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string | meta
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data Unit : Set where
unit : Unit
P : Unit → Set
P unit = Unit
postulate
Q : (u : Unit) → P u → Set
variable
u : Unit
p : P u
postulate
q : P u → Q u p
q' : (u : Unit) (p : P u) → P u → Q u p
q' u p = q {u} {p}
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data ℕ : Set where
zero : ℕ
suc : ℕ → ℕ
{-# BUILTIN NATURAL ℕ #-}
infixl 6 _+_
infix 6 _∸_
_+_ : ℕ → ℕ → ℕ
zero + n = n
suc m + n = suc (m + n)
_∸_ : ℕ → ℕ → ℕ
m ∸ zero = m
zero ∸ suc n = zero
suc m ∸ suc n = m ∸ n
should-be-rejected : ℕ
should-be-rejected = 1 + 0 ∸ 1
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-- {-# OPTIONS -v term:20 #-}
-- Andreas, 2011-04-19 (Agda list post by Leonard Rodriguez)
module TerminationSubExpression where
infixr 3 _⇨_
data Type : Set where
int : Type
_⇨_ : Type → Type → Type
test : Type → Type
test int = int
test (φ ⇨ int) = test φ
test (φ ⇨ (φ′ ⇨ φ″)) = test (φ′ ⇨ φ″)
-- this should terminate since rec. call on subterm
test' : Type → Type
test' int = int
test' (φ ⇨ int) = test' φ
test' (φ ⇨ φ′) = test' φ′
ok : Type → Type
ok int = int
ok (φ ⇨ φ′) with φ′
... | int = ok φ
... | (φ″ ⇨ φ‴) = ok (φ″ ⇨ φ‴)
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Sets.EquivalenceRelations
open import Setoids.Setoids
open import Functions.Definition
open import Groups.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Subgroups.Definition
open import Groups.Subgroups.Normal.Definition
module Groups.Cosets {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) {c : _} {pred : A → Set c} (subgrp : Subgroup G pred) where
open Equivalence (Setoid.eq S)
open import Groups.Lemmas G
open Group G
open Subgroup subgrp
cosetSetoid : Setoid A
Setoid._∼_ cosetSetoid g h = pred ((Group.inverse G h) + g)
Equivalence.reflexive (Setoid.eq cosetSetoid) = isSubset (symmetric (Group.invLeft G)) containsIdentity
Equivalence.symmetric (Setoid.eq cosetSetoid) yx = isSubset (transitive invContravariant (+WellDefined reflexive invInv)) (closedUnderInverse yx)
Equivalence.transitive (Setoid.eq cosetSetoid) yx zy = isSubset (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) reflexive)) (closedUnderPlus zy yx)
cosetGroup : normalSubgroup G subgrp → Group cosetSetoid _+_
Group.+WellDefined (cosetGroup norm) {m} {n} {x} {y} m=x n=y = ans
where
t : pred (inverse y + n)
t = n=y
u : pred (inverse x + m)
u = m=x
v : pred (m + inverse x)
v = isSubset (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) (norm u)
ans' : pred ((inverse y) + ((inverse x + m) + inverse (inverse y)))
ans' = norm u
ans'' : pred ((inverse y) + ((inverse x + m) + y))
ans'' = isSubset (+WellDefined reflexive (+WellDefined reflexive (invTwice y))) ans'
ans : pred (inverse (x + y) + (m + n))
ans = isSubset (transitive (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) +Associative)) reflexive) (symmetric +Associative))) (symmetric (+WellDefined invContravariant reflexive))) (closedUnderPlus ans'' t)
Group.0G (cosetGroup norm) = 0G
Group.inverse (cosetGroup norm) = inverse
Group.+Associative (cosetGroup norm) {a} {b} {c} = isSubset (symmetric (transitive (+WellDefined (inverseWellDefined (symmetric +Associative)) reflexive) (invLeft {a + (b + c)}))) containsIdentity
Group.identRight (cosetGroup norm) = isSubset (symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identRight))) containsIdentity
Group.identLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) containsIdentity
Group.invLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) containsIdentity
Group.invRight (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) containsIdentity
cosetGroupHom : (norm : normalSubgroup G subgrp) → GroupHom G (cosetGroup norm) id
GroupHom.groupHom (cosetGroupHom norm) = isSubset (symmetric (transitive (+WellDefined invContravariant reflexive) (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive invLeft)) reflexive) (transitive (+WellDefined identRight reflexive) invLeft))))) (Subgroup.containsIdentity subgrp)
GroupHom.wellDefined (cosetGroupHom norm) {x} {y} x=y = isSubset (symmetric (transitive (+WellDefined reflexive x=y) invLeft)) (Subgroup.containsIdentity subgrp)
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{-# BUILTIN NATURAL ℕ #-}
module the-naturals where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _∎)
infixl 6 _+_ _∸_
infixl 7 _*_
-- the naturals
data ℕ : Set where
zero : ℕ
suc : ℕ → ℕ
-- addition
_+_ : ℕ → ℕ → ℕ
zero + n = n
(suc m) + n = suc (m + n)
-- multiplication
_*_ : ℕ → ℕ → ℕ
zero * n = zero
(suc m) * n = n + (m * n)
-- monus ( subtraction for the naturals )
_∸_ : ℕ → ℕ → ℕ
m ∸ zero = m
zero ∸ suc n = zero
suc m ∸ suc n = m ∸ n
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module Category.Functor.Arr where
open import Agda.Primitive using (_⊔_)
open import Category.Functor using (RawFunctor ; module RawFunctor )
open import Category.Applicative using (RawApplicative; module RawApplicative)
open import Function using (_∘_)
open import Category.Functor.Lawful
open import Relation.Binary.PropositionalEquality using (refl)
Arr : ∀ {l₁ l₂} → Set l₁ → Set l₂ → Set (l₁ ⊔ l₂)
Arr A B = A → B
arrFunctor : ∀ {l₁ l₂} {B : Set l₁} → RawFunctor (Arr {l₁} {l₂} B)
arrFunctor = record { _<$>_ = λ z z₁ x → z (z₁ x) } -- auto-found
arrLawfulFunctor : ∀ {l₁ l₂} {B : Set l₁} → LawfulFunctorImp (arrFunctor {l₁} {l₂} {B})
arrLawfulFunctor = record
{ <$>-identity = refl
; <$>-compose = refl
}
arrApplicative : ∀ {l₁} {B : Set l₁} → RawApplicative (Arr {l₁} {l₁} B)
arrApplicative = record { pure = λ z x → z ; _⊛_ = λ z z₁ x → z x (z₁ x) } -- auto-found
arrLawfulApplicative : ∀ {l₁} {B : Set l₁} → LawfulApplicativeImp (arrApplicative {l₁} {B})
arrLawfulApplicative = record
{ ⊛-identity = refl
; ⊛-homomorphism = refl
; ⊛-interchange = refl
; ⊛-composition = refl
}
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Universe levels
------------------------------------------------------------------------
module Level where
-- Levels.
open import Agda.Primitive public
using (Level; _⊔_)
renaming (lzero to zero; lsuc to suc)
-- Lifting.
record Lift {a ℓ} (A : Set a) : Set (a ⊔ ℓ) where
constructor lift
field lower : A
open Lift public
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.NConnected
open import lib.types.Nat
open import lib.types.TLevel
open import lib.types.Empty
open import lib.types.Group
open import lib.types.Pi
open import lib.types.Pointed
open import lib.types.Paths
open import lib.types.Sigma
open import lib.types.Truncation
open import lib.cubical.Square
module lib.types.LoopSpace where
module _ {i} where
⊙Ω : Ptd i → Ptd i
⊙Ω (A , a) = ⊙[ (a == a) , idp ]
Ω : Ptd i → Type i
Ω = fst ∘ ⊙Ω
⊙Ω^ : (n : ℕ) → Ptd i → Ptd i
⊙Ω^ O X = X
⊙Ω^ (S n) X = ⊙Ω (⊙Ω^ n X)
Ω^ : (n : ℕ) → Ptd i → Type i
Ω^ n X = fst (⊙Ω^ n X)
idp^ : ∀ {i} (n : ℕ) {X : Ptd i} → Ω^ n X
idp^ n {X} = snd (⊙Ω^ n X)
{- for n ≥ 1, we have a group structure on the loop space -}
module _ {i} where
!^ : (n : ℕ) (t : n ≠ O) {X : Ptd i} → Ω^ n X → Ω^ n X
!^ O t = ⊥-rec (t idp)
!^ (S n) _ = !
conc^ : (n : ℕ) (t : n ≠ O) {X : Ptd i} → Ω^ n X → Ω^ n X → Ω^ n X
conc^ O t = ⊥-rec (t idp)
conc^ (S n) _ = _∙_
{- ap and ap2 for pointed functions -}
private
pt-lemma : ∀ {i} {A : Type i} {x y : A} (p : x == y)
→ ! p ∙ (idp ∙' p) == idp
pt-lemma idp = idp
⊙ap : ∀ {i j} {X : Ptd i} {Y : Ptd j}
→ fst (X ⊙→ Y) → fst (⊙Ω X ⊙→ ⊙Ω Y)
⊙ap (f , fpt) = ((λ p → ! fpt ∙ ap f p ∙' fpt) , pt-lemma fpt)
⊙ap2 : ∀ {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
→ fst (X ⊙× Y ⊙→ Z) → fst (⊙Ω X ⊙× ⊙Ω Y ⊙→ ⊙Ω Z)
⊙ap2 (f , fpt) = ((λ {(p , q) → ! fpt ∙ ap2 (curry f) p q ∙' fpt}) ,
pt-lemma fpt)
⊙ap-∘ : ∀ {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
(g : fst (Y ⊙→ Z)) (f : fst (X ⊙→ Y))
→ ⊙ap (g ⊙∘ f) == ⊙ap g ⊙∘ ⊙ap f
⊙ap-∘ (g , idp) (f , idp) = ⊙λ= (λ p → ap-∘ g f p) idp
⊙ap-idf : ∀ {i} {X : Ptd i} → ⊙ap (⊙idf X) == ⊙idf _
⊙ap-idf = ⊙λ= ap-idf idp
⊙ap2-fst : ∀ {i j} {X : Ptd i} {Y : Ptd j}
→ ⊙ap2 {X = X} {Y = Y} ⊙fst == ⊙fst
⊙ap2-fst = ⊙λ= (uncurry ap2-fst) idp
⊙ap2-snd : ∀ {i j} {X : Ptd i} {Y : Ptd j}
→ ⊙ap2 {X = X} {Y = Y} ⊙snd == ⊙snd
⊙ap2-snd = ⊙λ= (uncurry ap2-snd) idp
⊙ap-ap2 : ∀ {i j k l} {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}
(G : fst (Z ⊙→ W)) (F : fst (X ⊙× Y ⊙→ Z))
→ ⊙ap G ⊙∘ ⊙ap2 F == ⊙ap2 (G ⊙∘ F)
⊙ap-ap2 (g , idp) (f , idp) =
⊙λ= (uncurry (ap-ap2 g (curry f))) idp
⊙ap2-ap : ∀ {i j k l m}
{X : Ptd i} {Y : Ptd j} {U : Ptd k} {V : Ptd l} {Z : Ptd m}
(G : fst ((U ⊙× V) ⊙→ Z)) (F₁ : fst (X ⊙→ U)) (F₂ : fst (Y ⊙→ V))
→ ⊙ap2 G ⊙∘ pair⊙→ (⊙ap F₁) (⊙ap F₂) == ⊙ap2 (G ⊙∘ pair⊙→ F₁ F₂)
⊙ap2-ap (g , idp) (f₁ , idp) (f₂ , idp) =
⊙λ= (λ {(p , q) → ap2-ap-l (curry g) f₁ p (ap f₂ q)
∙ ap2-ap-r (λ x v → g (f₁ x , v)) f₂ p q})
idp
⊙ap2-diag : ∀ {i j} {X : Ptd i} {Y : Ptd j} (F : fst (X ⊙× X ⊙→ Y))
→ ⊙ap2 F ⊙∘ ⊙diag == ⊙ap (F ⊙∘ ⊙diag)
⊙ap2-diag (f , idp) = ⊙λ= (ap2-diag (curry f)) idp
{- ap and ap2 for higher loop spaces -}
ap^ : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j}
→ fst (X ⊙→ Y) → fst (⊙Ω^ n X ⊙→ ⊙Ω^ n Y)
ap^ O F = F
ap^ (S n) F = ⊙ap (ap^ n F)
ap2^ : ∀ {i j k} (n : ℕ) {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
→ fst ((X ⊙× Y) ⊙→ Z)
→ fst ((⊙Ω^ n X ⊙× ⊙Ω^ n Y) ⊙→ ⊙Ω^ n Z)
ap2^ O F = F
ap2^ (S n) F = ⊙ap2 (ap2^ n F)
ap^-idf : ∀ {i} (n : ℕ) {X : Ptd i} → ap^ n (⊙idf X) == ⊙idf _
ap^-idf O = idp
ap^-idf (S n) = ap ⊙ap (ap^-idf n) ∙ ⊙ap-idf
ap^-ap2^ : ∀ {i j k l} (n : ℕ) {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}
(G : fst (Z ⊙→ W)) (F : fst ((X ⊙× Y) ⊙→ Z))
→ ap^ n G ⊙∘ ap2^ n F == ap2^ n (G ⊙∘ F)
ap^-ap2^ O G F = idp
ap^-ap2^ (S n) G F = ⊙ap-ap2 (ap^ n G) (ap2^ n F) ∙ ap ⊙ap2 (ap^-ap2^ n G F)
ap2^-fst : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j}
→ ap2^ n {X} {Y} ⊙fst == ⊙fst
ap2^-fst O = idp
ap2^-fst (S n) = ap ⊙ap2 (ap2^-fst n) ∙ ⊙ap2-fst
ap2^-snd : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j}
→ ap2^ n {X} {Y} ⊙snd == ⊙snd
ap2^-snd O = idp
ap2^-snd (S n) = ap ⊙ap2 (ap2^-snd n) ∙ ⊙ap2-snd
ap2^-ap^ : ∀ {i j k l m} (n : ℕ)
{X : Ptd i} {Y : Ptd j} {U : Ptd k} {V : Ptd l} {Z : Ptd m}
(G : fst ((U ⊙× V) ⊙→ Z)) (F₁ : fst (X ⊙→ U)) (F₂ : fst (Y ⊙→ V))
→ ap2^ n G ⊙∘ pair⊙→ (ap^ n F₁) (ap^ n F₂) == ap2^ n (G ⊙∘ pair⊙→ F₁ F₂)
ap2^-ap^ O G F₁ F₂ = idp
ap2^-ap^ (S n) G F₁ F₂ =
⊙ap2-ap (ap2^ n G) (ap^ n F₁) (ap^ n F₂) ∙ ap ⊙ap2 (ap2^-ap^ n G F₁ F₂)
ap2^-diag : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j} (F : fst (X ⊙× X ⊙→ Y))
→ ap2^ n F ⊙∘ ⊙diag == ap^ n (F ⊙∘ ⊙diag)
ap2^-diag O F = idp
ap2^-diag (S n) F = ⊙ap2-diag (ap2^ n F) ∙ ap ⊙ap (ap2^-diag n F)
module _ {i} {X : Ptd i} where
{- Prove these as lemmas now
- so we don't have to deal with the n = O case later -}
conc^-unit-l : (n : ℕ) (t : n ≠ O) (q : Ω^ n X)
→ (conc^ n t (idp^ n) q) == q
conc^-unit-l O t _ = ⊥-rec (t idp)
conc^-unit-l (S n) _ _ = idp
conc^-unit-r : (n : ℕ) (t : n ≠ O) (q : Ω^ n X)
→ (conc^ n t q (idp^ n)) == q
conc^-unit-r O t = ⊥-rec (t idp)
conc^-unit-r (S n) _ = ∙-unit-r
conc^-assoc : (n : ℕ) (t : n ≠ O) (p q r : Ω^ n X)
→ conc^ n t (conc^ n t p q) r == conc^ n t p (conc^ n t q r)
conc^-assoc O t = ⊥-rec (t idp)
conc^-assoc (S n) _ = ∙-assoc
!^-inv-l : (n : ℕ) (t : n ≠ O) (p : Ω^ n X)
→ conc^ n t (!^ n t p) p == idp^ n
!^-inv-l O t = ⊥-rec (t idp)
!^-inv-l (S n) _ = !-inv-l
!^-inv-r : (n : ℕ) (t : n ≠ O) (p : Ω^ n X)
→ conc^ n t p (!^ n t p) == idp^ n
!^-inv-r O t = ⊥-rec (t idp)
!^-inv-r (S n) _ = !-inv-r
abstract
ap^-conc^ : ∀ {i j} (n : ℕ) (t : n ≠ O)
{X : Ptd i} {Y : Ptd j} (F : fst (X ⊙→ Y)) (p q : Ω^ n X)
→ fst (ap^ n F) (conc^ n t p q)
== conc^ n t (fst (ap^ n F) p) (fst (ap^ n F) q)
ap^-conc^ O t _ _ _ = ⊥-rec (t idp)
ap^-conc^ (S n) _ {X = X} {Y = Y} F p q =
! gpt ∙ ap g (p ∙ q) ∙' gpt
=⟨ ap-∙ g p q |in-ctx (λ w → ! gpt ∙ w ∙' gpt) ⟩
! gpt ∙ (ap g p ∙ ap g q) ∙' gpt
=⟨ lemma (ap g p) (ap g q) gpt ⟩
(! gpt ∙ ap g p ∙' gpt) ∙ (! gpt ∙ ap g q ∙' gpt) ∎
where
g : Ω^ n X → Ω^ n Y
g = fst (ap^ n F)
gpt : g (idp^ n) == idp^ n
gpt = snd (ap^ n F)
lemma : ∀ {i} {A : Type i} {x y : A}
→ (p q : x == x) (r : x == y)
→ ! r ∙ (p ∙ q) ∙' r == (! r ∙ p ∙' r) ∙ (! r ∙ q ∙' r)
lemma p q idp = idp
{- ap^ preserves (pointed) equivalences -}
module _ {i j} {X : Ptd i} {Y : Ptd j} where
is-equiv-ap^ : (n : ℕ) (F : fst (X ⊙→ Y)) (e : is-equiv (fst F))
→ is-equiv (fst (ap^ n F))
is-equiv-ap^ O F e = e
is-equiv-ap^ (S n) F e =
pre∙-is-equiv (! (snd (ap^ n F)))
∘ise post∙'-is-equiv (snd (ap^ n F))
∘ise snd (equiv-ap (_ , is-equiv-ap^ n F e) _ _)
equiv-ap^ : (n : ℕ) (F : fst (X ⊙→ Y)) (e : is-equiv (fst F))
→ Ω^ n X ≃ Ω^ n Y
equiv-ap^ n F e = (fst (ap^ n F) , is-equiv-ap^ n F e)
Ω^-level-in : ∀ {i} (m : ℕ₋₂) (n : ℕ) (X : Ptd i)
→ (has-level ((n -2) +2+ m) (fst X) → has-level m (Ω^ n X))
Ω^-level-in m O X pX = pX
Ω^-level-in m (S n) X pX =
Ω^-level-in (S m) n X
(transport (λ k → has-level k (fst X)) (! (+2+-βr (n -2) m)) pX)
(idp^ n) (idp^ n)
Ω^-conn-in : ∀ {i} (m : ℕ₋₂) (n : ℕ) (X : Ptd i)
→ (is-connected ((n -2) +2+ m) (fst X)) → is-connected m (Ω^ n X)
Ω^-conn-in m O X pX = pX
Ω^-conn-in m (S n) X pX =
path-conn $ Ω^-conn-in (S m) n X $
transport (λ k → is-connected k (fst X)) (! (+2+-βr (n -2) m)) pX
{- Eckmann-Hilton argument -}
module _ {i} {X : Ptd i} where
conc^2-comm : (α β : Ω^ 2 X) → conc^ 2 (ℕ-S≠O _) α β == conc^ 2 (ℕ-S≠O _) β α
conc^2-comm α β = ! (⋆2=conc^ α β) ∙ ⋆2=⋆'2 α β ∙ ⋆'2=conc^ α β
where
⋆2=conc^ : (α β : Ω^ 2 X) → α ⋆2 β == conc^ 2 (ℕ-S≠O _) α β
⋆2=conc^ α β = ap (λ π → π ∙ β) (∙-unit-r α)
⋆'2=conc^ : (α β : Ω^ 2 X) → α ⋆'2 β == conc^ 2 (ℕ-S≠O _) β α
⋆'2=conc^ α β = ap (λ π → β ∙ π) (∙-unit-r α)
{- Pushing truncation through loop space -}
module _ {i} where
Trunc-Ω^ : (m : ℕ₋₂) (n : ℕ) (X : Ptd i)
