e8{\u27e8h.fst, h.snd\u27e9}, \u03bb \u27e8_, _\u27e9, by simp\u27e9 else \u27e8\u2205, \u03bb \u27e8x, y\u27e9, h \u27e8x, y\u27e9\u27e9\n\ninstance set.fintype [fintype \u03b1] : fintype (set \u03b1) :=\n\u27e8(@finset.univ \u03b1 _).powerset.map \u27e8coe, coe_injective\u27e9, \u03bb s, begin\n  classical, refine mem_map.2 \u27e8finset.univ.filter s, mem_powerset.2 (subset_univ _), _\u27e9,\n  apply (coe_filter _ _).trans, rw [coe_univ, set.sep_univ], refl\nend\u27e9\n\ninstance pfun_fintype (p : Prop) [decidable p] (\u03b1 : p \u2192 Type*)\n  [\u03a0 hp, fintype (\u03b1 hp)] : fintype (\u03a0 hp : p, \u03b1 hp) :=\nif hp : p then fintype.of_equiv (\u03b1 hp) \u27e8\u03bb a _, a, \u03bb f, f hp, \u03bb _, rfl, \u03bb _, rfl\u27e9\n          else \u27e8singleton (\u03bb h, (hp h).elim), by simp [hp, function.funext_iff]\u27e9\n\n@[simp] lemma finset.univ_pi_univ {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*}\n  [decidable_eq \u03b1] [fintype \u03b1] [\u2200 a, fintype (\u03b2 a)] :\n  finset.univ.pi (\u03bb a : \u03b1, (finset.univ : finset (\u03b2 a))) = finset.univ :=\nby { ext, simp }\n\nlemma mem_image_univ_iff_mem_range\n  {\u03b1 \u03b2 : Type*} [fintype \u03b1] [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} {b : \u03b2} :\n  b \u2208 univ.image f \u2194 b \u2208 set.range f :=\nby simp\n\n\/-- An auxiliary function for `quotient.fin_choice`.  Given a\ncollection of setoids indexed by a type `\u03b9`, a (finite) list `l` of\nindices, and a function that for each `i \u2208 l` gives a term of the\ncorresponding quotient type, then there is a corresponding term in the\nquotient of the product of the setoids indexed by `l`. -\/\ndef quotient.fin_choice_aux {\u03b9 : Type*} [decidable_eq \u03b9]\n  {\u03b1 : \u03b9 \u2192 Type*} [S : \u2200 i, setoid (\u03b1 i)] :\n  \u03a0 (l : list \u03b9), (\u03a0 i \u2208 l, quotient (S i)) \u2192 @quotient (\u03a0 i \u2208 l, \u03b1 i) (by apply_instance)\n| []       f := \u27e6\u03bb i, false.elim\u27e7\n| (i :: l) f := begin\n  refine quotient.lift_on\u2082 (f i (list.mem_cons_self _ _))\n    (quotient.fin_choice_aux l (\u03bb j h, f j (list.mem_cons_of_mem _ h)))\n    _ _,\n  exact \u03bb a l, \u27e6\u03bb j h,\n    if e : j = i then by rw e; exact a else\n    l _ (h.resolve_left e)\u27e7,\n  refine \u03bb a\u2081 l\u2081 a\u2082 l\u2082 h\u2081 h\u2082, quotient.sound (\u03bb j h, _),\n  by_cases e : j = i; simp [e],\n  { subst j, exact h\u2081 },\n  { exact h\u2082 _ _ }\nend\n\ntheorem quotient.fin_choice_aux_eq {\u03b9 : Type*} [decidable_eq \u03b9]\n  {\u03b1 : \u03b9 \u2192 Type*} [S : \u2200 i, setoid (\u03b1 i)] :\n  \u2200 (l : list \u03b9) (f : \u03a0 i \u2208 l, \u03b1 i), quotient.fin_choice_aux l (\u03bb i h, \u27e6f i h\u27e7) = \u27e6f\u27e7\n| []       f := quotient.sound (\u03bb i h, h.elim)\n| (i :: l) f := begin\n  simp [quotient.fin_choice_aux, quotient.fin_choice_aux_eq l],\n  refine quotient.sound (\u03bb j h, _),\n  by_cases e : j = i; simp [e],\n  subst j, refl\nend\n\n\/-- Given a collection of setoids indexed by a fintype `\u03b9` and a\nfunction that for each `i : \u03b9` gives a term of the corresponding\nquotient type, then there is corresponding term in the quotient of the\nproduct of the setoids. -\/\ndef quotient.fin_choice {\u03b9 : Type*} [decidable_eq \u03b9] [fintype \u03b9]\n  {\u03b1 : \u03b9 \u2192 Type*} [S : \u2200 i, setoid (\u03b1 i)]\n  (f : \u03a0 i, quotient (S i)) : @quotient (\u03a0 i, \u03b1 i) (by apply_instance) :=\nquotient.lift_on (@quotient.rec_on _ _ (\u03bb l : multiset \u03b9,\n    @quotient (\u03a0 i \u2208 l, \u03b1 i) (by apply_instance))\n    finset.univ.1\n    (\u03bb l, quotient.fin_choice_aux l (\u03bb i _, f i))\n    (\u03bb a b h, begin\n      have := \u03bb a, quotient.fin_choice_aux_eq a (\u03bb i h, quotient.out (f i)),\n      simp [quotient.out_eq] at this,\n      simp [this],\n      let g := \u03bb a:multiset \u03b9, \u27e6\u03bb (i : \u03b9) (h : i \u2208 a), quotient.out (f i)\u27e7,\n      refine eq_of_heq ((eq_rec_heq _ _).trans (_ : g a == g b)),\n      congr' 1, exact quotient.sound h,\n    end))\n  (\u03bb f, \u27e6\u03bb i, f i (finset.mem_univ _)\u27e7)\n  (\u03bb a b h, quotient.sound $ \u03bb i, h _ _)\n\ntheorem quotient.fin_choice_eq {\u03b9 : Type*} [decidable_eq \u03b9] [fintype \u03b9]\n  {\u03b1 : \u03b9 \u2192 Type*} [\u2200 i, setoid (\u03b1 i)]\n  (f : \u03a0 i, \u03b1 i) : quotient.fin_choice (\u03bb i, \u27e6f i\u27e7) = \u27e6f\u27e7 :=\nbegin\n  let q, swap, change quotient.lift_on q _ _ = _,\n  have : q = \u27e6\u03bb i h, f i\u27e7,\n  { dsimp [q],\n    exact quotient.induction_on\n      (@finset.univ \u03b9 _).1 (\u03bb l, quotient.fin_choice_aux_eq _ _) },\n  simp [this], exact setoid.refl _\nend\n\nsection equiv\n\nopen list equiv equiv.perm\n\nvariables [decidable_eq \u03b1] [decidable_eq \u03b2]\n\n\/-- Given a list, produce a list of all permutations of its elements. -\/\ndef perms_of_list : list \u03b1 \u2192 list (perm \u03b1)\n| []       := [1]\n| (a :: l) := perms_of_list l ++ l.bind (\u03bb b, (perms_of_list l).map (\u03bb f, swap a b * f))\n\nlemma length_perms_of_list : \u2200 l : list \u03b1, length (perms_of_list l) = l.length!\n| []       := rfl\n| (a :: l) :=\nbegin\n  rw [length_cons, nat.factorial_succ],\n  simp [perms_of_list, length_bind, length_perms_of_list, function.comp, nat.succ_mul],\n  cc\nend\n\nlemma mem_perms_of_list_of_mem {l : list \u03b1} {f : perm \u03b1} (h : \u2200 x, f x \u2260 x \u2192 x \u2208 l) :\n  f \u2208 perms_of_list l :=\nbegin\n  induction l with a l IH generalizing f h,\n  { exact list.mem_singleton.2 (equiv.ext $ \u03bb x, decidable.by_contradiction $ h _) },\n  by_cases hfa : f a = a,\n  { refine mem_append_left _ (IH (\u03bb x hx, mem_of_ne_of_mem _ (h x hx))),\n    rintro rfl, exact hx hfa },\n  have hfa' : f (f a) \u2260 f a := mt (\u03bb h, f.injective h) hfa,\n  have : \u2200 (x : \u03b1), (swap a (f a) * f) x \u2260 x \u2192 x \u2208 l,\n  { intros x hx,\n    have hxa : x \u2260 a,\n    { rintro rfl, apply hx, simp only [mul_apply, swap_apply_right] },\n    refine list.mem_of_ne_of_mem hxa (h x (\u03bb h, _)),\n    simp only [h, mul_apply, swap_apply_def, mul_apply, ne.def, apply_eq_iff_eq] at hx;\n    split_ifs at hx, exacts [hxa (h.symm.trans h_1), hx h] },\n  suffices : f \u2208 perms_of_list l \u2228 \u2203 (b \u2208 l) (g \u2208 perms_of_list l), swap a b * g = f,\n  { simpa only [perms_of_list, exists_prop, list.mem_map, mem_append, list.mem_bind] },\n  refine or_iff_not_imp_left.2 (\u03bb hfl, \u27e8f a, _, swap a (f a) * f, IH this, _\u27e9),\n  { by_cases hffa : f (f a) = a,\n    { exact mem_of_ne_of_mem hfa (h _ (mt (\u03bb h, f.injective h) hfa)) },\n    { apply this,\n      simp only [mul_apply, swap_apply_def, mul_apply, ne.def, apply_eq_iff_eq],\n      split_ifs; cc } },\n  { rw [\u2190mul_assoc, mul_def (swap a (f a)) (swap a (f a)),\n        swap_swap, \u2190perm.one_def, one_mul] }\nend\n\nlemma mem_of_mem_perms_of_list :\n  \u2200 {l : list \u03b1} {f : perm \u03b1}, f \u2208 perms_of_list l \u2192 \u2200 {x}, f x \u2260 x \u2192 x \u2208 l\n| []       f h := have f = 1 := by simpa [perms_of_list] using h, by rw this; simp\n| (a :: l) f h :=\n(mem_append.1 h).elim\n  (\u03bb h x hx, mem_cons_of_mem _ (mem_of_mem_perms_of_list h hx))\n  (\u03bb h x hx,\n    let \u27e8y, hy, hy'\u27e9 := list.mem_bind.1 h in\n    let \u27e8g, hg\u2081, hg\u2082\u27e9 := list.mem_map.1 hy' in\n    if hxa : x = a then by simp [hxa]\n    else if hxy : x = y then mem_cons_of_mem _ $ by rwa hxy\n    else mem_cons_of_mem _ $\n    mem_of_mem_perms_of_list hg\u2081 $\n      by rw [eq_inv_mul_iff_mul_eq.2 hg\u2082, mul_apply, swap_inv, swap_apply_def];\n        split_ifs; cc)\n\nlemma mem_perms_of_list_iff {l : list \u03b1} {f : perm \u03b1} :\n  f \u2208 perms_of_list l \u2194 \u2200 {x}, f x \u2260 x \u2192 x \u2208 l :=\n\u27e8mem_of_mem_perms_of_list, mem_perms_of_list_of_mem\u27e9\n\nlemma nodup_perms_of_list : \u2200 {l : list \u03b1} (hl : l.nodup), (perms_of_list l).nodup\n| []       hl := by simp [perms_of_list]\n| (a :: l) hl :=\nhave hl' : l.nodup, from nodup_of_nodup_cons hl,\nhave hln' : (perms_of_list l).nodup, from nodup_perms_of_list hl',\nhave hmeml : \u2200 {f : perm \u03b1}, f \u2208 perms_of_list l \u2192 f a = a,\n  from \u03bb f hf, not_not.1 (mt (mem_of_mem_perms_of_list hf) (nodup_cons.1 hl).1),\nby rw [perms_of_list, list.nodup_append, list.nodup_bind, pairwise_iff_nth_le]; exact\n\u27e8hln', \u27e8\u03bb _ _, nodup_map (\u03bb _ _, mul_left_cancel) hln',\n  \u03bb i j hj hij x hx\u2081 hx\u2082,\n    let \u27e8f, hf\u27e9 := list.mem_map.1 hx\u2081 in\n    let \u27e8g, hg\u27e9 := list.mem_map.1 hx\u2082 in\n    have hix : x a = nth_le l i (lt_trans hij hj),\n      by rw [\u2190hf.2, mul_apply, hmeml hf.1, swap_apply_left],\n    have hiy : x a = nth_le l j hj,\n      by rw [\u2190 hg.2, mul_apply, hmeml hg.1, swap_apply_left],\n    absurd (hf.2.trans (hg.2.symm)) $\n      \u03bb h, ne_of_lt hij $ nodup_iff_nth_le_inj.1 hl' i j (lt_trans hij hj) hj $\n        by rw [\u2190 hix, hiy]\u27e9,\n  \u03bb f hf\u2081 hf\u2082,\n    let \u27e8x, hx, hx'\u27e9 := list.mem_bind.1 hf\u2082 in\n    let \u27e8g, hg\u27e9 := list.mem_map.1 hx' in\n    have hgxa : g\u207b\u00b9 x = a, from f.injective $\n      by rw [hmeml hf\u2081, \u2190 hg.2]; simp,\n    have hxa : x \u2260 a, from \u03bb h, (list.nodup_cons.1 hl).1 (h \u25b8 hx),\n    (list.nodup_cons.1 hl).1 $\n      hgxa \u25b8 mem_of_mem_perms_of_list hg.1 (by rwa [apply_inv_self, hgxa])\u27e9\n\n\/-- Given a finset, produce the finset of all permutations of its elements. -\/\ndef perms_of_finset (s : finset \u03b1) : finset (perm \u03b1) :=\nquotient.hrec_on s.1 (\u03bb l hl, \u27e8perms_of_list l, nodup_perms_of_list hl\u27e9)\n  (\u03bb a b hab, hfunext (congr_arg _ (quotient.sound hab))\n    (\u03bb ha hb _, heq_of_eq $ finset.ext $\n      by simp [mem_perms_of_list_iff, hab.mem_iff]))\n  s.2\n\nlemma mem_perms_of_finset_iff : \u2200 {s : finset \u03b1} {f : perm \u03b1},\n  f \u2208 perms_of_finset s \u2194 \u2200 {x}, f x \u2260 x \u2192 x \u2208 s :=\nby rintros \u27e8\u27e8l\u27e9, hs\u27e9 f; exact mem_perms_of_list_iff\n\nlemma card_perms_of_finset : \u2200 (s : finset \u03b1),\n  (perms_of_finset s).card = s.card! :=\nby rintros \u27e8\u27e8l\u27e9, hs\u27e9; exact length_perms_of_list l\n\n\/-- The collection of permutations of a fintype is a fintype. -\/\ndef fintype_perm [fintype \u03b1] : fintype (perm \u03b1) :=\n\u27e8perms_of_finset (@finset.univ \u03b1 _), by simp [mem_perms_of_finset_iff]\u27e9\n\ninstance [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u2243 \u03b2) :=\nif h : fintype.card \u03b2 = fintype.card \u03b1\nthen trunc.rec_on_subsingleton (fintype.trunc_equiv_fin \u03b1)\n  (\u03bb e\u03b1, trunc.rec_on_subsingleton (fintype.trunc_equiv_fin \u03b2)\n    (\u03bb e\u03b2, @fintype.of_equiv _ (perm \u03b1) fintype_perm\n      (equiv_congr (equiv.refl \u03b1) (e\u03b1.trans (eq.rec_on h e\u03b2.symm)) : (\u03b1 \u2243 \u03b1) \u2243 (\u03b1 \u2243 \u03b2))))\nelse \u27e8\u2205, \u03bb x, false.elim (h (fintype.card_eq.2 \u27e8x.symm\u27e9))\u27e9\n\nlemma fintype.card_perm [fintype \u03b1] : fintype.card (perm \u03b1) = (fintype.card \u03b1)! :=\nsubsingleton.elim (@fintype_perm \u03b1 _ _) (@equiv.fintype \u03b1 \u03b1 _ _ _ _) \u25b8\ncard_perms_of_finset _\n\nlemma fintype.card_equiv [fintype \u03b1] [fintype \u03b2] (e : \u03b1 \u2243 \u03b2) :\n  fintype.card (\u03b1 \u2243 \u03b2) = (fintype.card \u03b1)! :=\nfintype.card_congr (equiv_congr (equiv.refl \u03b1) e) \u25b8 fintype.card_perm\n\nlemma univ_eq_singleton_of_card_one {\u03b1} [fintype \u03b1] (x : \u03b1) (h : fintype.card \u03b1 = 1) :\n  (univ : finset \u03b1) = {x} :=\nbegin\n  symmetry,\n  apply eq_of_subset_of_card_le (subset_univ ({x})),\n  apply le_of_eq,\n  simp [h, finset.card_univ]\nend\n\nend equiv\n\nnamespace fintype\n\nsection choose\nopen fintype equiv\n\nvariables [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p]\n\n\/-- Given a fintype `\u03b1` and a predicate `p`, associate to a proof that there is a unique element of\n`\u03b1` satisfying `p` this unique element, as an element of the corresponding subtype. -\/\ndef choose_x (hp : \u2203! a : \u03b1, p a) : {a \/\/ p a} :=\n\u27e8finset.choose p univ (by simp; exact hp), finset.choose_property _ _ _\u27e9\n\n\/-- Given a fintype `\u03b1` and a predicate `p`, associate to a proof that there is a unique element of\n`\u03b1` satisfying `p` this unique element, as an element of `\u03b1`. -\/\ndef choose (hp : \u2203! a, p a) : \u03b1 := choose_x p hp\n\nlemma choose_spec (hp : \u2203! a, p a) : p (choose p hp) :=\n(choose_x p hp).property\n\nend choose\n\nsection bijection_inverse\nopen function\n\nvariables [fintype \u03b1] [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2}\n\n\/--\n`bij_inv f` is the unique inverse to a bijection `f`. This acts\n  as a computable alternative to `function.inv_fun`. -\/\ndef bij_inv (f_bij : bijective f) (b : \u03b2) : \u03b1 :=\nfintype.choose (\u03bb a, f a = b)\nbegin\n  rcases f_bij.right b with \u27e8a', fa_eq_b\u27e9,\n  rw \u2190 fa_eq_b,\n  exact \u27e8a', \u27e8rfl, (\u03bb a h, f_bij.left h)\u27e9\u27e9\nend\n\nlemma left_inverse_bij_inv (f_bij : bijective f) : left_inverse (bij_inv f_bij) f :=\n\u03bb a, f_bij.left (choose_spec (\u03bb a', f a' = f a) _)\n\nlemma right_inverse_bij_inv (f_bij : bijective f) : right_inverse (bij_inv f_bij) f :=\n\u03bb b, choose_spec (\u03bb a', f a' = b) _\n\nlemma bijective_bij_inv (f_bij : bijective f) : bijective (bij_inv f_bij) :=\n\u27e8(right_inverse_bij_inv _).injective, (left_inverse_bij_inv _).surjective\u27e9\n\nend bijection_inverse\n\nlemma well_founded_of_trans_of_irrefl [fintype \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n  [is_trans \u03b1 r] [is_irrefl \u03b1 r] : well_founded r :=\nby classical; exact\nhave \u2200 x y, r x y \u2192 (univ.filter (\u03bb z, r z x)).card < (univ.filter (\u03bb z, r z y)).card,\n  from \u03bb x y hxy, finset.card_lt_card $\n    by simp only [finset.lt_iff_ssubset.symm, lt_iff_le_not_le,\n      finset.le_iff_subset, finset.subset_iff, mem_filter, true_and, mem_univ, hxy];\n    exact \u27e8\u03bb z hzx, trans hzx hxy, not_forall_of_exists_not \u27e8x, not_imp.2 \u27e8hxy, irrefl x\u27e9\u27e9\u27e9,\nsubrelation.wf this (measure_wf _)\n\nlemma preorder.well_founded [fintype \u03b1] [preorder \u03b1] : well_founded ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\nwell_founded_of_trans_of_irrefl _\n\n@[instance, priority 10] lemma linear_order.is_well_order [fintype \u03b1] [linear_order \u03b1] :\n  is_well_order \u03b1 (<) :=\n{ wf := preorder.well_founded }\n\nend fintype\n\n\/-- A type is said to be infinite if it has no fintype instance.\n  Note that `infinite \u03b1` is equivalent to `is_empty (fintype \u03b1)`. -\/\nclass infinite (\u03b1 : Type*) : Prop :=\n(not_fintype : fintype \u03b1 \u2192 false)\n\nlemma not_fintype (\u03b1 : Type*) [h1 : infinite \u03b1] [h2 : fintype \u03b1] : false :=\ninfinite.not_fintype h2\n\nprotected lemma fintype.false {\u03b1 : Type*} [infinite \u03b1] (h : fintype \u03b1) : false :=\nnot_fintype \u03b1\n\nprotected lemma infinite.false {\u03b1 : Type*} [fintype \u03b1] (h : infinite \u03b1) : false :=\nnot_fintype \u03b1\n\n@[simp] lemma is_empty_fintype {\u03b1 : Type*} : is_empty (fintype \u03b1) \u2194 infinite \u03b1 :=\n\u27e8\u03bb \u27e8x\u27e9, \u27e8x\u27e9, \u03bb \u27e8x\u27e9, \u27e8x\u27e9\u27e9\n\n@[simp] lemma not_nonempty_fintype {\u03b1 : Type*} : \u00ac nonempty (fintype \u03b1) \u2194 infinite \u03b1 :=\nnot_nonempty_iff.trans is_empty_fintype\n\n\/-- A non-infinite type is a fintype. -\/\nnoncomputable def fintype_of_not_infinite {\u03b1 : Type*} (h : \u00ac infinite \u03b1) : fintype \u03b1 :=\n((not_iff_comm.mp not_nonempty_fintype).mp h).some\n\nlemma finset.exists_minimal {\u03b1 : Type*} [preorder \u03b1] (s : finset \u03b1) (h : s.nonempty) :\n  \u2203 m \u2208 s, \u2200 x \u2208 s, \u00ac (x < m) :=\nbegin\n  obtain \u27e8c, hcs : c \u2208 s\u27e9 := h,\n  have : well_founded (@has_lt.lt {x \/\/ x \u2208 s} _) := fintype.well_founded_of_trans_of_irrefl _,\n  obtain \u27e8\u27e8m, hms : m \u2208 s\u27e9, -, H\u27e9 := this.has_min set.univ \u27e8\u27e8c, hcs\u27e9, trivial\u27e9,\n  exact \u27e8m, hms, \u03bb x hx hxm, H \u27e8x, hx\u27e9 trivial hxm\u27e9,\nend\n\nlemma finset.exists_maximal {\u03b1 : Type*} [preorder \u03b1] (s : finset \u03b1) (h : s.nonempty) :\n  \u2203 m \u2208 s, \u2200 x \u2208 s, \u00ac (m < x) :=\n@finset.exists_minimal (order_dual \u03b1) _ s h\n\nnamespace infinite\n\nlemma exists_not_mem_finset [infinite \u03b1] (s : finset \u03b1) : \u2203 x, x \u2209 s :=\nnot_forall.1 $ \u03bb h, fintype.false \u27e8s, h\u27e9\n\n@[priority 100] -- see Note [lower instance priority]\ninstance (\u03b1 : Type*) [H : infinite \u03b1] : nontrivial \u03b1 :=\n\u27e8let \u27e8x, hx\u27e9 := exists_not_mem_finset (\u2205 : finset \u03b1) in\nlet \u27e8y, hy\u27e9 := exists_not_mem_finset ({x} : finset \u03b1) in\n\u27e8y, x, by simpa only [mem_singleton] using hy\u27e9\u27e9\n\nlemma nonempty (\u03b1 : Type*) [infinite \u03b1] : nonempty \u03b1 :=\nby apply_instance\n\nlemma of_injective [infinite \u03b2] (f : \u03b2 \u2192 \u03b1) (hf : injective f) : infinite \u03b1 :=\n\u27e8\u03bb I, by exactI (fintype.of_injective f hf).false\u27e9\n\nlemma of_surjective [infinite \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : surjective f) : infinite \u03b1 :=\n\u27e8\u03bb I, by { classical, exactI (fintype.of_surjective f hf).false }\u27e9\n\nprivate noncomputable def nat_embedding_aux (\u03b1 : Type*) [infinite \u03b1] : \u2115 \u2192 \u03b1\n| n := by letI := classical.dec_eq \u03b1; exact classical.some (exists_not_mem_finset\n  ((multiset.range n).pmap (\u03bb m (hm : m < n), nat_embedding_aux m)\n    (\u03bb _, multiset.mem_range.1)).to_finset)\n\nprivate lemma nat_embedding_aux_injective (\u03b1 : Type*) [infinite \u03b1] :\n  function.injective (nat_embedding_aux \u03b1) :=\nbegin\n  rintro m n h,\n  letI := classical.dec_eq \u03b1,\n  wlog hmlen : m \u2264 n using m n,\n  by_contradiction hmn,\n  have hmn : m < n, from lt_of_le_of_ne hmlen hmn,\n  refine (classical.some_spec (exists_not_mem_finset\n    ((multiset.range n).pmap (\u03bb m (hm : m < n), nat_embedding_aux \u03b1 m)\n      (\u03bb _, multiset.mem_range.1)).to_finset)) _,\n  refine multiset.mem_to_finset.2 (multiset.mem_pmap.2\n    \u27e8m, multiset.mem_range.2 hmn, _\u27e9),\n  rw [h, nat_embedding_aux]\nend\n\n\/-- Embedding of `\u2115` into an infinite type. -\/\nnoncomputable def nat_embedding (\u03b1 : Type*) [infinite \u03b1] : \u2115 \u21aa \u03b1 :=\n\u27e8_, nat_embedding_aux_injective \u03b1\u27e9\n\nlemma exists_subset_card_eq (\u03b1 : Type*) [infinite \u03b1] (n : \u2115) :\n  \u2203 s : finset \u03b1, s.card = n :=\n\u27e8(range n).map (nat_embedding \u03b1), by rw [card_map, card_range]\u27e9\n\nend infinite\n\nlemma not_injective_infinite_fintype [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u00ac injective f :=\n\u03bb hf, (fintype.of_injective f hf).false\n\n\/--\nThe pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are\ninfinitely many pigeons in finitely many pigeonholes, then there are at least two pigeons in the\nsame pigeonhole.\n\nSee also: `fintype.exists_ne_map_eq_of_card_lt`, `fintype.exists_infinite_fiber`.\n-\/\nlemma fintype.exists_ne_map_eq_of_infinite [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u2203 x y : \u03b1, x \u2260 y \u2227 f x = f y :=\nbegin\n  classical, by_contra hf, push_neg at hf,\n  apply not_injective_infinite_fintype f,\n  intros x y, contrapose, apply hf,\nend\n\n-- irreducible due to this conversation on Zulip:\n-- https:\/\/leanprover.zulipchat.com\/#narrow\/stream\/113488-general\/\n-- topic\/.60simp.60.20ignoring.20lemmas.3F\/near\/241824115\n\n@[irreducible]\ninstance function.embedding.is_empty {\u03b1 \u03b2} [infinite \u03b1] [fintype \u03b2] : is_empty (\u03b1 \u21aa \u03b2) :=\n  \u27e8\u03bb f, let \u27e8x, y, ne, feq\u27e9 := fintype.exists_ne_map_eq_of_infinite f in ne $ f.injective feq\u27e9\n\n@[priority 100]\nnoncomputable instance function.embedding.fintype' {\u03b1 \u03b2 : Type*} [fintype \u03b2] : fintype (\u03b1 \u21aa \u03b2) :=\nbegin\n  by_cases h : infinite \u03b1,\n  { resetI, apply_instance },\n  { have := fintype_of_not_infinite h, classical, apply_instance }\n  -- the `classical` generates `decidable_eq \u03b1\/\u03b2` instances, and resets instance cache\nend\n\n\/--\nThe strong pigeonhole principle for infinitely many pigeons in\nfinitely many pigeonholes.  If there are infinitely many pigeons in\nfinitely many pigeonholes, then there is a pigeonhole with infinitely\nmany pigeons.\n\nSee also: `fintype.exists_ne_map_eq_of_infinite`\n-\/\nlemma fintype.exists_infinite_fiber [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u2203 y : \u03b2, infinite (f \u207b\u00b9' {y}) :=\nbegin\n  classical,\n  by_contra hf,\n  push_neg at hf,\n\n  haveI := \u03bb y, fintype_of_not_infinite $ hf y,\n  let key : fintype \u03b1 :=\n  { elems := univ.bUnion (\u03bb (y : \u03b2), (f \u207b\u00b9' {y}).to_finset),\n    complete := by simp },\n  exact key.false,\nend\n\nlemma not_surjective_fintype_infinite [fintype \u03b1] [infinite \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  \u00ac surjective f :=\nassume (hf : surjective f),\nhave H : infinite \u03b1 := infinite.of_surjective f hf,\nby exactI not_fintype \u03b1\n\ninstance nat.infinite : infinite \u2115 :=\n\u27e8\u03bb \u27e8s, hs\u27e9, finset.not_mem_range_self $ s.subset_range_sup_succ (hs _)\u27e9\n\ninstance int.infinite : infinite \u2124 :=\ninfinite.of_injective int.of_nat (\u03bb _ _, int.of_nat.inj)\n\nsection trunc\n\n\/--\nFor `s : multiset \u03b1`, we can lift the existential statement that `\u2203 x, x \u2208 s` to a `trunc \u03b1`.\n-\/\ndef trunc_of_multiset_exists_mem {\u03b1} (s : multiset \u03b1) : (\u2203 x, x \u2208 s) \u2192 trunc \u03b1 :=\nquotient.rec_on_subsingleton s $ \u03bb l h,\n  match l, h with\n    | [],       _ := false.elim (by tauto)\n    | (a :: _), _ := trunc.mk a\n  end\n\n\/--\nA `nonempty` `fintype` constructively contains an element.\n-\/\ndef trunc_of_nonempty_fintype (\u03b1) [nonempty \u03b1] [fintype \u03b1] : trunc \u03b1 :=\ntrunc_of_multiset_exists_mem finset.univ.val (by simp)\n\n\/--\nA `fintype` with positive cardinality constructively contains an element.\n-\/\ndef trunc_of_card_pos {\u03b1} [fintype \u03b1] (h : 0 < fintype.card \u03b1) : trunc \u03b1 :=\nby { letI := (fintype.card_pos_iff.mp h), exact trunc_of_nonempty_fintype \u03b1 }\n\n\/--\nBy iterating over the elements of a fintype, we can lift an existential statement `\u2203 a, P a`\nto `trunc (\u03a3' a, P a)`, containing data.\n-\/\ndef trunc_sigma_of_exists {\u03b1} [fintype \u03b1] {P : \u03b1 \u2192 Prop} [decidable_pred P] (h : \u2203 a, P a) :\n  trunc (\u03a3' a, P a) :=\n@trunc_of_nonempty_fintype (\u03a3' a, P a) (exists.elim h $ \u03bb a ha, \u27e8\u27e8a, ha\u27e9\u27e9) _\n\nend trunc\n\nnamespace multiset\n\nvariables [fintype \u03b1] [decidable_eq \u03b1]\n\n@[simp] lemma count_univ (a : \u03b1) :\n  count a finset.univ.val = 1 :=\ncount_eq_one_of_mem finset.univ.nodup (finset.mem_univ _)\n\nend multiset\n\nnamespace fintype\n\n\/-- A recursor principle for finite types, analogous to `nat.rec`. It effectively says\nthat every `fintype` is either `empty` or `option \u03b1`, up to an `equiv`. -\/\ndef trunc_rec_empty_option {P : Type u \u2192 Sort v}\n  (of_equiv : \u2200 {\u03b1 \u03b2}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2)\n  (h_empty : P pempty)\n  (h_option : \u2200 {\u03b1} [fintype \u03b1] [decidable_eq \u03b1], P \u03b1 \u2192 P (option \u03b1))\n  (\u03b1 : Type u) [fintype \u03b1] [decidable_eq \u03b1] : trunc (P \u03b1) :=\nbegin\n  suffices : \u2200 n : \u2115, trunc (P (ulift $ fin n)),\n  { apply trunc.bind (this (fintype.card \u03b1)),\n    intro h,\n    apply trunc.map _ (fintype.trunc_equiv_fin \u03b1),\n    intro e,\n    exact of_equiv (equiv.ulift.trans e.symm) h },\n  intro n,\n  induction n with n ih,\n  { have : card pempty = card (ulift (fin 0)),\n    { simp only [card_fin, card_pempty, card_ulift] },\n    apply trunc.bind (trunc_equiv_of_card_eq this),\n    intro e,\n    apply trunc.mk,\n    refine of_equiv e h_empty, },\n  { have : card (option (ulift (fin n))) = card (ulift (fin n.succ)),\n    { simp only [card_fin, card_option, card_ulift] },\n    apply trunc.bind (trunc_equiv_of_card_eq this),\n    intro e,\n    apply trunc.map _ ih,\n    intro ih,\n    refine of_equiv e (h_option ih), },\nend\n\n\/-- An induction principle for finite types, analogous to `nat.rec`. It effectively says\nthat every `fintype` is either `empty` or `option \u03b1`, up to an `equiv`. -\/\nlemma induction_empty_option {P : Type u \u2192 Prop}\n  (of_equiv : \u2200 {\u03b1 \u03b2}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2)\n  (h_empty : P pempty)\n  (h_option : \u2200 {\u03b1} [fintype \u03b1], P \u03b1 \u2192 P (option \u03b1))\n  (\u03b1 : Type u) [fintype \u03b1] : P \u03b1 :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  obtain \u27e8p\u27e9 := trunc_rec_empty_option @of_equiv h_empty (\u03bb _ _ _, by exactI h_option) \u03b1,\n  exact p,\nend\n\nend fintype\n","avg_line_length":39.244639912,"max_line_length":100,"alphanum_fraction":0.6734233603}
-{"size":8281,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-\/\nimport set_theory.game.short\n\n\/-!\n# Games described via \"the state of the board\".\n\nWe provide a simple mechanism for constructing combinatorial (pre-)games, by describing\n\"the state of the board\", and providing an upper bound on the number of turns remaining.\n\n\n## Implementation notes\n\nWe're very careful to produce a computable definition, so small games can be evaluated\nusing `dec_trivial`. To achieve this, I've had to rely solely on induction on natural numbers:\nrelying on general well-foundedness seems to be poisonous to computation?\n\nSee `set_theory\/game\/domineering` for an example using this construction.\n-\/\n\nuniverse u\n\nnamespace pgame\n\n\/--\n`pgame_state S` describes how to interpret `s : S` as a state of a combinatorial game.\nUse `pgame.of s` or `game.of s` to construct the game.\n\n`pgame_state.L : S \u2192 finset S` and `pgame_state.R : S \u2192 finset S` describe the states reachable\nby a move by Left or Right. `pgame_state.turn_bound : S \u2192 \u2115` gives an upper bound on the number of\npossible turns remaining from this state.\n-\/\nclass state (S : Type u) :=\n(turn_bound : S \u2192 \u2115)\n(L : S \u2192 finset S)\n(R : S \u2192 finset S)\n(left_bound : \u2200 {s t : S} (m : t \u2208 L s), turn_bound t < turn_bound s)\n(right_bound : \u2200 {s t : S} (m : t \u2208 R s), turn_bound t < turn_bound s)\n\nopen state\n\nvariables {S : Type u} [state S]\n\nlemma turn_bound_ne_zero_of_left_move {s t : S} (m : t \u2208 L s) : turn_bound s \u2260 0 :=\nbegin\n  intro h,\n  have t := state.left_bound m,\n  rw h at t,\n  exact nat.not_succ_le_zero _ t,\nend\nlemma turn_bound_ne_zero_of_right_move {s t : S} (m : t \u2208 R s) : turn_bound s \u2260 0 :=\nbegin\n  intro h,\n  have t := state.right_bound m,\n  rw h at t,\n  exact nat.not_succ_le_zero _ t,\nend\n\nlemma turn_bound_of_left {s t : S} (m : t \u2208 L s) (n : \u2115) (h : turn_bound s \u2264 n + 1) :\n  turn_bound t \u2264 n :=\nnat.le_of_lt_succ (nat.lt_of_lt_of_le (left_bound m) h)\nlemma turn_bound_of_right {s t : S} (m : t \u2208 R s) (n : \u2115) (h : turn_bound s \u2264 n + 1) :\n  turn_bound t \u2264 n :=\nnat.le_of_lt_succ (nat.lt_of_lt_of_le (right_bound m) h)\n\n\/--\nConstruct a `pgame` from a state and a (not necessarily optimal) bound on the number of\nturns remaining.\n-\/\ndef of_aux : \u03a0 (n : \u2115) (s : S) (h : turn_bound s \u2264 n), pgame\n| 0 s h     := pgame.mk {t \/\/ t \u2208 L s} {t \/\/ t \u2208 R s}\n    (\u03bb t, begin exfalso, exact turn_bound_ne_zero_of_left_move t.2 (nonpos_iff_eq_zero.mp h) end)\n    (\u03bb t, begin exfalso, exact turn_bound_ne_zero_of_right_move t.2 (nonpos_iff_eq_zero.mp h) end)\n| (n+1) s h :=\n  pgame.mk {t \/\/ t \u2208 L s} {t \/\/ t \u2208 R s}\n    (\u03bb t, of_aux n t (turn_bound_of_left t.2 n h))\n    (\u03bb t, of_aux n t (turn_bound_of_right t.2 n h))\n\n\/-- Two different (valid) turn bounds give equivalent games. -\/\ndef of_aux_relabelling : \u03a0 (s : S) (n m : \u2115) (hn : turn_bound s \u2264 n) (hm : turn_bound s \u2264 m),\n  relabelling (of_aux n s hn) (of_aux m s hm)\n| s 0 0 hn hm :=\n  begin\n    dsimp [pgame.of_aux],\n    fsplit, refl, refl,\n    { intro i, dsimp at i, exfalso,\n      exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hn) },\n    { intro j, dsimp at j, exfalso,\n      exact turn_bound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hm) }\n  end\n| s 0 (m+1) hn hm :=\n  begin\n    dsimp [pgame.of_aux],\n    fsplit, refl, refl,\n    { intro i, dsimp at i, exfalso,\n      exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hn) },\n    { intro j, dsimp at j, exfalso,\n      exact turn_bound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hn) }\n  end\n| s (n+1) 0 hn hm :=\n  begin\n    dsimp [pgame.of_aux],\n    fsplit, refl, refl,\n    { intro i, dsimp at i, exfalso,\n      exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hm) },\n    { intro j, dsimp at j, exfalso,\n      exact turn_bound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hm) }\n  end\n| s (n+1) (m+1) hn hm :=\n  begin\n    dsimp [pgame.of_aux],\n    fsplit, refl, refl,\n    { intro i,\n      apply of_aux_relabelling, },\n    { intro j,\n      apply of_aux_relabelling, }\n  end\n\n\/-- Construct a combinatorial `pgame` from a state. -\/\ndef of (s : S) : pgame := of_aux (turn_bound s) s (refl _)\n\n\/--\nThe equivalence between `left_moves` for a `pgame` constructed using `of_aux _ s _`, and `L s`.\n-\/\ndef left_moves_of_aux (n : \u2115) {s : S} (h : turn_bound s \u2264 n) :\n  left_moves (of_aux n s h) \u2243 {t \/\/ t \u2208 L s} :=\nby induction n; refl\n\/--\nThe equivalence between `left_moves` for a `pgame` constructed using `of s`, and `L s`.\n-\/\ndef left_moves_of (s : S) : left_moves (of s) \u2243 {t \/\/ t \u2208 L s} :=\nleft_moves_of_aux _ _\n\/--\nThe equivalence between `right_moves` for a `pgame` constructed using `of_aux _ s _`, and `R s`.\n-\/\ndef right_moves_of_aux (n : \u2115) {s : S} (h : turn_bound s \u2264 n) :\n  right_moves (of_aux n s h) \u2243 {t \/\/ t \u2208 R s} :=\nby induction n; refl\n\/-- The equivalence between `right_moves` for a `pgame` constructed using `of s`, and `R s`. -\/\ndef right_moves_of (s : S) : right_moves (of s) \u2243 {t \/\/ t \u2208 R s} :=\nright_moves_of_aux _ _\n\n\/--\nThe relabelling showing `move_left` applied to a game constructed using `of_aux`\nhas itself been constructed using `of_aux`.\n-\/\ndef relabelling_move_left_aux (n : \u2115) {s : S} (h : turn_bound s \u2264 n)\n  (t : left_moves (of_aux n s h)) :\n  relabelling\n    (move_left (of_aux n s h) t)\n    (of_aux (n-1) (((left_moves_of_aux n h) t) : S)\n      ((turn_bound_of_left ((left_moves_of_aux n h) t).2 (n-1)\n        (nat.le_trans h le_tsub_add)))) :=\nbegin\n  induction n,\n  { have t' := (left_moves_of_aux 0 h) t,\n    exfalso, exact turn_bound_ne_zero_of_left_move t'.2 (nonpos_iff_eq_zero.mp h), },\n  { refl },\nend\n\/--\nThe relabelling showing `move_left` applied to a game constructed using `of`\nhas itself been constructed using `of`.\n-\/\ndef relabelling_move_left (s : S) (t : left_moves (of s)) :\n  relabelling\n    (move_left (of s) t)\n    (of (((left_moves_of s).to_fun t) : S)) :=\nbegin\n  transitivity,\n  apply relabelling_move_left_aux,\n  apply of_aux_relabelling,\nend\n\/--\nThe relabelling showing `move_right` applied to a game constructed using `of_aux`\nhas itself been constructed using `of_aux`.\n-\/\ndef relabelling_move_right_aux (n : \u2115) {s : S} (h : turn_bound s \u2264 n)\n  (t : right_moves (of_aux n s h)) :\n  relabelling\n    (move_right (of_aux n s h) t)\n    (of_aux (n-1) (((right_moves_of_aux n h) t) : S)\n      ((turn_bound_of_right ((right_moves_of_aux n h) t).2 (n-1)\n        (nat.le_trans h le_tsub_add)))) :=\nbegin\n  induction n,\n  { have t' := (right_moves_of_aux 0 h) t,\n    exfalso, exact turn_bound_ne_zero_of_right_move t'.2 (nonpos_iff_eq_zero.mp h), },\n  { refl },\nend\n\/--\nThe relabelling showing `move_right` applied to a game constructed using `of`\nhas itself been constructed using `of`.\n-\/\ndef relabelling_move_right (s : S) (t : right_moves (of s)) :\n  relabelling\n    (move_right (of s) t)\n    (of (((right_moves_of s).to_fun t) : S)) :=\nbegin\n  transitivity,\n  apply relabelling_move_right_aux,\n  apply of_aux_relabelling,\nend\n\ninstance fintype_left_moves_of_aux (n : \u2115) (s : S) (h : turn_bound s \u2264 n) :\n  fintype (left_moves (of_aux n s h)) :=\nbegin\n  apply fintype.of_equiv _ (left_moves_of_aux _ _).symm,\n  apply_instance,\nend\ninstance fintype_right_moves_of_aux (n : \u2115) (s : S) (h : turn_bound s \u2264 n) :\n  fintype (right_moves (of_aux n s h)) :=\nbegin\n  apply fintype.of_equiv _ (right_moves_of_aux _ _).symm,\n  apply_instance,\nend\n\ninstance short_of_aux : \u03a0 (n : \u2115) {s : S} (h : turn_bound s \u2264 n), short (of_aux n s h)\n| 0 s h :=\n  short.mk'\n  (\u03bb i, begin\n    have i := (left_moves_of_aux _ _).to_fun i,\n    exfalso,\n    exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp h),\n  end)\n  (\u03bb j, begin\n    have j := (right_moves_of_aux _ _).to_fun j,\n    exfalso,\n    exact turn_bound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp h),\n  end)\n| (n+1) s h :=\n  short.mk'\n  (\u03bb i, short_of_relabelling (relabelling_move_left_aux (n+1) h i).symm (short_of_aux n _))\n  (\u03bb j, short_of_relabelling (relabelling_move_right_aux (n+1) h j).symm (short_of_aux n _))\n\ninstance short_of (s : S) : short (of s) :=\nbegin\n  dsimp [pgame.of],\n  apply_instance\nend\n\nend pgame\n\nnamespace game\n\n\/-- Construct a combinatorial `game` from a state. -\/\ndef of {S : Type u} [pgame.state S] (s : S) : game := \u27e6pgame.of s\u27e7\n\nend game\n","avg_line_length":32.8611111111,"max_line_length":98,"alphanum_fraction":0.6717787707}
-{"size":15521,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Joseph Myers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Myers\n-\/\nimport linear_algebra.affine_space.independent\nimport linear_algebra.finite_dimensional\n\n\/-!\n# Finite-dimensional subspaces of affine spaces.\n\nThis file provides a few results relating to finite-dimensional\nsubspaces of affine spaces.\n\n## Main definitions\n\n* `collinear` defines collinear sets of points as those that span a\n  subspace of dimension at most 1.\n\n-\/\n\nnoncomputable theory\nopen_locale big_operators classical affine\n\nsection affine_space'\n\nvariables (k : Type*) {V : Type*} {P : Type*} [field k] [add_comm_group V] [module k V]\n          [affine_space V P]\nvariables {\u03b9 : Type*}\ninclude V\n\nopen affine_subspace finite_dimensional vector_space\n\n\/-- The `vector_span` of a finite set is finite-dimensional. -\/\nlemma finite_dimensional_vector_span_of_finite {s : set P} (h : set.finite s) :\n  finite_dimensional k (vector_span k s) :=\nspan_of_finite k $ h.vsub h\n\n\/-- The `vector_span` of a family indexed by a `fintype` is\nfinite-dimensional. -\/\ninstance finite_dimensional_vector_span_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P) :\n  finite_dimensional k (vector_span k (set.range p)) :=\nfinite_dimensional_vector_span_of_finite k (set.finite_range _)\n\n\/-- The `vector_span` of a subset of a family indexed by a `fintype`\nis finite-dimensional. -\/\ninstance finite_dimensional_vector_span_image_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P)\n  (s : set \u03b9) : finite_dimensional k (vector_span k (p '' s)) :=\nfinite_dimensional_vector_span_of_finite k ((set.finite.of_fintype _).image _)\n\n\/-- The direction of the affine span of a finite set is\nfinite-dimensional. -\/\nlemma finite_dimensional_direction_affine_span_of_finite {s : set P} (h : set.finite s) :\n  finite_dimensional k (affine_span k s).direction :=\n(direction_affine_span k s).symm \u25b8 finite_dimensional_vector_span_of_finite k h\n\n\/-- The direction of the affine span of a family indexed by a\n`fintype` is finite-dimensional. -\/\ninstance finite_dimensional_direction_affine_span_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P) :\n  finite_dimensional k (affine_span k (set.range p)).direction :=\nfinite_dimensional_direction_affine_span_of_finite k (set.finite_range _)\n\n\/-- The direction of the affine span of a subset of a family indexed\nby a `fintype` is finite-dimensional. -\/\ninstance finite_dimensional_direction_affine_span_image_of_fintype [fintype \u03b9] (p : \u03b9 \u2192 P)\n  (s : set \u03b9) : finite_dimensional k (affine_span k (p '' s)).direction :=\nfinite_dimensional_direction_affine_span_of_finite k ((set.finite.of_fintype _).image _)\n\nvariables {k}\n\n\/-- The `vector_span` of a finite subset of an affinely independent\nfamily has dimension one less than its cardinality. -\/\nlemma findim_vector_span_image_finset_of_affine_independent {p : \u03b9 \u2192 P}\n  (hi : affine_independent k p) {s : finset \u03b9} {n : \u2115} (hc : finset.card s = n + 1) :\n  findim k (vector_span k (p '' \u2191s)) = n :=\nbegin\n  have hi' := affine_independent_of_subset_affine_independent\n    (affine_independent_set_of_affine_independent hi) (set.image_subset_range p \u2191s),\n  have hc' : fintype.card (p '' \u2191s) = n + 1,\n  { rwa [set.card_image_of_injective \u2191s (injective_of_affine_independent hi), fintype.card_coe] },\n  have hn : (p '' \u2191s).nonempty,\n  { simp [hc, \u2190finset.card_pos] },\n  rcases hn with \u27e8p\u2081, hp\u2081\u27e9,\n  rw affine_independent_set_iff_linear_independent_vsub k hp\u2081 at hi',\n  have hfr : (p '' \u2191s \\ {p\u2081}).finite := ((set.finite_mem_finset _).image _).subset\n    (set.diff_subset _ _),\n  haveI := hfr.fintype,\n  have hf : set.finite ((\u03bb (p : P), p -\u1d65 p\u2081) '' (p '' \u2191s \\ {p\u2081})) := hfr.image _,\n  haveI := hf.fintype,\n  have hc : hf.to_finset.card = n,\n  { rw [hf.card_to_finset,\n        set.card_image_of_injective (p '' \u2191s \\ {p\u2081}) (vsub_left_injective _)],\n    have hd : insert p\u2081 (p '' \u2191s \\ {p\u2081}) = p '' \u2191s,\n    { rw [set.insert_diff_singleton, set.insert_eq_of_mem hp\u2081] },\n    have hc'' : fintype.card \u21a5(insert p\u2081 (p '' \u2191s \\ {p\u2081})) = n + 1,\n    { convert hc' },\n    rw set.card_insert (p '' \u2191s \\ {p\u2081}) (\u03bb h, ((set.mem_diff p\u2081).2 h).2 rfl) at hc'',\n    simpa using hc'' },\n  rw [vector_span_eq_span_vsub_set_right_ne k hp\u2081, findim_span_set_eq_card _ hi', \u2190hc],\n  congr\nend\n\n\/-- The `vector_span` of a finite affinely independent family has\ndimension one less than its cardinality. -\/\nlemma findim_vector_span_of_affine_independent [fintype \u03b9] {p : \u03b9 \u2192 P}\n  (hi : affine_independent k p) {n : \u2115} (hc : fintype.card \u03b9 = n + 1) :\n  findim k (vector_span k (set.range p)) = n :=\nbegin\n  rw \u2190finset.card_univ at hc,\n  rw [\u2190set.image_univ, \u2190finset.coe_univ],\n  exact findim_vector_span_image_finset_of_affine_independent hi hc\nend\n\n\/-- If the `vector_span` of a finite subset of an affinely independent\nfamily lies in a submodule with dimension one less than its\ncardinality, it equals that submodule. -\/\nlemma vector_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {s : finset \u03b9} {sm : submodule k V}\n  [finite_dimensional k sm] (hle : vector_span k (p '' \u2191s) \u2264 sm)\n  (hc : finset.card s = findim k sm + 1) : vector_span k (p '' \u2191s) = sm :=\neq_of_le_of_findim_eq hle $ findim_vector_span_image_finset_of_affine_independent hi hc\n\n\/-- If the `vector_span` of a finite affinely independent\nfamily lies in a submodule with dimension one less than its\ncardinality, it equals that submodule. -\/\nlemma vector_span_eq_of_le_of_affine_independent_of_card_eq_findim_add_one [fintype \u03b9]\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {sm : submodule k V} [finite_dimensional k sm]\n  (hle : vector_span k (set.range p) \u2264 sm) (hc : fintype.card \u03b9 = findim k sm + 1) :\n  vector_span k (set.range p) = sm :=\neq_of_le_of_findim_eq hle $ findim_vector_span_of_affine_independent hi hc\n\n\/-- If the `affine_span` of a finite subset of an affinely independent\nfamily lies in an affine subspace whose direction has dimension one\nless than its cardinality, it equals that subspace. -\/\nlemma affine_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {s : finset \u03b9} {sp : affine_subspace k P}\n  [finite_dimensional k sp.direction] (hle : affine_span k (p '' \u2191s) \u2264 sp)\n  (hc : finset.card s = findim k sp.direction + 1) : affine_span k (p '' \u2191s) = sp :=\nbegin\n  have hn : (p '' \u2191s).nonempty, { simp [hc, \u2190finset.card_pos] },\n  refine eq_of_direction_eq_of_nonempty_of_le _ ((affine_span_nonempty k _).2 hn) hle,\n  have hd := direction_le hle,\n  rw direction_affine_span at \u22a2 hd,\n  exact vector_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one hi hd hc\nend\n\n\/-- If the `affine_span` of a finite affinely independent family lies\nin an affine subspace whose direction has dimension one less than its\ncardinality, it equals that subspace. -\/\nlemma affine_span_eq_of_le_of_affine_independent_of_card_eq_findim_add_one [fintype \u03b9]\n  {p : \u03b9 \u2192 P} (hi : affine_independent k p) {sp : affine_subspace k P}\n  [finite_dimensional k sp.direction] (hle : affine_span k (set.range p) \u2264 sp)\n  (hc : fintype.card \u03b9 = findim k sp.direction + 1) : affine_span k (set.range p) = sp :=\nbegin\n  rw \u2190finset.card_univ at hc,\n  rw [\u2190set.image_univ, \u2190finset.coe_univ] at \u22a2 hle,\n  exact affine_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one hi hle hc\nend\n\n\/-- The `vector_span` of a finite affinely independent family whose\ncardinality is one more than that of the finite-dimensional space is\n`\u22a4`. -\/\nlemma vector_span_eq_top_of_affine_independent_of_card_eq_findim_add_one [finite_dimensional k V]\n  [fintype \u03b9] {p : \u03b9 \u2192 P} (hi : affine_independent k p) (hc : fintype.card \u03b9 = findim k V + 1) :\n  vector_span k (set.range p) = \u22a4 :=\neq_top_of_findim_eq $ findim_vector_span_of_affine_independent hi hc\n\n\/-- The `affine_span` of a finite affinely independent family whose\ncardinality is one more than that of the finite-dimensional space is\n`\u22a4`. -\/\nlemma affine_span_eq_top_of_affine_independent_of_card_eq_findim_add_one [finite_dimensional k V]\n  [fintype \u03b9] {p : \u03b9 \u2192 P} (hi : affine_independent k p) (hc : fintype.card \u03b9 = findim k V + 1) :\n  affine_span k (set.range p) = \u22a4 :=\nbegin\n  rw [\u2190findim_top, \u2190direction_top k V P] at hc,\n  exact affine_span_eq_of_le_of_affine_independent_of_card_eq_findim_add_one hi le_top hc\nend\n\nvariables (k)\n\n\/-- The `vector_span` of `n + 1` points in an indexed family has\ndimension at most `n`. -\/\nlemma findim_vector_span_image_finset_le (p : \u03b9 \u2192 P) (s : finset \u03b9) {n : \u2115}\n  (hc : finset.card s = n + 1) : findim k (vector_span k (p '' \u2191s)) \u2264 n :=\nbegin\n  have hn : (p '' \u2191s).nonempty,\n  { simp [hc, \u2190finset.card_pos] },\n  rcases hn with \u27e8p\u2081, hp\u2081\u27e9,\n  rw [vector_span_eq_span_vsub_set_right_ne k hp\u2081],\n  have hfp\u2081 : (p '' \u2191s \\ {p\u2081}).finite :=\n    ((finset.finite_to_set _).image _).subset (set.diff_subset _ _),\n  haveI := hfp\u2081.fintype,\n  have hf : ((\u03bb p, p -\u1d65 p\u2081) '' (p '' \u2191s \\ {p\u2081})).finite := hfp\u2081.image _,\n  haveI := hf.fintype,\n  convert le_trans (findim_span_le_card ((\u03bb p, p -\u1d65 p\u2081) '' (p '' \u2191s \\ {p\u2081}))) _,\n  have hm : p\u2081 \u2209 p '' \u2191s \\ {p\u2081}, by simp,\n  haveI := set.fintype_insert' (p '' \u2191s \\ {p\u2081}) hm,\n  rw [set.to_finset_card, set.card_image_of_injective (p '' \u2191s \\ {p\u2081}) (vsub_left_injective p\u2081),\n      \u2190add_le_add_iff_right 1, \u2190set.card_fintype_insert' _ hm],\n  have h : fintype.card (\u2191(s.image p) : set P) \u2264 n + 1,\n  { rw [fintype.card_coe, \u2190hc],\n    exact finset.card_image_le },\n  convert h,\n  simp [hp\u2081]\nend\n\n\/-- The `vector_span` of an indexed family of `n + 1` points has\ndimension at most `n`. -\/\nlemma findim_vector_span_range_le [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) : findim k (vector_span k (set.range p)) \u2264 n :=\nbegin\n  rw [\u2190set.image_univ, \u2190finset.coe_univ],\n  rw \u2190finset.card_univ at hc,\n  exact findim_vector_span_image_finset_le _ _ _ hc\nend\n\n\/-- `n + 1` points are affinely independent if and only if their\n`vector_span` has dimension `n`. -\/\nlemma affine_independent_iff_findim_vector_span_eq [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) :\n  affine_independent k p \u2194 findim k (vector_span k (set.range p)) = n :=\nbegin\n  have hn : nonempty \u03b9, by simp [\u2190fintype.card_pos_iff, hc],\n  cases hn with i\u2081,\n  rw [affine_independent_iff_linear_independent_vsub _ _ i\u2081,\n      linear_independent_iff_card_eq_findim_span, eq_comm,\n      vector_span_range_eq_span_range_vsub_right_ne k p i\u2081],\n  congr',\n  rw \u2190finset.card_univ at hc,\n  rw fintype.subtype_card,\n  simp [finset.filter_ne', finset.card_erase_of_mem, hc]\nend\n\n\/-- `n + 1` points are affinely independent if and only if their\n`vector_span` has dimension at least `n`. -\/\nlemma affine_independent_iff_le_findim_vector_span [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 1) :\n  affine_independent k p \u2194 n \u2264 findim k (vector_span k (set.range p)) :=\nbegin\n  rw affine_independent_iff_findim_vector_span_eq k p hc,\n  split,\n  { rintro rfl,\n    refl },\n  { exact \u03bb hle, le_antisymm (findim_vector_span_range_le k p hc) hle }\nend\n\n\/-- `n + 2` points are affinely independent if and only if their\n`vector_span` does not have dimension at most `n`. -\/\nlemma affine_independent_iff_not_findim_vector_span_le [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 2) :\n  affine_independent k p \u2194 \u00ac findim k (vector_span k (set.range p)) \u2264 n :=\nby rw [affine_independent_iff_le_findim_vector_span k p hc, \u2190nat.lt_iff_add_one_le, lt_iff_not_ge]\n\n\/-- `n + 2` points have a `vector_span` with dimension at most `n` if\nand only if they are not affinely independent. -\/\nlemma findim_vector_span_le_iff_not_affine_independent [fintype \u03b9] (p : \u03b9 \u2192 P) {n : \u2115}\n  (hc : fintype.card \u03b9 = n + 2) :\n  findim k (vector_span k (set.range p)) \u2264 n \u2194 \u00ac affine_independent k p :=\n(not_iff_comm.1 (affine_independent_iff_not_findim_vector_span_le k p hc).symm).symm\n\n\/-- A set of points is collinear if their `vector_span` has dimension\nat most `1`. -\/\ndef collinear (s : set P) : Prop := dim k (vector_span k s) \u2264 1\n\n\/-- The definition of `collinear`. -\/\nlemma collinear_iff_dim_le_one (s : set P) : collinear k s \u2194 dim k (vector_span k s) \u2264 1 :=\niff.rfl\n\n\/-- A set of points, whose `vector_span` is finite-dimensional, is\ncollinear if and only if their `vector_span` has dimension at most\n`1`. -\/\nlemma collinear_iff_findim_le_one (s : set P) [finite_dimensional k (vector_span k s)] :\n  collinear k s \u2194 findim k (vector_span k s) \u2264 1 :=\nbegin\n  have h := collinear_iff_dim_le_one k s,\n  rw \u2190findim_eq_dim at h,\n  exact_mod_cast h\nend\n\nvariables (P)\n\n\/-- The empty set is collinear. -\/\nlemma collinear_empty : collinear k (\u2205 : set P) :=\nbegin\n  rw [collinear_iff_dim_le_one, vector_span_empty],\n  simp\nend\n\nvariables {P}\n\n\/-- A single point is collinear. -\/\nlemma collinear_singleton (p : P) : collinear k ({p} : set P) :=\nbegin\n  rw [collinear_iff_dim_le_one, vector_span_singleton],\n  simp\nend\n\n\/-- Given a point `p\u2080` in a set of points, that set is collinear if and\nonly if the points can all be expressed as multiples of the same\nvector, added to `p\u2080`. -\/\nlemma collinear_iff_of_mem {s : set P} {p\u2080 : P} (h : p\u2080 \u2208 s) :\n  collinear k s \u2194 \u2203 v : V, \u2200 p \u2208 s, \u2203 r : k, p = r \u2022 v +\u1d65 p\u2080 :=\nbegin\n  simp_rw [collinear_iff_dim_le_one, dim_submodule_le_one_iff', submodule.le_span_singleton_iff],\n  split,\n  { rintro \u27e8v\u2080, hv\u27e9,\n    use v\u2080,\n    intros p hp,\n    obtain \u27e8r, hr\u27e9 := hv (p -\u1d65 p\u2080) (vsub_mem_vector_span k hp h),\n    use r,\n    rw eq_vadd_iff_vsub_eq,\n    exact hr.symm },\n  { rintro \u27e8v, hp\u2080v\u27e9,\n    use v,\n    intros w hw,\n    have hs : vector_span k s \u2264 k \u2219 v,\n    { rw [vector_span_eq_span_vsub_set_right k h, submodule.span_le, set.subset_def],\n      intros x hx,\n      rw [set_like.mem_coe, submodule.mem_span_singleton],\n      rw set.mem_image at hx,\n      rcases hx with \u27e8p, hp, rfl\u27e9,\n      rcases hp\u2080v p hp with \u27e8r, rfl\u27e9,\n      use r,\n      simp },\n    have hw' := set_like.le_def.1 hs hw,\n    rwa submodule.mem_span_singleton at hw' }\nend\n\n\/-- A set of points is collinear if and only if they can all be\nexpressed as multiples of the same vector, added to the same base\npoint. -\/\nlemma collinear_iff_exists_forall_eq_smul_vadd (s : set P) :\n  collinear k s \u2194 \u2203 (p\u2080 : P) (v : V), \u2200 p \u2208 s, \u2203 r : k, p = r \u2022 v +\u1d65 p\u2080 :=\nbegin\n  rcases set.eq_empty_or_nonempty s with rfl | \u27e8\u27e8p\u2081, hp\u2081\u27e9\u27e9,\n  { simp [collinear_empty] },\n  { rw collinear_iff_of_mem k hp\u2081,\n    split,\n    { exact \u03bb h, \u27e8p\u2081, h\u27e9 },\n    { rintros \u27e8p, v, hv\u27e9,\n      use v,\n      intros p\u2082 hp\u2082,\n      rcases hv p\u2082 hp\u2082 with \u27e8r, rfl\u27e9,\n      rcases hv p\u2081 hp\u2081 with \u27e8r\u2081, rfl\u27e9,\n      use r - r\u2081,\n      simp [vadd_assoc, \u2190add_smul] } }\nend\n\n\/-- Two points are collinear. -\/\nlemma collinear_insert_singleton (p\u2081 p\u2082 : P) : collinear k ({p\u2081, p\u2082} : set P) :=\nbegin\n  rw collinear_iff_exists_forall_eq_smul_vadd,\n  use [p\u2081, p\u2082 -\u1d65 p\u2081],\n  intros p hp,\n  rw [set.mem_insert_iff, set.mem_singleton_iff] at hp,\n  cases hp,\n  { use 0,\n    simp [hp] },\n  { use 1,\n    simp [hp] }\nend\n\n\/-- Three points are affinely independent if and only if they are not\ncollinear. -\/\nlemma affine_independent_iff_not_collinear (p : fin 3 \u2192 P) :\n  affine_independent k p \u2194 \u00ac collinear k (set.range p) :=\nby rw [collinear_iff_findim_le_one,\n       affine_independent_iff_not_findim_vector_span_le k p (fintype.card_fin 3)]\n\n\/-- Three points are collinear if and only if they are not affinely\nindependent. -\/\nlemma collinear_iff_not_affine_independent (p : fin 3 \u2192 P) :\n  collinear k (set.range p) \u2194 \u00ac affine_independent k p :=\nby rw [collinear_iff_findim_le_one,\n       findim_vector_span_le_iff_not_affine_independent k p (fintype.card_fin 3)]\n\nend affine_space'\n","avg_line_length":41.0608465608,"max_line_length":99,"alphanum_fraction":0.7080729334}
