Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
exact c1
case s1 z : String t : Var x : String c1 : Β¬z = x ⊒ Β¬z = x z : String t : Var x : String c1 : Β¬free_ z = free_ x s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x} ⊒ (free_ x).freeVarSet βŠ† t.freeVarSet βˆͺ (free_ x).freeVarSet \ {free_ z}
z : String t : Var x : String c1 : Β¬free_ z = free_ x s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x} ⊒ (free_ x).freeVarSet βŠ† t.freeVarSet βˆͺ (free_ x).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
simp only [s1]
z : String t : Var x : String c1 : Β¬free_ z = free_ x s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x} ⊒ (free_ x).freeVarSet βŠ† t.freeVarSet βˆͺ (free_ x).freeVarSet \ {free_ z}
z : String t : Var x : String c1 : Β¬free_ z = free_ x s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x} ⊒ (free_ x).freeVarSet βŠ† t.freeVarSet βˆͺ {free_ x}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
apply Finset.subset_union_right
z : String t : Var x : String c1 : Β¬free_ z = free_ x s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x} ⊒ (free_ x).freeVarSet βŠ† t.freeVarSet βˆͺ {free_ x}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
simp only [Var.subst]
z : String t : Var i : β„• ⊒ ((free_ z).subst t (bound_ i)).freeVarSet βŠ† t.freeVarSet βˆͺ (bound_ i).freeVarSet \ {free_ z}
z : String t : Var i : β„• ⊒ (bound_ i).freeVarSet βŠ† t.freeVarSet βˆͺ (bound_ i).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
conv => lhs simp only [Var.freeVarSet]
z : String t : Var i : β„• ⊒ (bound_ i).freeVarSet βŠ† t.freeVarSet βˆͺ (bound_ i).freeVarSet \ {free_ z}
z : String t : Var i : β„• ⊒ βˆ… βŠ† t.freeVarSet βˆͺ (bound_ i).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet
[474, 1]
[500, 9]
simp
z : String t : Var i : β„• ⊒ βˆ… βŠ† t.freeVarSet βˆͺ (bound_ i).freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
induction F
z : String t : Var F : Formula ⊒ (Formula.subst (free_ z) t F).freeVarSet βŠ† t.freeVarSet βˆͺ F.freeVarSet \ {free_ z}
case pred_ z : String t : Var a✝¹ : String a✝ : List Var ⊒ (Formula.subst (free_ z) t (pred_ a✝¹ a✝)).freeVarSet βŠ† t.freeVarSet βˆͺ (pred_ a✝¹ a✝).freeVarSet \ {free_ z} case not_ z : String t : Var a✝ : Formula a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet βŠ† t.freeVarSet βˆͺ a✝.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t a✝.not_).freeVarSet βŠ† t.freeVarSet βˆͺ a✝.not_.freeVarSet \ {free_ z} case imp_ z : String t : Var a✝¹ a✝ : Formula a_ih✝¹ : (Formula.subst (free_ z) t a✝¹).freeVarSet βŠ† t.freeVarSet βˆͺ a✝¹.freeVarSet \ {free_ z} a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet βŠ† t.freeVarSet βˆͺ a✝.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t (a✝¹.imp_ a✝)).freeVarSet βŠ† t.freeVarSet βˆͺ (a✝¹.imp_ a✝).freeVarSet \ {free_ z} case forall_ z : String t : Var a✝¹ : String a✝ : Formula a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet βŠ† t.freeVarSet βˆͺ a✝.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t (forall_ a✝¹ a✝)).freeVarSet βŠ† t.freeVarSet βˆͺ (forall_ a✝¹ a✝).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
case not_ phi phi_ih => simp only [Formula.subst] simp only [Formula.freeVarSet] exact phi_ih
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi.not_).freeVarSet βŠ† t.freeVarSet βˆͺ phi.not_.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
case forall_ x phi phi_ih => simp only [Formula.subst] simp only [Formula.freeVarSet] exact phi_ih
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet βŠ† t.freeVarSet βˆͺ (forall_ x phi).freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.subst]
z : String t : Var X : String vs : List Var ⊒ (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet βŠ† t.freeVarSet βˆͺ (pred_ X vs).freeVarSet \ {free_ z}
z : String t : Var X : String vs : List Var ⊒ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet βŠ† t.freeVarSet βˆͺ (pred_ X vs).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.freeVarSet]
z : String t : Var X : String vs : List Var ⊒ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet βŠ† t.freeVarSet βˆͺ (pred_ X vs).freeVarSet \ {free_ z}
z : String t : Var X : String vs : List Var ⊒ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp
z : String t : Var X : String vs : List Var ⊒ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
z : String t : Var X : String vs : List Var ⊒ βˆ€ a ∈ vs, ((free_ z).subst t a).freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
intro v a1
z : String t : Var X : String vs : List Var ⊒ βˆ€ a ∈ vs, ((free_ z).subst t a).freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ ((free_ z).subst t v).freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
trans Var.freeVarSet t βˆͺ Var.freeVarSet v \ {free_ z}
z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ ((free_ z).subst t v).freeVarSet βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ ((free_ z).subst t v).freeVarSet βŠ† t.freeVarSet βˆͺ v.freeVarSet \ {free_ z} z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ t.freeVarSet βˆͺ v.freeVarSet \ {free_ z} βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
exact VarSubstFreeVarSet z t v
z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ ((free_ z).subst t v).freeVarSet βŠ† t.freeVarSet βˆͺ v.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
apply Finset.union_subset_union_right
z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ t.freeVarSet βˆͺ v.freeVarSet \ {free_ z} βŠ† t.freeVarSet βˆͺ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
case h z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ v.freeVarSet \ {free_ z} βŠ† vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [← List.mem_toFinset] at a1
case h z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs ⊒ v.freeVarSet \ {free_ z} βŠ† vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