→ ⊙Trunc m (⊙Ω^ n X) == ⊙Ω^ n (⊙Trunc ((n -2) +2+ m) X)
Trunc-Ω^ m O X = idp
Trunc-Ω^ m (S n) X =
⊙Trunc m (⊙Ω^ (S n) X)
=⟨ ! (pair= (Trunc=-path [ _ ] [ _ ]) (↓-idf-ua-in _ idp)) ⟩
⊙Ω (⊙Trunc (S m) (⊙Ω^ n X))
=⟨ ap ⊙Ω (Trunc-Ω^ (S m) n X) ⟩
⊙Ω^ (S n) (⊙Trunc ((n -2) +2+ S m) X)
=⟨ +2+-βr (n -2) m |in-ctx (λ k → ⊙Ω^ (S n) (⊙Trunc k X)) ⟩
⊙Ω^ (S n) (⊙Trunc (S (n -2) +2+ m) X) ∎
Ω-Trunc-equiv : (m : ℕ₋₂) (X : Ptd i)
→ Ω (⊙Trunc (S m) X) ≃ Trunc m (Ω X)
Ω-Trunc-equiv m X = Trunc=-equiv [ snd X ] [ snd X ]
{- A loop space is a pregroup, and a group if it has the right level -}
module _ {i} (n : ℕ) (t : n ≠ O) (X : Ptd i) where
Ω^-group-structure : GroupStructure (Ω^ n X)
Ω^-group-structure = record {
ident = idp^ n;
inv = !^ n t;
comp = conc^ n t;
unitl = conc^-unit-l n t;
unitr = conc^-unit-r n t;
assoc = conc^-assoc n t;
invr = !^-inv-r n t;
invl = !^-inv-l n t
}
Ω^-Group : has-level ⟨ n ⟩ (fst X) → Group i
Ω^-Group pX = group
(Ω^ n X)
(Ω^-level-in ⟨0⟩ n X $
transport (λ t → has-level t (fst X)) (+2+-comm ⟨0⟩ (n -2)) pX)
Ω^-group-structure
{- Our definition of Ω^ builds up loops on the outside,
- but this is equivalent to building up on the inside -}
module _ {i} where
⊙Ω^-inner-path : (n : ℕ) (X : Ptd i)
→ ⊙Ω^ (S n) X == ⊙Ω^ n (⊙Ω X)
⊙Ω^-inner-path O X = idp
⊙Ω^-inner-path (S n) X = ap ⊙Ω (⊙Ω^-inner-path n X)
⊙Ω^-inner-out : (n : ℕ) (X : Ptd i)
→ fst (⊙Ω^ (S n) X ⊙→ ⊙Ω^ n (⊙Ω X))
⊙Ω^-inner-out O _ = (idf _ , idp)
⊙Ω^-inner-out (S n) X = ap^ 1 (⊙Ω^-inner-out n X)
Ω^-inner-out : (n : ℕ) (X : Ptd i)
→ (Ω^ (S n) X → Ω^ n (⊙Ω X))
Ω^-inner-out n X = fst (⊙Ω^-inner-out n X)
Ω^-inner-out-conc^ : (n : ℕ) (t : n ≠ O)
(X : Ptd i) (p q : Ω^ (S n) X)
→ Ω^-inner-out n X (conc^ (S n) (ℕ-S≠O _) p q)
== conc^ n t (Ω^-inner-out n X p) (Ω^-inner-out n X q)
Ω^-inner-out-conc^ O t X _ _ = ⊥-rec (t idp)
Ω^-inner-out-conc^ (S n) t X p q =
ap^-conc^ 1 (ℕ-S≠O _) (⊙Ω^-inner-out n X) p q
Ω^-inner-is-equiv : (n : ℕ) (X : Ptd i)
→ is-equiv (fst (⊙Ω^-inner-out n X))
Ω^-inner-is-equiv O X = is-eq (idf _) (idf _) (λ _ → idp) (λ _ → idp)
Ω^-inner-is-equiv (S n) X =
is-equiv-ap^ 1 (⊙Ω^-inner-out n X) (Ω^-inner-is-equiv n X)
Ω^-inner-equiv : (n : ℕ) (X : Ptd i) → Ω^ (S n) X ≃ Ω^ n (⊙Ω X)
Ω^-inner-equiv n X = _ , Ω^-inner-is-equiv n X
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open import Prelude
module Implicits.Syntax.Term where
open import Implicits.Syntax.Type
infixl 9 _[_] _·_
data Term (ν n : ℕ) : Set where
var : (x : Fin n) → Term ν n
Λ : Term (suc ν) n → Term ν n
λ' : Type ν → Term ν (suc n) → Term ν n
_[_] : Term ν n → Type ν → Term ν n
_·_ : Term ν n → Term ν n → Term ν n
-- rule abstraction and application
ρ : Type ν → Term ν (suc n) → Term ν n
_with'_ : Term ν n → Term ν n → Term ν n
-- implicit rule application
_⟨⟩ : Term ν n → Term ν n
ClosedTerm : Set
ClosedTerm = Term 0 0
-----------------------------------------------------------------------------
-- syntactic sugar
let'_∶_in'_ : ∀ {ν n} → Term ν n → Type ν → Term ν (suc n) → Term ν n
let' e₁ ∶ r in' e₂ = (λ' r e₂) · e₁
implicit_∶_in'_ : ∀ {ν n} → Term ν n → Type ν → Term ν (suc n) → Term ν n
implicit e₁ ∶ r in' e₂ = (ρ r e₂) with' e₁
¿_ : ∀ {ν n} → Type ν → Term ν n
¿ r = (ρ r (var zero)) ⟨⟩
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Library where
module Literals where open import Light.Literals public
module Data where
module Empty where open import Light.Library.Data.Empty public
module Either where open import Light.Library.Data.Either public
module Natural where open import Light.Library.Data.Natural public
module Unit where open import Light.Library.Data.Unit public
module Integer where open import Light.Library.Data.Integer public
module Boolean where open import Light.Library.Data.Boolean public
module Both where open import Light.Library.Data.Both public
module Product where open import Light.Library.Data.Product public
module These where open import Light.Library.Data.These public
module Relation where
module Binary where
open import Light.Library.Relation.Binary public
module Equality where
open import Light.Library.Relation.Binary.Equality public
module Decidable where open import Light.Library.Relation.Binary.Equality.Decidable public
module Decidable where open import Light.Library.Relation.Binary.Decidable public
open import Light.Library.Relation public
module Decidable where open import Light.Library.Relation.Decidable public
module Action where open import Light.Library.Action public
module Arithmetic where open import Light.Library.Arithmetic public
module Level where open import Light.Level public
module Subtyping where open import Light.Subtyping public
module Variable where
module Levels where open import Light.Variable.Levels public
module Sets where open import Light.Variable.Sets public
module Other {ℓ} (𝕒 : Set ℓ) where open import Light.Variable.Other 𝕒 public
module Package where open import Light.Package public
open Package using (Package) hiding (module Package) public
-- module Indexed where open import Light.Indexed
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{-# OPTIONS --safe #-}
module Cubical.Algebra.CommRing.Instances.Polynomials where
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.Polynomials
private
variable
ℓ : Level
Poly : (CommRing ℓ) → CommRing ℓ
Poly R = (PolyMod.Poly R) , str
where
open CommRingStr --(snd R)
str : CommRingStr (PolyMod.Poly R)
0r str = PolyMod.0P R
1r str = PolyMod.1P R
_+_ str = PolyMod._Poly+_ R
_·_ str = PolyMod._Poly*_ R
- str = PolyMod.Poly- R
isCommRing str = makeIsCommRing (PolyMod.isSetPoly R)
(PolyMod.Poly+Assoc R)
(PolyMod.Poly+Rid R)
(PolyMod.Poly+Inverses R)
(PolyMod.Poly+Comm R)
(PolyMod.Poly*Associative R)
(PolyMod.Poly*Rid R)
(PolyMod.Poly*LDistrPoly+ R)
(PolyMod.Poly*Commutative R)
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module empty where
open import level
----------------------------------------------------------------------
-- datatypes
----------------------------------------------------------------------
data ⊥ {ℓ : Level} : Set ℓ where
----------------------------------------------------------------------
-- syntax
----------------------------------------------------------------------
----------------------------------------------------------------------
-- theorems
----------------------------------------------------------------------
⊥-elim : ∀{ℓ} → ⊥ {ℓ} → ∀{ℓ'}{P : Set ℓ'} → P
⊥-elim ()
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-- Andreas, 2015-08-11, issue reported by G.Allais
-- The `a` record field of `Pack` is identified as a function
-- (coloured blue, put in a \AgdaFunction in the LaTeX backend)
-- when it should be coloured pink.
-- The problem does not show up when dropping the second record
-- type or removing the module declaration.
record Pack (A : Set) : Set where
field
a : A
record Packed {A : Set} (p : Pack A) : Set where
module PP = Pack p
module Synchronised {A : Set} {p : Pack A} (rel : Packed p) where
module M = Packed rel
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Data.Nat.GCD where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Induction.WellFounded
open import Cubical.Data.Fin
open import Cubical.Data.Sigma as Σ
open import Cubical.Data.NatPlusOne
open import Cubical.HITs.PropositionalTruncation as PropTrunc
open import Cubical.Data.Nat.Base
open import Cubical.Data.Nat.Properties
open import Cubical.Data.Nat.Order
open import Cubical.Data.Nat.Divisibility
private
variable
m n d : ℕ
-- common divisors
isCD : ℕ → ℕ → ℕ → Type₀
isCD m n d = (d ∣ m) × (d ∣ n)
isPropIsCD : isProp (isCD m n d)
isPropIsCD = isProp× isProp∣ isProp∣
symCD : isCD m n d → isCD n m d
symCD (d∣m , d∣n) = (d∣n , d∣m)
-- greatest common divisors
isGCD : ℕ → ℕ → ℕ → Type₀
isGCD m n d = (isCD m n d) × (∀ d' → isCD m n d' → d' ∣ d)
GCD : ℕ → ℕ → Type₀
GCD m n = Σ ℕ (isGCD m n)
isPropIsGCD : isProp (isGCD m n d)
isPropIsGCD = isProp× isPropIsCD (isPropΠ2 (λ _ _ → isProp∣))
isPropGCD : isProp (GCD m n)
isPropGCD (d , dCD , gr) (d' , d'CD , gr') =
Σ≡Prop (λ _ → isPropIsGCD) (antisym∣ (gr' d dCD) (gr d' d'CD))
symGCD : isGCD m n d → isGCD n m d
symGCD (dCD , gr) = symCD dCD , λ { d' d'CD → gr d' (symCD d'CD) }
divsGCD : m ∣ n → isGCD m n m
divsGCD p = (∣-refl refl , p) , λ { d (d∣m , _) → d∣m }
oneGCD : ∀ m → isGCD m 1 1
oneGCD m = symGCD (divsGCD (∣-oneˡ m))
-- The base case of the Euclidean algorithm
zeroGCD : ∀ m → isGCD m 0 m
zeroGCD m = divsGCD (∣-zeroʳ m)
private
lem₁ : prediv d (suc n) → prediv d (m % suc n) → prediv d m
lem₁ {d} {n} {m} (c₁ , p₁) (c₂ , p₂) = (q · c₁ + c₂) , p
where r = m % suc n; q = n%k≡n[modk] m (suc n) .fst
p = (q · c₁ + c₂) · d ≡⟨ sym (·-distribʳ (q · c₁) c₂ d) ⟩
(q · c₁) · d + c₂ · d ≡⟨ cong (_+ c₂ · d) (sym (·-assoc q c₁ d)) ⟩
q · (c₁ · d) + c₂ · d ≡[ i ]⟨ q · (p₁ i) + (p₂ i) ⟩
q · (suc n) + r ≡⟨ n%k≡n[modk] m (suc n) .snd ⟩
m ∎
lem₂ : prediv d (suc n) → prediv d m → prediv d (m % suc n)
lem₂ {d} {n} {m} (c₁ , p₁) (c₂ , p₂) = c₂ ∸ q · c₁ , p
where r = m % suc n; q = n%k≡n[modk] m (suc n) .fst
p = (c₂ ∸ q · c₁) · d ≡⟨ ∸-distribʳ c₂ (q · c₁) d ⟩
c₂ · d ∸ (q · c₁) · d ≡⟨ cong (c₂ · d ∸_) (sym (·-assoc q c₁ d)) ⟩
c₂ · d ∸ q · (c₁ · d) ≡[ i ]⟨ p₂ i ∸ q · (p₁ i) ⟩
m ∸ q · (suc n) ≡⟨ cong (_∸ q · (suc n)) (sym (n%k≡n[modk] m (suc n) .snd)) ⟩
(q · (suc n) + r) ∸ q · (suc n) ≡⟨ cong (_∸ q · (suc n)) (+-comm (q · (suc n)) r) ⟩
(r + q · (suc n)) ∸ q · (suc n) ≡⟨ ∸-cancelʳ r zero (q · (suc n)) ⟩
r ∎
-- The inductive step of the Euclidean algorithm
stepGCD : isGCD (suc n) (m % suc n) d
→ isGCD m (suc n) d
fst (stepGCD ((d∣n , d∣m%n) , gr)) = PropTrunc.map2 lem₁ d∣n d∣m%n , d∣n
snd (stepGCD ((d∣n , d∣m%n) , gr)) d' (d'∣m , d'∣n) = gr d' (d'∣n , PropTrunc.map2 lem₂ d'∣n d'∣m)
-- putting it all together using well-founded induction
euclid< : ∀ m n → n < m → GCD m n
euclid< = WFI.induction <-wellfounded λ {
m rec zero p → m , zeroGCD m ;
m rec (suc n) p → let d , dGCD = rec (suc n) p (m % suc n) (n%sk<sk m n)
in d , stepGCD dGCD }
euclid : ∀ m n → GCD m n
euclid m n with n ≟ m
... | lt p = euclid< m n p
... | gt p = Σ.map-snd symGCD (euclid< n m p)
... | eq p = m , divsGCD (∣-refl (sym p))
isContrGCD : ∀ m n → isContr (GCD m n)
isContrGCD m n = euclid m n , isPropGCD _
-- the gcd operator on ℕ
gcd : ℕ → ℕ → ℕ
gcd m n = euclid m n .fst
gcdIsGCD : ∀ m n → isGCD m n (gcd m n)
gcdIsGCD m n = euclid m n .snd
isGCD→gcd≡ : isGCD m n d → gcd m n ≡ d
isGCD→gcd≡ dGCD = cong fst (isContrGCD _ _ .snd (_ , dGCD))
gcd≡→isGCD : gcd m n ≡ d → isGCD m n d
gcd≡→isGCD p = subst (isGCD _ _) p (gcdIsGCD _ _)
-- multiplicative properties of the gcd
isCD-cancelʳ : ∀ k → isCD (m · suc k) (n · suc k) (d · suc k)
→ isCD m n d
isCD-cancelʳ k (dk∣mk , dk∣nk) = (∣-cancelʳ k dk∣mk , ∣-cancelʳ k dk∣nk)
isCD-multʳ : ∀ k → isCD m n d
→ isCD (m · k) (n · k) (d · k)
isCD-multʳ k (d∣m , d∣n) = (∣-multʳ k d∣m , ∣-multʳ k d∣n)
isGCD-cancelʳ : ∀ k → isGCD (m · suc k) (n · suc k) (d · suc k)
→ isGCD m n d
isGCD-cancelʳ {m} {n} {d} k (dCD , gr) =
isCD-cancelʳ k dCD , λ d' d'CD → ∣-cancelʳ k (gr (d' · suc k) (isCD-multʳ (suc k) d'CD))
gcd-factorʳ : ∀ m n k → gcd (m · k) (n · k) ≡ gcd m n · k
gcd-factorʳ m n zero = (λ i → gcd (0≡m·0 m (~ i)) (0≡m·0 n (~ i))) ∙ 0≡m·0 (gcd m n)
gcd-factorʳ m n (suc k) = sym p ∙ cong (_· suc k) (sym q)
where k∣gcd : suc k ∣ gcd (m · suc k) (n · suc k)
k∣gcd = gcdIsGCD (m · suc k) (n · suc k) .snd (suc k) (∣-right m , ∣-right n)
d' = ∣-untrunc k∣gcd .fst
p : d' · suc k ≡ gcd (m · suc k) (n · suc k)
p = ∣-untrunc k∣gcd .snd
d'GCD : isGCD m n d'
d'GCD = isGCD-cancelʳ _ (subst (isGCD _ _) (sym p) (gcdIsGCD (m · suc k) (n · suc k)))
q : gcd m n ≡ d'
q = isGCD→gcd≡ d'GCD
-- Q: Can this be proved directly? (i.e. without a transport)
isGCD-multʳ : ∀ k → isGCD m n d
→ isGCD (m · k) (n · k) (d · k)
isGCD-multʳ {m} {n} {d} k dGCD = gcd≡→isGCD (gcd-factorʳ m n k ∙ cong (_· k) r)
where r : gcd m n ≡ d
r = isGCD→gcd≡ dGCD
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{-# OPTIONS --prop --without-K --rewriting #-}
module Calf.Types.Bool where
open import Calf.Prelude
open import Calf.Metalanguage
open import Data.Bool public using (Bool; true; false; if_then_else_)
bool : tp pos
bool = U (meta Bool)
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{-# OPTIONS --safe #-}
module MissingDefinition where
T : Set -> Set
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{- Cubical Agda with K
This file demonstrates the incompatibility of the --cubical
and --with-K flags, relying on the well-known incosistency of K with
univalence.
The --safe flag can be used to prevent accidentally mixing such
incompatible flags.
-}
{-# OPTIONS --with-K #-}
module Cubical.WithK where
open import Cubical.Data.Equality
open import Cubical.Data.Bool
open import Cubical.Data.Empty
private
variable
ℓ : Level
A : Type ℓ
x y : A
uip : (prf : x ≡ x) → Path _ prf refl
uip refl i = refl
transport-uip : (prf : A ≡ A) → Path _ (transportPath (eqToPath prf) x) x
transport-uip {x = x} prf =
compPath (congPath (λ p → transportPath (eqToPath p) x) (uip prf)) (transportRefl x)
transport-not : Path _ (transportPath (eqToPath (pathToEq notEq)) true) false
transport-not = congPath (λ a → transportPath a true) (eqToPath-pathToEq notEq)
false-true : Path _ false true
false-true = compPath (symPath transport-not) (transport-uip (pathToEq notEq))
absurd : (X : Type) → X
absurd X = transportPath (congPath sel false-true) true
where
sel : Bool → Type
sel false = Bool
sel true = X
inconsistency : ⊥
inconsistency = absurd ⊥
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module WrongNamedArgument2 where
postulate
f : {A : Set₁} → A
test : Set
test = f {B = Set}
-- Unsolved meta.
-- It is not an error since A could be instantiated to a function type
-- accepting hidden argument with name B.