-{"size":17619,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nThis file is intended for Lean beginners. The goal is to demonstrate what it feels like to prove\nthings using Lean and mathlib. Complicated definitions and theory building are not covered.\nEverything is covered again more slowly and with exercises in the next files.\n-\/\n\n-- We want real numbers and their basic properties\nimport data.real.basic\n\n-- We want to be able to use Lean's built-in \"help\" functionality\nimport tactic.suggest\n\n-- We want to be able to define functions using the law of excluded middle\nnoncomputable theory\nopen_locale classical\n\n\n\/-\nOur first goal is to define the set of upper bounds of a set of real numbers.\nThis is already defined in mathlib (in a more general context), but we repeat\nit for the sake of exposition. Right-click \"upper_bounds\" below to get offered\nto jump to mathlib's version\n-\/\n#check upper_bounds\n\n\/-- The set of upper bounds of a set of real numbers \u211d -\/\ndef up_bounds (A : set \u211d) := { x : \u211d | \u2200 a \u2208 A, a \u2264 x}\n\n\/-- Predicate `is_max a A` means `a` is a maximum of `A` -\/\ndef is_max (a : \u211d) (A : set \u211d) := a \u2208 A \u2227 a \u2208 up_bounds A\n\n\/-\nIn the above definition, the symbol `\u2227` means \"and\". We also see the most\nvisible difference between set theoretic foundations and type theoretic ones\n(used by almost all proof assistants). In set theory, everything is a set, and the\nonly relation you get from foundations are `=` and `\u2208`. In type theory, there is\na meta-theoretic relation of \"typing\": `a : \u211d` reads \"`a` is a real number\" or,\nmore precisely, \"the type of `a` is `\u211d`\". Here \"meta-theoretic\" means this is not a\nstatement you can prove or disprove inside the theory, it's a fact that is true or\nnot. Here we impose this fact, in other circumstances, it would be checked by the\nLean kernel.\nBy contrast, `a \u2208 A` is a statement inside the theory. Here it's part of the\ndefinition, in other circumstances it could be something proven inside Lean.\n-\/\n\n\/- For illustrative purposes, we now define an infix version of the above predicate.\nIt will allow us to write `a is_a_max_of A`, which is closer to a sentence.\n-\/\ninfix ` is_a_max_of `:55 := is_max\n\n\/-\nLet's prove something now! A set of real numbers has at most one maximum. Here\neverything left of the final `:` is introducing the objects and assumption. The equality\n`x = y` right of the colon is the conclusion.\n-\/\nlemma unique_max (A : set \u211d) (x y : \u211d) (hx : x is_a_max_of A) (hy : y is_a_max_of A) : x = y :=\nbegin\n  -- We first break our assumptions in their two constituent pieces.\n  -- We are free to choose the name following `with`\n  cases hx with x_in x_up,\n  cases hy with y_in y_up,\n  -- Assumption `x_up` means x isn't less than elements of A, let's apply this to y\n  specialize x_up y,\n  -- Assumption `x_up` now needs the information that `y` is indeed in `A`.\n  specialize x_up y_in,\n  -- Let's do this quicker with roles swapped\n  specialize y_up x x_in,\n  -- We explained to Lean the idea of this proof.\n  -- Now we know `x \u2264 y` and `y \u2264 x`, and Lean shouldn't need more help.\n  -- `linarith` proves equalities and inequalities that follow linearly from\n  -- the assumption we have.\n  linarith,\nend\n\n\/-\nThe above proof is too long, even if you remove comments. We don't really need the\nunpacking steps at the beginning; we can access both parts of the assumption\n`hx : x is_a_max_of A` using shortcuts `hx.1` and `hx.2`. We can also improve\nreadability without assistance from the tactic state display, clearly announcing\nintermediate goals using `have`. This way we get to the following version of the\nsame proof.\n-\/\n\nexample (A : set \u211d) (x y : \u211d) (hx : x is_a_max_of A) (hy : y is_a_max_of A) : x = y :=\nbegin\n  have : x \u2264 y, from hy.2 x hx.1,\n  have : y \u2264 x, from hx.2 y hy.1,\n  linarith,\nend\n\n\/-\nNotice how mathematics based on type theory treats the assumption\n`\u2200 a \u2208 A, a \u2264 y` as a function turning an element `a` of `A` into the statement\n`a \u2264 y`. More precisely, this assumption is the abbreviation of\n`\u2200 a : \u211d, a \u2208 A \u2192 a \u2264 y`. The expression `hy.2 x` appearing in the above proof\nis then the statement `x \u2208 A \u2192 x \u2264 y`, which itself is a function turning a\nstatement `x \u2208 A` into `x \u2264 y` so that the full expression `hy.2 x hx.1` is\nindeed a proof of `x \u2264 y`.\n\nOne could argue a three-line-long proof of this lemma is still two lines too long.\nThis is debatable, but mathlib's style is to write very short proofs for trivial\nlemmas. Those proofs are not easy to read but they are meant to indicate that the\nproof is probably not worth reading.\n\nIn order to reach this stage, we need to know what `linarith` did for us. It invoked\nthe lemma `le_antisymm` which says: `x \u2264 y \u2192 y \u2264 x \u2192 x = y`. This arrow, which\nis used both for function and implication, is right associative. So the statement is\n`x \u2264 y \u2192 (y \u2264 x \u2192 x = y)` which reads: I will send a proof `p` of `x \u2264 y` to a function\nsending a proof `q'` of `y \u2264 x` to a proof of `x = y`. Hence `le_antisymm p q'` is a\nproof of `x = y`.\n\nUsing this we can get our one-line proof:\n-\/\n\nexample (A : set \u211d) (x y : \u211d) (hx : x is_a_max_of A) (hy : y is_a_max_of A) : x = y :=\nle_antisymm (hy.2 x hx.1) (hx.2 y hy.1)\n\n\/-\nSuch a proof is called a proof term (or a \"term mode\" proof). Notice it has no `begin`\nand `end`. It is directly the kind of low level proof that the Lean kernel is\nconsuming. Commands like `cases`, `specialize` or `linarith` are called tactics, they\nhelp users constructing proof terms that could be very tedious to write directly.\nThe most efficient proof style combines tactics with proof terms like our previous\n`have : x \u2264 y, from hy.2 x hx.1` where `hy.2 x hx.1` is a proof term embeded inside\na tactic mode proof.\n\nIn the remaining of this file, we'll be characterizing infima of sets of real numbers\nin term of sequences.\n-\/\n\n\/-- The set of lower bounds of a set of real numbers \u211d -\/\ndef low_bounds (A : set \u211d) := { x : \u211d | \u2200 a \u2208 A, x \u2264 a}\n\n\/-\nWe now define `a` is an infimum of `A`. Again there is already a more general version\nin mathlib.\n-\/\ndef is_inf (x : \u211d) (A : set \u211d) := x is_a_max_of (low_bounds A)\ninfix ` is_an_inf_of `:55 := is_inf\n\n\/-\nWe need to prove that any number which is greater than the infimum of A is greater\nthan some element of A.\n-\/\n\nlemma inf_lt {A : set \u211d} {x : \u211d} (hx : x is_an_inf_of A) :\n  \u2200 y, x < y \u2192 \u2203 a \u2208 A, a < y :=\nbegin\n  -- Let `y` be any real number.\n  intro y,\n  -- Let's prove the contrapositive\n  contrapose,\n  -- The symbol `\u00ac` means negation. Let's ask Lean to rewrite the goal without negation,\n  -- pushing negation through quantifiers and inequalities\n  push_neg,\n  -- Let's assume the premise, calling the assumption `h`\n  intro h,\n  -- `h` is exactly saying `y` is a lower bound of `A` so the second part of\n  -- the infimum assumption `hx` applied to `y` and `h` is exactly what we want.\n  exact hx.2 y h\nend\n\n\/-\nIn the above proof, the sequence `contrapose, push_neg` is so common that it can be\nabbreviated to `contrapose!`. With these commands, we enter the gray zone between\nproof checking and proof finding. Practical computer proof checking crucially needs\nthe computer to handle tedious proof steps. In the next proof, we'll start using\n`linarith` a bit more seriously, going one step further into automation.\n\nOur next real goal is to prove inequalities for limits of sequences. We extract the\nfollowing lemma: if `y \u2264 x + \u03b5` for all positive `\u03b5` then `y \u2264 x`.\n-\/\n\n\nlemma le_of_le_add_eps {x y : \u211d} : (\u2200 \u03b5 > 0, y \u2264 x + \u03b5) \u2192  y \u2264 x :=\nbegin\n  -- Let's prove the contrapositive, asking Lean to push negations right away.\n  contrapose!,\n  -- Assume `h : x < y`.\n  intro h,\n  -- We need to find `\u03b5` such that `\u03b5` is positive and `x + \u03b5 < y`.\n  -- Let's use `(y-x)\/2`\n  use ((y-x)\/2),\n  -- we now have two properties to prove. Let's do both in turn, using `linarith`\n  split,\n  linarith,\n  linarith,\nend\n\n\/-\nNote how `linarith` was used for both sub-goals at the end of the above proof.\nWe could have shortened that using the semi-colon combinator instead of comma,\nwriting `split ; linarith`.\n\nNext we will study a compressed version of that proof:\n-\/\n\nexample {x y : \u211d} : (\u2200 \u03b5 > 0, y \u2264 x + \u03b5) \u2192  y \u2264 x :=\nbegin\n  contrapose!,\n  exact assume h, \u27e8(y-x)\/2, by linarith, by linarith\u27e9,\nend\n\n\/-\nThe angle brackets `\u27e8` and `\u27e9` introduce compound data or proofs. A proof\nof a `\u2203 z, P z` statemement is composed of a witness `z\u2080` and a proof `h` of\n`P z\u2080`. The compound is denoted by `\u27e8z\u2080, h\u27e9`. In the example above, the predicate is\nitself compound, it is a conjunction `P z \u2227 Q z`. So the proof term should read\n`\u27e8z\u2080, \u27e8h\u2081, h\u2082\u27e9\u27e9` where `h\u2081` (resp. `h\u2082`) is a proof of `P z\u2080` (resp. `Q z\u2080`).\nBut these so-called \"anonymous constructor\" brackets are right-associative, so we can\nget rid of the nested brackets.\n\nThe keyword `by` introduces tactic mode inside term mode, it is a shorter version\nof the `begin`\/`end` pair, which is more convenient for single tactic blocks.\nIn this example, `begin` enters tactic mode, `exact` leaves it, `by` re-enters it.\n\nGoing all the way to a proof term would make the proof much longer, because we\ncrucially use automation with `contrapose!` and `linarith`. We can still get a one-line\nproof using curly braces to gather several tactic invocations, and the `by` abbreviation\ninstead of `begin`\/`end`:\n-\/\n\nexample {x y : \u211d} : (\u2200 \u03b5 > 0, y \u2264 x + \u03b5) \u2192  y \u2264 x :=\nby { contrapose!, exact assume h, \u27e8(y-x)\/2, by linarith, by linarith\u27e9 }\n\n\/-\nOne could argue that the above proof is a bit too terse, and we are relying too much\non linarith. Let's have more `linarith` calls for smaller steps. For the sake\nof (tiny) variation, we will also assume the premise and argue by contradiction\ninstead of contraposing.\n-\/\n\nexample {x y : \u211d} : (\u2200 \u03b5 > 0, y \u2264 x + \u03b5) \u2192  y \u2264 x :=\nbegin\n  intro h,\n  -- Assume the conclusion is false, and call this assumption H.\n  by_contradiction H,\n  push_neg at H,\n  -- Now let's compute.\n  have key := calc\n  -- Each line must end with a colon followed by a proof term\n  -- We want to specialize our assumption `h` to `\u03b5 = (y-x)\/2` but this is long to\n  -- type, so let's put a hole `_` that Lean will fill in by comparing the\n  -- statement we want to prove and our proof term with a hole. As usual,\n  -- positivity of `(y-x)\/2` is proved by `linarith`\n    y   \u2264 x + (y-x)\/2 : h _ (by linarith)\n    ... = x\/2 + y\/2   : by ring\n    ... < y           : by linarith,\n  -- our key now says `y < y` (notice how the sequence `\u2264`, `=`, `<` was correctly\n  -- merged into a `<`). Let `linarith` find the desired contradiction now.\n  linarith,\n  -- alternatively, we could have provided the proof term\n  -- `exact lt_irrefl y key`\nend\n\n\/-\nNow we are ready for some analysis. Let's set up notation for absolute value\n-\/\n\nlocal notation `|`x`|` := abs x\n\n\/-\nAnd let's define convergence of sequences of real numbers (of course there is\na much more general definition in mathlib).\n-\/\n\n\/-- The sequence `u` tends to `l` -\/\ndef limit (u : \u2115 \u2192 \u211d) (l : \u211d) := \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - l| \u2264 \u03b5\n\n\/-\nIn the above definition, `u n` denotes the n-th term of the sequence. We can\nadd parentheses to get `u(n)` but we try to avoid parentheses because they pile up\nvery quickly\n-\/\n\n-- If y \u2264 u n for all n and u n goes to x then y \u2264 x\nlemma le_lim {x y : \u211d} {u : \u2115 \u2192 \u211d} (hu : limit u x) (ineq : \u2200 n, y \u2264 u n) : y \u2264 x :=\nbegin\n  -- Let's apply our previous lemma\n  apply le_of_le_add_eps,\n  -- We need to prove y \u2264 x + \u03b5 for all positive \u03b5.\n  -- Let \u03b5 be any positive real\n  intros \u03b5 \u03b5_pos,\n  -- we now specialize our limit assumption to this `\u03b5`, and immediately\n  -- fix a `N` as promised by the definition.\n  cases hu \u03b5 \u03b5_pos with N HN,\n  -- Now we only need to compute until reaching the conclusion\n  calc\n  y \u2264 u N             : ineq N\n  ... = x + (u N - x) : by linarith\n    -- We'll need `add_le_add` which says `a \u2264 b` and `c \u2264 d` implies `a + c \u2264 b + d`\n    -- We need a lemma saying `z \u2264 |z|`. Because we don't know the name of this lemma,\n    -- let's use `library_search`. Because searching through the library is slow,\n    -- Lean will write what it found in the Lean message window when cursor is on\n    -- that line, so that we can replace it by the lemma. We see `le_abs_self`, which\n    -- says `a \u2264 |a|`, exactly what we're looking for.\n  ... \u2264 x + |u N - x| : add_le_add (by linarith) (by library_search)\n  ... \u2264 x + \u03b5         : add_le_add (by linarith) (HN N (by linarith)),\nend\n\n\/-\nThe next lemma has been extracted from the main proof in order to discuss numbers.\nIn ordinary maths, we know that \u2115 is *not* contained in `\u211d`, whatever the\nconstruction of real numbers that we use. For instance a natural number is not\nan equivalence class of Cauchy sequences. But it's very easy to\npretend otherwise. Formal maths requires slightly more care. In the statement below,\nthe \"type ascription\" `(n + 1 : \u211d)` forces Lean to convert the natural number\n`n+1` into a real number.  The \"inclusion\" map will be displayed in tactic state\nas `\u2191`. There are various lemmas asserting this map is compatible with addition and\nmonotone, but we don't want to bother writing their names. The `norm_cast`\ntactic is designed to wisely apply those lemmas for us.\n-\/\n\nlemma inv_succ_pos : \u2200 n : \u2115, 1\/(n+1 : \u211d) > 0 :=\nbegin\n  -- Let `n` be any integer\n  intro n,\n  -- Since we don't know the name of the relevant lemma, asserting that the inverse of\n  -- a positive number is positive, let's state that is suffices\n  -- to prove that `n+1`, seen as a real number, is positive, and ask `library_search`\n  suffices : (n + 1 : \u211d) > 0,\n  { library_search },\n  -- Now we want to reduce to a statement about natural numbers, not real numbers\n  -- coming from natural numbers.\n  norm_cast,\n  -- and then get the usual help from `linarith`\n  linarith,\nend\n\n\/-\nThat was a pretty long proof for an obvious fact. And stating it as a lemma feels\nstupid, so let's find a way to write it on one line in case we want to include it\nin some other proof without stating a lemma. First the `library_search` call\nabove displays the name of the relevant lemma: `one_div_pos`. We can also\nreplace the `linarith` call on the last line by `library_search` to learn the name\nof the lemma `nat.succ_pos` asserting that the successor of a natural number is\npositive. There is also a variant on `norm_cast` that combines it with `exact`.\nThe term mode analogue of `intro` is `\u03bb`. We get down to:\n-\/\n\nexample : \u2200 n : \u2115, 1\/(n+1 : \u211d) > 0 :=\n\u03bb n, one_div_pos.mpr (by exact_mod_cast nat.succ_pos n)\n\n\/-\nThe next proof uses mostly known things, so we will commment only new aspects.\n-\/\n\nlemma limit_inv_succ : \u2200 \u03b5 > 0, \u2203 N : \u2115, \u2200 n \u2265 N, 1\/(n + 1 : \u211d) \u2264 \u03b5 :=\nbegin\n  intros \u03b5 \u03b5_pos,\n  suffices : \u2203 N : \u2115, 1\/\u03b5 \u2264 N,\n  { -- Because we didn't provide a name for the above statement, Lean called it `this`.\n    -- Let's fix an `N` that works.\n    cases this with N HN,\n    use N,\n    intros n Hn,\n    -- Now we want to rewrite the goal using lemmas\n    -- `div_le_iff' : 0 < b \u2192  (a \/ b \u2264 c \u2194 a \u2264 b * c)`\n    -- `div_le_iff : 0 < b \u2192  (a \/ b \u2264 c \u2194 a \u2264 c * b)`\n    -- the second one will be rewritten from right to left, as indicated by `\u2190`.\n    -- Lean will create a side goal for the required positivity assumption that\n    -- we don't provide for `div_le_iff'`.\n    rw [div_le_iff', \u2190 div_le_iff \u03b5_pos],\n    -- We want to replace assumption `Hn` by its real counter-part so that\n    -- linarith can find what it needs.\n    replace Hn : (N : \u211d) \u2264 n, exact_mod_cast Hn,\n    linarith,\n    -- we are still left with the positivity assumption, but already discussed\n    -- how to prove it in the preceding lemma\n    exact_mod_cast nat.succ_pos n },\n  -- Now we need to prove that sufficient statement.\n  -- We want to use that `\u211d` is archimedean. So we start typing\n  -- `exact archimedean_` and hit Ctrl-space to see what completion Lean proposes\n  -- the lemma `archimedean_iff_nat_le` sounds promising. We select the left to\n  -- right implication using `.1`. This a generic lemma for fields equiped with\n  -- a linear (ie total) order. We need to provide a proof that `\u211d` is indeed\n  -- archimedean. This is done using the `apply_instance` tactic that will be\n  -- covered elsewhere.\n  exact archimedean_iff_nat_le.1 (by apply_instance) (1\/\u03b5),\nend\n\n\/-\nWe can now put all pieces together, with almost no new things to explain.\n-\/\n\nlemma inf_seq (A : set \u211d) (x : \u211d) :\n  (x is_an_inf_of A) \u2194 (x \u2208 low_bounds A \u2227 \u2203 u : \u2115 \u2192 \u211d, limit u x \u2227 \u2200 n, u n \u2208 A ) :=\nbegin\n  split,\n  { intro h,\n    split,\n    { exact h.1 },\n    -- On the next line, we don't need to tell Lean to treat `n+1` as a real number because\n    -- we add `x` to it, so Lean knows there is only one way to make sense of this expression.\n    have key : \u2200 n : \u2115, \u2203 a \u2208 A, a < x + 1\/(n+1),\n    { intro n,\n      -- we can use the lemma we proved above\n      apply inf_lt h,\n      -- and another one we proved!\n      have : 0 < 1\/(n+1 : \u211d), from inv_succ_pos n,\n      linarith },\n    -- Now we need to use axiom of (countable) choice\n    choose u hu using key,\n    use u,\n    split,\n    { intros \u03b5 \u03b5_pos,\n      -- again we use a lemma we proved, specializing it to our fixed `\u03b5`, and fixing a `N`\n      cases limit_inv_succ \u03b5 \u03b5_pos with N H,\n      use N,\n      intros n hn,\n      have : x \u2264 u n, from h.1 _ (hu n).1,\n      have := calc\n        u n < x + 1\/(n + 1) : (hu n).2\n        ... \u2264 x + \u03b5         : add_le_add (le_refl x) (H n hn),\n      rw abs_of_nonneg ; linarith },\n    { intro n,\n      exact (hu n).1 } },\n  { intro h,\n    -- Assumption `h` is made of nested compound statements. We can use the\n    -- recursive version of `cases` to unpack it in one go.\n    rcases h with \u27e8x_min, u, lim, huA\u27e9,\n    split,\n    exact x_min,\n    intros y y_mino,\n    apply le_lim lim,\n    intro n,\n    exact y_mino (u n) (huA n) },\nend\n\n\n","avg_line_length":40.879350348,"max_line_length":96,"alphanum_fraction":0.6785856178}
-{"size":643,"ext":"lean","lang":"Lean","max_stars_count":1538.0,"content":"-- From @joehendrix\n-- The imul doesn't type check as Lean won't try to coerce from a reg (bv 64) to a expr (bv ?u)\n\ninductive MCType\n| bv : Nat \u2192 MCType\n\nopen MCType\n\ninductive Reg : MCType \u2192 Type\n| rax : Reg (bv 64)\n\ninductive Expr : MCType \u2192 Type\n| r : \u2200{tp:MCType}, Reg tp \u2192 Expr tp\n| sextC {s:Nat} (x : Expr (bv s)) (t:Nat) : Expr (bv t)\n\ninstance reg_is_expr {tp:MCType} : HasCoe (Reg tp) (Expr tp) := \u27e8Expr.r\u27e9\n\ndef bvmul {w:Nat} (x y : Expr (bv w)) : Expr (bv w) := x\n\ndef sext {s:Nat} (x : Expr (bv s)) (t:Nat) : Expr (bv t) := Expr.sextC x t\n\ndef imul {u:Nat} (e:Expr (bv 64)) : Expr (bv 128) :=\n  bvmul (sext Reg.rax 128) (sext e _)\n","avg_line_length":26.7916666667,"max_line_length":95,"alphanum_fraction":0.6065318818}
-{"size":6909,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2018 Kevin Buzzard and Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Patrick Massot.\n\nThis file is to a certain extent based on `quotient_module.lean` by Johannes H\u00f6lzl.\n-\/\nimport group_theory.coset\n\nuniverses u v\n\nnamespace quotient_group\n\nvariables {G : Type u} [group G] (N : set G) [normal_subgroup N] {H : Type v} [group H]\n\n@[to_additive quotient_add_group.add_group]\ninstance : group (quotient N) :=\n{ one := (1 : G),\n  mul := quotient.map\u2082' (*)\n  (\u03bb a\u2081 b\u2081 hab\u2081 a\u2082 b\u2082 hab\u2082,\n    ((is_subgroup.mul_mem_cancel_left N (is_subgroup.inv_mem hab\u2082)).1\n        (by rw [mul_inv_rev, mul_inv_rev, \u2190 mul_assoc (a\u2082\u207b\u00b9 * a\u2081\u207b\u00b9),\n          mul_assoc _ b\u2082, \u2190 mul_assoc b\u2082, mul_inv_self, one_mul, mul_assoc (a\u2082\u207b\u00b9)];\n          exact normal_subgroup.normal _ hab\u2081 _))),\n  mul_assoc := \u03bb a b c, quotient.induction_on\u2083' a b c\n    (\u03bb a b c, congr_arg mk (mul_assoc a b c)),\n  one_mul := \u03bb a, quotient.induction_on' a\n    (\u03bb a, congr_arg mk (one_mul a)),\n  mul_one := \u03bb a, quotient.induction_on' a\n    (\u03bb a, congr_arg mk (mul_one a)),\n  inv := \u03bb a, quotient.lift_on' a (\u03bb a, ((a\u207b\u00b9 : G) : quotient N))\n    (\u03bb a b hab, quotient.sound' begin\n      show a\u207b\u00b9\u207b\u00b9 * b\u207b\u00b9 \u2208 N,\n      rw \u2190 mul_inv_rev,\n      exact is_subgroup.inv_mem (is_subgroup.mem_norm_comm hab)\n    end),\n  mul_left_inv := \u03bb a, quotient.induction_on' a\n    (\u03bb a, congr_arg mk (mul_left_inv a)) }\n\n@[to_additive quotient_add_group.is_add_group_hom]\ninstance : is_group_hom (mk : G \u2192 quotient N) := { map_mul := \u03bb _ _, rfl }\n\n@[simp, to_additive quotient_add_group.ker_mk]\nlemma ker_mk :\n  is_group_hom.ker (quotient_group.mk : G \u2192 quotient_group.quotient N) = N :=\nbegin\n  ext g,\n  rw [is_group_hom.mem_ker, eq_comm],\n  show (((1 : G) : quotient_group.quotient N)) = g \u2194 _,\n  rw [quotient_group.eq, one_inv, one_mul],\nend\n\n@[to_additive quotient_add_group.add_comm_group]\ninstance {G : Type*} [comm_group G] (s : set G) [is_subgroup s] : comm_group (quotient s) :=\n{ mul_comm := \u03bb a b, quotient.induction_on\u2082' a b\n    (\u03bb a b, congr_arg mk (mul_comm a b)),\n  ..@quotient_group.group _ _ s (normal_subgroup_of_comm_group s) }\n\n@[simp, to_additive quotient_add_group.coe_zero]\nlemma coe_one : ((1 : G) : quotient N) = 1 := rfl\n\n@[simp, to_additive quotient_add_group.coe_add]\nlemma coe_mul (a b : G) : ((a * b : G) : quotient N) = a * b := rfl\n\n@[simp, to_additive quotient_add_group.coe_neg]\nlemma coe_inv (a : G) : ((a\u207b\u00b9 : G) : quotient N) = a\u207b\u00b9 := rfl\n\n@[simp] lemma coe_pow (a : G) (n : \u2115) : ((a ^ n : G) : quotient N) = a ^ n :=\n(monoid_hom.of mk).map_pow a n\n\n@[simp] lemma coe_gpow (a : G) (n : \u2124) : ((a ^ n : G) : quotient N) = a ^ n :=\n(monoid_hom.of mk).map_gpow a n\n\nlocal notation ` Q ` := quotient N\n\n@[to_additive quotient_add_group.lift]\ndef lift (\u03c6 : G \u2192 H) [is_group_hom \u03c6] (HN : \u2200x\u2208N, \u03c6 x = 1) (q : Q) : H :=\nq.lift_on' \u03c6 $ assume a b (hab : a\u207b\u00b9 * b \u2208 N),\n(calc \u03c6 a = \u03c6 a * 1           : (mul_one _).symm\n...       = \u03c6 a * \u03c6 (a\u207b\u00b9 * b) : HN (a\u207b\u00b9 * b) hab \u25b8 rfl\n...       = \u03c6 (a * (a\u207b\u00b9 * b)) : (is_mul_hom.map_mul \u03c6 a (a\u207b\u00b9 * b)).symm\n...       = \u03c6 b               : by rw mul_inv_cancel_left)\n\n@[simp, to_additive quotient_add_group.lift_mk]\nlemma lift_mk {\u03c6 : G \u2192 H} [is_group_hom \u03c6] (HN : \u2200x\u2208N, \u03c6 x = 1) (g : G) :\n  lift N \u03c6 HN (g : Q) = \u03c6 g := rfl\n\n@[simp, to_additive quotient_add_group.lift_mk']\nlemma lift_mk' {\u03c6 : G \u2192 H} [is_group_hom \u03c6] (HN : \u2200x\u2208N, \u03c6 x = 1) (g : G) :\n  lift N \u03c6 HN (mk g : Q) = \u03c6 g := rfl\n\n@[to_additive quotient_add_group.map]\ndef map (M : set H) [normal_subgroup M] (f : G \u2192 H) [is_group_hom f] (h : N \u2286 f \u207b\u00b9' M) :\n  quotient N \u2192 quotient M :=\nbegin\n  haveI : is_group_hom ((mk : H \u2192 quotient M) \u2218 f) := is_group_hom.comp _ _,\n  refine quotient_group.lift N (mk \u2218 f) _,\n  assume x hx,\n  refine quotient_group.eq.2 _,\n  rw [mul_one, is_subgroup.inv_mem_iff],\n  exact h hx,\nend\n\nvariables (\u03c6 : G \u2192 H) [is_group_hom \u03c6] (HN : \u2200x\u2208N, \u03c6 x = 1)\n\n@[to_additive quotient_add_group.is_add_group_hom_quotient_lift]\ninstance is_group_hom_quotient_lift  :\n  is_group_hom (lift N \u03c6 HN) :=\n{ map_mul := \u03bb q r, quotient.induction_on\u2082' q r $ is_mul_hom.map_mul \u03c6 }\n\n@[to_additive quotient_add_group.map_is_add_group_hom]\ninstance map_is_group_hom (M : set H) [normal_subgroup M]\n(f : G \u2192 H) [is_group_hom f] (h : N \u2286 f \u207b\u00b9' M) : is_group_hom (map N M f h) :=\n@quotient_group.is_group_hom_quotient_lift _ _ _ _ _ _ _ (is_group_hom.comp _ _) _\n\nopen function is_group_hom\n\n\/-- The induced map from the quotient by the kernel to the codomain. -\/\n@[to_additive quotient_add_group.ker_lift]\ndef ker_lift : quotient (ker \u03c6) \u2192 H :=\nlift _ \u03c6 $ \u03bb g, (mem_ker \u03c6).mp\n\n@[simp, to_additive quotient_add_group.ker_lift_mk]\nlemma ker_lift_mk (g : G) : (ker_lift \u03c6) g = \u03c6 g :=\nlift_mk _ _ _\n\n@[simp, to_additive quotient_add_group.ker_lift_mk']\nlemma ker_lift_mk' (g : G) : (ker_lift \u03c6) (mk g) = \u03c6 g :=\nlift_mk' _ _ _\n\n@[to_additive quotient_add_group.ker_lift_is_add_group_hom]\ninstance ker_lift_is_group_hom : is_group_hom (ker_lift \u03c6) :=\nquotient_group.is_group_hom_quotient_lift _ _ _\n\n@[to_additive quotient_add_group.injective_ker_lift]\nlemma injective_ker_lift : injective (ker_lift \u03c6) :=\nassume a b, quotient.induction_on\u2082' a b $ assume a b (h : \u03c6 a = \u03c6 b), quotient.sound' $\nshow a\u207b\u00b9 * b \u2208 ker \u03c6, by rw [mem_ker \u03c6,\n  is_mul_hom.map_mul \u03c6, \u2190 h, is_group_hom.map_inv \u03c6, inv_mul_self]\n\n--@[to_additive quotient_add_group.quotient_ker_equiv_range]\nnoncomputable def quotient_ker_equiv_range : (quotient (ker \u03c6)) \u2243 set.range \u03c6 :=\n@equiv.of_bijective _ (set.range \u03c6) (\u03bb x, \u27e8lift (ker \u03c6) \u03c6\n  (by simp [mem_ker]) x, by exact quotient.induction_on' x (\u03bb x, \u27e8x, rfl\u27e9)\u27e9)\n\u27e8\u03bb a b h, injective_ker_lift _ (subtype.mk.inj h),\n  \u03bb \u27e8x, y, hy\u27e9, \u27e8mk y, subtype.eq hy\u27e9\u27e9\n\nnoncomputable def quotient_ker_equiv_of_surjective (h\u03c6 : function.surjective \u03c6) :\n  (quotient (ker \u03c6)) \u2243 H :=\ncalc (quotient_group.quotient (is_group_hom.ker \u03c6)) \u2243 set.range \u03c6 : quotient_ker_equiv_range _\n... \u2243 H : \u27e8\u03bb a, a.1, \u03bb b, \u27e8b, h\u03c6 b\u27e9, \u03bb \u27e8_, _\u27e9, rfl, \u03bb _, rfl\u27e9\n\nend quotient_group\n\nnamespace quotient_add_group\nopen is_add_group_hom\n\nvariables {G : Type u} [_root_.add_group G] (N : set G) [normal_add_subgroup N] {H : Type v} [_root_.add_group H]\nvariables (\u03c6 : G \u2192 H) [_root_.is_add_group_hom \u03c6]\n\nnoncomputable def quotient_ker_equiv_range : (quotient (ker \u03c6)) \u2243 set.range \u03c6 :=\n@quotient_group.quotient_ker_equiv_range (multiplicative G) _ (multiplicative H) _ \u03c6\n  (multiplicative.is_group_hom _)\n\nnoncomputable def quotient_ker_equiv_of_surjective (h\u03c6 : function.surjective \u03c6) :\n  (quotient (ker \u03c6)) \u2243 H :=\n@quotient_group.quotient_ker_equiv_of_surjective (multiplicative G) _ (multiplicative H) _ \u03c6\n  (multiplicative.is_group_hom _) h\u03c6\n\nattribute [to_additive quotient_add_group.quotient_ker_equiv_range] quotient_group.quotient_ker_equiv_range\nattribute [to_additive quotient_add_group.quotient_ker_equiv_of_surjective] quotient_group.quotient_ker_equiv_of_surjective\n\nend quotient_add_group\n","avg_line_length":39.7068965517,"max_line_length":123,"alphanum_fraction":0.6854827037}
-{"size":92791,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Scott Morrison\n-\/\nimport data.finset.preimage\nimport algebra.indicator_function\n\n\/-!\n# Type of functions with finite support\n\nFor any type `\u03b1` and a type `M` with zero, we define the type `finsupp \u03b1 M` (notation: `\u03b1 \u2192\u2080 M`)\nof finitely supported functions from `\u03b1` to `M`, i.e. the functions which are zero everywhere\non `\u03b1` except on a finite set.\n\nFunctions with finite support are used (at least) in the following parts of the library:\n\n* `monoid_algebra R M` and `add_monoid_algebra R M` are defined as `M \u2192\u2080 R`;\n\n* polynomials and multivariate polynomials are defined as `add_monoid_algebra`s, hence they use\n  `finsupp` under the hood;\n\n* the linear combination of a family of vectors `v i` with coefficients `f i` (as used, e.g., to\n  define linearly independent family `linear_independent`) is defined as a map\n  `finsupp.total : (\u03b9 \u2192 M) \u2192 (\u03b9 \u2192\u2080 R) \u2192\u2097[R] M`.\n\nSome other constructions are naturally equivalent to `\u03b1 \u2192\u2080 M` with some `\u03b1` and `M` but are defined\nin a different way in the library:\n\n* `multiset \u03b1 \u2243+ \u03b1 \u2192\u2080 \u2115`;\n* `free_abelian_group \u03b1 \u2243+ \u03b1 \u2192\u2080 \u2124`.\n\nMost of the theory assumes that the range is a commutative additive monoid. This gives us the big\nsum operator as a powerful way to construct `finsupp` elements.\n\nMany constructions based on `\u03b1 \u2192\u2080 M` use `semireducible` type tags to avoid reusing unwanted type\ninstances. E.g., `monoid_algebra`, `add_monoid_algebra`, and types based on these two have\nnon-pointwise multiplication.\n\n## Notations\n\nThis file adds `\u03b1 \u2192\u2080 M` as a global notation for `finsupp \u03b1 M`. We also use the following convention\nfor `Type*` variables in this file\n\n* `\u03b1`, `\u03b2`, `\u03b3`: types with no additional structure that appear as the first argument to `finsupp`\n  somewhere in the statement;\n\n* `\u03b9` : an auxiliary index type;\n\n* `M`, `M'`, `N`, `P`: types with `has_zero` or `(add_)(comm_)monoid` structure; `M` is also used\n  for a (semi)module over a (semi)ring.\n\n* `G`, `H`: groups (commutative or not, multiplicative or additive);\n\n* `R`, `S`: (semi)rings.\n\n## TODO\n\n* This file is currently ~2K lines long, so possibly it should be splitted into smaller chunks;\n\n* Add the list of definitions and important lemmas to the module docstring.\n\n## Implementation notes\n\nThis file is a `noncomputable theory` and uses classical logic throughout.\n\n## Notation\n\nThis file defines `\u03b1 \u2192\u2080 \u03b2` as notation for `finsupp \u03b1 \u03b2`.\n\n-\/\n\nnoncomputable theory\nopen_locale classical big_operators\n\nopen finset\n\nvariables {\u03b1 \u03b2 \u03b3 \u03b9 M M' N P G H R S : Type*}\n\n\/-- `finsupp \u03b1 M`, denoted `\u03b1 \u2192\u2080 M`, is the type of functions `f : \u03b1 \u2192 M` such that\n  `f x = 0` for all but finitely many `x`. -\/\nstructure finsupp (\u03b1 : Type*) (M : Type*) [has_zero M] :=\n(support            : finset \u03b1)\n(to_fun             : \u03b1 \u2192 M)\n(mem_support_to_fun : \u2200a, a \u2208 support \u2194 to_fun a \u2260 0)\n\ninfixr ` \u2192\u2080 `:25 := finsupp\n\nnamespace finsupp\n\n\/-! ### Basic declarations about `finsupp` -\/\n\nsection basic\nvariable [has_zero M]\n\ninstance : has_coe_to_fun (\u03b1 \u2192\u2080 M) := \u27e8\u03bb _, \u03b1 \u2192 M, to_fun\u27e9\n\n@[simp] lemma coe_mk (f : \u03b1 \u2192 M) (s : finset \u03b1) (h : \u2200 a, a \u2208 s \u2194 f a \u2260 0) :\n  \u21d1(\u27e8s, f, h\u27e9 : \u03b1 \u2192\u2080 M) = f := rfl\n\ninstance : has_zero (\u03b1 \u2192\u2080 M) := \u27e8\u27e8\u2205, 0, \u03bb _, \u27e8false.elim, \u03bb H, H rfl\u27e9\u27e9\u27e9\n\n@[simp] lemma coe_zero : \u21d1(0 : \u03b1 \u2192\u2080 M) = 0 := rfl\nlemma zero_apply {a : \u03b1} : (0 : \u03b1 \u2192\u2080 M) a = 0 := rfl\n@[simp] lemma support_zero : (0 : \u03b1 \u2192\u2080 M).support = \u2205 := rfl\n\ninstance : inhabited (\u03b1 \u2192\u2080 M) := \u27e80\u27e9\n\n@[simp] lemma mem_support_iff {f : \u03b1 \u2192\u2080 M} : \u2200{a:\u03b1}, a \u2208 f.support \u2194 f a \u2260 0 :=\nf.mem_support_to_fun\n\n@[simp, norm_cast] lemma fun_support_eq (f : \u03b1 \u2192\u2080 M) : function.support f = f.support :=\nset.ext $ \u03bb x, mem_support_iff.symm\n\nlemma not_mem_support_iff {f : \u03b1 \u2192\u2080 M} {a} : a \u2209 f.support \u2194 f a = 0 :=\nnot_iff_comm.1 mem_support_iff.symm\n\nlemma coe_fn_injective : @function.injective (\u03b1 \u2192\u2080 M) (\u03b1 \u2192 M) coe_fn\n| \u27e8s, f, hf\u27e9 \u27e8t, g, hg\u27e9 h :=\n  begin\n    change f = g at h, subst h,\n    have : s = t, { ext a, exact (hf a).trans (hg a).symm },\n    subst this\n  end\n\n@[simp, norm_cast] lemma coe_fn_inj {f g : \u03b1 \u2192\u2080 M} : (f : \u03b1 \u2192 M) = g \u2194 f = g :=\ncoe_fn_injective.eq_iff\n\n@[simp, norm_cast] lemma coe_eq_zero {f : \u03b1 \u2192\u2080 M} : (f : \u03b1 \u2192 M) = 0 \u2194 f = 0 :=\nby rw [\u2190 coe_zero, coe_fn_inj]\n\n@[ext] lemma ext {f g : \u03b1 \u2192\u2080 M} (h : \u2200a, f a = g a) : f = g := coe_fn_injective (funext h)\n\nlemma ext_iff {f g : \u03b1 \u2192\u2080 M} : f = g \u2194 (\u2200a:\u03b1, f a = g a) :=\n\u27e8by rintros rfl a; refl, ext\u27e9\n\nlemma ext_iff' {f g : \u03b1 \u2192\u2080 M} : f = g \u2194 f.support = g.support \u2227 \u2200 x \u2208 f.support, f x = g x :=\n\u27e8\u03bb h, h \u25b8 \u27e8rfl, \u03bb _ _, rfl\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, ext $ \u03bb a,\n  if h : a \u2208 f.support then h\u2082 a h else\n    have hf : f a = 0, from not_mem_support_iff.1 h,\n    have hg : g a = 0, by rwa [h\u2081, not_mem_support_iff] at h,\n    by rw [hf, hg]\u27e9\n\nlemma congr_fun {f g : \u03b1 \u2192\u2080 M} (h : f = g) (a : \u03b1) : f a = g a :=\ncongr_fun (congr_arg finsupp.to_fun h) a\n\n@[simp] lemma support_eq_empty {f : \u03b1 \u2192\u2080 M} : f.support = \u2205 \u2194 f = 0 :=\nby exact_mod_cast @function.support_eq_empty_iff _ _ _ f\n\nlemma support_nonempty_iff {f : \u03b1 \u2192\u2080 M} : f.support.nonempty \u2194 f \u2260 0 :=\nby simp only [finsupp.support_eq_empty, finset.nonempty_iff_ne_empty, ne.def]\n\nlemma nonzero_iff_exists {f : \u03b1 \u2192\u2080 M} : f \u2260 0 \u2194 \u2203 a : \u03b1, f a \u2260 0 :=\nby simp [\u2190 finsupp.support_eq_empty, finset.eq_empty_iff_forall_not_mem]\n\nlemma card_support_eq_zero {f : \u03b1 \u2192\u2080 M} : card f.support = 0 \u2194 f = 0 :=\nby simp\n\ninstance [decidable_eq \u03b1] [decidable_eq M] : decidable_eq (\u03b1 \u2192\u2080 M) :=\nassume f g, decidable_of_iff (f.support = g.support \u2227 (\u2200a\u2208f.support, f a = g a)) ext_iff'.symm\n\nlemma finite_support (f : \u03b1 \u2192\u2080 M) : set.finite (function.support f) :=\nf.fun_support_eq.symm \u25b8 f.support.finite_to_set\n\nlemma support_subset_iff {s : set \u03b1} {f : \u03b1 \u2192\u2080 M} :\n  \u2191f.support \u2286 s \u2194 (\u2200a\u2209s, f a = 0) :=\nby simp only [set.subset_def, mem_coe, mem_support_iff];\n   exact forall_congr (assume a, not_imp_comm)\n\n\/-- Given `fintype \u03b1`, `equiv_fun_on_fintype` is the `equiv` between `\u03b1 \u2192\u2080 \u03b2` and `\u03b1 \u2192 \u03b2`.\n  (All functions on a finite type are finitely supported.) -\/\n@[simps] def equiv_fun_on_fintype [fintype \u03b1] : (\u03b1 \u2192\u2080 M) \u2243 (\u03b1 \u2192 M) :=\n\u27e8\u03bbf a, f a, \u03bbf, mk (finset.univ.filter $ \u03bba, f a \u2260 0) f (by simp only [true_and, finset.mem_univ,\n  iff_self, finset.mem_filter, finset.filter_congr_decidable, forall_true_iff]),\n  begin intro f, ext a, refl end,\n  begin intro f, ext a, refl end\u27e9\n\nend basic\n\n\/-! ### Declarations about `single` -\/\n\nsection single\nvariables [has_zero M] {a a' : \u03b1} {b : M}\n\n\/-- `single a b` is the finitely supported function which has\n  value `b` at `a` and zero otherwise. -\/\ndef single (a : \u03b1) (b : M) : \u03b1 \u2192\u2080 M :=\n\u27e8if b = 0 then \u2205 else {a}, \u03bb a', if a = a' then b else 0, \u03bb a', begin\n  by_cases hb : b = 0; by_cases a = a';\n    simp only [hb, h, if_pos, if_false, mem_singleton],\n  { exact \u27e8false.elim, \u03bb H, H rfl\u27e9 },\n  { exact \u27e8false.elim, \u03bb H, H rfl\u27e9 },\n  { exact \u27e8\u03bb _, hb, \u03bb _, rfl\u27e9 },\n  { exact \u27e8\u03bb H _, h H.symm, \u03bb H, (H rfl).elim\u27e9 }\nend\u27e9\n\nlemma single_apply [decidable (a = a')] : single a b a' = if a = a' then b else 0 :=\nby convert rfl\n\nlemma single_eq_indicator : \u21d1(single a b) = set.indicator {a} (\u03bb _, b) :=\nby { ext, simp [single_apply, set.indicator, @eq_comm _ a] }\n\n@[simp] lemma single_eq_same : (single a b : \u03b1 \u2192\u2080 M) a = b :=\nif_pos rfl\n\n@[simp] lemma single_eq_of_ne (h : a \u2260 a') : (single a b : \u03b1 \u2192\u2080 M) a' = 0 :=\nif_neg h\n\nlemma single_eq_update : \u21d1(single a b) = function.update 0 a b :=\nby rw [single_eq_indicator, \u2190 set.piecewise_eq_indicator, set.piecewise_singleton]\n\nlemma single_eq_pi_single : \u21d1(single a b) = pi.single a b :=\nsingle_eq_update\n\n@[simp] lemma single_zero : (single a 0 : \u03b1 \u2192\u2080 M) = 0 :=\ncoe_fn_injective $ by simpa only [single_eq_update, coe_zero]\n  using function.update_eq_self a (0 : \u03b1 \u2192 M)\n\nlemma single_of_single_apply (a a' : \u03b1) (b : M) :\n  single a ((single a' b) a) = single a' (single a' b) a :=\nbegin\n  rw [single_apply, single_apply],\n  ext,\n  split_ifs,\n  { rw h, },\n  { rw [zero_apply, single_apply, if_t_t], },\nend\n\nlemma support_single_ne_zero (hb : b \u2260 0) : (single a b).support = {a} :=\nif_neg hb\n\nlemma support_single_subset : (single a b).support \u2286 {a} :=\nshow ite _ _ _ \u2286 _, by split_ifs; [exact empty_subset _, exact subset.refl _]\n\nlemma single_apply_mem (x) : single a b x \u2208 ({0, b} : set M) :=\nby rcases em (a = x) with (rfl|hx); [simp, simp [single_eq_of_ne hx]]\n\nlemma range_single_subset : set.range (single a b) \u2286 {0, b} :=\nset.range_subset_iff.2 single_apply_mem\n\n\/-- `finsupp.single a b` is injective in `b`. For the statement that it is injective in `a`, see\n`finsupp.single_left_injective` -\/\nlemma single_injective (a : \u03b1) : function.injective (single a : M \u2192 \u03b1 \u2192\u2080 M) :=\nassume b\u2081 b\u2082 eq,\nhave (single a b\u2081 : \u03b1 \u2192\u2080 M) a = (single a b\u2082 : \u03b1 \u2192\u2080 M) a, by rw eq,\nby rwa [single_eq_same, single_eq_same] at this\n\nlemma single_apply_eq_zero {a x : \u03b1} {b : M} : single a b x = 0 \u2194 (x = a \u2192 b = 0) :=\nby simp [single_eq_indicator]\n\nlemma mem_support_single (a a' : \u03b1) (b : M) :\n  a \u2208 (single a' b).support \u2194 a = a' \u2227 b \u2260 0 :=\nby simp [single_apply_eq_zero, not_or_distrib]\n\nlemma eq_single_iff {f : \u03b1 \u2192\u2080 M} {a b} : f = single a b \u2194 f.support \u2286 {a} \u2227 f a = b :=\nbegin\n  refine \u27e8\u03bb h, h.symm \u25b8 \u27e8support_single_subset, single_eq_same\u27e9, _\u27e9,\n  rintro \u27e8h, rfl\u27e9,\n  ext x,\n  by_cases hx : a = x; simp only [hx, single_eq_same, single_eq_of_ne, ne.def, not_false_iff],\n  exact not_mem_support_iff.1 (mt (\u03bb hx, (mem_singleton.1 (h hx)).symm) hx)\nend\n\nlemma single_eq_single_iff (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : M) :\n  single a\u2081 b\u2081 = single a\u2082 b\u2082 \u2194 ((a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082) \u2228 (b\u2081 = 0 \u2227 b\u2082 = 0)) :=\nbegin\n  split,\n  { assume eq,\n    by_cases a\u2081 = a\u2082,\n    { refine or.inl \u27e8h, _\u27e9,\n      rwa [h, (single_injective a\u2082).eq_iff] at eq },\n    { rw [ext_iff] at eq,\n      have h\u2081 := eq a\u2081,\n      have h\u2082 := eq a\u2082,\n      simp only [single_eq_same, single_eq_of_ne h, single_eq_of_ne (ne.symm h)] at h\u2081 h\u2082,\n      exact or.inr \u27e8h\u2081, h\u2082.symm\u27e9 } },\n  { rintros (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9),\n    { refl },\n    { rw [single_zero, single_zero] } }\nend\n\n\/-- `finsupp.single a b` is injective in `a`. For the statement that it is injective in `b`, see\n`finsupp.single_injective` -\/\nlemma single_left_injective (h : b \u2260 0) : function.injective (\u03bb a : \u03b1, single a b) :=\n\u03bb a a' H, (((single_eq_single_iff _ _ _ _).mp H).resolve_right $ \u03bb hb, h hb.1).left\n\nlemma single_left_inj (h : b \u2260 0) : single a b = single a' b \u2194 a = a' :=\n(single_left_injective h).eq_iff\n\nlemma support_single_ne_bot (i : \u03b1) (h : b \u2260 0) :\n  (single i b).support \u2260 \u22a5 :=\nbegin\n  have : i \u2208 (single i b).support := by simpa using h,\n  intro H,\n  simpa [H]\nend\n\nlemma support_single_disjoint {b' : M} (hb : b \u2260 0) (hb' : b' \u2260 0) {i j : \u03b1} :\n  disjoint (single i b).support (single j b').support \u2194 i \u2260 j :=\nby simpa [support_single_ne_zero, hb, hb'] using ne_comm\n\n@[simp] lemma single_eq_zero : single a b = 0 \u2194 b = 0 :=\nby simp [ext_iff, single_eq_indicator]\n\nlemma single_swap (a\u2081 a\u2082 : \u03b1) (b : M) : single a\u2081 b a\u2082 = single a\u2082 b a\u2081 :=\nby simp only [single_apply]; ac_refl\n\ninstance [nonempty \u03b1] [nontrivial M] : nontrivial (\u03b1 \u2192\u2080 M) :=\nbegin\n  inhabit \u03b1,\n  rcases exists_ne (0 : M) with \u27e8x, hx\u27e9,\n  exact nontrivial_of_ne (single (default \u03b1) x) 0 (mt single_eq_zero.1 hx)\nend\n\nlemma unique_single [unique \u03b1] (x : \u03b1 \u2192\u2080 M) : x = single (default \u03b1) (x (default \u03b1)) :=\next $ unique.forall_iff.2 single_eq_same.symm\n\nlemma unique_ext [unique \u03b1] {f g : \u03b1 \u2192\u2080 M} (h : f (default \u03b1) = g (default \u03b1)) : f = g :=\next $ \u03bb a, by rwa [unique.eq_default a]\n\nlemma unique_ext_iff [unique \u03b1] {f g : \u03b1 \u2192\u2080 M} : f = g \u2194  f (default \u03b1) = g (default \u03b1) :=\n\u27e8\u03bb h, h \u25b8 rfl, unique_ext\u27e9\n\n@[simp] lemma unique_single_eq_iff [unique \u03b1] {b' : M} :\n  single a b = single a' b' \u2194 b = b' :=\nby rw [unique_ext_iff, unique.eq_default a, unique.eq_default a', single_eq_same, single_eq_same]\n\nlemma support_eq_singleton {f : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  f.support = {a} \u2194 f a \u2260 0 \u2227 f = single a (f a) :=\n\u27e8\u03bb h, \u27e8mem_support_iff.1 $ h.symm \u25b8 finset.mem_singleton_self a,\n  eq_single_iff.2 \u27e8subset_of_eq h, rfl\u27e9\u27e9, \u03bb h, h.2.symm \u25b8 support_single_ne_zero h.1\u27e9\n\nlemma support_eq_singleton' {f : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  f.support = {a} \u2194 \u2203 b \u2260 0, f = single a b :=\n\u27e8\u03bb h, let h := support_eq_singleton.1 h in \u27e8_, h.1, h.2\u27e9,\n  \u03bb \u27e8b, hb, hf\u27e9, hf.symm \u25b8 support_single_ne_zero hb\u27e9\n\nlemma card_support_eq_one {f : \u03b1 \u2192\u2080 M} : card f.support = 1 \u2194 \u2203 a, f a \u2260 0 \u2227 f = single a (f a) :=\nby simp only [card_eq_one, support_eq_singleton]\n\nlemma card_support_eq_one' {f : \u03b1 \u2192\u2080 M} : card f.support = 1 \u2194 \u2203 a (b \u2260 0), f = single a b :=\nby simp only [card_eq_one, support_eq_singleton']\n\n@[simp] lemma equiv_fun_on_fintype_single [fintype \u03b1] (x : \u03b1) (m : M) :\n  (@finsupp.equiv_fun_on_fintype \u03b1 M _ _) (finsupp.single x m) = pi.single x m :=\nby { ext, simp [finsupp.single_eq_pi_single, finsupp.equiv_fun_on_fintype], }\n\n@[simp] lemma equiv_fun_on_fintype_symm_single [fintype \u03b1] (x : \u03b1) (m : M) :\n  (@finsupp.equiv_fun_on_fintype \u03b1 M _ _).symm (pi.single x m) = finsupp.single x m :=\nby { ext, simp [finsupp.single_eq_pi_single, finsupp.equiv_fun_on_fintype], }\n\nend single\n\n\/-! ### Declarations about `on_finset` -\/\n\nsection on_finset\nvariables [has_zero M]\n\n\/-- `on_finset s f hf` is the finsupp function representing `f` restricted to the finset `s`.