Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
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https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact h1 x a1
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length x : VarName a1 : x ∈ xs ⊒ V' x = V (Οƒ x)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ I.pred_var_ X (List.map V' xs) ↔ Holds D I V E (pred_var_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ I.pred_var_ X (List.map V' xs) ↔ I.pred_var_ X (List.map V (List.map Οƒ xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ I.pred_var_ X (List.map V' xs) ↔ I.pred_var_ X (List.map V (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ I.pred_var_ X (List.map V' xs) ↔ I.pred_var_ X (List.map (V ∘ Οƒ) xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
congr! 1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ I.pred_var_ X (List.map V' xs) ↔ I.pred_var_ X (List.map (V ∘ Οƒ) xs)
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ List.map V' xs = List.map (V ∘ Οƒ) xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [List.map_eq_map_iff]
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ List.map V' xs = List.map (V ∘ Οƒ) xs
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ βˆ€ x ∈ xs, V' x = (V ∘ Οƒ) x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro x a1
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true ⊒ βˆ€ x ∈ xs, V' x = (V ∘ Οƒ) x
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true x : VarName a1 : x ∈ xs ⊒ V' x = (V ∘ Οƒ) x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true x : VarName a1 : x ∈ xs ⊒ V' x = (V ∘ Οƒ) x
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true x : VarName a1 : x ∈ xs ⊒ V' x = V (Οƒ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact h1 x a1
case a.h.e'_4 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ if h : (Ο„ X xs.length).isSome = true then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x c1 : Β¬(Ο„ X xs.length).isSome = true x : VarName a1 : x ∈ xs ⊒ V' x = V (Οƒ x)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [isFreeIn] at h1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), isFreeIn x_1 (eq_ x y) β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ x y) ↔ Holds D I V E (subAux c Ο„ Οƒ (eq_ x y))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ x y) ↔ Holds D I V E (subAux c Ο„ Οƒ (eq_ x y))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ x y) ↔ Holds D I V E (subAux c Ο„ Οƒ (eq_ x y))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ x y) ↔ Holds D I V E (eq_ (Οƒ x) (Οƒ y))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ x y) ↔ Holds D I V E (eq_ (Οƒ x) (Οƒ y))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
have s1 : V' x = V (Οƒ x)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
case s1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ V' x = V (Οƒ x) D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h1
case s1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ V' x = V (Οƒ x) D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
case s1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ x = x ∨ x = y D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp
case s1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ x = x ∨ x = y D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [s1]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' x = V' y ↔ V (Οƒ x) = V (Οƒ y)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
have s2 : V' y = V (Οƒ y)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
case s2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' y = V (Οƒ y) D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h1
case s2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ V' y = V (Οƒ y) D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
case s2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ y = x ∨ y = y D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp
case s2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) ⊒ y = x ∨ y = y D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [s2]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x y : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), x_1 = x ∨ x_1 = y β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 s1 : V' x = V (Οƒ x) s2 : V' y = V (Οƒ y) ⊒ V (Οƒ x) = V' y ↔ V (Οƒ x) = V (Οƒ y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x false_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ false_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E false_ ↔ Holds D I V E (subAux c Ο„ Οƒ false_)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x false_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ false_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E false_ ↔ Holds D I V E false_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x false_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ false_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E false_ ↔ Holds D I V E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [isFreeIn] at h1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi.not_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.not_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi.not_)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.not_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi.not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [predVarSet] at h2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.not_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi.not_)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi.not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi.not_)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi).not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ phi).not_
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Β¬Holds D (I' D I V'' E Ο„) V' E phi ↔ Β¬Holds D I V E (subAux c Ο„ Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
congr! 1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Β¬Holds D (I' D I V'' E Ο„) V' E phi ↔ Β¬Holds D I V E (subAux c Ο„ Οƒ phi)
case a.h.e'_1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact phi_ih V V' Οƒ h1 h2
case a.h.e'_1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [isFreeIn] at h1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (phi.iff_ psi) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.iff_ psi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E (subAux c Ο„ Οƒ (phi.iff_ psi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.iff_ psi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E (subAux c Ο„ Οƒ (phi.iff_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [predVarSet] at h2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.iff_ psi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E (subAux c Ο„ Οƒ (phi.iff_ psi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E (subAux c Ο„ Οƒ (phi.iff_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E (subAux c Ο„ Οƒ (phi.iff_ psi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E ((subAux c Ο„ Οƒ phi).iff_ (subAux c Ο„ Οƒ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (phi.iff_ psi) ↔ Holds D I V E ((subAux c Ο„ Οƒ phi).iff_ (subAux c Ο„ Οƒ psi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D (I' D I V'' E Ο„) V' E psi) ↔ (Holds D I V E (subAux c Ο„ Οƒ phi) ↔ Holds D I V E (subAux c Ο„ Οƒ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
congr! 1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D (I' D I V'' E Ο„) V' E psi) ↔ (Holds D I V E (subAux c Ο„ Οƒ phi) ↔ Holds D I V E (subAux c Ο„ Οƒ psi))
case a.h.e'_1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi) case a.h.e'_2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply phi_ih V V' Οƒ
case a.h.e'_1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)
case a.h.e'_1.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x) case a.h.e'_1.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro x a1
case a.h.e'_1.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)
case a.h.e'_1.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ V' x = V (Οƒ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h1
case a.h.e'_1.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ V' x = V (Οƒ x)
case a.h.e'_1.a.h1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ isFreeIn x phi ∨ isFreeIn x psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