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Argument information used in the reflection machinery
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Reflection.Argument.Information where
open import Data.Product
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open import Reflection.Argument.Relevance as Relevance using (Relevance)
open import Reflection.Argument.Visibility as Visibility using (Visibility)
------------------------------------------------------------------------
-- Re-exporting the builtins publically
open import Agda.Builtin.Reflection public using (ArgInfo)
open ArgInfo public
------------------------------------------------------------------------
-- Operations
visibility : ArgInfo → Visibility
visibility (arg-info v _) = v
relevance : ArgInfo → Relevance
relevance (arg-info _ r) = r
------------------------------------------------------------------------
-- Decidable equality
arg-info-injective₁ : ∀ {v r v′ r′} → arg-info v r ≡ arg-info v′ r′ → v ≡ v′
arg-info-injective₁ refl = refl
arg-info-injective₂ : ∀ {v r v′ r′} → arg-info v r ≡ arg-info v′ r′ → r ≡ r′
arg-info-injective₂ refl = refl
arg-info-injective : ∀ {v r v′ r′} → arg-info v r ≡ arg-info v′ r′ → v ≡ v′ × r ≡ r′
arg-info-injective = < arg-info-injective₁ , arg-info-injective₂ >
_≟_ : DecidableEquality ArgInfo
arg-info v r ≟ arg-info v′ r′ =
Dec.map′ (uncurry (cong₂ arg-info))
arg-info-injective
(v Visibility.≟ v′ ×-dec r Relevance.≟ r′)
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{-# OPTIONS --rewriting #-}
module FFI.Data.Vector where
open import Agda.Builtin.Equality using (_≡_)
open import Agda.Builtin.Equality.Rewrite using ()
open import Agda.Builtin.Int using (Int; pos; negsuc)
open import Agda.Builtin.Nat using (Nat)
open import Agda.Builtin.Bool using (Bool; false; true)
open import FFI.Data.HaskellInt using (HaskellInt; haskellIntToInt; intToHaskellInt)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import Properties.Equality using (_≢_)
{-# FOREIGN GHC import qualified Data.Vector #-}
postulate Vector : Set → Set
{-# POLARITY Vector ++ #-}
{-# COMPILE GHC Vector = type Data.Vector.Vector #-}
postulate
empty : ∀ {A} → (Vector A)
null : ∀ {A} → (Vector A) → Bool
unsafeHead : ∀ {A} → (Vector A) → A
unsafeTail : ∀ {A} → (Vector A) → (Vector A)
length : ∀ {A} → (Vector A) → Nat
lookup : ∀ {A} → (Vector A) → Nat → (Maybe A)
snoc : ∀ {A} → (Vector A) → A → (Vector A)
{-# COMPILE GHC empty = \_ -> Data.Vector.empty #-}
{-# COMPILE GHC null = \_ -> Data.Vector.null #-}
{-# COMPILE GHC unsafeHead = \_ -> Data.Vector.unsafeHead #-}
{-# COMPILE GHC unsafeTail = \_ -> Data.Vector.unsafeTail #-}
{-# COMPILE GHC length = \_ -> (fromIntegral . Data.Vector.length) #-}
{-# COMPILE GHC lookup = \_ v -> ((v Data.Vector.!?) . fromIntegral) #-}
{-# COMPILE GHC snoc = \_ -> Data.Vector.snoc #-}
postulate length-empty : ∀ {A} → (length (empty {A}) ≡ 0)
postulate lookup-empty : ∀ {A} n → (lookup (empty {A}) n ≡ nothing)
postulate lookup-snoc : ∀ {A} (x : A) (v : Vector A) → (lookup (snoc v x) (length v) ≡ just x)
postulate lookup-length : ∀ {A} (v : Vector A) → (lookup v (length v) ≡ nothing)
postulate lookup-snoc-empty : ∀ {A} (x : A) → (lookup (snoc empty x) 0 ≡ just x)
postulate lookup-snoc-not : ∀ {A n} (x : A) (v : Vector A) → (n ≢ length v) → (lookup v n ≡ lookup (snoc v x) n)
{-# REWRITE length-empty lookup-snoc lookup-length lookup-snoc-empty lookup-empty #-}
head : ∀ {A} → (Vector A) → (Maybe A)
head vec with null vec
head vec | false = just (unsafeHead vec)
head vec | true = nothing
tail : ∀ {A} → (Vector A) → Vector A
tail vec with null vec
tail vec | false = unsafeTail vec
tail vec | true = empty
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open import Function using (_∘_)
open import Category.Functor
open import Category.Monad
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Props as FinProps using ()
open import Data.Maybe as Maybe using (Maybe; maybe; just; nothing)
open import Data.Nat using (ℕ; zero; suc)
open import Data.Product using (Σ; ∃; _,_; proj₁; proj₂) renaming (_×_ to _∧_)
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_])
open import Data.Vec as Vec using (Vec; []; _∷_; head; tail)
open import Data.Vec.Equality as VecEq
open import Relation.Nullary using (Dec; yes; no; ¬_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as PropEq using (_≡_; _≢_; refl; sym; trans; cong; cong₂; inspect; Reveal_is_; [_])
module Unification.Correctness (Symbol : ℕ -> Set) (decEqSym : ∀ {k} (f g : Symbol k) → Dec (f ≡ g)) where
open import Unification Symbol decEqSym
open RawFunctor {{...}}
open DecSetoid {{...}} using (_≟_)
private maybeFunctor = Maybe.functor
private finDecSetoid : ∀ {n} → DecSetoid _ _
finDecSetoid {n} = FinProps.decSetoid n
-- * proving correctness of replacement function
mutual
-- | proof that var is the identity of replace
replace-thm₁ : ∀ {n} (t : Term n) → replace var t ≡ t
replace-thm₁ (var x) = refl
replace-thm₁ (con s ts) = cong (con s) (replaceChildren-thm₁ ts)
-- | proof that var is the identity of replaceChildren
replaceChildren-thm₁ : ∀ {n k} (ts : Vec (Term n) k) → replaceChildren var ts ≡ ts
replaceChildren-thm₁ [] = refl
replaceChildren-thm₁ (t ∷ ts) rewrite replace-thm₁ t = cong (_∷_ _) (replaceChildren-thm₁ ts)
-- * proving correctness of substitution/replacement composition
-- | proof that `var ∘ _` is the identity of ◇
compose-thm₁
: ∀ {m n l} (f : Fin m → Term n) (r : Fin l → Fin m) (t : Term l)
→ f ◇ (var ∘ r) ≡ f ∘ r
compose-thm₁ f r t = refl
mutual
-- | proof that _◇_ behaves as composition of replacements
compose-thm₂
: ∀ {m n l} (f : Fin m → Term n) (g : Fin l → Term m) (t : Term l)
→ replace (f ◇ g) t ≡ replace f (replace g t)
compose-thm₂ f g (var x) = refl
compose-thm₂ f g (con s ts) = cong (con s) (composeChildren-thm₂ f g ts)
-- | proof that _◇_ behaves as composition of replacements
composeChildren-thm₂
: ∀ {m n l k} (f : Fin m → Term n) (g : Fin l → Term m) (ts : Vec (Term l) k)
→ replaceChildren (f ◇ g) ts ≡ replaceChildren f (replaceChildren g ts)
composeChildren-thm₂ f g [] = refl
composeChildren-thm₂ f g (t ∷ ts) rewrite compose-thm₂ f g t = cong (_∷_ _) (composeChildren-thm₂ f g ts)
-- * proving correctness of thick and thin
-- | predecessor function over finite numbers
pred : ∀ {n} → Fin (suc (suc n)) → Fin (suc n)
pred zero = zero
pred (suc x) = x
-- | proof of injectivity of thin
thin-injective
: ∀ {n} (x : Fin (suc n)) (y z : Fin n)
→ thin x y ≡ thin x z → y ≡ z
thin-injective {zero} zero () _ _
thin-injective {zero} (suc _) () _ _
thin-injective {suc _} zero zero zero refl = refl
thin-injective {suc _} zero zero (suc _) ()
thin-injective {suc _} zero (suc _) zero ()
thin-injective {suc _} zero (suc y) (suc .y) refl = refl
thin-injective {suc _} (suc _) zero zero refl = refl
thin-injective {suc _} (suc _) zero (suc _) ()
thin-injective {suc _} (suc _) (suc _) zero ()
thin-injective {suc n} (suc x) (suc y) (suc z) p
= cong suc (thin-injective x y z (cong pred p))
-- | proof that thin x will never map anything to x
thinxy≢x
: ∀ {n} (x : Fin (suc n)) (y : Fin n)
→ thin x y ≢ x
thinxy≢x zero zero ()
thinxy≢x zero (suc _) ()
thinxy≢x (suc _) zero ()
thinxy≢x (suc x) (suc y) p
= thinxy≢x x y (cong pred p)
-- | proof that `thin x` reaches all y where x ≢ y
x≢y→thinxz≡y
: ∀ {n} (x y : Fin (suc n))
→ x ≢ y → ∃ (λ z → thin x z ≡ y)
x≢y→thinxz≡y zero zero 0≢0 with 0≢0 refl
x≢y→thinxz≡y zero zero 0≢0 | ()
x≢y→thinxz≡y {zero} (suc ()) _ _
x≢y→thinxz≡y {zero} zero (suc ()) _
x≢y→thinxz≡y {suc _} zero (suc y) _ = y , refl
x≢y→thinxz≡y {suc _} (suc x) zero _ = zero , refl
x≢y→thinxz≡y {suc _} (suc x) (suc y) sx≢sy
= (suc (proj₁ prf)) , (lem x y (proj₁ prf) (proj₂ prf))
where
x≢y = sx≢sy ∘ cong suc
prf = x≢y→thinxz≡y x y x≢y
lem : ∀ {n} (x y : Fin (suc n)) (z : Fin n)
→ thin x z ≡ y → thin (suc x) (suc z) ≡ suc y
lem zero zero _ ()
lem zero (suc .z) z refl = refl
lem (suc _) zero zero refl = refl
lem (suc _) zero (suc _) ()
lem (suc _) (suc _) zero ()
lem (suc x) (suc .(thin x z)) (suc z) refl = refl
-- | proof that thick x composed with thin x is the identity
thickx∘thinx≡yes
: ∀ {n} (x : Fin (suc n)) (y : Fin n)
→ thick x (thin x y) ≡ just y
thickx∘thinx≡yes zero zero = refl
thickx∘thinx≡yes zero (suc _) = refl
thickx∘thinx≡yes (suc _) zero = refl
thickx∘thinx≡yes (suc x) (suc y) = cong (_<$>_ suc) (thickx∘thinx≡yes x y)
-- | proof that `thin` is a partial inverse of `thick`
thin≡thick⁻¹
: ∀ {n} (x : Fin (suc n)) (y : Fin n) (z : Fin (suc n))
→ thin x y ≡ z
→ thick x z ≡ just y
thin≡thick⁻¹ x y z p with p
thin≡thick⁻¹ x y .(thin x y) _ | refl = thickx∘thinx≡yes x y
-- | proof that `thick x x` returns nothing
thickxx≡no
: ∀ {n} (x : Fin (suc n))
→ thick x x ≡ nothing
thickxx≡no zero = refl
thickxx≡no {zero} (suc ())
thickxx≡no {suc n} (suc x)
= cong (maybe (λ x → just (suc x)) nothing) (thickxx≡no x)
-- | proof that `thick x y` returns something when x ≢ y
x≢y→thickxy≡yes
: ∀ {n} (x y : Fin (suc n))
→ x ≢ y → ∃ (λ z → thick x y ≡ just z)
x≢y→thickxy≡yes zero zero 0≢0 with 0≢0 refl
x≢y→thickxy≡yes zero zero 0≢0 | ()
x≢y→thickxy≡yes zero (suc y) p = y , refl
x≢y→thickxy≡yes {zero} (suc ()) _ _
x≢y→thickxy≡yes {suc n} (suc x) zero _ = zero , refl
x≢y→thickxy≡yes {suc n} (suc x) (suc y) sx≢sy
= (suc (proj₁ prf)) , (cong (_<$>_ suc) (proj₂ prf))
where
x≢y = sx≢sy ∘ cong suc
prf = x≢y→thickxy≡yes {n} x y x≢y
-- | proof that `thick` is the partial inverse of `thin`
thick≡thin⁻¹ : ∀ {n} (x y : Fin (suc n)) (r : Maybe (Fin n))
→ thick x y ≡ r
→ x ≡ y ∧ r ≡ nothing
⊎ ∃ (λ z → thin x z ≡ y ∧ r ≡ just z)
thick≡thin⁻¹ x y _ thickxy≡r with x ≟ y | thickxy≡r
thick≡thin⁻¹ x .x .(thick x x) _ | yes refl | refl
= inj₁ (refl , thickxx≡no x)
thick≡thin⁻¹ x y .(thick x y) _ | no x≢y | refl
= inj₂ (proj₁ prf₁ , (proj₂ prf₁) , prf₂)
where
prf₁ = x≢y→thinxz≡y x y x≢y
prf₂ = thin≡thick⁻¹ x (proj₁ prf₁) y (proj₂ prf₁)
-- | proof that if check returns nothing, checkChildren will too
check≡no→checkChildren≡no
: ∀ {n} (x : Fin (suc n)) (s : Symbol (suc n)) (ts : Vec (Term (suc n)) (suc n))
→ check x (con s ts) ≡ nothing → checkChildren x ts ≡ nothing
check≡no→checkChildren≡no x s ts p with checkChildren x ts
check≡no→checkChildren≡no x s ts p | nothing = refl
check≡no→checkChildren≡no x s ts () | just _
-- | proof that if check returns something, checkChildren will too
check≡yes→checkChildren≡yes
: ∀ {n} (x : Fin (suc n)) (s : Symbol (suc n)) (ts : Vec (Term (suc n)) (suc n)) (ts' : Vec (Term n) (suc n))
→ check x (con s ts) ≡ just (con s ts') → checkChildren x ts ≡ just ts'
check≡yes→checkChildren≡yes x s ts ts' p with checkChildren x ts
check≡yes→checkChildren≡yes x s ts ts' refl | just .ts' = refl
check≡yes→checkChildren≡yes x s ts ts' () | nothing
-- | occurs predicate that is only inhabited if x occurs in t
mutual
data Occurs {n : ℕ} (x : Fin n) : Term n → Set where
Here : Occurs x (var x)
Further : ∀ {k ts} {s : Symbol k} → OccursChildren x {k} ts → Occurs x (con s ts)
data OccursChildren {n : ℕ} (x : Fin n) : {k : ℕ} → Vec (Term n) k → Set where
Here : ∀ {k t ts} → Occurs x t → OccursChildren x {suc k} (t ∷ ts)
Further : ∀ {k t ts} → OccursChildren x {k} ts → OccursChildren x {suc k} (t ∷ ts)
-- | proof of decidability for the occurs predicate
mutual
occurs? : ∀ {n} (x : Fin n) (t : Term n) → Dec (Occurs x t)
occurs? x₁ (var x₂) with x₁ ≟ x₂
occurs? .x₂ (var x₂) | yes refl = yes Here
occurs? x₁ (var x₂) | no x₁≢x₂ = no (x₁≢x₂ ∘ lem x₁ x₂)
where
lem : ∀ {n} (x y : Fin n) → Occurs x (var y) → x ≡ y
lem zero zero _ = refl
lem zero (suc _) ()
lem (suc x) zero ()
lem (suc x) (suc .x) Here = refl
occurs? x₁ (con s ts) with occursChildren? x₁ ts
occurs? x₁ (con s ts) | yes x₁∈ts = yes (Further x₁∈ts)
occurs? x₁ (con s ts) | no x₁∉ts = no (x₁∉ts ∘ lem x₁)
where
lem : ∀ {n s ts} (x : Fin n) → Occurs x (con s ts) → OccursChildren x ts
lem x (Further x₂) = x₂
occursChildren? : ∀ {n k} (x : Fin n) (ts : Vec (Term n) k) → Dec (OccursChildren x ts)
occursChildren? x₁ [] = no (λ ())
occursChildren? x₁ (t ∷ ts) with occurs? x₁ t
occursChildren? x₁ (t ∷ ts) | yes h = yes (Here h)
occursChildren? x₁ (t ∷ ts) | no ¬h with occursChildren? x₁ ts
occursChildren? x₁ (t ∷ ts) | no ¬h | yes f = yes (Further f)
occursChildren? x₁ (t ∷ ts) | no ¬h | no ¬f = no lem
where
lem : OccursChildren x₁ (t ∷ ts) → ⊥
lem (Here p) = ¬h p
lem (Further p) = ¬f p
-- * proving correctness of check
mutual
-- | proving that if x occurs in t, check returns nothing
occurs→check≡no
: ∀ {n} (x : Fin (suc n)) (t : Term (suc n))
→ Occurs x t → check x t ≡ nothing
occurs→check≡no x .(Unification.var x) Here
rewrite thickxx≡no x = refl
occurs→check≡no x .(Unification.con s ts) (Further {k} {ts} {s} p)
rewrite occursChildren→checkChildren≡no x ts p = refl
-- | proving that if x occurs in ts, checkChildren returns nothing
occursChildren→checkChildren≡no
: ∀ {n k} (x : Fin (suc n)) (ts : Vec (Term (suc n)) k)
→ OccursChildren x ts → checkChildren x ts ≡ nothing
occursChildren→checkChildren≡no x .(t ∷ ts) (Here {k} {t} {ts} p)
rewrite occurs→check≡no x t p = refl
occursChildren→checkChildren≡no x .(t ∷ ts) (Further {k} {t} {ts} p)
with check x t
... | just _ rewrite occursChildren→checkChildren≡no x ts p = refl
... | nothing rewrite occursChildren→checkChildren≡no x ts p = refl
mutual
-- | proof that if check x t returns nothing, x occurs in t
check≡no→occurs
: ∀ {n} (x : Fin (suc n)) (t : Term (suc n))
→ check x t ≡ nothing → Occurs x t
check≡no→occurs x₁ (var x₂) p with x₁ ≟ x₂
check≡no→occurs .x₂ (var x₂) p | yes refl = Here
check≡no→occurs x₁ (var x₂) p | no x₁≢x₂ = ⊥-elim (lem₂ p)
where
lem₁ : ∃ (λ z → thick x₁ x₂ ≡ just z)
lem₁ = x≢y→thickxy≡yes x₁ x₂ x₁≢x₂
lem₂ : var <$> thick x₁ x₂ ≡ nothing → ⊥
lem₂ rewrite proj₂ lem₁ = λ ()
check≡no→occurs {n} x₁ (con s ts) p
= Further (checkChildren≡no→occursChildren x₁ ts (lem p))
where
lem : con s <$> checkChildren x₁ ts ≡ nothing → checkChildren x₁ ts ≡ nothing
lem p with checkChildren x₁ ts | inspect (checkChildren x₁) ts
lem () | just _ | [ eq ]
lem p | nothing | [ eq ] = refl