case h z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ v.freeVarSet \ {free_ z} βŠ† vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
apply Finset.sdiff_subset_sdiff
case h z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ v.freeVarSet \ {free_ z} βŠ† vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
case h.hst z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ v.freeVarSet βŠ† vs.toFinset.biUnion Var.freeVarSet case h.hvu z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ {free_ z} βŠ† {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
apply Finset.subset_biUnion_of_mem Var.freeVarSet a1
case h.hst z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ v.freeVarSet βŠ† vs.toFinset.biUnion Var.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp
case h.hvu z : String t : Var X : String vs : List Var v : Var a1 : v ∈ vs.toFinset ⊒ {free_ z} βŠ† {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.subst]
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi.not_).freeVarSet βŠ† t.freeVarSet βˆͺ phi.not_.freeVarSet \ {free_ z}
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).not_.freeVarSet βŠ† t.freeVarSet βˆͺ phi.not_.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.freeVarSet]
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).not_.freeVarSet βŠ† t.freeVarSet βˆͺ phi.not_.freeVarSet \ {free_ z}
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
exact phi_ih
z : String t : Var phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.subst]
z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t (phi.imp_ psi)).freeVarSet βŠ† t.freeVarSet βˆͺ (phi.imp_ psi).freeVarSet \ {free_ z}
z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet βŠ† t.freeVarSet βˆͺ (phi.imp_ psi).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.freeVarSet]
z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet βŠ† t.freeVarSet βˆͺ (phi.imp_ psi).freeVarSet \ {free_ z}
z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βˆͺ (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ (phi.freeVarSet βˆͺ psi.freeVarSet) \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
apply Finset.union_subset_left_right_diff
z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βˆͺ (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ (phi.freeVarSet βˆͺ psi.freeVarSet) \ {free_ z}
case h1 z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} case h2 z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
exact phi_ih
case h1 z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
exact psi_ih
case h2 z : String t : Var phi psi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} psi_ih : (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t psi).freeVarSet βŠ† t.freeVarSet βˆͺ psi.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.subst]
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet βŠ† t.freeVarSet βˆͺ (forall_ x phi).freeVarSet \ {free_ z}
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet βŠ† t.freeVarSet βˆͺ (forall_ x phi).freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
simp only [Formula.freeVarSet]
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet βŠ† t.freeVarSet βˆͺ (forall_ x phi).freeVarSet \ {free_ z}
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet
[503, 1]
[536, 17]
exact phi_ih
z : String t : Var x : String phi : Formula phi_ih : (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z} ⊒ (Formula.subst (free_ z) t phi).freeVarSet βŠ† t.freeVarSet βˆͺ phi.freeVarSet \ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
cases v
z : String t v : Var ⊒ v.freeVarSet \ {free_ z} βŠ† ((free_ z).subst t v).freeVarSet
case free_ z : String t : Var a✝ : String ⊒ (free_ a✝).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (free_ a✝)).freeVarSet case bound_ z : String t : Var a✝ : β„• ⊒ (bound_ a✝).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (bound_ a✝)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
case free_ x => simp only [Var.subst] split_ifs case pos c1 => simp only [c1] conv => lhs simp only [Var.freeVarSet] simp case neg c1 => simp only [Var.freeVarSet] exact Finset.sdiff_subset {free_ x} {free_ z}
z : String t : Var x : String ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (free_ x)).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
case bound_ i => conv => lhs simp only [Var.freeVarSet]
z : String t : Var i : β„• ⊒ (bound_ i).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (bound_ i)).freeVarSet
z : String t : Var i : β„• ⊒ βˆ… \ {free_ z} βŠ† ((free_ z).subst t (bound_ i)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
simp only [Var.subst]
z : String t : Var x : String ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (free_ x)).freeVarSet
z : String t : Var x : String ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† (if free_ z = free_ x then t else free_ x).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
split_ifs
z : String t : Var x : String ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† (if free_ z = free_ x then t else free_ x).freeVarSet
case pos z : String t : Var x : String h✝ : free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† t.freeVarSet case neg z : String t : Var x : String h✝ : Β¬free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† (free_ x).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
case pos c1 => simp only [c1] conv => lhs simp only [Var.freeVarSet] simp
z : String t : Var x : String c1 : free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† t.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
case neg c1 => simp only [Var.freeVarSet] exact Finset.sdiff_subset {free_ x} {free_ z}
z : String t : Var x : String c1 : Β¬free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† (free_ x).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
simp only [c1]