left
case a.h.e'_1.a.h1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ isFreeIn x phi ∨ isFreeIn x psi
case a.h.e'_1.a.h1.a.h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ isFreeIn x phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact a1
case a.h.e'_1.a.h1.a.h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x phi ⊒ isFreeIn x phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro x a1
case a.h.e'_1.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x
case a.h.e'_1.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V'' x = V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h2
case a.h.e'_1.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V'' x = V x
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Finset.mem_biUnion, Finset.mem_union] at a1
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply Exists.elim a1
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ βˆ€ (a : PredName Γ— β„•), a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a β†’ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro a a2
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ βˆ€ (a : PredName Γ— β„•), a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a β†’ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Finset.mem_biUnion, Finset.mem_union]
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ βˆƒ a, (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply Exists.intro a
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ βˆƒ a, (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
tauto
case a.h.e'_1.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ phi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ phi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply psi_ih V V' Οƒ
case a.h.e'_2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)
case a.h.e'_2.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x) case a.h.e'_2.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro x a1
case a.h.e'_2.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)
case a.h.e'_2.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ V' x = V (Οƒ x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h1
case a.h.e'_2.a.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ V' x = V (Οƒ x)
case a.h.e'_2.a.h1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ isFreeIn x phi ∨ isFreeIn x psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
right
case a.h.e'_2.a.h1.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ isFreeIn x phi ∨ isFreeIn x psi
case a.h.e'_2.a.h1.a.h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ isFreeIn x psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact a1
case a.h.e'_2.a.h1.a.h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : isFreeIn x psi ⊒ isFreeIn x psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro x a1
case a.h.e'_2.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x
case a.h.e'_2.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V'' x = V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h2
case a.h.e'_2.a.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V'' x = V x
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Finset.mem_biUnion, Finset.mem_union] at a1
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply Exists.elim a1
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ βˆ€ (a : PredName Γ— β„•), a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a β†’ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro a a2
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a ⊒ βˆ€ (a : PredName Γ— β„•), a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a β†’ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Finset.mem_biUnion, Finset.mem_union]
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„)
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ βˆƒ a, (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply Exists.intro a
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ βˆƒ a, (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
tauto
case a.h.e'_2.a.h2.a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) phi psi : Formula phi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) psi_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x psi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ psi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E psi ↔ Holds D I V E (subAux c Ο„ Οƒ psi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x phi ∨ isFreeIn x psi β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (phi.predVarSet βˆͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : βˆƒ a ∈ psi.predVarSet, x ∈ predVarFreeVarSet Ο„ a a : PredName Γ— β„• a2 : a ∈ psi.predVarSet ∧ x ∈ predVarFreeVarSet Ο„ a ⊒ (a ∈ phi.predVarSet ∨ a ∈ psi.predVarSet) ∧ x ∈ predVarFreeVarSet Ο„ a
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [isFreeIn] at h1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), isFreeIn x_1 (exists_ x phi) β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ x phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ x phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [predVarSet] at h2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x_1 ∈ (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = V x_1 ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ x phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ x phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ x phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [I']
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (Interpretation.usingPred D I fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds) V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Interpretation.usingPred]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (Interpretation.usingPred D I fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds) V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } V' E (exists_ x phi) ↔ Holds D I V E (exists_ (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
first | apply forall_congr' | apply exists_congr
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x a) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) a) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro d
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x a) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) a) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply ih
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
case h.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D ⊒ βˆ€ (x_1 : VarName), isFreeIn x_1 phi β†’ Function.updateITE V' x d x_1 = Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) x_1) case h.h2 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D ⊒ βˆ€ x_1 ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x_1 = Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d x_1
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply forall_congr'
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (βˆ€ (d : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x a) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) a) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply exists_congr
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X ds.length).get β‹―).1 ds) E ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } (Function.updateITE V' x a) E phi ↔ Holds D I (Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) a) E (subAux c Ο„ (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x)) phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
intro v a1
case h.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D ⊒ βˆ€ (x_1 : VarName), isFreeIn x_1 phi β†’ Function.updateITE V' x d x_1 = Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) x_1)
case h.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi ⊒ Function.updateITE V' x d v = Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
split_ifs
case h.h1 D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi ⊒ Function.updateITE V' x d v = Function.updateITE V (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) d (Function.updateITE Οƒ x (if x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else x) v)