-- | proof that if checkChildren x ts returns nothing, x occurs in ts
checkChildren≡no→occursChildren
: ∀ {n k} (x : Fin (suc n)) (ts : Vec (Term (suc n)) k)
→ checkChildren x ts ≡ nothing → OccursChildren x ts
checkChildren≡no→occursChildren x [] ()
checkChildren≡no→occursChildren x (t ∷ ts) p with check x t | inspect (check x) t
... | nothing | [ e₁ ] = Here (check≡no→occurs x t e₁)
... | just _ | [ e₁ ] with checkChildren x ts | inspect (checkChildren x) ts
... | nothing | [ e₂ ] = Further (checkChildren≡no→occursChildren x ts e₂)
checkChildren≡no→occursChildren x (t ∷ ts) () | just _ | [ e₁ ] | just _ | [ e₂ ]
-- | proof that if check returns just, x does not occur in t
check≡yes→¬occurs
: ∀ {n} (x : Fin (suc n)) (t : Term (suc n)) (t' : Term n)
→ check x t ≡ just t' → ¬ (Occurs x t)
check≡yes→¬occurs x t t' p₁ x∈t with occurs→check≡no x t x∈t
check≡yes→¬occurs x t t' p₁ _ | p₂ with check x t
check≡yes→¬occurs x t t' p₁ _ | () | just _
check≡yes→¬occurs x t t' () _ | p₂ | nothing
-- | proof that x does not occur in t, check returns just
¬occurs→check≡yes
: ∀ {n} (x : Fin (suc n)) (t : Term (suc n))
→ ¬ (Occurs x t) → ∃ (λ t' → check x t ≡ just t')
¬occurs→check≡yes x t x∉t with check x t | inspect (check x) t
¬occurs→check≡yes x t x∉t | nothing | [ eq ] with x∉t (check≡no→occurs x t eq)
¬occurs→check≡yes x t x∉t | nothing | [ eq ] | ()
¬occurs→check≡yes x t x∉t | just t' | [ eq ] = t' , refl
-- * proving correctness of _for_
-- | proof that if there is nothing to unify, _for_ is the identity
for-thm₁
: ∀ {n} (t : Term n) (x : Fin (suc n)) (y : Fin n)
→ (t for x) (thin x y) ≡ var y
for-thm₁ t x y rewrite thickx∘thinx≡yes x y = refl
mutual
-- | proof that if there is something to unify, _for_ unifies
for-thm₂
: ∀ {n} (x : Fin (suc n)) (t : Term (suc n)) (t' : Term n)
→ check x t ≡ just t' → replace (t' for x) t ≡ (t' for x) x
for-thm₂ x (var y) _ _ with x ≟ y
for-thm₂ .y (var y) _ _ | yes refl = refl
for-thm₂ x (var y) _ _ | no x≢y
with thick x y | x≢y→thickxy≡yes x y x≢y
| thick x x | thickxx≡no x
for-thm₂ x (var y) .(var z) refl | no _
| .(just z) | z , refl
| .nothing | refl = refl
for-thm₂ x (con s ts) _ _ with checkChildren x ts | inspect (checkChildren x) ts
for-thm₂ x (con s ts) _ () | nothing | _
for-thm₂ x (con s ts) .(con s ts') refl | just ts' | [ checkChildren≡yes ]
rewrite thickxx≡no x = cong (con s) (forChildren-thm₂ x s ts ts' checkChildren≡yes)
forChildren-thm₂ : ∀ {n k} -> (x : Fin (suc n)) (s : Symbol k)
(ts : Vec (Term (suc n)) k) (ts' : Vec (Term n) k) ->
checkChildren x ts ≡ just ts' ->
replaceChildren (con s ts' for x) ts ≡ ts'
forChildren-thm₂ x s [] [] eq rewrite thickxx≡no x = refl
forChildren-thm₂ x s (t1 ∷ ts) (t2 ∷ ts') eq
with check x t1 | inspect (check x) t1 | checkChildren x ts | inspect (checkChildren x) ts
forChildren-thm₂ x s (t1 ∷ ts) (t2 ∷ ts') refl | just .t2 | [ eq1 ] | just .ts' | [ eq2 ]
= cong₂ _∷_ {!!} {!!}
where
lemma₁ = for-thm₂ x t1 t2 eq1
forChildren-thm₂ x s (t1 ∷ ts) (t2 ∷ ts') () | just x₁ | _ | nothing | _
forChildren-thm₂ x s (t1 ∷ ts) (t2 ∷ ts') () | nothing | _ | cs | _
-- * proving correctness of apply, concat and compose
++-thm₁ : ∀ {m n} (s : Subst m n) → nil ++ s ≡ s
++-thm₁ nil = refl
++-thm₁ (snoc s t x) = cong (λ s → snoc s t x) (++-thm₁ s)
mutual
replace-var-id : ∀ {m} (t : Term m) -> replace var t ≡ t
replace-var-id (Unification.var x) = refl
replace-var-id (Unification.con s ts) = cong (con s) (replaceChildren-var-id ts)
replaceChildren-var-id : ∀ {m n} -> (ts : Vec (Term m) n) -> replaceChildren var ts ≡ ts
replaceChildren-var-id [] = refl
replaceChildren-var-id (x ∷ ts) = cong₂ _∷_ (replace-var-id x) (replaceChildren-var-id ts)
mutual
replace-var-id' : ∀ {n m} (f : Fin n -> Term m) (t : Term n) ->
replace (\x -> replace var (f x)) t ≡ replace f t
replace-var-id' f (Unification.var x) = replace-var-id (f x)
replace-var-id' f (Unification.con s ts) = cong (con s) (replaceChildren-var-id' f ts)
replaceChildren-var-id' : ∀ {m n k} -> (f : Fin m -> Term k) (ts : Vec (Term m) n) ->
replaceChildren (\x -> replace var (f x)) ts ≡ replaceChildren f ts
replaceChildren-var-id' f [] = refl
replaceChildren-var-id' f (x ∷ ts) = cong₂ _∷_ (replace-var-id' f x) (replaceChildren-var-id' f ts)
++-lem₁ : ∀ {m n} (s : Subst m n) (t : Term (suc m)) (t' : Term m) (x : Fin (suc m)) ->
replace (apply s) (replace (t' for x) t) ≡ replace (\x' -> replace (apply s) (_for_ t' x x')) t
++-lem₁ Unification.nil t t' x
rewrite replace-var-id (replace (t' for x) t)
| replace-var-id' (t' for x) t = refl
++-lem₁ (Unification.snoc s t x) t₁ t' x₁ = {!!}
++-lem₂
: ∀ {l m n} (s₁ : Subst m n) (s₂ : Subst l m) (t : Term l)
→ replace (apply (s₁ ++ s₂)) t ≡ replace (apply s₁) (replace (apply s₂) t)
++-lem₂ s₁ nil t rewrite replace-thm₁ t = refl
++-lem₂ {.(suc k)} {m} {n} s₁ (snoc {k} s₂ t₂ x) t = {!!}
where
lem = ++-lem₂ s₁ s₂ (replace (t₂ for x) t)
++-thm₂
: ∀ {l m n} (s₁ : Subst m n) (s₂ : Subst l m) (x : Fin l)
→ apply (s₁ ++ s₂) x ≡ (apply s₁ ◇ apply s₂) x
++-thm₂ s₁ nil x = refl
++-thm₂ s₁ (snoc s₂ t y) x with thick y x
++-thm₂ s₁ (snoc s₂ t y) x | just t' = ++-thm₂ s₁ s₂ t'
++-thm₂ s₁ (snoc s₂ t y) x | nothing = ++-lem₂ s₁ s₂ t
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.ZCohomology.RingStructure.GradedCommutativity where
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Function
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.GroupoidLaws hiding (assoc)
open import Cubical.Foundations.Path
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Nat
open import Cubical.Data.Int
renaming (_+_ to _ℤ+_ ; _·_ to _ℤ∙_ ; +Comm to +ℤ-comm ; ·Comm to ∙-comm ; +Assoc to ℤ+-assoc ; -_ to -ℤ_)
hiding (_+'_ ; +'≡+)
open import Cubical.Data.Sigma
open import Cubical.Data.Sum
open import Cubical.HITs.SetTruncation as ST
open import Cubical.HITs.Truncation as T
open import Cubical.HITs.S1 hiding (_·_)
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp
open import Cubical.Homotopy.Loopspace
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.GroupStructure
open import Cubical.ZCohomology.RingStructure.CupProduct
open import Cubical.ZCohomology.RingStructure.RingLaws
open import Cubical.ZCohomology.Properties
private
variable
ℓ : Level
open PlusBis
natTranspLem : ∀ {ℓ} {A B : ℕ → Type ℓ} {n m : ℕ} (a : A n)
(f : (n : ℕ) → (a : A n) → B n) (p : n ≡ m)
→ f m (subst A p a) ≡ subst B p (f n a)
natTranspLem {A = A} {B = B} a f p = sym (substCommSlice A B f p a)
transp0₁ : (n : ℕ) → subst coHomK (+'-comm 1 (suc n)) (0ₖ _) ≡ 0ₖ _
transp0₁ zero = refl
transp0₁ (suc n) = refl
transp0₂ : (n m : ℕ) → subst coHomK (+'-comm (suc (suc n)) (suc m)) (0ₖ _) ≡ 0ₖ _
transp0₂ n zero = refl
transp0₂ n (suc m) = refl
-- Recurring expressions
private
ΩKn+1→Ω²Kn+2 : {k : ℕ} → typ (Ω (coHomK-ptd k)) → typ ((Ω^ 2) (coHomK-ptd (suc k)))
ΩKn+1→Ω²Kn+2 x = sym (Kn→ΩKn+10ₖ _) ∙∙ cong (Kn→ΩKn+1 _) x ∙∙ Kn→ΩKn+10ₖ _
ΩKn+1→Ω²Kn+2' : {k : ℕ} → Kn→ΩKn+1 k (0ₖ k) ≡ Kn→ΩKn+1 k (0ₖ k) → typ ((Ω^ 2) (coHomK-ptd (suc k)))
ΩKn+1→Ω²Kn+2' p = sym (Kn→ΩKn+10ₖ _) ∙∙ p ∙∙ Kn→ΩKn+10ₖ _
Kn→Ω²Kn+2 : {k : ℕ} → coHomK k → typ ((Ω^ 2) (coHomK-ptd (2 + k)))
Kn→Ω²Kn+2 x = ΩKn+1→Ω²Kn+2 (Kn→ΩKn+1 _ x)
-- Definition of of -ₖ'ⁿ̇*ᵐ
-- This definition is introduced to facilite the proofs
-ₖ'-helper : {k : ℕ} (n m : ℕ)
→ isEvenT n ⊎ isOddT n → isEvenT m ⊎ isOddT m
→ coHomKType k → coHomKType k
-ₖ'-helper {k = zero} n m (inl x₁) q x = x
-ₖ'-helper {k = zero} n m (inr x₁) (inl x₂) x = x
-ₖ'-helper {k = zero} n m (inr x₁) (inr x₂) x = 0 - x
-ₖ'-helper {k = suc zero} n m p q base = base
-ₖ'-helper {k = suc zero} n m (inl x) q (loop i) = loop i
-ₖ'-helper {k = suc zero} n m (inr x) (inl x₁) (loop i) = loop i
-ₖ'-helper {k = suc zero} n m (inr x) (inr x₁) (loop i) = loop (~ i)
-ₖ'-helper {k = suc (suc k)} n m p q north = north
-ₖ'-helper {k = suc (suc k)} n m p q south = north
-ₖ'-helper {k = suc (suc k)} n m (inl x) q (merid a i) =
(merid a ∙ sym (merid (ptSn (suc k)))) i
-ₖ'-helper {k = suc (suc k)} n m (inr x) (inl x₁) (merid a i) =
(merid a ∙ sym (merid (ptSn (suc k)))) i
-ₖ'-helper {k = suc (suc k)} n m (inr x) (inr x₁) (merid a i) =
(merid a ∙ sym (merid (ptSn (suc k)))) (~ i)
-ₖ'-gen : {k : ℕ} (n m : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ coHomK k → coHomK k
-ₖ'-gen {k = zero} = -ₖ'-helper {k = zero}
-ₖ'-gen {k = suc k} n m p q = T.map (-ₖ'-helper {k = suc k} n m p q)
-- -ₖ'ⁿ̇*ᵐ
-ₖ'^_·_ : {k : ℕ} (n m : ℕ) → coHomK k → coHomK k
-ₖ'^_·_ {k = k} n m = -ₖ'-gen n m (evenOrOdd n) (evenOrOdd m)
-- cohomology version
-ₕ'^_·_ : {k : ℕ} {A : Type ℓ} (n m : ℕ) → coHom k A → coHom k A
-ₕ'^_·_ n m = ST.map λ f x → (-ₖ'^ n · m) (f x)
-- -ₖ'ⁿ̇*ᵐ = -ₖ' for n m odd
-ₖ'-gen-inr≡-ₖ' : {k : ℕ} (n m : ℕ) (p : _) (q : _) (x : coHomK k)
→ -ₖ'-gen n m (inr p) (inr q) x ≡ (-ₖ x)
-ₖ'-gen-inr≡-ₖ' {k = zero} n m p q _ = refl
-ₖ'-gen-inr≡-ₖ' {k = suc zero} n m p q =
T.elim ((λ _ → isOfHLevelTruncPath))
λ { base → refl
; (loop i) → refl}
-ₖ'-gen-inr≡-ₖ' {k = suc (suc k)} n m p q =
T.elim ((λ _ → isOfHLevelTruncPath))
λ { north → refl
; south → refl
; (merid a i) k → ∣ symDistr (merid (ptSn _)) (sym (merid a)) (~ k) (~ i) ∣ₕ}
-- -ₖ'ⁿ̇*ᵐ x = x for n even
-ₖ'-gen-inl-left : {k : ℕ} (n m : ℕ) (p : _) (q : _) (x : coHomK k)
→ -ₖ'-gen n m (inl p) q x ≡ x
-ₖ'-gen-inl-left {k = zero} n m p q x = refl
-ₖ'-gen-inl-left {k = suc zero} n m p q =
T.elim (λ _ → isOfHLevelTruncPath)
λ { base → refl ; (loop i) → refl}
-ₖ'-gen-inl-left {k = suc (suc k)} n m p q =
T.elim (λ _ → isOfHLevelPath (4 + k) (isOfHLevelTrunc (4 + k)) _ _)
λ { north → refl
; south → cong ∣_∣ₕ (merid (ptSn _))
; (merid a i) k → ∣ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i ∣ₕ}
-- -ₖ'ⁿ̇*ᵐ x = x for m even
-ₖ'-gen-inl-right : {k : ℕ} (n m : ℕ) (p : _) (q : _) (x : coHomK k)
→ -ₖ'-gen n m p (inl q) x ≡ x
-ₖ'-gen-inl-right {k = zero} n m (inl x₁) q x = refl
-ₖ'-gen-inl-right {k = zero} n m (inr x₁) q x = refl
-ₖ'-gen-inl-right {k = suc zero} n m (inl x₁) q =
T.elim (λ _ → isOfHLevelTruncPath)
λ { base → refl ; (loop i) → refl}
-ₖ'-gen-inl-right {k = suc zero} n m (inr x₁) q =
T.elim (λ _ → isOfHLevelTruncPath)
λ { base → refl ; (loop i) → refl}
-ₖ'-gen-inl-right {k = suc (suc k)} n m (inl x₁) q =
T.elim (λ _ → isOfHLevelTruncPath)
λ { north → refl
; south → cong ∣_∣ₕ (merid (ptSn _))
; (merid a i) k → ∣ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i ∣ₕ}
-ₖ'-gen-inl-right {k = suc (suc k)} n m (inr x₁) q =
T.elim (λ _ → isOfHLevelTruncPath)
λ { north → refl
; south → cong ∣_∣ₕ (merid (ptSn _))
; (merid a i) k → ∣ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i ∣ₕ}
-ₖ'-gen² : {k : ℕ} (n m : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ (x : coHomK k) → -ₖ'-gen n m p q (-ₖ'-gen n m p q x) ≡ x
-ₖ'-gen² {k = zero} n m (inl x₁) q x = refl
-ₖ'-gen² {k = zero} n m (inr x₁) (inl x₂) x = refl
-ₖ'-gen² {k = zero} n m (inr x₁) (inr x₂) x =
cong (pos 0 -_) (-AntiComm (pos 0) x)
∙∙ -AntiComm (pos 0) (-ℤ (x - pos 0))
∙∙ h x
where
h : (x : _) → -ℤ (-ℤ (x - pos 0) - pos 0) ≡ x
h (pos zero) = refl
h (pos (suc n)) = refl
h (negsuc n) = refl
-ₖ'-gen² {k = suc k} n m (inl x₁) q x i =
-ₖ'-gen-inl-left n m x₁ q (-ₖ'-gen-inl-left n m x₁ q x i) i
-ₖ'-gen² {k = suc k} n m (inr x₁) (inl x₂) x i =
-ₖ'-gen-inl-right n m (inr x₁) x₂ (-ₖ'-gen-inl-right n m (inr x₁) x₂ x i) i
-ₖ'-gen² {k = suc k} n m (inr x₁) (inr x₂) x =
(λ i → -ₖ'-gen-inr≡-ₖ' n m x₁ x₂ (-ₖ'-gen-inr≡-ₖ' n m x₁ x₂ x i) i) ∙ -ₖ^2 x
-ₖ'-genIso : {k : ℕ} (n m : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ Iso (coHomK k) (coHomK k)
Iso.fun (-ₖ'-genIso {k = k} n m p q) = -ₖ'-gen n m p q
Iso.inv (-ₖ'-genIso {k = k} n m p q) = -ₖ'-gen n m p q
Iso.rightInv (-ₖ'-genIso {k = k} n m p q) = -ₖ'-gen² n m p q
Iso.leftInv (-ₖ'-genIso {k = k} n m p q) = -ₖ'-gen² n m p q
-- action of cong on -ₖ'ⁿ̇*ᵐ
cong-ₖ'-gen-inr : {k : ℕ} (n m : ℕ) (p : _) (q : _) (P : Path (coHomK (2 + k)) (0ₖ _) (0ₖ _))
→ cong (-ₖ'-gen n m (inr p) (inr q)) P ≡ sym P
cong-ₖ'-gen-inr {k = k} n m p q P = code≡sym (0ₖ _) P
where
code : (x : coHomK (2 + k)) → 0ₖ _ ≡ x → x ≡ 0ₖ _
code = T.elim (λ _ → isOfHLevelΠ (4 + k) λ _ → isOfHLevelTruncPath)
λ { north → cong (-ₖ'-gen n m (inr p) (inr q))
; south P → cong ∣_∣ₕ (sym (merid (ptSn _))) ∙ (cong (-ₖ'-gen n m (inr p) (inr q)) P)
; (merid a i) → t a i}
where
t : (a : S₊ (suc k)) → PathP (λ i → 0ₖ (2 + k) ≡ ∣ merid a i ∣ₕ → ∣ merid a i ∣ₕ ≡ 0ₖ (2 + k))
(cong (-ₖ'-gen n m (inr p) (inr q)))
(λ P → cong ∣_∣ₕ (sym (merid (ptSn _))) ∙ (cong (-ₖ'-gen n m (inr p) (inr q)) P))
t a = toPathP (funExt λ P → cong (transport (λ i → ∣ merid a i ∣ ≡ 0ₖ (suc (suc k))))
(cong (cong (-ₖ'-gen n m (inr p) (inr q))) (λ i → (transp (λ j → 0ₖ (suc (suc k)) ≡ ∣ merid a (~ j ∧ ~ i) ∣) i
(compPath-filler P (λ j → ∣ merid a (~ j) ∣ₕ) i))))
∙∙ cong (transport (λ i → ∣ merid a i ∣ ≡ 0ₖ (suc (suc k)))) (congFunct (-ₖ'-gen n m (inr p) (inr q)) P (sym (cong ∣_∣ₕ (merid a))))
∙∙ (λ j → transp (λ i → ∣ merid a (i ∨ j) ∣ ≡ 0ₖ (suc (suc k))) j
(compPath-filler' (cong ∣_∣ₕ (sym (merid a)))
(cong (-ₖ'-gen n m (inr p) (inr q)) P
∙ cong (-ₖ'-gen n m (inr p) (inr q)) (sym (cong ∣_∣ₕ (merid a)))) j))
∙∙ (λ i → sym (cong ∣_∣ₕ (merid a))
∙ isCommΩK (2 + k) (cong (-ₖ'-gen n m (inr p) (inr q)) P)
(cong (-ₖ'-gen n m (inr p) (inr q)) (sym (cong ∣_∣ₕ (merid a)))) i)
∙∙ (λ j → (λ i → ∣ merid a (~ i ∨ j) ∣)
∙ (cong ∣_∣ₕ (compPath-filler' (merid a) (sym (merid (ptSn _))) (~ j)) ∙ (λ i → -ₖ'-gen n m (inr p) (inr q) (P i))))
∙ sym (lUnit _))
code≡sym : (x : coHomK (2 + k)) → (p : 0ₖ _ ≡ x) → code x p ≡ sym p
code≡sym x = J (λ x p → code x p ≡ sym p) refl
cong-cong-ₖ'-gen-inr : {k : ℕ} (n m : ℕ) (p : _) (q : _)
(P : Square (refl {x = 0ₖ (suc (suc k))}) refl refl refl)
→ cong (cong (-ₖ'-gen n m (inr p) (inr q))) P ≡ sym P
cong-cong-ₖ'-gen-inr n m p q P =
rUnit _
∙∙ (λ k → (λ i → cong-ₖ'-gen-inr n m p q refl (i ∧ k))
∙∙ (λ i → cong-ₖ'-gen-inr n m p q (P i) k)
∙∙ λ i → cong-ₖ'-gen-inr n m p q refl (~ i ∧ k))
∙∙ (λ k → transportRefl refl k
∙∙ cong sym P
∙∙ transportRefl refl k)
∙∙ sym (rUnit (cong sym P))
∙∙ sym (sym≡cong-sym P)
Kn→ΩKn+1-ₖ'' : {k : ℕ} (n m : ℕ) (p : _) (q : _) (x : coHomK k)