\n  The function needs to be `0` outside of `s`. Use this when the set needs to be filtered anyways,\n  otherwise a better set representation is often available. -\/\ndef on_finset (s : finset \u03b1) (f : \u03b1 \u2192 M) (hf : \u2200a, f a \u2260 0 \u2192 a \u2208 s) : \u03b1 \u2192\u2080 M :=\n\u27e8s.filter (\u03bba, f a \u2260 0), f, by simpa\u27e9\n\n@[simp] lemma on_finset_apply {s : finset \u03b1} {f : \u03b1 \u2192 M} {hf a} :\n  (on_finset s f hf : \u03b1 \u2192\u2080 M) a = f a :=\nrfl\n\n@[simp] lemma support_on_finset_subset {s : finset \u03b1} {f : \u03b1 \u2192 M} {hf} :\n  (on_finset s f hf).support \u2286 s :=\nfilter_subset _ _\n\n@[simp] lemma mem_support_on_finset\n  {s : finset \u03b1} {f : \u03b1 \u2192 M} (hf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s) {a : \u03b1} :\n  a \u2208 (finsupp.on_finset s f hf).support \u2194 f a \u2260 0 :=\nby rw [finsupp.mem_support_iff, finsupp.on_finset_apply]\n\nlemma support_on_finset\n  {s : finset \u03b1} {f : \u03b1 \u2192 M} (hf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s) :\n  (finsupp.on_finset s f hf).support = s.filter (\u03bb a, f a \u2260 0) :=\nrfl\n\nend on_finset\n\nsection of_support_finite\n\nvariables [has_zero M]\n\n\/-- The natural `finsupp` induced by the function `f` given that it has finite support. -\/\nnoncomputable def of_support_finite\n  (f : \u03b1 \u2192 M) (hf : (function.support f).finite) : \u03b1 \u2192\u2080 M :=\n{ support := hf.to_finset,\n  to_fun := f,\n  mem_support_to_fun := \u03bb _, hf.mem_to_finset }\n\nlemma of_support_finite_coe {f : \u03b1 \u2192 M} {hf : (function.support f).finite} :\n  (of_support_finite f hf : \u03b1 \u2192 M) = f := rfl\n\ninstance : can_lift (\u03b1 \u2192 M) (\u03b1 \u2192\u2080 M) :=\n{ coe := coe_fn,\n  cond := \u03bb f, (function.support f).finite,\n  prf := \u03bb f hf, \u27e8of_support_finite f hf, rfl\u27e9 }\n\nend of_support_finite\n\n\/-! ### Declarations about `map_range` -\/\n\nsection map_range\nvariables [has_zero M] [has_zero N] [has_zero P]\n\n\/-- The composition of `f : M \u2192 N` and `g : \u03b1 \u2192\u2080 M` is\n`map_range f hf g : \u03b1 \u2192\u2080 N`, well-defined when `f 0 = 0`.\n\nThis preserves the structure on `f`, and exists in various bundled forms for when `f` is itself\nbundled:\n\n* `finsupp.map_range.zero_hom`\n* `finsupp.map_range.add_monoid_hom`\n* `finsupp.map_range.add_equiv`\n* `finsupp.map_range.linear_map`\n* `finsupp.map_range.linear_equiv`\n-\/\ndef map_range (f : M \u2192 N) (hf : f 0 = 0) (g : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 N :=\non_finset g.support (f \u2218 g) $\n  assume a, by rw [mem_support_iff, not_imp_not]; exact \u03bb H, (congr_arg f H).trans hf\n\n@[simp] lemma map_range_apply {f : M \u2192 N} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} {a : \u03b1} :\n  map_range f hf g a = f (g a) :=\nrfl\n\n@[simp] lemma map_range_zero {f : M \u2192 N} {hf : f 0 = 0} : map_range f hf (0 : \u03b1 \u2192\u2080 M) = 0 :=\next $ \u03bb a, by simp only [hf, zero_apply, map_range_apply]\n\n@[simp] lemma map_range_id (g : \u03b1 \u2192\u2080 M) : map_range id rfl g = g :=\next $ \u03bb _, rfl\n\nlemma map_range_comp\n  (f : N \u2192 P) (hf : f 0 = 0) (f\u2082 : M \u2192 N) (hf\u2082 : f\u2082 0 = 0) (h : (f \u2218 f\u2082) 0 = 0) (g : \u03b1 \u2192\u2080 M) :\n  map_range (f \u2218 f\u2082) h g = map_range f hf (map_range f\u2082 hf\u2082 g) :=\next $ \u03bb _, rfl\n\nlemma support_map_range {f : M \u2192 N} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} :\n  (map_range f hf g).support \u2286 g.support :=\nsupport_on_finset_subset\n\n@[simp] lemma map_range_single {f : M \u2192 N} {hf : f 0 = 0} {a : \u03b1} {b : M} :\n  map_range f hf (single a b) = single a (f b) :=\next $ \u03bb a', show f (ite _ _ _) = ite _ _ _, by split_ifs; [refl, exact hf]\n\nend map_range\n\n\/-! ### Declarations about `emb_domain` -\/\n\nsection emb_domain\nvariables [has_zero M] [has_zero N]\n\n\/-- Given `f : \u03b1 \u21aa \u03b2` and `v : \u03b1 \u2192\u2080 M`, `emb_domain f v : \u03b2 \u2192\u2080 M`\nis the finitely supported function whose value at `f a : \u03b2` is `v a`.\nFor a `b : \u03b2` outside the range of `f`, it is zero. -\/\ndef emb_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\nbegin\n  refine \u27e8v.support.map f, \u03bba\u2082,\n    if h : a\u2082 \u2208 v.support.map f then v (v.support.choose (\u03bba\u2081, f a\u2081 = a\u2082) _) else 0, _\u27e9,\n  { rcases finset.mem_map.1 h with \u27e8a, ha, rfl\u27e9,\n    exact exists_unique.intro a \u27e8ha, rfl\u27e9 (assume b \u27e8_, hb\u27e9, f.injective hb) },\n  { assume a\u2082,\n    split_ifs,\n    { simp only [h, true_iff, ne.def],\n      rw [\u2190 not_mem_support_iff, not_not],\n      apply finset.choose_mem },\n    { simp only [h, ne.def, ne_self_iff_false] } }\nend\n\n@[simp] lemma support_emb_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) :\n  (emb_domain f v).support = v.support.map f :=\nrfl\n\n@[simp] lemma emb_domain_zero (f : \u03b1 \u21aa \u03b2) : (emb_domain f 0 : \u03b2 \u2192\u2080 M) = 0 :=\nrfl\n\n@[simp] lemma emb_domain_apply (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  emb_domain f v (f a) = v a :=\nbegin\n  change dite _ _ _ = _,\n  split_ifs; rw [finset.mem_map' f] at h,\n  { refine congr_arg (v : \u03b1 \u2192 M) (f.inj' _),\n    exact finset.choose_property (\u03bba\u2081, f a\u2081 = f a) _ _ },\n  { exact (not_mem_support_iff.1 h).symm }\nend\n\nlemma emb_domain_notin_range (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) (a : \u03b2) (h : a \u2209 set.range f) :\n  emb_domain f v a = 0 :=\nbegin\n  refine dif_neg (mt (assume h, _) h),\n  rcases finset.mem_map.1 h with \u27e8a, h, rfl\u27e9,\n  exact set.mem_range_self a\nend\n\nlemma emb_domain_injective (f : \u03b1 \u21aa \u03b2) :\n  function.injective (emb_domain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)) :=\n\u03bb l\u2081 l\u2082 h, ext $ \u03bb a, by simpa only [emb_domain_apply] using ext_iff.1 h (f a)\n\n@[simp] lemma emb_domain_inj {f : \u03b1 \u21aa \u03b2} {l\u2081 l\u2082 : \u03b1 \u2192\u2080 M} :\n  emb_domain f l\u2081 = emb_domain f l\u2082 \u2194 l\u2081 = l\u2082 :=\n(emb_domain_injective f).eq_iff\n\n@[simp] lemma emb_domain_eq_zero {f : \u03b1 \u21aa \u03b2} {l : \u03b1 \u2192\u2080 M} :\n  emb_domain f l = 0 \u2194 l = 0 :=\n(emb_domain_injective f).eq_iff' $ emb_domain_zero f\n\nlemma emb_domain_map_range\n  (f : \u03b1 \u21aa \u03b2) (g : M \u2192 N) (p : \u03b1 \u2192\u2080 M) (hg : g 0 = 0) :\n  emb_domain f (map_range g hg p) = map_range g hg (emb_domain f p) :=\nbegin\n  ext a,\n  by_cases a \u2208 set.range f,\n  { rcases h with \u27e8a', rfl\u27e9,\n    rw [map_range_apply, emb_domain_apply, emb_domain_apply, map_range_apply] },\n  { rw [map_range_apply, emb_domain_notin_range, emb_domain_notin_range, \u2190 hg]; assumption }\nend\n\nlemma single_of_emb_domain_single\n  (l : \u03b1 \u2192\u2080 M) (f : \u03b1 \u21aa \u03b2) (a : \u03b2) (b : M) (hb : b \u2260 0)\n  (h : l.emb_domain f = single a b) :\n  \u2203 x, l = single x b \u2227 f x = a :=\nbegin\n  have h_map_support : finset.map f (l.support) = {a},\n    by rw [\u2190support_emb_domain, h, support_single_ne_zero hb]; refl,\n  have ha : a \u2208 finset.map f (l.support),\n    by simp only [h_map_support, finset.mem_singleton],\n  rcases finset.mem_map.1 ha with \u27e8c, hc\u2081, hc\u2082\u27e9,\n  use c,\n  split,\n  { ext d,\n    rw [\u2190 emb_domain_apply f l, h],\n    by_cases h_cases : c = d,\n    { simp only [eq.symm h_cases, hc\u2082, single_eq_same] },\n    { rw [single_apply, single_apply, if_neg, if_neg h_cases],\n      by_contra hfd,\n      exact h_cases (f.injective (hc\u2082.trans hfd)) } },\n  { exact hc\u2082 }\nend\n\nend emb_domain\n\n\/-! ### Declarations about `zip_with` -\/\n\nsection zip_with\nvariables [has_zero M] [has_zero N] [has_zero P]\n\n\/-- `zip_with f hf g\u2081 g\u2082` is the finitely supported function satisfying\n  `zip_with f hf g\u2081 g\u2082 a = f (g\u2081 a) (g\u2082 a)`, and it is well-defined when `f 0 0 = 0`. -\/\ndef zip_with (f : M \u2192 N \u2192 P) (hf : f 0 0 = 0) (g\u2081 : \u03b1 \u2192\u2080 M) (g\u2082 : \u03b1 \u2192\u2080 N) : (\u03b1 \u2192\u2080 P) :=\non_finset (g\u2081.support \u222a g\u2082.support) (\u03bba, f (g\u2081 a) (g\u2082 a)) $ \u03bb a H,\nbegin\n  simp only [mem_union, mem_support_iff, ne], rw [\u2190 not_and_distrib],\n  rintro \u27e8h\u2081, h\u2082\u27e9, rw [h\u2081, h\u2082] at H, exact H hf\nend\n\n@[simp] lemma zip_with_apply\n  {f : M \u2192 N \u2192 P} {hf : f 0 0 = 0} {g\u2081 : \u03b1 \u2192\u2080 M} {g\u2082 : \u03b1 \u2192\u2080 N} {a : \u03b1} :\n  zip_with f hf g\u2081 g\u2082 a = f (g\u2081 a) (g\u2082 a) :=\nrfl\n\nlemma support_zip_with [D : decidable_eq \u03b1] {f : M \u2192 N \u2192 P} {hf : f 0 0 = 0}\n  {g\u2081 : \u03b1 \u2192\u2080 M} {g\u2082 : \u03b1 \u2192\u2080 N} : (zip_with f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nby rw subsingleton.elim D; exact support_on_finset_subset\n\nend zip_with\n\n\/-! ### Declarations about `erase` -\/\n\nsection erase\n\nvariables [has_zero M]\n\n\/-- `erase a f` is the finitely supported function equal to `f` except at `a` where it is equal to\n  `0`. -\/\ndef erase (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M :=\n\u27e8f.support.erase a, (\u03bba', if a' = a then 0 else f a'),\n  assume a', by rw [mem_erase, mem_support_iff]; split_ifs;\n    [exact \u27e8\u03bb H _, H.1 h, \u03bb H, (H rfl).elim\u27e9,\n    exact and_iff_right h]\u27e9\n\n@[simp] lemma support_erase {a : \u03b1} {f : \u03b1 \u2192\u2080 M} :\n  (f.erase a).support = f.support.erase a :=\nrfl\n\n@[simp] lemma erase_same {a : \u03b1} {f : \u03b1 \u2192\u2080 M} : (f.erase a) a = 0 :=\nif_pos rfl\n\n@[simp] lemma erase_ne {a a' : \u03b1} {f : \u03b1 \u2192\u2080 M} (h : a' \u2260 a) : (f.erase a) a' = f a' :=\nif_neg h\n\n@[simp] lemma erase_single {a : \u03b1} {b : M} : (erase a (single a b)) = 0 :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, erase_same], refl },\n  { rw [erase_ne hs], exact single_eq_of_ne (ne.symm hs) }\nend\n\nlemma erase_single_ne {a a' : \u03b1} {b : M} (h : a \u2260 a') : (erase a (single a' b)) = single a' b :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, erase_same, single_eq_of_ne (h.symm)] },\n  { rw [erase_ne hs] }\nend\n\n@[simp] lemma erase_zero (a : \u03b1) : erase a (0 : \u03b1 \u2192\u2080 M) = 0 :=\nby rw [\u2190 support_eq_empty, support_erase, support_zero, erase_empty]\n\nend erase\n\n\/-!\n### Declarations about `sum` and `prod`\n\nIn most of this section, the domain `\u03b2` is assumed to be an `add_monoid`.\n-\/\n\nsection sum_prod\n\n-- [to_additive sum] for finsupp.prod doesn't work, the equation lemmas are not generated\n\/-- `sum f g` is the sum of `g a (f a)` over the support of `f`. -\/\ndef sum [has_zero M] [add_comm_monoid N] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : N :=\n\u2211 a in f.support, g a (f a)\n\n\/-- `prod f g` is the product of `g a (f a)` over the support of `f`. -\/\n@[to_additive]\ndef prod [has_zero M] [comm_monoid N] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : N :=\n\u220f a in f.support, g a (f a)\n\nvariables [has_zero M] [has_zero M'] [comm_monoid N]\n\n@[to_additive]\nlemma prod_of_support_subset (f : \u03b1 \u2192\u2080 M) {s : finset \u03b1}\n  (hs : f.support \u2286 s) (g : \u03b1 \u2192 M \u2192 N) (h : \u2200 i \u2208 s, g i 0 = 1) :\n  f.prod g = \u220f x in s, g x (f x) :=\nfinset.prod_subset hs $ \u03bb x hxs hx, h x hxs \u25b8 congr_arg (g x) $ not_mem_support_iff.1 hx\n\n@[to_additive]\nlemma prod_fintype [fintype \u03b1] (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) (h : \u2200 i, g i 0 = 1) :\n  f.prod g = \u220f i, g i (f i) :=\nf.prod_of_support_subset (subset_univ _) g (\u03bb x _, h x)\n\n@[simp, to_additive]\nlemma prod_single_index {a : \u03b1} {b : M} {h : \u03b1 \u2192 M \u2192 N} (h_zero : h a 0 = 1) :\n  (single a b).prod h = h a b :=\ncalc (single a b).prod h = \u220f x in {a}, h x (single a b x) :\n  prod_of_support_subset _ support_single_subset h $ \u03bb x hx, (mem_singleton.1 hx).symm \u25b8 h_zero\n... = h a b : by simp\n\n@[to_additive]\nlemma prod_map_range_index {f : M \u2192 M'} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} {h : \u03b1 \u2192 M' \u2192 N}\n  (h0 : \u2200a, h a 0 = 1) : (map_range f hf g).prod h = g.prod (\u03bba b, h a (f b)) :=\nfinset.prod_subset support_map_range $ \u03bb _ _ H,\n  by rw [not_mem_support_iff.1 H, h0]\n\n@[simp, to_additive]\nlemma prod_zero_index {h : \u03b1 \u2192 M \u2192 N} : (0 : \u03b1 \u2192\u2080 M).prod h = 1 := rfl\n\n@[to_additive]\nlemma prod_comm (f : \u03b1 \u2192\u2080 M) (g : \u03b2 \u2192\u2080 M') (h : \u03b1 \u2192 M \u2192 \u03b2 \u2192 M' \u2192 N) :\n  f.prod (\u03bb x v, g.prod (\u03bb x' v', h x v x' v')) = g.prod (\u03bb x' v', f.prod (\u03bb x v, h x v x' v')) :=\nfinset.prod_comm\n\n@[simp, to_additive]\nlemma prod_ite_eq [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 M) (a : \u03b1) (b : \u03b1 \u2192 M \u2192 N) :\n  f.prod (\u03bb x v, ite (a = x) (b x v) 1) = ite (a \u2208 f.support) (b a (f a)) 1 :=\nby { dsimp [finsupp.prod], rw f.support.prod_ite_eq, }\n\n@[simp] lemma sum_ite_self_eq\n  [decidable_eq \u03b1] {N : Type*} [add_comm_monoid N] (f : \u03b1 \u2192\u2080 N) (a : \u03b1) :\n  f.sum (\u03bb x v, ite (a = x) v 0) = f a :=\nby { convert f.sum_ite_eq a (\u03bb x, id), simp [ite_eq_right_iff.2 eq.symm] }\n\n\/-- A restatement of `prod_ite_eq` with the equality test reversed. -\/\n@[simp, to_additive \"A restatement of `sum_ite_eq` with the equality test reversed.\"]\nlemma prod_ite_eq' [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 M) (a : \u03b1) (b : \u03b1 \u2192 M \u2192 N) :\n  f.prod (\u03bb x v, ite (x = a) (b x v) 1) = ite (a \u2208 f.support) (b a (f a)) 1 :=\nby { dsimp [finsupp.prod], rw f.support.prod_ite_eq', }\n\n@[simp] lemma sum_ite_self_eq'\n  [decidable_eq \u03b1] {N : Type*} [add_comm_monoid N] (f : \u03b1 \u2192\u2080 N) (a : \u03b1) :\n  f.sum (\u03bb x v, ite (x = a) v 0) = f a :=\nby { convert f.sum_ite_eq' a (\u03bb x, id), simp [ite_eq_right_iff.2 eq.symm] }\n\n@[simp] lemma prod_pow [fintype \u03b1] (f : \u03b1 \u2192\u2080 \u2115) (g : \u03b1 \u2192 N) :\n  f.prod (\u03bb a b, g a ^ b) = \u220f a, g a ^ (f a) :=\nf.prod_fintype _ $ \u03bb a, pow_zero _\n\n\/-- If `g` maps a second argument of 0 to 1, then multiplying it over the\nresult of `on_finset` is the same as multiplying it over the original\n`finset`. -\/\n@[to_additive \"If `g` maps a second argument of 0 to 0, summing it over the\nresult of `on_finset` is the same as summing it over the original\n`finset`.\"]\nlemma on_finset_prod {s : finset \u03b1} {f : \u03b1 \u2192 M} {g : \u03b1 \u2192 M \u2192 N}\n    (hf : \u2200a, f a \u2260 0 \u2192 a \u2208 s) (hg : \u2200 a, g a 0 = 1) :\n  (on_finset s f hf).prod g = \u220f a in s, g a (f a) :=\nfinset.prod_subset support_on_finset_subset $ by simp [*] { contextual := tt }\n\nend sum_prod\n\n\/-!\n### Additive monoid structure on `\u03b1 \u2192\u2080 M`\n-\/\n\nsection add_zero_class\n\nvariables [add_zero_class M]\n\ninstance : has_add (\u03b1 \u2192\u2080 M) := \u27e8zip_with (+) (add_zero 0)\u27e9\n\n@[simp] lemma coe_add (f g : \u03b1 \u2192\u2080 M) : \u21d1(f + g) = f + g := rfl\nlemma add_apply (g\u2081 g\u2082 : \u03b1 \u2192\u2080 M) (a : \u03b1) : (g\u2081 + g\u2082) a = g\u2081 a + g\u2082 a := rfl\n\nlemma support_add [decidable_eq \u03b1] {g\u2081 g\u2082 : \u03b1 \u2192\u2080 M} :\n  (g\u2081 + g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nsupport_zip_with\n\nlemma support_add_eq [decidable_eq \u03b1] {g\u2081 g\u2082 : \u03b1 \u2192\u2080 M} (h : disjoint g\u2081.support g\u2082.support) :\n  (g\u2081 + g\u2082).support = g\u2081.support \u222a g\u2082.support :=\nle_antisymm support_zip_with $ assume a ha,\n(finset.mem_union.1 ha).elim\n  (assume ha, have a \u2209 g\u2082.support, from disjoint_left.1 h ha,\n    by simp only [mem_support_iff, not_not] at *;\n    simpa only [add_apply, this, add_zero])\n  (assume ha, have a \u2209 g\u2081.support, from disjoint_right.1 h ha,\n    by simp only [mem_support_iff, not_not] at *;\n    simpa only [add_apply, this, zero_add])\n\n@[simp] lemma single_add {a : \u03b1} {b\u2081 b\u2082 : M} : single a (b\u2081 + b\u2082) = single a b\u2081 + single a b\u2082 :=\next $ assume a',\nbegin\n  by_cases h : a = a',\n  { rw [h, add_apply, single_eq_same, single_eq_same, single_eq_same] },\n  { rw [add_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h, zero_add] }\nend\n\ninstance : add_zero_class (\u03b1 \u2192\u2080 M) :=\n{ zero      := 0,\n  add       := (+),\n  zero_add  := assume \u27e8s, f, hf\u27e9, ext $ assume a, zero_add _,\n  add_zero  := assume \u27e8s, f, hf\u27e9, ext $ assume a, add_zero _ }\n\n\/-- `finsupp.single` as an `add_monoid_hom`.\n\nSee `finsupp.lsingle` for the stronger version as a linear map.\n-\/\n@[simps] def single_add_hom (a : \u03b1) : M \u2192+ \u03b1 \u2192\u2080 M :=\n\u27e8single a, single_zero, \u03bb _ _, single_add\u27e9\n\n\/-- Evaluation of a function `f : \u03b1 \u2192\u2080 M` at a point as an additive monoid homomorphism.\n\nSee `finsupp.lapply` for the stronger version as a linear map. -\/\n@[simps apply]\ndef apply_add_hom (a : \u03b1) : (\u03b1 \u2192\u2080 M) \u2192+ M := \u27e8\u03bb g, g a, zero_apply, \u03bb _ _, add_apply _ _ _\u27e9\n\nlemma single_add_erase (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : single a (f a) + f.erase a = f :=\next $ \u03bb a',\nif h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, add_zero]\nelse by simp only [add_apply, single_eq_of_ne h, zero_add, erase_ne (ne.symm h)]\n\nlemma erase_add_single (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : f.erase a + single a (f a) = f :=\next $ \u03bb a',\nif h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, zero_add]\nelse by simp only [add_apply, single_eq_of_ne h, add_zero, erase_ne (ne.symm h)]\n\n@[simp] lemma erase_add (a : \u03b1) (f f' : \u03b1 \u2192\u2080 M) : erase a (f + f') = erase a f + erase a f' :=\nbegin\n  ext s, by_cases hs : s = a,\n  { rw [hs, add_apply, erase_same, erase_same, erase_same, add_zero] },\n  rw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply],\nend\n\n@[elab_as_eliminator]\nprotected theorem induction {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (ha : \u2200a b (f : \u03b1 \u2192\u2080 M), a \u2209 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)) :\n  p f :=\nsuffices \u2200s (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f, from this _ _ rfl,\nassume s, finset.induction_on s (\u03bb f hf, by rwa [support_eq_empty.1 hf]) $\nassume a s has ih f hf,\nsuffices p (single a (f a) + f.erase a), by rwa [single_add_erase] at this,\nbegin\n  apply ha,\n  { rw [support_erase, mem_erase], exact \u03bb H, H.1 rfl },\n  { rw [\u2190 mem_support_iff, hf], exact mem_insert_self _ _ },\n  { apply ih _ _,\n    rw [support_erase, hf, finset.erase_insert has] }\nend\n\nlemma induction\u2082 {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (ha : \u2200a b (f : \u03b1 \u2192\u2080 M), a \u2209 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)) :\n  p f :=\nsuffices \u2200s (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f, from this _ _ rfl,\nassume s, finset.induction_on s (\u03bb f hf, by rwa [support_eq_empty.1 hf]) $\nassume a s has ih f hf,\nsuffices p (f.erase a + single a (f a)), by rwa [erase_add_single] at this,\nbegin\n  apply ha,\n  { rw [support_erase, mem_erase], exact \u03bb H, H.1 rfl },\n  { rw [\u2190 mem_support_iff, hf], exact mem_insert_self _ _ },\n  { apply ih _ _,\n    rw [support_erase, hf, finset.erase_insert has] }\nend\n\nlemma induction_linear {p : (\u03b1 \u2192\u2080 M) \u2192 Prop} (f : \u03b1 \u2192\u2080 M)\n  (h0 : p 0) (hadd : \u2200 f g : \u03b1 \u2192\u2080 M, p f \u2192 p g \u2192 p (f + g)) (hsingle : \u2200 a b, p (single a b)) :\n  p f :=\ninduction\u2082 f h0 (\u03bb a b f _ _ w, hadd _ _ w (hsingle _ _))\n\n@[simp] lemma add_closure_Union_range_single :\n  add_submonoid.closure (\u22c3 a : \u03b1, set.range (single a : M \u2192 \u03b1 \u2192\u2080 M)) = \u22a4 :=\ntop_unique $ \u03bb x hx, finsupp.induction x (add_submonoid.zero_mem _) $\n  \u03bb a b f ha hb hf, add_submonoid.add_mem _\n    (add_submonoid.subset_closure $ set.mem_Union.2 \u27e8a, set.mem_range_self _\u27e9) hf\n\n\/-- If two additive homomorphisms from `\u03b1 \u2192\u2080 M` are equal on each `single a b`, then\nthey are equal. -\/\nlemma add_hom_ext [add_zero_class N] \u2983f g : (\u03b1 \u2192\u2080 M) \u2192+ N\u2984\n  (H : \u2200 x y, f (single x y) = g (single x y)) :\n  f = g :=\nbegin\n  refine add_monoid_hom.eq_of_eq_on_mdense add_closure_Union_range_single (\u03bb f hf, _),\n  simp only [set.mem_Union, set.mem_range] at hf,\n  rcases hf with \u27e8x, y, rfl\u27e9,\n  apply H\nend\n\n\/-- If two additive homomorphisms from `\u03b1 \u2192\u2080 M` are equal on each `single a b`, then\nthey are equal.\n\nWe formulate this using equality of `add_monoid_hom`s so that `ext` tactic can apply a type-specific\nextensionality lemma after this one.  E.g., if the fiber `M` is `\u2115` or `\u2124`, then it suffices to\nverify `f (single a 1) = g (single a 1)`. -\/\n@[ext] lemma add_hom_ext' [add_zero_class N] \u2983f g : (\u03b1 \u2192\u2080 M) \u2192+ N\u2984\n  (H : \u2200 x, f.comp (single_add_hom x) = g.comp (single_add_hom x)) :\n  f = g :=\nadd_hom_ext $ \u03bb x, add_monoid_hom.congr_fun (H x)\n\nlemma mul_hom_ext [mul_one_class N] \u2983f g : multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\u2984\n  (H : \u2200 x y, f (multiplicative.of_add $ single x y) = g (multiplicative.of_add $ single x y)) :\n  f = g :=\nmonoid_hom.ext $ add_monoid_hom.congr_fun $\n  @add_hom_ext \u03b1 M (additive N) _ _ f.to_additive'' g.to_additive'' H\n\n@[ext] lemma mul_hom_ext' [mul_one_class N] {f g : multiplicative (\u03b1 \u2192\u2080 M) \u2192* N}\n  (H : \u2200 x, f.comp (single_add_hom x).to_multiplicative =\n    g.comp (single_add_hom x).to_multiplicative) :\n  f = g :=\nmul_hom_ext $ \u03bb x, monoid_hom.congr_fun (H x)\n\nlemma map_range_add [add_zero_class N]\n  {f : M \u2192 N} {hf : f 0 = 0} (hf' : \u2200 x y, f (x + y) = f x + f y) (v\u2081 v\u2082 : \u03b1 \u2192\u2080 M) :\n  map_range f hf (v\u2081 + v\u2082) = map_range f hf v\u2081 + map_range f hf v\u2082 :=\next $ \u03bb a, by simp only [hf', add_apply, map_range_apply]\n\nend add_zero_class\n\nsection add_monoid\n\nvariables [add_monoid M]\n\ninstance : add_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_monoid .\n  zero      := 0,\n  add       := (+),\n  add_assoc := assume \u27e8s, f, hf\u27e9 \u27e8t, g, hg\u27e9 \u27e8u, h, hh\u27e9, ext $ assume a, add_assoc _ _ _,\n  nsmul := \u03bb n v, v.map_range ((\u2022) n) (nsmul_zero _),\n  nsmul_zero' := \u03bb v, by { ext i, simp },\n  nsmul_succ' := \u03bb n v, by { ext i, simp [nat.succ_eq_one_add, add_nsmul] },\n  .. finsupp.add_zero_class }\n\nend add_monoid\n\nend finsupp\n\n@[to_additive]\nlemma mul_equiv.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]\n  (h : N \u2243* P) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\n@[to_additive]\nlemma monoid_hom.map_finsupp_prod [has_zero M] [comm_monoid N] [comm_monoid P]\n  (h : N \u2192* P) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\nlemma ring_hom.map_finsupp_sum [has_zero M] [semiring R] [semiring S]\n  (h : R \u2192+* S) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 R) : h (f.sum g) = f.sum (\u03bb a b, h (g a b)) :=\nh.map_sum _ _\n\nlemma ring_hom.map_finsupp_prod [has_zero M] [comm_semiring R] [comm_semiring S]\n  (h : R \u2192+* S) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 R) : h (f.prod g) = f.prod (\u03bb a b, h (g a b)) :=\nh.map_prod _ _\n\n@[to_additive]\nlemma monoid_hom.coe_finsupp_prod [has_zero \u03b2] [monoid N] [comm_monoid P]\n  (f : \u03b1 \u2192\u2080 \u03b2) (g : \u03b1 \u2192 \u03b2 \u2192 N \u2192* P) :\n  \u21d1(f.prod g) = f.prod (\u03bb i fi, g i fi) :=\nmonoid_hom.coe_prod _ _\n\n@[simp, to_additive]\nlemma monoid_hom.finsupp_prod_apply [has_zero \u03b2] [monoid N] [comm_monoid P]\n  (f : \u03b1 \u2192\u2080 \u03b2) (g : \u03b1 \u2192 \u03b2 \u2192 N \u2192* P) (x : N) :\n  f.prod g x = f.prod (\u03bb i fi, g i fi x) :=\nmonoid_hom.finset_prod_apply _ _ _\n\nnamespace finsupp\n\nsection nat_sub\ninstance nat_sub : has_sub (\u03b1 \u2192\u2080 \u2115) := \u27e8zip_with (\u03bb m n, m - n) (nat.sub_zero 0)\u27e9\n\n@[simp] lemma coe_nat_sub (g\u2081 g\u2082 : \u03b1 \u2192\u2080 \u2115) : \u21d1(g\u2081 - g\u2082) = g\u2081 - g\u2082 := rfl\nlemma nat_sub_apply (g\u2081 g\u2082 : \u03b1 \u2192\u2080 \u2115) (a : \u03b1) : (g\u2081 - g\u2082) a = g\u2081 a - g\u2082 a := rfl\n\n@[simp] lemma single_nat_sub {a : \u03b1} {n\u2081 n\u2082 : \u2115} : single a (n\u2081 - n\u2082) = single a n\u2081 - single a n\u2082 :=\nbegin\n  ext f,\n  by_cases h : (a = f),\n  { rw [h, nat_sub_apply, single_eq_same, single_eq_same, single_eq_same] },\n  rw [nat_sub_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h]\nend\n\n-- These next two lemmas are used in developing\n-- the partial derivative on `mv_polynomial`.\n\nlemma sub_single_one_add {a : \u03b1} {u u' : \u03b1 \u2192\u2080 \u2115} (h : u a \u2260 0) :\n  u - single a 1 + u' = u + u' - single a 1 :=\nbegin\n  ext b,\n  rw [add_apply, nat_sub_apply, nat_sub_apply, add_apply],\n  by_cases h : a = b,\n  { rw [\u2190h, single_eq_same], cases (u a), { contradiction }, { simp }, },\n  { simp [h], }\nend\n\nlemma add_sub_single_one {a : \u03b1} {u u' : \u03b1 \u2192\u2080 \u2115} (h : u' a \u2260 0) :\n  u + (u' - single a 1) = u + u' - single a 1 :=\nbegin\n  ext b,\n  rw [add_apply, nat_sub_apply, nat_sub_apply, add_apply],\n  by_cases h : a = b,\n  { rw [\u2190h, single_eq_same], cases (u' a), { contradiction }, { simp }, },\n  { simp [h], }\nend\n\n@[simp] lemma nat_zero_sub (f : \u03b1 \u2192\u2080 \u2115) : 0 - f = 0 := ext $ \u03bb x, nat.zero_sub _\n\nend nat_sub\n\ninstance [add_comm_monoid M] : add_comm_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_comm := assume \u27e8s, f, _\u27e9 \u27e8t, g, _\u27e9, ext $ assume a, add_comm _ _,\n  .. finsupp.add_monoid }\n\ninstance [add_group G] : has_sub (\u03b1 \u2192\u2080 G) := \u27e8zip_with has_sub.sub (sub_zero _)\u27e9\n\ninstance [add_group G] : add_group (\u03b1 \u2192\u2080 G) :=\n{ neg            := map_range (has_neg.neg) neg_zero,\n  sub            := has_sub.sub,\n  sub_eq_add_neg := \u03bb x y, ext (\u03bb i, sub_eq_add_neg _ _),\n  add_left_neg   := assume \u27e8s, f, _\u27e9, ext $ assume x, add_left_neg _,\n  gsmul := \u03bb n v, v.map_range ((\u2022) n) (gsmul_zero _),\n  gsmul_zero' := \u03bb v, by { ext i, simp },\n  gsmul_succ' := \u03bb n v, by { ext i, simp [nat.succ_eq_one_add, add_gsmul] },\n  gsmul_neg' := \u03bb n v, by { ext i, simp only [nat.succ_eq_add_one, map_range_apply,\n    gsmul_neg_succ_of_nat, int.coe_nat_succ, neg_inj,\n    add_gsmul, add_nsmul, one_gsmul, gsmul_coe_nat, one_nsmul] },\n  .. finsupp.add_monoid }\n\ninstance [add_comm_group G] : add_comm_group (\u03b1 \u2192\u2080 G) :=\n{ add_comm := add_comm, ..finsupp.add_group }\n\nlemma single_multiset_sum [add_comm_monoid M] (s : multiset M) (a : \u03b1) :\n  single a s.sum = (s.map (single a)).sum :=\nmultiset.induction_on s single_zero $ \u03bb a s ih,\nby rw [multiset.sum_cons, single_add, ih, multiset.map_cons, multiset.sum_cons]\n\nlemma single_finset_sum [add_comm_monoid M] (s : finset \u03b9) (f : \u03b9 \u2192 M) (a : \u03b1) :\n  single a (\u2211 b in s, f b) = \u2211 b in s, single a (f b) :=\nbegin\n  transitivity,\n  apply single_multiset_sum,\n  rw [multiset.map_map],\n  refl\nend\n\nlemma single_sum [has_zero M] [add_comm_monoid N] (s : \u03b9 \u2192\u2080 M) (f : \u03b9 \u2192 M \u2192 N) (a : \u03b1) :\n  single a (s.sum f) = s.sum (\u03bbd c, single a (f d c)) :=\nsingle_finset_sum _ _ _\n\n@[to_additive]\nlemma prod_neg_index [add_group G] [comm_monoid M] {g : \u03b1 \u2192\u2080 G} {h : \u03b1 \u2192 G \u2192 M}\n  (h0 : \u2200a, h a 0 = 1) :\n  (-g).prod h = g.prod (\u03bba b, h a (- b)) :=\nprod_map_range_index h0\n\n@[simp] lemma coe_neg [add_group G] (g : \u03b1 \u2192\u2080 G) : \u21d1(-g) = -g := rfl\nlemma neg_apply [add_group G] (g : \u03b1 \u2192\u2080 G) (a : \u03b1) : (- g) a = - g a := rfl\n\n@[simp] lemma coe_sub [add_group G] (g\u2081 g\u2082 : \u03b1 \u2192\u2080 G) : \u21d1(g\u2081 - g\u2082) = g\u2081 - g\u2082 := rfl\nlemma sub_apply [add_group G] (g\u2081 g\u2082 : \u03b1 \u2192\u2080 G) (a : \u03b1) : (g\u2081 - g\u2082) a = g\u2081 a - g\u2082 a := rfl\n\n@[simp] lemma support_neg [add_group G] {f : \u03b1 \u2192\u2080 G} : support (-f) = support f :=\nfinset.subset.antisymm\n  support_map_range\n  (calc support f = support (- (- f)) : congr_arg support (neg_neg _).symm\n     ... \u2286 support (- f) : support_map_range)\n\n@[simp] lemma sum_apply [has_zero M] [add_comm_monoid N]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 \u03b2 \u2192\u2080 N} {a\u2082 : \u03b2} :\n  (f.sum g) a\u2082 = f.sum (\u03bba\u2081 b, g a\u2081 b a\u2082) :=\n(apply_add_hom a\u2082 : (\u03b2 \u2192\u2080 N) \u2192+ _).map_sum _ _\n\nlemma support_sum [decidable_eq \u03b2] [has_zero M] [add_comm_monoid N]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 (\u03b2 \u2192\u2080 N)} :\n  (f.sum g).support \u2286 f.support.bUnion (\u03bba, (g a (f a)).support) :=\nhave \u2200 c, f.sum (\u03bb a b, g a b c) \u2260 0 \u2192 (\u2203 a, f a \u2260 0 \u2227 \u00ac (g a (f a)) c = 0),\n  from assume a\u2081 h,\n  let \u27e8a, ha, ne\u27e9 := finset.exists_ne_zero_of_sum_ne_zero h in\n  \u27e8a, mem_support_iff.mp ha, ne\u27e9,\nby simpa only [finset.subset_iff, mem_support_iff, finset.mem_bUnion, sum_apply, exists_prop]\n\n@[simp] lemma sum_zero [has_zero M] [add_comm_monoid N] {f : \u03b1 \u2192\u2080 M} :\n  f.sum (\u03bba b, (0 : N)) = 0 :=\nfinset.sum_const_zero\n\n@[simp, to_additive]\nlemma prod_mul  [has_zero M] [comm_monoid N] {f : \u03b1 \u2192\u2080 M} {h\u2081 h\u2082 : \u03b1 \u2192 M \u2192 N} :\n  f.prod (\u03bba b, h\u2081 a b * h\u2082 a b) = f.prod h\u2081 * f.prod h\u2082 :=\nfinset.prod_mul_distrib\n\n@[simp, to_additive]\nlemma prod_inv [has_zero M] [comm_group G] {f : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 G} : f.prod (\u03bba b, (h a b)\u207b\u00b9) = (f.prod h)\u207b\u00b9 :=\n(((monoid_hom.id G)\u207b\u00b9).map_prod _ _).symm\n\n@[simp] lemma sum_sub [has_zero M] [add_comm_group G] {f : \u03b1 \u2192\u2080 M}\n  {h\u2081 h\u2082 : \u03b1 \u2192 M \u2192 G} :\n  f.sum (\u03bba b, h\u2081 a b - h\u2082 a b) = f.sum h\u2081 - f.sum h\u2082 :=\nfinset.sum_sub_distrib\n\n@[to_additive]\nlemma prod_add_index [add_comm_monoid M] [comm_monoid N] {f g : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  (f + g).prod h = f.prod h * g.prod h :=\nhave hf : f.prod h = \u220f a in f.support \u222a g.support, h a (f a),\n  from f.prod_of_support_subset (subset_union_left _ _) _ $ \u03bb a ha, h_zero a,\nhave hg : g.prod h = \u220f a in f.support \u222a g.support, h a (g a),\n  from g.prod_of_support_subset (subset_union_right _ _) _ $ \u03bb a ha, h_zero a,\nhave hfg : (f + g).prod h = \u220f a in f.support \u222a g.support, h a ((f + g) a),\n  from (f + g).prod_of_support_subset support_add _ $ \u03bb a ha, h_zero a,\nby simp only [*, add_apply, prod_mul_distrib]\n\n@[simp]\nlemma sum_add_index' [add_comm_monoid M] [add_comm_monoid N] {f g : \u03b1 \u2192\u2080 M} (h : \u03b1 \u2192 M \u2192+ N) :\n  (f + g).sum (\u03bb x, h x) = f.sum (\u03bb x, h x) + g.sum (\u03bb x, h x) :=\nsum_add_index (\u03bb a, (h a).map_zero) (\u03bb a, (h a).map_add)\n\n@[simp]\nlemma prod_add_index' [add_comm_monoid M] [comm_monoid N] {f g : \u03b1 \u2192\u2080 M}\n  (h : \u03b1 \u2192 multiplicative M \u2192* N) :\n  (f + g).prod (\u03bb a b, h a (multiplicative.of_add b)) =\n    f.prod (\u03bb a b, h a (multiplicative.of_add b)) * g.prod (\u03bb a b, h a (multiplicative.of_add b)) :=\nprod_add_index (\u03bb a, (h a).map_one) (\u03bb a, (h a).map_mul)\n\n\/-- The canonical isomorphism between families of additive monoid homomorphisms `\u03b1 \u2192 (M \u2192+ N)`\nand monoid homomorphisms `(\u03b1 \u2192\u2080 M) \u2192+ N`. -\/\ndef lift_add_hom [add_comm_monoid M] [add_comm_monoid N] : (\u03b1 \u2192 M \u2192+ N) \u2243+ ((\u03b1 \u2192\u2080 M) \u2192+ N) :=\n{ to_fun := \u03bb F,\n  { to_fun := \u03bb f, f.sum (\u03bb x, F x),\n    map_zero' := finset.sum_empty,\n    map_add' := \u03bb _ _, sum_add_index (\u03bb x, (F x).map_zero) (\u03bb x, (F x).map_add) },\n  inv_fun := \u03bb F x, F.comp $ single_add_hom x,\n  left_inv := \u03bb F, by { ext, simp },\n  right_inv := \u03bb F, by { ext, simp },\n  map_add' := \u03bb F G, by { ext, simp } }\n\n@[simp] lemma lift_add_hom_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : \u03b1 \u2192 M \u2192+ N) (f : \u03b1 \u2192\u2080 M) :\n  lift_add_hom F f = f.sum (\u03bb x, F x) :=\nrfl\n\n@[simp] lemma lift_add_hom_symm_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : (\u03b1 \u2192\u2080 M) \u2192+ N) (x : \u03b1) :\n  lift_add_hom.symm F x = F.comp (single_add_hom x) :=\nrfl\n\nlemma lift_add_hom_symm_apply_apply [add_comm_monoid M] [add_comm_monoid N]\n  (F : (\u03b1 \u2192\u2080 M) \u2192+ N) (x : \u03b1) (y : M) :\n  lift_add_hom.symm F x y = F (single x y) :=\nrfl\n\n@[simp] lemma lift_add_hom_single_add_hom [add_comm_monoid M] :\n  lift_add_hom (single_add_hom : \u03b1 \u2192 M \u2192+ \u03b1 \u2192\u2080 M) = add_monoid_hom.id _ :=\nlift_add_hom.to_equiv.apply_eq_iff_eq_symm_apply.2 rfl\n\n@[simp] lemma sum_single [add_comm_monoid M] (f : \u03b1 \u2192\u2080 M) :\n  f.sum single = f :=\nadd_monoid_hom.congr_fun lift_add_hom_single_add_hom f\n\n@[simp] lemma lift_add_hom_apply_single [add_comm_monoid M] [add_comm_monoid N]\n  (f : \u03b1 \u2192 M \u2192+ N) (a : \u03b1) (b : M) :\n  lift_add_hom f (single a b) = f a b :=\nsum_single_index (f a).map_zero\n\n@[simp] lemma lift_add_hom_comp_single [add_comm_monoid M] [add_comm_monoid N] (f : \u03b1 \u2192 M \u2192+ N)\n  (a : \u03b1) :\n  (lift_add_hom f).comp (single_add_hom a) = f a :=\nadd_monoid_hom.ext $ \u03bb b, lift_add_hom_apply_single f a b\n\nlemma comp_lift_add_hom [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n  (g : N \u2192+ P) (f : \u03b1 \u2192 M \u2192+ N) :\n  g.comp (lift_add_hom f) = lift_add_hom (\u03bb a, g.comp (f a)) :=\nlift_add_hom.symm_apply_eq.1 $ funext $ \u03bb a,\n  by rw [lift_add_hom_symm_apply, add_monoid_hom.comp_assoc, lift_add_hom_comp_single]\n\nlemma sum_sub_index [add_comm_group \u03b2] [add_comm_group \u03b3] {f g : \u03b1 \u2192\u2080 \u03b2}\n  {h : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (h_sub : \u2200a b\u2081 b\u2082, h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082) :\n  (f - g).sum h = f.sum h - g.sum h :=\n(lift_add_hom (\u03bb a, add_monoid_hom.of_map_sub (h a) (h_sub a))).map_sub f g\n\n@[to_additive]\nlemma prod_emb_domain [has_zero M] [comm_monoid N] {v : \u03b1 \u2192\u2080 M} {f : \u03b1 \u21aa \u03b2} {g : \u03b2 \u2192 M \u2192 N} :\n  (v.emb_domain f).prod g = v.prod (\u03bb a b, g (f a) b) :=\nbegin\n  rw [prod, prod, support_emb_domain, finset.prod_map],\n  simp_rw emb_domain_apply,\nend\n\n@[to_additive]\nlemma prod_finset_sum_index [add_comm_monoid M] [comm_monoid N]\n  {s : finset \u03b9} {g : \u03b9 \u2192 \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  \u220f i in s, (g i).prod h = (\u2211 i in s, g i).prod h :=\nfinset.induction_on s rfl $ \u03bb a s has ih,\nby rw [prod_insert has, ih, sum_insert has, prod_add_index h_zero h_add]\n\n@[to_additive]\nlemma prod_sum_index\n  [add_comm_monoid M] [add_comm_monoid N] [comm_monoid P]\n  {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 \u03b2 \u2192\u2080 N}\n  {h : \u03b2 \u2192 N \u2192 P} (h_zero : \u2200a, h a 0 = 1) (h_add : \u2200a b\u2081 b\u2082, h a (b\u2081 + b\u2082) = h a b\u2081 * h a b\u2082) :\n  (f.sum g).prod h = f.prod (\u03bba b, (g a b).prod h) :=\n(prod_finset_sum_index h_zero h_add).symm\n\nlemma multiset_sum_sum_index\n  [add_comm_monoid M] [add_comm_monoid N]\n  (f : multiset (\u03b1 \u2192\u2080 M)) (h : \u03b1 \u2192 M \u2192 N)\n  (h\u2080 : \u2200a, h a 0 = 0) (h\u2081 : \u2200 (a : \u03b1) (b\u2081 b\u2082 : M), h a (b\u2081 + b\u2082) = h a b\u2081 + h a b\u2082) :\n  (f.sum.sum h) = (f.map $ \u03bbg:\u03b1 \u2192\u2080 M, g.sum h).sum :=\nmultiset.induction_on f rfl $ assume a s ih,\nby rw [multiset.sum_cons, multiset.map_cons, multiset.sum_cons, sum_add_index h\u2080 h\u2081, ih]\n\nlemma support_sum_eq_bUnion {\u03b1 : Type*} {\u03b9 : Type*} {M : Type*} [add_comm_monoid M]\n  {g : \u03b9 \u2192 \u03b1 \u2192\u2080 M} (s : finset \u03b9) (h : \u2200 i\u2081 i\u2082, i\u2081 \u2260 i\u2082 \u2192 disjoint (g i\u2081).support (g i\u2082).support) :\n  (\u2211 i in s, g i).support = s.bUnion (\u03bb i, (g i).support) :=\nbegin\n  apply finset.induction_on s,\n  { simp },\n  { intros i s hi,\n    simp only [hi, sum_insert, not_false_iff, bUnion_insert],\n    intro hs,\n    rw [finsupp.support_add_eq, hs],\n    rw [hs],\n    intros x hx,\n    simp only [mem_bUnion, exists_prop, inf_eq_inter, ne.def, mem_inter] at hx,\n    obtain \u27e8hxi, j, hj, hxj\u27e9 := hx,\n    have hn : i \u2260 j := \u03bb H, hi (H.symm \u25b8 hj),\n    apply h _ _ hn,\n    simp [hxi, hxj] }\nend\n\nlemma multiset_map_sum [has_zero M] {f : \u03b1 \u2192\u2080 M} {m : \u03b2 \u2192 \u03b3} {h : \u03b1 \u2192 M \u2192 multiset \u03b2} :\n  multiset.map m (f.sum h) = f.sum (\u03bba b, (h a b).map m) :=\n(f.support.sum_hom _).symm\n\nlemma multiset_sum_sum [has_zero M] [add_comm_monoid N] {f : \u03b1 \u2192\u2080 M} {h : \u03b1 \u2192 M \u2192 multiset N} :\n  multiset.sum (f.sum h) = f.sum (\u03bba b, multiset.sum (h a b)) :=\n(f.support.sum_hom multiset.sum).symm\n\nsection map_range\n\nsection zero_hom\nvariables [has_zero M] [has_zero N] [has_zero P]\n\n\/-- Composition with a fixed zero-preserving homomorphism is itself an zero-preserving homomorphism\non functions. -\/\n@[simps]\ndef map_range.zero_hom (f : zero_hom M N) : zero_hom (\u03b1 \u2192\u2080 M) (\u03b1 \u2192\u2080 N) :=\n{ to_fun := (map_range f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 (\u03b1 \u2192\u2080 N)),\n  map_zero' := map_range_zero }\n\n@[simp]\nlemma map_range.zero_hom_id :\n  map_range.zero_hom (zero_hom.id M) = zero_hom.id (\u03b1 \u2192\u2080 M) := zero_hom.ext map_range_id\n\nlemma map_range.zero_hom_comp (f : zero_hom N P) (f\u2082 : zero_hom M N) :\n  (map_range.zero_hom (f.comp f\u2082) : zero_hom (\u03b1 \u2192\u2080 _) _) =\n    (map_range.zero_hom f).comp (map_range.zero_hom f\u2082) :=\nzero_hom.ext $ map_range_comp _ _ _ _ _\n\nend zero_hom\n\nsection add_monoid_hom\nvariables [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n\n\/--\nComposition with a fixed additive homomorphism is itself an additive homomorphism on functions.\n-\/\n@[simps]\ndef map_range.add_monoid_hom (f : M \u2192+ N) : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b1 \u2192\u2080 N) :=\n{ to_fun := (map_range f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 (\u03b1 \u2192\u2080 N)),\n  map_zero' := map_range_zero,\n  map_add' := \u03bb a b, map_range_add f.map_add _ _ }\n\n@[simp]\nlemma map_range.add_monoid_hom_id :\n  map_range.add_monoid_hom (add_monoid_hom.id M) = add_monoid_hom.id (\u03b1 \u2192\u2080 M) :=\nadd_monoid_hom.ext map_range_id\n\nlemma map_range.add_monoid_hom_comp (f : N \u2192+ P) (f\u2082 : M \u2192+ N) :\n  (map_range.add_monoid_hom (f.comp f\u2082) : (\u03b1 \u2192\u2080 _) \u2192+ _) =\n    (map_range.add_monoid_hom f).comp (map_range.add_monoid_hom f\u2082) :=\nadd_monoid_hom.ext $ map_range_comp _ _ _ _ _\n\nlemma map_range_multiset_sum (f : M \u2192+ N) (m : multiset (\u03b1 \u2192\u2080 M)) :\n  map_range f f.map_zero m.sum = (m.map $ \u03bbx, map_range f f.map_zero x).sum :=\n(map_range.add_monoid_hom f : (\u03b1 \u2192\u2080 _) \u2192+ _).map_multiset_sum _\n\nlemma map_range_finset_sum (f : M \u2192+ N) (s : finset \u03b9) (g : \u03b9 \u2192 (\u03b1 \u2192\u2080 M))  :\n  map_range f f.map_zero (\u2211 x in s, g x) = \u2211 x in s, map_range f f.map_zero (g x) :=\n(map_range.add_monoid_hom f : (\u03b1 \u2192\u2080 _) \u2192+ _).map_sum _ _\n\n\n\/-- `finsupp.map_range.add_monoid_hom` as an equiv. -\/\n@[simps apply]\ndef map_range.add_equiv (f : M \u2243+ N) : (\u03b1 \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 N) :=\n{ to_fun := (map_range f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 (\u03b1 \u2192\u2080 N)),\n  inv_fun := (map_range f.symm f.symm.map_zero : (\u03b1 \u2192\u2080 N) \u2192 (\u03b1 \u2192\u2080 M)),\n  left_inv := \u03bb x, begin\n    rw \u2190map_range_comp _ _ _ _; simp_rw add_equiv.symm_comp_self,\n    { exact map_range_id _ },\n    { refl },\n  end,\n  right_inv := \u03bb x, begin\n    rw \u2190map_range_comp _ _ _ _; simp_rw add_equiv.self_comp_symm,\n    { exact map_range_id _ },\n    { refl },\n  end,\n  ..(map_range.add_monoid_hom f.to_add_monoid_hom) }\n\n@[simp]\nlemma map_range.add_equiv_refl :\n  map_range.add_equiv (add_equiv.refl M) = add_equiv.refl (\u03b1 \u2192\u2080 M) :=\nadd_equiv.ext map_range_id\n\nlemma map_range.add_equiv_trans (f : M \u2243+ N) (f\u2082 : N \u2243+ P) :\n  (map_range.add_equiv (f.trans f\u2082) : (\u03b1 \u2192\u2080 _) \u2243+ _) =\n    (map_range.add_equiv f).trans (map_range.add_equiv f\u2082) :=\nadd_equiv.ext $ map_range_comp _ _ _ _ _\n\nlemma map_range.add_equiv_symm (f : M \u2243+ N) :\n  ((map_range.add_equiv f).symm : (\u03b1 \u2192\u2080 _) \u2243+ _) = map_range.add_equiv f.symm :=\nadd_equiv.ext $ \u03bb x, rfl\n\nend add_monoid_hom\n\nend map_range\n\n\/-! ### Declarations about `map_domain` -\/\n\nsection map_domain\nvariables [add_comm_monoid M] {v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M}\n\n\/-- Given `f : \u03b1 \u2192 \u03b2` and `v : \u03b1 \u2192\u2080 M`, `map_domain f v : \u03b2 \u2192\u2080 M`\n  is the finitely supported function whose value at `a : \u03b2` is the sum\n  of `v x` over all `x` such that `f x = a`. -\/\ndef map_domain (f : \u03b1 \u2192 \u03b2) (v : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\nv.sum $ \u03bba, single (f a)\n\nlemma map_domain_apply {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) (x : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  map_domain f x (f a) = x a :=\nbegin\n  rw [map_domain, sum_apply, sum, finset.sum_eq_single a, single_eq_same],\n  { assume b _ hba, exact single_eq_of_ne (hf.ne hba) },\n  { assume h, rw [not_mem_support_iff.1 h, single_zero, zero_apply] }\nend\n\nlemma map_domain_notin_range {f : \u03b1 \u2192 \u03b2} (x : \u03b1 \u2192\u2080 M) (a : \u03b2) (h : a \u2209 set.range f) :\n  map_domain f x a = 0 :=\nbegin\n  rw [map_domain, sum_apply, sum],\n  exact finset.sum_eq_zero\n    (assume a' h', single_eq_of_ne $ assume eq, h $ eq \u25b8 set.mem_range_self _)\nend\n\n@[simp]\nlemma map_domain_id : map_domain id v = v :=\nsum_single _\n\nlemma map_domain_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} :\n  map_domain (g \u2218 f) v = map_domain g (map_domain f v) :=\nbegin\n  refine ((sum_sum_index _ _).trans _).symm,\n  { intros, exact single_zero },\n  { intros, exact single_add },\n  refine sum_congr rfl (\u03bb _ _, sum_single_index _),\n  { exact single_zero }\nend\n\n@[simp]\nlemma map_domain_single {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : M} : map_domain f (single a b) = single (f a) b :=\nsum_single_index single_zero\n\n@[simp] lemma map_domain_zero {f : \u03b1 \u2192 \u03b2} : map_domain f (0 : \u03b1 \u2192\u2080 M) = (0 : \u03b2 \u2192\u2080 M) :=\nsum_zero_index\n\nlemma map_domain_congr {f g : \u03b1 \u2192 \u03b2} (h : \u2200x\u2208v.support, f x = g x) :\n  v.map_domain f = v.map_domain g :=\nfinset.sum_congr rfl $ \u03bb _ H, by simp only [h _ H]\n\nlemma map_domain_add {f : \u03b1 \u2192 \u03b2} : map_domain f (v\u2081 + v\u2082) = map_domain f v\u2081 + map_domain f v\u2082 :=\nsum_add_index (\u03bb _, single_zero) (\u03bb _ _ _, single_add)\n\n@[simp] lemma map_domain_equiv_apply {f : \u03b1 \u2243 \u03b2} (x : \u03b1 \u2192\u2080 M) (a : \u03b2) :\n  map_domain f x a = x (f.symm a) :=\nbegin\n  conv_lhs { rw \u2190f.apply_symm_apply a },\n  exact map_domain_apply f.injective _ _,\nend\n\n\/-- `finsupp.map_domain` is an `add_monoid_hom`. -\/\n@[simps]\ndef map_domain.add_monoid_hom (f : \u03b1 \u2192 \u03b2) : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b2 \u2192\u2080 M) :=\n{ to_fun := map_domain f,\n  map_zero' := map_domain_zero,\n  map_add' := \u03bb _ _, map_domain_add}\n\n@[simp]\nlemma map_domain.add_monoid_hom_id : map_domain.add_monoid_hom id = add_monoid_hom.id (\u03b1 \u2192\u2080 M) :=\nadd_monoid_hom.ext $ \u03bb _, map_domain_id\n\nlemma map_domain.add_monoid_hom_comp (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n  (map_domain.add_monoid_hom (f \u2218 g) : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b3 \u2192\u2080 M)) =\n    (map_domain.add_monoid_hom f).comp (map_domain.add_monoid_hom g) :=\nadd_monoid_hom.ext $ \u03bb _, map_domain_comp\n\nlemma map_domain_finset_sum {f : \u03b1 \u2192 \u03b2} {s : finset \u03b9} {v : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n  map_domain f (\u2211 i in s, v i) = \u2211 i in s, map_domain f (v i) :=\n(map_domain.add_monoid_hom f : (\u03b1 \u2192\u2080 M) \u2192+ \u03b2 \u2192\u2080 M).map_sum _ _\n\nlemma map_domain_sum [has_zero N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 N} {v : \u03b1 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n  map_domain f (s.sum v) = s.sum (\u03bba b, map_domain f (v a b)) :=\n(map_domain.add_monoid_hom f : (\u03b1 \u2192\u2080 M) \u2192+ \u03b2 \u2192\u2080 M).map_finsupp_sum _ _\n\nlemma map_domain_support [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M} :\n  (s.map_domain f).support \u2286 s.support.image f :=\nfinset.subset.trans support_sum $\n  finset.subset.trans (finset.bUnion_mono $ assume a ha, support_single_subset) $\n  by rw [finset.bUnion_singleton]; exact subset.refl _\n\n@[to_additive]\nlemma prod_map_domain_index [comm_monoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M}\n  {h : \u03b2 \u2192 M \u2192 N} (h_zero : \u2200b, h b 0 = 1) (h_add : \u2200b m\u2081 m\u2082, h b (m\u2081 + m\u2082) = h b m\u2081 * h b m\u2082) :\n  (map_domain f s).prod h = s.prod (\u03bba m, h (f a) m) :=\n(prod_sum_index h_zero h_add).trans $ prod_congr rfl $ \u03bb _ _, prod_single_index (h_zero _)\n\n\/--\nA version of `sum_map_domain_index` that takes a bundled `add_monoid_hom`,\nrather than separate linearity hypotheses.\n-\/\n-- Note that in `prod_map_domain_index`, `M` is still an additive monoid,\n-- so there is no analogous version in terms of `monoid_hom`.