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi h✝ : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ Function.updateITE V' x d v = Function.updateITE V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) d (Function.updateITE Οƒ x (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) v) case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi h✝ : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ Function.updateITE V' x d v = Function.updateITE V x d (Function.updateITE Οƒ x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c1 => simp only [Function.updateITE] split_ifs case _ c2 c3 => rfl case _ c2 c3 => contradiction case _ c2 c3 => obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Οƒ x x) (freeVarSet phi)) βˆͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο„))) simp only [← c3] at s1 simp only [Finset.mem_union] at s1 simp only [isFreeIn_iff_mem_freeVarSet] at a1 obtain s2 := Finset.mem_image_of_mem (Function.updateITE Οƒ x x) a1 simp only [Function.updateITE] at s2 simp only [if_neg c2] at s2 exfalso apply s1 left exact s2 case _ c2 c3 => apply h1 tauto
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ Function.updateITE V' x d v = Function.updateITE V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) d (Function.updateITE Οƒ x (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Function.updateITE]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ Function.updateITE V' x d v = Function.updateITE V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) d (Function.updateITE Οƒ x (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) v)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ (if v = x then d else V' v) = if (if v = x then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else Οƒ v) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) then d else V (if v = x then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
split_ifs
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ (if v = x then d else V' v) = if (if v = x then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else Οƒ v) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) then d else V (if v = x then fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) else Οƒ v)
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) h✝¹ : v = x h✝ : fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = d case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) h✝¹ : v = x h✝ : Β¬fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„))) case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) h✝¹ : Β¬v = x h✝ : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = d case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) h✝¹ : Β¬v = x h✝ : ¬σ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = V (Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c2 c3 => rfl
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x c3 : fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = d
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c2 c3 => contradiction
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x c3 : Β¬fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c2 c3 => obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Οƒ x x) (freeVarSet phi)) βˆͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο„))) simp only [← c3] at s1 simp only [Finset.mem_union] at s1 simp only [isFreeIn_iff_mem_freeVarSet] at a1 obtain s2 := Finset.mem_image_of_mem (Function.updateITE Οƒ x x) a1 simp only [Function.updateITE] at s2 simp only [if_neg c2] at s2 exfalso apply s1 left exact s2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = d
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c2 c3 => apply h1 tauto
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : ¬σ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = V (Οƒ v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
rfl
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x c3 : fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = d
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
contradiction
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x c3 : Β¬fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ d = V (fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Οƒ x x) (freeVarSet phi)) βˆͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο„)))
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [← c3] at s1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Οƒ v βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Finset.mem_union] at s1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Οƒ v βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [isFreeIn_iff_mem_freeVarSet] at a1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
obtain s2 := Finset.mem_image_of_mem (Function.updateITE Οƒ x x) a1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Function.updateITE Οƒ x x v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Function.updateITE] at s2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Function.updateITE Οƒ x x v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : (if v = x then x else Οƒ v) ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [if_neg c2] at s2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : (if v = x then x else Οƒ v) ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exfalso
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ V' v = d
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply s1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ False
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
left
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
exact s2
case h D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) s1 : Β¬(Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ∨ Οƒ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) a1 : v ∈ phi.freeVarSet s2 : Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet ⊒ Οƒ v ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
apply h1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : ¬σ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ V' v = V (Οƒ v)
case a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : ¬σ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ Β¬v = x ∧ isFreeIn v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
tauto
case a D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x ∈ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : ¬σ v = fresh x c (Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„)) ⊒ Β¬v = x ∧ isFreeIn v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Function.updateITE]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ Function.updateITE V' x d v = Function.updateITE V x d (Function.updateITE Οƒ x x v)
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
by_cases c2 : v = x
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v)
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v) case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [if_pos c2]
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v)
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x ⊒ d = if True then d else V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : v = x ⊒ d = if True then d else V x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [if_neg c2]
case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x ⊒ (if v = x then d else V' v) = if (if v = x then x else Οƒ v) = x then d else V (if v = x then x else Οƒ v)
case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x ⊒ V' v = if Οƒ v = x then d else V (Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
split_ifs
case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x ⊒ V' v = if Οƒ v = x then d else V (Οƒ v)
case pos D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x h✝ : Οƒ v = x ⊒ V' v = d case neg D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x h✝ : ¬σ v = x ⊒ V' v = V (Οƒ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
case _ c3 => obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Οƒ x x) c phi simp only [s1] at c1 simp only [← c3] at c1 simp only [Finset.mem_union] at c1 simp only [isFreeIn_iff_mem_freeVarSet] at a1 obtain s2 := Finset.mem_image_of_mem (Function.updateITE Οƒ (Οƒ v) (Οƒ v)) a1 simp only [Function.updateITE] at s2 simp only [ite_self] at s2 exfalso apply c1 left exact s2
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : isFreeIn v phi c1 : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet βˆͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c2 : Β¬v = x c3 : Οƒ v = x ⊒ V' v = d
no goals