z : String t : Var x : String c1 : free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† t.freeVarSet
z : String t : Var x : String c1 : free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ x} βŠ† t.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
conv => lhs simp only [Var.freeVarSet]
z : String t : Var x : String c1 : free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ x} βŠ† t.freeVarSet
z : String t : Var x : String c1 : free_ z = free_ x ⊒ {free_ x} \ {free_ x} βŠ† t.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
simp
z : String t : Var x : String c1 : free_ z = free_ x ⊒ {free_ x} \ {free_ x} βŠ† t.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
simp only [Var.freeVarSet]
z : String t : Var x : String c1 : Β¬free_ z = free_ x ⊒ (free_ x).freeVarSet \ {free_ z} βŠ† (free_ x).freeVarSet
z : String t : Var x : String c1 : Β¬free_ z = free_ x ⊒ {free_ x} \ {free_ z} βŠ† {free_ x}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
exact Finset.sdiff_subset {free_ x} {free_ z}
z : String t : Var x : String c1 : Β¬free_ z = free_ x ⊒ {free_ x} \ {free_ z} βŠ† {free_ x}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarSubstFreeVarSet'
[540, 1]
[562, 33]
conv => lhs simp only [Var.freeVarSet]
z : String t : Var i : β„• ⊒ (bound_ i).freeVarSet \ {free_ z} βŠ† ((free_ z).subst t (bound_ i)).freeVarSet
z : String t : Var i : β„• ⊒ βˆ… \ {free_ z} βŠ† ((free_ z).subst t (bound_ i)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
induction F
z : String t : Var F : Formula ⊒ F.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t F).freeVarSet
case pred_ z : String t : Var a✝¹ : String a✝ : List Var ⊒ (pred_ a✝¹ a✝).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (pred_ a✝¹ a✝)).freeVarSet case not_ z : String t : Var a✝ : Formula a_ih✝ : a✝.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t a✝).freeVarSet ⊒ a✝.not_.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t a✝.not_).freeVarSet case imp_ z : String t : Var a✝¹ a✝ : Formula a_ih✝¹ : a✝¹.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t a✝¹).freeVarSet a_ih✝ : a✝.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t a✝).freeVarSet ⊒ (a✝¹.imp_ a✝).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (a✝¹.imp_ a✝)).freeVarSet case forall_ z : String t : Var a✝¹ : String a✝ : Formula a_ih✝ : a✝.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t a✝).freeVarSet ⊒ (forall_ a✝¹ a✝).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (forall_ a✝¹ a✝)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
case pred_ X vs => simp only [Formula.subst] simp only [Formula.freeVarSet] induction vs case nil => simp case cons hd tl ih => simp have s1 : (Var.freeVarSet hd βˆͺ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) βˆͺ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z}) exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z} simp only [s1] exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
z : String t : Var X : String vs : List Var ⊒ (pred_ X vs).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
case not_ phi phi_ih => simp only [Formula.subst] simp only [Formula.freeVarSet] exact phi_ih
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.not_.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi.not_).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
case forall_ x phi phi_ih => simp only [Formula.subst] simp only [Formula.freeVarSet] exact phi_ih
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ (forall_ x phi).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.subst]
z : String t : Var X : String vs : List Var ⊒ (pred_ X vs).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet
z : String t : Var X : String vs : List Var ⊒ (pred_ X vs).freeVarSet \ {free_ z} βŠ† (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.freeVarSet]
z : String t : Var X : String vs : List Var ⊒ (pred_ X vs).freeVarSet \ {free_ z} βŠ† (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet
z : String t : Var X : String vs : List Var ⊒ vs.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
induction vs
z : String t : Var X : String vs : List Var ⊒ vs.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet
case nil z : String t : Var X : String ⊒ [].toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet case cons z : String t : Var X : String head✝ : Var tail✝ : List Var tail_ih✝ : tail✝.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tail✝).toFinset.biUnion Var.freeVarSet ⊒ (head✝ :: tail✝).toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) (head✝ :: tail✝)).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
case nil => simp
z : String t : Var X : String ⊒ [].toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
case cons hd tl ih => simp have s1 : (Var.freeVarSet hd βˆͺ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) βˆͺ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z}) exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z} simp only [s1] exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd :: tl).toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) (hd :: tl)).toFinset.biUnion Var.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp
z : String t : Var X : String ⊒ [].toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd :: tl).toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) (hd :: tl)).toFinset.biUnion Var.freeVarSet
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
have s1 : (Var.freeVarSet hd βˆͺ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) βˆͺ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z})
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
case s1 z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z}
case s1 z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [s1]
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