\n@[simp]\nlemma sum_map_domain_index_add_monoid_hom [add_comm_monoid N] {f : \u03b1 \u2192 \u03b2}\n  {s : \u03b1 \u2192\u2080 M} (h : \u03b2 \u2192 M \u2192+ N) :\n  (map_domain f s).sum (\u03bb b m, h b m) = s.sum (\u03bb a m, h (f a) m) :=\n@sum_map_domain_index _ _ _ _ _ _ _ _\n  (\u03bb b m, h b m)\n  (\u03bb b, (h b).map_zero)\n  (\u03bb b m\u2081 m\u2082, (h b).map_add _ _)\n\nlemma emb_domain_eq_map_domain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) :\n  emb_domain f v = map_domain f v :=\nbegin\n  ext a,\n  by_cases a \u2208 set.range f,\n  { rcases h with \u27e8a, rfl\u27e9,\n    rw [map_domain_apply f.injective, emb_domain_apply] },\n  { rw [map_domain_notin_range, emb_domain_notin_range]; assumption }\nend\n\n@[to_additive]\nlemma prod_map_domain_index_inj [comm_monoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M}\n  {h : \u03b2 \u2192 M \u2192 N} (hf : function.injective f) :\n  (s.map_domain f).prod h = s.prod (\u03bba b, h (f a) b) :=\nby rw [\u2190function.embedding.coe_fn_mk f hf, \u2190emb_domain_eq_map_domain, prod_emb_domain]\n\nlemma map_domain_injective {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  function.injective (map_domain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)) :=\nbegin\n  assume v\u2081 v\u2082 eq, ext a,\n  have : map_domain f v\u2081 (f a) = map_domain f v\u2082 (f a), { rw eq },\n  rwa [map_domain_apply hf, map_domain_apply hf] at this,\nend\n\nlemma map_domain.add_monoid_hom_comp_map_range [add_comm_monoid N] (f : \u03b1 \u2192 \u03b2) (g : M \u2192+ N) :\n  (map_domain.add_monoid_hom f).comp (map_range.add_monoid_hom g) =\n    (map_range.add_monoid_hom g).comp (map_domain.add_monoid_hom f) :=\nby { ext, simp }\n\n\/-- When `g` preserves addition, `map_range` and `map_domain` commute. -\/\nlemma map_domain_map_range [add_comm_monoid N] (f : \u03b1 \u2192 \u03b2) (v : \u03b1 \u2192\u2080 M) (g : M \u2192 N)\n  (h0 : g 0 = 0) (hadd : \u2200 x y, g (x + y) = g x + g y) :\n  map_domain f (map_range g h0 v) = map_range g h0 (map_domain f v) :=\nlet g' : M \u2192+ N := { to_fun := g, map_zero' := h0, map_add' := hadd} in\nadd_monoid_hom.congr_fun (map_domain.add_monoid_hom_comp_map_range f g') v\n\nend map_domain\n\n\/-! ### Declarations about `comap_domain` -\/\n\nsection comap_domain\n\n\/-- Given `f : \u03b1 \u2192 \u03b2`, `l : \u03b2 \u2192\u2080 M` and a proof `hf` that `f` is injective on\nthe preimage of `l.support`, `comap_domain f l hf` is the finitely supported function\nfrom `\u03b1` to `M` given by composing `l` with `f`. -\/\ndef comap_domain [has_zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : set.inj_on f (f \u207b\u00b9' \u2191l.support)) :\n  \u03b1 \u2192\u2080 M :=\n{ support := l.support.preimage f hf,\n  to_fun := (\u03bb a, l (f a)),\n  mem_support_to_fun :=\n    begin\n      intros a,\n      simp only [finset.mem_def.symm, finset.mem_preimage],\n      exact l.mem_support_to_fun (f a),\n    end }\n\n@[simp]\nlemma comap_domain_apply [has_zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M)\n  (hf : set.inj_on f (f \u207b\u00b9' \u2191l.support)) (a : \u03b1) :\n  comap_domain f l hf a = l (f a) :=\nrfl\n\nlemma sum_comap_domain [has_zero M] [add_comm_monoid N]\n  (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (g : \u03b2 \u2192 M \u2192 N)\n  (hf : set.bij_on f (f \u207b\u00b9' \u2191l.support) \u2191l.support) :\n  (comap_domain f l hf.inj_on).sum (g \u2218 f) = l.sum g :=\nbegin\n  simp only [sum, comap_domain_apply, (\u2218)],\n  simp [comap_domain, finset.sum_preimage_of_bij f _ _ (\u03bb x, g x (l x))],\nend\n\nlemma eq_zero_of_comap_domain_eq_zero [add_comm_monoid M]\n  (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : set.bij_on f (f \u207b\u00b9' \u2191l.support) \u2191l.support) :\n   comap_domain f l hf.inj_on = 0 \u2192 l = 0 :=\nbegin\n  rw [\u2190 support_eq_empty, \u2190 support_eq_empty, comap_domain],\n  simp only [finset.ext_iff, finset.not_mem_empty, iff_false, mem_preimage],\n  assume h a ha,\n  cases hf.2.2 ha with b hb,\n  exact h b (hb.2.symm \u25b8 ha)\nend\n\nlemma map_domain_comap_domain [add_comm_monoid M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M)\n  (hf : function.injective f) (hl : \u2191l.support \u2286 set.range f):\n  map_domain f (comap_domain f l (hf.inj_on _)) = l :=\nbegin\n  ext a,\n  by_cases h_cases: a \u2208 set.range f,\n  { rcases set.mem_range.1 h_cases with \u27e8b, hb\u27e9,\n    rw [hb.symm, map_domain_apply hf, comap_domain_apply] },\n  { rw map_domain_notin_range _ _ h_cases,\n    by_contra h_contr,\n    apply h_cases (hl $ finset.mem_coe.2 $ mem_support_iff.2 $ \u03bb h, h_contr h.symm) }\nend\n\nend comap_domain\n\n\/-! ### Declarations about `equiv_congr_left` -\/\n\nsection equiv_congr_left\n\nvariable [has_zero M]\n\n\/-- Given `f : \u03b1 \u2243 \u03b2`, we can map `l : \u03b1 \u2192\u2080 M` to  `equiv_map_domain f l : \u03b2 \u2192\u2080 M` (computably)\nby mapping the support forwards and the function backwards. -\/\ndef equiv_map_domain (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\n{ support := l.support.map f.to_embedding,\n  to_fun := \u03bb a, l (f.symm a),\n  mem_support_to_fun := \u03bb a, by simp only [finset.mem_map_equiv, mem_support_to_fun]; refl }\n\n@[simp] lemma equiv_map_domain_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) (b : \u03b2) :\n  equiv_map_domain f l b = l (f.symm b) := rfl\n\nlemma equiv_map_domain_symm_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b2 \u2192\u2080 M) (a : \u03b1) :\n  equiv_map_domain f.symm l a = l (f a) := rfl\n\n@[simp] lemma equiv_map_domain_refl (l : \u03b1 \u2192\u2080 M) : equiv_map_domain (equiv.refl _) l = l :=\nby ext x; refl\n\nlemma equiv_map_domain_refl' : equiv_map_domain (equiv.refl _) = @id (\u03b1 \u2192\u2080 M) :=\nby ext x; refl\n\nlemma equiv_map_domain_trans (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) (l : \u03b1 \u2192\u2080 M) :\n  equiv_map_domain (f.trans g) l = equiv_map_domain g (equiv_map_domain f l) := by ext x; refl\n\nlemma equiv_map_domain_trans' (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) :\n  @equiv_map_domain _ _ M _ (f.trans g) = equiv_map_domain g \u2218 equiv_map_domain f := by ext x; refl\n\n@[simp] lemma equiv_map_domain_single (f : \u03b1 \u2243 \u03b2) (a : \u03b1) (b : M) :\n  equiv_map_domain f (single a b) = single (f a) b :=\nby ext x; simp only [single_apply, equiv.apply_eq_iff_eq_symm_apply, equiv_map_domain_apply]; congr\n\n@[simp] lemma equiv_map_domain_zero {f : \u03b1 \u2243 \u03b2} : equiv_map_domain f (0 : \u03b1 \u2192\u2080 M) = (0 : \u03b2 \u2192\u2080 M) :=\nby ext x; simp only [equiv_map_domain_apply, coe_zero, pi.zero_apply]\n\nlemma equiv_map_domain_eq_map_domain {M} [add_comm_monoid M] (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) :\n  equiv_map_domain f l = map_domain f l := by ext x; simp [map_domain_equiv_apply]\n\n\/-- Given `f : \u03b1 \u2243 \u03b2`, the finitely supported function spaces are also in bijection:\n`(\u03b1 \u2192\u2080 M) \u2243 (\u03b2 \u2192\u2080 M)`.\n\nThis is the finitely-supported version of `equiv.Pi_congr_left`. -\/\ndef equiv_congr_left (f : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192\u2080 M) \u2243 (\u03b2 \u2192\u2080 M) :=\nby refine \u27e8equiv_map_domain f, equiv_map_domain f.symm, \u03bb f, _, \u03bb f, _\u27e9;\n  ext x; simp only [equiv_map_domain_apply, equiv.symm_symm,\n    equiv.symm_apply_apply, equiv.apply_symm_apply]\n\n@[simp] lemma equiv_congr_left_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) :\n  equiv_congr_left f l = equiv_map_domain f l := rfl\n\n@[simp] lemma equiv_congr_left_symm (f : \u03b1 \u2243 \u03b2) :\n  (@equiv_congr_left _ _ M _ f).symm = equiv_congr_left f.symm := rfl\n\nend equiv_congr_left\n\n\/-! ### Declarations about `filter` -\/\n\nsection filter\nsection has_zero\nvariables [has_zero M] (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M)\n\n\/-- `filter p f` is the function which is `f a` if `p a` is true and 0 otherwise. -\/\ndef filter (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M :=\n{ to_fun := \u03bb a, if p a then f a else 0,\n  support := f.support.filter (\u03bb a, p a),\n  mem_support_to_fun := \u03bb a, by split_ifs; { simp only [h, mem_filter, mem_support_iff], tauto } }\n\nlemma filter_apply (a : \u03b1) [D : decidable (p a)] : f.filter p a = if p a then f a else 0 :=\nby rw subsingleton.elim D; refl\n\nlemma filter_eq_indicator : \u21d1(f.filter p) = set.indicator {x | p x} f := rfl\n\n@[simp] lemma filter_apply_pos {a : \u03b1} (h : p a) : f.filter p a = f a :=\nif_pos h\n\n@[simp] lemma filter_apply_neg {a : \u03b1} (h : \u00ac p a) : f.filter p a = 0 :=\nif_neg h\n\n@[simp] lemma support_filter [D : decidable_pred p] : (f.filter p).support = f.support.filter p :=\nby rw subsingleton.elim D; refl\n\nlemma filter_zero : (0 : \u03b1 \u2192\u2080 M).filter p = 0 :=\nby rw [\u2190 support_eq_empty, support_filter, support_zero, finset.filter_empty]\n\n@[simp] lemma filter_single_of_pos\n  {a : \u03b1} {b : M} (h : p a) : (single a b).filter p = single a b :=\ncoe_fn_injective $ by simp [filter_eq_indicator, set.subset_def, mem_support_single, h]\n\n@[simp] lemma filter_single_of_neg\n  {a : \u03b1} {b : M} (h : \u00ac p a) : (single a b).filter p = 0 :=\next $ by simp [filter_eq_indicator, single_apply_eq_zero, @imp.swap (p _), h]\n\nend has_zero\n\nlemma filter_pos_add_filter_neg [add_zero_class M] (f : \u03b1 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n  f.filter p + f.filter (\u03bba, \u00ac p a) = f :=\ncoe_fn_injective $ set.indicator_self_add_compl {x | p x} f\n\nend filter\n\n\/-! ### Declarations about `frange` -\/\n\nsection frange\nvariables [has_zero M]\n\n\/-- `frange f` is the image of `f` on the support of `f`. -\/\ndef frange (f : \u03b1 \u2192\u2080 M) : finset M := finset.image f f.support\n\ntheorem mem_frange {f : \u03b1 \u2192\u2080 M} {y : M} :\n  y \u2208 f.frange \u2194 y \u2260 0 \u2227 \u2203 x, f x = y :=\nfinset.mem_image.trans\n\u27e8\u03bb \u27e8x, hx1, hx2\u27e9, \u27e8hx2 \u25b8 mem_support_iff.1 hx1, x, hx2\u27e9,\n\u03bb \u27e8hy, x, hx\u27e9, \u27e8x, mem_support_iff.2 (hx.symm \u25b8 hy), hx\u27e9\u27e9\n\ntheorem zero_not_mem_frange {f : \u03b1 \u2192\u2080 M} : (0:M) \u2209 f.frange :=\n\u03bb H, (mem_frange.1 H).1 rfl\n\ntheorem frange_single {x : \u03b1} {y : M} : frange (single x y) \u2286 {y} :=\n\u03bb r hr, let \u27e8t, ht1, ht2\u27e9 := mem_frange.1 hr in ht2 \u25b8\n  (by rw single_apply at ht2 \u22a2; split_ifs at ht2 \u22a2; [exact finset.mem_singleton_self _, cc])\n\nend frange\n\n\/-! ### Declarations about `subtype_domain` -\/\n\nsection subtype_domain\n\n\nsection zero\n\nvariables [has_zero M] {p : \u03b1 \u2192 Prop}\n\n\/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -\/\ndef subtype_domain (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : (subtype p \u2192\u2080 M) :=\n\u27e8f.support.subtype p, f \u2218 coe, \u03bb a, by simp only [mem_subtype, mem_support_iff]\u27e9\n\n@[simp] lemma support_subtype_domain [D : decidable_pred p] {f : \u03b1 \u2192\u2080 M} :\n  (subtype_domain p f).support = f.support.subtype p :=\nby rw subsingleton.elim D; refl\n\n@[simp] lemma subtype_domain_apply {a : subtype p} {v : \u03b1 \u2192\u2080 M} :\n  (subtype_domain p v) a = v (a.val) :=\nrfl\n\n@[simp] lemma subtype_domain_zero : subtype_domain p (0 : \u03b1 \u2192\u2080 M) = 0 :=\nrfl\n\nlemma subtype_domain_eq_zero_iff' {f : \u03b1 \u2192\u2080 M} :\n  f.subtype_domain p = 0 \u2194 \u2200 x, p x \u2192 f x = 0 :=\nby simp_rw [\u2190 support_eq_empty, support_subtype_domain, subtype_eq_empty, not_mem_support_iff]\n\nlemma subtype_domain_eq_zero_iff {f : \u03b1 \u2192\u2080 M} (hf : \u2200 x \u2208 f.support , p x) :\n  f.subtype_domain p = 0 \u2194 f = 0 :=\nsubtype_domain_eq_zero_iff'.trans \u27e8\u03bb H, ext $ \u03bb x,\n  if hx : p x then H x hx else not_mem_support_iff.1 $ mt (hf x) hx, \u03bb H x _, by simp [H]\u27e9\n\n@[to_additive]\nlemma prod_subtype_domain_index [comm_monoid N] {v : \u03b1 \u2192\u2080 M}\n  {h : \u03b1 \u2192 M \u2192 N} (hp : \u2200x\u2208v.support, p x) :\n  (v.subtype_domain p).prod (\u03bba b, h a b) = v.prod h :=\nprod_bij (\u03bbp _, p.val)\n  (\u03bb _, mem_subtype.1)\n  (\u03bb _ _, rfl)\n  (\u03bb _ _ _ _, subtype.eq)\n  (\u03bb b hb, \u27e8\u27e8b, hp b hb\u27e9, mem_subtype.2 hb, rfl\u27e9)\n\nend zero\n\nsection add_zero_class\nvariables [add_zero_class M] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 M}\n\n@[simp] lemma subtype_domain_add {v v' : \u03b1 \u2192\u2080 M} :\n  (v + v').subtype_domain p = v.subtype_domain p + v'.subtype_domain p :=\next $ \u03bb _, rfl\n\ninstance subtype_domain.is_add_monoid_hom :\n  is_add_monoid_hom (subtype_domain p : (\u03b1 \u2192\u2080 M) \u2192 subtype p \u2192\u2080 M) :=\n{ map_add := \u03bb _ _, subtype_domain_add, map_zero := subtype_domain_zero }\n\n\/-- `finsupp.filter` as an `add_monoid_hom`. -\/\ndef filter_add_hom (p : \u03b1 \u2192 Prop) : (\u03b1 \u2192\u2080 M) \u2192+ (\u03b1 \u2192\u2080 M) :=\n{ to_fun := filter p,\n  map_zero' := filter_zero p,\n  map_add' := \u03bb f g, coe_fn_injective $ set.indicator_add {x | p x} f g }\n\n@[simp] lemma filter_add {v v' : \u03b1 \u2192\u2080 M} : (v + v').filter p = v.filter p + v'.filter p :=\n(filter_add_hom p).map_add v v'\n\nend add_zero_class\n\nsection comm_monoid\nvariables [add_comm_monoid M] {p : \u03b1 \u2192 Prop}\n\nlemma subtype_domain_sum {s : finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n  (\u2211 c in s, h c).subtype_domain p = \u2211 c in s, (h c).subtype_domain p :=\neq.symm (s.sum_hom _)\n\nlemma subtype_domain_finsupp_sum [has_zero N] {s : \u03b2 \u2192\u2080 N} {h : \u03b2 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n  (s.sum h).subtype_domain p = s.sum (\u03bbc d, (h c d).subtype_domain p) :=\nsubtype_domain_sum\n\nlemma filter_sum (s : finset \u03b9) (f : \u03b9 \u2192 \u03b1 \u2192\u2080 M) :\n  (\u2211 a in s, f a).filter p = \u2211 a in s, filter p (f a) :=\n(filter_add_hom p : (\u03b1 \u2192\u2080 M) \u2192+ _).map_sum f s\n\nlemma filter_eq_sum (p : \u03b1 \u2192 Prop) [D : decidable_pred p] (f : \u03b1 \u2192\u2080 M) :\n  f.filter p = \u2211 i in f.support.filter p, single i (f i) :=\n(f.filter p).sum_single.symm.trans $ finset.sum_congr (by rw subsingleton.elim D; refl) $\n  \u03bb x hx, by rw [filter_apply_pos _ _ (mem_filter.1 hx).2]\n\nend comm_monoid\n\nsection group\nvariables [add_group G] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 G}\n\n@[simp] lemma subtype_domain_neg : (- v).subtype_domain p = - v.subtype_domain p :=\next $ \u03bb _, rfl\n\n@[simp] lemma subtype_domain_sub :\n  (v - v').subtype_domain p = v.subtype_domain p - v'.subtype_domain p :=\next $ \u03bb _, rfl\n\n@[simp] lemma single_neg {a : \u03b1} {b : G} : single a (-b) = -single a b :=\n(single_add_hom a : G \u2192+ _).map_neg b\n\n@[simp] lemma single_sub {a : \u03b1} {b\u2081 b\u2082 : G} : single a (b\u2081 - b\u2082) = single a b\u2081 - single a b\u2082 :=\n(single_add_hom a : G \u2192+ _).map_sub b\u2081 b\u2082\n\nend group\n\nend subtype_domain\n\n\/-! ### Declarations relating `finsupp` to `multiset` -\/\n\nsection multiset\n\n\/-- Given `f : \u03b1 \u2192\u2080 \u2115`, `f.to_multiset` is the multiset with multiplicities given by the values of\n`f` on the elements of `\u03b1`. We define this function as an `add_equiv`. -\/\ndef to_multiset : (\u03b1 \u2192\u2080 \u2115) \u2243+ multiset \u03b1 :=\n{ to_fun := \u03bb f, f.sum (\u03bba n, n \u2022 {a}),\n  inv_fun := \u03bb s, \u27e8s.to_finset, \u03bb a, s.count a, \u03bb a, by simp\u27e9,\n  left_inv := \u03bb f, ext $ \u03bb a,\n    suffices (if f a = 0 then 0 else f a) = f a,\n    by simpa [finsupp.sum, multiset.count_sum', multiset.count_cons],\n    by split_ifs with h; [rw h, refl],\n  right_inv := \u03bb s, by simp [finsupp.sum],\n  map_add' := \u03bb f g, sum_add_index (\u03bb a, zero_nsmul _) (\u03bb a, add_nsmul _) }\n\nlemma to_multiset_zero : (0 : \u03b1 \u2192\u2080 \u2115).to_multiset = 0 :=\nrfl\n\nlemma to_multiset_add (m n : \u03b1 \u2192\u2080 \u2115) :\n  (m + n).to_multiset = m.to_multiset + n.to_multiset :=\nto_multiset.map_add m n\n\nlemma to_multiset_apply (f : \u03b1 \u2192\u2080 \u2115) : f.to_multiset = f.sum (\u03bb a n, n \u2022 {a}) := rfl\n\n@[simp]\nlemma to_multiset_symm_apply (s : multiset \u03b1) (x : \u03b1) :\n  finsupp.to_multiset.symm s x = s.count x :=\nrfl\n\n@[simp] lemma to_multiset_single (a : \u03b1) (n : \u2115) : to_multiset (single a n) = n \u2022 {a} :=\nby rw [to_multiset_apply, sum_single_index]; apply zero_nsmul\n\nlemma to_multiset_sum {\u03b9 : Type*} {f : \u03b9 \u2192 \u03b1 \u2192\u2080 \u2115} (s : finset \u03b9) :\n  finsupp.to_multiset (\u2211 i in s, f i) = \u2211 i in s, finsupp.to_multiset (f i) :=\nadd_equiv.map_sum _ _ _\n\nlemma to_multiset_sum_single {\u03b9 : Type*} (s : finset \u03b9) (n : \u2115) :\n  finsupp.to_multiset (\u2211 i in s, single i n) = n \u2022 s.val :=\nby simp_rw [to_multiset_sum, finsupp.to_multiset_single, multiset.singleton_eq_singleton,\n            sum_nsmul, sum_multiset_singleton]\n\nlemma card_to_multiset (f : \u03b1 \u2192\u2080 \u2115) : f.to_multiset.card = f.sum (\u03bba, id) :=\nby simp [to_multiset_apply, add_monoid_hom.map_finsupp_sum, function.id_def]\n\nlemma to_multiset_map (f : \u03b1 \u2192\u2080 \u2115) (g : \u03b1 \u2192 \u03b2) :\n  f.to_multiset.map g = (f.map_domain g).to_multiset :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.map_zero, map_domain_zero, to_multiset_zero] },\n  { assume a n f _ _ ih,\n    rw [to_multiset_add, multiset.map_add, ih, map_domain_add, map_domain_single,\n      to_multiset_single, to_multiset_add, to_multiset_single,\n      is_add_monoid_hom.map_nsmul (multiset.map g)],\n    refl }\nend\n\n@[simp] lemma prod_to_multiset [comm_monoid M] (f : M \u2192\u2080 \u2115) :\n  f.to_multiset.prod = f.prod (\u03bba n, a ^ n) :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.prod_zero, finsupp.prod_zero_index] },\n  { assume a n f _ _ ih,\n    rw [to_multiset_add, multiset.prod_add, ih, to_multiset_single, finsupp.prod_add_index,\n      finsupp.prod_single_index, multiset.prod_nsmul, multiset.singleton_eq_singleton,\n      multiset.prod_singleton],\n    { exact pow_zero a },\n    { exact pow_zero },\n    { exact pow_add  } }\nend\n\n@[simp] lemma to_finset_to_multiset [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 \u2115) :\n  f.to_multiset.to_finset = f.support :=\nbegin\n  refine f.induction _ _,\n  { rw [to_multiset_zero, multiset.to_finset_zero, support_zero] },\n  { assume a n f ha hn ih,\n    rw [to_multiset_add, multiset.to_finset_add, ih, to_multiset_single, support_add_eq,\n      support_single_ne_zero hn, multiset.to_finset_nsmul _ _ hn,\n      multiset.singleton_eq_singleton, multiset.to_finset_cons, multiset.to_finset_zero],\n    refl,\n    refine disjoint.mono_left support_single_subset _,\n    rwa [finset.singleton_disjoint] }\nend\n\n@[simp] lemma count_to_multiset [decidable_eq \u03b1] (f : \u03b1 \u2192\u2080 \u2115) (a : \u03b1) :\n  f.to_multiset.count a = f a :=\ncalc f.to_multiset.count a = f.sum (\u03bbx n, (n \u2022 {x} : multiset \u03b1).count a) :\n    (f.support.sum_hom $ multiset.count a).symm\n  ... = f.sum (\u03bbx n, n * ({x} : multiset \u03b1).count a) : by simp only [multiset.count_nsmul]\n  ... = f.sum (\u03bbx n, n * (x ::\u2098 0 : multiset \u03b1).count a) : rfl\n  ... = f a * (a ::\u2098 0 : multiset \u03b1).count a : sum_eq_single _\n    (\u03bb a' _ H, by simp only [multiset.count_cons_of_ne (ne.symm H), multiset.count_zero, mul_zero])\n    (\u03bb H, by simp only [not_mem_support_iff.1 H, zero_mul])\n  ... = f a : by simp only [multiset.count_singleton, mul_one]\n\nlemma mem_support_multiset_sum [add_comm_monoid M]\n  {s : multiset (\u03b1 \u2192\u2080 M)} (a : \u03b1) :\n  a \u2208 s.sum.support \u2192 \u2203f\u2208s, a \u2208 (f : \u03b1 \u2192\u2080 M).support :=\nmultiset.induction_on s false.elim\n  begin\n    assume f s ih ha,\n    by_cases a \u2208 f.support,\n    { exact \u27e8f, multiset.mem_cons_self _ _, h\u27e9 },\n    { simp only [multiset.sum_cons, mem_support_iff, add_apply,\n        not_mem_support_iff.1 h, zero_add] at ha,\n      rcases ih (mem_support_iff.2 ha) with \u27e8f', h\u2080, h\u2081\u27e9,\n      exact \u27e8f', multiset.mem_cons_of_mem h\u2080, h\u2081\u27e9 }\n  end\n\nlemma mem_support_finset_sum [add_comm_monoid M]\n  {s : finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} (a : \u03b1) (ha : a \u2208 (\u2211 c in s, h c).support) :\n  \u2203 c \u2208 s, a \u2208 (h c).support :=\nlet \u27e8f, hf, hfa\u27e9 := mem_support_multiset_sum a ha in\nlet \u27e8c, hc, eq\u27e9 := multiset.mem_map.1 hf in\n\u27e8c, hc, eq.symm \u25b8 hfa\u27e9\n\n@[simp] lemma mem_to_multiset (f : \u03b1 \u2192\u2080 \u2115) (i : \u03b1) :\n  i \u2208 f.to_multiset \u2194 i \u2208 f.support :=\nby rw [\u2190 multiset.count_ne_zero, finsupp.count_to_multiset, finsupp.mem_support_iff]\n\nend multiset\n\n\/-! ### Declarations about `curry` and `uncurry` -\/\n\nsection curry_uncurry\n\nvariables [add_comm_monoid M] [add_comm_monoid N]\n\n\/-- Given a finitely supported function `f` from a product type `\u03b1 \u00d7 \u03b2` to `\u03b3`,\n`curry f` is the \"curried\" finitely supported function from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `\u03b3`. -\/\nprotected def curry (f : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M) : \u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M) :=\nf.sum $ \u03bbp c, single p.1 (single p.2 c)\n\n@[simp] lemma curry_apply (f : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M) (x : \u03b1) (y : \u03b2) :\n  f.curry x y = f (x, y) :=\nbegin\n  have : \u2200 (b : \u03b1 \u00d7 \u03b2), single b.fst (single b.snd (f b)) x y = if b = (x, y) then f b else 0,\n  { rintros \u27e8b\u2081, b\u2082\u27e9,\n    simp [single_apply, ite_apply, prod.ext_iff, ite_and],\n    split_ifs; simp [single_apply, *] },\n  rw [finsupp.curry, sum_apply, sum_apply, finsupp.sum, finset.sum_eq_single, this, if_pos rfl],\n  { intros b hb b_ne, rw [this b, if_neg b_ne] },\n  { intros hxy, rw [this (x, y), if_pos rfl, not_mem_support_iff.mp hxy] }\nend\n\nlemma sum_curry_index (f : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M) (g : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N)\n  (hg\u2080 : \u2200 a b, g a b 0 = 0) (hg\u2081 : \u2200a b c\u2080 c\u2081, g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081) :\n  f.curry.sum (\u03bba f, f.sum (g a)) = f.sum (\u03bbp c, g p.1 p.2 c) :=\nbegin\n  rw [finsupp.curry],\n  transitivity,\n  { exact sum_sum_index (assume a, sum_zero_index)\n      (assume a b\u2080 b\u2081, sum_add_index (assume a, hg\u2080 _ _) (assume c d\u2080 d\u2081, hg\u2081 _ _ _ _)) },\n  congr, funext p c,\n  transitivity,\n  { exact sum_single_index sum_zero_index },\n  exact sum_single_index (hg\u2080 _ _)\nend\n\n\/-- Given a finitely supported function `f` from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `M`,\n`uncurry f` is the \"uncurried\" finitely supported function from `\u03b1 \u00d7 \u03b2` to `M`. -\/\nprotected def uncurry (f : \u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M)) : (\u03b1 \u00d7 \u03b2) \u2192\u2080 M :=\nf.sum $ \u03bba g, g.sum $ \u03bbb c, single (a, b) c\n\n\/-- `finsupp_prod_equiv` defines the `equiv` between `((\u03b1 \u00d7 \u03b2) \u2192\u2080 M)` and `(\u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M))` given by\ncurrying and uncurrying. -\/\ndef finsupp_prod_equiv : ((\u03b1 \u00d7 \u03b2) \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M)) :=\nby refine \u27e8finsupp.curry, finsupp.uncurry, \u03bb f, _, \u03bb f, _\u27e9; simp only [\n  finsupp.curry, finsupp.uncurry, sum_sum_index, sum_zero_index, sum_add_index,\n  sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff,\n  forall_3_true_iff, prod.mk.eta, (single_sum _ _ _).symm, sum_single]\n\nlemma filter_curry (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n  (f.filter (\u03bba:\u03b1\u00d7\u03b2, p a.1)).curry = f.curry.filter p :=\nbegin\n  rw [finsupp.curry, finsupp.curry, finsupp.sum, finsupp.sum, filter_sum, support_filter,\n    sum_filter],\n  refine finset.sum_congr rfl _,\n  rintros \u27e8a\u2081, a\u2082\u27e9 ha,\n  dsimp only,\n  split_ifs,\n  { rw [filter_apply_pos, filter_single_of_pos]; exact h },\n  { rwa [filter_single_of_neg] }\nend\n\nlemma support_curry [decidable_eq \u03b1] (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) :\n  f.curry.support \u2286 f.support.image prod.fst :=\nbegin\n  rw \u2190 finset.bUnion_singleton,\n  refine finset.subset.trans support_sum _,\n  refine finset.bUnion_mono (assume a _, support_single_subset)\nend\n\nend curry_uncurry\n\nsection sum\n\n\/-- `finsupp.sum_elim f g` maps `inl x` to `f x` and `inr y` to `g y`. -\/\ndef sum_elim {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) : \u03b1 \u2295 \u03b2 \u2192\u2080 \u03b3 :=\non_finset\n  ((f.support.map \u27e8_, sum.inl_injective\u27e9) \u222a g.support.map \u27e8_, sum.inr_injective\u27e9)\n  (sum.elim f g)\n  (\u03bb ab h, by { cases ab with a b; simp only [sum.elim_inl, sum.elim_inr] at h; simpa })\n\n@[simp] lemma coe_sum_elim {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) : \u21d1(sum_elim f g) = sum.elim f g := rfl\n\nlemma sum_elim_apply {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b1 \u2295 \u03b2) : sum_elim f g x = sum.elim f g x := rfl\n\nlemma sum_elim_inl {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b1) : sum_elim f g (sum.inl x) = f x := rfl\n\nlemma sum_elim_inr {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b2) : sum_elim f g (sum.inr x) = g x := rfl\n\n\/-- The equivalence between `(\u03b1 \u2295 \u03b2) \u2192\u2080 \u03b3` and `(\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)`.\n\nThis is the `finsupp` version of `equiv.sum_arrow_equiv_prod_arrow`. -\/\n@[simps apply symm_apply]\ndef sum_finsupp_equiv_prod_finsupp {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3] :\n  ((\u03b1 \u2295 \u03b2) \u2192\u2080 \u03b3) \u2243 (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3) :=\n{ to_fun := \u03bb f,\n    \u27e8f.comap_domain sum.inl (sum.inl_injective.inj_on _),\n     f.comap_domain sum.inr (sum.inr_injective.inj_on _)\u27e9,\n  inv_fun := \u03bb fg, sum_elim fg.1 fg.2,\n  left_inv := \u03bb f, by { ext ab, cases ab with a b; simp },\n  right_inv := \u03bb fg, by { ext; simp } }\n\nlemma fst_sum_finsupp_equiv_prod_finsupp {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : (\u03b1 \u2295 \u03b2) \u2192\u2080 \u03b3) (x : \u03b1) :\n  (sum_finsupp_equiv_prod_finsupp f).1 x = f (sum.inl x) :=\nrfl\n\nlemma snd_sum_finsupp_equiv_prod_finsupp {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (f : (\u03b1 \u2295 \u03b2) \u2192\u2080 \u03b3) (y : \u03b2) :\n  (sum_finsupp_equiv_prod_finsupp f).2 y = f (sum.inr y) :=\nrfl\n\nlemma sum_finsupp_equiv_prod_finsupp_symm_inl {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (fg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)) (x : \u03b1) :\n  (sum_finsupp_equiv_prod_finsupp.symm fg) (sum.inl x) = fg.1 x :=\nrfl\n\nlemma sum_finsupp_equiv_prod_finsupp_symm_inr {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3]\n  (fg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)) (y : \u03b2) :\n  (sum_finsupp_equiv_prod_finsupp.symm fg) (sum.inr y) = fg.2 y :=\nrfl\n\nvariables [add_monoid M]\n\n\/-- The additive equivalence between `(\u03b1 \u2295 \u03b2) \u2192\u2080 M` and `(\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)`.\n\nThis is the `finsupp` version of `equiv.sum_arrow_equiv_prod_arrow`. -\/\n@[simps apply symm_apply] def sum_finsupp_add_equiv_prod_finsupp {\u03b1 \u03b2 : Type*} :\n  ((\u03b1 \u2295 \u03b2) \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) :=\n{ map_add' :=\n    by { intros, ext;\n          simp only [equiv.to_fun_as_coe, prod.fst_add, prod.snd_add, add_apply,\n              snd_sum_finsupp_equiv_prod_finsupp, fst_sum_finsupp_equiv_prod_finsupp] },\n  .. sum_finsupp_equiv_prod_finsupp }\n\nlemma fst_sum_finsupp_add_equiv_prod_finsupp {\u03b1 \u03b2 : Type*}\n  (f : (\u03b1 \u2295 \u03b2) \u2192\u2080 M) (x : \u03b1) :\n  (sum_finsupp_add_equiv_prod_finsupp f).1 x = f (sum.inl x) :=\nrfl\n\nlemma snd_sum_finsupp_add_equiv_prod_finsupp {\u03b1 \u03b2 : Type*}\n  (f : (\u03b1 \u2295 \u03b2) \u2192\u2080 M) (y : \u03b2) :\n  (sum_finsupp_add_equiv_prod_finsupp f).2 y = f (sum.inr y) :=\nrfl\n\nlemma sum_finsupp_add_equiv_prod_finsupp_symm_inl {\u03b1 \u03b2 : Type*}\n  (fg : (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)) (x : \u03b1) :\n  (sum_finsupp_add_equiv_prod_finsupp.symm fg) (sum.inl x) = fg.1 x :=\nrfl\n\nlemma sum_finsupp_add_equiv_prod_finsupp_symm_inr {\u03b1 \u03b2 : Type*}\n  (fg : (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)) (y : \u03b2) :\n  (sum_finsupp_add_equiv_prod_finsupp.symm fg) (sum.inr y) = fg.2 y :=\nrfl\n\nend sum\n\nsection\nvariables [group G] [mul_action G \u03b1] [add_comm_monoid M]\n\n\/--\nScalar multiplication by a group element g,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain.\n-\/\ndef comap_has_scalar : has_scalar G (\u03b1 \u2192\u2080 M) :=\n{ smul := \u03bb g f, f.comap_domain (\u03bb a, g\u207b\u00b9 \u2022 a)\n  (\u03bb a a' m m' h, by simpa [\u2190mul_smul] using (congr_arg (\u03bb a, g \u2022 a) h)) }\n\nlocal attribute [instance] comap_has_scalar\n\n\/--\nScalar multiplication by a group element,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain,\nis multiplicative in g.\n-\/\ndef comap_mul_action : mul_action G (\u03b1 \u2192\u2080 M) :=\n{ one_smul := \u03bb f, by { ext, dsimp [(\u2022)], simp, },\n  mul_smul := \u03bb g g' f, by { ext, dsimp [(\u2022)], simp [mul_smul], }, }\n\nlocal attribute [instance] comap_mul_action\n\n\/--\nScalar multiplication by a group element,\ngiven by precomposition with the action of g\u207b\u00b9 on the domain,\nis additive in the second argument.\n-\/\ndef comap_distrib_mul_action :\n  distrib_mul_action G (\u03b1 \u2192\u2080 M) :=\n{ smul_zero := \u03bb g, by { ext, dsimp [(\u2022)], simp, },\n  smul_add := \u03bb g f f', by { ext, dsimp [(\u2022)], simp, }, }\n\n\/--\nScalar multiplication by a group element on finitely supported functions on a group,\ngiven by precomposition with the action of g\u207b\u00b9. -\/\ndef comap_distrib_mul_action_self :\n  distrib_mul_action G (G \u2192\u2080 M) :=\n@finsupp.comap_distrib_mul_action G M G _ (monoid.to_mul_action G) _\n\n@[simp]\nlemma comap_smul_single (g : G) (a : \u03b1) (b : M) :\n  g \u2022 single a b = single (g \u2022 a) b :=\nbegin\n  ext a',\n  dsimp [(\u2022)],\n  by_cases h : g \u2022 a = a',\n  { subst h, simp [\u2190mul_smul], },\n  { simp [single_eq_of_ne h], rw [single_eq_of_ne],\n    rintro rfl, simpa [\u2190mul_smul] using h, }\nend\n\n@[simp]\nlemma comap_smul_apply (g : G) (f : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n  (g \u2022 f) a = f (g\u207b\u00b9 \u2022 a) := rfl\n\nend\n\nsection\ninstance [monoid R] [add_monoid M] [distrib_mul_action R M] : has_scalar R (\u03b1 \u2192\u2080 M) :=\n\u27e8\u03bba v, v.map_range ((\u2022) a) (smul_zero _)\u27e9\n\n\/-!\nThroughout this section, some `monoid` and `semiring` arguments are specified with `{}` instead of\n`[]`. See note [implicit instance arguments].\n-\/\n\n@[simp] lemma coe_smul {_ : monoid R} [add_monoid M] [distrib_mul_action R M]\n  (b : R) (v : \u03b1 \u2192\u2080 M) : \u21d1(b \u2022 v) = b \u2022 v := rfl\nlemma smul_apply {_ : monoid R} [add_monoid M] [distrib_mul_action R M]\n  (b : R) (v : \u03b1 \u2192\u2080 M) (a : \u03b1) : (b \u2022 v) a = b \u2022 (v a) := rfl\n\nvariables (\u03b1 M)\n\ninstance [monoid R] [add_monoid M] [distrib_mul_action R M] : distrib_mul_action R (\u03b1 \u2192\u2080 M) :=\n{ smul      := (\u2022),\n  smul_add  := \u03bb a x y, ext $ \u03bb _, smul_add _ _ _,\n  one_smul  := \u03bb x, ext $ \u03bb _, one_smul _ _,\n  mul_smul  := \u03bb r s x, ext $ \u03bb _, mul_smul _ _ _,\n  smul_zero := \u03bb x, ext $ \u03bb _, smul_zero _ }\n\ninstance [monoid R] [monoid S] [add_monoid M] [distrib_mul_action R M] [distrib_mul_action S M]\n  [has_scalar R S] [is_scalar_tower R S M] :\n  is_scalar_tower R S (\u03b1 \u2192\u2080 M) :=\n{ smul_assoc := \u03bb r s a, ext $ \u03bb _, smul_assoc _ _ _ }\n\ninstance [monoid R] [monoid S] [add_monoid M] [distrib_mul_action R M] [distrib_mul_action S M]\n  [smul_comm_class R S M] :\n  smul_comm_class R S (\u03b1 \u2192\u2080 M) :=\n{ smul_comm := \u03bb r s a, ext $ \u03bb _, smul_comm _ _ _ }\n\ninstance [semiring R] [add_comm_monoid M] [module R M] : module R (\u03b1 \u2192\u2080 M) :=\n{ smul      := (\u2022),\n  zero_smul := \u03bb x, ext $ \u03bb _, zero_smul _ _,\n  add_smul  := \u03bb a x y, ext $ \u03bb _, add_smul _ _ _,\n  .. finsupp.distrib_mul_action \u03b1 M }\n\nvariables {\u03b1 M} {R}\n\nlemma support_smul {_ : monoid R} [add_monoid M] [distrib_mul_action R M] {b : R} {g : \u03b1 \u2192\u2080 M} :\n  (b \u2022 g).support \u2286 g.support :=\n\u03bb a, by { simp only [smul_apply, mem_support_iff, ne.def], exact mt (\u03bb h, h.symm \u25b8 smul_zero _) }\n\nsection\n\nvariables {p : \u03b1 \u2192 Prop}\n\n@[simp] lemma filter_smul {_ : monoid R} [add_monoid M] [distrib_mul_action R M]\n  {b : R} {v : \u03b1 \u2192\u2080 M} : (b \u2022 v).filter p = b \u2022 v.filter p :=\ncoe_fn_injective $ set.indicator_smul {x | p x} b v\n\nend\n\nlemma map_domain_smul {_ : monoid R} [add_comm_monoid M] [distrib_mul_action R M]\n   {f : \u03b1 \u2192 \u03b2} (b : R) (v : \u03b1 \u2192\u2080 M) : map_domain f (b \u2022 v) = b \u2022 map_domain f v :=\nbegin\n  change map_domain f (map_range _ _ _) = map_range _ _ _,\n  apply finsupp.induction v, { simp only [map_domain_zero, map_range_zero] },\n  intros a b v' hv\u2081 hv\u2082 IH,\n  rw [map_range_add, map_domain_add, IH, map_domain_add, map_range_add,\n    map_range_single, map_domain_single, map_domain_single, map_range_single];\n  apply smul_add\nend\n\n@[simp] lemma smul_single {_ : monoid R} [add_monoid M] [distrib_mul_action R M]\n  (c : R) (a : \u03b1) (b : M) : c \u2022 finsupp.single a b = finsupp.single a (c \u2022 b) :=\nmap_range_single\n\n@[simp] lemma smul_single' {_ : semiring R}\n  (c : R) (a : \u03b1) (b : R) : c \u2022 finsupp.single a b = finsupp.single a (c * b) :=\nsmul_single _ _ _\n\nlemma map_range_smul {_ : monoid R} [add_monoid M] [distrib_mul_action R M]\n  [add_monoid N] [distrib_mul_action R N]\n  {f : M \u2192 N} {hf : f 0 = 0} (c : R) (v : \u03b1 \u2192\u2080 M) (hsmul : \u2200 x, f (c \u2022 x) = c \u2022 f x) :\n  map_range f hf (c \u2022 v) = c \u2022 map_range f hf v :=\nbegin\n  erw \u2190map_range_comp,\n  have : (f \u2218 (\u2022) c) = ((\u2022) c \u2218 f) := funext hsmul,\n  simp_rw this,\n  apply map_range_comp,\n  rw [function.comp_apply, smul_zero, hf],\nend\n\nlemma smul_single_one [semiring R] (a : \u03b1) (b : R) : b \u2022 single a 1 = single a b :=\nby rw [smul_single, smul_eq_mul, mul_one]\n\nend\n\nlemma sum_smul_index [semiring R] [add_comm_monoid M] {g : \u03b1 \u2192\u2080 R} {b : R} {h : \u03b1 \u2192 R \u2192 M}\n  (h0 : \u2200i, h i 0 = 0) : (b \u2022 g).sum h = g.sum (\u03bbi a, h i (b * a)) :=\nfinsupp.sum_map_range_index h0\n\nlemma sum_smul_index' [monoid R] [add_monoid M] [distrib_mul_action R M] [add_comm_monoid N]\n  {g : \u03b1 \u2192\u2080 M} {b : R} {h : \u03b1 \u2192 M \u2192 N} (h0 : \u2200i, h i 0 = 0) :\n  (b \u2022 g).sum h = g.sum (\u03bbi c, h i (b \u2022 c)) :=\nfinsupp.sum_map_range_index h0\n\n\/-- A version of `finsupp.sum_smul_index'` for bundled additive maps. -\/\nlemma sum_smul_index_add_monoid_hom\n  [monoid R] [add_monoid M] [add_comm_monoid N] [distrib_mul_action R M]\n  {g : \u03b1 \u2192\u2080 M} {b : R} {h : \u03b1 \u2192 M \u2192+ N} :\n  (b \u2022 g).sum (\u03bb a, h a) = g.sum (\u03bb i c, h i (b \u2022 c)) :=\nsum_map_range_index (\u03bb i, (h i).map_zero)\n\ninstance [semiring R] [add_comm_monoid M] [module R M] {\u03b9 : Type*}\n  [no_zero_smul_divisors R M] : no_zero_smul_divisors R (\u03b9 \u2192\u2080 M) :=\n\u27e8\u03bb c f h, or_iff_not_imp_left.mpr (\u03bb hc, finsupp.ext\n  (\u03bb i, (smul_eq_zero.mp (finsupp.ext_iff.mp h i)).resolve_left hc))\u27e9\n\nsection\nvariables [has_zero R]\n\n\/-- The `finsupp` version of `pi.unique`. -\/\ninstance unique_of_right [subsingleton R] : unique (\u03b1 \u2192\u2080 R) :=\n{ uniq := \u03bb l, ext $ \u03bb i, subsingleton.elim _ _,\n  .. finsupp.inhabited }\n\n\/-- The `finsupp` version of `pi.unique_of_is_empty`. -\/\ninstance unique_of_left [is_empty \u03b1] : unique (\u03b1 \u2192\u2080 R) :=\n{ uniq := \u03bb l, ext is_empty_elim,\n  .. finsupp.inhabited }\n\nend\n\n\/-- Given an `add_comm_monoid M` and `s : set \u03b1`, `restrict_support_equiv s M` is the `equiv`\nbetween the subtype of finitely supported functions with support contained in `s` and\nthe type of finitely supported functions from `s`. -\/\ndef restrict_support_equiv (s : set \u03b1) (M : Type*) [add_comm_monoid M] :\n  {f : \u03b1 \u2192\u2080 M \/\/ \u2191f.support \u2286 s } \u2243 (s \u2192\u2080 M) :=\nbegin\n  refine \u27e8\u03bbf, subtype_domain (\u03bbx, x \u2208 s) f.1, \u03bb f, \u27e8f.map_domain subtype.val, _\u27e9, _, _\u27e9,\n  { refine set.subset.trans (finset.coe_subset.2 map_domain_support) _,\n    rw [finset.coe_image, set.image_subset_iff],\n    exact assume x hx, x.2 },\n  { rintros \u27e8f, hf\u27e9,\n    apply subtype.eq,\n    ext a,\n    dsimp only,\n    refine classical.by_cases (assume h : a \u2208 set.range (subtype.val : s \u2192 \u03b1), _) (assume h, _),\n    { rcases h with \u27e8x, rfl\u27e9,\n      rw [map_domain_apply subtype.val_injective, subtype_domain_apply] },\n    { convert map_domain_notin_range _ _ h,\n      rw [\u2190 not_mem_support_iff],\n      refine mt _ h,\n      exact assume ha, \u27e8\u27e8a, hf ha\u27e9, rfl\u27e9 } },\n  { assume f,\n    ext \u27e8a, ha\u27e9,\n    dsimp only,\n    rw [subtype_domain_apply, map_domain_apply subtype.val_injective] }\nend\n\n\/-- Given `add_comm_monoid M` and `e : \u03b1 \u2243 \u03b2`, `dom_congr e` is the corresponding `equiv` between\n`\u03b1 \u2192\u2080 M` and `\u03b2 \u2192\u2080 M`.\n\nThis is `finsupp.equiv_congr_left` as an `add_equiv`. -\/\n@[simps apply]\nprotected def dom_congr [add_comm_monoid M] (e : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192\u2080 M) \u2243+ (\u03b2 \u2192\u2080 M) :=\n{ to_fun := equiv_map_domain e,\n  inv_fun := equiv_map_domain e.symm,\n  left_inv := \u03bb v, begin\n    simp only [\u2190 equiv_map_domain_trans, equiv.trans_symm],\n    exact equiv_map_domain_refl _\n  end,\n  right_inv := begin\n    assume v,\n    simp only [\u2190 equiv_map_domain_trans, equiv.symm_trans],\n    exact equiv_map_domain_refl _\n  end,\n  map_add' := \u03bb a b, by simp only [equiv_map_domain_eq_map_domain]; exact map_domain_add }\n\n@[simp] lemma dom_congr_refl [add_comm_monoid M] :\n  finsupp.dom_congr (equiv.refl \u03b1) = add_equiv.refl (\u03b1 \u2192\u2080 M) :=\nadd_equiv.ext $ \u03bb _, equiv_map_domain_refl _\n\n@[simp] lemma dom_congr_symm [add_comm_monoid M] (e : \u03b1 \u2243 \u03b2) :\n  (finsupp.dom_congr e).symm = (finsupp.dom_congr e.symm : (\u03b2 \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 M)):=\nadd_equiv.ext $ \u03bb _, rfl\n\n@[simp] lemma dom_congr_trans [add_comm_monoid M] (e : \u03b1 \u2243 \u03b2) (f : \u03b2 \u2243 \u03b3) :\n  (finsupp.dom_congr e).trans (finsupp.dom_congr f) =\n    (finsupp.dom_congr (e.trans f) : (\u03b1 \u2192\u2080 M) \u2243+ _) :=\nadd_equiv.ext $ \u03bb _, (equiv_map_domain_trans _ _ _).symm\n\nend finsupp\n\nnamespace finsupp\n\n\/-! ### Declarations about sigma types -\/\n\nsection sigma\n\nvariables {\u03b1s : \u03b9 \u2192 Type*} [has_zero M] (l : (\u03a3 i, \u03b1s i) \u2192\u2080 M)\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `M` and\nan index element `i : \u03b9`, `split l i` is the `i`th component of `l`,\na finitely supported function from `as i` to `M`.\n\nThis is the `finsupp` version of `sigma.curry`.\n-\/\ndef split (i : \u03b9) : \u03b1s i \u2192\u2080 M :=\nl.comap_domain (sigma.mk i) (\u03bb x1 x2 _ _ hx, heq_iff_eq.1 (sigma.mk.inj hx).2)\n\nlemma split_apply (i : \u03b9) (x : \u03b1s i) : split l i x = l \u27e8i, x\u27e9 :=\nbegin\n  dunfold split,\n  rw comap_domain_apply\nend\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `\u03b2`,\n`split_support l` is the finset of indices in `\u03b9` that appear in the support of `l`. -\/\ndef split_support : finset \u03b9 := l.support.image sigma.fst\n\nlemma mem_split_support_iff_nonzero (i : \u03b9) :\n  i \u2208 split_support l \u2194 split l i \u2260 0 :=\nbegin\n  rw [split_support, mem_image, ne.def, \u2190 support_eq_empty, \u2190 ne.def,\n    \u2190 finset.nonempty_iff_ne_empty, split, comap_domain, finset.nonempty],\n  simp only [exists_prop, finset.mem_preimage, exists_and_distrib_right, exists_eq_right,\n    mem_support_iff, sigma.exists, ne.def]\nend\n\n\/-- Given `l`, a finitely supported function from the sigma type `\u03a3 i, \u03b1s i` to `\u03b2` and\nan `\u03b9`-indexed family `g` of functions from `(\u03b1s i \u2192\u2080 \u03b2)` to `\u03b3`, `split_comp` defines a\nfinitely supported function from the index type `\u03b9` to `\u03b3` given by composing `g i` with\n`split l i`. -\/\ndef split_comp [has_zero N] (g : \u03a0 i, (\u03b1s i \u2192\u2080 M) \u2192 N)\n  (hg : \u2200 i x, x = 0 \u2194 g i x = 0) : \u03b9 \u2192\u2080 N :=\n{ support := split_support l,\n  to_fun := \u03bb i, g i (split l i),\n  mem_support_to_fun :=\n  begin\n    intros i,\n    rw [mem_split_support_iff_nonzero, not_iff_not, hg],\n  end }\n\nlemma sigma_support : l.support = l.split_support.sigma (\u03bb i, (l.split i).support) :=\nby simp only [finset.ext_iff, split_support, split, comap_domain, mem_image,\n  mem_preimage, sigma.forall, mem_sigma]; tauto\n\nlemma sigma_sum [add_comm_monoid N] (f : (\u03a3 (i : \u03b9), \u03b1s i) \u2192 M \u2192 N) :\n  l.sum f = \u2211 i in split_support l, (split l i).sum (\u03bb (a : \u03b1s i) b, f \u27e8i, a\u27e9 b) :=\nby simp only [sum, sigma_support, sum_sigma, split_apply]\n\nvariables {\u03b7 : Type*} [fintype \u03b7] {\u03b9s : \u03b7 \u2192 Type*} [has_zero \u03b1]\n\n\/-- On a `fintype \u03b7`, `finsupp.split` is an equivalence between `(\u03a3 (j : \u03b7), \u03b9s j) \u2192\u2080 \u03b1`\nand `\u03a0 j, (\u03b9s j \u2192\u2080 \u03b1)`.\n\nThis is the `finsupp` version of `equiv.Pi_curry`. -\/\nnoncomputable def sigma_finsupp_equiv_pi_finsupp :\n  ((\u03a3 j, \u03b9s j) \u2192\u2080 \u03b1) \u2243 \u03a0 j, (\u03b9s j \u2192\u2080 \u03b1) :=\n{ to_fun := split,\n  inv_fun := \u03bb f, on_finset\n    (finset.univ.sigma (\u03bb j, (f j).support))\n    (\u03bb ji, f ji.1 ji.2)\n    (\u03bb g hg, finset.mem_sigma.mpr \u27e8finset.mem_univ _, mem_support_iff.mpr hg\u27e9),\n  left_inv := \u03bb f, by { ext, simp [split] },\n  right_inv := \u03bb f, by { ext, simp [split] } }\n\n@[simp] lemma sigma_finsupp_equiv_pi_finsupp_apply\n  (f : (\u03a3 j, \u03b9s j) \u2192\u2080 \u03b1) (j i) :\nsigma_finsupp_equiv_pi_finsupp f j i = f \u27e8j, i\u27e9 := rfl\n\n\/-- On a `fintype \u03b7`, `finsupp.split` is an additive equivalence between\n`(\u03a3 (j : \u03b7), \u03b9s j) \u2192\u2080 \u03b1` and `\u03a0 j, (\u03b9s j \u2192\u2080 \u03b1)`.\n\nThis is the `add_equiv` version of `finsupp.sigma_finsupp_equiv_pi_finsupp`.\n-\/\nnoncomputable def sigma_finsupp_add_equiv_pi_finsupp\n  {\u03b1 : Type*} {\u03b9s : \u03b7 \u2192 Type*} [add_monoid \u03b1] :\n  ((\u03a3 j, \u03b9s j) \u2192\u2080 \u03b1) \u2243+ \u03a0 j, (\u03b9s j \u2192\u2080 \u03b1) :=\n{ map_add' := \u03bb f g, by { ext, simp },\n  .. sigma_finsupp_equiv_pi_finsupp }\n\n@[simp] lemma sigma_finsupp_add_equiv_pi_finsupp_apply\n  {\u03b1 : Type*} {\u03b9s : \u03b7 \u2192 Type*} [add_monoid \u03b1] (f : (\u03a3 j, \u03b9s j) \u2192\u2080 \u03b1) (j i) :\nsigma_finsupp_add_equiv_pi_finsupp f j i = f \u27e8j, i\u27e9 := rfl\n\nend sigma\n\nend finsupp\n\n\/-! ### Declarations relating `multiset` to `finsupp` -\/\n\nnamespace multiset\n\n\/-- Given a multiset `s`, `s.to_finsupp` returns the finitely supported function on `\u2115` given by\nthe multiplicities of the elements of `s`. -\/\ndef to_finsupp : multiset \u03b1 \u2243+ (\u03b1 \u2192\u2080 \u2115) := finsupp.to_multiset.symm\n\n@[simp] lemma to_finsupp_support [D : decidable_eq \u03b1] (s : multiset \u03b1) :\n  s.to_finsupp.support = s.to_finset :=\nby rw subsingleton.elim D; refl\n\n@[simp] lemma to_finsupp_apply [D : decidable_eq \u03b1] (s : multiset \u03b1) (a : \u03b1) :\n  to_finsupp s a = s.count a :=\nby rw subsingleton.elim D; refl\n\nlemma to_finsupp_zero : to_finsupp (0 : multiset \u03b1) = 0 := add_equiv.map_zero _\n\nlemma to_finsupp_add (s t : multiset \u03b1) :\n  to_finsupp (s + t) = to_finsupp s + to_finsupp t :=\nto_finsupp.map_add s t\n\n@[simp] lemma to_finsupp_singleton (a : \u03b1) : to_finsupp (a ::\u2098 0) = finsupp.single a 1 :=\nfinsupp.to_multiset.symm_apply_eq.2 $ by simp\n\n@[simp] lemma to_finsupp_to_multiset (s : multiset \u03b1) :\n  s.to_finsupp.to_multiset = s :=\nfinsupp.to_multiset.apply_symm_apply s\n\nlemma to_finsupp_eq_iff {s : multiset \u03b1} {f : \u03b1 \u2192\u2080 \u2115} : s.to_finsupp = f \u2194 s = f.to_multiset :=\nfinsupp.to_multiset.symm_apply_eq\n\nend multiset\n\n@[simp] lemma finsupp.to_multiset_to_finsupp (f : \u03b1 \u2192\u2080 \u2115) :\n  f.to_multiset.to_finsupp = f :=\nfinsupp.to_multiset.symm_apply_apply f\n\n\/-! ### Declarations about order(ed) instances on `finsupp` -\/\n\nnamespace finsupp\n\ninstance [preorder M] [has_zero M] : preorder (\u03b1 \u2192\u2080 M) :=\n{ le := \u03bb f g, \u2200 s, f s \u2264 g s,\n  le_refl := \u03bb f s, le_refl _,\n  le_trans := \u03bb f g h Hfg Hgh s, le_trans (Hfg s) (Hgh s) }\n\ninstance [partial_order M] [has_zero M] : partial_order (\u03b1 \u2192\u2080 M) :=\n{ le_antisymm := \u03bb f g hfg hgf, ext $ \u03bb s, le_antisymm (hfg s) (hgf s),\n  .. finsupp.preorder }\n\ninstance [ordered_cancel_add_comm_monoid M] : ordered_cancel_add_comm_monoid (\u03b1 \u2192\u2080 M) :=\n{ add_le_add_left := \u03bb a b h c s, add_le_add_left (h s) (c s),\n  le_of_add_le_add_left := \u03bb a b c h s, le_of_add_le_add_left (h s),\n  add_left_cancel := \u03bb a b c h, ext $ \u03bb s, add_left_cancel (ext_iff.1 h s),\n  .. finsupp.add_comm_monoid, .. finsupp.partial_order }\n\nlemma le_def [preorder M] [has_zero M] {f g : \u03b1 \u2192\u2080 M} : f \u2264 g \u2194 \u2200 x, f x \u2264 g x := iff.rfl\n\nlemma le_iff [canonically_ordered_add_monoid M] (f g : \u03b1 \u2192\u2080 M) :\n  f \u2264 g \u2194 \u2200 s \u2208 f.support, f s \u2264 g s :=\n\u27e8\u03bb h s hs, h s,\n\u03bb h s, if H : s \u2208 f.support then h s H else (not_mem_support_iff.1 H).symm \u25b8 zero_le (g s)\u27e9\n\ninstance decidable_le [canonically_ordered_add_monoid M] [decidable_rel (@has_le.le M _)] :\n  decidable_rel (@has_le.le (\u03b1 \u2192\u2080 M) _) :=\n\u03bb f g, decidable_of_iff _ (le_iff f g).symm\n\n@[simp] lemma add_eq_zero_iff [canonically_ordered_add_monoid M] (f g : \u03b1 \u2192\u2080 M) :\n  f + g = 0 \u2194 f = 0 \u2227 g = 0 :=\nby simp [ext_iff, forall_and_distrib]\n\n\/-- `finsupp.to_multiset` as an order isomorphism. -\/\ndef order_iso_multiset : (\u03b1 \u2192\u2080 \u2115) \u2243o multiset \u03b1 :=\n{ to_equiv := to_multiset.to_equiv,\n  map_rel_iff' := \u03bb f g, by simp [multiset.le_iff_count, le_def] }\n\n@[simp] lemma coe_order_iso_multiset : \u21d1(@order_iso_multiset \u03b1) = to_multiset := rfl\n\n@[simp] lemma coe_order_iso_multiset_symm :\n  \u21d1(@order_iso_multiset \u03b1).symm = multiset.to_finsupp := rfl\n\nlemma to_multiset_strict_mono : strict_mono (@to_multiset \u03b1) :=\norder_iso_multiset.strict_mono\n\nlemma sum_id_lt_of_lt (m n : \u03b1 \u2192\u2080 \u2115) (h : m < n) :\n  m.sum (\u03bb _, id) < n.sum (\u03bb _, id) :=\nbegin\n  rw [\u2190 card_to_multiset, \u2190 card_to_multiset],\n  apply multiset.card_lt_of_lt,\n  exact to_multiset_strict_mono h\nend\n\nvariable (\u03b1)\n\n\/-- The order on `\u03c3 \u2192\u2080 \u2115` is well-founded.-\/\nlemma lt_wf : well_founded (@has_lt.lt (\u03b1 \u2192\u2080 \u2115) _) :=\nsubrelation.wf (sum_id_lt_of_lt) $ inv_image.wf _ nat.lt_wf\n\nvariable {\u03b1}\n\n@[simp] lemma nat_add_sub_cancel (f g : \u03b1 \u2192\u2080 \u2115) : f + g - g = f :=\next $ \u03bb a, nat.add_sub_cancel _ _\n\n@[simp] lemma nat_add_sub_cancel_left (f g : \u03b1 \u2192\u2080 \u2115) : f + g - f = g :=\next $ \u03bb a, nat.add_sub_cancel_left _ _\n\nlemma nat_add_sub_of_le {f g : \u03b1 \u2192\u2080 \u2115} (h : f \u2264 g) : f + (g - f) = g :=\next $ \u03bb a, nat.add_sub_of_le (h a)\n\nlemma nat_sub_add_cancel {f g : \u03b1 \u2192\u2080 \u2115} (h : f \u2264 g) : g - f + f = g :=\next $ \u03bb a, nat.sub_add_cancel (h a)\n\ninstance : canonically_ordered_add_monoid (\u03b1 \u2192\u2080 \u2115) :=\n{ bot := 0,\n  bot_le := \u03bb f s, zero_le (f s),\n  le_iff_exists_add := \u03bb f g, \u27e8\u03bb H, \u27e8g - f, (nat_add_sub_of_le H).symm\u27e9,\n    \u03bb \u27e8c, hc\u27e9, hc.symm \u25b8 \u03bb x, by simp\u27e9,\n .. (infer_instance : ordered_add_comm_monoid (\u03b1 \u2192\u2080 \u2115)) }\n\nend finsupp\n\nnamespace multiset\n\nlemma to_finsuppstrict_mono : strict_mono (@to_finsupp \u03b1) :=\nfinsupp.order_iso_multiset.symm.strict_mono\n\nend multiset\n","avg_line_length":37.4761712439,"max_line_length":100,"alphanum_fraction":0.6379498012}