→ Kn→ΩKn+1 k (-ₖ'-gen n m (inr p) (inr q) x) ≡ sym (Kn→ΩKn+1 k x)
Kn→ΩKn+1-ₖ'' n m p q x = cong (Kn→ΩKn+1 _) (-ₖ'-gen-inr≡-ₖ' n m p q x) ∙ Kn→ΩKn+1-ₖ _ x
transpΩ² : {n m : ℕ} (p q : n ≡ m) → (P : _)
→ transport (λ i → refl {x = 0ₖ (p i)} ≡ refl {x = 0ₖ (p i)}) P
≡ transport (λ i → refl {x = 0ₖ (q i)} ≡ refl {x = 0ₖ (q i)}) P
transpΩ² p q P k = subst (λ n → refl {x = 0ₖ n} ≡ refl {x = 0ₖ n}) (isSetℕ _ _ p q k) P
-- Some technical lemmas about Kn→Ω²Kn+2 and its interaction with -ₖ'ⁿ̇*ᵐ and transports
-- TODO : Check if this can be cleaned up more by having more general lemmas
private
lem₁ : (n : ℕ) (a : _)
→ (cong (cong (subst coHomK (+'-comm (suc zero) (suc (suc n)))))
(Kn→Ω²Kn+2 ∣ a ∣ₕ))
≡ ΩKn+1→Ω²Kn+2
(sym (transp0₁ n) ∙∙ cong (subst coHomK (+'-comm (suc zero) (suc n))) (Kn→ΩKn+1 (suc n) ∣ a ∣ₕ) ∙∙ transp0₁ n)
lem₁ zero a =
(λ k i j → transportRefl (Kn→Ω²Kn+2 ∣ a ∣ₕ i j) k)
∙ cong ΩKn+1→Ω²Kn+2 λ k → rUnit (λ i → transportRefl (Kn→ΩKn+1 1 ∣ a ∣ i) (~ k)) k
lem₁ (suc n) a =
(λ k → transp (λ i → refl {x = 0ₖ (+'-comm 1 (suc (suc (suc n))) (i ∨ ~ k))}
≡ refl {x = 0ₖ (+'-comm 1 (suc (suc (suc n))) (i ∨ ~ k))}) (~ k)
(λ i j → transp (λ i → coHomK (+'-comm 1 (suc (suc (suc n))) (i ∧ ~ k))) k
(Kn→Ω²Kn+2 ∣ a ∣ₕ i j)))
∙∙ transpΩ² (+'-comm 1 (suc (suc (suc n))))
(cong suc (+'-comm (suc zero) (suc (suc n))))
(Kn→Ω²Kn+2 ∣ a ∣ₕ)
∙∙ sym (natTranspLem {A = λ n → 0ₖ n ≡ 0ₖ n}
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣)
(λ _ → ΩKn+1→Ω²Kn+2)
(+'-comm 1 (suc (suc n))))
∙∙ cong ΩKn+1→Ω²Kn+2
(λ k → transp (λ i → 0ₖ (+'-comm (suc zero) (suc (suc n)) (i ∨ k))
≡ 0ₖ (+'-comm (suc zero) (suc (suc n)) (i ∨ k))) k
(λ i → transp (λ i → coHomK (+'-comm (suc zero) (suc (suc n)) (i ∧ k))) (~ k)
(Kn→ΩKn+1 _ ∣ a ∣ₕ i)))
∙∙ cong ΩKn+1→Ω²Kn+2 (rUnit (cong (subst coHomK (+'-comm (suc zero) (suc (suc n))))
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ)))
lem₂ : (n : ℕ) (a : _) (p : _) (q : _)
→ (cong (cong (-ₖ'-gen (suc (suc n)) (suc zero) p q
∘ (subst coHomK (+'-comm 1 (suc (suc n))))))
(Kn→Ω²Kn+2 (∣ a ∣ₕ)))
≡ ΩKn+1→Ω²Kn+2
(sym (transp0₁ n)
∙∙ cong (subst coHomK (+'-comm (suc zero) (suc n)))
(cong (-ₖ'-gen (suc (suc n)) (suc zero) p q)
(Kn→ΩKn+1 (suc n) ∣ a ∣ₕ))
∙∙ transp0₁ n)
lem₂ n a (inl x) (inr y) =
(λ k i j → (-ₖ'-gen-inl-left (suc (suc n)) 1 x (inr y) (
subst coHomK (+'-comm 1 (suc (suc n)))
(Kn→Ω²Kn+2 ∣ a ∣ₕ i j))) k)
∙∙ lem₁ n a
∙∙ cong ΩKn+1→Ω²Kn+2 (cong (sym (transp0₁ n) ∙∙_∙∙ transp0₁ n)
λ k i → subst coHomK (+'-comm 1 (suc n))
(-ₖ'-gen-inl-left (suc (suc n)) 1 x (inr y) (Kn→ΩKn+1 (suc n) ∣ a ∣ i) (~ k)))
lem₂ n a (inr x) (inr y) =
cong-cong-ₖ'-gen-inr (suc (suc n)) 1 x y
(cong
(cong
(subst coHomK (+'-comm 1 (suc (suc n)))))
(Kn→Ω²Kn+2 ∣ a ∣ₕ))
∙∙ cong sym (lem₁ n a)
∙∙ λ k → ΩKn+1→Ω²Kn+2
(sym (transp0₁ n) ∙∙
cong (subst coHomK (+'-comm 1 (suc n)))
(cong-ₖ'-gen-inr (suc (suc n)) 1 x y
(Kn→ΩKn+1 (suc n) ∣ a ∣) (~ k))
∙∙ transp0₁ n)
lem₃ : (n m : ℕ) (q : _) (p : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n))) (x : _)
→ (((sym (cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))
∙∙ (λ j → -ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n)) (Kn→ΩKn+1 _ x j)))
∙∙ cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))))
≡ (Kn→ΩKn+1 _ (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x)))
lem₃ n m q p x =
sym (cong-∙∙ (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (sym (transp0₂ m n))
(λ j → subst coHomK (+'-comm (suc (suc m)) (suc n)) (Kn→ΩKn+1 _ x j))
(transp0₂ m n))
∙ h n m p q x
where
help : (n m : ℕ) (x : _)
→ ((sym (transp0₂ m n))
∙∙ (λ j → subst coHomK (+'-comm (suc (suc m)) (suc n))
(Kn→ΩKn+1 (suc (suc (m + n))) x j))
∙∙ transp0₂ m n)
≡ Kn→ΩKn+1 (suc (n + suc m))
(subst coHomK (cong suc (+-comm (suc m) n)) x)
help zero m x =
sym (rUnit _)
∙∙ (λ k i → transp (λ i → coHomK (+'-comm (suc (suc m)) 1 (i ∨ k))) k
(Kn→ΩKn+1 _
(transp (λ i → coHomK (predℕ (+'-comm (suc (suc m)) 1 (i ∧ k)))) (~ k) x) i))
∙∙ cong (Kn→ΩKn+1 _)
λ k → subst coHomK (isSetℕ _ _ (cong predℕ (+'-comm (suc (suc m)) 1))
(cong suc (+-comm (suc m) zero)) k) x
help (suc n) m x =
sym (rUnit _)
∙∙ ((λ k i → transp (λ i → coHomK (+'-comm (suc (suc m)) (suc (suc n)) (i ∨ k))) k
(Kn→ΩKn+1 _
(transp (λ i → coHomK (predℕ (+'-comm (suc (suc m)) (suc (suc n)) (i ∧ k)))) (~ k) x) i)))
∙∙ cong (Kn→ΩKn+1 _)
(λ k → subst coHomK (isSetℕ _ _ (cong predℕ (+'-comm (suc (suc m)) (suc (suc n))))
(cong suc (+-comm (suc m) (suc n))) k) x)
h : (n m : ℕ) (p : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n))) (q : _) (x : _)
→ cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q)
(sym (transp0₂ m n)
∙∙ (λ j → subst coHomK (+'-comm (suc (suc m)) (suc n))
(Kn→ΩKn+1 (suc (suc (m + n))) x j))
∙∙ transp0₂ m n)
≡ Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x))
h n m (inl p) (inl q) x =
(λ k → cong (-ₖ'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr p) k) (inl q))
(help n m x k))
∙∙ ((λ k i → -ₖ'-gen-inl-right (suc n) (suc (suc m)) (inr p) q (help n m x i1 i) k))
∙∙ λ i → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-right (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inr p) (evenOrOdd (suc n)) i) q
(subst coHomK (cong suc (+-comm (suc m) n)) x) (~ i))
h n m (inl p) (inr q) x =
(λ k → cong (-ₖ'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr p) k) (inr q))
(help n m x k))
∙∙ cong-ₖ'-gen-inr (suc n) (suc (suc m)) p q (help n m x i1)
∙∙ sym (Kn→ΩKn+1-ₖ'' (suc n) (suc (suc m)) p q
(subst coHomK (λ i → suc (+-comm (suc m) n i)) x))
∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inr p) (evenOrOdd (suc n)) k) (inr q)
(subst coHomK (cong suc (+-comm (suc m) n)) x))
h n m (inr p) (inl q) x =
(λ k → cong (-ₖ'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl p) k) (inl q))
(help n m x k))
∙∙ (λ k i → -ₖ'-gen-inl-left (suc n) (suc (suc m)) p (inl q) (help n m x i1 i) k)
∙∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x) (~ k))
h n m (inr p) (inr q) x =
(λ k → cong (-ₖ'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl p) k) (inr q))
(help n m x k))
∙∙ (λ k i → -ₖ'-gen-inl-left (suc n) (suc (suc m)) p (inr q) (help n m x i1 i) k)
∙∙ cong (Kn→ΩKn+1 (suc (n + suc m)))
(sym (-ₖ'-gen-inl-left (suc n) (suc (suc m)) p (inr q)
(subst coHomK (λ i → suc (+-comm (suc m) n i)) x)))
∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inl p) (evenOrOdd (suc n)) k) (inr q)
(subst coHomK (cong suc (+-comm (suc m) n)) x))
lem₄ : (n m : ℕ) (q : _) (p : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n))) (a : _) (b : _)
→ cong (Kn→ΩKn+1 _) (((sym (cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))
∙∙ (λ j → -ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n))
(_⌣ₖ_ {n = suc (suc m)} {m = (suc n)} ∣ merid b j ∣ₕ ∣ a ∣)))
∙∙ cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))))
≡ cong (Kn→ΩKn+1 _) (Kn→ΩKn+1 _ (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_⌣ₖ_ {n = suc m} {m = (suc n)} ∣ b ∣ₕ ∣ a ∣))))
lem₄ n m q p a b = cong (cong (Kn→ΩKn+1 _)) (lem₃ n m q p (_⌣ₖ_ {n = suc m} {m = (suc n)} ∣ b ∣ₕ ∣ a ∣))
lem₅ : (n m : ℕ) (p : _) (q : _) (a : _) (b : _)
→ cong (cong (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
∘ (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))))
(ΩKn+1→Ω²Kn+2 (sym (cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m))
∙∙ (λ i → -ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_⌣ₖ_ {n = suc (suc n)} {m = suc m} ∣ merid a i ∣ₕ ∣ b ∣ₕ)))
∙∙ cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m)))
≡ Kn→Ω²Kn+2 (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m))) (_⌣ₖ_ {n = suc n} {m = suc m} ∣ a ∣ₕ ∣ b ∣ₕ))))
lem₅ n m p q a b =
cong (cong (cong (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
∘ (subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))))
(cong (sym (Kn→ΩKn+10ₖ _) ∙∙_∙∙ Kn→ΩKn+10ₖ _)
(lem₄ m n p q b a))
∙ help p q (_⌣ₖ_ {n = suc n} {m = suc m} ∣ a ∣ ∣ b ∣)
where
annoying : (x : _)
→ cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))
(Kn→Ω²Kn+2 (subst coHomK (cong suc (+-comm (suc n) m)) x))
≡ Kn→Ω²Kn+2 (subst coHomK (cong suc (sym (+-suc n m))) x)
annoying x =
((λ k → transp (λ i → refl {x = 0ₖ ((+'-comm (suc (suc m)) (suc (suc n))) (i ∨ ~ k))}
≡ refl {x = 0ₖ ((+'-comm (suc (suc m)) (suc (suc n))) (i ∨ ~ k))}) (~ k)
λ i j → transp (λ i → coHomK (+'-comm (suc (suc m)) (suc (suc n)) (i ∧ ~ k))) k
(Kn→Ω²Kn+2 (subst coHomK (cong suc (+-comm (suc n) m)) x) i j)))
∙∙ cong (transport (λ i → refl {x = 0ₖ ((+'-comm (suc (suc m)) (suc (suc n))) i)}
≡ refl {x = 0ₖ ((+'-comm (suc (suc m)) (suc (suc n))) i)}))
(natTranspLem {A = coHomK} x (λ _ → Kn→Ω²Kn+2) (cong suc (+-comm (suc n) m)))
∙∙ sym (substComposite (λ n → refl {x = 0ₖ n} ≡ refl {x = 0ₖ n})
(cong (suc ∘ suc ∘ suc) (+-comm (suc n) m)) (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→Ω²Kn+2 x))
∙∙ (λ k → subst (λ n → refl {x = 0ₖ n} ≡ refl {x = 0ₖ n})
(isSetℕ _ _
(cong (suc ∘ suc ∘ suc) (+-comm (suc n) m) ∙ (+'-comm (suc (suc m)) (suc (suc n))))
(cong (suc ∘ suc ∘ suc) (sym (+-suc n m))) k)
(Kn→Ω²Kn+2 x))
∙∙ sym (natTranspLem {A = coHomK} x (λ _ → Kn→Ω²Kn+2) (cong suc (sym (+-suc n m))))
help : (p : _) (q : _) (x : _) →
cong (cong (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
∘ subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))
(Kn→Ω²Kn+2 (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (+-comm (suc n) m)) x)))
≡ Kn→Ω²Kn+2
(-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m))) x)))
help (inl x) (inl y) z =
(λ k i j →
-ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
((ΩKn+1→Ω²Kn+2
(Kn→ΩKn+1 _ (-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (+-comm (suc n) m)) z) k))) i j)) k)
∙∙ annoying z
∙∙ cong Kn→Ω²Kn+2
λ k → (-ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (sym (+-suc n m))) z) (~ k)) (~ k))
help (inl x) (inr y) z =
(λ k i j →
-ₖ'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→Ω²Kn+2 (-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (+-comm (suc n) m)) z) k) i j)) k)
∙∙ annoying z
∙∙ cong Kn→Ω²Kn+2
(λ k → (-ₖ'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (sym (+-suc n m))) z) (~ k)) (~ k)))
help (inr x) (inl y) z =
(λ k i j → -ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→Ω²Kn+2
(-ₖ'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inr y) k) (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z)) i j)) k)
∙∙ cong (cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))) ∘ ΩKn+1→Ω²Kn+2)
(Kn→ΩKn+1-ₖ'' (suc m) (suc (suc n)) y x
(subst coHomK (cong suc (+-comm (suc n) m)) z))
∙∙ cong sym (annoying z)
∙∙ cong ΩKn+1→Ω²Kn+2 (sym (Kn→ΩKn+1-ₖ'' (suc m) (suc (suc n)) y x
(subst coHomK (cong suc (sym (+-suc n m))) z)))
∙∙ cong Kn→Ω²Kn+2 λ k → (-ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(-ₖ'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inr y) (~ k)) (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z)) (~ k))
help (inr x) (inr y) z =
(λ k → cong-cong-ₖ'-gen-inr (suc (suc n)) (suc (suc m)) x y
(λ i j → subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→Ω²Kn+2
(-ₖ'-gen (suc m) (suc (suc n))
(isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inl y) k) (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z)) i j)) k)
∙∙ cong (sym ∘ cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))) ∘ Kn→Ω²Kn+2)
(-ₖ'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z))
∙∙ cong sym (annoying z)
∙∙ cong (sym ∘ Kn→Ω²Kn+2)
(sym (-ₖ'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z)))
∙∙ cong ΩKn+1→Ω²Kn+2
λ k → (Kn→ΩKn+1-ₖ'' (suc (suc n)) (suc (suc m)) x y
(-ₖ'-gen (suc m) (suc (suc n)) (
isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inl y) (~ k)) (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z))) (~ k)
lem₆ : (n m : ℕ) (p : _) (q : _) (a : _) (b : _)
→ flipSquare
(Kn→Ω²Kn+2 (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_⌣ₖ_ {n = suc m} {m = (suc n)} ∣ b ∣ₕ ∣ a ∣))))
≡ Kn→Ω²Kn+2
(-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(-ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) (suc n)) (∣ b ∣ₕ ⌣ₖ ∣ a ∣ₕ))))))
lem₆ n m p q a b =
sym (sym≡flipSquare (Kn→Ω²Kn+2 (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_⌣ₖ_ {n = suc m} {m = (suc n)} ∣ b ∣ₕ ∣ a ∣)))))
∙ cong ΩKn+1→Ω²Kn+2
(help₁
(subst coHomK (cong suc (+-comm (suc m) n))
(_⌣ₖ_ {n = suc m} {m = (suc n)} ∣ b ∣ ∣ a ∣)) p q
∙ cong (Kn→ΩKn+1 _) (sym (help₂ (∣ b ∣ ⌣ₖ ∣ a ∣))))
where
help₁ : (x : _) (p : _) (q : _)
→ sym (Kn→ΩKn+1 (suc (n + suc m)) (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q x))
≡ Kn→ΩKn+1 (suc (n + suc m)) ((-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(-ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m)) x))))
help₁ z (inl x) (inl y) =
cong (λ x → sym (Kn→ΩKn+1 (suc (n + suc m)) x))
(-ₖ'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) y z)
∙∙ sym (Kn→ΩKn+1-ₖ'' (suc n) (suc m) x y z)
∙∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(-ₖ'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inr x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) z) (~ k)) (~ k))
help₁ z (inl x) (inr y) =
(λ k → sym (Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr x) k) (inr y) z)))
∙∙ cong sym (Kn→ΩKn+1-ₖ'' (suc n) (suc (suc m)) x y z)
∙∙ cong (Kn→ΩKn+1 (suc (n + suc m))) (sym (-ₖ'-gen-inl-right (suc n) (suc m) (inr x) y z))
∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(-ₖ'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(-ₖ'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inr x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) z) (~ k)) (~ k))
help₁ z (inr x) (inl y) =
cong (λ x → sym (Kn→ΩKn+1 (suc (n + suc m)) x))
(-ₖ'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) y z)
∙∙ (λ k → Kn→ΩKn+1-ₖ'' (suc m) (suc (suc n)) y x
(-ₖ'-gen-inl-left (suc n) (suc m) x (inr y) z (~ k)) (~ k))
∙∙ λ k → Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(-ₖ'-gen (suc m) (suc (suc n))
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) (inr x)
(-ₖ'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inl x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) z)) (~ k))
help₁ z (inr x) (inr y) =
((λ k → sym (Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl x) k) (inr y) z))))