z : String t : Var X : String hd : Var tl : List Var ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet s1 : (hd.freeVarSet βˆͺ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} = hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊒ hd.freeVarSet \ {free_ z} βˆͺ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} βŠ† ((free_ z).subst t hd).freeVarSet βˆͺ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.subst]
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.not_.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi.not_).freeVarSet
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.not_.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).not_.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.freeVarSet]
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.not_.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).not_.freeVarSet
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
exact phi_ih
z : String t : Var phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.subst]
z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ (phi.imp_ psi).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (phi.imp_ psi)).freeVarSet
z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ (phi.imp_ psi).freeVarSet \ {free_ z} βŠ† ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.freeVarSet]
z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ (phi.imp_ psi).freeVarSet \ {free_ z} βŠ† ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet
z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ (phi.freeVarSet βˆͺ psi.freeVarSet) \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet βˆͺ (Formula.subst (free_ z) t psi).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
apply Finset.diff_union_subset
z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ (phi.freeVarSet βˆͺ psi.freeVarSet) \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet βˆͺ (Formula.subst (free_ z) t psi).freeVarSet
case h1 z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet case h2 z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
apply phi_ih
case h1 z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
apply psi_ih
case h2 z : String t : Var phi psi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet psi_ih : psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet ⊒ psi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t psi).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.subst]
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ (forall_ x phi).freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ (forall_ x phi).freeVarSet \ {free_ z} βŠ† (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
simp only [Formula.freeVarSet]
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ (forall_ x phi).freeVarSet \ {free_ z} βŠ† (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaSubstFreeVarSet'
[565, 1]
[600, 17]
exact phi_ih
z : String t : Var x : String phi : Formula phi_ih : phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet ⊒ phi.freeVarSet \ {free_ z} βŠ† (Formula.subst (free_ z) t phi).freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
cases v
v : Var Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) v) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) v)
case free_ Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z a✝ : String ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (free_ a✝)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (free_ a✝)) case bound_ Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z a✝ : β„• ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (bound_ a✝)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
case free_ x => conv => lhs simp only [Var.open] simp only [Var.substFun] simp only [str_fun_to_var_fun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z x : String ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (free_ x)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (free_ x))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z x : String ⊒ free_ (Οƒ x) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (free_ x))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
case bound_ i => conv => lhs simp only [Var.open] split_ifs case pos c1 => simp only [Var.substFun] simp only [Var.open] simp only [if_pos c1] case pos c1 c2 => simp only [Var.substFun] simp only [str_fun_to_var_fun] simp only [Var.open] simp only [if_neg c1] simp only [if_pos c2] simp only [h1] case neg c1 c2 => simp only [Var.substFun] simp only [Var.open] simp only [if_neg c1] simp only [if_neg c2]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (bound_ i)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
conv => lhs simp only [Var.open] simp only [Var.substFun] simp only [str_fun_to_var_fun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z x : String ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (free_ x)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (free_ x))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z x : String ⊒ free_ (Οƒ x) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (free_ x))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
conv => lhs simp only [Var.open]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (Var.open j (free_ z) (bound_ i)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
split_ifs
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
case pos Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• h✝ : i < j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i)) case pos Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• h✝¹ : Β¬i < j h✝ : i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i)) case neg Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• h✝¹ : Β¬i < j h✝ : Β¬i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ (i - 1)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
case pos c1 => simp only [Var.substFun] simp only [Var.open] simp only [if_pos c1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