-{"size":13364,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-\/\nimport data.polynomial.degree.card_pow_degree\nimport field_theory.finite.basic\nimport number_theory.class_number.admissible_absolute_value\n\n\/-!\n# Admissible absolute values on polynomials\nThis file defines an admissible absolute value\n`polynomial.card_pow_degree_is_admissible` which we use to show the class number\nof the ring of integers of a function field is finite.\n\n## Main results\n\n* `polynomial.card_pow_degree_is_admissible` shows `card_pow_degree`,\n  mapping `p : polynomial \ud835\udd3d_q` to `q ^ degree p`, is admissible\n-\/\n\nnamespace polynomial\n\nopen absolute_value real\n\nvariables {Fq : Type*} [field Fq] [fintype Fq]\n\n\/-- If `A` is a family of enough low-degree polynomials over a finite field, there is a\npair of equal elements in `A`. -\/\nlemma exists_eq_polynomial {d : \u2115} {m : \u2115} (hm : fintype.card Fq ^ d \u2264 m) (b : polynomial Fq)\n  (hb : nat_degree b \u2264 d) (A : fin m.succ \u2192 polynomial Fq) (hA : \u2200 i, degree (A i) < degree b) :\n  \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 A i\u2081 = A i\u2080 :=\nbegin\n  -- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,\n  -- there must be two elements of A with the same coefficients at\n  -- `0`, ... `degree b - 1` \u2264 `d - 1`.\n  -- In other words, the following map is not injective:\n  set f : fin m.succ \u2192 (fin d \u2192 Fq) := \u03bb i j, (A i).coeff j,\n  have : fintype.card (fin d \u2192 Fq) < fintype.card (fin m.succ),\n  { simpa using lt_of_le_of_lt hm (nat.lt_succ_self m) },\n  -- Therefore, the differences have all coefficients higher than `deg b - d` equal.\n  obtain \u27e8i\u2080, i\u2081, i_ne, i_eq\u27e9 := fintype.exists_ne_map_eq_of_card_lt f this,\n  use [i\u2080, i\u2081, i_ne],\n  ext j,\n  -- The coefficients higher than `deg b` are the same because they are equal to 0.\n  by_cases hbj : degree b \u2264 j,\n  { rw [coeff_eq_zero_of_degree_lt (lt_of_lt_of_le (hA _) hbj),\n        coeff_eq_zero_of_degree_lt (lt_of_lt_of_le (hA _) hbj)] },\n  -- So we only need to look for the coefficients between `0` and `deg b`.\n  rw not_le at hbj,\n  apply congr_fun i_eq.symm \u27e8j, _\u27e9,\n  exact lt_of_lt_of_le (coe_lt_degree.mp hbj) hb\nend\n\n\/-- If `A` is a family of enough low-degree polynomials over a finite field,\nthere is a pair of elements in `A` (with different indices but not necessarily\ndistinct), such that their difference has small degree. -\/\nlemma exists_approx_polynomial_aux {d : \u2115} {m : \u2115} (hm : fintype.card Fq ^ d \u2264 m)\n  (b : polynomial Fq) (A : fin m.succ \u2192 polynomial Fq) (hA : \u2200 i, degree (A i) < degree b) :\n  \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 degree (A i\u2081 - A i\u2080) < \u2191(nat_degree b - d) :=\nbegin\n  have hb : b \u2260 0,\n  { rintro rfl,\n    specialize hA 0,\n    rw degree_zero at hA,\n    exact not_lt_of_le bot_le hA },\n  -- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,\n  -- there must be two elements of A with the same coefficients at\n  -- `degree b - 1`, ... `degree b - d`.\n  -- In other words, the following map is not injective:\n  set f : fin m.succ \u2192 (fin d \u2192 Fq) := \u03bb i j, (A i).coeff (nat_degree b - j.succ),\n  have : fintype.card (fin d \u2192 Fq) < fintype.card (fin m.succ),\n  { simpa using lt_of_le_of_lt hm (nat.lt_succ_self m) },\n  -- Therefore, the differences have all coefficients higher than `deg b - d` equal.\n  obtain \u27e8i\u2080, i\u2081, i_ne, i_eq\u27e9 := fintype.exists_ne_map_eq_of_card_lt f this,\n  use [i\u2080, i\u2081, i_ne],\n  refine (degree_lt_iff_coeff_zero _ _).mpr (\u03bb j hj, _),\n  -- The coefficients higher than `deg b` are the same because they are equal to 0.\n  by_cases hbj : degree b \u2264 j,\n  { refine coeff_eq_zero_of_degree_lt (lt_of_lt_of_le _ hbj),\n    exact lt_of_le_of_lt (degree_sub_le _ _) (max_lt (hA _) (hA _)) },\n  -- So we only need to look for the coefficients between `deg b - d` and `deg b`.\n  rw [coeff_sub, sub_eq_zero],\n  rw [not_le, degree_eq_nat_degree hb, with_bot.coe_lt_coe] at hbj,\n  have hj : nat_degree b - j.succ < d,\n  { by_cases hd : nat_degree b < d,\n    { exact lt_of_le_of_lt (nat.sub_le_self _ _) hd },\n    { rw not_lt at hd,\n      have := lt_of_le_of_lt hj (nat.lt_succ_self j),\n      rwa [nat.sub_lt_iff hd hbj] at this } },\n  have : j = b.nat_degree - (nat_degree b - j.succ).succ,\n  { rw [\u2190 nat.succ_sub hbj, nat.succ_sub_succ, nat.sub_sub_self hbj.le] },\n  convert congr_fun i_eq.symm \u27e8nat_degree b - j.succ, hj\u27e9\nend\n\n\/-- If `A` is a family of enough low-degree polynomials over a finite field,\nthere is a pair of elements in `A` (with different indices but not necessarily\ndistinct), such that the difference of their remainders is close together. -\/\nlemma exists_approx_polynomial {b : polynomial Fq} (hb : b \u2260 0)\n  {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5)\n  (A : fin (fintype.card Fq ^ nat_ceil (- log \u03b5 \/ log (fintype.card Fq))).succ \u2192 polynomial Fq) :\n  \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 (card_pow_degree (A i\u2081 % b - A i\u2080 % b) : \u211d) < card_pow_degree b \u2022 \u03b5 :=\nbegin\n  have hb\u03b5 : 0 < card_pow_degree b \u2022 \u03b5,\n  { rw [algebra.smul_def, ring_hom.eq_int_cast],\n    exact mul_pos (int.cast_pos.mpr (absolute_value.pos _ hb)) h\u03b5 },\n  have one_lt_q : 1 < fintype.card Fq := fintype.one_lt_card,\n  have one_lt_q' : (1 : \u211d) < fintype.card Fq, { assumption_mod_cast },\n  have q_pos : 0 < fintype.card Fq, { linarith },\n  have q_pos' : (0 : \u211d) < fintype.card Fq, { assumption_mod_cast },\n  -- If `b` is already small enough, then the remainders are equal and we are done.\n  by_cases le_b : b.nat_degree \u2264 nat_ceil (-log \u03b5 \/ log \u2191(fintype.card Fq)),\n  { obtain \u27e8i\u2080, i\u2081, i_ne, mod_eq\u27e9 := exists_eq_polynomial (le_refl _) b le_b (\u03bb i, A i % b)\n      (\u03bb i, euclidean_domain.mod_lt (A i) hb),\n    refine \u27e8i\u2080, i\u2081, i_ne, _\u27e9,\n    simp only at mod_eq,\n    rwa [mod_eq, sub_self, absolute_value.map_zero, int.cast_zero] },\n  -- Otherwise, it suffices to choose two elements whose difference is of small enough degree.\n  rw not_le at le_b,\n  obtain \u27e8i\u2080, i\u2081, i_ne, deg_lt\u27e9 := exists_approx_polynomial_aux (le_refl _) b (\u03bb i, A i % b)\n    (\u03bb i, euclidean_domain.mod_lt (A i) hb),\n  simp only at deg_lt,\n  use [i\u2080, i\u2081, i_ne],\n  -- Again, if the remainders are equal we are done.\n  by_cases h : A i\u2081 % b = A i\u2080 % b,\n  { rwa [h, sub_self, absolute_value.map_zero, int.cast_zero] },\n  have h' : A i\u2081 % b - A i\u2080 % b \u2260 0 := mt sub_eq_zero.mp h,\n  -- If the remainders are not equal, we'll show their difference is of small degree.\n  -- In particular, we'll show the degree is less than the following:\n  suffices : (nat_degree (A i\u2081 % b - A i\u2080 % b) : \u211d) <\n    b.nat_degree + log \u03b5 \/ log (fintype.card Fq),\n  { rwa [\u2190 real.log_lt_log_iff (int.cast_pos.mpr (card_pow_degree.pos h')) hb\u03b5,\n        card_pow_degree_nonzero _ h', card_pow_degree_nonzero _ hb,\n        algebra.smul_def, ring_hom.eq_int_cast,\n        int.cast_pow, int.cast_coe_nat, int.cast_pow, int.cast_coe_nat,\n        log_mul (pow_ne_zero _ q_pos'.ne') h\u03b5.ne',\n        \u2190 rpow_nat_cast, \u2190 rpow_nat_cast, log_rpow q_pos', log_rpow q_pos',\n        \u2190 lt_div_iff (log_pos one_lt_q'), add_div, mul_div_cancel _ (log_pos one_lt_q').ne'] },\n  -- And that result follows from manipulating the result from `exists_approx_polynomial_aux`\n  -- to turn the `- ceil (- stuff)` into `+ stuff`.\n  refine lt_of_lt_of_le (nat.cast_lt.mpr (with_bot.coe_lt_coe.mp _)) _,\n  swap, { convert deg_lt, rw degree_eq_nat_degree h' },\n  rw [\u2190 sub_neg_eq_add, neg_div],\n  refine le_trans _ (sub_le_sub_left (le_nat_ceil _) (b.nat_degree : \u211d)),\n  rw \u2190 neg_div,\n  exact le_of_eq (nat.cast_sub le_b.le)\nend\n\n\/-- If `x` is close to `y` and `y` is close to `z`, then `x` and `z` are at least as close. -\/\nlemma card_pow_degree_anti_archimedean {x y z : polynomial Fq} {a : \u2124}\n  (hxy : card_pow_degree (x - y) < a) (hyz : card_pow_degree (y - z) < a) :\n  card_pow_degree (x - z) < a :=\nbegin\n  have ha : 0 < a := lt_of_le_of_lt (absolute_value.nonneg _ _) hxy,\n  by_cases hxy' : x = y,\n  { rwa hxy' },\n  by_cases hyz' : y = z,\n  { rwa \u2190 hyz' },\n  by_cases hxz' : x = z,\n  { rwa [hxz', sub_self, absolute_value.map_zero] },\n  rw [\u2190 ne.def, \u2190 sub_ne_zero] at hxy' hyz' hxz',\n  refine lt_of_le_of_lt _ (max_lt hxy hyz),\n  rw [card_pow_degree_nonzero _ hxz', card_pow_degree_nonzero _ hxy',\n      card_pow_degree_nonzero _ hyz'],\n  have : (1 : \u2124) \u2264 fintype.card Fq, { exact_mod_cast (@fintype.one_lt_card Fq _ _).le },\n  simp only [int.cast_pow, int.cast_coe_nat, le_max_iff],\n  refine or.imp (pow_le_pow this) (pow_le_pow this) _,\n  rw [nat_degree_le_iff_degree_le, nat_degree_le_iff_degree_le, \u2190 le_max_iff,\n      \u2190 degree_eq_nat_degree hxy', \u2190 degree_eq_nat_degree hyz'],\n  convert degree_add_le (x - y) (y - z) using 2,\n  exact (sub_add_sub_cancel _ _ _).symm\nend\n\n\/-- A slightly stronger version of `exists_partition` on which we perform induction on `n`:\nfor all `\u03b5 > 0`, we can partition the remainders of any family of polynomials `A`\ninto equivalence classes, where the equivalence(!) relation is \"closer than `\u03b5`\". -\/\nlemma exists_partition_polynomial_aux (n : \u2115) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5)\n  {b : polynomial Fq} (hb : b \u2260 0) (A : fin n \u2192 polynomial Fq) :\n  \u2203 (t : fin n \u2192 fin (fintype.card Fq ^ nat_ceil (-log \u03b5 \/ log \u2191(fintype.card Fq)))),\n  \u2200 (i\u2080 i\u2081 : fin n),\n  t i\u2080 = t i\u2081 \u2194 (card_pow_degree (A i\u2081 % b - A i\u2080 % b) : \u211d) < card_pow_degree b \u2022 \u03b5 :=\nbegin\n  have hb\u03b5 : 0 < card_pow_degree b \u2022 \u03b5,\n  { rw [algebra.smul_def, ring_hom.eq_int_cast],\n    exact mul_pos (int.cast_pos.mpr (absolute_value.pos _ hb)) h\u03b5 },\n  -- We go by induction on the size `A`.\n  induction n with n ih,\n  { refine \u27e8fin_zero_elim, fin_zero_elim\u27e9 },\n\n  -- Show `anti_archimedean` also holds for real distances.\n  have anti_archim' : \u2200 {i j k} {\u03b5 : \u211d}, (card_pow_degree (A i % b - A j % b) : \u211d) < \u03b5 \u2192\n    (card_pow_degree (A j % b - A k % b) : \u211d) < \u03b5 \u2192 (card_pow_degree (A i % b - A k % b) : \u211d) < \u03b5,\n  { intros i j k \u03b5,\n    rw [\u2190 lt_ceil, \u2190 lt_ceil, \u2190 lt_ceil],\n    exact card_pow_degree_anti_archimedean },\n\n  obtain \u27e8t', ht'\u27e9 := ih (fin.tail A),\n  -- We got rid of `A 0`, so determine the index `j` of the partition we'll re-add it to.\n  suffices : \u2203 j,\n    \u2200 i, t' i = j \u2194 (card_pow_degree (A 0 % b - A i.succ % b) : \u211d) < card_pow_degree b \u2022 \u03b5,\n  { obtain \u27e8j, hj\u27e9 := this,\n    refine \u27e8fin.cons j t', \u03bb i\u2080 i\u2081, _\u27e9,\n    refine fin.cases _ (\u03bb i\u2080, _) i\u2080; refine fin.cases _ (\u03bb i\u2081, _) i\u2081,\n    { simpa using hb\u03b5 },\n    { rw [fin.cons_succ, fin.cons_zero, eq_comm, absolute_value.map_sub],\n      exact hj i\u2081 },\n    { rw [fin.cons_succ, fin.cons_zero],\n      exact hj i\u2080 },\n    { rw [fin.cons_succ, fin.cons_succ],\n      exact ht' i\u2080 i\u2081 } },\n  -- `exists_approx_polynomial` guarantees that we can insert `A 0` into some partition `j`,\n  -- but not that `j` is uniquely defined (which is needed to keep the induction going).\n  obtain \u27e8j, hj\u27e9 : \u2203 j, \u2200 (i : fin n), t' i = j \u2192\n    (card_pow_degree (A 0 % b - A i.succ % b) : \u211d) < card_pow_degree b \u2022 \u03b5,\n  { by_contra this, push_neg at this,\n    obtain \u27e8j\u2080, j\u2081, j_ne, approx\u27e9 := exists_approx_polynomial hb h\u03b5\n      (fin.cons (A 0) (\u03bb j, A (fin.succ (classical.some (this j))))),\n    revert j_ne approx,\n    refine fin.cases _ (\u03bb j\u2080, _) j\u2080; refine fin.cases (\u03bb j_ne approx, _) (\u03bb j\u2081 j_ne approx, _) j\u2081,\n    { exact absurd rfl j_ne },\n    { rw [fin.cons_succ, fin.cons_zero, \u2190 not_le, absolute_value.map_sub] at approx,\n      have := (classical.some_spec (this j\u2081)).2,\n      contradiction },\n    { rw [fin.cons_succ, fin.cons_zero, \u2190 not_le] at approx,\n      have := (classical.some_spec (this j\u2080)).2,\n      contradiction },\n    { rw [fin.cons_succ, fin.cons_succ] at approx,\n      rw [ne.def, fin.succ_inj] at j_ne,\n      have : j\u2080 = j\u2081 :=\n        (classical.some_spec (this j\u2080)).1.symm.trans\n        (((ht' (classical.some (this j\u2080)) (classical.some (this j\u2081))).mpr approx).trans\n        (classical.some_spec (this j\u2081)).1),\n      contradiction } },\n  -- However, if one of those partitions `j` is inhabited by some `i`, then this `j` works.\n  by_cases exists_nonempty_j : \u2203 j, (\u2203 i, t' i = j) \u2227\n    \u2200 i, t' i = j \u2192 (card_pow_degree (A 0 % b - A i.succ % b) : \u211d) < card_pow_degree b \u2022 \u03b5,\n  { obtain \u27e8j, \u27e8i, hi\u27e9, hj\u27e9 := exists_nonempty_j,\n    refine \u27e8j, \u03bb i', \u27e8hj i', \u03bb hi', trans ((ht' _ _).mpr _) hi\u27e9\u27e9,\n    apply anti_archim' _ hi',\n    rw absolute_value.map_sub,\n    exact hj _ hi },\n  -- And otherwise, we can just take any `j`, since those are empty.\n  refine \u27e8j, \u03bb i, \u27e8hj i, \u03bb hi, _\u27e9\u27e9,\n  have := exists_nonempty_j \u27e8t' i, \u27e8i, rfl\u27e9, \u03bb i' hi', anti_archim' hi ((ht' _ _).mp hi')\u27e9,\n  contradiction\nend\n\n\/-- For all `\u03b5 > 0`, we can partition the remainders of any family of polynomials `A`\ninto classes, where all remainders in a class are close together. -\/\nlemma exists_partition_polynomial (n : \u2115) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5)\n  {b : polynomial Fq} (hb : b \u2260 0) (A : fin n \u2192 polynomial Fq) :\n  \u2203 (t : fin n \u2192 fin (fintype.card Fq ^ nat_ceil (-log \u03b5 \/ log \u2191(fintype.card Fq)))),\n    \u2200 (i\u2080 i\u2081 : fin n), t i\u2080 = t i\u2081 \u2192\n      (card_pow_degree (A i\u2081 % b - A i\u2080 % b) : \u211d) < card_pow_degree b \u2022 \u03b5 :=\nbegin\n  obtain \u27e8t, ht\u27e9 := exists_partition_polynomial_aux n h\u03b5 hb A,\n  exact \u27e8t, \u03bb i\u2080 i\u2081 hi, (ht i\u2080 i\u2081).mp hi\u27e9\nend\n\n\/-- `\u03bb p, fintype.card Fq ^ degree p` is an admissible absolute value.\nWe set `q ^ degree 0 = 0`. -\/\nnoncomputable def card_pow_degree_is_admissible :\n  is_admissible (card_pow_degree : absolute_value (polynomial Fq) \u2124) :=\n{ card := \u03bb \u03b5, fintype.card Fq ^ (nat_ceil (- log \u03b5 \/ log (fintype.card Fq))),\n  exists_partition' := \u03bb n \u03b5 h\u03b5 b hb, exists_partition_polynomial n h\u03b5 hb,\n  .. @card_pow_degree_is_euclidean Fq _ _ }\n\nend polynomial\n","avg_line_length":49.6802973978,"max_line_length":98,"alphanum_fraction":0.6576623765}
-{"size":2199,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright \u00a9 2019 Fran\u00e7ois G. Dorais. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n-\/\n\nimport .groupoids.paths\n\nnamespace tactic\n\nmeta def match_ne' : expr \u2192 tactic (expr \u00d7 expr)\n| `(%%a \u2260 %%b) := return (a,b)\n| `(\u00ac %%a = %%b) := return (a,b)\n| _ := failed\n\nmeta def eq_groupoid : groupoid (\u03bb x y : expr, tactic expr) :=\n{ id := mk_eq_refl\n, inv := \u03bb _ _ g, g >>= mk_eq_symm\n, comp := \u03bb _ _ _ g h, g >>= \u03bb g, h >>= \u03bb h, mk_eq_trans g h\n}\n\nmeta def eq_state := path_state eq_groupoid\n\nmeta def eq_gather : list expr \u2192 tactic eq_state\n| [] := return []\n| (e :: es) :=\n  do {\n    `(%%a = %%b) \u2190 infer_type e,\n    eq_gather es >>= \u03bb ps, return $ path_state.intro (path.mk eq_groupoid a b (return e)) ps\n  } <|> eq_gather es\n\nmeta def eq_find (eqs : eq_state) : expr \u2192 expr \u2192 tactic expr := \n\u03bb a b, eqs.find a b >>= \u03bb e, e\n\nmeta def eq_solve (eqs : eq_state) : tactic unit :=\ndo {\n  `(%%a = %%b) \u2190 target,\n  eq_find eqs a b >>= exact\n}\n\nmeta def eq_rewrite : eq_state \u2192 tactic unit\n| [] := skip\n| (p :: ps) := repeat (p.map >>= \u03bb e, rewrite_target e {symm:=tt}) >> eq_rewrite ps\n\nmeta def eq_contra (eqs : eq_state) : list expr \u2192 tactic unit\n| [] := failed\n| (h :: hs) :=\n  do {\n    (a,b) \u2190 infer_type h >>= match_ne',\n    e \u2190 eq_find eqs a b,\n    to_expr ``(absurd %%e %%h) >>= exact\n  } <|> eq_contra hs\n\nmeta def eq_path_to_string : path eq_groupoid \u2192 tactic string\n| \u27e8_,a,b,e\u27e9 := e >>= \u03bb e, return $ to_string a ++ \" = \" ++ to_string b ++ \" := \" ++ to_string e\n\nmeta def eq_state_to_string : eq_state \u2192 tactic string\n| [] := return \"\"\n| (p :: ps) := eq_path_to_string p >>= \u03bb s, eq_state_to_string ps >>= \u03bb t, return $ s ++ \"; \" ++ t\n\nmeta def eq_trace (eqs : eq_state) : tactic unit :=\neq_state_to_string eqs >>= trace\n\nmeta def interactive.eq_trace : tactic unit := \nlocal_context >>= eq_gather >>= eq_trace\n\nmeta def interactive.eq_solve : tactic unit := \nlocal_context >>= eq_gather >>= eq_solve\n\nmeta def interactive.eq_rewrite : tactic unit := \nlocal_context >>= eq_gather >>= eq_rewrite\n\nmeta def interactive.eq_contradiction : tactic unit :=\ndo {\n  by_contradiction,\n  ctx \u2190 local_context,\n  eqs \u2190 eq_gather ctx,\n  eq_contra eqs ctx\n}\n\nend tactic","avg_line_length":27.4875,"max_line_length":98,"alphanum_fraction":0.6261937244}
-{"size":2929,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"\/-\nCopyright (c) 2020 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth\n-\/\nimport analysis.normed_space.hahn_banach\n\n\/-!\n# The topological dual of a normed space\n\nIn this file we define the topological dual of a normed space, and the bounded linear map from\na normed space into its double dual.\n\nWe also prove that, for base field the real numbers, this map is an isometry.  (TODO:  the same for\nthe complex numbers.)\n-\/\n\nnoncomputable theory\n\nnamespace normed_space\n\nsection general\nvariables (\ud835\udd5c : Type*) [nondiscrete_normed_field \ud835\udd5c]\nvariables (E : Type*) [normed_group E] [normed_space \ud835\udd5c E]\n\n\/-- The topological dual of a normed space `E`. -\/\n@[derive [has_coe_to_fun, normed_group, normed_space \ud835\udd5c]] def dual := E \u2192L[\ud835\udd5c] \ud835\udd5c\n\ninstance : inhabited (dual \ud835\udd5c E) := \u27e80\u27e9\n\n\/-- The inclusion of a normed space in its double (topological) dual. -\/\ndef inclusion_in_double_dual' (x : E) : (dual \ud835\udd5c (dual \ud835\udd5c E)) :=\nlinear_map.mk_continuous\n  { to_fun := \u03bb f, f x,\n    map_add'    := by simp,\n    map_smul'   := by simp }\n  \u2225x\u2225\n  (\u03bb f, by { rw mul_comm, exact f.le_op_norm x } )\n\n@[simp] lemma dual_def (x : E) (f : dual \ud835\udd5c E) :\n  ((inclusion_in_double_dual' \ud835\udd5c E) x) f = f x := rfl\n\nlemma double_dual_bound (x : E) : \u2225(inclusion_in_double_dual' \ud835\udd5c E) x\u2225 \u2264 \u2225x\u2225 :=\nbegin\n  apply continuous_linear_map.op_norm_le_bound,\n  { simp },\n  { intros f, rw mul_comm, exact f.le_op_norm x, }\nend\n\n\/-- The inclusion of a normed space in its double (topological) dual, considered\n   as a bounded linear map. -\/\ndef inclusion_in_double_dual : E \u2192L[\ud835\udd5c] (dual \ud835\udd5c (dual \ud835\udd5c E)) :=\nlinear_map.mk_continuous\n  { to_fun := \u03bb (x : E), (inclusion_in_double_dual' \ud835\udd5c E) x,\n    map_add'    := \u03bb x y, by { ext, simp },\n    map_smul'   := \u03bb (c : \ud835\udd5c) x, by { ext, simp } }\n  1\n  (\u03bb x, by { convert double_dual_bound _ _ _, simp } )\n\nend general\n\nsection real\nvariables (E : Type*) [normed_group E] [normed_space \u211d E]\n\n\/-- If one controls the norm of every `f x`, then one controls the norm of `x`.\n    Compare `continuous_linear_map.op_norm_le_bound`. -\/\nlemma norm_le_dual_bound (x : E) {M : \u211d} (hMp: 0 \u2264 M) (hM : \u2200 (f: dual \u211d E), \u2225f x\u2225 \u2264 M * \u2225f\u2225) :\n  \u2225x\u2225 \u2264 M :=\nbegin\n  classical,\n  by_cases h : x = 0,\n  { simp only [h, hMp, norm_zero] },\n  { cases exists_dual_vector x h with f hf,\n    calc \u2225x\u2225 = f x : hf.2.symm\n    ... \u2264 \u2225f x\u2225 : le_max_left (f x) (-f x)\n    ... \u2264 M * \u2225f\u2225 : hM f\n    ... = M : by rw [ hf.1, mul_one ] }\nend\n\n\/-- The inclusion of a real normed space in its double dual is an isometry onto its image.-\/\nlemma inclusion_in_double_dual_isometry (x : E) : \u2225inclusion_in_double_dual \u211d E x\u2225 = \u2225x\u2225 :=\nbegin\n  refine le_antisymm_iff.mpr \u27e8double_dual_bound \u211d E x, _\u27e9,\n  rw continuous_linear_map.norm_def,\n  apply real.lb_le_Inf _ continuous_linear_map.bounds_nonempty,\n  intros c,\n  simpa using norm_le_dual_bound E x,\nend\n\n-- TODO: This is also true over \u2102.\n\nend real\n\nend normed_space\n","avg_line_length":30.8315789474,"max_line_length":99,"alphanum_fraction":0.6725844998}
-{"size":2779,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Marc Huisinga. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Marc Huisinga, Wojciech Nawrocki\n-\/\nimport Lean.Data.Lsp.Capabilities\nimport Lean.Data.Lsp.Workspace\nimport Lean.Data.Json\n\n\/-! Functionality to do with initializing and shutting down\nthe server (\"General Messages\" section of LSP spec). -\/\n\nnamespace Lean\nnamespace Lsp\n\nopen Json\n\nstructure ClientInfo where\n  name : String\n  version? : Option String := none\n  deriving ToJson, FromJson\n\ninductive Trace where\n  | off\n  | messages\n  | verbose\n\ninstance : FromJson Trace := \u27e8fun j =>\n  match j.getStr? with\n  | some \"off\"      => Trace.off\n  | some \"messages\" => Trace.messages\n  | some \"verbose\"  => Trace.verbose\n  | _               => none\u27e9\n\ninstance Trace.hasToJson : ToJson Trace :=\n\u27e8fun\n  | Trace.off => \"off\"\n  | Trace.messages => \"messages\"\n  | Trace.verbose => \"verbose\"\u27e9\n\nstructure InitializeParams where\n  processId? : Option Int := none\n  clientInfo? : Option ClientInfo := none\n  \/- We don't support the deprecated rootPath\n  (rootPath? : Option String) -\/\n  rootUri? : Option String := none\n  initializationOptions? : Option Json := none\n  capabilities : ClientCapabilities\n  \/- If omitted, we default to off. -\/\n  trace : Trace := Trace.off\n  workspaceFolders? : Option (Array WorkspaceFolder) := none\n  deriving ToJson\n\ninstance : FromJson InitializeParams := \u27e8fun j => do\n  \/- Many of these params can be null instead of not present.\n  For ease of implementation, we're liberal:\n  missing params, wrong json types and null all map to none,\n  even if LSP sometimes only allows some subset of these.\n  In cases where LSP makes a meaningful distinction\n  between different kinds of missing values, we'll\n  follow accordingly. -\/\n  let processId? := j.getObjValAs? Int \"processId\"\n  let clientInfo? := j.getObjValAs? ClientInfo \"clientInfo\"\n  let rootUri? := j.getObjValAs? String \"rootUri\"\n  let initializationOptions? := j.getObjVal? \"initializationOptions\"\n  let capabilities \u2190 j.getObjValAs? ClientCapabilities \"capabilities\"\n  let trace := (j.getObjValAs? Trace \"trace\").getD Trace.off\n  let workspaceFolders? := j.getObjValAs? (Array WorkspaceFolder) \"workspaceFolders\"\n  pure \u27e8processId?, clientInfo?, rootUri?, initializationOptions?, capabilities, trace, workspaceFolders?\u27e9\u27e9\n\ninductive InitializedParams where\n  | mk\n\ninstance : FromJson InitializedParams :=\n  \u27e8fun _ => InitializedParams.mk\u27e9\n\ninstance : ToJson InitializedParams :=\n  \u27e8fun _ => Json.null\u27e9\n\nstructure ServerInfo where\n  name : String\n  version? : Option String := none\n  deriving ToJson, FromJson\n\nstructure InitializeResult where\n  capabilities : ServerCapabilities\n  serverInfo? : Option ServerInfo := none\n  deriving ToJson, FromJson\n\nend Lsp\nend Lean\n","avg_line_length":29.8817204301,"max_line_length":107,"alphanum_fraction":0.7297589061}
-{"size":38020,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-\/\nimport Lean.Parser.Term\nimport Lean.Meta.Closure\nimport Lean.Meta.Check\nimport Lean.PrettyPrinter.Delaborator.Options\nimport Lean.Elab.Command\nimport Lean.Elab.Match\nimport Lean.Elab.DefView\nimport Lean.Elab.PreDefinition\nimport Lean.Elab.DeclarationRange\n\nnamespace Lean.Elab\nopen Lean.Parser.Term\n\n\/- DefView after elaborating the header. -\/\nstructure DefViewElabHeader where\n  ref           : Syntax\n  modifiers     : Modifiers\n  kind          : DefKind\n  shortDeclName : Name\n  declName      : Name\n  levelNames    : List Name\n  binderIds     : Array Syntax\n  numParams     : Nat\n  type          : Expr -- including the parameters\n  valueStx      : Syntax\n  deriving Inhabited\n\nnamespace Term\nopen Meta\n\nprivate def checkModifiers (m\u2081 m\u2082 : Modifiers) : TermElabM Unit := do\n  unless m\u2081.isUnsafe == m\u2082.isUnsafe do\n    throwError \"cannot mix unsafe and safe definitions\"\n  unless m\u2081.isNoncomputable == m\u2082.isNoncomputable do\n    throwError \"cannot mix computable and non-computable definitions\"\n  unless m\u2081.isPartial == m\u2082.isPartial do\n    throwError \"cannot mix partial and non-partial definitions\"\n\nprivate def checkKinds (k\u2081 k\u2082 : DefKind) : TermElabM Unit := do\n  unless k\u2081.isExample == k\u2082.isExample do\n    throwError \"cannot mix examples and definitions\" -- Reason: we should discard examples\n  unless k\u2081.isTheorem == k\u2082.isTheorem do\n    throwError \"cannot mix theorems and definitions\" -- Reason: we will eventually elaborate theorems in `Task`s.\n\nprivate def check (prevHeaders : Array DefViewElabHeader) (newHeader : DefViewElabHeader) : TermElabM Unit := do\n  if newHeader.kind.isTheorem && newHeader.modifiers.isUnsafe then\n    throwError \"'unsafe' theorems are not allowed\"\n  if newHeader.kind.isTheorem && newHeader.modifiers.isPartial then\n    throwError \"'partial' theorems are not allowed, 'partial' is a code generation directive\"\n  if newHeader.kind.isTheorem && newHeader.modifiers.isNoncomputable then\n    throwError \"'theorem' subsumes 'noncomputable', code is not generated for theorems\"\n  if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isUnsafe then\n    throwError \"'noncomputable unsafe' is not allowed\"\n  if newHeader.modifiers.isNoncomputable && newHeader.modifiers.isPartial then\n    throwError \"'noncomputable partial' is not allowed\"\n  if newHeader.modifiers.isPartial && newHeader.modifiers.isUnsafe then\n    throwError \"'unsafe' subsumes 'partial'\"\n  if h : 0 < prevHeaders.size then\n    let firstHeader := prevHeaders.get \u27e80, h\u27e9\n    try\n      unless newHeader.levelNames == firstHeader.levelNames do\n        throwError \"universe parameters mismatch\"\n      checkModifiers newHeader.modifiers firstHeader.modifiers\n      checkKinds newHeader.kind firstHeader.kind\n    catch\n       | Exception.error ref msg => throw (Exception.error ref m!\"invalid mutually recursive definitions, {msg}\")\n       | ex => throw ex\n  else\n    pure ()\n\nprivate def registerFailedToInferDefTypeInfo (type : Expr) (ref : Syntax) : TermElabM Unit :=\n  registerCustomErrorIfMVar type ref \"failed to infer definition type\"\n\n\/--\n  Return `some [b, c]` if the given `views` are representing a declaration of the form\n  ```\n  constant a b c : Nat\n  ```  -\/\nprivate def isMultiConstant? (views : Array DefView) : Option (List Name) :=\n  if views.size == 1 &&\n     views[0].kind == DefKind.opaque &&\n     views[0].binders.getArgs.size > 0 &&\n     views[0].binders.getArgs.all (\u00b7.getKind == ``Parser.Term.simpleBinder) then\n    some <| (views[0].binders.getArgs.toList.map (fun stx => stx[0].getArgs.toList.map (\u00b7.getId))).join\n  else\n    none\n\nprivate def getPendindMVarErrorMessage (views : Array DefView) : String :=\n  match isMultiConstant? views with\n  | some ids =>\n    let idsStr := \", \".intercalate <| ids.map fun id => s!\"`{id}`\"\n    let paramsStr := \", \".intercalate <| ids.map fun id => s!\"`({id} : _)`\"\n    s!\"\\nrecall that you cannot declare multiple constants in a single declaration. The identifier(s) {idsStr} are being interpreted as parameters {paramsStr}\"\n  | none =>\n    \"\\nwhen the resulting type of a declaration is explicitly provided, all holes (e.g., `_`) in the header are resolved before the declaration body is processed\"\n\nprivate def elabHeaders (views : Array DefView) : TermElabM (Array DefViewElabHeader) := do\n  let expandedDeclIds \u2190 views.mapM fun view => withRef view.ref do\n    Term.expandDeclId (\u2190 getCurrNamespace) (\u2190 getLevelNames) view.declId view.modifiers\n  withAutoBoundImplicitForbiddenPred (fun n => expandedDeclIds.any (\u00b7.shortName == n)) do\n    let mut headers := #[]\n    for view in views, \u27e8shortDeclName, declName, levelNames\u27e9 in expandedDeclIds do\n      let newHeader \u2190 withRef view.ref do\n        addDeclarationRanges declName view.ref\n        applyAttributesAt declName view.modifiers.attrs AttributeApplicationTime.beforeElaboration\n        withDeclName declName <| withAutoBoundImplicit <| withLevelNames levelNames <|\n          elabBindersEx view.binders.getArgs fun xs => do\n            let refForElabFunType := view.value\n            let type \u2190 match view.type? with\n              | some typeStx =>\n                let type \u2190 elabType typeStx\n                registerFailedToInferDefTypeInfo type typeStx\n                pure type\n              | none =>\n                let hole := mkHole refForElabFunType\n                let type \u2190 elabType hole\n                trace[Elab.definition] \">> type: {type}\\n{type.mvarId!}\"\n                registerFailedToInferDefTypeInfo type refForElabFunType\n                pure type\n            Term.synthesizeSyntheticMVarsNoPostponing\n            let (binderIds, xs) := xs.unzip\n            -- TODO: add forbidden predicate using `shortDeclName` from `views`\n            let xs \u2190 addAutoBoundImplicits xs\n            let type \u2190 mkForallFVars' xs type\n            let type \u2190 instantiateMVars type\n            let levelNames \u2190 getLevelNames\n            if view.type?.isSome then\n              let pendingMVarIds \u2190 getMVars type\n              discard <| logUnassignedUsingErrorInfos pendingMVarIds <|\n                getPendindMVarErrorMessage views\n            let newHeader := {\n              ref           := view.ref,\n              modifiers     := view.modifiers,\n              kind          := view.kind,\n              shortDeclName := shortDeclName,\n              declName      := declName,\n              levelNames    := levelNames,\n              binderIds     := binderIds,\n              numParams     := xs.size,\n              type          := type,\n              valueStx      := view.value : DefViewElabHeader }\n            check headers newHeader\n            return newHeader\n      headers := headers.push newHeader\n    return headers\n\nprivate partial def withFunLocalDecls {\u03b1} (headers : Array DefViewElabHeader) (k : Array Expr \u2192 TermElabM \u03b1) : TermElabM \u03b1 :=\n  let rec loop (i : Nat) (fvars : Array Expr) := do\n    if h : i < headers.size then\n      let header := headers.get \u27e8i, h\u27e9\n      if header.modifiers.isNonrec then\n        loop (i+1) fvars\n      else\n        withLocalDecl header.shortDeclName BinderInfo.auxDecl header.type fun fvar => loop (i+1) (fvars.push fvar)\n    else\n      k fvars\n  loop 0 #[]\n\nprivate def expandWhereStructInst : Macro\n  | `(Parser.Command.whereStructInst|where $[$decls:letDecl$[;]?]*) => do\n    let letIdDecls \u2190 decls.mapM fun stx => match stx with\n      | `(letDecl|$_decl:letPatDecl)  => Macro.throwErrorAt stx \"patterns are not allowed here\"\n      | `(letDecl|$decl:letEqnsDecl) => expandLetEqnsDecl decl\n      | `(letDecl|$decl:letIdDecl)   => pure decl\n      | _                               => Macro.throwUnsupported\n    let structInstFields \u2190 letIdDecls.mapM fun\n      | stx@`(letIdDecl|$id:ident $[$binders]* $[: $ty?]? := $val) => withRef stx do\n        let mut val := val\n        if let some ty := ty? then\n          val \u2190 `(($val : $ty))\n        val \u2190 if binders.size > 0 then `(fun $[$binders]* => $val:term) else pure val\n        `(structInstField|$id:ident := $val)\n      | _ => Macro.throwUnsupported\n    `({ $[$structInstFields,]* })\n  | _ => Macro.throwUnsupported\n\n\/-\nRecall that\n```\ndef declValSimple    := leading_parser \" :=\\n\" >> termParser >> optional Term.whereDecls\ndef declValEqns      := leading_parser Term.matchAltsWhereDecls\ndef declVal          := declValSimple <|> declValEqns <|> Term.whereDecls\n```\n-\/\nprivate def declValToTerm (declVal : Syntax) : MacroM Syntax := withRef declVal do\n  if declVal.isOfKind ``Lean.Parser.Command.declValSimple then\n    expandWhereDeclsOpt declVal[2] declVal[1]\n  else if declVal.isOfKind ``Lean.Parser.Command.declValEqns then\n    expandMatchAltsWhereDecls declVal[0]\n  else if declVal.isOfKind ``Lean.Parser.Command.whereStructInst then\n    expandWhereStructInst declVal\n  else if declVal.isMissing then\n    Macro.throwErrorAt declVal \"declaration body is missing\"\n  else\n    Macro.throwErrorAt declVal \"unexpected declaration body\"\n\nprivate def elabFunValues (headers : Array DefViewElabHeader) : TermElabM (Array Expr) :=\n  headers.mapM fun header => withDeclName header.declName <| withLevelNames header.levelNames do\n    let valStx \u2190 liftMacroM <| declValToTerm header.valueStx\n    forallBoundedTelescope header.type header.numParams fun xs type => do\n      -- Add new info nodes for new fvars. The server will detect all fvars of a binder by the binder's source location.\n      for i in [0:header.binderIds.size] do\n        -- skip auto-bound prefix in `xs`\n        addLocalVarInfo header.binderIds[i] xs[header.numParams - header.binderIds.size + i]\n      let val \u2190 elabTermEnsuringType valStx type\n      mkLambdaFVars xs val\n\nprivate def collectUsed (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)\n    : StateRefT CollectFVars.State MetaM Unit := do\n  headers.forM fun header => collectUsedFVars header.type\n  values.forM collectUsedFVars\n  toLift.forM fun letRecToLift => do\n    collectUsedFVars letRecToLift.type\n    collectUsedFVars letRecToLift.val\n\nprivate def removeUnusedVars (vars : Array Expr) (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)\n    : TermElabM (LocalContext \u00d7 LocalInstances \u00d7 Array Expr) := do\n  let (_, used) \u2190 (collectUsed headers values toLift).run {}\n  removeUnused vars used\n\nprivate def withUsed {\u03b1} (vars : Array Expr) (headers : Array DefViewElabHeader) (values : Array Expr) (toLift : List LetRecToLift)\n    (k : Array Expr \u2192 TermElabM \u03b1) : TermElabM \u03b1 := do\n  let (lctx, localInsts, vars) \u2190 removeUnusedVars vars headers values toLift\n  withLCtx lctx localInsts <| k vars\n\nprivate def isExample (views : Array DefView) : Bool :=\n  views.any (\u00b7.kind.isExample)\n\nprivate def isTheorem (views : Array DefView) : Bool :=\n  views.any (\u00b7.kind.isTheorem)\n\nprivate def instantiateMVarsAtHeader (header : DefViewElabHeader) : TermElabM DefViewElabHeader := do\n  let type \u2190 instantiateMVars header.type\n  pure { header with type := type }\n\nprivate def instantiateMVarsAtLetRecToLift (toLift : LetRecToLift) : TermElabM LetRecToLift := do\n  let type \u2190 instantiateMVars toLift.type\n  let val \u2190 instantiateMVars toLift.val\n  pure { toLift with type := type, val := val }\n\nprivate def typeHasRecFun (type : Expr) (funFVars : Array Expr) (letRecsToLift : List LetRecToLift) : Option FVarId :=\n  let occ? := type.find? fun e => match e with\n    | Expr.fvar fvarId _ => funFVars.contains e || letRecsToLift.any fun toLift => toLift.fvarId == fvarId\n    | _ => false\n  match occ? with\n  | some (Expr.fvar fvarId _) => some fvarId\n  | _ => none\n\nprivate def getFunName (fvarId : FVarId) (letRecsToLift : List LetRecToLift) : TermElabM Name := do\n  match (\u2190 findLocalDecl? fvarId) with\n  | some decl => pure decl.userName\n  | none =>\n    \/- Recall that the FVarId of nested let-recs are not in the current local context. -\/\n    match letRecsToLift.findSome? fun toLift => if toLift.fvarId == fvarId then some toLift.shortDeclName else none with\n    | none   => throwError \"unknown function\"\n    | some n => pure n\n\n\/-\nEnsures that the of let-rec definition types do not contain functions being defined.\nIn principle, this test can be improved. We could perform it after we separate the set of functions is strongly connected components.\nHowever, this extra complication doesn't seem worth it.\n-\/\nprivate def checkLetRecsToLiftTypes (funVars : Array Expr) (letRecsToLift : List LetRecToLift) : TermElabM Unit :=\n  letRecsToLift.forM fun toLift =>\n    match typeHasRecFun toLift.type funVars letRecsToLift with\n    | none        => pure ()\n    | some fvarId => do\n      let fnName \u2190 getFunName fvarId letRecsToLift\n      throwErrorAt toLift.ref \"invalid type in 'let rec', it uses '{fnName}' which is being defined simultaneously\"\n\nnamespace MutualClosure\n\n\/- A mapping from FVarId to Set of FVarIds. -\/\nabbrev UsedFVarsMap := FVarIdMap FVarIdSet\n\n\/-\nCreate the `UsedFVarsMap` mapping that takes the variable id for the mutually recursive functions being defined to the set of\nfree variables in its definition.\n\nFor `mainFVars`, this is just the set of section variables `sectionVars` used.\nFor nested let-rec functions, we collect their free variables.\n\nRecall that a `let rec` expressions are encoded as follows in the elaborator.\n```lean\nlet rec\n  f : A := t,\n  g : B := s;\nbody\n```\nis encoded as\n```lean\nlet f : A := ?m\u2081;\nlet g : B := ?m\u2082;\nbody\n```\nwhere `?m\u2081` and `?m\u2082` are synthetic opaque metavariables. That are assigned by this module.\nWe may have nested `let rec`s.\n```lean\nlet rec f : A :=\n    let rec g : B := t;\n    s;\nbody\n```\nis encoded as\n```lean\nlet f : A := ?m\u2081;\nbody\n```\nand the body of `f` is stored the field `val` of a `LetRecToLift`. For the example above,\nwe would have a `LetRecToLift` containing:\n```\n{\n  mvarId := m\u2081,\n  val    := `(let g : B := ?m\u2082; body)\n  ...\n}\n```\nNote that `g` is not a free variable at `(let g : B := ?m\u2082; body)`. We recover the fact that\n`f` depends on `g` because it contains `m\u2082`\n-\/\nprivate def mkInitialUsedFVarsMap (mctx : MetavarContext) (sectionVars : Array Expr) (mainFVarIds : Array FVarId) (letRecsToLift : Array LetRecToLift)\n    : UsedFVarsMap := Id.run do\n  let mut sectionVarSet := {}\n  for var in sectionVars do\n    sectionVarSet := sectionVarSet.insert var.fvarId!\n  let mut usedFVarMap := {}\n  for mainFVarId in mainFVarIds do\n    usedFVarMap := usedFVarMap.insert mainFVarId sectionVarSet\n  for toLift in letRecsToLift do\n    let state := Lean.collectFVars {} toLift.val\n    let state := Lean.collectFVars state toLift.type\n    let mut set := state.fvarSet\n    \/- toLift.val may contain metavariables that are placeholders for nested let-recs. We should collect the fvarId\n       for the associated let-rec because we need this information to compute the fixpoint later. -\/\n    let mvarIds := (toLift.val.collectMVars {}).result\n    for mvarId in mvarIds do\n      match letRecsToLift.findSome? fun (toLift : LetRecToLift) => if toLift.mvarId == mctx.getDelayedRoot mvarId then some toLift.fvarId else none with\n      | some fvarId => set := set.insert fvarId\n      | none        => pure ()\n    usedFVarMap := usedFVarMap.insert toLift.fvarId set\n  pure usedFVarMap\n\n\/-\nThe let-recs may invoke each other. Example:\n```\nlet rec\n  f (x : Nat) := g x + y\n  g : Nat \u2192 Nat\n    | 0   => 1\n    | x+1 => f x + z\n```\n`y` is free variable in `f`, and `z` is a free variable in `g`.\nTo close `f` and `g`, `y` and `z` must be in the closure of both.\nThat is, we need to generate the top-level definitions.\n```\ndef f (y z x : Nat) := g y z x + y\ndef g (y z : Nat) : Nat \u2192 Nat\n  | 0 => 1\n  | x+1 => f y z x + z\n```\n-\/\nnamespace FixPoint\n\nstructure State where\n  usedFVarsMap : UsedFVarsMap := {}\n  modified     : Bool         := false\n\nabbrev M := ReaderT (Array FVarId) $ StateM State\n\nprivate def isModified : M Bool := do pure (\u2190 get).modified\nprivate def resetModified : M Unit := modify fun s => { s with modified := false }\nprivate def markModified : M Unit := modify fun s => { s with modified := true }\nprivate def getUsedFVarsMap : M UsedFVarsMap := do pure (\u2190 get).usedFVarsMap\nprivate def modifyUsedFVars (f : UsedFVarsMap \u2192 UsedFVarsMap) : M Unit := modify fun s => { s with usedFVarsMap := f s.usedFVarsMap }\n\n-- merge s\u2082 into s\u2081\nprivate def merge (s\u2081 s\u2082 : FVarIdSet) : M FVarIdSet :=\n  s\u2082.foldM (init := s\u2081) fun s\u2081 k => do\n    if s\u2081.contains k then\n      return s\u2081\n    else\n      markModified\n      return s\u2081.insert k\n\nprivate def updateUsedVarsOf (fvarId : FVarId) : M Unit := do\n  let usedFVarsMap \u2190 getUsedFVarsMap\n  match usedFVarsMap.find? fvarId with\n  | none         => pure ()\n  | some fvarIds =>\n    let fvarIdsNew \u2190 fvarIds.foldM (init := fvarIds) fun fvarIdsNew fvarId' =>\n      if fvarId == fvarId' then\n        pure fvarIdsNew\n      else\n        match usedFVarsMap.find? fvarId' with\n        | none => pure fvarIdsNew\n          \/- We are being sloppy here `otherFVarIds` may contain free variables that are\n             not in the context of the let-rec associated with fvarId.\n             We filter these out-of-context free variables later. -\/\n        | some otherFVarIds => merge fvarIdsNew otherFVarIds\n    modifyUsedFVars fun usedFVars => usedFVars.insert fvarId fvarIdsNew\n\nprivate partial def fixpoint : Unit \u2192 M Unit\n  | _ => do\n    resetModified\n    let letRecFVarIds \u2190 read\n    letRecFVarIds.forM updateUsedVarsOf\n    if (\u2190 isModified) then\n      fixpoint ()\n\ndef run (letRecFVarIds : Array FVarId) (usedFVarsMap : UsedFVarsMap) : UsedFVarsMap :=\n  let (_, s) := ((fixpoint ()).run letRecFVarIds).run { usedFVarsMap := usedFVarsMap }\n  s.usedFVarsMap\n\nend FixPoint\n\nabbrev FreeVarMap := FVarIdMap (Array FVarId)\n\nprivate def mkFreeVarMap\n    (mctx : MetavarContext) (sectionVars : Array Expr) (mainFVarIds : Array FVarId)\n    (recFVarIds : Array FVarId) (letRecsToLift : Array LetRecToLift) : FreeVarMap := Id.run do\n  let usedFVarsMap  := mkInitialUsedFVarsMap mctx sectionVars mainFVarIds letRecsToLift\n  let letRecFVarIds := letRecsToLift.map fun toLift => toLift.fvarId\n  let usedFVarsMap  := FixPoint.run letRecFVarIds usedFVarsMap\n  let mut freeVarMap := {}\n  for toLift in letRecsToLift do\n    let lctx       := toLift.lctx\n    let fvarIdsSet := (usedFVarsMap.find? toLift.fvarId).get!\n    let fvarIds    := fvarIdsSet.fold (init := #[]) fun fvarIds fvarId =>\n      if lctx.contains fvarId && !recFVarIds.contains fvarId then\n        fvarIds.push fvarId\n      else\n        fvarIds\n    freeVarMap := freeVarMap.insert toLift.fvarId fvarIds\n  return freeVarMap\n\nstructure ClosureState where\n  newLocalDecls : Array LocalDecl := #[]\n  localDecls    : Array LocalDecl := #[]\n  newLetDecls   : Array LocalDecl := #[]\n  exprArgs      : Array Expr      := #[]\n\nprivate def pickMaxFVar? (lctx : LocalContext) (fvarIds : Array FVarId) : Option FVarId :=\n  fvarIds.getMax? fun fvarId\u2081 fvarId\u2082 => (lctx.get! fvarId\u2081).index < (lctx.get! fvarId\u2082).index\n\nprivate def preprocess (e : Expr) : TermElabM Expr := do\n  let e \u2190 instantiateMVars e\n  -- which let-decls are dependent. We say a let-decl is dependent if its lambda abstraction is type incorrect.\n  Meta.check e\n  pure e\n\n\/- Push free variables in `s` to `toProcess` if they are not already there. -\/\nprivate def pushNewVars (toProcess : Array FVarId) (s : CollectFVars.State) : Array FVarId :=\n  s.fvarSet.fold (init := toProcess) fun toProcess fvarId =>\n    if toProcess.contains fvarId then toProcess else toProcess.push fvarId\n\nprivate def pushLocalDecl (toProcess : Array FVarId) (fvarId : FVarId) (userName : Name) (type : Expr) (bi := BinderInfo.default)\n    : StateRefT ClosureState TermElabM (Array FVarId) := do\n  let type \u2190 preprocess type\n  modify fun s => { s with\n    newLocalDecls := s.newLocalDecls.push <| LocalDecl.cdecl default fvarId userName type bi,\n    exprArgs      := s.exprArgs.push (mkFVar fvarId)\n  }\n  return pushNewVars toProcess (collectFVars {} type)\n\nprivate partial def mkClosureForAux (toProcess : Array FVarId) : StateRefT ClosureState TermElabM Unit := do\n  let lctx \u2190 getLCtx\n  match pickMaxFVar? lctx toProcess with\n  | none        => pure ()\n  | some fvarId =>\n    trace[Elab.definition.mkClosure] \"toProcess: {toProcess.map mkFVar}, maxVar: {mkFVar fvarId}\"\n    let toProcess := toProcess.erase fvarId\n    let localDecl \u2190 getLocalDecl fvarId\n    match localDecl with\n    | LocalDecl.cdecl _ _ userName type bi =>\n      let toProcess \u2190 pushLocalDecl toProcess fvarId userName type bi\n      mkClosureForAux toProcess\n    | LocalDecl.ldecl _ _ userName type val _ =>\n      let zetaFVarIds \u2190 getZetaFVarIds\n      if !zetaFVarIds.contains fvarId then\n        \/- Non-dependent let-decl. See comment at src\/Lean\/Meta\/Closure.lean -\/\n        let toProcess \u2190 pushLocalDecl toProcess fvarId userName type\n        mkClosureForAux toProcess\n      else\n        \/- Dependent let-decl. -\/\n        let type \u2190 preprocess type\n        let val  \u2190 preprocess val\n        modify fun s => { s with\n          newLetDecls   := s.newLetDecls.push <| LocalDecl.ldecl default fvarId userName type val false,\n          \/- We don't want to interleave let and lambda declarations in our closure. So, we expand any occurrences of fvarId\n             at `newLocalDecls` and `localDecls` -\/\n          newLocalDecls := s.newLocalDecls.map (replaceFVarIdAtLocalDecl fvarId val),\n          localDecls := s.localDecls.map (replaceFVarIdAtLocalDecl fvarId val)\n        }\n        mkClosureForAux (pushNewVars toProcess (collectFVars (collectFVars {} type) val))\n\nprivate partial def mkClosureFor (freeVars : Array FVarId) (localDecls : Array LocalDecl) : TermElabM ClosureState := do\n  let (_, s) \u2190 (mkClosureForAux freeVars).run { localDecls := localDecls }\n  pure { s with\n    newLocalDecls := s.newLocalDecls.reverse,\n    newLetDecls   := s.newLetDecls.reverse,\n    exprArgs      := s.exprArgs.reverse\n  }\n\nstructure LetRecClosure where\n  ref        : Syntax\n  localDecls : Array LocalDecl\n  closed     : Expr -- expression used to replace occurrences of the let-rec FVarId\n  toLift     : LetRecToLift\n\nprivate def mkLetRecClosureFor (toLift : LetRecToLift) (freeVars : Array FVarId) : TermElabM LetRecClosure := do\n  let lctx := toLift.lctx\n  withLCtx lctx toLift.localInstances do\n  lambdaTelescope toLift.val fun xs val => do\n    let type \u2190 instantiateForall toLift.type xs\n    let lctx \u2190 getLCtx\n    let s \u2190 mkClosureFor freeVars <| xs.map fun x => lctx.get! x.fvarId!\n    let type := Closure.mkForall s.localDecls <| Closure.mkForall s.newLetDecls type\n    let val  := Closure.mkLambda s.localDecls <| Closure.mkLambda s.newLetDecls val\n    let c    := mkAppN (Lean.mkConst toLift.declName) s.exprArgs\n    assignExprMVar toLift.mvarId c\n    return {\n      ref        := toLift.ref\n      localDecls := s.newLocalDecls\n      closed     := c\n      toLift     := { toLift with val := val, type := type }\n    }\n\nprivate def mkLetRecClosures (sectionVars : Array Expr) (mainFVarIds : Array FVarId) (recFVarIds : Array FVarId) (letRecsToLift : Array LetRecToLift) : TermElabM (List LetRecClosure) := do\n  -- Compute the set of free variables (excluding `recFVarIds`) for each let-rec.