∙∙ cong sym (cong (Kn→ΩKn+1 (suc (n + suc m))) (-ₖ'-gen-inl-left (suc n) (suc (suc m)) x (inr y) z))
∙∙ (λ k → sym (Kn→ΩKn+1 (suc (n + suc m))
(-ₖ'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(-ₖ'-gen-inl-right (suc n) (suc m) (inl x) y z (~ k)) (~ k))))
∙ λ k → Kn→ΩKn+1-ₖ'' (suc (suc n)) (suc (suc m)) x y
(-ₖ'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) (inr x)
(-ₖ'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inl x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) z)) (~ k)
help₂ : (x : _) →
(-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(-ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) (suc n)) x)))))
≡ -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(-ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (cong suc (+-comm (suc m) n)) x)))
help₂ x =
(λ k → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(transp (λ i → coHomK ((cong suc (sym (+-suc n m))) (i ∨ k))) k
(-ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(transp (λ i → coHomK ((cong suc (sym (+-suc n m))) (i ∧ k))) (~ k)
(subst coHomK (+'-comm (suc m) (suc n)) x))))))
∙ cong (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
∘ -ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
∘ -ₖ'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m)))
(sym (substComposite coHomK (+'-comm (suc m) (suc n)) ((cong suc (sym (+-suc n m)))) x)
∙ λ k → subst coHomK (isSetℕ _ _ (+'-comm (suc m) (suc n) ∙ cong suc (sym (+-suc n m)))
((cong suc (+-comm (suc m) n))) k) x)
lem₇ : (n : ℕ) (a : _) (p : _) (q : _)
→ ((λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(transp0₁ n (~ i))))
∙∙ (λ i j → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc n)) (Kn→ΩKn+1 (suc n) ∣ a ∣ₕ i))) j)
∙∙ (λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(transp0₁ n i))))
≡ (cong (Kn→ΩKn+1 (suc (suc (n + zero))))
(sym (transp0₁ n)
∙∙ sym (cong (subst coHomK (+'-comm (suc zero) (suc n)))
(cong (-ₖ'-gen (suc (suc n)) (suc zero) p q) (Kn→ΩKn+1 (suc n) ∣ a ∣ₕ)))
∙∙ transp0₁ n))
lem₇ zero a (inl x) (inr tt) =
(λ k → rUnit ((cong (Kn→ΩKn+1 _) (cong-ₖ'-gen-inr (suc zero) (suc zero) tt tt
(λ i → (subst coHomK (+'-comm (suc zero) (suc zero))
(Kn→ΩKn+1 (suc zero) ∣ a ∣ₕ i))) k))) (~ k))
∙ λ k → ((cong (Kn→ΩKn+1 (suc (suc zero)))
(rUnit (λ i → subst coHomK (+'-comm (suc zero) (suc zero))
(-ₖ'-gen-inl-left (suc (suc zero)) (suc zero) tt (inr tt)
(Kn→ΩKn+1 (suc zero) ∣ a ∣ₕ (~ i)) (~ k))) k)))
lem₇ (suc n) a (inl x) (inr tt) =
((λ k → rUnit (cong (Kn→ΩKn+1 _)
(λ i → -ₖ'-gen (suc (suc n)) (suc zero)
(isPropEvenOrOdd n (evenOrOdd (suc (suc n))) (inr x) k) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ i)))) (~ k)))
∙∙ (((λ k → ((cong (Kn→ΩKn+1 _) (cong-ₖ'-gen-inr (suc (suc n)) (suc zero) x tt
(λ i → (subst coHomK (+'-comm (suc zero) (suc (suc n)))
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ i))) k))))))
∙∙ λ k → ((cong (Kn→ΩKn+1 (suc (suc (suc n + zero))))
(rUnit (λ i → subst coHomK (+'-comm (suc zero) (suc (suc n)))
(-ₖ'-gen-inl-left (suc (suc (suc n))) (suc zero) x (inr tt)
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ (~ i)) (~ k))) k)))
lem₇ (suc n) a (inr x) (inr tt) =
(λ k → rUnit (λ i j → Kn→ΩKn+1 _
(-ₖ'-gen (suc (suc n)) (suc zero)
(isPropEvenOrOdd (suc (suc n)) (evenOrOdd (suc (suc n))) (inl x) k) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ i))) j) (~ k))
∙∙ (λ k i j → Kn→ΩKn+1 _ (-ₖ'-gen-inl-left (suc (suc n)) (suc zero) x (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ i)) k) j)
∙∙ λ k → cong (Kn→ΩKn+1 _)
(rUnit (sym (cong (subst coHomK (+'-comm (suc zero) (suc (suc n))))
(cong-ₖ'-gen-inr (suc (suc (suc n))) (suc zero) x tt (Kn→ΩKn+1 (suc (suc n)) ∣ a ∣ₕ) (~ k)))) k)
-- ∣ a ∣ ⌣ₖ ∣ b ∣ ≡ -ₖ'ⁿ*ᵐ (∣ b ∣ ⌣ₖ ∣ a ∣) for n ≥ 1, m = 1
gradedComm'-elimCase-left : (n : ℕ) (p : _) (q : _) (a : S₊ (suc n)) (b : S¹) →
(_⌣ₖ_ {n = suc n} {m = (suc zero)} ∣ a ∣ₕ ∣ b ∣ₕ)
≡ (-ₖ'-gen (suc n) (suc zero) p q)
(subst coHomK (+'-comm (suc zero) (suc n))
(_⌣ₖ_ {n = suc zero} {m = suc n} ∣ b ∣ₕ ∣ a ∣ₕ))
gradedComm'-elimCase-left zero (inr tt) (inr tt) a b =
proof a b
∙ cong (-ₖ'-gen 1 1 (inr tt) (inr tt))
(sym (transportRefl ((_⌣ₖ_ {n = suc zero} {m = suc zero} ∣ b ∣ ∣ a ∣))))
where
help : flipSquare (ΩKn+1→Ω²Kn+2' (λ j i → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ loop i ∣ₕ ∣ loop j ∣ₕ)) ≡
cong (cong (-ₖ'-gen 1 1 (inr tt) (inr tt)))
(ΩKn+1→Ω²Kn+2' (λ i j → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ loop j ∣ₕ ∣ loop i ∣ₕ))
help = sym (sym≡flipSquare _)
∙ sym (cong-cong-ₖ'-gen-inr 1 1 tt tt
(ΩKn+1→Ω²Kn+2' (λ i j → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ loop j ∣ ∣ loop i ∣)))
proof : (a b : S¹) → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ a ∣ₕ ∣ b ∣ₕ ≡
-ₖ'-gen 1 1 (inr tt) (inr tt) (_⌣ₖ_ {n = suc zero} {m = suc zero} ∣ b ∣ ∣ a ∣)
proof base base = refl
proof base (loop i) k = -ₖ'-gen 1 1 (inr tt) (inr tt) (Kn→ΩKn+10ₖ _ (~ k) i)
proof (loop i) base k = Kn→ΩKn+10ₖ _ k i
proof (loop i) (loop j) k =
hcomp (λ r → λ { (i = i0) → -ₖ'-gen 1 1 (inr tt) (inr tt) (Kn→ΩKn+10ₖ _ (~ k ∨ ~ r) j)
; (i = i1) → -ₖ'-gen 1 1 (inr tt) (inr tt) (Kn→ΩKn+10ₖ _ (~ k ∨ ~ r) j)
; (j = i0) → Kn→ΩKn+10ₖ _ (k ∨ ~ r) i
; (j = i1) → Kn→ΩKn+10ₖ _ (k ∨ ~ r) i
; (k = i0) → doubleCompPath-filler
(sym (Kn→ΩKn+10ₖ _))
(λ j i → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ loop i ∣ₕ ∣ loop j ∣ₕ)
(Kn→ΩKn+10ₖ _) (~ r) j i
; (k = i1) → (-ₖ'-gen 1 1 (inr tt) (inr tt)
(doubleCompPath-filler
(sym (Kn→ΩKn+10ₖ _))
(λ i j → _⌣ₖ_ {n = suc zero} {m = suc zero} ∣ loop j ∣ₕ ∣ loop i ∣ₕ)
(Kn→ΩKn+10ₖ _) (~ r) i j))})
(help k i j)
gradedComm'-elimCase-left (suc n) p q north b =
cong (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
(sym (⌣ₖ-0ₖ _ (suc (suc n)) ∣ b ∣ₕ))
gradedComm'-elimCase-left (suc n) p q south b =
cong (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
((sym (⌣ₖ-0ₖ _ (suc (suc n)) ∣ b ∣ₕ)) ∙ λ i → ∣ b ∣ ⌣ₖ ∣ merid (ptSn (suc n)) i ∣ₕ)
gradedComm'-elimCase-left (suc n) p q (merid a i) base k =
hcomp (λ j → λ {(i = i0) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
(0ₖ _)
; (i = i1) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
(compPath-filler (sym (⌣ₖ-0ₖ _ (suc (suc n)) ∣ base ∣ₕ))
(λ i → ∣ base ∣ ⌣ₖ ∣ merid a i ∣ₕ) j k)
; (k = i0) → _⌣ₖ_ {n = suc (suc n)} {m = suc zero} ∣ merid a i ∣ₕ ∣ base ∣ₕ
; (k = i1) → -ₖ'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(∣ base ∣ₕ ⌣ₖ ∣ merid a i ∣ₕ))})
(hcomp (λ j → λ {(i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (k = i0) → (sym (Kn→ΩKn+10ₖ _)
∙ (λ j → Kn→ΩKn+1 _
(sym (gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base
∙ cong (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)) (transp0₁ n)) j))) j i
; (k = i1) → ∣ north ∣})
∣ north ∣)
gradedComm'-elimCase-left (suc n) p q (merid a i) (loop j) k =
hcomp (λ r →
λ { (i = i0) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
(sym (⌣ₖ-0ₖ _ (suc (suc n)) ∣ (loop j) ∣ₕ) k)
; (i = i1) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
(compPath-filler (sym (⌣ₖ-0ₖ _ (suc (suc n)) ∣ (loop j) ∣ₕ))
(λ i → ∣ loop j ∣ ⌣ₖ ∣ merid (ptSn (suc n)) i ∣ₕ) r k)
; (k = i0) → _⌣ₖ_ {n = suc (suc n)} {m = suc zero} ∣ merid a i ∣ₕ ∣ loop j ∣ₕ
; (k = i1) → -ₖ'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_⌣ₖ_ {n = suc zero} {m = suc (suc n)} ∣ loop j ∣ₕ
∣ compPath-filler (merid a) (sym (merid (ptSn (suc n)))) (~ r) i ∣ₕ))})
(hcomp (λ r →
λ { (i = i0) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
∣ rCancel (merid (ptSn (suc (suc n)))) (~ k ∧ r) j ∣
; (i = i1) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
(subst coHomK (+'-comm 1 (suc (suc n)))))
∣ rCancel (merid (ptSn (suc (suc n)))) (~ k ∧ r) j ∣
; (k = i0) → help₂ r i j
; (k = i1) → -ₖ'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_⌣ₖ_ {n = suc zero} {m = suc (suc n)} ∣ loop j ∣ₕ
(Kn→ΩKn+1 _ ∣ a ∣ₕ i)))})
(-ₖ'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_⌣ₖ_ {n = suc zero} {m = suc (suc n)} ∣ loop j ∣ₕ
(Kn→ΩKn+1 _ ∣ a ∣ₕ i)))))
where
P : Path _ (Kn→ΩKn+1 (suc (suc (n + 0))) (0ₖ _))
(Kn→ΩKn+1 (suc (suc (n + 0))) (_⌣ₖ_ {n = (suc n)} {m = suc zero} ∣ a ∣ ∣ base ∣))
P i = Kn→ΩKn+1 (suc (suc (n + 0)))
((sym (gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base
∙ cong (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)) (transp0₁ n)) i))
help₁ : (P ∙∙ ((λ i j → _⌣ₖ_ {n = suc (suc n)} {m = suc zero} ∣ merid a j ∣ₕ ∣ loop i ∣ₕ)) ∙∙ sym P)
≡ ((λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0₁ n (~ i))))
∙∙ (λ i j → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm 1 (suc n)) (∣ loop i ∣ₕ ⌣ₖ ∣ a ∣ₕ))) j)
∙∙ (λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0₁ n i))))
help₁ k i j =
((λ i → (Kn→ΩKn+1 (suc (suc (n + 0))))
(compPath-filler'
((gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base))
(cong (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt))
(transp0₁ n)) (~ k) (~ i)))
∙∙ (λ i j → (Kn→ΩKn+1 _
(gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a (loop i) k) j))
∙∙ λ i → (Kn→ΩKn+1 (suc (suc (n + 0))))
(compPath-filler'
((gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base))
(cong (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt))
(transp0₁ n)) (~ k) i)) i j
help₂ : I → I → I → coHomK _
help₂ r i j =
hcomp (λ k →
λ { (i = i0) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
subst coHomK (+'-comm 1 (suc (suc n))))
∣ rCancel (merid (ptSn (suc (suc n)))) (~ k ∨ r) j ∣
; (i = i1) → (-ₖ'-gen (suc (suc n)) 1 p q ∘
subst coHomK (+'-comm 1 (suc (suc n))))
∣ rCancel (merid (ptSn (suc (suc n)))) (~ k ∨ r) j ∣
; (j = i0) → compPath-filler (sym (Kn→ΩKn+10ₖ (suc (suc (n + 0)))))
P k r i
; (j = i1) → compPath-filler (sym (Kn→ΩKn+10ₖ (suc (suc (n + 0)))))
P k r i
; (r = i0) → -ₖ'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(doubleCompPath-filler (sym (Kn→ΩKn+10ₖ _))
(λ i j → _⌣ₖ_ {n = suc zero} {m = suc (suc n)} ∣ loop j ∣ₕ
(Kn→ΩKn+1 (suc n) ∣ a ∣ₕ i)) (Kn→ΩKn+10ₖ _) (~ k) i j))
; (r = i1) → doubleCompPath-filler P
(λ i j → _⌣ₖ_ {n = suc (suc n)} {m = suc zero} ∣ merid a j ∣ₕ ∣ loop i ∣ₕ)
(sym P) (~ k) j i})
(hcomp (λ k →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → (Kn→ΩKn+10ₖ (suc (suc (n + 0)))) (~ r) i
; (j = i1) → (Kn→ΩKn+10ₖ (suc (suc (n + 0)))) (~ r) i
; (r = i0) → lem₂ n a p q (~ k) i j
; (r = i1) → help₁ (~ k) j i})
(hcomp (λ k →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → Kn→ΩKn+10ₖ (suc (suc (n + 0))) (~ r) i
; (j = i1) → Kn→ΩKn+10ₖ (suc (suc (n + 0))) (~ r) i
; (r = i0) → flipSquare≡cong-sym (flipSquare (ΩKn+1→Ω²Kn+2
(sym (transp0₁ n)
∙∙ cong (subst coHomK (+'-comm 1 (suc n)))
(cong (-ₖ'-gen (suc (suc n)) 1 p q)
(Kn→ΩKn+1 (suc n) ∣ a ∣ₕ))
∙∙ transp0₁ n))) (~ k) i j
; (r = i1) → ((λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0₁ n (~ i))))
∙∙ (λ i j → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm 1 (suc n)) (Kn→ΩKn+1 (suc n) ∣ a ∣ₕ i))) j)
∙∙ (λ i → Kn→ΩKn+1 _ (-ₖ'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0₁ n i)))) j i})
(hcomp (λ k →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → Kn→ΩKn+10ₖ (suc (suc (n + 0))) (~ r ∧ k) i
; (j = i1) → Kn→ΩKn+10ₖ (suc (suc (n + 0))) (~ r ∧ k) i
; (r = i0) → doubleCompPath-filler
(sym (Kn→ΩKn+10ₖ _))
(cong (Kn→ΩKn+1 (suc (suc (n + 0))))
(sym (transp0₁ n)
∙∙ sym (cong (subst coHomK (+'-comm 1 (suc n)))
(cong (-ₖ'-gen (suc (suc n)) 1 p q)
(Kn→ΩKn+1 (suc n) ∣ a ∣ₕ)))
∙∙ transp0₁ n))
(Kn→ΩKn+10ₖ _) k j i
; (r = i1) → lem₇ n a p q (~ k) j i})
(lem₇ n a p q i1 j i))))
-- ∣ a ∣ ⌣ₖ ∣ b ∣ ≡ -ₖ'ⁿ*ᵐ (∣ b ∣ ⌣ₖ ∣ a ∣) for all n, m ≥ 1
gradedComm'-elimCase : (k n m : ℕ) (term : n + m ≡ k) (p : _) (q : _) (a : _) (b : _) →
(_⌣ₖ_ {n = suc n} {m = (suc m)} ∣ a ∣ₕ ∣ b ∣ₕ)
≡ (-ₖ'-gen (suc n) (suc m) p q)
(subst coHomK (+'-comm (suc m) (suc n))
(_⌣ₖ_ {n = suc m} {m = suc n} ∣ b ∣ₕ ∣ a ∣ₕ))
gradedComm'-elimCase k zero zero term p q a b = gradedComm'-elimCase-left zero p q a b
gradedComm'-elimCase k zero (suc m) term (inr tt) q a b =
help q
∙ sym (cong (-ₖ'-gen 1 (suc (suc m)) (inr tt) q
∘ (subst coHomK (+'-comm (suc (suc m)) 1)))
(gradedComm'-elimCase-left (suc m) q (inr tt) b a))
where
help : (q : _) → ∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ ≡
-ₖ'-gen 1 (suc (suc m)) (inr tt) q
(subst coHomK (+'-comm (suc (suc m)) 1)
(-ₖ'-gen (suc (suc m)) 1 q (inr tt)
(subst coHomK (+'-comm 1 (suc (suc m))) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ))))
help (inl x) =
(sym (transportRefl _)
∙ (λ i → subst coHomK (isSetℕ _ _ refl (+'-comm 1 (suc (suc m)) ∙ +'-comm (suc (suc m)) 1) i)
(∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ)))
∙∙ substComposite coHomK
(+'-comm 1 (suc (suc m)))
(+'-comm (suc (suc m)) 1)
((∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ))
∙∙ λ i → -ₖ'-gen-inl-right (suc zero) (suc (suc m)) (inr tt) x
((subst coHomK (+'-comm (suc (suc m)) 1)
(-ₖ'-gen-inl-left (suc (suc m)) 1 x (inr tt)
(subst coHomK (+'-comm 1 (suc (suc m))) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ)) (~ i)))) (~ i)
help (inr x) =
(sym (transportRefl _)
∙∙ (λ k → subst coHomK (isSetℕ _ _ refl (+'-comm 1 (suc (suc m)) ∙ +'-comm (suc (suc m)) 1) k) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ))
∙∙ sym (-ₖ^2 (subst coHomK (+'-comm 1 (suc (suc m)) ∙ +'-comm (suc (suc m)) 1) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ))))
∙∙ (λ i → -ₖ'-gen-inr≡-ₖ' 1 (suc (suc m)) tt x
(-ₖ'-gen-inr≡-ₖ' (suc (suc m)) 1 x tt
(substComposite coHomK (+'-comm 1 (suc (suc m))) (+'-comm (suc (suc m)) 1) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ) i)
(~ i)) (~ i))
∙∙ λ i → (-ₖ'-gen 1 (suc (suc m)) (inr tt) (inr x)
(transp (λ j → coHomK ((+'-comm (suc (suc m)) 1) (j ∨ ~ i))) (~ i)
(-ₖ'-gen (suc (suc m)) 1 (inr x) (inr tt)
(transp (λ j → coHomK ((+'-comm (suc (suc m)) 1) (j ∧ ~ i))) i
((subst coHomK (+'-comm 1 (suc (suc m))) (∣ a ∣ₕ ⌣ₖ ∣ b ∣ₕ)))))))
gradedComm'-elimCase k (suc n) zero term p q a b =
gradedComm'-elimCase-left (suc n) p q a b
gradedComm'-elimCase zero (suc n) (suc m) term p q a b =
⊥.rec (snotz (sym (+-suc n m) ∙ cong predℕ term))
gradedComm'-elimCase (suc zero) (suc n) (suc m) term p q a b =