case pos c1 c2 => simp only [Var.substFun] simp only [str_fun_to_var_fun] simp only [Var.open] simp only [if_neg c1] simp only [if_pos c2] simp only [h1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
case neg c1 c2 => simp only [Var.substFun] simp only [Var.open] simp only [if_neg c1] simp only [if_neg c2]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ (i - 1)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.substFun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ bound_ i = Var.open j (free_ z) (bound_ i)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.open]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ bound_ i = Var.open j (free_ z) (bound_ i)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ bound_ i = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [if_pos c1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : i < j ⊒ bound_ i = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.substFun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ str_fun_to_var_fun Οƒ (free_ z) = Var.open j (free_ z) (bound_ i)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [str_fun_to_var_fun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ str_fun_to_var_fun Οƒ (free_ z) = Var.open j (free_ z) (bound_ i)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = Var.open j (free_ z) (bound_ i)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.open]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = Var.open j (free_ z) (bound_ i)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [if_neg c1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = if i = j then free_ z else bound_ (i - 1)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [if_pos c2]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = if i = j then free_ z else bound_ (i - 1)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = free_ z
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [h1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : i = j ⊒ free_ (Οƒ z) = free_ z
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.substFun]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ Var.substFun (str_fun_to_var_fun Οƒ) (bound_ (i - 1)) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun Οƒ) (bound_ i))
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = Var.open j (free_ z) (bound_ i)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [Var.open]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = Var.open j (free_ z) (bound_ i)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [if_neg c1]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = if i = j then free_ z else bound_ (i - 1)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.SubOpenVar
[611, 1]
[647, 28]
simp only [if_neg c2]
Οƒ : String β†’ String j : β„• z : String h1 : Οƒ z = z i : β„• c1 : Β¬i < j c2 : Β¬i = j ⊒ bound_ (i - 1) = if i = j then free_ z else bound_ (i - 1)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
funext v
D : Type V : VarAssignment D j : β„• zs : List String d : D ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d = shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)
case h D : Type V : VarAssignment D j : β„• zs : List String d : D v : Var ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
simp
case h D : Type V : VarAssignment D j : β„• zs : List String d : D v : Var ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) v
case h D : Type V : VarAssignment D j : β„• zs : List String d : D v : Var ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = shift D V d (Var.openList (j + 1) (List.map free_ zs) v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
cases v
case h D : Type V : VarAssignment D j : β„• zs : List String d : D v : Var ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = shift D V d (Var.openList (j + 1) (List.map free_ zs) v)
case h.free_ D : Type V : VarAssignment D j : β„• zs : List String d : D a✝ : String ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ a✝) = shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ a✝)) case h.bound_ D : Type V : VarAssignment D j : β„• zs : List String d : D a✝ : β„• ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ a✝) = shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
case _ x => simp only [Var.openList] simp only [shift] simp
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ x))
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ V (Var.openList j (List.map free_ zs) (free_ x)) = V (free_ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
case _ i => cases i case zero => simp only [shift] simp only [Var.openList] simp case succ i => simp only [shift] simp only [Var.openList] simp split case _ c1 => have s1 : i + 1 < j + 1 linarith simp only [if_pos s1] case _ c1 => have s1 : Β¬ i + 1 < j + 1 linarith simp only [if_neg s1] split case _ c2 => simp case _ c2 => simp
D : Type V : VarAssignment D j : β„• zs : List String d : D i : β„• ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ i) = shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ i))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
simp only [Var.openList]
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ x))
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (free_ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
simp only [shift]
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (free_ x)
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ (V ∘ Var.openList j (List.map free_ zs)) (free_ x) = V (free_ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.ShiftVarOpenList
[815, 1]
[854, 17]
simp
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ (V ∘ Var.openList j (List.map free_ zs)) (free_ x) = V (free_ x)
D : Type V : VarAssignment D j : β„• zs : List String d : D x : String ⊒ V (Var.openList j (List.map free_ zs) (free_ x)) = V (free_ x)