\n  let mut letRecsToLift := letRecsToLift\n  let mut freeVarMap    := mkFreeVarMap (\u2190 getMCtx) sectionVars mainFVarIds recFVarIds letRecsToLift\n  let mut result := #[]\n  for i in [:letRecsToLift.size] do\n    if letRecsToLift[i].val.hasExprMVar then\n      -- This can happen when this particular let-rec has nested let-rec that have been resolved in previous iterations.\n      -- This code relies on the fact that nested let-recs occur before the outer most let-recs at `letRecsToLift`.\n      -- Unresolved nested let-recs appear as metavariables before they are resolved. See `assignExprMVar` at `mkLetRecClosureFor`\n      let valNew \u2190 instantiateMVars letRecsToLift[i].val\n      letRecsToLift := letRecsToLift.modify i fun t => { t with val := valNew }\n      -- We have to recompute the `freeVarMap` in this case. This overhead should not be an issue in practice.\n      freeVarMap := mkFreeVarMap (\u2190 getMCtx) sectionVars mainFVarIds recFVarIds letRecsToLift\n    let toLift := letRecsToLift[i]\n    result := result.push (\u2190 mkLetRecClosureFor toLift (freeVarMap.find? toLift.fvarId).get!)\n  return result.toList\n\n\/- Mapping from FVarId of mutually recursive functions being defined to \"closure\" expression. -\/\nabbrev Replacement := FVarIdMap Expr\n\ndef insertReplacementForMainFns (r : Replacement) (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainFVars : Array Expr) : Replacement :=\n  mainFVars.size.fold (init := r) fun i r =>\n    r.insert mainFVars[i].fvarId! (mkAppN (Lean.mkConst mainHeaders[i].declName) sectionVars)\n\n\ndef insertReplacementForLetRecs (r : Replacement) (letRecClosures : List LetRecClosure) : Replacement :=\n  letRecClosures.foldl (init := r) fun r c =>\n    r.insert c.toLift.fvarId c.closed\n\ndef Replacement.apply (r : Replacement) (e : Expr) : Expr :=\n  e.replace fun e => match e with\n    | Expr.fvar fvarId _ => match r.find? fvarId with\n      | some c => some c\n      | _      => none\n    | _ => none\n\ndef pushMain (preDefs : Array PreDefinition) (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainVals : Array Expr)\n    : TermElabM (Array PreDefinition) :=\n  mainHeaders.size.foldM (init := preDefs) fun i preDefs => do\n    let header := mainHeaders[i]\n    let val  \u2190 mkLambdaFVars sectionVars mainVals[i]\n    let type \u2190 mkForallFVars sectionVars header.type\n    return preDefs.push {\n      ref         := getDeclarationSelectionRef header.ref\n      kind        := header.kind\n      declName    := header.declName\n      levelParams := [], -- we set it later\n      modifiers   := header.modifiers\n      type        := type\n      value       := val\n    }\n\ndef pushLetRecs (preDefs : Array PreDefinition) (letRecClosures : List LetRecClosure) (kind : DefKind) (modifiers : Modifiers) : Array PreDefinition :=\n  letRecClosures.foldl (init := preDefs) fun preDefs c =>\n    let type := Closure.mkForall c.localDecls c.toLift.type\n    let val  := Closure.mkLambda c.localDecls c.toLift.val\n    preDefs.push {\n      ref         := c.ref\n      kind        := kind\n      declName    := c.toLift.declName\n      levelParams := [] -- we set it later\n      modifiers   := { modifiers with attrs := c.toLift.attrs }\n      type        := type\n      value       := val\n    }\n\ndef getKindForLetRecs (mainHeaders : Array DefViewElabHeader) : DefKind :=\n  if mainHeaders.any fun h => h.kind.isTheorem then DefKind.\u00abtheorem\u00bb\n  else DefKind.\u00abdef\u00bb\n\ndef getModifiersForLetRecs (mainHeaders : Array DefViewElabHeader) : Modifiers := {\n  isNoncomputable := mainHeaders.any fun h => h.modifiers.isNoncomputable\n  recKind         := if mainHeaders.any fun h => h.modifiers.isPartial then RecKind.partial else RecKind.default\n  isUnsafe        := mainHeaders.any fun h => h.modifiers.isUnsafe\n}\n\n\/-\n- `sectionVars`:   The section variables used in the `mutual` block.\n- `mainHeaders`:   The elaborated header of the top-level definitions being defined by the mutual block.\n- `mainFVars`:     The auxiliary variables used to represent the top-level definitions being defined by the mutual block.\n- `mainVals`:      The elaborated value for the top-level definitions\n- `letRecsToLift`: The let-rec's definitions that need to be lifted\n-\/\ndef main (sectionVars : Array Expr) (mainHeaders : Array DefViewElabHeader) (mainFVars : Array Expr) (mainVals : Array Expr) (letRecsToLift : List LetRecToLift)\n    : TermElabM (Array PreDefinition) := do\n  -- Store in recFVarIds the fvarId of every function being defined by the mutual block.\n  let letRecsToLift := letRecsToLift.toArray\n  let mainFVarIds := mainFVars.map Expr.fvarId!\n  let recFVarIds  := (letRecsToLift.map fun toLift => toLift.fvarId) ++ mainFVarIds\n  resetZetaFVarIds\n  withTrackingZeta do\n    -- By checking `toLift.type` and `toLift.val` we populate `zetaFVarIds`. See comments at `src\/Lean\/Meta\/Closure.lean`.\n    let letRecsToLift \u2190 letRecsToLift.mapM fun toLift => withLCtx toLift.lctx toLift.localInstances do\n      Meta.check toLift.type\n      Meta.check toLift.val\n      return { toLift with val := (\u2190 instantiateMVars toLift.val), type := (\u2190 instantiateMVars toLift.type) }\n    let letRecClosures \u2190 mkLetRecClosures sectionVars mainFVarIds recFVarIds letRecsToLift\n    -- mkLetRecClosures assign metavariables that were placeholders for the lifted declarations.\n    let mainVals    \u2190 mainVals.mapM (instantiateMVars \u00b7)\n    let mainHeaders \u2190 mainHeaders.mapM instantiateMVarsAtHeader\n    let letRecClosures \u2190 letRecClosures.mapM fun closure => do pure { closure with toLift := (\u2190 instantiateMVarsAtLetRecToLift closure.toLift) }\n    -- Replace fvarIds for functions being defined with closed terms\n    let r              := insertReplacementForMainFns {} sectionVars mainHeaders mainFVars\n    let r              := insertReplacementForLetRecs r letRecClosures\n    let mainVals       := mainVals.map r.apply\n    let mainHeaders    := mainHeaders.map fun h => { h with type := r.apply h.type }\n    let letRecClosures := letRecClosures.map fun c => { c with toLift := { c.toLift with type := r.apply c.toLift.type, val := r.apply c.toLift.val } }\n    let letRecKind     := getKindForLetRecs mainHeaders\n    let letRecMods     := getModifiersForLetRecs mainHeaders\n    pushMain (pushLetRecs #[] letRecClosures letRecKind letRecMods) sectionVars mainHeaders mainVals\n\nend MutualClosure\n\nprivate def getAllUserLevelNames (headers : Array DefViewElabHeader) : List Name :=\n  if h : 0 < headers.size then\n    -- Recall that all top-level functions must have the same levels. See `check` method above\n    (headers.get \u27e80, h\u27e9).levelNames\n  else\n    []\n\n\/-- Eagerly convert universe metavariables occurring in theorem headers to universe parameters. -\/\nprivate def levelMVarToParamHeaders (views : Array DefView) (headers : Array DefViewElabHeader) : TermElabM (Array DefViewElabHeader) := do\n  let rec process : StateRefT Nat TermElabM (Array DefViewElabHeader) := do\n    let mut newHeaders := #[]\n    for view in views, header in headers do\n      if view.kind.isTheorem then\n        newHeaders := newHeaders.push { header with type := (\u2190 levelMVarToParam' header.type) }\n      else\n        newHeaders := newHeaders.push header\n    return newHeaders\n  let newHeaders \u2190 (process).run' 1\n  newHeaders.mapM fun header => return { header with type := (\u2190 instantiateMVars header.type) }\n\n\/-- Result for `mkInst?` -\/\nstructure MkInstResult where\n  instVal   : Expr\n  instType  : Expr\n  outParams : Array Expr := #[]\n\n\/--\n  Construct an instance for `className out\u2081 ... out\u2099 type`.\n  The method support classes with a prefix of `outParam`s (e.g. `MonadReader`). -\/\nprivate partial def mkInst? (className : Name) (type : Expr) : MetaM (Option MkInstResult) := do\n  let rec go? (instType instTypeType : Expr) (outParams : Array Expr) : MetaM (Option MkInstResult) := do\n    let instTypeType \u2190 whnfD instTypeType\n    unless instTypeType.isForall do\n      return none\n    let d := instTypeType.bindingDomain!\n    if d.isOutParam then\n      let mvar \u2190 mkFreshExprMVar d\n      go? (mkApp instType mvar) (instTypeType.bindingBody!.instantiate1 mvar) (outParams.push mvar)\n    else\n      unless (\u2190 isDefEqGuarded (\u2190 inferType type) d) do\n        return none\n      let instType \u2190 instantiateMVars (mkApp instType type)\n      let instVal \u2190 synthInstance instType\n      return some { instVal, instType, outParams }\n  let instType \u2190 mkConstWithFreshMVarLevels className\n  go? instType (\u2190 inferType instType) #[]\n\ndef processDefDeriving (className : Name) (declName : Name) : TermElabM Bool := do\n  try\n    let ConstantInfo.defnInfo info \u2190 getConstInfo declName | return false\n    let some result \u2190 mkInst? className info.value | return false\n    let instTypeNew := mkApp result.instType.appFn! (Lean.mkConst declName (info.levelParams.map mkLevelParam))\n    Meta.check instTypeNew\n    let instName \u2190 liftMacroM <| mkUnusedBaseName (declName.appendBefore \"inst\" |>.appendAfter className.getString!)\n    addAndCompile <| Declaration.defnDecl {\n      name        := instName\n      levelParams := info.levelParams\n      type        := (\u2190 instantiateMVars instTypeNew)\n      value       := (\u2190 instantiateMVars result.instVal)\n      hints       := info.hints\n      safety      := info.safety\n    }\n    addInstance instName AttributeKind.global (eval_prio default)\n    return true\n  catch _ =>\n    return false\n\n\/-- Remove auxiliary match discriminant let-declarations. -\/\ndef eraseAuxDiscr (e : Expr) : CoreM Expr := do\n  Core.transform e fun e => match e with\n    | Expr.letE n _ v b .. =>\n      if isAuxDiscrName n then\n        return TransformStep.visit (b.instantiate1 v)\n      else\n        return TransformStep.visit e\n    | e => return TransformStep.visit e\n\npartial def checkForHiddenUnivLevels (allUserLevelNames : List Name) (preDefs : Array PreDefinition) : TermElabM Unit :=\n  unless (\u2190 MonadLog.hasErrors) do\n    -- We do not report this kind of error if the declaration already contains errors\n    let mut sTypes : CollectLevelParams.State := {}\n    let mut sValues : CollectLevelParams.State := {}\n    for preDef in preDefs do\n      sTypes  := collectLevelParams sTypes preDef.type\n      sValues := collectLevelParams sValues preDef.value\n    if sValues.params.all fun u => sTypes.params.contains u || allUserLevelNames.contains u then\n      -- If all universe level occurring in values also occur in types or explicitly provided universes, then everything is fine\n      -- and we just return\n      return ()\n    let checkPreDef (preDef : PreDefinition) : TermElabM Unit :=\n      -- Otherwise, we try to produce an error message containing the expression with the offending universe\n      let rec visitLevel (u : Level) : ReaderT Expr TermElabM Unit := do\n        match u with\n        | .succ u _ => visitLevel u\n        | .imax u v _ | .max u v _ => visitLevel u; visitLevel v\n        | .param n _ =>\n          unless sTypes.visitedLevel.contains u || allUserLevelNames.contains n do\n            let parent \u2190 withOptions (fun o => pp.universes.set o true) do addMessageContext m!\"{indentExpr (\u2190 read)}\"\n            let body \u2190 withOptions (fun o => pp.letVarTypes.setIfNotSet (pp.funBinderTypes.setIfNotSet o true) true) do addMessageContext m!\"{indentExpr preDef.value}\"\n            throwError \"invalid occurrence of universe level '{u}' at '{preDef.declName}', it does not occur at the declaration type, nor it is explicit universe level provided by the user, occurring at expression{parent}\\nat declaration body{body}\"\n        | _ => pure ()\n      let rec visit (e : Expr) : ReaderT Expr (MonadCacheT ExprStructEq Unit TermElabM) Unit := do\n        checkCache { val := e : ExprStructEq } fun _ => do\n          match e with\n          | .forallE n d b c | .lam n d b c => visit d e; withLocalDecl n c.binderInfo d fun x => visit (b.instantiate1 x) e\n          | .letE n t v b _  => visit t e; visit v e; withLetDecl n t v fun x => visit (b.instantiate1 x) e\n          | .app ..        => e.withApp fun f args => do visit f e; args.forM fun arg => visit arg e\n          | .mdata _ b _   => visit b e\n          | .proj _ _ b _  => visit b e\n          | .sort u _      => visitLevel u (\u2190 read)\n          | .const _ us _  => us.forM (visitLevel \u00b7 (\u2190 read))\n          | _              => pure ()\n      visit preDef.value preDef.value |>.run {}\n    for preDef in preDefs do\n      checkPreDef preDef\n\ndef elabMutualDef (vars : Array Expr) (views : Array DefView) (hints : TerminationHints) : TermElabM Unit :=\n  if isExample views then\n    withoutModifyingEnv go\n  else\n    go\nwhere\n  go := do\n    let scopeLevelNames \u2190 getLevelNames\n    let headers \u2190 elabHeaders views\n    let headers \u2190 levelMVarToParamHeaders views headers\n    let allUserLevelNames := getAllUserLevelNames headers\n    withFunLocalDecls headers fun funFVars => do\n      for view in views, funFVar in funFVars do\n        addLocalVarInfo view.declId funFVar\n      let values \u2190 elabFunValues headers\n      Term.synthesizeSyntheticMVarsNoPostponing\n      let values \u2190 values.mapM (instantiateMVars \u00b7)\n      let headers \u2190 headers.mapM instantiateMVarsAtHeader\n      let letRecsToLift \u2190 getLetRecsToLift\n      let letRecsToLift \u2190 letRecsToLift.mapM instantiateMVarsAtLetRecToLift\n      checkLetRecsToLiftTypes funFVars letRecsToLift\n      withUsed vars headers values letRecsToLift fun vars => do\n        let preDefs \u2190 MutualClosure.main vars headers funFVars values letRecsToLift\n        for preDef in preDefs do\n          trace[Elab.definition] \"{preDef.declName} : {preDef.type} :=\\n{preDef.value}\"\n        let preDefs \u2190 levelMVarToParamPreDecls preDefs\n        let preDefs \u2190 instantiateMVarsAtPreDecls preDefs\n        let preDefs \u2190 fixLevelParams preDefs scopeLevelNames allUserLevelNames\n        let preDefs \u2190 preDefs.mapM fun preDef =>\n          if preDef.kind.isTheorem || preDef.kind.isExample then\n            return preDef\n          else\n            return { preDef with value := (\u2190 eraseAuxDiscr preDef.value) }\n        checkForHiddenUnivLevels allUserLevelNames preDefs\n        addPreDefinitions preDefs hints\n        processDeriving headers\n\n  processDeriving (headers : Array DefViewElabHeader) := do\n    for header in headers, view in views do\n      if let some classNamesStx := view.deriving? then\n        for classNameStx in classNamesStx do\n          let className \u2190 resolveGlobalConstNoOverload classNameStx\n          withRef classNameStx do\n            unless (\u2190 processDefDeriving className header.declName) do\n              throwError \"failed to synthesize instance '{className}' for '{header.declName}'\"\n\nend Term\nnamespace Command\n\ndef elabMutualDef (ds : Array Syntax) (hints : TerminationHints) : CommandElabM Unit := do\n  let views \u2190 ds.mapM fun d => do\n    let modifiers \u2190 elabModifiers d[0]\n    if ds.size > 1 && modifiers.isNonrec then\n      throwErrorAt d \"invalid use of 'nonrec' modifier in 'mutual' block\"\n    mkDefView modifiers d[1]\n  runTermElabM none fun vars => Term.elabMutualDef vars views hints\n\nend Command\nend Lean.Elab\n","avg_line_length":46.2530413625,"max_line_length":249,"alphanum_fraction":0.6929773803}
-{"size":963,"ext":"lean","lang":"Lean","max_stars_count":1538.0,"content":"def m1 : IO Bool :=\npure true\n\ndef p (x : Nat) : Bool :=\nx > 0\n\ndef tst1 : IO Bool :=\ntrue <&&> m1\n\ndef tst2 (x : Nat) : IO Bool :=\nx = 0 <&&> m1\n\ndef tst3 (x : Nat) : IO Unit := do\nif \u2190 (m1 <&&> x > 0) then\n  throw $ IO.userError \"test\"\n\ndef tst4 (x : Nat) : IO Unit := do\nif x > 0 && (\u2190 m1) then\n  throw $ IO.userError \"test\"\n\ndef tst5 (x : Nat) : IO Unit := do\nif \u2190 (p x <&&> m1) then\n  throw $ IO.userError \"test\"\n\ndef tst6 (x : Nat) : IO Unit := do\nif \u2190 (p x <&&> id m1) then\n  throw $ IO.userError \"test\"\n\ndef tst7 (x : Nat) : IO Unit := do\nif (\u2190 m1) && x > 0 then\n  throw $ IO.userError \"test\"\n\ndef tst8 (x : Nat) : IO Unit := do\nif x > 0 && (\u2190 m1) then\n  throw $ IO.userError \"test\"\n\ndef tst9 (x : Nat) : IO Unit := do\nif p x && (\u2190 m1) then\n  throw $ IO.userError \"test\"\n\ndef tst10 (x : Nat) : IO Unit := do\nif p x && (\u2190 id m1) then\n  throw $ IO.userError \"test\"\n\ndef tst11 (x : Nat) : IO Unit := do\nif p x && id (\u2190 m1) then\n  throw $ IO.userError \"test\"\n","avg_line_length":20.0625,"max_line_length":35,"alphanum_fraction":0.5420560748}
-{"size":24608,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2014 Robert Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Lewis, Leonardo de Moura, Mario Carneiro\n-\/\nimport algebra.ordered_ring\nimport algebra.field\n\nset_option default_priority 100 -- see Note [default priority]\nset_option old_structure_cmd true\n\nvariable {\u03b1 : Type*}\n\n@[protect_proj] class linear_ordered_field (\u03b1 : Type*) extends linear_ordered_ring \u03b1, field \u03b1\n\nsection linear_ordered_field\nvariables [linear_ordered_field \u03b1] {a b c d e : \u03b1}\n\nlemma mul_zero_lt_mul_inv_of_pos (h : 0 < a) : a * 0 < a * (1 \/ a) :=\ncalc a * 0 = 0           : by rw mul_zero\n       ... < 1           : zero_lt_one\n       ... = a * a\u207b\u00b9     : eq.symm (mul_inv_cancel (ne.symm (ne_of_lt h)))\n       ... = a * (1 \/ a) : by rw inv_eq_one_div\n\nlemma mul_zero_lt_mul_inv_of_neg (h : a < 0) : a * 0 < a * (1 \/ a) :=\ncalc  a * 0 = 0           : by rw mul_zero\n        ... < 1           : zero_lt_one\n        ... = a * a\u207b\u00b9     : eq.symm (mul_inv_cancel (ne_of_lt h))\n        ... = a * (1 \/ a) : by rw inv_eq_one_div\n\nlemma one_div_pos_of_pos (h : 0 < a) : 0 < 1 \/ a :=\nlt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos h) (le_of_lt h)\n\nlemma pos_of_one_div_pos (h : 0 < 1 \/ a) : 0 < a :=\none_div_one_div a \u25b8 one_div_pos_of_pos h\n\nlemma one_div_neg_of_neg (h : a < 0) : 1 \/ a < 0 :=\ngt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg h) (le_of_lt h)\n\nlemma neg_of_one_div_neg (h : 1 \/ a < 0) : a < 0 :=\none_div_one_div a \u25b8 one_div_neg_of_neg h\n\nlemma le_mul_of_ge_one_right (hb : b \u2265 0) (h : a \u2265 1) : b \u2264 b * a :=\nsuffices b * 1 \u2264 b * a, by rwa mul_one at this,\nmul_le_mul_of_nonneg_left h hb\n\nlemma le_mul_of_ge_one_left (hb : b \u2265 0) (h : a \u2265 1) : b \u2264 a * b :=\nby rw mul_comm; exact le_mul_of_ge_one_right hb h\n\nlemma lt_mul_of_gt_one_right (hb : b > 0) (h : a > 1) : b < b * a :=\nsuffices b * 1 < b * a, by rwa mul_one at this,\nmul_lt_mul_of_pos_left h hb\n\nlemma one_le_div_of_le (a : \u03b1) {b : \u03b1} (hb : b > 0) (h : b \u2264 a) : 1 \u2264 a \/ b :=\nhave hb'   : b \u2260 0,     from ne.symm (ne_of_lt hb),\nhave hbinv : 1 \/ b > 0, from  one_div_pos_of_pos hb,\ncalc\n   1  = b * (1 \/ b)  : eq.symm (mul_one_div_cancel hb')\n   ... \u2264 a * (1 \/ b) : mul_le_mul_of_nonneg_right h (le_of_lt hbinv)\n   ... = a \/ b       : eq.symm $ div_eq_mul_one_div a b\n\nlemma le_of_one_le_div (a : \u03b1) {b : \u03b1} (hb : b > 0) (h : 1 \u2264 a \/ b) : b \u2264 a :=\nhave hb'   : b \u2260 0,     from ne.symm (ne_of_lt hb),\ncalc\n   b   \u2264 b * (a \/ b) : le_mul_of_ge_one_right (le_of_lt hb) h\n   ... = a           : by rw [mul_div_cancel' _ hb']\n\nlemma one_lt_div_of_lt (a : \u03b1) {b : \u03b1} (hb : b > 0) (h : b < a) : 1 < a \/ b :=\nhave hb' : b \u2260 0, from ne.symm (ne_of_lt hb),\nhave hbinv : 1 \/ b > 0, from  one_div_pos_of_pos hb, calc\n     1 = b * (1 \/ b) : eq.symm (mul_one_div_cancel hb')\n   ... < a * (1 \/ b) : mul_lt_mul_of_pos_right h hbinv\n   ... = a \/ b       : eq.symm $ div_eq_mul_one_div a b\n\nlemma lt_of_one_lt_div (a : \u03b1) {b : \u03b1} (hb : b > 0) (h : 1 < a \/ b) : b < a :=\nhave hb' : b \u2260 0, from ne.symm (ne_of_lt hb),\ncalc\n   b   < b * (a \/ b) : lt_mul_of_gt_one_right hb h\n   ... = a           : by rw [mul_div_cancel' _ hb']\n\n-- the following lemmas amount to four iffs, for <, \u2264, \u2265, >.\n\nlemma mul_le_of_le_div (hc : 0 < c) (h : a \u2264 b \/ c) : a * c \u2264 b :=\ndiv_mul_cancel b (ne.symm (ne_of_lt hc)) \u25b8 mul_le_mul_of_nonneg_right h (le_of_lt hc)\n\nlemma le_div_of_mul_le (hc : 0 < c) (h : a * c \u2264 b) : a \u2264 b \/ c :=\ncalc\n    a   = a * c * (1 \/ c) : mul_mul_div a (ne.symm (ne_of_lt hc))\n    ... \u2264 b * (1 \/ c)     : mul_le_mul_of_nonneg_right h (le_of_lt (one_div_pos_of_pos hc))\n    ... = b \/ c           : eq.symm $ div_eq_mul_one_div b c\n\nlemma mul_lt_of_lt_div (hc : 0 < c) (h : a < b \/ c) : a * c < b :=\ndiv_mul_cancel b (ne.symm (ne_of_lt hc)) \u25b8 mul_lt_mul_of_pos_right h hc\n\nlemma lt_div_of_mul_lt (hc : 0 < c) (h : a * c < b) : a < b \/ c :=\ncalc\n   a   = a * c * (1 \/ c) : mul_mul_div a (ne.symm (ne_of_lt hc))\n   ... < b * (1 \/ c)     : mul_lt_mul_of_pos_right h (one_div_pos_of_pos hc)\n   ... = b \/ c           : eq.symm $ div_eq_mul_one_div b c\n\nlemma mul_le_of_div_le_of_neg (hc : c < 0) (h : b \/ c \u2264 a) : a * c \u2264 b :=\ndiv_mul_cancel b (ne_of_lt hc) \u25b8 mul_le_mul_of_nonpos_right h (le_of_lt hc)\n\nlemma div_le_of_mul_le_of_neg (hc : c < 0) (h : a * c \u2264 b) : b \/ c \u2264 a :=\ncalc\n   a   = a * c * (1 \/ c) : mul_mul_div a (ne_of_lt hc)\n    ... \u2265 b * (1 \/ c)     : mul_le_mul_of_nonpos_right h (le_of_lt (one_div_neg_of_neg hc))\n    ... = b \/ c           : eq.symm $ div_eq_mul_one_div b c\n\nlemma mul_lt_of_gt_div_of_neg (hc : c < 0) (h : a > b \/ c) : a * c < b :=\ndiv_mul_cancel b (ne_of_lt hc) \u25b8 mul_lt_mul_of_neg_right h hc\n\nlemma div_lt_of_mul_lt_of_pos (hc : c > 0) (h : b < a * c) : b \/ c < a :=\ncalc\n   a   = a * c * (1 \/ c) : mul_mul_div a (ne_of_gt hc)\n   ... > b * (1 \/ c)     : mul_lt_mul_of_pos_right h (one_div_pos_of_pos hc)\n   ... = b \/ c           : eq.symm $ div_eq_mul_one_div b c\n\nlemma div_lt_of_mul_gt_of_neg (hc : c < 0) (h : a * c < b) : b \/ c < a :=\ncalc\n   a   = a * c * (1 \/ c) : mul_mul_div a (ne_of_lt hc)\n   ... > b * (1 \/ c)     : mul_lt_mul_of_neg_right h (one_div_neg_of_neg hc)\n   ... = b \/ c           : eq.symm $ div_eq_mul_one_div b c\n\nlemma div_le_of_le_mul (hb : b > 0) (h : a \u2264 b * c) : a \/ b \u2264 c :=\ncalc\n   a \/ b = a * (1 \/ b)       : div_eq_mul_one_div a b\n     ... \u2264 (b * c) * (1 \/ b) : mul_le_mul_of_nonneg_right h (le_of_lt (one_div_pos_of_pos hb))\n     ... = (b * c) \/ b       : eq.symm $ div_eq_mul_one_div (b * c) b\n     ... = c                 : by rw [mul_div_cancel_left _ (ne.symm (ne_of_lt hb))]\n\nlemma le_mul_of_div_le (hc : c > 0) (h : a \/ c \u2264 b) : a \u2264 b * c :=\ncalc\n   a = a \/ c * c : by rw (div_mul_cancel _ (ne.symm (ne_of_lt hc)))\n ... \u2264 b * c     : mul_le_mul_of_nonneg_right h (le_of_lt hc)\n\n  -- following these in the isabelle file, there are 8 biconditionals for the above with - signs\n  -- skipping for now\n\nlemma mul_sub_mul_div_mul_neg (hc : c \u2260 0) (hd : d \u2260 0) (h : a \/ c < b \/ d) :\n      (a * d - b * c) \/ (c * d) < 0 :=\nhave h1 : a \/ c - b \/ d < 0, from calc\n  a \/ c - b \/ d < b \/ d - b \/ d : sub_lt_sub_right h _\n            ... = 0             : by rw sub_self,\ncalc\n    0 > a \/ c - b \/ d             : h1\n  ... = (a * d - c * b) \/ (c * d) : div_sub_div _ _ hc hd\n  ... = (a * d - b * c) \/ (c * d) : by rw (mul_comm b c)\n\nlemma mul_sub_mul_div_mul_nonpos (hc : c \u2260 0) (hd : d \u2260 0) (h : a \/ c \u2264 b \/ d) :\n      (a * d - b * c) \/ (c * d) \u2264 0 :=\nhave h1 : a \/ c - b \/ d \u2264 0, from calc\n    a \/ c - b \/ d \u2264 b \/ d - b \/ d : sub_le_sub_right h _\n              ... = 0             : by rw sub_self,\ncalc\n    0 \u2265 a \/ c - b \/ d : h1\n  ... = (a * d - c * b) \/ (c * d) : div_sub_div _ _ hc hd\n  ... = (a * d - b * c) \/ (c * d) : by rw (mul_comm b c)\n\nlemma div_lt_div_of_mul_sub_mul_div_neg (hc : c \u2260 0) (hd : d \u2260 0)\n      (h : (a * d - b * c) \/ (c * d) < 0) : a \/ c < b \/ d :=\nhave (a * d - c * b) \/ (c * d) < 0,     by rwa [mul_comm c b],\nhave a \/ c - b \/ d < 0,                 by rwa [div_sub_div _ _ hc hd],\nhave a \/ c - b \/ d + b \/ d < 0 + b \/ d, from add_lt_add_right this _,\nby rwa [zero_add, sub_eq_add_neg, neg_add_cancel_right] at this\n\n\nlemma div_le_div_of_mul_sub_mul_div_nonpos (hc : c \u2260 0) (hd : d \u2260 0)\n      (h : (a * d - b * c) \/ (c * d) \u2264 0) : a \/ c \u2264 b \/ d :=\nhave (a * d - c * b) \/ (c * d) \u2264 0,     by rwa [mul_comm c b],\nhave a \/ c - b \/ d \u2264 0,                 by rwa [div_sub_div _ _ hc hd],\nhave a \/ c - b \/ d + b \/ d \u2264 0 + b \/ d, from add_le_add_right this _,\nby rwa [zero_add, sub_eq_add_neg, neg_add_cancel_right] at this\n\n\nlemma div_pos_of_pos_of_pos : 0 < a \u2192 0 < b \u2192 0 < a \/ b :=\nbegin\n  intros,\n  rw div_eq_mul_one_div,\n  apply mul_pos,\n  assumption,\n  apply one_div_pos_of_pos,\n  assumption\nend\n\nlemma div_nonneg_of_nonneg_of_pos : 0 \u2264 a \u2192 0 < b \u2192 0 \u2264 a \/ b :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_nonneg, assumption,\n  apply le_of_lt,\n  apply one_div_pos_of_pos,\n  assumption\nend\n\nlemma div_neg_of_neg_of_pos : a < 0 \u2192 0 < b \u2192 a \/ b < 0 :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_neg_of_neg_of_pos,\n  assumption,\n  apply one_div_pos_of_pos,\n  assumption\nend\n\nlemma div_nonpos_of_nonpos_of_pos : a \u2264 0 \u2192 0 < b \u2192 a \/ b \u2264 0 :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_nonpos_of_nonpos_of_nonneg,\n  assumption,\n  apply le_of_lt,\n  apply one_div_pos_of_pos,\n  assumption\nend\n\nlemma div_neg_of_pos_of_neg : 0 < a \u2192 b < 0 \u2192 a \/ b < 0 :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_neg_of_pos_of_neg,\n  assumption,\n  apply one_div_neg_of_neg,\n  assumption\nend\n\nlemma div_nonpos_of_nonneg_of_neg : 0 \u2264 a \u2192 b < 0 \u2192 a \/ b \u2264 0 :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_nonpos_of_nonneg_of_nonpos,\n  assumption,\n  apply le_of_lt,\n  apply one_div_neg_of_neg,\n  assumption\nend\n\nlemma div_pos_of_neg_of_neg : a < 0 \u2192 b < 0 \u2192 0 < a \/ b :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_pos_of_neg_of_neg,\n  assumption,\n  apply one_div_neg_of_neg,\n  assumption\nend\n\nlemma div_nonneg_of_nonpos_of_neg : a \u2264 0 \u2192 b < 0 \u2192 0 \u2264 a \/ b :=\nbegin\n  intros, rw div_eq_mul_one_div,\n  apply mul_nonneg_of_nonpos_of_nonpos,\n  assumption,\n  apply le_of_lt,\n  apply one_div_neg_of_neg,\n  assumption\nend\n\nlemma div_lt_div_of_lt_of_pos (h : a < b) (hc : 0 < c) : a \/ c < b \/ c :=\nbegin\n  intros,\n  rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],\n  exact mul_lt_mul_of_pos_right h (one_div_pos_of_pos hc)\nend\n\nlemma div_le_div_of_le_of_pos (h : a \u2264 b) (hc : 0 < c) : a \/ c \u2264 b \/ c :=\nbegin\n  rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],\n  exact mul_le_mul_of_nonneg_right h (le_of_lt (one_div_pos_of_pos hc))\nend\n\nlemma div_lt_div_of_lt_of_neg (h : b < a) (hc : c < 0) : a \/ c < b \/ c :=\nbegin\n  rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],\n  exact mul_lt_mul_of_neg_right h (one_div_neg_of_neg hc)\nend\n\nlemma div_le_div_of_le_of_neg (h : b \u2264 a) (hc : c < 0) : a \/ c \u2264 b \/ c :=\nbegin\n  rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],\n  exact mul_le_mul_of_nonpos_right h (le_of_lt (one_div_neg_of_neg hc))\nend\n\nlemma add_halves (a : \u03b1) : a \/ 2 + a \/ 2 = a :=\nby { rw [div_add_div_same, \u2190 two_mul, mul_div_cancel_left], exact two_ne_zero }\n\nlemma sub_self_div_two (a : \u03b1) : a - a \/ 2 = a \/ 2 :=\nsuffices a \/ 2 + a \/ 2 - a \/ 2 = a \/ 2, by rwa add_halves at this,\nby rw [add_sub_cancel]\n\nlemma add_midpoint {a b : \u03b1} (h : a < b) : a + (b - a) \/ 2 < b :=\nbegin\n  rw [\u2190 div_sub_div_same, sub_eq_add_neg, add_comm (b\/2), \u2190 add_assoc, \u2190 sub_eq_add_neg],\n  apply add_lt_of_lt_sub_right,\n  rw [sub_self_div_two, sub_self_div_two],\n  apply div_lt_div_of_lt_of_pos h two_pos\nend\n\nlemma div_two_sub_self (a : \u03b1) : a \/ 2 - a = - (a \/ 2) :=\nsuffices a \/ 2 - (a \/ 2 + a \/ 2) = - (a \/ 2), by rwa add_halves at this,\nby rw [sub_add_eq_sub_sub, sub_self, zero_sub]\n\nlemma add_self_div_two (a : \u03b1) : (a + a) \/ 2 = a :=\neq.symm\n  (iff.mpr (eq_div_iff_mul_eq _ _ (ne_of_gt (add_pos (@zero_lt_one \u03b1 _) zero_lt_one)))\n           (begin unfold bit0, rw [left_distrib, mul_one] end))\n\nlemma mul_le_mul_of_mul_div_le {a b c d : \u03b1} (h : a * (b \/ c) \u2264 d) (hc : c > 0) : b * a \u2264 d * c :=\nbegin\n  rw [\u2190 mul_div_assoc] at h, rw [mul_comm b],\n  apply le_mul_of_div_le hc h\nend\n\nlemma div_two_lt_of_pos {a : \u03b1} (h : a > 0) : a \/ 2 < a :=\nsuffices a \/ (1 + 1) < a, begin unfold bit0, assumption end,\nhave ha : a \/ 2 > 0, from div_pos_of_pos_of_pos h (add_pos zero_lt_one zero_lt_one),\ncalc\n      a \/ 2 < a \/ 2 + a \/ 2 : lt_add_of_pos_left _ ha\n        ... = a             : add_halves a\n\nlemma div_mul_le_div_mul_of_div_le_div_pos {a b c d e : \u03b1} (h : a \/ b \u2264 c \/ d)\n      (he : e > 0) : a \/ (b * e) \u2264 c \/ (d * e) :=\nbegin\n  have h\u2081 := div_mul_eq_div_mul_one_div a b e,\n  have h\u2082 := div_mul_eq_div_mul_one_div c d e,\n  rw [h\u2081, h\u2082],\n  apply mul_le_mul_of_nonneg_right h,\n  apply le_of_lt,\n  apply one_div_pos_of_pos he\nend\n\nlemma exists_add_lt_and_pos_of_lt {a b : \u03b1} (h : b < a) : \u2203 c : \u03b1, b + c < a \u2227 0 < c :=\nbegin\n  apply exists.intro ((a - b) \/ (1 + 1)),\n  split,\n  {have h2 : a + a > (b + b) + (a - b),\n    calc\n      a + a > b + a             : add_lt_add_right h _\n        ... = b + a + b - b     : by rw add_sub_cancel\n        ... = b + b + a - b     : by simp [add_comm, add_left_comm]\n        ... = (b + b) + (a - b) : by rw add_sub,\n   have h3 : (a + a) \/ 2 > ((b + b) + (a - b)) \/ 2,\n     exact div_lt_div_of_lt_of_pos h2 two_pos,\n   rw [one_add_one_eq_two, sub_eq_add_neg],\n   rw [add_self_div_two, \u2190 div_add_div_same, add_self_div_two, sub_eq_add_neg] at h3,\n   exact h3},\n  exact div_pos_of_pos_of_pos (sub_pos_of_lt h) two_pos\nend\n\nlemma le_of_forall_sub_le {a b : \u03b1} (h : \u2200 \u03b5 > 0, b - \u03b5 \u2264 a) : b \u2264 a :=\nbegin\n  apply le_of_not_gt,\n  intro hb,\n  cases exists_add_lt_and_pos_of_lt hb with c hc,\n  have  hc' := h c (and.right hc),\n  apply (not_le_of_gt (and.left hc)) (le_add_of_sub_right_le hc')\nend\n\nlemma one_div_lt_one_div_of_lt {a b : \u03b1} (ha : 0 < a) (h : a < b) : 1 \/ b < 1 \/ a :=\nbegin\n  apply lt_div_of_mul_lt ha,\n  rw [mul_comm, \u2190 div_eq_mul_one_div],\n  apply div_lt_of_mul_lt_of_pos (lt_trans ha h),\n  rwa [one_mul]\nend\n\nlemma one_div_le_one_div_of_le {a b : \u03b1} (ha : 0 < a) (h : a \u2264 b) : 1 \/ b \u2264 1 \/ a :=\n(lt_or_eq_of_le h).elim\n  (\u03bb h, le_of_lt $ one_div_lt_one_div_of_lt ha h)\n  (\u03bb h, by rw [h])\n\nlemma one_div_lt_one_div_of_lt_of_neg {a b : \u03b1} (hb : b < 0) (h : a < b) : 1 \/ b < 1 \/ a :=\nbegin\n  apply div_lt_of_mul_gt_of_neg hb,\n  rw [mul_comm, \u2190 div_eq_mul_one_div],\n  apply div_lt_of_mul_gt_of_neg (lt_trans h hb),\n  rwa [one_mul]\nend\n\nlemma one_div_le_one_div_of_le_of_neg {a b : \u03b1} (hb : b < 0) (h : a \u2264 b) : 1 \/ b \u2264 1 \/ a :=\n(lt_or_eq_of_le h).elim\n  (\u03bb h, le_of_lt $ one_div_lt_one_div_of_lt_of_neg hb h)\n  (\u03bb h, by rw [h])\n\nlemma le_of_one_div_le_one_div {a b : \u03b1} (h : 0 < a) (hl : 1 \/ a \u2264 1 \/ b) : b \u2264 a :=\nle_of_not_gt $ \u03bb hn, not_lt_of_ge hl $ one_div_lt_one_div_of_lt h hn\n\nlemma le_of_one_div_le_one_div_of_neg {a b : \u03b1} (h : b < 0) (hl : 1 \/ a \u2264 1 \/ b) : b \u2264 a :=\nle_of_not_gt $ \u03bb hn, not_lt_of_ge hl $ one_div_lt_one_div_of_lt_of_neg h hn\n\nlemma lt_of_one_div_lt_one_div {a b : \u03b1} (h : 0 < a) (hl : 1 \/ a < 1 \/ b) : b < a :=\nlt_of_not_ge $ \u03bb hn, not_le_of_gt hl $ one_div_le_one_div_of_le h hn\n\nlemma lt_of_one_div_lt_one_div_of_neg {a b : \u03b1} (h : b < 0) (hl : 1 \/ a < 1 \/ b) : b < a :=\nlt_of_not_ge $ \u03bb hn, not_le_of_gt hl $ one_div_le_one_div_of_le_of_neg h hn\n\nlemma one_div_le_of_one_div_le_of_pos {a b : \u03b1} (ha : a > 0) (h : 1 \/ a \u2264 b) : 1 \/ b \u2264 a :=\nbegin\n  rw [\u2190 one_div_one_div a],\n  apply one_div_le_one_div_of_le _ h,\n  apply one_div_pos_of_pos ha\nend\n\nlemma one_div_le_of_one_div_le_of_neg {a b : \u03b1} (hb : b < 0) (h : 1 \/ a \u2264 b) : 1 \/ b \u2264 a :=\nle_of_not_gt $ \u03bb hl, begin\n  have : a < 0, from lt_trans hl (one_div_neg_of_neg hb),\n  rw \u2190 one_div_one_div a at hl,\n  exact not_lt_of_ge h (lt_of_one_div_lt_one_div_of_neg hb hl)\nend\n\nlemma one_lt_one_div {a : \u03b1} (h1 : 0 < a) (h2 : a < 1) : 1 < 1 \/ a :=\nsuffices 1 \/ 1 < 1 \/ a, by rwa one_div_one at this,\none_div_lt_one_div_of_lt h1 h2\n\nlemma one_le_one_div {a : \u03b1} (h1 : 0 < a) (h2 : a \u2264 1) : 1 \u2264 1 \/ a :=\nsuffices 1 \/ 1 \u2264 1 \/ a, by rwa one_div_one at this,\none_div_le_one_div_of_le h1 h2\n\nlemma one_div_lt_neg_one {a : \u03b1} (h1 : a < 0) (h2 : -1 < a) : 1 \/ a < -1 :=\nsuffices 1 \/ a < 1 \/ -1, by rwa one_div_neg_one_eq_neg_one at this,\none_div_lt_one_div_of_lt_of_neg h1 h2\n\nlemma one_div_le_neg_one {a : \u03b1} (h1 : a < 0) (h2 : -1 \u2264 a) : 1 \/ a \u2264 -1 :=\nsuffices 1 \/ a \u2264 1 \/ -1, by rwa one_div_neg_one_eq_neg_one at this,\none_div_le_one_div_of_le_of_neg h1 h2\n\nlemma div_lt_div_of_pos_of_lt_of_pos (hb : 0 < b) (h : b < a) (hc : 0 < c) : c \/ a < c \/ b :=\nbegin\n  apply lt_of_sub_neg,\n  rw [div_eq_mul_one_div, div_eq_mul_one_div c b, \u2190 mul_sub_left_distrib],\n  apply mul_neg_of_pos_of_neg,\n  exact hc,\n  apply sub_neg_of_lt,\n  apply one_div_lt_one_div_of_lt; assumption,\nend\n\nlemma div_mul_le_div_mul_of_div_le_div_pos' (h : a \/ b \u2264 c \/ d)\n          (he : e > 0) : a \/ (b * e) \u2264 c \/ (d * e) :=\nbegin\n  rw [div_mul_eq_div_mul_one_div, div_mul_eq_div_mul_one_div],\n  apply mul_le_mul_of_nonneg_right h,\n  apply le_of_lt,\n  apply one_div_pos_of_pos he\nend\n\nlemma div_pos : 0 < a \u2192 0 < b \u2192 0 < a \/ b := div_pos_of_pos_of_pos\n\n@[simp] lemma inv_pos : \u2200 {a : \u03b1}, 0 < a\u207b\u00b9 \u2194 0 < a :=\nsuffices \u2200 a : \u03b1, 0 < a \u2192 0 < a\u207b\u00b9,\nfrom \u03bb a, \u27e8\u03bb h, inv_inv'' a \u25b8 this _ h, this a\u27e9,\n\u03bb a, one_div_eq_inv a \u25b8 one_div_pos_of_pos\n\n@[simp] lemma inv_lt_zero : \u2200 {a : \u03b1}, a\u207b\u00b9 < 0 \u2194 a < 0 :=\nsuffices \u2200 a : \u03b1, a < 0 \u2192 a\u207b\u00b9 < 0,\nfrom \u03bb a, \u27e8\u03bb h, inv_inv'' a \u25b8 this _ h, this a\u27e9,\n\u03bb a, one_div_eq_inv a \u25b8 one_div_neg_of_neg\n\n@[simp] lemma inv_nonneg : 0 \u2264 a\u207b\u00b9 \u2194 0 \u2264 a :=\nle_iff_le_iff_lt_iff_lt.2 inv_lt_zero\n\n@[simp] lemma inv_nonpos : a\u207b\u00b9 \u2264 0 \u2194 a \u2264 0 :=\nle_iff_le_iff_lt_iff_lt.2 inv_pos\n\nlemma one_le_div_iff_le (hb : 0 < b) : 1 \u2264 a \/ b \u2194 b \u2264 a :=\n\u27e8le_of_one_le_div a hb, one_le_div_of_le a hb\u27e9\n\nlemma one_lt_div_iff_lt (hb : 0 < b) : 1 < a \/ b \u2194 b < a :=\n\u27e8lt_of_one_lt_div a hb, one_lt_div_of_lt a hb\u27e9\n\nlemma div_le_one_iff_le (hb : 0 < b) : a \/ b \u2264 1 \u2194 a \u2264 b :=\nle_iff_le_iff_lt_iff_lt.2 (one_lt_div_iff_lt hb)\n\nlemma div_lt_one_iff_lt (hb : 0 < b) : a \/ b < 1 \u2194 a < b :=\nlt_iff_lt_of_le_iff_le (one_le_div_iff_le hb)\n\nlemma le_div_iff (hc : 0 < c) : a \u2264 b \/ c \u2194 a * c \u2264 b :=\n\u27e8mul_le_of_le_div hc, le_div_of_mul_le hc\u27e9\n\nlemma le_div_iff' (hc : 0 < c) : a \u2264 b \/ c \u2194 c * a \u2264 b :=\nby rw [mul_comm, le_div_iff hc]\n\nlemma div_le_iff (hb : 0 < b) : a \/ b \u2264 c \u2194 a \u2264 c * b :=\n\u27e8le_mul_of_div_le hb, by rw [mul_comm]; exact div_le_of_le_mul hb\u27e9\n\nlemma div_le_iff' (hb : 0 < b) : a \/ b \u2264 c \u2194 a \u2264 b * c :=\nby rw [mul_comm, div_le_iff hb]\n\nlemma lt_div_iff (hc : 0 < c) : a < b \/ c \u2194 a * c < b :=\n\u27e8mul_lt_of_lt_div hc, lt_div_of_mul_lt hc\u27e9\n\nlemma lt_div_iff' (hc : 0 < c) : a < b \/ c \u2194 c * a < b :=\nby rw [mul_comm, lt_div_iff hc]\n\nlemma div_le_iff_of_neg (hc : c < 0) : b \/ c \u2264 a \u2194 a * c \u2264 b :=\n\u27e8mul_le_of_div_le_of_neg hc, div_le_of_mul_le_of_neg hc\u27e9\n\nlemma le_div_iff_of_neg (hc : c < 0) : a \u2264 b \/ c \u2194 b \u2264 a * c :=\nby rw [\u2190 neg_neg c, mul_neg_eq_neg_mul_symm, div_neg, le_neg,\n    div_le_iff (neg_pos.2 hc), neg_mul_eq_neg_mul_symm]\n\nlemma div_lt_iff (hc : 0 < c) : b \/ c < a \u2194 b < a * c :=\nlt_iff_lt_of_le_iff_le (le_div_iff hc)\n\nlemma div_lt_iff' (hc : 0 < c) : b \/ c < a \u2194 b < c * a :=\nby rw [mul_comm, div_lt_iff hc]\n\nlemma div_lt_iff_of_neg (hc : c < 0) : b \/ c < a \u2194 a * c < b :=\n\u27e8mul_lt_of_gt_div_of_neg hc, div_lt_of_mul_gt_of_neg hc\u27e9\n\nlemma inv_le_inv (ha : 0 < a) (hb : 0 < b) : a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a :=\nby rw [inv_eq_one_div, div_le_iff ha,\n       \u2190 div_eq_inv_mul, one_le_div_iff_le hb]\n\nlemma inv_le (ha : 0 < a) (hb : 0 < b) : a\u207b\u00b9 \u2264 b \u2194 b\u207b\u00b9 \u2264 a :=\nby rw [\u2190 inv_le_inv hb (inv_pos.2 ha), inv_inv']\n\nlemma le_inv (ha : 0 < a) (hb : 0 < b) : a \u2264 b\u207b\u00b9 \u2194 b \u2264 a\u207b\u00b9 :=\nby rw [\u2190 inv_le_inv (inv_pos.2 hb) ha, inv_inv']\n\nlemma one_div_le_one_div (ha : 0 < a) (hb : 0 < b) : 1 \/ a \u2264 1 \/ b \u2194 b \u2264 a :=\nby simpa [one_div_eq_inv] using inv_le_inv ha hb\n\nlemma inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a\u207b\u00b9 < b\u207b\u00b9 \u2194 b < a :=\nlt_iff_lt_of_le_iff_le (inv_le_inv hb ha)\n\nlemma inv_lt (ha : 0 < a) (hb : 0 < b) : a\u207b\u00b9 < b \u2194 b\u207b\u00b9 < a :=\nlt_iff_lt_of_le_iff_le (le_inv hb ha)\n\nlemma one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 \/ a < b \u2194 1 \/ b < a :=\n(one_div_eq_inv a).symm \u25b8 (one_div_eq_inv b).symm \u25b8 inv_lt ha hb\n\nlemma lt_inv (ha : 0 < a) (hb : 0 < b) : a < b\u207b\u00b9 \u2194 b < a\u207b\u00b9 :=\nlt_iff_lt_of_le_iff_le (inv_le hb ha)\n\nlemma one_div_lt_one_div (ha : 0 < a) (hb : 0 < b) : 1 \/ a < 1 \/ b \u2194 b < a :=\nlt_iff_lt_of_le_iff_le (one_div_le_one_div hb ha)\n\nlemma div_nonneg : 0 \u2264 a \u2192 0 < b \u2192 0 \u2264 a \/ b := div_nonneg_of_nonneg_of_pos\n\nlemma div_lt_div_right (hc : 0 < c) : a \/ c < b \/ c \u2194 a < b :=\n\u27e8lt_imp_lt_of_le_imp_le (\u03bb h, div_le_div_of_le_of_pos h hc),\n \u03bb h, div_lt_div_of_lt_of_pos h hc\u27e9\n\nlemma div_le_div_right (hc : 0 < c) : a \/ c \u2264 b \/ c \u2194 a \u2264 b :=\nle_iff_le_iff_lt_iff_lt.2 (div_lt_div_right hc)\n\nlemma div_lt_div_right_of_neg (hc : c < 0) : a \/ c < b \/ c \u2194 b < a :=\n\u27e8lt_imp_lt_of_le_imp_le (\u03bb h, div_le_div_of_le_of_neg h hc),\n \u03bb h, div_lt_div_of_lt_of_neg h hc\u27e9\n\nlemma div_le_div_right_of_neg (hc : c < 0) : a \/ c \u2264 b \/ c \u2194 b \u2264 a :=\nle_iff_le_iff_lt_iff_lt.2 (div_lt_div_right_of_neg hc)\n\nlemma div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a \/ b < a \/ c \u2194 c < b :=\n(mul_lt_mul_left ha).trans (inv_lt_inv hb hc)\n\nlemma div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a \/ b \u2264 a \/ c \u2194 c \u2264 b :=\nle_iff_le_iff_lt_iff_lt.2 (div_lt_div_left ha hc hb)\n\nlemma div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) :\n  a \/ b < c \/ d \u2194 a * d < c * b :=\nby rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]\n\nlemma div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a \/ b \u2264 c \/ d \u2194 a * d \u2264 c * b :=\nby rw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0]\n\nlemma div_le_div (hc : 0 \u2264 c) (hac : a \u2264 c) (hd : 0 < d) (hbd : d \u2264 b) : a \/ b \u2264 c \/ d :=\nbegin\n  rw div_le_div_iff (lt_of_lt_of_le hd hbd) hd,\n  exact mul_le_mul hac hbd (le_of_lt hd) hc\nend\n\nlemma div_lt_div (hac : a < c) (hbd : d \u2264 b) (c0 : 0 \u2264 c) (d0 : 0 < d) :\n  a \/ b < c \/ d :=\n(div_lt_div_iff (lt_of_lt_of_le d0 hbd) d0).2 (mul_lt_mul hac hbd d0 c0)\n\nlemma div_lt_div' (hac : a \u2264 c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) :\n  a \/ b < c \/ d :=\n(div_lt_div_iff (lt_trans d0 hbd) d0).2 (mul_lt_mul' hac hbd (le_of_lt d0) c0)\n\nlemma monotone.div_const {\u03b2 : Type*} [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} (hf : monotone f)\n  {c : \u03b1} (hc : 0 \u2264 c) :\n  monotone (\u03bb x, (f x) \/ c) :=\nhf.mul_const (inv_nonneg.2 hc)\n\nlemma strict_mono.div_const {\u03b2 : Type*} [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} (hf : strict_mono f)\n  {c : \u03b1} (hc : 0 < c) :\n  strict_mono (\u03bb x, (f x) \/ c) :=\nhf.mul_const (inv_pos.2 hc)\n\nlemma half_pos {a : \u03b1} (h : 0 < a) : 0 < a \/ 2 := div_pos h two_pos\n\nlemma one_half_pos : (0:\u03b1) < 1 \/ 2 := half_pos zero_lt_one\n\nlemma half_lt_self : 0 < a \u2192 a \/ 2 < a := div_two_lt_of_pos\n\nlemma one_half_lt_one : (1 \/ 2 : \u03b1) < 1 := half_lt_self zero_lt_one\n\ninstance linear_ordered_field.to_densely_ordered : densely_ordered \u03b1 :=\n{ dense := assume a\u2081 a\u2082 h, \u27e8(a\u2081 + a\u2082) \/ 2,\n  calc a\u2081 = (a\u2081 + a\u2081) \/ 2 : (add_self_div_two a\u2081).symm\n    ... < (a\u2081 + a\u2082) \/ 2 : div_lt_div_of_lt_of_pos (add_lt_add_left h _) two_pos,\n  calc (a\u2081 + a\u2082) \/ 2 < (a\u2082 + a\u2082) \/ 2 : div_lt_div_of_lt_of_pos (add_lt_add_right h _) two_pos\n    ... = a\u2082 : add_self_div_two a\u2082\u27e9 }\n\ninstance linear_ordered_field.to_no_top_order : no_top_order \u03b1 :=\n{ no_top := assume a, \u27e8a + 1, lt_add_of_le_of_pos (le_refl a) zero_lt_one \u27e9 }\n\ninstance linear_ordered_field.to_no_bot_order : no_bot_order \u03b1 :=\n{ no_bot := assume a, \u27e8a + -1,\n    add_lt_of_le_of_neg (le_refl _) (neg_lt_of_neg_lt $ by simp [zero_lt_one]) \u27e9 }\n\nlemma inv_lt_one (ha : 1 < a) : a\u207b\u00b9 < 1 :=\nby rw [inv_eq_one_div]; exact div_lt_of_mul_lt_of_pos (lt_trans zero_lt_one ha) (by simp *)\n\nlemma one_lt_inv (h\u2081 : 0 < a) (h\u2082 : a < 1) : 1 < a\u207b\u00b9 :=\nby rw [inv_eq_one_div, lt_div_iff h\u2081]; simp [h\u2082]\n\nlemma inv_le_one (ha : 1 \u2264 a) : a\u207b\u00b9 \u2264 1 :=\nby rw [inv_eq_one_div]; exact div_le_of_le_mul (lt_of_lt_of_le zero_lt_one ha) (by simp *)\n\nlemma one_le_inv (ha0 : 0 < a) (ha : a \u2264 1) : 1 \u2264 a\u207b\u00b9 :=\nle_of_mul_le_mul_left (by simpa [mul_inv_cancel (ne.symm (ne_of_lt ha0))]) ha0\n\nlemma mul_self_inj_of_nonneg (a0 : 0 \u2264 a) (b0 : 0 \u2264 b) : a * a = b * b \u2194 a = b :=\n(mul_self_eq_mul_self_iff a b).trans $ or_iff_left_of_imp $\n\u03bb h, by subst a; rw [le_antisymm (neg_nonneg.1 a0) b0, neg_zero]\n\nlemma div_le_div_of_le_left (ha : 0 \u2264 a) (hc : 0 < c) (h : c \u2264 b) :\n  a \/ b \u2264 a \/ c :=\nby haveI := classical.dec_eq \u03b1; exact\nif ha0 : a = 0 then by simp [ha0]\nelse (div_le_div_left (lt_of_le_of_ne ha (ne.symm ha0)) (lt_of_lt_of_le hc h) hc).2 h\n\nlemma inv_le_inv_of_le (hb : 0 < b) (h : b \u2264 a) : a\u207b\u00b9 \u2264 b\u207b\u00b9 :=\nbegin\n  rw [inv_eq_one_div, inv_eq_one_div],\n  exact one_div_le_one_div_of_le hb h\nend\n\nlemma div_nonneg' (ha : 0 \u2264 a) (hb : 0 \u2264 b) : 0 \u2264 a \/ b :=\n(lt_or_eq_of_le hb).elim (div_nonneg ha) (\u03bb h, by simp [h.symm])\n\nlemma div_le_div_of_le_of_nonneg (hab : a \u2264 b) (hc : 0 \u2264 c) :\n  a \/ c \u2264 b \/ c :=\nmul_le_mul_of_nonneg_right hab (inv_nonneg.2 hc)\n\nend linear_ordered_field\n\n@[protect_proj] class discrete_linear_ordered_field (\u03b1 : Type*)\n  extends linear_ordered_field \u03b1, decidable_linear_ordered_comm_ring \u03b1\n\nsection discrete_linear_ordered_field\nvariables [discrete_linear_ordered_field \u03b1]\n\nlemma abs_div (a b : \u03b1) : abs (a \/ b) = abs a \/ abs b :=\ndecidable.by_cases\n  (assume h : b = 0, by rw [h, abs_zero, div_zero, div_zero, abs_zero])\n  (assume h : b \u2260 0,\n   have h\u2081 : abs b \u2260 0, from\n     assume h\u2082, h (eq_zero_of_abs_eq_zero h\u2082),\n   eq_div_of_mul_eq _ _ h\u2081\n   (show abs (a \/ b) * abs b = abs a, by rw [\u2190 abs_mul, div_mul_cancel _ h]))\n\nlemma abs_one_div (a : \u03b1) : abs (1 \/ a) = 1 \/ abs a :=\nby rw [abs_div, abs_of_nonneg (zero_le_one : 1 \u2265 (0 : \u03b1))]\n\nlemma abs_inv (a : \u03b1) : abs a\u207b\u00b9 = (abs a)\u207b\u00b9 :=\nhave h : abs (1 \/ a) = 1 \/ abs a,\n  begin rw [abs_div, abs_of_nonneg], exact zero_le_one end,\nby simp [*] at *\n\nend discrete_linear_ordered_field\n","avg_line_length":36.8935532234,"max_line_length":98,"alphanum_fraction":0.6148407022}
-{"size":392,"ext":"lean","lang":"Lean","max_stars_count":130.0,"content":"inductive foo {A : Type} : A \u2192 Type :=\nmk : \u03a0 a : A, foo a\n\nexample (A : Type) (B : A \u2192 Type) (a : A) (H : foo a) (Hb : B a) : A :=\nbegin\n  cases H,\n  state,\n  assumption\nend\n\ninductive foo\u2082 {A : Type} : A \u2192 A \u2192 Type :=\nmk : \u03a0 a b : A, foo\u2082 a b\n\nexample (A : Type) (B : A \u2192 Type) (f : A \u2192 A) (a : A) (H : foo\u2082 (f a) a) (Hb : H = H) (Hc : a = a) : A :=\nbegin\n  cases H,\n  state,\n  exact a\nend\n","avg_line_length":19.6,"max_line_length":105,"alphanum_fraction":0.487244898}