⊥.rec (snotz (sym (+-suc n m) ∙ cong predℕ term))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north north = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north south = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north (merid a i) r =
-ₖ'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((sym (Kn→ΩKn+10ₖ _)
∙ cong (Kn→ΩKn+1 _)
(cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0₂ n m))
∙ sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p a north))) r i))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south north = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south south = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south (merid a i) r =
-ₖ'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((sym (Kn→ΩKn+10ₖ _)
∙ cong (Kn→ΩKn+1 _)
(cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0₂ n m))
∙ sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p a south))) r i))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) north r =
(cong (Kn→ΩKn+1 (suc (suc (n + suc m))))
(gradedComm'-elimCase (suc k) n (suc m) (cong predℕ term) (evenOrOdd (suc n)) q a north
∙ cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))
∙' Kn→ΩKn+10ₖ _) r i
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) south r =
(cong (Kn→ΩKn+1 (suc (suc (n + suc m))))
(gradedComm'-elimCase (suc k) n (suc m) (cong predℕ term) (evenOrOdd (suc n)) q a south
∙ cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n))
∙' Kn→ΩKn+10ₖ _) r i
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) (merid b j) r =
hcomp (λ l →
λ { (i = i0) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((compPath-filler (sym (Kn→ΩKn+10ₖ _))
(cong (Kn→ΩKn+1 _)
(cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0₂ n m))
∙ sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p b north))) l r j)))
; (i = i1) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((compPath-filler (sym (Kn→ΩKn+10ₖ _))
(cong (Kn→ΩKn+1 _)
(cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0₂ n m))
∙ sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p b south))) l r j)))
; (r = i0) → help₂ l i j
; (r = i1) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(help₁ l i j))})
(hcomp (λ l →
λ { (i = i0) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→ΩKn+10ₖ _ (~ r ∨ ~ l) j))
; (i = i1) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(Kn→ΩKn+10ₖ _ (~ r ∨ ~ l) j))
; (j = i0) → Kn→ΩKn+10ₖ _ r i
; (j = i1) → Kn→ΩKn+10ₖ _ r i
; (r = i0) → lem₄ n m q p a b (~ l) j i
; (r = i1) → -ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(doubleCompPath-filler
(sym (Kn→ΩKn+10ₖ _))
(λ i j → Kn→ΩKn+1 _ ((sym (cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m))
∙∙ (λ i → -ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_⌣ₖ_ {n = suc (suc n)} {m = suc m} ∣ merid a i ∣ₕ ∣ b ∣ₕ)))
∙∙ cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m)) i) j)
(Kn→ΩKn+10ₖ _) (~ l) i j))})
(hcomp (λ l →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → Kn→ΩKn+10ₖ _ r i
; (j = i1) → Kn→ΩKn+10ₖ _ r i
; (r = i0) → lem₄ n m q p a b i1 j i
; (r = i1) → lem₅ n m p q a b (~ l) i j})
(hcomp (λ l →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → Kn→ΩKn+10ₖ _ (r ∨ ~ l) i
; (j = i1) → Kn→ΩKn+10ₖ _ (r ∨ ~ l) i
; (r = i0) → doubleCompPath-filler
(sym (Kn→ΩKn+10ₖ _))
(lem₄ n m q p a b i1)
(Kn→ΩKn+10ₖ _) (~ l) j i
; (r = i1) → Kn→Ω²Kn+2 (-ₖ'-gen (suc (suc n)) (suc (suc m)) p q
(-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(gradedComm'-elimCase k n m
(+-comm n m ∙∙ cong predℕ (+-comm (suc m) n) ∙∙ cong (predℕ ∘ predℕ) term)
(evenOrOdd (suc n)) (evenOrOdd (suc m)) a b (~ l))))) i j})
(lem₆ n m p q a b r i j))))
where
help₁ : I → I → I → coHomK _
help₁ l i j =
Kn→ΩKn+1 _
(hcomp (λ r
→ λ { (i = i0) → compPath-filler' (cong ((-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p)) (sym (transp0₂ n m)))
(sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p b north)) r l
; (i = i1) → compPath-filler' (cong ((-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p)) (sym (transp0₂ n m)))
(sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p b south)) r l
; (l = i0) → doubleCompPath-filler (sym (cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m)))
(λ i → -ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_⌣ₖ_ {n = suc (suc n)} {m = suc m} ∣ merid a i ∣ₕ ∣ b ∣ₕ)))
(cong (-ₖ'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0₂ n m)) r i
; (l = i1) → _⌣ₖ_ {n = suc m} {m = suc (suc n)} ∣ b ∣ₕ ∣ merid a i ∣ₕ})
(gradedComm'-elimCase (suc k) m (suc n) (+-suc m n ∙ +-comm (suc m) n ∙ cong predℕ term)
(evenOrOdd (suc m)) p b (merid a i) (~ l))) j
help₂ : I → I → I → coHomK _
help₂ l i j =
hcomp (λ r →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) →
Kn→ΩKn+1 (suc (suc (n + suc m)))
(compPath-filler (gradedComm'-elimCase (suc k) n (suc m)
(cong predℕ term) (evenOrOdd (suc n)) q a north)
(cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n)) r (~ l)) i
; (j = i1) →
Kn→ΩKn+1 (suc (suc (n + suc m)))
(compPath-filler (gradedComm'-elimCase (suc k) n (suc m)
(cong predℕ term) (evenOrOdd (suc n)) q a south)
(cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n)) r (~ l)) i
; (l = i0) →
Kn→ΩKn+1 _
(doubleCompPath-filler (sym (cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n)))
(λ j → -ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n))
(_⌣ₖ_ {n = suc (suc m)} {m = (suc n)} ∣ merid b j ∣ₕ ∣ a ∣)))
(cong (-ₖ'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0₂ m n)) r j) i
; (l = i1) → Kn→ΩKn+1 _ (_⌣ₖ_ {n = (suc n)} {m = suc (suc m)} ∣ a ∣ ∣ merid b j ∣ₕ) i})
(hcomp (λ r →
λ { (i = i0) → ∣ north ∣
; (i = i1) → ∣ north ∣
; (j = i0) → Kn→ΩKn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predℕ term) (evenOrOdd (suc n)) q a north (~ l ∨ ~ r)) i
; (j = i1) → Kn→ΩKn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predℕ term) (evenOrOdd (suc n)) q a south (~ l ∨ ~ r)) i
; (l = i0) → Kn→ΩKn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predℕ term) (evenOrOdd (suc n)) q a (merid b j) i1) i
; (l = i1) → Kn→ΩKn+1 _ (gradedComm'-elimCase (suc k) n (suc m) (cong predℕ term)
(evenOrOdd (suc n)) q a (merid b j) (~ r)) i})
(Kn→ΩKn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m) (cong predℕ term)
(evenOrOdd (suc n)) q a (merid b j) i1) i))
private
coherence-transp : (n m : ℕ) (p : _) (q : _)
→ -ₖ'-gen (suc n) (suc m) p q
(subst coHomK (+'-comm (suc m) (suc n)) (0ₖ (suc m +' suc n))) ≡ 0ₖ _
coherence-transp zero zero p q = refl
coherence-transp zero (suc m) p q = refl
coherence-transp (suc n) zero p q = refl
coherence-transp (suc n) (suc m) p q = refl
gradedComm'-⌣ₖ∙ : (n m : ℕ) (p : _) (q : _) (a : _)
→ ⌣ₖ∙ (suc n) (suc m) a
≡ ((λ b → -ₖ'-gen (suc n) (suc m) p q (subst coHomK (+'-comm (suc m) (suc n)) (b ⌣ₖ a)))
, (cong (-ₖ'-gen (suc n) (suc m) p q)
(cong (subst coHomK (+'-comm (suc m) (suc n)))
(0ₖ-⌣ₖ (suc m) (suc n) a))
∙ coherence-transp n m p q))
gradedComm'-⌣ₖ∙ n m p q =
T.elim (λ _ → isOfHLevelPath (3 + n) ((isOfHLevel↑∙ (suc n) m)) _ _)
λ a → →∙Homogeneous≡ (isHomogeneousKn _) (funExt λ b → funExt⁻ (cong fst (f₁≡f₂ b)) a)
where
f₁ : coHomK (suc m) → S₊∙ (suc n) →∙ coHomK-ptd (suc n +' suc m)
fst (f₁ b) a = _⌣ₖ_ {n = suc n} {m = suc m} ∣ a ∣ₕ b
snd (f₁ b) = 0ₖ-⌣ₖ (suc n) (suc m) b
f₂ : coHomK (suc m) → S₊∙ (suc n) →∙ coHomK-ptd (suc n +' suc m)
fst (f₂ b) a =
-ₖ'-gen (suc n) (suc m) p q (subst coHomK (+'-comm (suc m) (suc n))
(_⌣ₖ_ {n = suc m} {m = suc n} b ∣ a ∣ₕ))
snd (f₂ b) =
(cong (-ₖ'-gen (suc n) (suc m) p q)
(cong (subst coHomK (+'-comm (suc m) (suc n)))
(⌣ₖ-0ₖ (suc m) (suc n) b))
∙ coherence-transp n m p q)
f₁≡f₂ : (b : _) → f₁ b ≡ f₂ b
f₁≡f₂ =
T.elim (λ _ → isOfHLevelPath (3 + m)
(subst (isOfHLevel (3 + m))
(λ i → S₊∙ (suc n) →∙ coHomK-ptd (+'-comm (suc n) (suc m) (~ i)))
(isOfHLevel↑∙' (suc m) n)) _ _)
λ b → →∙Homogeneous≡ (isHomogeneousKn _)
(funExt λ a → gradedComm'-elimCase (n + m) n m refl p q a b)
-- Finally, graded commutativity:
gradedComm'-⌣ₖ : (n m : ℕ) (a : coHomK n) (b : coHomK m)
→ a ⌣ₖ b ≡ (-ₖ'^ n · m) (subst coHomK (+'-comm m n) (b ⌣ₖ a))
gradedComm'-⌣ₖ zero zero a b = sym (transportRefl _) ∙ cong (transport refl) (comm-·₀ a b)
gradedComm'-⌣ₖ zero (suc m) a b =
sym (transportRefl _)
∙∙ (λ k → subst coHomK (isSetℕ _ _ refl (+'-comm (suc m) zero) k) (b ⌣ₖ a))
∙∙ sym (-ₖ'-gen-inl-left zero (suc m) tt (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) zero) (b ⌣ₖ a)))
gradedComm'-⌣ₖ (suc n) zero a b =
sym (transportRefl _)
∙∙ ((λ k → subst coHomK (isSetℕ _ _ refl (+'-comm zero (suc n)) k) (b ⌣ₖ a)))
∙∙ sym (-ₖ'-gen-inl-right (suc n) zero (evenOrOdd (suc n)) tt
(subst coHomK (+'-comm zero (suc n)) (b ⌣ₖ a)))
gradedComm'-⌣ₖ (suc n) (suc m) a b =
funExt⁻ (cong fst (gradedComm'-⌣ₖ∙ n m (evenOrOdd (suc n)) (evenOrOdd (suc m)) a)) b
gradedComm'-⌣ : {A : Type ℓ} (n m : ℕ) (a : coHom n A) (b : coHom m A)
→ a ⌣ b ≡ (-ₕ'^ n · m) (subst (λ n → coHom n A) (+'-comm m n) (b ⌣ a))
gradedComm'-⌣ n m =
ST.elim2 (λ _ _ → isOfHLevelPath 2 squash₂ _ _)
λ f g →
cong ∣_∣₂ (funExt (λ x →
gradedComm'-⌣ₖ n m (f x) (g x)
∙ cong ((-ₖ'^ n · m) ∘ (subst coHomK (+'-comm m n)))
λ i → g (transportRefl x (~ i)) ⌣ₖ f (transportRefl x (~ i))))
-----------------------------------------------------------------------------
-- The previous code introduces another - to facilitate proof
-- This a reformulation with the usual -ₕ' definition (the one of the ring) of the results
-ₕ^-gen : {k : ℕ} → {A : Type ℓ} → (n m : ℕ)
→ (p : isEvenT n ⊎ isOddT n)
→ (q : isEvenT m ⊎ isOddT m)
→ (a : coHom k A) → coHom k A
-ₕ^-gen n m (inl p) q a = a
-ₕ^-gen n m (inr p) (inl q) a = a
-ₕ^-gen n m (inr p) (inr q) a = -ₕ a
-ₕ^_·_ : {k : ℕ} → {A : Type ℓ} → (n m : ℕ) → (a : coHom k A) → coHom k A
-ₕ^_·_ n m a = -ₕ^-gen n m (evenOrOdd n) (evenOrOdd m) a
-ₕ^-gen-eq : ∀ {ℓ} {k : ℕ} {A : Type ℓ} (n m : ℕ)
→ (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
→ (x : coHom k A)
→ -ₕ^-gen n m p q x ≡ (ST.map λ f x → (-ₖ'-gen n m p q) (f x)) x
-ₕ^-gen-eq {k = k} n m (inl p) q = ST.elim (λ _ → isSetPathImplicit) λ f → cong ∣_∣₂ (funExt λ x → sym (-ₖ'-gen-inl-left n m p q (f x)))
-ₕ^-gen-eq {k = k} n m (inr p) (inl q) = ST.elim (λ _ → isSetPathImplicit) λ f → cong ∣_∣₂ (funExt λ z → sym (-ₖ'-gen-inl-right n m (inr p) q (f z)))
-ₕ^-gen-eq {k = k} n m (inr p) (inr q) = ST.elim (λ _ → isSetPathImplicit) λ f → cong ∣_∣₂ (funExt λ z → sym (-ₖ'-gen-inr≡-ₖ' n m p q (f z)))
-ₕ^-eq : ∀ {ℓ} {k : ℕ} {A : Type ℓ} (n m : ℕ) → (a : coHom k A)
→ (-ₕ^ n · m) a ≡ (-ₕ'^ n · m) a
-ₕ^-eq n m a = -ₕ^-gen-eq n m (evenOrOdd n) (evenOrOdd m) a
gradedComm-⌣ : ∀ {ℓ} {A : Type ℓ} (n m : ℕ) (a : coHom n A) (b : coHom m A)
→ a ⌣ b ≡ (-ₕ^ n · m) (subst (λ n → coHom n A) (+'-comm m n) (b ⌣ a))
gradedComm-⌣ n m a b = (gradedComm'-⌣ n m a b) ∙ (sym (-ₕ^-eq n m (subst (λ n₁ → coHom n₁ _) (+'-comm m n) (b ⌣ a))))
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Divisibility
------------------------------------------------------------------------
module Data.Nat.Divisibility where
open import Data.Nat as Nat
open import Data.Nat.DivMod
import Data.Nat.Properties as NatProp
open import Data.Fin as Fin using (Fin; zero; suc)
import Data.Fin.Properties as FP
open NatProp.SemiringSolver
open import Algebra
private
module CS = CommutativeSemiring NatProp.commutativeSemiring
open import Data.Product
open import Relation.Nullary
open import Relation.Binary
import Relation.Binary.PartialOrderReasoning as PartialOrderReasoning
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; _≢_; refl; sym; cong; subst)
open import Function
-- m ∣ n is inhabited iff m divides n. Some sources, like Hardy and
-- Wright's "An Introduction to the Theory of Numbers", require m to
-- be non-zero. However, some things become a bit nicer if m is
-- allowed to be zero. For instance, _∣_ becomes a partial order, and
-- the gcd of 0 and 0 becomes defined.
infix 4 _∣_
data _∣_ : ℕ → ℕ → Set where
divides : {m n : ℕ} (q : ℕ) (eq : n ≡ q * m) → m ∣ n
-- Extracts the quotient.
quotient : ∀ {m n} → m ∣ n → ℕ
quotient (divides q _) = q
-- If m divides n, and n is positive, then m ≤ n.
∣⇒≤ : ∀ {m n} → m ∣ suc n → m ≤ suc n
∣⇒≤ (divides zero ())
∣⇒≤ {m} {n} (divides (suc q) eq) = begin
m ≤⟨ NatProp.m≤m+n m (q * m) ⟩
suc q * m ≡⟨ sym eq ⟩
suc n ∎
where open ≤-Reasoning
-- _∣_ is a partial order.
poset : Poset _ _ _
poset = record
{ Carrier = ℕ
; _≈_ = _≡_
; _≤_ = _∣_
; isPartialOrder = record
{ isPreorder = record
{ isEquivalence = PropEq.isEquivalence
; reflexive = reflexive
; trans = trans
}
; antisym = antisym
}
}
where
module DTO = DecTotalOrder Nat.decTotalOrder
open PropEq.≡-Reasoning
reflexive : _≡_ ⇒ _∣_
reflexive {n} refl = divides 1 (sym $ proj₁ CS.*-identity n)
antisym : Antisymmetric _≡_ _∣_
antisym (divides {n = zero} q₁ eq₁) (divides {n = n₂} q₂ eq₂) = begin
n₂ ≡⟨ eq₂ ⟩
q₂ * 0 ≡⟨ CS.*-comm q₂ 0 ⟩
0 ∎
antisym (divides {n = n₁} q₁ eq₁) (divides {n = zero} q₂ eq₂) = begin
0 ≡⟨ CS.*-comm 0 q₁ ⟩
q₁ * 0 ≡⟨ sym eq₁ ⟩
n₁ ∎
antisym (divides {n = suc n₁} q₁ eq₁) (divides {n = suc n₂} q₂ eq₂) =
DTO.antisym (∣⇒≤ (divides q₁ eq₁)) (∣⇒≤ (divides q₂ eq₂))
trans : Transitive _∣_
trans (divides q₁ refl) (divides q₂ refl) =
divides (q₂ * q₁) (sym (CS.*-assoc q₂ q₁ _))
module ∣-Reasoning = PartialOrderReasoning poset
renaming (_≤⟨_⟩_ to _∣⟨_⟩_; _≈⟨_⟩_ to _≡⟨_⟩_)
private module P = Poset poset
-- 1 divides everything.
1∣_ : ∀ n → 1 ∣ n
1∣ n = divides n (sym $ proj₂ CS.*-identity n)
-- Everything divides 0.
_∣0 : ∀ n → n ∣ 0
n ∣0 = divides 0 refl
-- 0 only divides 0.
0∣⇒≡0 : ∀ {n} → 0 ∣ n → n ≡ 0
0∣⇒≡0 {n} 0∣n = P.antisym (n ∣0) 0∣n
-- Only 1 divides 1.
∣1⇒≡1 : ∀ {n} → n ∣ 1 → n ≡ 1
∣1⇒≡1 {n} n∣1 = P.antisym n∣1 (1∣ n)
-- If i divides m and n, then i divides their sum.
∣-+ : ∀ {i m n} → i ∣ m → i ∣ n → i ∣ m + n
∣-+ (divides {m = i} q refl) (divides q' refl) =
divides (q + q') (sym $ proj₂ CS.distrib i q q')
-- If i divides m and n, then i divides their difference.
∣-∸ : ∀ {i m n} → i ∣ m + n → i ∣ m → i ∣ n
∣-∸ (divides {m = i} q' eq) (divides q refl) =
divides (q' ∸ q)
(sym $ NatProp.im≡jm+n⇒[i∸j]m≡n q' q i _ $ sym eq)
-- A simple lemma: n divides kn.