-{"size":10916,"ext":"hlean","lang":"Lean","max_stars_count":40.0,"content":"\/-\nCopyright (c) 2017 Floris van Doorn and Ulrik Buchholtz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Ulrik Buchholtz\n\nTruncatedness and truncation of spectra\n-\/\n\nimport .basic\nopen int trunc eq is_trunc lift unit pointed equiv is_equiv algebra EM trunc_index\n\nnamespace spectrum\n\n\/- interactions of ptrunc \/ trunc with maxm2 -\/\ndefinition ptrunc_maxm2_change_int {k l : \u2124} (p : k = l) (X : Type*)\n  : ptrunc (maxm2 k) X \u2243* ptrunc (maxm2 l) X :=\nptrunc_change_index (ap maxm2 p) X\n\ndefinition is_trunc_maxm2_change_int {k l : \u2124} (X : pType) (p : k = l)\n  : is_trunc (maxm2 k) X \u2192 is_trunc (maxm2 l) X :=\nby induction p; exact id\n\ndefinition is_trunc_maxm2_loop {k : \u2124} {A : Type*} (H : is_trunc (maxm2 (k+1)) A) :\n  is_trunc (maxm2 k) (\u03a9 A) :=\nbegin\n  induction k with k k,\n    apply is_trunc_loop, exact H,\n  apply is_contr_loop,\n  cases k with k,\n  { exact H },\n  { apply is_trunc_succ, apply is_trunc_succ, exact H }\nend\n\ndefinition loop_ptrunc_maxm2_pequiv {k : \u2124} {l : \u2115\u208b\u2082} (p : maxm2 (k+1) = l) (X : Type*) :\n  \u03a9 (ptrunc l X) \u2243* ptrunc (maxm2 k) (\u03a9 X) :=\nbegin\n  induction p,\n  induction k with k k,\n  { exact loop_ptrunc_pequiv k X },\n  { refine pequiv_of_is_contr _ _ _ !is_trunc_trunc,\n    apply is_contr_loop,\n    cases k with k,\n    { change is_set (trunc 0 X), apply _ },\n    { change is_set (trunc -2 X), apply _ }}\nend\n\ndefinition loop_ptrunc_maxm2_pequiv_ptrunc_elim' {k : \u2124} {l : \u2115\u208b\u2082} (p : maxm2 (k+1) = l)\n  {A B : Type*} (f : A \u2192* B) {H : is_trunc l B} :\n  \u03a9\u2192 (ptrunc.elim l f) \u2218* (loop_ptrunc_maxm2_pequiv p A)\u207b\u00b9\u1d49* ~*\n  @ptrunc.elim (maxm2 k) _ _ (is_trunc_maxm2_loop (is_trunc_of_eq p\u207b\u00b9 H)) (\u03a9\u2192 f) :=\nbegin\n  induction p, induction k with k k,\n  { refine pwhisker_right _ (ap1_phomotopy _) \u2b1d* @(ap1_ptrunc_elim k f) H,\n    apply ptrunc_elim_phomotopy2, reflexivity },\n  { apply phomotopy_of_is_contr_cod_pmap, exact is_trunc_maxm2_loop H }\nend\n\ndefinition loop_ptrunc_maxm2_pequiv_ptrunc_elim {k : \u2124} {l : \u2115\u208b\u2082} (p : maxm2 (k+1) = l)\n  {A B : Type*} (f : A \u2192* B) {H1 : is_trunc ((maxm2 k).+1) B } {H2 : is_trunc l B} :\n  \u03a9\u2192 (ptrunc.elim l f) \u2218* (loop_ptrunc_maxm2_pequiv p A)\u207b\u00b9\u1d49* ~* ptrunc.elim (maxm2 k) (\u03a9\u2192 f) :=\nbegin\n  induction p, induction k with k k: esimp at H1,\n  { refine pwhisker_right _ (ap1_phomotopy _) \u2b1d* ap1_ptrunc_elim k f,\n    apply ptrunc_elim_phomotopy2, reflexivity },\n  { apply phomotopy_of_is_contr_cod }\nend\n\ndefinition loop_ptrunc_maxm2_pequiv_ptr {k : \u2124} {l : \u2115\u208b\u2082} (p : maxm2 (k+1) = l) (A : Type*) :\n  \u03a9\u2192 (ptr l A) ~* (loop_ptrunc_maxm2_pequiv p A)\u207b\u00b9\u1d49* \u2218* ptr (maxm2 k) (\u03a9 A) :=\nbegin\n  induction p, induction k with k k,\n  { exact ap1_ptr k A },\n  { apply phomotopy_pinv_left_of_phomotopy, apply phomotopy_of_is_contr_cod_pmap,\n    apply is_trunc_trunc }\nend\n\ndefinition is_trunc_of_is_trunc_maxm2 (k : \u2124) (X : Type)\n  : is_trunc (maxm2 k) X \u2192 is_trunc (max0 k) X :=\n\u03bb H, @is_trunc_of_le X _ _ (maxm2_le_maxm0 k) H\n\ndefinition ptrunc_maxm2_pred {n m : \u2124} (A : Type*) (p : n - 1 = m) :\n  ptrunc (maxm2 m) A \u2243* ptrunc (trunc_index.pred (maxm2 n)) A :=\nbegin\n  cases n with n, cases n with n, apply pequiv_of_is_contr,\n        induction p, apply is_trunc_trunc,\n      apply is_contr_ptrunc_minus_one,\n    exact ptrunc_change_index (ap maxm2 (p\u207b\u00b9 \u2b1d !add_sub_cancel)) A,\n  exact ptrunc_change_index (ap maxm2 p\u207b\u00b9) A\nend\n\ndefinition ptrunc_maxm2_pred_nat {n : \u2115} {m l : \u2124} (A : Type*)\n  (p : nat.succ n = l) (q : pred l = m) (r : maxm2 m = trunc_index.pred (maxm2 (nat.succ n))) :\n  @ptrunc_maxm2_pred (nat.succ n) m A (ap pred p \u2b1d q) ~* ptrunc_change_index r A :=\nbegin\n  have ap maxm2 ((ap pred p \u2b1d q)\u207b\u00b9 \u2b1d add_sub_cancel n 1) = r, from !is_set.elim,\n  induction this, reflexivity\nend\n\n\/- truncatedness of spectra -\/\ndefinition is_strunc [reducible] (k : \u2124) (E : spectrum) : Type :=\n\u03a0 (n : \u2124), is_trunc (maxm2 (k + n)) (E n)\n\ndefinition is_strunc_change_int {k l : \u2124} (E : spectrum) (p : k = l)\n  : is_strunc k E \u2192 is_strunc l E :=\nbegin induction p, exact id end\n\ndefinition is_strunc_of_le {k l : \u2124} (E : spectrum) (H : k \u2264 l)\n  : is_strunc k E \u2192 is_strunc l E :=\nbegin\n  intro T, intro n, exact is_trunc_of_le (E n) (maxm2_monotone (algebra.add_le_add_right H n)) _\nend\n\ndefinition is_strunc_pequiv_closed {k : \u2124} {E F : spectrum} (H : \u03a0n, E n \u2243* F n)\n  (H2 : is_strunc k E) : is_strunc k F :=\n\u03bbn, is_trunc_equiv_closed (maxm2 (k + n)) (H n) _\n\ndefinition is_strunc_sunit (n : \u2124) : is_strunc n sunit :=\nbegin\n  intro k, apply is_trunc_lift, apply is_trunc_unit\nend\n\ndefinition is_contr_of_is_strunc (n : \u2124) {m : \u2124} (E : spectrum) (H : is_strunc n E)\n  (Hmn : m < -n) : is_contr (E m) :=\nbegin\n  refine transport (\u03bbn, is_trunc n (E m)) _ (H m),\n  apply maxm2_eq_minus_two,\n  exact lt_of_lt_of_le (add_lt_add_left Hmn n) (le_of_eq !add.right_inv)\nend\n\nopen option\ndefinition is_strunc_add_point_spectrum {X : Type} {Y : X \u2192 spectrum} {s\u2080 : \u2124}\n  (H : \u03a0x, is_strunc s\u2080 (Y x)) : \u03a0(x : X\u208a), is_strunc s\u2080 (add_point_spectrum Y x)\n| (some x) := proof H x qed\n| none     := begin intro k, apply is_trunc_lift, apply is_trunc_unit end\n\ndefinition is_strunc_EM_spectrum (G : AbGroup)\n  : is_strunc 0 (EM_spectrum G) :=\nbegin\n  intro n, induction n with n n,\n  { -- case \u2265 0\n    apply is_trunc_maxm2_change_int (EM G n) (zero_add n)\u207b\u00b9,\n    apply is_trunc_EM },\n  { change is_contr (EM_spectrum G (-[1+n])),\n    induction n with n IH,\n    { -- case = -1\n      apply is_contr_loop, exact is_trunc_EM G 0 },\n    { -- case < -1\n      apply is_trunc_loop, apply is_trunc_succ, exact IH }}\nend\n\ndefinition trivial_shomotopy_group_of_is_strunc (E : spectrum)\n  {n k : \u2124} (K : is_strunc n E) (H : n < k)\n  : is_contr (\u03c0\u209b[k] E) :=\nlet m := n + (2 - k) in\nhave I : m < 2, from\n  calc\n    m = (2 - k) + n : int.add_comm n (2 - k)\n  ... < (2 - k) + k : add_lt_add_left H (2 - k)\n  ... = 2           : sub_add_cancel 2 k,\n@trivial_homotopy_group_of_is_trunc (E (2 - k)) (max0 m) 2\n  (is_trunc_of_is_trunc_maxm2 m (E (2 - k)) (K (2 - k)))\n  (nat.succ_le_succ (max0_le_of_le (le_sub_one_of_lt I)))\n\n\/- truncation of spectra -\/\ndefinition strunc [constructor] (k : \u2124) (E : spectrum) : spectrum :=\nspectrum.MK (\u03bb(n : \u2124), ptrunc (maxm2 (k + n)) (E n))\n            (\u03bb(n : \u2124), ptrunc_pequiv_ptrunc (maxm2 (k + n)) (equiv_glue E n)\n              \u2b1de* (loop_ptrunc_maxm2_pequiv (ap maxm2 (add.assoc k n 1)) (E (n+1)))\u207b\u00b9\u1d49*)\n\ndefinition strunc_change_int [constructor] {k l : \u2124} (E : spectrum) (p : k = l) :\n  strunc k E \u2192\u209b strunc l E :=\nbegin induction p, reflexivity end\n\ndefinition is_strunc_strunc (k : \u2124) (E : spectrum)\n  : is_strunc k (strunc k E) :=\n\u03bb n, is_trunc_trunc (maxm2 (k + n)) (E n)\n\ndefinition is_strunc_strunc_pred (X : spectrum) (k : \u2124) : is_strunc k (strunc (k - 1) X) :=\n\u03bbn, @(is_trunc_of_le _ (maxm2_monotone (add_le_add_right (sub_one_le k) n))) !is_strunc_strunc\n\ndefinition is_strunc_strunc_of_is_strunc (k : \u2124) {l : \u2124} {E : spectrum} (H : is_strunc l E)\n  : is_strunc l (strunc k E) :=\n\u03bb n, !is_trunc_trunc_of_is_trunc\n\ndefinition str [constructor] (k : \u2124) (E : spectrum) : E \u2192\u209b strunc k E :=\nsmap.mk (\u03bb n, ptr (maxm2 (k + n)) (E n))\n  abstract begin\n    intro n,\n    apply psquare_of_phomotopy,\n    refine !passoc \u2b1d* pwhisker_left _ !ptr_natural \u2b1d* _,\n    refine !passoc\u207b\u00b9* \u2b1d* pwhisker_right _ !loop_ptrunc_maxm2_pequiv_ptr\u207b\u00b9*,\n  end end\n\ndefinition strunc_elim [constructor] {k : \u2124} {E F : spectrum} (f : E \u2192\u209b F)\n  (H : is_strunc k F) : strunc k E \u2192\u209b F :=\nsmap.mk (\u03bbn, ptrunc.elim (maxm2 (k + n)) (f n))\n  abstract begin\n    intro n,\n    apply psquare_of_phomotopy,\n    symmetry,\n    refine !passoc\u207b\u00b9* \u2b1d* pwhisker_right _ !loop_ptrunc_maxm2_pequiv_ptrunc_elim' \u2b1d* _,\n    refine @(ptrunc_elim_ptrunc_functor _ _ _) _ \u2b1d* _,\n    refine _ \u2b1d* @(ptrunc_elim_pcompose _ _ _) _ _,\n      apply is_trunc_maxm2_loop,\n      refine is_trunc_of_eq _ (H (n+1)), exact ap maxm2 (add.assoc k n 1)\u207b\u00b9,\n    apply ptrunc_elim_phomotopy2,\n    apply phomotopy_of_psquare,\n    apply ptranspose,\n    apply smap.glue_square\n  end end\n\ndefinition strunc_functor [constructor] (k : \u2124) {E F : spectrum} (f : E \u2192\u209b F) :\n  strunc k E \u2192\u209b strunc k F :=\nstrunc_elim (str k F \u2218\u209b f) (is_strunc_strunc k F)\n\n\/- truncated spectra -\/\nstructure truncspectrum (n : \u2124) :=\n  (carrier : spectrum)\n  (struct : is_strunc n carrier)\n\nnotation n `-spectrum` := truncspectrum n\n\nattribute truncspectrum.carrier [coercion]\n\ndefinition genspectrum_of_truncspectrum [coercion] (n : \u2124) : n-spectrum \u2192 gen_spectrum +\u2124 :=\n\u03bb E, truncspectrum.carrier E\n\n\/- Comment (by Floris):\n\n  I think we should really not bundle truncated spectra up,\n  unless we are interested in the type of truncated spectra (e.g. when proving n-spectrum \u2243 ...).\n  Properties of truncated a spectrum should just be stated with two assumptions\n  (X : spectrum) (H : is_strunc n X)\n-\/\n\n\/- truncatedness of spi and sp_cotensor assuming the domain has a level of connectedness -\/\nsection\n\n  open is_conn\n\n  definition is_conn_maxm1_of_maxm2 (A : Type*) (n : \u2124)\n    : is_conn (maxm2 n) A \u2192 is_conn (maxm1m1 n).+1 A :=\n  begin\n    intro H, induction n with n n,\n    { exact H },\n    { exact is_conn_minus_one A (tr pt) }\n  end\n\n  definition is_trunc_maxm2_of_maxm1 (A : Type*) (n : \u2124)\n    : is_trunc (maxm1m1 n).+1 A \u2192 is_trunc (maxm2 n) A :=\n  begin\n    intro H, induction n with n n,\n    { exact H},\n    { apply is_contr_of_merely_prop,\n      { exact H },\n      { exact tr pt } }\n  end\n\n  variables (A : Type*) (n : \u2124) [H : is_conn (maxm2 n) A]\n  include H\n\n  definition is_trunc_maxm2_ppi (k l : \u2124) (H3 : l \u2264 n+1+k) (P : A \u2192 Type*)\n    (H2 : \u03a0a, is_trunc (maxm2 l) (P a)) : is_trunc (maxm2 k) (\u03a0*(a : A), P a) :=\n  is_trunc_maxm2_of_maxm1 (\u03a0*(a : A), P a) k\n    (@is_trunc_ppi_of_is_conn A (maxm1m1 n) (is_conn_maxm1_of_maxm2 A n H) (maxm1m1 k) _\n      (le.trans (maxm2_monotone H3) (maxm2_le n k)) P H2)\n\n  definition is_strunc_spi_of_is_conn (k l : \u2124) (H3 : l \u2264 n+1+k) (P : A \u2192 spectrum)\n    (H2 : \u03a0a, is_strunc l (P a)) : is_strunc k (spi A P) :=\n  begin\n    intro m, unfold spi,\n    exact is_trunc_maxm2_ppi A n (k+m) _ (le.trans (add_le_add_right H3 _)\n            (le_of_eq (add.assoc (n+1) k m))) (\u03bb a, P a m) (\u03bba, H2 a m)\n  end\n\nend\n\ndefinition is_strunc_spi_of_le {A : Type*} (k n : \u2124) (H : n \u2264 k) (P : A \u2192 spectrum)\n  (H2 : \u03a0a, is_strunc n (P a)) : is_strunc k (spi A P) :=\nbegin\n  assert K : n \u2264 -[1+ 0] + 1 + k,\n  { krewrite (int.zero_add k), exact H },\n  { exact @is_strunc_spi_of_is_conn A (-[1+ 0]) (is_conn.is_conn_minus_two A) k _ K P H2 }\nend\n\ndefinition is_strunc_spi {A : Type*} (n : \u2124) (P : A \u2192 spectrum) (H : \u03a0a, is_strunc n (P a))\n  : is_strunc n (spi A P) :=\nis_strunc_spi_of_le n n !le.refl P H\n\ndefinition is_strunc_sp_cotensor (n : \u2124) (A : Type*) {Y : spectrum} (H : is_strunc n Y)\n  : is_strunc n (sp_cotensor A Y) :=\nis_strunc_pequiv_closed (\u03bbn, !pppi_pequiv_ppmap) (is_strunc_spi n (\u03bba, Y) (\u03bba, H))\n\ndefinition is_strunc_sp_ucotensor (n : \u2124) (A : Type) {Y : spectrum} (H : is_strunc n Y)\n  : is_strunc n (sp_ucotensor A Y) :=\n\u03bbk, !pi.is_trunc_arrow\n\nend spectrum\n","avg_line_length":36.508361204,"max_line_length":97,"alphanum_fraction":0.6567423965}
-{"size":49227,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-\/\nimport order.bounds\n\n\/-!\n# Theory of complete lattices\n\n## Main definitions\n\n* `Sup` and `Inf` are the supremum and the infimum of a set;\n* `supr (f : \u03b9 \u2192 \u03b1)` and `infi (f : \u03b9 \u2192 \u03b1)` are indexed supremum and infimum of a function,\n  defined as `Sup` and `Inf` of the range of this function;\n* `class complete_lattice`: a bounded lattice such that `Sup s` is always the least upper boundary\n  of `s` and `Inf s` is always the greatest lower boundary of `s`;\n* `class complete_linear_order`: a linear ordered complete lattice.\n\n## Naming conventions\n\nWe use `Sup`\/`Inf`\/`supr`\/`infi` for the corresponding functions in the statement. Sometimes we\nalso use `bsupr`\/`binfi` for \"bounded\" supremum or infimum, i.e. one of `\u2a06 i \u2208 s, f i`,\n`\u2a06 i (hi : p i), f i`, or more generally `\u2a06 i (hi : p i), f i hi`.\n\n## Notation\n\n* `\u2a06 i, f i` : `supr f`, the supremum of the range of `f`;\n* `\u2a05 i, f i` : `infi f`, the infimum of the range of `f`.\n-\/\n\nset_option old_structure_cmd true\nopen set\n\nvariables {\u03b1 \u03b2 \u03b2\u2082 : Type*} {\u03b9 \u03b9\u2082 : Sort*}\n\n\/-- class for the `Sup` operator -\/\nclass has_Sup (\u03b1 : Type*) := (Sup : set \u03b1 \u2192 \u03b1)\n\/-- class for the `Inf` operator -\/\nclass has_Inf (\u03b1 : Type*) := (Inf : set \u03b1 \u2192 \u03b1)\n\nexport has_Sup (Sup) has_Inf (Inf)\n\n\/-- Supremum of a set -\/\nadd_decl_doc has_Sup.Sup\n\/-- Infimum of a set -\/\nadd_decl_doc has_Inf.Inf\n\n\/-- Indexed supremum -\/\ndef supr [has_Sup \u03b1] {\u03b9} (s : \u03b9 \u2192 \u03b1) : \u03b1 := Sup (range s)\n\/-- Indexed infimum -\/\ndef infi [has_Inf \u03b1] {\u03b9} (s : \u03b9 \u2192 \u03b1) : \u03b1 := Inf (range s)\n\n@[priority 50] instance has_Inf_to_nonempty (\u03b1) [has_Inf \u03b1] : nonempty \u03b1 := \u27e8Inf \u2205\u27e9\n@[priority 50] instance has_Sup_to_nonempty (\u03b1) [has_Sup \u03b1] : nonempty \u03b1 := \u27e8Sup \u2205\u27e9\n\nnotation `\u2a06` binders `, ` r:(scoped f, supr f) := r\nnotation `\u2a05` binders `, ` r:(scoped f, infi f) := r\n\ninstance (\u03b1) [has_Inf \u03b1] : has_Sup (order_dual \u03b1) := \u27e8(Inf : set \u03b1 \u2192 \u03b1)\u27e9\ninstance (\u03b1) [has_Sup \u03b1] : has_Inf (order_dual \u03b1) := \u27e8(Sup : set \u03b1 \u2192 \u03b1)\u27e9\n\n\/--\nNote that we rarely use `complete_semilattice_Sup`\n(in fact, any such object is always a `complete_lattice`, so it's usually best to start there).\n\nNevertheless it is sometimes a useful intermediate step in constructions.\n-\/\nclass complete_semilattice_Sup (\u03b1 : Type*) extends partial_order \u03b1, has_Sup \u03b1 :=\n(le_Sup : \u2200s, \u2200a\u2208s, a \u2264 Sup s)\n(Sup_le : \u2200s a, (\u2200b\u2208s, b \u2264 a) \u2192 Sup s \u2264 a)\n\nsection\nvariables [complete_semilattice_Sup \u03b1] {s t : set \u03b1} {a b : \u03b1}\n\n@[ematch] theorem le_Sup : a \u2208 s \u2192 a \u2264 Sup s := complete_semilattice_Sup.le_Sup s a\n\ntheorem Sup_le : (\u2200b\u2208s, b \u2264 a) \u2192 Sup s \u2264 a := complete_semilattice_Sup.Sup_le s a\n\nlemma is_lub_Sup (s : set \u03b1) : is_lub s (Sup s) := \u27e8assume x, le_Sup, assume x, Sup_le\u27e9\n\nlemma is_lub.Sup_eq (h : is_lub s a) : Sup s = a := (is_lub_Sup s).unique h\n\ntheorem le_Sup_of_le (hb : b \u2208 s) (h : a \u2264 b) : a \u2264 Sup s :=\nle_trans h (le_Sup hb)\n\ntheorem Sup_le_Sup (h : s \u2286 t) : Sup s \u2264 Sup t :=\n(is_lub_Sup s).mono (is_lub_Sup t) h\n\n@[simp] theorem Sup_le_iff : Sup s \u2264 a \u2194 (\u2200b \u2208 s, b \u2264 a) :=\nis_lub_le_iff (is_lub_Sup s)\n\nlemma le_Sup_iff :\n  a \u2264 Sup s \u2194 (\u2200 b, (\u2200 x \u2208 s, x \u2264 b) \u2192 a \u2264 b) :=\n\u27e8\u03bb h b hb, le_trans h (Sup_le hb), \u03bb hb, hb _ (\u03bb x, le_Sup)\u27e9\n\ntheorem Sup_le_Sup_of_forall_exists_le (h : \u2200 x \u2208 s, \u2203 y \u2208 t, x \u2264 y) : Sup s \u2264 Sup t :=\nle_of_forall_le' begin\n  simp only [Sup_le_iff],\n  introv h\u2080 h\u2081,\n  rcases h _ h\u2081 with \u27e8y,hy,hy'\u27e9,\n  solve_by_elim [le_trans hy']\nend\n\n-- We will generalize this to conditionally complete lattices in `cSup_singleton`.\ntheorem Sup_singleton {a : \u03b1} : Sup {a} = a :=\nis_lub_singleton.Sup_eq\n\nend\n\n\/--\nNote that we rarely use `complete_semilattice_Inf`\n(in fact, any such object is always a `complete_lattice`, so it's usually best to start there).\n\nNevertheless it is sometimes a useful intermediate step in constructions.\n-\/\nclass complete_semilattice_Inf (\u03b1 : Type*) extends partial_order \u03b1, has_Inf \u03b1 :=\n(Inf_le : \u2200s, \u2200a\u2208s, Inf s \u2264 a)\n(le_Inf : \u2200s a, (\u2200b\u2208s, a \u2264 b) \u2192 a \u2264 Inf s)\n\n\nsection\nvariables [complete_semilattice_Inf \u03b1] {s t : set \u03b1} {a b : \u03b1}\n\n@[ematch] theorem Inf_le : a \u2208 s \u2192 Inf s \u2264 a := complete_semilattice_Inf.Inf_le s a\n\ntheorem le_Inf : (\u2200b\u2208s, a \u2264 b) \u2192 a \u2264 Inf s := complete_semilattice_Inf.le_Inf s a\n\nlemma is_glb_Inf (s : set \u03b1) : is_glb s (Inf s) := \u27e8assume a, Inf_le, assume a, le_Inf\u27e9\n\nlemma is_glb.Inf_eq (h : is_glb s a) : Inf s = a := (is_glb_Inf s).unique h\n\ntheorem Inf_le_of_le (hb : b \u2208 s) (h : b \u2264 a) : Inf s \u2264 a :=\nle_trans (Inf_le hb) h\n\ntheorem Inf_le_Inf (h : s \u2286 t) : Inf t \u2264 Inf s :=\n(is_glb_Inf s).mono (is_glb_Inf t) h\n\n@[simp] theorem le_Inf_iff : a \u2264 Inf s \u2194 (\u2200b \u2208 s, a \u2264 b) :=\nle_is_glb_iff (is_glb_Inf s)\n\nlemma Inf_le_iff :\n  Inf s \u2264 a \u2194 (\u2200 b, (\u2200 x \u2208 s, b \u2264 x) \u2192 b \u2264 a) :=\n\u27e8\u03bb h b hb, le_trans (le_Inf hb) h, \u03bb hb, hb _ (\u03bb x, Inf_le)\u27e9\n\ntheorem Inf_le_Inf_of_forall_exists_le (h : \u2200 x \u2208 s, \u2203 y \u2208 t, y \u2264 x) : Inf t \u2264 Inf s :=\nle_of_forall_le begin\n  simp only [le_Inf_iff],\n  introv h\u2080 h\u2081,\n  rcases h _ h\u2081 with \u27e8y,hy,hy'\u27e9,\n  solve_by_elim [le_trans _ hy']\nend\n\n-- We will generalize this to conditionally complete lattices in `cInf_singleton`.\ntheorem Inf_singleton {a : \u03b1} : Inf {a} = a :=\nis_glb_singleton.Inf_eq\n\nend\n\n\/-- A complete lattice is a bounded lattice which\n  has suprema and infima for every subset. -\/\n@[protect_proj]\nclass complete_lattice (\u03b1 : Type*) extends\n  bounded_lattice \u03b1, complete_semilattice_Sup \u03b1, complete_semilattice_Inf \u03b1.\n\n\/-- Create a `complete_lattice` from a `partial_order` and `Inf` function\nthat returns the greatest lower bound of a set. Usually this constructor provides\npoor definitional equalities.  If other fields are known explicitly, they should be\nprovided; for example, if `inf` is known explicitly, construct the `complete_lattice`\ninstance as\n```\ninstance : complete_lattice my_T :=\n{ inf := better_inf,\n  le_inf := ...,\n  inf_le_right := ...,\n  inf_le_left := ...\n  -- don't care to fix sup, Sup, bot, top\n  ..complete_lattice_of_Inf my_T _ }\n```\n-\/\ndef complete_lattice_of_Inf (\u03b1 : Type*) [H1 : partial_order \u03b1]\n  [H2 : has_Inf \u03b1] (is_glb_Inf : \u2200 s : set \u03b1, is_glb s (Inf s)) :\n  complete_lattice \u03b1 :=\n{ bot := Inf univ,\n  bot_le := \u03bb x, (is_glb_Inf univ).1 trivial,\n  top := Inf \u2205,\n  le_top := \u03bb a, (is_glb_Inf \u2205).2 $ by simp,\n  sup := \u03bb a b, Inf {x | a \u2264 x \u2227 b \u2264 x},\n  inf := \u03bb a b, Inf {a, b},\n  le_inf := \u03bb a b c hab hac, by { apply (is_glb_Inf _).2, simp [*] },\n  inf_le_right := \u03bb a b, (is_glb_Inf _).1 $ mem_insert_of_mem _ $ mem_singleton _,\n  inf_le_left := \u03bb a b, (is_glb_Inf _).1 $ mem_insert _ _,\n  sup_le := \u03bb a b c hac hbc, (is_glb_Inf _).1 $ by simp [*],\n  le_sup_left := \u03bb a b, (is_glb_Inf _).2 $ \u03bb x, and.left,\n  le_sup_right := \u03bb a b, (is_glb_Inf _).2 $ \u03bb x, and.right,\n  le_Inf := \u03bb s a ha, (is_glb_Inf s).2 ha,\n  Inf_le := \u03bb s a ha, (is_glb_Inf s).1 ha,\n  Sup := \u03bb s, Inf (upper_bounds s),\n  le_Sup := \u03bb s a ha, (is_glb_Inf (upper_bounds s)).2 $ \u03bb b hb, hb ha,\n  Sup_le := \u03bb s a ha, (is_glb_Inf (upper_bounds s)).1 ha,\n  .. H1, .. H2 }\n\n\/--\nAny `complete_semilattice_Inf` is in fact a `complete_lattice`.\n\nNote that this construction has bad definitional properties:\nsee the doc-string on `complete_lattice_of_Inf`.\n-\/\ndef complete_lattice_of_complete_semilattice_Inf (\u03b1 : Type*) [complete_semilattice_Inf \u03b1] :\n  complete_lattice \u03b1 :=\ncomplete_lattice_of_Inf \u03b1 (\u03bb s, is_glb_Inf s)\n\n\/-- Create a `complete_lattice` from a `partial_order` and `Sup` function\nthat returns the least upper bound of a set. Usually this constructor provides\npoor definitional equalities.  If other fields are known explicitly, they should be\nprovided; for example, if `inf` is known explicitly, construct the `complete_lattice`\ninstance as\n```\ninstance : complete_lattice my_T :=\n{ inf := better_inf,\n  le_inf := ...,\n  inf_le_right := ...,\n  inf_le_left := ...\n  -- don't care to fix sup, Inf, bot, top\n  ..complete_lattice_of_Sup my_T _ }\n```\n-\/\ndef complete_lattice_of_Sup (\u03b1 : Type*) [H1 : partial_order \u03b1]\n  [H2 : has_Sup \u03b1] (is_lub_Sup : \u2200 s : set \u03b1, is_lub s (Sup s)) :\n  complete_lattice \u03b1 :=\n{ top := Sup univ,\n  le_top := \u03bb x, (is_lub_Sup univ).1 trivial,\n  bot := Sup \u2205,\n  bot_le := \u03bb x, (is_lub_Sup \u2205).2 $ by simp,\n  sup := \u03bb a b, Sup {a, b},\n  sup_le := \u03bb a b c hac hbc, (is_lub_Sup _).2 (by simp [*]),\n  le_sup_left := \u03bb a b, (is_lub_Sup _).1 $ mem_insert _ _,\n  le_sup_right := \u03bb a b, (is_lub_Sup _).1 $ mem_insert_of_mem _ $ mem_singleton _,\n  inf := \u03bb a b, Sup {x | x \u2264 a \u2227 x \u2264 b},\n  le_inf := \u03bb a b c hab hac, (is_lub_Sup _).1 $ by simp [*],\n  inf_le_left := \u03bb a b, (is_lub_Sup _).2 (\u03bb x, and.left),\n  inf_le_right := \u03bb a b, (is_lub_Sup _).2 (\u03bb x, and.right),\n  Inf := \u03bb s, Sup (lower_bounds s),\n  Sup_le := \u03bb s a ha, (is_lub_Sup s).2 ha,\n  le_Sup := \u03bb s a ha, (is_lub_Sup s).1 ha,\n  Inf_le := \u03bb s a ha, (is_lub_Sup (lower_bounds s)).2 (\u03bb b hb, hb ha),\n  le_Inf := \u03bb s a ha, (is_lub_Sup (lower_bounds s)).1 ha,\n  .. H1, .. H2 }\n\n\/--\nAny `complete_semilattice_Sup` is in fact a `complete_lattice`.\n\nNote that this construction has bad definitional properties:\nsee the doc-string on `complete_lattice_of_Sup`.\n-\/\ndef complete_lattice_of_complete_semilattice_Sup (\u03b1 : Type*) [complete_semilattice_Sup \u03b1] :\n  complete_lattice \u03b1 :=\ncomplete_lattice_of_Sup \u03b1 (\u03bb s, is_lub_Sup s)\n\n\/-- A complete linear order is a linear order whose lattice structure is complete. -\/\nclass complete_linear_order (\u03b1 : Type*) extends complete_lattice \u03b1, linear_order \u03b1\n\nnamespace order_dual\nvariable (\u03b1)\n\ninstance [complete_lattice \u03b1] : complete_lattice (order_dual \u03b1) :=\n{ le_Sup := @complete_lattice.Inf_le \u03b1 _,\n  Sup_le := @complete_lattice.le_Inf \u03b1 _,\n  Inf_le := @complete_lattice.le_Sup \u03b1 _,\n  le_Inf := @complete_lattice.Sup_le \u03b1 _,\n  .. order_dual.bounded_lattice \u03b1, ..order_dual.has_Sup \u03b1, ..order_dual.has_Inf \u03b1 }\n\ninstance [complete_linear_order \u03b1] : complete_linear_order (order_dual \u03b1) :=\n{ .. order_dual.complete_lattice \u03b1, .. order_dual.linear_order \u03b1 }\n\nend order_dual\n\nsection\nvariables [complete_lattice \u03b1] {s t : set \u03b1} {a b : \u03b1}\n\ntheorem Inf_le_Sup (hs : s.nonempty) : Inf s \u2264 Sup s :=\nis_glb_le_is_lub (is_glb_Inf s) (is_lub_Sup s) hs\n\ntheorem Sup_union {s t : set \u03b1} : Sup (s \u222a t) = Sup s \u2294 Sup t :=\n((is_lub_Sup s).union (is_lub_Sup t)).Sup_eq\n\ntheorem Sup_inter_le {s t : set \u03b1} : Sup (s \u2229 t) \u2264 Sup s \u2293 Sup t :=\nby finish\n\/-\n  Sup_le (assume a \u27e8a_s, a_t\u27e9, le_inf (le_Sup a_s) (le_Sup a_t))\n-\/\n\ntheorem Inf_union {s t : set \u03b1} : Inf (s \u222a t) = Inf s \u2293 Inf t :=\n((is_glb_Inf s).union (is_glb_Inf t)).Inf_eq\n\ntheorem le_Inf_inter {s t : set \u03b1} : Inf s \u2294 Inf t \u2264 Inf (s \u2229 t) :=\n@Sup_inter_le (order_dual \u03b1) _ _ _\n\n@[simp] theorem Sup_empty : Sup \u2205 = (\u22a5 : \u03b1) :=\n(@is_lub_empty \u03b1 _).Sup_eq\n\n@[simp] theorem Inf_empty : Inf \u2205 = (\u22a4 : \u03b1) :=\n(@is_glb_empty \u03b1 _).Inf_eq\n\n@[simp] theorem Sup_univ : Sup univ = (\u22a4 : \u03b1) :=\n(@is_lub_univ \u03b1 _).Sup_eq\n\n@[simp] theorem Inf_univ : Inf univ = (\u22a5 : \u03b1) :=\n(@is_glb_univ \u03b1 _).Inf_eq\n\n-- TODO(Jeremy): get this automatically\n@[simp] theorem Sup_insert {a : \u03b1} {s : set \u03b1} : Sup (insert a s) = a \u2294 Sup s :=\n((is_lub_Sup s).insert a).Sup_eq\n\n@[simp] theorem Inf_insert {a : \u03b1} {s : set \u03b1} : Inf (insert a s) = a \u2293 Inf s :=\n((is_glb_Inf s).insert a).Inf_eq\n\ntheorem Sup_le_Sup_of_subset_insert_bot (h : s \u2286 insert \u22a5 t) : Sup s \u2264 Sup t :=\nle_trans (Sup_le_Sup h) (le_of_eq (trans Sup_insert bot_sup_eq))\n\ntheorem Inf_le_Inf_of_subset_insert_top (h : s \u2286 insert \u22a4 t) : Inf t \u2264 Inf s :=\nle_trans (le_of_eq (trans top_inf_eq.symm Inf_insert.symm)) (Inf_le_Inf h)\n\ntheorem Sup_pair {a b : \u03b1} : Sup {a, b} = a \u2294 b :=\n(@is_lub_pair \u03b1 _ a b).Sup_eq\n\ntheorem Inf_pair {a b : \u03b1} : Inf {a, b} = a \u2293 b :=\n(@is_glb_pair \u03b1 _ a b).Inf_eq\n\n@[simp] theorem Inf_eq_top : Inf s = \u22a4 \u2194 (\u2200a\u2208s, a = \u22a4) :=\niff.intro\n  (assume h a ha, top_unique $ h \u25b8 Inf_le ha)\n  (assume h, top_unique $ le_Inf $ assume a ha, top_le_iff.2 $ h a ha)\n\nlemma eq_singleton_top_of_Inf_eq_top_of_nonempty {s : set \u03b1}\n  (h_inf : Inf s = \u22a4) (hne : s.nonempty) : s = {\u22a4} :=\nby { rw set.eq_singleton_iff_nonempty_unique_mem, rw Inf_eq_top at h_inf, exact \u27e8hne, h_inf\u27e9, }\n\n@[simp] theorem Sup_eq_bot : Sup s = \u22a5 \u2194 (\u2200a\u2208s, a = \u22a5) :=\n@Inf_eq_top (order_dual \u03b1) _ _\n\nlemma eq_singleton_bot_of_Sup_eq_bot_of_nonempty {s : set \u03b1}\n  (h_sup : Sup s = \u22a5) (hne : s.nonempty) : s = {\u22a5} :=\nby { rw set.eq_singleton_iff_nonempty_unique_mem, rw Sup_eq_bot at h_sup, exact \u27e8hne, h_sup\u27e9, }\n\n\/--Introduction rule to prove that `b` is the supremum of `s`: it suffices to check that `b`\nis larger than all elements of `s`, and that this is not the case of any `w<b`.\nSee `cSup_eq_of_forall_le_of_forall_lt_exists_gt` for a version in conditionally complete\nlattices. -\/\ntheorem Sup_eq_of_forall_le_of_forall_lt_exists_gt (_ : \u2200a\u2208s, a \u2264 b)\n  (H : \u2200w, w < b \u2192 (\u2203a\u2208s, w < a)) : Sup s = b :=\nhave bdd_above s := \u27e8b, by assumption\u27e9,\nhave (Sup s < b) \u2228 (Sup s = b) := lt_or_eq_of_le (Sup_le \u2039\u2200a\u2208s, a \u2264 b\u203a),\nhave \u00ac(Sup s < b) :=\n  assume: Sup s < b,\n  let \u27e8a, _, _\u27e9 := (H (Sup s) \u2039Sup s < b\u203a) in  \/- a \u2208 s, Sup s < a-\/\n  have Sup s < Sup s := lt_of_lt_of_le \u2039Sup s < a\u203a (le_Sup \u2039a \u2208 s\u203a),\n  show false, by finish [lt_irrefl (Sup s)],\nshow Sup s = b, by finish\n\n\/--Introduction rule to prove that `b` is the infimum of `s`: it suffices to check that `b`\nis smaller than all elements of `s`, and that this is not the case of any `w>b`.\nSee `cInf_eq_of_forall_ge_of_forall_gt_exists_lt` for a version in conditionally complete\nlattices. -\/\ntheorem Inf_eq_of_forall_ge_of_forall_gt_exists_lt (_ : \u2200a\u2208s, b \u2264 a)\n  (H : \u2200w, b < w \u2192 (\u2203a\u2208s, a < w)) : Inf s = b :=\n@Sup_eq_of_forall_le_of_forall_lt_exists_gt (order_dual \u03b1) _ _ \u2039_\u203a \u2039_\u203a \u2039_\u203a\n\nend\n\nsection complete_linear_order\nvariables [complete_linear_order \u03b1] {s t : set \u03b1} {a b : \u03b1}\n\nlemma Inf_lt_iff : Inf s < b \u2194 (\u2203a\u2208s, a < b) :=\nis_glb_lt_iff (is_glb_Inf s)\n\nlemma lt_Sup_iff : b < Sup s \u2194 (\u2203a\u2208s, b < a) :=\nlt_is_lub_iff (is_lub_Sup s)\n\nlemma Sup_eq_top : Sup s = \u22a4 \u2194 (\u2200b<\u22a4, \u2203a\u2208s, b < a) :=\niff.intro\n  (assume (h : Sup s = \u22a4) b hb, by rwa [\u2190h, lt_Sup_iff] at hb)\n  (assume h, top_unique $ le_of_not_gt $ assume h',\n    let \u27e8a, ha, h\u27e9 := h _ h' in\n    lt_irrefl a $ lt_of_le_of_lt (le_Sup ha) h)\n\nlemma Inf_eq_bot : Inf s = \u22a5 \u2194 (\u2200b>\u22a5, \u2203a\u2208s, a < b) :=\n@Sup_eq_top (order_dual \u03b1) _ _\n\nlemma lt_supr_iff {f : \u03b9 \u2192 \u03b1} : a < supr f \u2194 (\u2203i, a < f i) :=\nlt_Sup_iff.trans exists_range_iff\n\nlemma infi_lt_iff {f : \u03b9 \u2192 \u03b1} : infi f < a \u2194 (\u2203i, f i < a) :=\nInf_lt_iff.trans exists_range_iff\n\nend complete_linear_order\n\n\/-\n### supr & infi\n-\/\n\nsection\nvariables [complete_lattice \u03b1] {s t : \u03b9 \u2192 \u03b1} {a b : \u03b1}\n\n-- TODO: this declaration gives error when starting smt state\n--@[ematch]\ntheorem le_supr (s : \u03b9 \u2192 \u03b1) (i : \u03b9) : s i \u2264 supr s :=\nle_Sup \u27e8i, rfl\u27e9\n\n@[ematch] theorem le_supr' (s : \u03b9 \u2192 \u03b1) (i : \u03b9) : (: s i \u2264 supr s :) :=\nle_Sup \u27e8i, rfl\u27e9\n\n\/- TODO: this version would be more powerful, but, alas, the pattern matcher\n   doesn't accept it.\n@[ematch] theorem le_supr' (s : \u03b9 \u2192 \u03b1) (i : \u03b9) : (: s i :) \u2264 (: supr s :) :=\nle_Sup \u27e8i, rfl\u27e9\n-\/\n\nlemma is_lub_supr : is_lub (range s) (\u2a06j, s j) := is_lub_Sup _\n\nlemma is_lub.supr_eq (h : is_lub (range s) a) : (\u2a06j, s j) = a := h.Sup_eq\n\nlemma is_glb_infi : is_glb (range s) (\u2a05j, s j) := is_glb_Inf _\n\nlemma is_glb.infi_eq (h : is_glb (range s) a) : (\u2a05j, s j) = a := h.Inf_eq\n\ntheorem le_supr_of_le (i : \u03b9) (h : a \u2264 s i) : a \u2264 supr s :=\nle_trans h (le_supr _ i)\n\ntheorem le_bsupr {p : \u03b9 \u2192 Prop} {f : \u03a0 i (h : p i), \u03b1} (i : \u03b9) (hi : p i) :\n  f i hi \u2264 \u2a06 i hi, f i hi :=\nle_supr_of_le i $ le_supr (f i) hi\n\ntheorem le_bsupr_of_le {p : \u03b9 \u2192 Prop} {f : \u03a0 i (h : p i), \u03b1} (i : \u03b9) (hi : p i) (h : a \u2264 f i hi) :\n  a \u2264 \u2a06 i hi, f i hi :=\nle_trans h (le_bsupr i hi)\n\ntheorem supr_le (h : \u2200i, s i \u2264 a) : supr s \u2264 a :=\nSup_le $ assume b \u27e8i, eq\u27e9, eq \u25b8 h i\n\ntheorem bsupr_le {p : \u03b9 \u2192 Prop} {f : \u03a0 i (h : p i), \u03b1} (h : \u2200 i hi, f i hi \u2264 a) :\n  (\u2a06 i (hi : p i), f i hi) \u2264 a :=\nsupr_le $ \u03bb i, supr_le $ h i\n\ntheorem bsupr_le_supr (p : \u03b9 \u2192 Prop) (f : \u03b9 \u2192 \u03b1) : (\u2a06 i (H : p i), f i) \u2264 \u2a06 i, f i :=\nbsupr_le (\u03bb i hi, le_supr f i)\n\ntheorem supr_le_supr (h : \u2200i, s i \u2264 t i) : supr s \u2264 supr t :=\nsupr_le $ assume i, le_supr_of_le i (h i)\n\ntheorem supr_le_supr2 {t : \u03b9\u2082 \u2192 \u03b1} (h : \u2200i, \u2203j, s i \u2264 t j) : supr s \u2264 supr t :=\nsupr_le $ assume j, exists.elim (h j) le_supr_of_le\n\ntheorem bsupr_le_bsupr {p : \u03b9 \u2192 Prop} {f g : \u03a0 i (hi : p i), \u03b1} (h : \u2200 i hi, f i hi \u2264 g i hi) :\n  (\u2a06 i hi, f i hi) \u2264 \u2a06 i hi, g i hi :=\nbsupr_le $ \u03bb i hi, le_trans (h i hi) (le_bsupr i hi)\n\ntheorem supr_le_supr_const (h : \u03b9 \u2192 \u03b9\u2082) : (\u2a06 i:\u03b9, a) \u2264 (\u2a06 j:\u03b9\u2082, a) :=\nsupr_le $ le_supr _ \u2218 h\n\ntheorem bsupr_le_bsupr' {p q : \u03b9 \u2192 Prop} (hpq : \u2200 i, p i \u2192 q i) {f : \u03b9 \u2192 \u03b1} :\n  (\u2a06 i (hpi : p i), f i) \u2264 \u2a06 i (hqi : q i), f i :=\nsupr_le_supr $ \u03bb i, supr_le_supr_const (hpq i)\n\n@[simp] theorem supr_le_iff : supr s \u2264 a \u2194 (\u2200i, s i \u2264 a) :=\n(is_lub_le_iff is_lub_supr).trans forall_range_iff\n\ntheorem supr_lt_iff : supr s < a \u2194 \u2203 b < a, \u2200 i, s i \u2264 b :=\n\u27e8\u03bb h, \u27e8supr s, h, \u03bb i, le_supr s i\u27e9, \u03bb \u27e8b, hba, hsb\u27e9, (supr_le hsb).trans_lt hba\u27e9\n\ntheorem Sup_eq_supr {s : set \u03b1} : Sup s = (\u2a06a \u2208 s, a) :=\nle_antisymm\n  (Sup_le $ assume b h, le_supr_of_le b $ le_supr _ h)\n  (supr_le $ assume b, supr_le $ assume h, le_Sup h)\n\nlemma Sup_eq_supr' {\u03b1} [has_Sup \u03b1] (s : set \u03b1) : Sup s = \u2a06 x : s, (x : \u03b1) :=\nby rw [supr, subtype.range_coe]\n\nlemma Sup_sUnion {s : set (set \u03b1)} :\n  Sup (\u22c3\u2080 s) = \u2a06 (t \u2208 s), Sup t :=\nbegin\n  apply le_antisymm,\n  { apply Sup_le (\u03bb b hb, _),\n    rcases hb with \u27e8t, ts, bt\u27e9,\n    apply le_trans _ (le_supr _ t),\n    exact le_trans (le_Sup bt) (le_supr _ ts), },\n  { apply supr_le (\u03bb t, _),\n    exact supr_le (\u03bb ts, Sup_le_Sup (\u03bb x xt, \u27e8t, ts, xt\u27e9)) }\nend\n\nlemma le_supr_iff : (a \u2264 supr s) \u2194 (\u2200 b, (\u2200 i, s i \u2264 b) \u2192 a \u2264 b) :=\n\u27e8\u03bb h b hb, le_trans h (supr_le hb), \u03bb h, h _ $ \u03bb i, le_supr s i\u27e9\n\nlemma monotone.le_map_supr [complete_lattice \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f) :\n  (\u2a06 i, f (s i)) \u2264 f (supr s) :=\nsupr_le $ \u03bb i, hf $ le_supr _ _\n\nlemma monotone.le_map_supr2 [complete_lattice \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f)\n  {\u03b9' : \u03b9 \u2192 Sort*} (s : \u03a0 i, \u03b9' i \u2192 \u03b1) :\n  (\u2a06 i (h : \u03b9' i), f (s i h)) \u2264 f (\u2a06 i (h : \u03b9' i), s i h) :=\ncalc (\u2a06 i h, f (s i h)) \u2264 (\u2a06 i, f (\u2a06 h, s i h)) :\n  supr_le_supr $ \u03bb i, hf.le_map_supr\n... \u2264 f (\u2a06 i (h : \u03b9' i), s i h) : hf.le_map_supr\n\nlemma monotone.le_map_Sup [complete_lattice \u03b2] {s : set \u03b1} {f : \u03b1 \u2192 \u03b2} (hf : monotone f) :\n  (\u2a06a\u2208s, f a) \u2264 f (Sup s) :=\nby rw [Sup_eq_supr]; exact hf.le_map_supr2 _\n\nlemma supr_comp_le {\u03b9' : Sort*} (f : \u03b9' \u2192 \u03b1) (g : \u03b9 \u2192 \u03b9') :\n  (\u2a06 x, f (g x)) \u2264 \u2a06 y, f y :=\nsupr_le_supr2 $ \u03bb x, \u27e8_, le_refl _\u27e9\n\nlemma monotone.supr_comp_eq [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} (hf : monotone f)\n  {s : \u03b9 \u2192 \u03b2} (hs : \u2200 x, \u2203 i, x \u2264 s i) :\n  (\u2a06 x, f (s x)) = \u2a06 y, f y :=\nle_antisymm (supr_comp_le _ _) (supr_le_supr2 $ \u03bb x, (hs x).imp $ \u03bb i hi, hf hi)\n\nlemma function.surjective.supr_comp {\u03b1 : Type*} [has_Sup \u03b1] {f : \u03b9 \u2192 \u03b9\u2082}\n  (hf : function.surjective f) (g : \u03b9\u2082 \u2192 \u03b1) :\n  (\u2a06 x, g (f x)) = \u2a06 y, g y :=\nby simp only [supr, hf.range_comp]\n\nlemma supr_congr {\u03b1 : Type*} [has_Sup \u03b1] {f : \u03b9 \u2192 \u03b1} {g : \u03b9\u2082 \u2192 \u03b1} (h : \u03b9 \u2192 \u03b9\u2082)\n  (h1 : function.surjective h) (h2 : \u2200 x, g (h x) = f x) : (\u2a06 x, f x) = \u2a06 y, g y :=\nby { convert h1.supr_comp g, exact (funext h2).symm }\n\n-- TODO: finish doesn't do well here.\n@[congr] theorem supr_congr_Prop {\u03b1 : Type*} [has_Sup \u03b1] {p q : Prop} {f\u2081 : p \u2192 \u03b1} {f\u2082 : q \u2192 \u03b1}\n  (pq : p \u2194 q) (f : \u2200x, f\u2081 (pq.mpr x) = f\u2082 x) : supr f\u2081 = supr f\u2082 :=\nbegin\n  have := propext pq, subst this,\n  congr' with x,\n  apply f\nend\n\ntheorem infi_le (s : \u03b9 \u2192 \u03b1) (i : \u03b9) : infi s \u2264 s i :=\nInf_le \u27e8i, rfl\u27e9\n\n@[ematch] theorem infi_le' (s : \u03b9 \u2192 \u03b1) (i : \u03b9) : (: infi s \u2264 s i :) :=\nInf_le \u27e8i, rfl\u27e9\n\ntheorem infi_le_of_le (i : \u03b9) (h : s i \u2264 a) : infi s \u2264 a :=\nle_trans (infi_le _ i) h\n\ntheorem binfi_le {p : \u03b9 \u2192 Prop} {f : \u03a0 i (hi : p i), \u03b1} (i : \u03b9) (hi : p i) :\n  (\u2a05 i hi, f i hi) \u2264 f i hi :=\ninfi_le_of_le i $ infi_le (f i) hi\n\ntheorem binfi_le_of_le {p : \u03b9 \u2192 Prop} {f : \u03a0 i (hi : p i), \u03b1} (i : \u03b9) (hi : p i) (h : f i hi \u2264 a) :\n  (\u2a05 i hi, f i hi) \u2264 a :=\nle_trans (binfi_le i hi) h\n\ntheorem le_infi (h : \u2200i, a \u2264 s i) : a \u2264 infi s :=\nle_Inf $ assume b \u27e8i, eq\u27e9, eq \u25b8 h i\n\ntheorem le_binfi {p : \u03b9 \u2192 Prop} {f : \u03a0 i (h : p i), \u03b1} (h : \u2200 i hi, a \u2264 f i hi) :\n  a \u2264 \u2a05 i hi, f i hi :=\nle_infi $ \u03bb i, le_infi $ h i\n\ntheorem infi_le_binfi (p : \u03b9 \u2192 Prop) (f : \u03b9 \u2192 \u03b1) : (\u2a05 i, f i) \u2264 \u2a05 i (H : p i), f i :=\nle_binfi (\u03bb i hi, infi_le f i)\n\ntheorem infi_le_infi (h : \u2200i, s i \u2264 t i) : infi s \u2264 infi t :=\nle_infi $ assume i, infi_le_of_le i (h i)\n\ntheorem infi_le_infi2 {t : \u03b9\u2082 \u2192 \u03b1} (h : \u2200j, \u2203i, s i \u2264 t j) : infi s \u2264 infi t :=\nle_infi $ assume j, exists.elim (h j) infi_le_of_le\n\ntheorem binfi_le_binfi {p : \u03b9 \u2192 Prop} {f g : \u03a0 i (h : p i), \u03b1} (h : \u2200 i hi, f i hi \u2264 g i hi) :\n  (\u2a05 i hi, f i hi) \u2264 \u2a05 i hi, g i hi :=\nle_binfi $ \u03bb i hi, le_trans (binfi_le i hi) (h i hi)\n\ntheorem infi_le_infi_const (h : \u03b9\u2082 \u2192 \u03b9) : (\u2a05 i:\u03b9, a) \u2264 (\u2a05 j:\u03b9\u2082, a) :=\nle_infi $ infi_le _ \u2218 h\n\n@[simp] theorem le_infi_iff : a \u2264 infi s \u2194 (\u2200i, a \u2264 s i) :=\n\u27e8assume : a \u2264 infi s, assume i, le_trans this (infi_le _ _), le_infi\u27e9\n\ntheorem Inf_eq_infi {s : set \u03b1} : Inf s = (\u2a05a \u2208 s, a) :=\n@Sup_eq_supr (order_dual \u03b1) _ _\n\ntheorem Inf_eq_infi' {\u03b1} [has_Inf \u03b1] (s : set \u03b1) : Inf s = \u2a05 a : s, a :=\n@Sup_eq_supr' (order_dual \u03b1) _ _\n\nlemma monotone.map_infi_le [complete_lattice \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f) :\n  f (infi s) \u2264 (\u2a05 i, f (s i)) :=\nle_infi $ \u03bb i, hf $ infi_le _ _\n\nlemma monotone.map_infi2_le [complete_lattice \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f)\n  {\u03b9' : \u03b9 \u2192 Sort*} (s : \u03a0 i, \u03b9' i \u2192 \u03b1) :\n  f (\u2a05 i (h : \u03b9' i), s i h) \u2264 (\u2a05 i (h : \u03b9' i), f (s i h)) :=\n@monotone.le_map_supr2 (order_dual \u03b1) (order_dual \u03b2) _ _ _ f hf.order_dual _ _\n\nlemma monotone.map_Inf_le [complete_lattice \u03b2] {s : set \u03b1} {f : \u03b1 \u2192 \u03b2} (hf : monotone f) :\n  f (Inf s) \u2264 \u2a05 a\u2208s, f a :=\nby rw [Inf_eq_infi]; exact hf.map_infi2_le _\n\nlemma le_infi_comp {\u03b9' : Sort*} (f : \u03b9' \u2192 \u03b1) (g : \u03b9 \u2192 \u03b9') :\n  (\u2a05 y, f y) \u2264 \u2a05 x, f (g x) :=\ninfi_le_infi2 $ \u03bb x, \u27e8_, le_refl _\u27e9\n\nlemma monotone.infi_comp_eq [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} (hf : monotone f)\n  {s : \u03b9 \u2192 \u03b2} (hs : \u2200 x, \u2203 i, s i \u2264 x) :\n  (\u2a05 x, f (s x)) = \u2a05 y, f y :=\nle_antisymm (infi_le_infi2 $ \u03bb x, (hs x).imp $ \u03bb i hi, hf hi) (le_infi_comp _ _)\n\nlemma function.surjective.infi_comp {\u03b1 : Type*} [has_Inf \u03b1] {f : \u03b9 \u2192 \u03b9\u2082}\n  (hf : function.surjective f) (g : \u03b9\u2082 \u2192 \u03b1) :\n  (\u2a05 x, g (f x)) = \u2a05 y, g y :=\n@function.surjective.supr_comp _ _ (order_dual \u03b1) _ f hf g\n\nlemma infi_congr {\u03b1 : Type*} [has_Inf \u03b1] {f : \u03b9 \u2192 \u03b1} {g : \u03b9\u2082 \u2192 \u03b1} (h : \u03b9 \u2192 \u03b9\u2082)\n  (h1 : function.surjective h) (h2 : \u2200 x, g (h x) = f x) : (\u2a05 x, f x) = \u2a05 y, g y :=\n@supr_congr _ _ (order_dual \u03b1) _ _ _ h h1 h2\n\n@[congr] theorem infi_congr_Prop {\u03b1 : Type*} [has_Inf \u03b1] {p q : Prop} {f\u2081 : p \u2192 \u03b1} {f\u2082 : q \u2192 \u03b1}\n  (pq : p \u2194 q) (f : \u2200x, f\u2081 (pq.mpr x) = f\u2082 x) : infi f\u2081 = infi f\u2082 :=\n@supr_congr_Prop (order_dual \u03b1) _ p q f\u2081 f\u2082 pq f\n\nlemma supr_const_le {x : \u03b1} : (\u2a06 (h : \u03b9), x) \u2264 x :=\nsupr_le (\u03bb _, le_rfl)\n\nlemma le_infi_const {x : \u03b1} : x \u2264 (\u2a05 (h : \u03b9), x) :=\nle_infi (\u03bb _, le_rfl)\n\n-- We will generalize this to conditionally complete lattices in `cinfi_const`.\ntheorem infi_const [nonempty \u03b9] {a : \u03b1} : (\u2a05 b:\u03b9, a) = a :=\nby rw [infi, range_const, Inf_singleton]\n\n-- We will generalize this to conditionally complete lattices in `csupr_const`.\ntheorem supr_const [nonempty \u03b9] {a : \u03b1} : (\u2a06 b:\u03b9, a) = a :=\n@infi_const (order_dual \u03b1) _ _ _ _\n\n@[simp] lemma infi_top : (\u2a05i:\u03b9, \u22a4 : \u03b1) = \u22a4 :=\ntop_unique $ le_infi $ assume i, le_refl _\n\n@[simp] lemma supr_bot : (\u2a06i:\u03b9, \u22a5 : \u03b1) = \u22a5 :=\n@infi_top (order_dual \u03b1) _ _\n\n@[simp] lemma infi_eq_top : infi s = \u22a4 \u2194 (\u2200i, s i = \u22a4) :=\nInf_eq_top.trans forall_range_iff\n\n@[simp] lemma supr_eq_bot : supr s = \u22a5 \u2194 (\u2200i, s i = \u22a5) :=\nSup_eq_bot.trans forall_range_iff\n\n@[simp] lemma infi_pos {p : Prop} {f : p \u2192 \u03b1} (hp : p) : (\u2a05 h : p, f h) = f hp :=\nle_antisymm (infi_le _ _) (le_infi $ assume h, le_refl _)\n\n@[simp] lemma infi_neg {p : Prop} {f : p \u2192 \u03b1} (hp : \u00ac p) : (\u2a05 h : p, f h) = \u22a4 :=\nle_antisymm le_top $ le_infi $ assume h, (hp h).elim\n\n@[simp] lemma supr_pos {p : Prop} {f : p \u2192 \u03b1} (hp : p) : (\u2a06 h : p, f h) = f hp :=\nle_antisymm (supr_le $ assume h, le_refl _) (le_supr _ _)\n\n@[simp] lemma supr_neg {p : Prop} {f : p \u2192 \u03b1} (hp : \u00ac p) : (\u2a06 h : p, f h) = \u22a5 :=\nle_antisymm (supr_le $ assume h, (hp h).elim) bot_le\n\n\/--Introduction rule to prove that `b` is the supremum of `f`: it suffices to check that `b`\nis larger than `f i` for all `i`, and that this is not the case of any `w<b`.\nSee `csupr_eq_of_forall_le_of_forall_lt_exists_gt` for a version in conditionally complete\nlattices. -\/\ntheorem supr_eq_of_forall_le_of_forall_lt_exists_gt {f : \u03b9 \u2192 \u03b1} (h\u2081 : \u2200 i, f i \u2264 b)\n  (h\u2082 : \u2200 w, w < b \u2192 (\u2203 i, w < f i)) : (\u2a06 (i : \u03b9), f i) = b :=\nSup_eq_of_forall_le_of_forall_lt_exists_gt (forall_range_iff.mpr h\u2081)\n  (\u03bb w hw, exists_range_iff.mpr $ h\u2082 w hw)\n\n\/--Introduction rule to prove that `b` is the infimum of `f`: it suffices to check that `b`\nis smaller than `f i` for all `i`, and that this is not the case of any `w>b`.\nSee `cinfi_eq_of_forall_ge_of_forall_gt_exists_lt` for a version in conditionally complete\nlattices. -\/\ntheorem infi_eq_of_forall_ge_of_forall_gt_exists_lt {f : \u03b9 \u2192 \u03b1} (h\u2081 : \u2200 i, b \u2264 f i)\n(h\u2082 : \u2200 w, b < w \u2192 (\u2203 i, f i < w)) : (\u2a05 (i : \u03b9), f i) = b :=\n@supr_eq_of_forall_le_of_forall_lt_exists_gt (order_dual \u03b1) _ _ _ \u2039_\u203a \u2039_\u203a \u2039_\u203a\n\nlemma supr_eq_dif {p : Prop} [decidable p] (a : p \u2192 \u03b1) :\n  (\u2a06h:p, a h) = (if h : p then a h else \u22a5) :=\nby by_cases p; simp [h]\n\nlemma supr_eq_if {p : Prop} [decidable p] (a : \u03b1) :\n  (\u2a06h:p, a) = (if p then a else \u22a5) :=\nsupr_eq_dif (\u03bb _, a)\n\nlemma infi_eq_dif {p : Prop} [decidable p] (a : p \u2192 \u03b1) :\n  (\u2a05h:p, a h) = (if h : p then a h else \u22a4) :=\n@supr_eq_dif (order_dual \u03b1) _ _ _ _\n\nlemma infi_eq_if {p : Prop} [decidable p] (a : \u03b1) :\n  (\u2a05h:p, a) = (if p then a else \u22a4) :=\ninfi_eq_dif (\u03bb _, a)\n\n-- TODO: should this be @[simp]?\ntheorem infi_comm {f : \u03b9 \u2192 \u03b9\u2082 \u2192 \u03b1} : (\u2a05i, \u2a05j, f i j) = (\u2a05j, \u2a05i, f i j) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume j, infi_le_of_le j $ infi_le _ i)\n  (le_infi $ assume j, le_infi $ assume i, infi_le_of_le i $ infi_le _ j)\n\n\/- TODO: this is strange. In the proof below, we get exactly the desired\n   among the equalities, but close does not get it.\nbegin\n  apply @le_antisymm,\n    simp, intros,\n    begin [smt]\n      ematch, ematch, ematch, trace_state, have := le_refl (f i_1 i),\n      trace_state, close\n    end\nend\n-\/\n\n-- TODO: should this be @[simp]?\ntheorem supr_comm {f : \u03b9 \u2192 \u03b9\u2082 \u2192 \u03b1} : (\u2a06i, \u2a06j, f i j) = (\u2a06j, \u2a06i, f i j) :=\n@infi_comm (order_dual \u03b1) _ _ _ _\n\n@[simp] theorem infi_infi_eq_left {b : \u03b2} {f : \u03a0x:\u03b2, x = b \u2192 \u03b1} :\n  (\u2a05x, \u2a05h:x = b, f x h) = f b rfl :=\nle_antisymm\n  (infi_le_of_le b $ infi_le _ rfl)\n  (le_infi $ assume b', le_infi $ assume eq, match b', eq with ._, rfl := le_refl _ end)\n\n@[simp] theorem infi_infi_eq_right {b : \u03b2} {f : \u03a0x:\u03b2, b = x \u2192 \u03b1} :\n  (\u2a05x, \u2a05h:b = x, f x h) = f b rfl :=\nle_antisymm\n  (infi_le_of_le b $ infi_le _ rfl)\n  (le_infi $ assume b', le_infi $ assume eq, match b', eq with ._, rfl := le_refl _ end)\n\n@[simp] theorem supr_supr_eq_left {b : \u03b2} {f : \u03a0x:\u03b2, x = b \u2192 \u03b1} :\n  (\u2a06x, \u2a06h : x = b, f x h) = f b rfl :=\n@infi_infi_eq_left (order_dual \u03b1) _ _ _ _\n\n@[simp] theorem supr_supr_eq_right {b : \u03b2} {f : \u03a0x:\u03b2, b = x \u2192 \u03b1} :\n  (\u2a06x, \u2a06h : b = x, f x h) = f b rfl :=\n@infi_infi_eq_right (order_dual \u03b1) _ _ _ _\n\nattribute [ematch] le_refl\n\ntheorem infi_subtype {p : \u03b9 \u2192 Prop} {f : subtype p \u2192 \u03b1} : (\u2a05 x, f x) = (\u2a05 i (h:p i), f \u27e8i, h\u27e9) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume : p i, infi_le _ _)\n  (le_infi $ assume \u27e8i, h\u27e9, infi_le_of_le i $ infi_le _ _)\n\nlemma infi_subtype' {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1} :\n  (\u2a05 i (h : p i), f i h) = (\u2a05 x : subtype p, f x x.property) :=\n(@infi_subtype _ _ _ p (\u03bb x, f x.val x.property)).symm\n\nlemma infi_subtype'' {\u03b9} (s : set \u03b9) (f : \u03b9 \u2192 \u03b1) :\n  (\u2a05 i : s, f i) = \u2a05 (t : \u03b9) (H : t \u2208 s), f t :=\ninfi_subtype\n\ntheorem infi_inf_eq {f g : \u03b9 \u2192 \u03b1} : (\u2a05 x, f x \u2293 g x) = (\u2a05 x, f x) \u2293 (\u2a05 x, g x) :=\nle_antisymm\n  (le_inf\n    (le_infi $ assume i, infi_le_of_le i inf_le_left)\n    (le_infi $ assume i, infi_le_of_le i inf_le_right))\n  (le_infi $ assume i, le_inf\n    (inf_le_of_left_le $ infi_le _ _)\n    (inf_le_of_right_le $ infi_le _ _))\n\n\/- TODO: here is another example where more flexible pattern matching\n   might help.