∣-* : ∀ k {n} → n ∣ k * n
∣-* k = divides k refl
-- If i divides j, then ki divides kj.
*-cong : ∀ {i j} k → i ∣ j → k * i ∣ k * j
*-cong {i} {j} k (divides q eq) = divides q lemma
where
open PropEq.≡-Reasoning
lemma = begin
k * j ≡⟨ cong (_*_ k) eq ⟩
k * (q * i) ≡⟨ solve 3 (λ k q i → k :* (q :* i)
:= q :* (k :* i))
refl k q i ⟩
q * (k * i) ∎
-- If ki divides kj, and k is positive, then i divides j.
/-cong : ∀ {i j} k → suc k * i ∣ suc k * j → i ∣ j
/-cong {i} {j} k (divides q eq) = divides q lemma
where
open PropEq.≡-Reasoning
k′ = suc k
lemma = NatProp.cancel-*-right j (q * i) (begin
j * k′ ≡⟨ CS.*-comm j k′ ⟩
k′ * j ≡⟨ eq ⟩
q * (k′ * i) ≡⟨ solve 3 (λ q k i → q :* (k :* i)
:= q :* i :* k)
refl q k′ i ⟩
q * i * k′ ∎)
-- If the remainder after division is non-zero, then the divisor does
-- not divide the dividend.
nonZeroDivisor-lemma
: ∀ m q (r : Fin (1 + m)) → Fin.toℕ r ≢ 0 →
¬ (1 + m) ∣ (Fin.toℕ r + q * (1 + m))
nonZeroDivisor-lemma m zero r r≢zero (divides zero eq) = r≢zero $ begin
Fin.toℕ r
≡⟨ sym $ proj₁ CS.*-identity (Fin.toℕ r) ⟩
1 * Fin.toℕ r
≡⟨ eq ⟩
0
∎
where open PropEq.≡-Reasoning
nonZeroDivisor-lemma m zero r r≢zero (divides (suc q) eq) =
NatProp.¬i+1+j≤i m $ begin
m + suc (q * suc m)
≡⟨ solve 2 (λ m q → m :+ (con 1 :+ q) := con 1 :+ m :+ q)
refl m (q * suc m) ⟩
suc (m + q * suc m)
≡⟨ sym eq ⟩
1 * Fin.toℕ r
≡⟨ proj₁ CS.*-identity (Fin.toℕ r) ⟩
Fin.toℕ r
≤⟨ ≤-pred $ FP.bounded r ⟩
m
∎
where open ≤-Reasoning
nonZeroDivisor-lemma m (suc q) r r≢zero d =
nonZeroDivisor-lemma m q r r≢zero (∣-∸ d' P.refl)
where
lem = solve 3 (λ m r q → r :+ (m :+ q) := m :+ (r :+ q))
refl (suc m) (Fin.toℕ r) (q * suc m)
d' = subst (λ x → (1 + m) ∣ x) lem d
-- Divisibility is decidable.
_∣?_ : Decidable _∣_
zero ∣? zero = yes (0 ∣0)
zero ∣? suc n = no ((λ ()) ∘′ 0∣⇒≡0)
suc m ∣? n with n divMod suc m
suc m ∣? .(q * suc m) | result q zero refl =
yes $ divides q refl
suc m ∣? .(1 + Fin.toℕ r + q * suc m) | result q (suc r) refl =
no $ nonZeroDivisor-lemma m q (suc r) (λ())
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------------------------------------------------------------------------
-- An alternative (non-standard) classical definition of weak
-- bisimilarity
------------------------------------------------------------------------
-- This definition is based on the function "wb" in Section 6.5.1 of
-- Pous and Sangiorgi's "Enhancements of the bisimulation proof
-- method".
{-# OPTIONS --sized-types #-}
open import Labelled-transition-system
module Bisimilarity.Weak.Alternative.Classical {ℓ} (lts : LTS ℓ) where
open import Prelude
import Bisimilarity.Classical
open LTS lts
-- We get weak bisimilarity by instantiating strong bisimilarity with
-- a different LTS.
private
module WB = Bisimilarity.Classical (weak lts)
open WB public
using (⟪_,_⟫)
renaming ( Bisimulation to Weak-bisimulation
; Bisimilarity′ to Weak-bisimilarity′
; Bisimilarity to Weak-bisimilarity
; _∼_ to _≈_
)
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------------------------------------------------------------------------------
-- Properties related with the group commutator
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module GroupTheory.Commutator.PropertiesATP where
open import GroupTheory.Base
open import GroupTheory.Commutator
------------------------------------------------------------------------------
-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising
-- Company, 2nd edition, 1960. p. 99.
postulate commutatorInverse : ∀ a b → [ a , b ] · [ b , a ] ≡ ε
{-# ATP prove commutatorInverse #-}
-- If the commutator is associative, then commutator of any two
-- elements lies in the center of the group, i.e. a [b,c] = [b,c] a.
-- From: TPTP 6.4.0 problem GRP/GRP024-5.p.
postulate commutatorAssocCenter : (∀ a b c → commutatorAssoc a b c) →
(∀ a b c → a · [ b , c ] ≡ [ b , c ] · a)
{-# ATP prove commutatorAssocCenter #-}
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-- 2014-01-01 Andreas, test case constructed by Christian Sattler
{-# OPTIONS --allow-unsolved-metas #-}
-- unguarded recursive record
record R : Set where
constructor cons
field
r : R
postulate F : (R → Set) → Set
q : (∀ P → F P) → (∀ P → F P)
q h P = h (λ {(cons x) → {!!}})
-- ISSUE WAS: Bug in implementation of eta-expansion of projected var,
-- leading to loop in Agda.
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Examples of format strings and printf
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module README.Text.Printf where
open import Data.Nat.Base
open import Data.Char.Base
open import Data.List.Base
open import Data.String.Base
open import Data.Sum.Base
open import Relation.Binary.PropositionalEquality
------------------------------------------------------------------------
-- Format strings
open import Text.Format
-- We can specify a format by writing a string which will get interpreted
-- by a lexer into a list of formatting directives.
-- The specification types are always started with a '%' character:
-- Integers (%d or %i)
-- Naturals (%u)
-- Floats (%f)
-- Chars (%c)
-- Strings (%s)
-- Anything which is not a type specification is a raw string to be spliced
-- in the output of printf.
-- For instance the following format alternates types and raw strings
_ : lexer "%s: %u + %u ≡ %u"
≡ inj₂ (`String ∷ Raw ": " ∷ `ℕ ∷ Raw " + " ∷ `ℕ ∷ Raw " ≡ " ∷ `ℕ ∷ [])
_ = refl
-- Lexing can fail. There are two possible errors:
-- If we start a specification type with a '%' but the string ends then
-- we get an UnexpectedEndOfString error
_ : lexer "%s: %u + %u ≡ %"
≡ inj₁ (UnexpectedEndOfString "%s: %u + %u ≡ %")
_ = refl
-- If we start a specification type with a '%' and the following character
-- does not correspond to an existing type, we get an InvalidType error
-- together with a focus highlighting the position of the problematic type.
_ : lexer "%s: %u + %a ≡ %u"
≡ inj₁ (InvalidType "%s: %u + %" 'a' " ≡ %u")
_ = refl
------------------------------------------------------------------------
-- Printf
open import Text.Printf
-- printf is a function which takes a format string as an argument and
-- returns a function expecting a value for each type specification present
-- in the format and returns a string splicing in these values into the
-- format string.
-- For instance `printf "%s: %u + %u ≡ %u"` is a
-- `String → ℕ → ℕ → ℕ → String` function.
_ : String → ℕ → ℕ → ℕ → String
_ = printf "%s: %u + %u ≡ %u"
_ : printf "%s: %u + %u ≡ %u" "example" 3 2 5
≡ "example: 3 + 2 ≡ 5"
_ = refl
-- If the format string str is invalid then `printf str` will have type
-- `Error e` where `e` is the lexing error.
_ : Text.Printf.Error (UnexpectedEndOfString "%s: %u + %u ≡ %")
_ = printf "%s: %u + %u ≡ %"
_ : Text.Printf.Error (InvalidType "%s: %u + %" 'a' " ≡ %u")
_ = printf "%s: %u + %a ≡ %u"
-- Trying to pass arguments to such an ̀Error` type will lead to a
-- unification error which hopefully makes the problem clear e.g.
-- `printf "%s: %u + %a ≡ %u" "example" 3 2 5` fails with the error:
-- Text.Printf.Error (InvalidType "%s: %u + %" 'a' " ≡ %u") should be
-- a function type, but it isn't
-- when checking that "example" 3 2 5 are valid arguments to a
-- function of type Text.Printf.Printf (lexer "%s: %u + %a ≡ %u")
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module Inference-of-implicit-function-space where
postulate
_⇔_ : Set → Set → Set
equivalence : {A B : Set} → (A → B) → (B → A) → A ⇔ B
A : Set
P : Set
P = {x : A} → A ⇔ A
works : P ⇔ P
works = equivalence (λ r {x} → r {x = x}) (λ r {x} → r {x = x})
works₂ : P ⇔ P
works₂ = equivalence {A = P} (λ r {x} → r {x = x}) (λ r {y} → r {y})
fails : P ⇔ P
fails = equivalence (λ r {x} → r {x = x}) (λ r {y} → r {y})
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Transp {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.Typed.Properties
open import Definition.Typed.Weakening as T hiding (wk; wkTerm; wkEqTerm)
open import Definition.Typed.RedSteps
open import Definition.LogicalRelation
open import Definition.LogicalRelation.ShapeView
open import Definition.LogicalRelation.Irrelevance as I
open import Definition.LogicalRelation.Weakening
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Application
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.Properties
open import Definition.LogicalRelation.Substitution.Irrelevance as S
open import Definition.LogicalRelation.Substitution.Reflexivity
open import Definition.LogicalRelation.Substitution.Introductions.Sigma
open import Definition.LogicalRelation.Substitution.Introductions.Fst
open import Definition.LogicalRelation.Substitution.Introductions.Pi
open import Definition.LogicalRelation.Substitution.Introductions.Lambda
open import Definition.LogicalRelation.Substitution.Introductions.Application
open import Definition.LogicalRelation.Substitution.Introductions.Cast
open import Definition.LogicalRelation.Substitution.Introductions.Id
open import Definition.LogicalRelation.Substitution.Introductions.SingleSubst
open import Definition.LogicalRelation.Substitution.MaybeEmbed
open import Definition.LogicalRelation.Substitution.Escape
open import Definition.LogicalRelation.Substitution.Introductions.Universe
open import Definition.LogicalRelation.Substitution.Reduction
open import Definition.LogicalRelation.Substitution.Weakening
open import Definition.LogicalRelation.Substitution.ProofIrrelevance
open import Tools.Product
import Tools.PropositionalEquality as PE
IdSymᵗᵛ : ∀ {A l t u e Γ}
([Γ] : ⊩ᵛ Γ)
([U] : Γ ⊩ᵛ⟨ ∞ ⟩ U l ^ [ ! , next l ] / [Γ])
([AU] : Γ ⊩ᵛ⟨ ∞ ⟩ A ∷ U l ^ [ ! , next l ] / [Γ] / [U])
([A] : Γ ⊩ᵛ⟨ ∞ ⟩ A ^ [ ! , ι l ] / [Γ])
([t] : Γ ⊩ᵛ⟨ ∞ ⟩ t ∷ A ^ [ ! , ι l ] / [Γ] / [A])
([u] : Γ ⊩ᵛ⟨ ∞ ⟩ u ∷ A ^ [ ! , ι l ] / [Γ] / [A])
([Id] : Γ ⊩ᵛ⟨ ∞ ⟩ Id A t u ^ [ % , ι l ] / [Γ]) →
([Idinv] : Γ ⊩ᵛ⟨ ∞ ⟩ Id A u t ^ [ % , ι l ] / [Γ]) →
([e] : Γ ⊩ᵛ⟨ ∞ ⟩ e ∷ Id A t u ^ [ % , ι l ] / [Γ] / [Id] ) →
Γ ⊩ᵛ⟨ ∞ ⟩ Idsym A t u e ∷ Id A u t ^ [ % , ι l ] / [Γ] / [Idinv]
IdSymᵗᵛ {A} {l} {t} {u} {e} {Γ} [Γ] [U] [AU] [A] [t] [u] [Id] [Idinv] [e] = validityIrr {A = Id A u t} {t = Idsym A t u e} [Γ] [Idinv] λ {Δ} {σ} ⊢Δ [σ] →
PE.subst (λ X → Δ ⊢ X ∷ subst σ (Id A u t) ^ [ % , ι l ] ) (PE.sym (subst-Idsym σ A t u e))
(Idsymⱼ {A = subst σ A} {x = subst σ t} {y = subst σ u} (escapeTerm (proj₁ ([U] {Δ} {σ} ⊢Δ [σ])) (proj₁ ([AU] ⊢Δ [σ])))
(escapeTerm (proj₁ ([A] {Δ} {σ} ⊢Δ [σ])) (proj₁ ([t] ⊢Δ [σ])))
(escapeTerm (proj₁ ([A] {Δ} {σ} ⊢Δ [σ])) (proj₁ ([u] ⊢Δ [σ])))
(escapeTerm (proj₁ ([Id] {Δ} {σ} ⊢Δ [σ])) (proj₁ ([e] ⊢Δ [σ]))))
abstract
transpᵗᵛ : ∀ {A P l t s u e Γ}
([Γ] : ⊩ᵛ Γ)
([A] : Γ ⊩ᵛ⟨ ∞ ⟩ A ^ [ ! , l ] / [Γ])
([P] : Γ ∙ A ^ [ ! , l ] ⊩ᵛ⟨ ∞ ⟩ P ^ [ % , l ] / (_∙_ {A = A} [Γ] [A]))
([t] : Γ ⊩ᵛ⟨ ∞ ⟩ t ∷ A ^ [ ! , l ] / [Γ] / [A])
([s] : Γ ⊩ᵛ⟨ ∞ ⟩ s ∷ P [ t ] ^ [ % , l ] / [Γ] / substS {A} {P} {t} [Γ] [A] [P] [t])
([u] : Γ ⊩ᵛ⟨ ∞ ⟩ u ∷ A ^ [ ! , l ] / [Γ] / [A])
([Id] : Γ ⊩ᵛ⟨ ∞ ⟩ Id A t u ^ [ % , l ] / [Γ]) →
([e] : Γ ⊩ᵛ⟨ ∞ ⟩ e ∷ Id A t u ^ [ % , l ] / [Γ] / [Id] ) →
Γ ⊩ᵛ⟨ ∞ ⟩ transp A P t s u e ∷ P [ u ] ^ [ % , l ] / [Γ] / substS {A} {P} {u} [Γ] [A] [P] [u]
transpᵗᵛ {A} {P} {l} {t} {s} {u} {e} {Γ} [Γ] [A] [P] [t] [s] [u] [Id] [e] =
validityIrr {A = P [ u ]} {t = transp A P t s u e } [Γ] (substS {A} {P} {u} [Γ] [A] [P] [u]) λ {Δ} {σ} ⊢Δ [σ] →
let [liftσ] = liftSubstS {F = A} [Γ] ⊢Δ [A] [σ]
[A]σ = proj₁ ([A] {Δ} {σ} ⊢Δ [σ])
[P[t]]σ = I.irrelevance′ (singleSubstLift P t) (proj₁ (substS {A} {P} {t} [Γ] [A] [P] [t] {Δ} {σ} ⊢Δ [σ]))
X = transpⱼ (escape [A]σ) (escape (proj₁ ([P] {Δ ∙ subst σ A ^ [ ! , l ]} {liftSubst σ} (⊢Δ ∙ (escape [A]σ)) [liftσ])))
(escapeTerm [A]σ (proj₁ ([t] ⊢Δ [σ]))) (escapeTerm [P[t]]σ (I.irrelevanceTerm′ (singleSubstLift P t) PE.refl PE.refl (proj₁ (substS {A} {P} {t} [Γ] [A] [P] [t] {Δ} {σ} ⊢Δ [σ])) [P[t]]σ (proj₁ ([s] ⊢Δ [σ]))))
(escapeTerm [A]σ (proj₁ ([u] ⊢Δ [σ]))) (escapeTerm (proj₁ ([Id] {Δ} {σ} ⊢Δ [σ])) (proj₁ ([e] ⊢Δ [σ])))
in PE.subst (λ X → Δ ⊢ transp (subst σ A) ( subst (liftSubst σ) P) (subst σ t) (subst σ s) (subst σ u) (subst σ e) ∷ X ^ [ % , l ] ) (PE.sym (singleSubstLift P u)) X
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--------------------------------------------------------------------------------
-- This is part of Agda Inference Systems
{-# OPTIONS --sized-types --guardedness #-}
open import Data.Product
open import Data.Vec
open import Codata.Colist as Colist
open import Agda.Builtin.Equality
open import Size
open import Codata.Thunk
open import Data.Fin
open import Data.Nat
open import Data.Maybe
open import Examples.Colists.Auxiliary.Colist_member
open import is-lib.InfSys
module Examples.Colists.member {A : Set} where
U = A × Colist A ∞
data memberRN : Set where
mem-h mem-t : memberRN
mem-h-r : FinMetaRule U
mem-h-r .Ctx = A × Thunk (Colist A) ∞
mem-h-r .comp (x , xs) =
[] ,
----------------
(x , x ∷ xs)
mem-t-r : FinMetaRule U
mem-t-r .Ctx = A × A × Thunk (Colist A) ∞
mem-t-r .comp (x , y , xs) =
((x , xs .force) ∷ []) ,
----------------
(x , y ∷ xs)
memberIS : IS U
memberIS .Names = memberRN
memberIS .rules mem-h = from mem-h-r
memberIS .rules mem-t = from mem-t-r
_member_ : A → Colist A ∞ → Set
x member xs = Ind⟦ memberIS ⟧ (x , xs)
memSpec : U → Set
memSpec (x , xs) = Σ[ i ∈ ℕ ] (Colist.lookup i xs ≡ just x)
memSpecClosed : ISClosed memberIS memSpec
memSpecClosed mem-h _ _ = zero , refl
memSpecClosed mem-t _ pr =
let (i , proof) = pr Fin.zero in
(suc i) , proof
memberSound : ∀{x xs} → x member xs → memSpec (x , xs)
memberSound = ind[ memberIS ] memSpec memSpecClosed
-- Completeness using memSpec does not terminate
-- Product implemented as record. Record projections do not decrease
memSpec' : U → ℕ → Set
memSpec' (x , xs) i = Colist.lookup i xs ≡ just x
memberCompl : ∀{x xs i} → memSpec' (x , xs) i → x member xs
memberCompl {.x} {x ∷ _} {zero} refl = apply-ind mem-h _ λ ()
memberCompl {x} {y ∷ xs} {suc i} eq = apply-ind mem-t _ λ{zero → memberCompl eq}
memberComplete : ∀{x xs} → memSpec (x , xs) → x member xs
memberComplete (i , eq) = memberCompl eq
{- Correctness wrt to Agda DataType -}
∈-sound : ∀{x xs} → x ∈ xs → x member xs
∈-sound here = apply-ind mem-h _ λ ()
∈-sound (there mem) = apply-ind mem-t _ λ{zero → ∈-sound mem}
∈-complete : ∀{x xs} → x member xs → x ∈ xs
∈-complete (fold (mem-h , _ , refl , _)) = here
∈-complete (fold (mem-t , _ , refl , prem)) = there (∈-complete (prem zero)) | {
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{-# OPTIONS --cubical-compatible #-}
open import Common.Prelude
open import Common.Equality
open import Common.Product
data _≅_ {A : Set} (a : A) : {B : Set} (b : B) → Set₁ where
refl : a ≅ a
data D : Bool → Set where
x : D true
y : D false
P : Set -> Set₁
P S = Σ S (\s → s ≅ x)
pbool : P (D true)
pbool = _ , refl
¬pfin : P (D false) → ⊥
¬pfin (y , ())
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