\n\nbegin\n  apply @le_antisymm,\n  safe, pose h := f a \u2293 g a, begin [smt] ematch, ematch  end\nend\n-\/\n\nlemma infi_inf [h : nonempty \u03b9] {f : \u03b9 \u2192 \u03b1} {a : \u03b1} : (\u2a05x, f x) \u2293 a = (\u2a05 x, f x \u2293 a) :=\nby rw [infi_inf_eq, infi_const]\n\nlemma inf_infi [nonempty \u03b9] {f : \u03b9 \u2192 \u03b1} {a : \u03b1} : a \u2293 (\u2a05x, f x) = (\u2a05 x, a \u2293 f x) :=\nby rw [inf_comm, infi_inf]; simp [inf_comm]\n\nlemma binfi_inf {p : \u03b9 \u2192 Prop} {f : \u03a0 i (hi : p i), \u03b1} {a : \u03b1} (h : \u2203 i, p i) :\n  (\u2a05i (h : p i), f i h) \u2293 a = (\u2a05 i (h : p i), f i h \u2293 a) :=\nby haveI : nonempty {i \/\/ p i} := (let \u27e8i, hi\u27e9 := h in \u27e8\u27e8i, hi\u27e9\u27e9);\n  rw [infi_subtype', infi_subtype', infi_inf]\n\nlemma inf_binfi {p : \u03b9 \u2192 Prop} {f : \u03a0 i (hi : p i), \u03b1} {a : \u03b1} (h : \u2203 i, p i) :\n  a \u2293 (\u2a05i (h : p i), f i h) = (\u2a05 i (h : p i), a \u2293 f i h) :=\nby simpa only [inf_comm] using binfi_inf h\n\ntheorem supr_sup_eq {f g : \u03b9 \u2192 \u03b1} : (\u2a06 x, f x \u2294 g x) = (\u2a06 x, f x) \u2294 (\u2a06 x, g x) :=\n@infi_inf_eq (order_dual \u03b1) \u03b9 _ _ _\n\nlemma supr_sup [h : nonempty \u03b9] {f : \u03b9 \u2192 \u03b1} {a : \u03b1} : (\u2a06 x, f x) \u2294 a = (\u2a06 x, f x \u2294 a) :=\n@infi_inf (order_dual \u03b1) _ _ _ _ _\n\nlemma sup_supr [nonempty \u03b9] {f : \u03b9 \u2192 \u03b1} {a : \u03b1} : a \u2294 (\u2a06 x, f x) = (\u2a06 x, a \u2294 f x) :=\n@inf_infi (order_dual \u03b1) _ _ _ _ _\n\n\/-! ### `supr` and `infi` under `Prop` -\/\n\n@[simp] theorem infi_false {s : false \u2192 \u03b1} : infi s = \u22a4 :=\nle_antisymm le_top (le_infi $ assume i, false.elim i)\n\n@[simp] theorem supr_false {s : false \u2192 \u03b1} : supr s = \u22a5 :=\nle_antisymm (supr_le $ assume i, false.elim i) bot_le\n\ntheorem infi_true {s : true \u2192 \u03b1} : infi s = s trivial :=\ninfi_pos trivial\n\ntheorem supr_true {s : true \u2192 \u03b1} : supr s = s trivial :=\nsupr_pos trivial\n\n@[simp] theorem infi_exists {p : \u03b9 \u2192 Prop} {f : Exists p \u2192 \u03b1} :\n  (\u2a05 x, f x) = (\u2a05 i, \u2a05 h:p i, f \u27e8i, h\u27e9) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume : p i, infi_le _ _)\n  (le_infi $ assume \u27e8i, h\u27e9, infi_le_of_le i $ infi_le _ _)\n\n@[simp] theorem supr_exists {p : \u03b9 \u2192 Prop} {f : Exists p \u2192 \u03b1} :\n  (\u2a06 x, f x) = (\u2a06 i, \u2a06 h:p i, f \u27e8i, h\u27e9) :=\n@infi_exists (order_dual \u03b1) _ _ _ _\n\ntheorem infi_and {p q : Prop} {s : p \u2227 q \u2192 \u03b1} : infi s = (\u2a05 h\u2081 h\u2082, s \u27e8h\u2081, h\u2082\u27e9) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume j, infi_le _ _)\n  (le_infi $ assume \u27e8i, h\u27e9, infi_le_of_le i $ infi_le _ _)\n\n\/-- The symmetric case of `infi_and`, useful for rewriting into a infimum over a conjunction -\/\nlemma infi_and' {p q : Prop} {s : p \u2192 q \u2192 \u03b1} :\n  (\u2a05 (h\u2081 : p) (h\u2082 : q), s h\u2081 h\u2082) = \u2a05 (h : p \u2227 q), s h.1 h.2 :=\nby { symmetry, exact infi_and }\n\ntheorem supr_and {p q : Prop} {s : p \u2227 q \u2192 \u03b1} : supr s = (\u2a06 h\u2081 h\u2082, s \u27e8h\u2081, h\u2082\u27e9) :=\n@infi_and (order_dual \u03b1) _ _ _ _\n\n\/-- The symmetric case of `supr_and`, useful for rewriting into a supremum over a conjunction -\/\nlemma supr_and' {p q : Prop} {s : p \u2192 q \u2192 \u03b1} :\n  (\u2a06 (h\u2081 : p) (h\u2082 : q), s h\u2081 h\u2082) = \u2a06 (h : p \u2227 q), s h.1 h.2 :=\nby { symmetry, exact supr_and }\n\ntheorem infi_or {p q : Prop} {s : p \u2228 q \u2192 \u03b1} :\n  infi s = (\u2a05 h : p, s (or.inl h)) \u2293 (\u2a05 h : q, s (or.inr h)) :=\nle_antisymm\n  (le_inf\n    (infi_le_infi2 $ assume j, \u27e8_, le_refl _\u27e9)\n    (infi_le_infi2 $ assume j, \u27e8_, le_refl _\u27e9))\n  (le_infi $ assume i, match i with\n  | or.inl i := inf_le_of_left_le $ infi_le _ _\n  | or.inr j := inf_le_of_right_le $ infi_le _ _\n  end)\n\ntheorem supr_or {p q : Prop} {s : p \u2228 q \u2192 \u03b1} :\n  (\u2a06 x, s x) = (\u2a06 i, s (or.inl i)) \u2294 (\u2a06 j, s (or.inr j)) :=\n@infi_or (order_dual \u03b1) _ _ _ _\n\nlemma Sup_range {\u03b1 : Type*} [has_Sup \u03b1] {f : \u03b9 \u2192 \u03b1} : Sup (range f) = supr f := rfl\n\nlemma Inf_range {\u03b1 : Type*} [has_Inf \u03b1] {f : \u03b9 \u2192 \u03b1} : Inf (range f) = infi f := rfl\n\nlemma supr_range' {\u03b1} [has_Sup \u03b1] (g : \u03b2 \u2192 \u03b1) (f : \u03b9 \u2192 \u03b2) :\n  (\u2a06 b : range f, g b) = \u2a06 i, g (f i) :=\nby rw [supr, supr, \u2190 image_eq_range, \u2190 range_comp]\n\nlemma infi_range' {\u03b1} [has_Inf \u03b1] (g : \u03b2 \u2192 \u03b1) (f : \u03b9 \u2192 \u03b2) :\n  (\u2a05 b : range f, g b) = \u2a05 i, g (f i) :=\n@supr_range' _ _ (order_dual \u03b1) _ _ _\n\nlemma infi_range {g : \u03b2 \u2192 \u03b1} {f : \u03b9 \u2192 \u03b2} : (\u2a05b\u2208range f, g b) = (\u2a05i, g (f i)) :=\nby rw [\u2190 infi_subtype'', infi_range']\n\nlemma supr_range {g : \u03b2 \u2192 \u03b1} {f : \u03b9 \u2192 \u03b2} : (\u2a06b\u2208range f, g b) = (\u2a06i, g (f i)) :=\n@infi_range (order_dual \u03b1) _ _ _ _ _\n\ntheorem Inf_image' {\u03b1} [has_Inf \u03b1] {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} :\n  Inf (f '' s) = (\u2a05 a : s, f a) :=\nby rw [infi, image_eq_range]\n\ntheorem Sup_image' {\u03b1} [has_Sup \u03b1] {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} :\n  Sup (f '' s) = (\u2a06 a : s, f a) :=\n@Inf_image' _ (order_dual \u03b1) _ _ _\n\ntheorem Inf_image {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} : Inf (f '' s) = (\u2a05 a \u2208 s, f a) :=\nby rw [\u2190 infi_subtype'', Inf_image']\n\ntheorem Sup_image {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} : Sup (f '' s) = (\u2a06 a \u2208 s, f a) :=\n@Inf_image (order_dual \u03b1) _ _ _ _\n\n\/-\n### supr and infi under set constructions\n-\/\n\ntheorem infi_emptyset {f : \u03b2 \u2192 \u03b1} : (\u2a05 x \u2208 (\u2205 : set \u03b2), f x) = \u22a4 :=\nby simp\n\ntheorem supr_emptyset {f : \u03b2 \u2192 \u03b1} : (\u2a06 x \u2208 (\u2205 : set \u03b2), f x) = \u22a5 :=\nby simp\n\ntheorem infi_univ {f : \u03b2 \u2192 \u03b1} : (\u2a05 x \u2208 (univ : set \u03b2), f x) = (\u2a05 x, f x) :=\nby simp\n\ntheorem supr_univ {f : \u03b2 \u2192 \u03b1} : (\u2a06 x \u2208 (univ : set \u03b2), f x) = (\u2a06 x, f x) :=\nby simp\n\ntheorem infi_union {f : \u03b2 \u2192 \u03b1} {s t : set \u03b2} : (\u2a05 x \u2208 s \u222a t, f x) = (\u2a05x\u2208s, f x) \u2293 (\u2a05x\u2208t, f x) :=\nby simp only [\u2190 infi_inf_eq, infi_or]\n\nlemma infi_split (f : \u03b2 \u2192 \u03b1) (p : \u03b2 \u2192 Prop) :\n  (\u2a05 i, f i) = (\u2a05 i (h : p i), f i) \u2293 (\u2a05 i (h : \u00ac p i), f i) :=\nby simpa [classical.em] using @infi_union _ _ _ f {i | p i} {i | \u00ac p i}\n\nlemma infi_split_single (f : \u03b2 \u2192 \u03b1) (i\u2080 : \u03b2) :\n  (\u2a05 i, f i) = f i\u2080 \u2293 (\u2a05 i (h : i \u2260 i\u2080), f i) :=\nby convert infi_split _ _; simp\n\ntheorem infi_le_infi_of_subset {f : \u03b2 \u2192 \u03b1} {s t : set \u03b2} (h : s \u2286 t) :\n  (\u2a05 x \u2208 t, f x) \u2264 (\u2a05 x \u2208 s, f x) :=\nby rw [(union_eq_self_of_subset_left h).symm, infi_union]; exact inf_le_left\n\ntheorem supr_union {f : \u03b2 \u2192 \u03b1} {s t : set \u03b2} : (\u2a06 x \u2208 s \u222a t, f x) = (\u2a06x\u2208s, f x) \u2294 (\u2a06x\u2208t, f x) :=\n@infi_union (order_dual \u03b1) _ _ _ _ _\n\nlemma supr_split (f : \u03b2 \u2192 \u03b1) (p : \u03b2 \u2192 Prop) :\n  (\u2a06 i, f i) = (\u2a06 i (h : p i), f i) \u2294 (\u2a06 i (h : \u00ac p i), f i) :=\n@infi_split (order_dual \u03b1) _ _ _ _\n\nlemma supr_split_single (f : \u03b2 \u2192 \u03b1) (i\u2080 : \u03b2) :\n  (\u2a06 i, f i) = f i\u2080 \u2294 (\u2a06 i (h : i \u2260 i\u2080), f i) :=\n@infi_split_single (order_dual \u03b1) _ _ _ _\n\ntheorem supr_le_supr_of_subset {f : \u03b2 \u2192 \u03b1} {s t : set \u03b2} (h : s \u2286 t) :\n  (\u2a06 x \u2208 s, f x) \u2264 (\u2a06 x \u2208 t, f x) :=\n@infi_le_infi_of_subset (order_dual \u03b1) _ _ _ _ _ h\n\ntheorem infi_insert {f : \u03b2 \u2192 \u03b1} {s : set \u03b2} {b : \u03b2} :\n  (\u2a05 x \u2208 insert b s, f x) = f b \u2293 (\u2a05x\u2208s, f x) :=\neq.trans infi_union $ congr_arg (\u03bbx:\u03b1, x \u2293 (\u2a05x\u2208s, f x)) infi_infi_eq_left\n\ntheorem supr_insert {f : \u03b2 \u2192 \u03b1} {s : set \u03b2} {b : \u03b2} :\n  (\u2a06 x \u2208 insert b s, f x) = f b \u2294 (\u2a06x\u2208s, f x) :=\neq.trans supr_union $ congr_arg (\u03bbx:\u03b1, x \u2294 (\u2a06x\u2208s, f x)) supr_supr_eq_left\n\ntheorem infi_singleton {f : \u03b2 \u2192 \u03b1} {b : \u03b2} : (\u2a05 x \u2208 (singleton b : set \u03b2), f x) = f b :=\nby simp\n\ntheorem infi_pair {f : \u03b2 \u2192 \u03b1} {a b : \u03b2} : (\u2a05 x \u2208 ({a, b} : set \u03b2), f x) = f a \u2293 f b :=\nby rw [infi_insert, infi_singleton]\n\ntheorem supr_singleton {f : \u03b2 \u2192 \u03b1} {b : \u03b2} : (\u2a06 x \u2208 (singleton b : set \u03b2), f x) = f b :=\n@infi_singleton (order_dual \u03b1) _ _ _ _\n\ntheorem supr_pair {f : \u03b2 \u2192 \u03b1} {a b : \u03b2} : (\u2a06 x \u2208 ({a, b} : set \u03b2), f x) = f a \u2294 f b :=\nby rw [supr_insert, supr_singleton]\n\nlemma infi_image {\u03b3} {f : \u03b2 \u2192 \u03b3} {g : \u03b3 \u2192 \u03b1} {t : set \u03b2} :\n  (\u2a05 c \u2208 f '' t, g c) = (\u2a05 b \u2208 t, g (f b)) :=\nby rw [\u2190 Inf_image, \u2190 Inf_image, \u2190 image_comp]\n\nlemma supr_image {\u03b3} {f : \u03b2 \u2192 \u03b3} {g : \u03b3 \u2192 \u03b1} {t : set \u03b2} :\n  (\u2a06 c \u2208 f '' t, g c) = (\u2a06 b \u2208 t, g (f b)) :=\n@infi_image (order_dual \u03b1) _ _ _ _ _ _\n\n\/-!\n### `supr` and `infi` under `Type`\n-\/\n\ntheorem infi_of_empty' (h : \u03b9 \u2192 false) {s : \u03b9 \u2192 \u03b1} : infi s = \u22a4 :=\ntop_unique (le_infi $ assume i, (h i).elim)\n\ntheorem supr_of_empty' (h : \u03b9 \u2192 false) {s : \u03b9 \u2192 \u03b1} : supr s = \u22a5 :=\nbot_unique (supr_le $ assume i, (h i).elim)\n\ntheorem infi_of_empty (h : \u00acnonempty \u03b9) {s : \u03b9 \u2192 \u03b1} : infi s = \u22a4 :=\ninfi_of_empty' (\u03bb i, h \u27e8i\u27e9)\n\ntheorem supr_of_empty (h : \u00acnonempty \u03b9) {s : \u03b9 \u2192 \u03b1} : supr s = \u22a5 :=\nsupr_of_empty' (\u03bb i, h \u27e8i\u27e9)\n\n@[simp] theorem infi_empty {s : empty \u2192 \u03b1} : infi s = \u22a4 :=\ninfi_of_empty nonempty_empty\n\n@[simp] theorem supr_empty {s : empty \u2192 \u03b1} : supr s = \u22a5 :=\nsupr_of_empty nonempty_empty\n\nlemma supr_bool_eq {f : bool \u2192 \u03b1} : (\u2a06b:bool, f b) = f tt \u2294 f ff :=\nle_antisymm\n  (supr_le $ assume b, match b with tt := le_sup_left | ff := le_sup_right end)\n  (sup_le (le_supr _ _) (le_supr _ _))\n\nlemma infi_bool_eq {f : bool \u2192 \u03b1} : (\u2a05b:bool, f b) = f tt \u2293 f ff :=\n@supr_bool_eq (order_dual \u03b1) _ _\n\nlemma is_glb_binfi {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} : is_glb (f '' s) (\u2a05 x \u2208 s, f x) :=\nby simpa only [range_comp, subtype.range_coe, infi_subtype'] using @is_glb_infi \u03b1 s _ (f \u2218 coe)\n\ntheorem supr_subtype {p : \u03b9 \u2192 Prop} {f : subtype p \u2192 \u03b1} : (\u2a06 x, f x) = (\u2a06 i (h:p i), f \u27e8i, h\u27e9) :=\n@infi_subtype (order_dual \u03b1) _ _ _ _\n\nlemma supr_subtype' {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1} :\n  (\u2a06 i (h : p i), f i h) = (\u2a06 x : subtype p, f x x.property) :=\n(@supr_subtype _ _ _ p (\u03bb x, f x.val x.property)).symm\n\nlemma supr_subtype'' {\u03b9} (s : set \u03b9) (f : \u03b9 \u2192 \u03b1) :\n  (\u2a06 i : s, f i) = \u2a06 (t : \u03b9) (H : t \u2208 s), f t :=\nsupr_subtype\n\nlemma is_lub_bsupr {s : set \u03b2} {f : \u03b2 \u2192 \u03b1} : is_lub (f '' s) (\u2a06 x \u2208 s, f x) :=\nby simpa only [range_comp, subtype.range_coe, supr_subtype'] using @is_lub_supr \u03b1 s _ (f \u2218 coe)\n\ntheorem infi_sigma {p : \u03b2 \u2192 Type*} {f : sigma p \u2192 \u03b1} : (\u2a05 x, f x) = (\u2a05 i (h:p i), f \u27e8i, h\u27e9) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume : p i, infi_le _ _)\n  (le_infi $ assume \u27e8i, h\u27e9, infi_le_of_le i $ infi_le _ _)\n\ntheorem supr_sigma {p : \u03b2 \u2192 Type*} {f : sigma p \u2192 \u03b1} : (\u2a06 x, f x) = (\u2a06 i (h:p i), f \u27e8i, h\u27e9) :=\n@infi_sigma (order_dual \u03b1) _ _ _ _\n\ntheorem infi_prod {\u03b3 : Type*} {f : \u03b2 \u00d7 \u03b3 \u2192 \u03b1} : (\u2a05 x, f x) = (\u2a05 i j, f (i, j)) :=\nle_antisymm\n  (le_infi $ assume i, le_infi $ assume j, infi_le _ _)\n  (le_infi $ assume \u27e8i, h\u27e9, infi_le_of_le i $ infi_le _ _)\n\ntheorem supr_prod {\u03b3 : Type*} {f : \u03b2 \u00d7 \u03b3 \u2192 \u03b1} : (\u2a06 x, f x) = (\u2a06 i j, f (i, j)) :=\n@infi_prod (order_dual \u03b1) _ _ _ _\n\ntheorem infi_sum {\u03b3 : Type*} {f : \u03b2 \u2295 \u03b3 \u2192 \u03b1} :\n  (\u2a05 x, f x) = (\u2a05 i, f (sum.inl i)) \u2293 (\u2a05 j, f (sum.inr j)) :=\nle_antisymm\n  (le_inf\n    (infi_le_infi2 $ assume i, \u27e8_, le_refl _\u27e9)\n    (infi_le_infi2 $ assume j, \u27e8_, le_refl _\u27e9))\n  (le_infi $ assume s, match s with\n  | sum.inl i := inf_le_of_left_le $ infi_le _ _\n  | sum.inr j := inf_le_of_right_le $ infi_le _ _\n  end)\n\ntheorem supr_sum {\u03b3 : Type*} {f : \u03b2 \u2295 \u03b3 \u2192 \u03b1} :\n  (\u2a06 x, f x) = (\u2a06 i, f (sum.inl i)) \u2294 (\u2a06 j, f (sum.inr j)) :=\n@infi_sum (order_dual \u03b1) _ _ _ _\n\ntheorem supr_option (f : option \u03b2 \u2192 \u03b1) :\n  (\u2a06 o, f o) = f none \u2294 \u2a06 b, f (option.some b) :=\neq_of_forall_ge_iff $ \u03bb c, by simp only [supr_le_iff, sup_le_iff, option.forall]\n\ntheorem infi_option (f : option \u03b2 \u2192 \u03b1) :\n  (\u2a05 o, f o) = f none \u2293 \u2a05 b, f (option.some b) :=\n@supr_option (order_dual \u03b1) _ _ _\n\n\/-!\n### `supr` and `infi` under `\u2115`\n-\/\n\nlemma supr_ge_eq_supr_nat_add {u : \u2115 \u2192 \u03b1} (n : \u2115) : (\u2a06 i \u2265 n, u i) = \u2a06 i, u (i + n) :=\nbegin\n  apply le_antisymm;\n  simp only [supr_le_iff],\n  { exact \u03bb i hi, le_Sup \u27e8i - n, by { dsimp only, rw nat.sub_add_cancel hi }\u27e9 },\n  { exact \u03bb i, le_Sup \u27e8i + n, supr_pos (nat.le_add_left _ _)\u27e9 }\nend\n\nlemma infi_ge_eq_infi_nat_add {u : \u2115 \u2192 \u03b1} (n : \u2115) : (\u2a05 i \u2265 n, u i) = \u2a05 i, u (i + n) :=\n@supr_ge_eq_supr_nat_add (order_dual \u03b1) _ _ _\n\nlemma monotone.supr_nat_add {f : \u2115 \u2192 \u03b1} (hf : monotone f) (k : \u2115) :\n  (\u2a06 n, f (n + k)) = \u2a06 n, f n :=\nle_antisymm (supr_le (\u03bb i, (le_refl _).trans (le_supr _ (i + k))))\n    (supr_le_supr (\u03bb i, hf (nat.le_add_right i k)))\n\n@[simp] lemma supr_infi_ge_nat_add (f : \u2115 \u2192 \u03b1) (k : \u2115) :\n  (\u2a06 n, \u2a05 i \u2265 n, f (i + k)) = \u2a06 n, \u2a05 i \u2265 n, f i :=\nbegin\n  have hf : monotone (\u03bb n, \u2a05 i \u2265 n, f i),\n    from \u03bb n m hnm, le_infi (\u03bb i, (infi_le _ i).trans (le_infi (\u03bb h, infi_le _ (hnm.trans h)))),\n  rw \u2190monotone.supr_nat_add hf k,\n  { simp_rw [infi_ge_eq_infi_nat_add, \u2190nat.add_assoc], },\nend\n\nlemma sup_supr_nat_succ (u : \u2115 \u2192 \u03b1) : u 0 \u2294 (\u2a06 i, u (i + 1)) = \u2a06 i, u i :=\nbegin\n  refine eq_of_forall_ge_iff (\u03bb c, _),\n  simp only [sup_le_iff, supr_le_iff],\n  refine \u27e8\u03bb h, _, \u03bb h, \u27e8h _, \u03bb i, h _\u27e9\u27e9,\n  rintro (_|i),\n  exacts [h.1, h.2 i]\nend\n\nlemma inf_infi_nat_succ (u : \u2115 \u2192 \u03b1) : u 0 \u2293 (\u2a05 i, u (i + 1)) = \u2a05 i, u i :=\n@sup_supr_nat_succ (order_dual \u03b1) _ u\n\nend\n\nsection complete_linear_order\nvariables [complete_linear_order \u03b1]\n\nlemma supr_eq_top (f : \u03b9 \u2192 \u03b1) : supr f = \u22a4 \u2194 (\u2200b<\u22a4, \u2203i, b < f i) :=\nby simp only [\u2190 Sup_range, Sup_eq_top, set.exists_range_iff]\n\nlemma infi_eq_bot (f : \u03b9 \u2192 \u03b1) : infi f = \u22a5 \u2194 (\u2200b>\u22a5, \u2203i, f i < b) :=\nby simp only [\u2190 Inf_range, Inf_eq_bot, set.exists_range_iff]\n\nend complete_linear_order\n\n\/-!\n### Instances\n-\/\n\ninstance Prop.complete_lattice : complete_lattice Prop :=\n{ Sup    := \u03bbs, \u2203a\u2208s, a,\n  le_Sup := assume s a h p, \u27e8a, h, p\u27e9,\n  Sup_le := assume s a h \u27e8b, h', p\u27e9, h b h' p,\n  Inf    := \u03bbs, \u2200a:Prop, a\u2208s \u2192 a,\n  Inf_le := assume s a h p, p a h,\n  le_Inf := assume s a h p b hb, h b hb p,\n  .. Prop.bounded_distrib_lattice }\n\n@[simp] lemma Inf_Prop_eq {s : set Prop} : Inf s = (\u2200p \u2208 s, p) := rfl\n\n@[simp] lemma Sup_Prop_eq {s : set Prop} : Sup s = (\u2203p \u2208 s, p) := rfl\n\n@[simp] lemma infi_Prop_eq {\u03b9 : Sort*} {p : \u03b9 \u2192 Prop} : (\u2a05i, p i) = (\u2200i, p i) :=\nle_antisymm (assume h i, h _ \u27e8i, rfl\u27e9 ) (assume h p \u27e8i, eq\u27e9, eq \u25b8 h i)\n\n@[simp] lemma supr_Prop_eq {\u03b9 : Sort*} {p : \u03b9 \u2192 Prop} : (\u2a06i, p i) = (\u2203i, p i) :=\nle_antisymm (\u03bb \u27e8q, \u27e8i, (eq : p i = q)\u27e9, hq\u27e9, \u27e8i, eq.symm \u25b8 hq\u27e9) (\u03bb \u27e8i, hi\u27e9, \u27e8p i, \u27e8i, rfl\u27e9, hi\u27e9)\n\ninstance pi.has_Sup {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u03a0 i, has_Sup (\u03b2 i)] : has_Sup (\u03a0 i, \u03b2 i) :=\n\u27e8\u03bb s i, \u2a06 f : s, (f : \u03a0 i, \u03b2 i) i\u27e9\n\ninstance pi.has_Inf {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u03a0 i, has_Inf (\u03b2 i)] : has_Inf (\u03a0 i, \u03b2 i) :=\n\u27e8\u03bb s i, \u2a05 f : s, (f : \u03a0 i, \u03b2 i) i\u27e9\n\ninstance pi.complete_lattice {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u2200 i, complete_lattice (\u03b2 i)] :\n  complete_lattice (\u03a0 i, \u03b2 i) :=\n{ Sup := Sup,\n  Inf := Inf,\n  le_Sup := \u03bb s f hf i, le_supr (\u03bb f : s, (f : \u03a0 i, \u03b2 i) i) \u27e8f, hf\u27e9,\n  Inf_le := \u03bb s f hf i, infi_le (\u03bb f : s, (f : \u03a0 i, \u03b2 i) i) \u27e8f, hf\u27e9,\n  Sup_le := \u03bb s f hf i, supr_le $ \u03bb g, hf g g.2 i,\n  le_Inf := \u03bb s f hf i, le_infi $ \u03bb g, hf g g.2 i,\n  .. pi.bounded_lattice }\n\nlemma Inf_apply {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u03a0 i, has_Inf (\u03b2 i)]\n  {s : set (\u03a0a, \u03b2 a)} {a : \u03b1} :\n  (Inf s) a = (\u2a05 f : s, (f : \u03a0a, \u03b2 a) a) :=\nrfl\n\n@[simp] lemma infi_apply {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} {\u03b9 : Sort*} [\u03a0 i, has_Inf (\u03b2 i)]\n  {f : \u03b9 \u2192 \u03a0a, \u03b2 a} {a : \u03b1} :\n  (\u2a05i, f i) a = (\u2a05i, f i a) :=\nby rw [infi, Inf_apply, infi, infi, \u2190 image_eq_range (\u03bb f : \u03a0 i, \u03b2 i, f a) (range f), \u2190 range_comp]\n\nlemma Sup_apply {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u03a0 i, has_Sup (\u03b2 i)] {s : set (\u03a0a, \u03b2 a)} {a : \u03b1} :\n  (Sup s) a = (\u2a06f:s, (f : \u03a0a, \u03b2 a) a) :=\nrfl\n\nlemma unary_relation_Sup_iff {\u03b1 : Type*} (s : set (\u03b1 \u2192 Prop)) {a : \u03b1} :\n  Sup s a \u2194 \u2203 (r : \u03b1 \u2192 Prop), r \u2208 s \u2227 r a :=\nby { change (\u2203 _, _) \u2194 _, simp [-eq_iff_iff] }\n\nlemma binary_relation_Sup_iff {\u03b1 \u03b2 : Type*} (s : set (\u03b1 \u2192 \u03b2 \u2192 Prop)) {a : \u03b1} {b : \u03b2} :\n  Sup s a b \u2194 \u2203 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2227 r a b :=\nby { change (\u2203 _, _) \u2194 _, simp [-eq_iff_iff] }\n\n@[simp] lemma supr_apply {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} {\u03b9 : Sort*} [\u03a0 i, has_Sup (\u03b2 i)]\n  {f : \u03b9 \u2192 \u03a0a, \u03b2 a} {a : \u03b1} :\n  (\u2a06i, f i) a = (\u2a06i, f i a) :=\n@infi_apply \u03b1 (\u03bb i, order_dual (\u03b2 i)) _ _ f a\n\nsection complete_lattice\nvariables [preorder \u03b1] [complete_lattice \u03b2]\n\ntheorem monotone_Sup_of_monotone {s : set (\u03b1 \u2192 \u03b2)} (m_s : \u2200f\u2208s, monotone f) : monotone (Sup s) :=\nassume x y h, supr_le $ \u03bb f, le_supr_of_le f $ m_s f f.2 h\n\ntheorem monotone_Inf_of_monotone {s : set (\u03b1 \u2192 \u03b2)} (m_s : \u2200f\u2208s, monotone f) : monotone (Inf s) :=\nassume x y h, le_infi $ \u03bb f, infi_le_of_le f $ m_s f f.2 h\n\nend complete_lattice\n\nnamespace prod\nvariables (\u03b1 \u03b2)\n\ninstance [has_Inf \u03b1] [has_Inf \u03b2] : has_Inf (\u03b1 \u00d7 \u03b2) :=\n\u27e8\u03bbs, (Inf (prod.fst '' s), Inf (prod.snd '' s))\u27e9\n\ninstance [has_Sup \u03b1] [has_Sup \u03b2] : has_Sup (\u03b1 \u00d7 \u03b2) :=\n\u27e8\u03bbs, (Sup (prod.fst '' s), Sup (prod.snd '' s))\u27e9\n\ninstance [complete_lattice \u03b1] [complete_lattice \u03b2] : complete_lattice (\u03b1 \u00d7 \u03b2) :=\n{ le_Sup := assume s p hab, \u27e8le_Sup $ mem_image_of_mem _ hab, le_Sup $ mem_image_of_mem _ hab\u27e9,\n  Sup_le := assume s p h,\n    \u27e8 Sup_le $ ball_image_of_ball $ assume p hp, (h p hp).1,\n      Sup_le $ ball_image_of_ball $ assume p hp, (h p hp).2\u27e9,\n  Inf_le := assume s p hab, \u27e8Inf_le $ mem_image_of_mem _ hab, Inf_le $ mem_image_of_mem _ hab\u27e9,\n  le_Inf := assume s p h,\n    \u27e8 le_Inf $ ball_image_of_ball $ assume p hp, (h p hp).1,\n      le_Inf $ ball_image_of_ball $ assume p hp, (h p hp).2\u27e9,\n  .. prod.bounded_lattice \u03b1 \u03b2,\n  .. prod.has_Sup \u03b1 \u03b2,\n  .. prod.has_Inf \u03b1 \u03b2 }\n\nend prod\n\nsection complete_lattice\nvariables [complete_lattice \u03b1] {a : \u03b1} {s : set \u03b1}\n\n\/-- This is a weaker version of `sup_Inf_eq` -\/\nlemma sup_Inf_le_infi_sup :\n  a \u2294 Inf s \u2264 (\u2a05 b \u2208 s, a \u2294 b) :=\nle_infi $ assume i, le_infi $ assume h, sup_le_sup_left (Inf_le h) _\n\n\/-- This is a weaker version of `Inf_sup_eq` -\/\nlemma Inf_sup_le_infi_sup :\n  Inf s \u2294 a \u2264 (\u2a05 b \u2208 s, b \u2294 a) :=\nle_infi $ assume i, le_infi $ assume h, sup_le_sup_right (Inf_le h) _\n\n\/-- This is a weaker version of `inf_Sup_eq` -\/\nlemma supr_inf_le_inf_Sup :\n  (\u2a06 b \u2208 s, a \u2293 b) \u2264 a \u2293 Sup s :=\nsupr_le $ assume i, supr_le $ assume h, inf_le_inf_left _ (le_Sup h)\n\n\/-- This is a weaker version of `Sup_inf_eq` -\/\nlemma supr_inf_le_Sup_inf :\n  (\u2a06 b \u2208 s, b \u2293 a) \u2264 Sup s \u2293 a :=\nsupr_le $ assume i, supr_le $ assume h, inf_le_inf_right _ (le_Sup h)\n\nlemma disjoint_Sup_left {a : set \u03b1} {b : \u03b1} (d : disjoint (Sup a) b) {i} (hi : i \u2208 a) :\n  disjoint i b :=\n(supr_le_iff.mp (supr_le_iff.mp (supr_inf_le_Sup_inf.trans (d : _)) i : _) hi : _)\n\nlemma disjoint_Sup_right {a : set \u03b1} {b : \u03b1} (d : disjoint b (Sup a)) {i} (hi : i \u2208 a) :\n  disjoint b i :=\n(supr_le_iff.mp (supr_le_iff.mp (supr_inf_le_inf_Sup.trans (d : _)) i : _) hi : _)\n\nend complete_lattice\n\nnamespace complete_lattice\nvariables [complete_lattice \u03b1]\n\n\/-- An independent set of elements in a complete lattice is one in which every element is disjoint\n  from the `Sup` of the rest. -\/\ndef set_independent (s : set \u03b1) : Prop := \u2200 \u2983a\u2984, a \u2208 s \u2192 disjoint a (Sup (s \\ {a}))\n\nvariables {s : set \u03b1} (hs : set_independent s)\n\n@[simp]\nlemma set_independent_empty : set_independent (\u2205 : set \u03b1) :=\n\u03bb x hx, (set.not_mem_empty x hx).elim\n\ntheorem set_independent.mono {t : set \u03b1} (hst : t \u2286 s) :\n  set_independent t :=\n\u03bb a ha, (hs (hst ha)).mono_right (Sup_le_Sup (diff_subset_diff_left hst))\n\n\/-- If the elements of a set are independent, then any pair within that set is disjoint. -\/\nlemma set_independent.disjoint {x y : \u03b1} (hx : x \u2208 s) (hy : y \u2208 s) (h : x \u2260 y) : disjoint x y :=\ndisjoint_Sup_right (hs hx) ((mem_diff y).mpr \u27e8hy, by simp [h.symm]\u27e9)\n\ninclude hs\n\n\/-- If the elements of a set are independent, then any element is disjoint from the `Sup` of some\nsubset of the rest. -\/\nlemma set_independent.disjoint_Sup {x : \u03b1} {y : set \u03b1} (hx : x \u2208 s) (hy : y \u2286 s) (hxy : x \u2209 y) :\n  disjoint x (Sup y) :=\nbegin\n  have := (hs.mono $ insert_subset.mpr \u27e8hx, hy\u27e9) (mem_insert x _),\n  rw [insert_diff_of_mem _ (mem_singleton _), diff_singleton_eq_self hxy] at this,\n  exact this,\nend\n\nomit hs\n\n\/-- An independent indexed family of elements in a complete lattice is one in which every element\n  is disjoint from the `supr` of the rest.\n\n  Example: an indexed family of non-zero elements in a\n  vector space is linearly independent iff the indexed family of subspaces they generate is\n  independent in this sense.\n\n  Example: an indexed family of submodules of a module is independent in this sense if\n  and only the natural map from the direct sum of the submodules to the module is injective. -\/\ndef independent {\u03b9 : Sort*} {\u03b1 : Type*} [complete_lattice \u03b1] (t : \u03b9 \u2192 \u03b1) : Prop :=\n\u2200 i : \u03b9, disjoint (t i) (\u2a06 (j \u2260 i), t j)\n\nlemma set_independent_iff {\u03b1 : Type*} [complete_lattice \u03b1] (s : set \u03b1) :\n  set_independent s \u2194 independent (coe : s \u2192 \u03b1) :=\nbegin\n  simp_rw [independent, set_independent, set_coe.forall, Sup_eq_supr],\n  apply forall_congr, intro a, apply forall_congr, intro ha,\n  congr' 2,\n  convert supr_subtype.symm,\n  simp [supr_and],\nend\n\nvariables {t : \u03b9 \u2192 \u03b1} (ht : independent t)\n\ntheorem independent_def : independent t \u2194 \u2200 i : \u03b9, disjoint (t i) (\u2a06 (j \u2260 i), t j) :=\niff.rfl\n\ntheorem independent_def' {\u03b9 : Type*} {t : \u03b9 \u2192 \u03b1} :\n  independent t \u2194 \u2200 i, disjoint (t i) (Sup (t '' {j | j \u2260 i})) :=\nby {simp_rw Sup_image, refl}\n\ntheorem independent_def'' {\u03b9 : Type*} {t : \u03b9 \u2192 \u03b1} :\n  independent t \u2194 \u2200 i, disjoint (t i) (Sup {a | \u2203 j \u2260 i, t j = a}) :=\nby {rw independent_def', tidy}\n\n@[simp]\nlemma independent_empty (t : empty \u2192 \u03b1) : independent t.\n\n@[simp]\nlemma independent_pempty (t : pempty \u2192 \u03b1) : independent t.\n\n\/-- If the elements of a set are independent, then any pair within that set is disjoint. -\/\nlemma independent.disjoint {x y : \u03b9} (h : x \u2260 y) : disjoint (t x) (t y) :=\ndisjoint_Sup_right (ht x) \u27e8y, by simp [h.symm]\u27e9\n\nlemma independent.mono {\u03b9 : Type*} {\u03b1 : Type*} [complete_lattice \u03b1]\n  {s t : \u03b9 \u2192 \u03b1} (hs : independent s) (hst : t \u2264 s) :\n  independent t :=\n\u03bb i, (hs i).mono (hst i) (supr_le_supr $ \u03bb j, supr_le_supr $ \u03bb _, hst j)\n\n\/-- Composing an indepedent indexed family with an injective function on the index results in\nanother indepedendent indexed family. -\/\nlemma independent.comp {\u03b9 \u03b9' : Sort*} {\u03b1 : Type*} [complete_lattice \u03b1]\n  {s : \u03b9 \u2192 \u03b1} (hs : independent s) (f : \u03b9' \u2192 \u03b9) (hf : function.injective f) :\n  independent (s \u2218 f) :=\n\u03bb i, (hs (f i)).mono_right begin\n  refine (supr_le_supr $ \u03bb i, _).trans (supr_comp_le _ f),\n  exact supr_le_supr_const hf.ne,\nend\n\n\/-- If the elements of a set are independent, then any element is disjoint from the `supr` of some\nsubset of the rest. -\/\nlemma independent.disjoint_bsupr {\u03b9 : Type*} {\u03b1 : Type*} [complete_lattice \u03b1]\n  {t : \u03b9 \u2192 \u03b1} (ht : independent t) {x : \u03b9} {y : set \u03b9} (hx : x \u2209 y) :\n  disjoint (t x) (\u2a06 i \u2208 y, t i) :=\ndisjoint.mono_right (bsupr_le_bsupr' $ \u03bb i hi, (ne_of_mem_of_not_mem hi hx : _)) (ht x)\n\nend complete_lattice\n","avg_line_length":37.0127819549,"max_line_length":99,"alphanum_fraction":0.5982286144}
-{"size":2028,"ext":"lean","lang":"Lean","max_stars_count":1.0,"content":"import GMLInit.Logic.Basic\n\nnamespace Function\n\nabbrev curry {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) (a : \u03b1) (b : \u03b2) := f (a,b)\n\nabbrev uncurry {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (ab : \u03b1 \u00d7 \u03b2) := f ab.fst ab.snd\n\nabbrev dcurry {\u03b1} {\u03b2 : \u03b1 \u2192 Sort _} {\u03b3 : (a : \u03b1) \u00d7 \u03b2 a \u2192 Sort _} (f : (p : (a : \u03b1) \u00d7 \u03b2 a) \u2192 \u03b3 p) (a : \u03b1) (b : \u03b2 a) : \u03b3 \u27e8a, b\u27e9 := f \u27e8a, b\u27e9\n\nabbrev duncurry {\u03b1} {\u03b2 : \u03b1 \u2192 Sort _} {\u03b3 : {a : \u03b1} \u2192 \u03b2 a \u2192 Sort _} (f : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 \u03b3 b) (ab : (a : \u03b1) \u00d7 \u03b2 a) : \u03b3 ab.snd := f ab.fst ab.snd\n\nclass InjectiveT {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) : Prop where\n  protected inj {a\u2081 a\u2082} : f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n\nexport InjectiveT (inj)\n\nabbrev injection {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) [InjectiveT f] {lhs rhs : \u03b1} : f lhs = f rhs \u2192 lhs = rhs := inj\n\ntheorem injectionIff {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) [InjectiveT f] {a\u2081 a\u2082 : \u03b1} : f a\u2081 = f a\u2082 \u2194 a\u2081 = a\u2082 := Iff.intro inj (congrArg f)\n\ntheorem injectionEq {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) [InjectiveT f] {a\u2081 a\u2082 : \u03b1} : (f a\u2081 = f a\u2082) = (a\u2081 = a\u2082) := propext (injectionIff f)\n\nclass Injective {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) extends InjectiveT f : Prop\n\nprotected instance InjectiveT.comp {\u03b1 \u03b2 \u03b3} (g : \u03b1 \u2192 \u03b2) (f : \u03b2 \u2192 \u03b3) [Injective f] [InjectiveT g] : InjectiveT (f $ g .) where\n  inj {x\u2081 x\u2082} h := inj (inj h)\n\ninstance (\u03b1) : Injective (id : \u03b1 \u2192 \u03b1) where inj := id\ninstance (\u03b1 \u03b2) : Injective (Sum.inl : \u03b1 \u2192 Sum \u03b1 \u03b2) where inj := Sum.inl.inj\ninstance (\u03b1 \u03b2) : Injective (Sum.inr : \u03b2 \u2192 Sum \u03b1 \u03b2) where inj := Sum.inr.inj\ninstance (\u03b1 \u03b2) (a : \u03b1) : Injective ((a,.) : \u03b2 \u2192 \u03b1 \u00d7 \u03b2) where inj := \u03bb h => (Prod.mk.inj h).right\ninstance (\u03b1 \u03b2) (b : \u03b2) : Injective ((.,b) : \u03b1 \u2192 \u03b1 \u00d7 \u03b2) where inj := \u03bb h => (Prod.mk.inj h).left\ninstance (a : \u03b1) : Injective (a::.) where inj := \u03bb h => (List.cons.inj h).2\ninstance (as : List \u03b1) : Injective (.::as) where inj := \u03bb h => (List.cons.inj h).left\ninstance (\u03b1) : Injective (some : \u03b1 \u2192 Option \u03b1) where inj := Option.some.inj\ninstance : Injective Nat.succ where inj := Nat.succ.inj\ninstance (n : Nat) : Injective (n+.: Nat \u2192 Nat) where inj := Nat.add_left_cancel\ninstance (n : Nat) : Injective (.+n: Nat \u2192 Nat) where inj := Nat.add_right_cancel\n\nend Function\n","avg_line_length":48.2857142857,"max_line_length":148,"alphanum_fraction":0.5665680473}
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-{"size":110,"ext":"lean","lang":"Lean","max_stars_count":2232.0,"content":"#check expr\n#check expr ff\ndef f (x := 3) (y : nat) := y\n#check f 3 4\nset_option pp.implicit true\n#check expr\n","avg_line_length":15.7142857143,"max_line_length":29,"alphanum_fraction":0.6545454545}
-{"size":5530,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-\/\nimport data.finset.lattice\nimport data.set.pairwise\nimport order.hom.basic\n\n\/-!\n# The monotone sequence of partial supremums of a sequence\n\nWe define `partial_sups : (\u2115 \u2192 \u03b1) \u2192 \u2115 \u2192o \u03b1` inductively. For `f : \u2115 \u2192 \u03b1`, `partial_sups f` is\nthe sequence `f 0 `, `f 0 \u2294 f 1`, `f 0 \u2294 f 1 \u2294 f 2`, ... The point of this definition is that\n* it doesn't need a `\u2a06`, as opposed to `\u2a06 (i \u2264 n), f i`.\n* it doesn't need a `\u22a5`, as opposed to `(finset.range (n + 1)).sup f`.\n* it avoids needing to prove that `finset.range (n + 1)` is nonempty to use `finset.sup'`.\n\nEquivalence with those definitions is shown by `partial_sups_eq_bsupr`, `partial_sups_eq_sup_range`,\n`partial_sups_eq_sup'_range` and respectively.\n\n## Notes\n\nOne might dispute whether this sequence should start at `f 0` or `\u22a5`. We choose the former because :\n* Starting at `\u22a5` requires... having a bottom element.\n* `\u03bb f n, (finset.range n).sup f` is already effectively the sequence starting at `\u22a5`.\n* If we started at `\u22a5` we wouldn't have the Galois insertion. See `partial_sups.gi`.\n\n## TODO\n\nOne could generalize `partial_sups` to any locally finite bot preorder domain, in place of `\u2115`.\nNecessary for the TODO in the module docstring of `order.disjointed`.\n-\/\n\nvariables {\u03b1 : Type*}\n\nsection semilattice_sup\nvariables [semilattice_sup \u03b1]\n\n\/-- The monotone sequence whose value at `n` is the supremum of the `f m` where `m \u2264 n`. -\/\ndef partial_sups (f : \u2115 \u2192 \u03b1) : \u2115 \u2192o \u03b1 :=\n\u27e8@nat.rec (\u03bb _, \u03b1) (f 0) (\u03bb (n : \u2115) (a : \u03b1), a \u2294 f (n + 1)),\n  monotone_nat_of_le_succ (\u03bb n, le_sup_left)\u27e9\n\n@[simp] lemma partial_sups_zero (f : \u2115 \u2192 \u03b1) : partial_sups f 0 = f 0 := rfl\n@[simp] lemma partial_sups_succ (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n  partial_sups f (n + 1) = partial_sups f n \u2294 f (n + 1) := rfl\n\nlemma le_partial_sups_of_le (f : \u2115 \u2192 \u03b1) {m n : \u2115} (h : m \u2264 n) :\n  f m \u2264 partial_sups f n :=\nbegin\n  induction n with n ih,\n  { cases h, apply le_refl, },\n  { cases h with h h,\n    { exact le_sup_right, },\n    { exact (ih h).trans le_sup_left, } },\nend\n\nlemma le_partial_sups (f : \u2115 \u2192 \u03b1) :\n  f \u2264 partial_sups f :=\n\u03bb n, le_partial_sups_of_le f le_rfl\n\nlemma partial_sups_le (f : \u2115 \u2192 \u03b1) (n : \u2115)\n  (a : \u03b1) (w : \u2200 m, m \u2264 n \u2192 f m \u2264 a) : partial_sups f n \u2264 a :=\nbegin\n  induction n with n ih,\n  { apply w 0 le_rfl, },\n  { exact sup_le (ih (\u03bb m p, w m (nat.le_succ_of_le p))) (w (n + 1) le_rfl) }\nend\n\nlemma monotone.partial_sups_eq {f : \u2115 \u2192 \u03b1} (hf : monotone f) :\n  (partial_sups f : \u2115 \u2192 \u03b1) = f :=\nbegin\n  ext n,\n  induction n with n ih,\n  { refl },\n  { rw [partial_sups_succ, ih, sup_eq_right.2 (hf (nat.le_succ _))] }\nend\n\nlemma partial_sups_mono : monotone (partial_sups : (\u2115 \u2192 \u03b1) \u2192 \u2115 \u2192o \u03b1) :=\nbegin\n  rintro f g h n,\n  induction n with n ih,\n  { exact h 0 },\n  { exact sup_le_sup ih (h _) }\nend\n\n\/-- `partial_sups` forms a Galois insertion with the coercion from monotone functions to functions.\n-\/\ndef partial_sups.gi : galois_insertion (partial_sups : (\u2115 \u2192 \u03b1) \u2192 \u2115 \u2192o \u03b1) coe_fn :=\n{ choice := \u03bb f h, \u27e8f, begin\n    convert (partial_sups f).monotone,\n    exact (le_partial_sups f).antisymm h,\n  end\u27e9,\n  gc := \u03bb f g, begin\n    refine \u27e8(le_partial_sups f).trans, \u03bb h, _\u27e9,\n    convert partial_sups_mono h,\n    exact order_hom.ext _ _ g.monotone.partial_sups_eq.symm,\n  end,\n  le_l_u := \u03bb f, le_partial_sups f,\n  choice_eq := \u03bb f h, order_hom.ext _ _ ((le_partial_sups f).antisymm h) }\n\nlemma partial_sups_eq_sup'_range (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n  partial_sups f n = (finset.range (n + 1)).sup' \u27e8n, finset.self_mem_range_succ n\u27e9 f :=\nbegin\n  induction n with n ih,\n  { simp },\n  { dsimp [partial_sups] at ih \u22a2,\n    simp_rw @finset.range_succ n.succ,\n    rw [ih, finset.sup'_insert, sup_comm] }\nend\n\nend semilattice_sup\n\nlemma partial_sups_eq_sup_range [semilattice_sup \u03b1] [order_bot \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n  partial_sups f n = (finset.range (n + 1)).sup f :=\nbegin\n  induction n with n ih,\n  { simp },\n  { dsimp [partial_sups] at ih \u22a2,\n    rw [finset.range_succ, finset.sup_insert, sup_comm, ih] }\nend\n\n\/- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as\nsubmodules. -\/\nlemma partial_sups_disjoint_of_disjoint [distrib_lattice \u03b1] [order_bot \u03b1]\n  (f : \u2115 \u2192 \u03b1) (h : pairwise (disjoint on f)) {m n : \u2115} (hmn : m < n) :\n  disjoint (partial_sups f m) (f n) :=\nbegin\n  induction m with m ih,\n  { exact h 0 n hmn.ne, },\n  { rw [partial_sups_succ, disjoint_sup_left],\n    exact \u27e8ih (nat.lt_of_succ_lt hmn), h (m + 1) n hmn.ne\u27e9 }\nend\n\nsection complete_lattice\nvariables [complete_lattice \u03b1]\n\nlemma partial_sups_eq_bsupr (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n  partial_sups f n = \u2a06 (i \u2264 n), f i :=\nbegin\n  rw [partial_sups_eq_sup_range, finset.sup_eq_supr],\n  congr,\n  ext a,\n  exact supr_congr_Prop (by rw [finset.mem_range, nat.lt_succ_iff]) (\u03bb _, rfl),\nend\n\n@[simp] lemma supr_partial_sups_eq (f : \u2115 \u2192 \u03b1) :\n  (\u2a06 n, partial_sups f n) = \u2a06 n, f n :=\nbegin\n  refine (supr_le $ \u03bb n, _).antisymm (supr_le_supr $ le_partial_sups f),\n  rw partial_sups_eq_bsupr,\n  exact bsupr_le_supr _ _,\nend\n\nlemma supr_le_supr_of_partial_sups_le_partial_sups {f g : \u2115 \u2192 \u03b1}\n  (h : partial_sups f \u2264 partial_sups g) :\n  (\u2a06 n, f n) \u2264 \u2a06 n, g n :=\nbegin\n  rw [\u2190supr_partial_sups_eq f, \u2190supr_partial_sups_eq g],\n  exact supr_le_supr h,\nend\n\nlemma supr_eq_supr_of_partial_sups_eq_partial_sups {f g : \u2115 \u2192 \u03b1}\n  (h : partial_sups f = partial_sups g) :\n  (\u2a06 n, f n) = \u2a06 n, g n :=\nby simp_rw [\u2190supr_partial_sups_eq f, \u2190supr_partial_sups_eq g, h]\n\nend complete_lattice\n","avg_line_length":32.5294117647,"max_line_length":100,"alphanum_fraction":0.6650994575}
-{"size":4118,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import Lean.Util.Trace\n\nset_option linter.all true\n\ndef explicitelyUsedVariable (x : Nat) : Nat :=\n  x\n\ntheorem implicitlyUsedVariable : P \u2227 Q \u2192 Q := by\n  intro HPQ\n  have HQ : Q := by exact And.right HPQ\n  assumption\n\naxiom axiomVariable (x : Prop) : True\n\ndef unusedVariables (x : Nat) : Nat :=\n  let y := 5\n  3\n\ndef usedAndUnusedVariables : Nat :=\n  let x : Nat :=\n    let x := 5\n    3\n  x\n\ndef unusedWhereVariable : Nat :=\n  3\nwhere\n  x := 5\n\ndef unusedWhereArgument : Nat :=\n  f 2\nwhere\n  f (x : Nat) := 3\n\ndef unusedWhereFunction : Nat :=\n  2\nwhere\n  f (x : Nat) := 3\n\ndef unusedFunctionArgument : Nat :=\n  (fun x => 3) (x := 2)\n\ndef unusedTypedFunctionArgument : Nat :=\n  (fun (x : Nat) => 3) 2\n\ndef pattern (x y : Option Nat) : Nat :=\n  match x with\n  | some z =>\n    match y with\n    | some z => 1\n    | none => 0\n  | none => 0\n\ndef patternLet (x : Option Nat) : Nat :=\n  if let some y := x then\n    0\n  else\n    1\n\ndef patternMatches (x : Option Nat) : Nat :=\n  if x matches some y then\n    0\n  else\n    1\n\ndef implicitVariables {\u03b1 : Type} [inst : ToString \u03b1] : Nat := 4\n\ndef autoImplicitVariable [Inhabited \u03b1] := 5\n\ndef unusedArrow : (x : Nat) \u2192 Nat := fun x => x\n\ndef mutVariable (x : Nat) : Nat := Id.run <| do\n  let mut y := 5\n  if x == 5 then\n    y := 3\n  y\n\ndef mutVariableDo (list : List Nat) : Nat := Id.run <| do\n  let mut sum := 0\n  for elem in list do\n    sum := sum + elem\n  return sum\n\ndef mutVariableDo2 (list : List Nat) : Nat := Id.run <| do\n  let mut sum := 0\n  for _ in list do\n    sum := sum.add 1\n  return sum\n\n\ndef unusedVariablesPattern (_x : Nat) : Nat :=\n  let _y := 5\n  3\n\nset_option linter.unusedVariables false in\ndef nolintUnusedVariables (x : Nat) : Nat :=\n  let y := 5\n  3\n\nset_option linter.all false in\ndef nolintAll (x : Nat) : Nat :=\n  let y := 5\n  3\n\nset_option linter.all false in\nset_option linter.unusedVariables true in\ndef lintUnusedVariables (x : Nat) : Nat :=\n  let y := 5\n  3\n\n\nset_option linter.unusedVariables.funArgs false in\ndef nolintFunArgs (w : Nat) : Nat :=\n  let a := 5\n  let f (x : Nat) := 3\n  let g := fun (y : Nat) => 3\n  f <| g <| h <| 2\nwhere\n  h (z : Nat) := 3\n\nset_option linter.unusedVariables.patternVars false in\ndef nolintPatternVars (x : Option (Option Nat)) : Nat :=\n  match x with\n  | some (some y) => (fun z => 1) 2\n  | _ => 0\n\nset_option linter.unusedVariables.patternVars false in\ntheorem nolintPatternVarsInduction (n : Nat) : True := by\n  induction n with\n  | zero => exact True.intro\n  | succ m =>\n    have h : True := by simp\n    exact True.intro\n\n\ninductive Foo (\u03b1 : Type)\n  | foo (x : Nat) (y : Nat)\n\nstructure Bar (\u03b1 : Type) where\n  bar (x : Nat) : Nat\n  bar' (x : Nat) : Nat := 3\n\nclass Baz (\u03b1 : Type) where\n  baz (x : Nat) : Nat\n  baz' (x : Nat) : Nat :=\n    let y := 5\n    3\n\ninstance instBaz (\u03b1 \u03b2 : Type) : Baz \u03b1 where\n  baz (x : Nat) := 5\n\n\nstructure State where\n  fieldA : Nat\n  fieldB : Nat\n\nabbrev M := StateT State Id\n\ndef modifyState : M Unit := do\n  let s \u2190 get\n  modify fun s => { s with fieldA := s.fieldA + 1 }\n\ndef modifyState' : M Unit := do\n  modify fun s => { s with fieldA := 1}\n\ndef modifyStateUnnecessaryWith : M Unit := do\n  modify fun s => { s with fieldA := 1, fieldB := 2 }\n\n\ndef universeParam.{u} (T : Type u) (t : T) : T := t\n\n\nopen Lean in\ninitialize tc : Unit \u2190 registerTraceClass `Baz\n\nregister_option opt : Nat := {\n  defValue := 3\n  descr := \"test option\"\n}\n\n\nopaque foo (x : Nat) : Nat\nopaque foo' (x : Nat) : Nat :=\n  let y := 5\n  3\nvariable (bar)\nvariable (bar' : (x : Nat) \u2192 Nat)\nvariable {\u03b1 \u03b2} [inst : ToString \u03b1]\n\n@[specialize]\ndef specializeDef (x : Nat) : Nat := 3\n\n@[implementedBy specializeDef]\ndef implementedByDef (x : Nat) : Nat :=\n  let y := 3\n  5\n\n@[extern \"test\"]\ndef externDef (x : Nat) : Nat :=\n  let y := 3\n  5\n\n@[extern \"test\"]\nopaque externConst (x : Nat) : Nat :=\n  let y := 3\n  5\n\ntheorem not_eq_zero_of_lt (h : b < a) : a \u2260 0 := by -- *not* unused\n  cases a\n  exact absurd h (Nat.not_lt_zero _)\n  apply Nat.noConfusion\n\n-- should not be reported either\nexample (a : Nat) : Nat := _\nexample (a : Nat) : Nat := sorry\nexample (a : sorry) : Nat := 0\nexample (a : Nat) : Nat := by\n","avg_line_length":18.8899082569,"max_line_length":67,"alphanum_fraction":0.6058766391}
-{"size":334,"ext":"lean","lang":"Lean","max_stars_count":2232.0,"content":"\nnamespace tst\nvariable {A : Type}\n\nattribute [reducible]\ndefinition foo\u2081 (a b c : A) (H\u2081 : a = b) (H\u2082 : c = b) : a = c :=\neq.trans H\u2081 (eq.symm H\u2082)\n\nlemma foo\u2082 (f : A \u2192 A \u2192 A) (a b c : A) (H\u2081 : a = b) (H\u2082 : c = b) : f a = f c :=\neq.symm H\u2082 \u25b8 H\u2081 \u25b8 rfl\n\n#check foo\u2081\n#check foo\u2082\n\nend tst\n\n#check tst.foo\u2081\n#check tst.foo\u2082\n#print tst.foo\u2081\n","avg_line_length":16.7,"max_line_length":79,"alphanum_fraction":0.5688622754}
-{"size":2250,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"universe u v\n\ntheorem eqLitOfSize0 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 0) : a = #[] :=\na.toArrayLitEq 0 hsz\n\ntheorem eqLitOfSize1 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 1) : a = #[a.getLit 0 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 1 hsz\n\ntheorem eqLitOfSize2 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 2) : a = #[a.getLit 0 hsz (ofDecideEqTrue rfl), a.getLit 1 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 2 hsz\n\ntheorem eqLitOfSize3 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 3) :\n  a = #[a.getLit 0 hsz (ofDecideEqTrue rfl), a.getLit 1 hsz (ofDecideEqTrue rfl), a.getLit 2 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 3 hsz\n\n\/-\nMatcher for the following patterns\n```\n| #[]           => _\n| #[a\u2081]         => _\n| #[a\u2081, a\u2082, a\u2083] => _\n| a             => _\n``` -\/\ndef matchArrayLit {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v) (a : Array \u03b1)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    : C a :=\nif h : a.size = 0 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2081 ()) _ (a.toArrayLitEq 0 h).symm\nelse if h : a.size = 1 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2082 (a.getLit 0 h (ofDecideEqTrue rfl))) _ (a.toArrayLitEq 1 h).symm\nelse if h : a.size = 3 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2083 (a.getLit 0 h (ofDecideEqTrue rfl)) (a.getLit 1 h (ofDecideEqTrue rfl)) (a.getLit 2 h (ofDecideEqTrue rfl))) _ (a.toArrayLitEq 3 h).symm\nelse\n  h\u2084 a\n\n\/- Equational lemmas that should be generated automatically. -\/\ntheorem matchArrayLit.eq1 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    : matchArrayLit C #[] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2081 () :=\nrfl\n\ntheorem matchArrayLit.eq2 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    (a\u2081 : \u03b1)\n    : matchArrayLit C #[a\u2081] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2082 a\u2081 :=\nrfl\n\ntheorem matchArrayLit.eq3 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    (a\u2081 a\u2082 a\u2083 : \u03b1)\n    : matchArrayLit C #[a\u2081, a\u2082, a\u2083] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2083 a\u2081 a\u2082 a\u2083 :=\nrfl\n","avg_line_length":34.6153846154,"max_line_length":171,"alphanum_fraction":0.5253333333}
-{"size":842,"ext":"lean","lang":"Lean","max_stars_count":13.0,"content":"inductive itreeF (E: Type -> Type) (R: Type) (itree : Type) :=\n| RetF: (r : R) -> itreeF E R itree\n| TauF: (t : itree) -> itreeF E R itree\n| VisF: (X : Type) -> (e : E X)  -> (k : X -> itree) -> itreeF E R itree\n\n\/--\/\nThis presents the traces of effectful programs as trees where\neach node issues a command c from signature C and receives a\nsuitable response in R c. For more on this way of looking at\ninteraction, see papers by Peter Hancock, Anton Setzer, and\nothers.\n-\/\n\nInductive FreeMonad (C:Type) (R : C -> Type) (X:Type): Type:=\n| ret: X -> FreeMonad C R X\n| eff: forall c:C, R c -> FreeMonad C R X.\n\n\n\/-\nA more expensive, but perhaps more rewarding approach is to\ndefine a *universe* of strictly positive functors, interpreted\nvia an inductive family. You can read more about this approach\nin papers by Peter Morris and colleagues.\n-\/","avg_line_length":35.0833333333,"max_line_length":72,"alphanum_fraction":0.6876484561}
-{"size":2252,"ext":"lean","lang":"Lean","max_stars_count":null,"content":"import Std\nopen Std\n\ndef check (b : Bool) : IO Unit := do\nunless b do IO.println \"ERROR\"\n\ndef sz {\u03b1 \u03b2 : Type} {cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering} (m : RBMap \u03b1 \u03b2 cmp) : Nat :=\nm.fold (fun sz _ _ => sz+1) 0\n\ndef depth {\u03b1 \u03b2 : Type} {cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering} (m : RBMap \u03b1 \u03b2 cmp) : Nat :=\nm.depth Nat.max\n\ndef tst1 : IO Unit :=\ndo let Map := RBMap String Nat compare\n   let m : Map := {}\n   let m := m.insert \"hello\" 0\n   let m := m.insert \"world\" 1\n   check (m.find? \"hello\" == some 0)\n   check (m.find? \"world\" == some 1)\n   let m := m.erase \"hello\"\n   check (m.find? \"hello\" == none)\n   check (m.find? \"world\" == some 1)\n   pure ()\n\ndef tst2 : IO Unit :=\ndo let Map := RBMap Nat Nat compare\n   let m : Map := {}\n   let n : Nat := 10000\n   let mut m := n.fold (fun i (m : Map) => m.insert i (i*10)) m\n   check (m.all (fun k v => v == k*10))\n   check (sz m == n)\n   IO.println (\">> \" ++ toString (depth m) ++ \", \" ++ toString (sz m))\n   for i in [:n\/2] do\n     m := m.erase (2*i)\n   check (m.all (fun k v => v == k*10))\n   check (sz m == n \/ 2)\n   IO.println (\">> \" ++ toString (depth m) ++ \", \" ++ toString (sz m))\n   pure ()\n\nabbrev Map := RBMap Nat Nat compare\n\ndef mkRandMap (max : Nat) : Nat \u2192 Map \u2192 Array (Nat \u00d7 Nat) \u2192 IO (Map \u00d7 Array (Nat \u00d7 Nat))\n| 0,     m, a => pure (m, a)\n| n+1,   m, a => do\n  let k \u2190 IO.rand 0 max\n  let v \u2190 IO.rand 0 max\n  if m.find? k == none then do\n    let m := m.insert k v\n    let a := a.push (k, v)\n    mkRandMap max n m a\n  else\n    mkRandMap max n m a\n\ndef tst3 (seed : Nat) (n : Nat) (max : Nat) : IO Unit :=\ndo IO.setRandSeed seed\n   let mut (m, a) \u2190 mkRandMap max n {} Array.empty\n   check (sz m == a.size)\n   check (a.all (fun \u27e8k, v\u27e9 => m.find? k == some v))\n   IO.println (\"tst3 size: \" ++ toString a.size)\n   let mut i := 0\n   for (k, b) in a do\n     if i % 2 == 0 then\n       m := m.erase k\n     i := i + 1\n   check (sz m == a.size \/ 2)\n   let i := 0\n   for (k, v) in a do\n     if i % 2 == 1 then\n       check (m.find? k == some v)\n     i := i + 1\n   IO.println (\"tst3 after, depth: \" ++ toString (depth m) ++ \", size: \" ++ toString (sz m))\n   pure ()\n\ndef main (xs : List String) : IO Unit :=\ntst1 *> tst2 *>\ntst3 1 1000 20000 *>\ntst3 2 1000 40000 *>\ntst3 3 100  4000 *>\ntst3 4 5000 100000 *>\ntst3 5 1000 40000 *>\npure ()\n","avg_line_length":27.4634146341,"max_line_length":92,"alphanum_fraction":0.526642984}
-{"size":2926,"ext":"hlean","lang":"Lean","max_stars_count":null,"content":"\/-\nCopyright (c) 2015 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Floris van Doorn\n\nTheorems about algebra specific to HoTT\n-\/\n\nimport .group arity types.pi prop_trunc types.unit .bundled\n\nopen equiv eq is_trunc unit\n\nnamespace algebra\n\n  definition trivial_group [constructor] : group unit :=\n  group.mk (\u03bbx y, star) _ (\u03bbx y z, idp) star (unit.rec idp) (unit.rec idp) (\u03bbx, star) (\u03bbx, idp)\n\n  definition Trivial_group [constructor] : Group :=\n  Group.mk _ trivial_group\n\n  abbreviation G0 := Trivial_group\n\n  open Group has_mul has_inv\n  -- we prove under which conditions two groups are equal\n\n  -- group and has_mul are classes. So, lean does not automatically generate\n  -- coercions between them anymore.\n  -- So, an application such as (@mul A G g h) in the following definition\n  -- is type incorrect if the coercion is not added (explicitly or implicitly).\n  definition group.to_has_mul {A : Type} (H : group A) : has_mul A := _\n  local attribute group.to_has_mul group.to_has_inv [coercion]\n\n  universe variable l\n  variables {A B : Type.{l}}\n  definition group_eq {G H : group A} (same_mul' : \u03a0(g h : A), @mul A G g h = @mul A H g h)\n    : G = H :=\n  begin\n    have foo : \u03a0(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,\n      from \u03bbg, !mul_inv_cancel_right\u207b\u00b9,\n    cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,\n    cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,\n    rewrite [\u2191[semigroup.to_has_mul,group.to_has_inv] at (same_mul',foo)],\n    have same_mul : Gm = Hm, from eq_of_homotopy2 same_mul',\n    cases same_mul,\n    have same_one : G1 = H1, from calc\n      G1 = Hm G1 H1 : Hh3\n     ... = H1 : Gh2,\n    have same_inv : Gi = Hi, from eq_of_homotopy (take g, calc\n      Gi g = Hm (Hm (Gi g) g) (Hi g) : foo\n       ... = Hm G1 (Hi g) : by rewrite Gh4\n       ... = Hi g : Gh2),\n    cases same_one, cases same_inv,\n    have ps  : Gs  = Hs,  from !is_prop.elim,\n    have ph1 : Gh1 = Hh1, from !is_prop.elim,\n    have ph2 : Gh2 = Hh2, from !is_prop.elim,\n    have ph3 : Gh3 = Hh3, from !is_prop.elim,\n    have ph4 : Gh4 = Hh4, from !is_prop.elim,\n    cases ps, cases ph1, cases ph2, cases ph3, cases ph4, reflexivity\n  end\n\n  definition group_pathover {G : group A} {H : group B} {p : A = B}\n    (resp_mul : \u03a0(g h : A), cast p (g * h) = cast p g * cast p h) : G =[p] H :=\n  begin\n    induction p,\n    apply pathover_idp_of_eq, exact group_eq (resp_mul)\n  end\n\n  definition Group_eq_of_eq {G H : Group} (p : carrier G = carrier H)\n    (resp_mul : \u03a0(g h : G), cast p (g * h) = cast p g * cast p h) : G = H :=\n  begin\n    cases G with Gc G, cases H with Hc H,\n    apply (apd011 mk p),\n    exact group_pathover resp_mul\n  end\n\n  definition Group_eq {G H : Group} (f : carrier G \u2243 carrier H)\n    (resp_mul : \u03a0(g h : G), f (g * h) = f g * f h) : G = H :=\n  Group_eq_of_eq (ua f) (\u03bbg h, !cast_ua \u2b1d resp_mul g h \u2b1d ap011 mul !cast_ua\u207b\u00b9 !cast_ua\u207b\u00b9)\n\nend algebra\n","avg_line_length":36.1234567901,"max_line_length":95,"alphanum_fraction":0.6418318524}