Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files
xet
 Community
1
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stringclasses
147 values
commit
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7
101
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94
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https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
simp only [← s1]
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (Ο„ X ds.length).1; let H := (Ο„ X ds.length).2; if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F) ⊒ Holds D I V E (replace Ο„ F)
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (Ο„ X ds.length).1; let H := (Ο„ X ds.length).2; if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
apply h2
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (Ο„ X ds.length).1; let H := (Ο„ X ds.length).2; if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E F
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]
induction F generalizing V V' Οƒ
D : Type I : Interpretation D V V' V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) Οƒ : VarName β†’ VarName F : Formula h1 : βˆ€ (x : VarName), isFreeIn x F β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ F.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E F ↔ Holds D I V E (subAux c Ο„ Οƒ F)
case pred_const_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (pred_const_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_const_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_const_ a✝¹ a✝)) case pred_var_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ : PredName a✝ : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (pred_var_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_var_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ a✝¹ a✝)) case eq_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ a✝ : VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (eq_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (eq_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (eq_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (eq_ a✝¹ a✝)) case true_ 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 true_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ true_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E true_ ↔ Holds D I V E (subAux c Ο„ Οƒ true_) case 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 (subAux c Ο„ Οƒ false_) case not_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x a✝.not_ β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ a✝.not_.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E a✝.not_ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝.not_) case imp_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝¹ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝¹.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝¹ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (a✝¹.imp_ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (a✝¹.imp_ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (a✝¹.imp_ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (a✝¹.imp_ a✝)) case and_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝¹ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝¹.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝¹ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (a✝¹.and_ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (a✝¹.and_ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (a✝¹.and_ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (a✝¹.and_ a✝)) case or_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝¹ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝¹.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝¹ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (a✝¹.or_ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (a✝¹.or_ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (a✝¹.or_ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (a✝¹.or_ a✝)) case iff_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝¹ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝¹.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝¹ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝¹)) a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (a✝¹.iff_ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (a✝¹.iff_ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (a✝¹.iff_ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (a✝¹.iff_ a✝)) case forall_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (forall_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (forall_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (forall_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (forall_ a✝¹ a✝)) case exists_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x a✝ β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ a✝.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E a✝ ↔ Holds D I V E (subAux c Ο„ Οƒ a✝)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (exists_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (exists_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (exists_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (exists_ a✝¹ a✝)) case def_ D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) a✝¹ : DefName a✝ : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ a✝¹ a✝) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ a✝¹ a✝).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (def_ a✝¹ a✝) ↔ Holds D I V E (subAux c Ο„ Οƒ (def_ a✝¹ 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]
case pred_const_ X xs => simp only [isFreeIn] at h1 simp only [subAux] simp only [Holds] simp only [I'] simp only [Interpretation.usingPred] simp congr! 1 simp only [List.map_eq_map_iff] intro x a1 simp exact h1 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (pred_const_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_const_ X xs))
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 eq_ x y => simp only [isFreeIn] at h1 simp only [subAux] simp only [Holds] have s1 : V' x = V (Οƒ x) apply h1 simp simp only [s1] have s2 : V' y = V (Οƒ y) apply h1 simp 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), 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))
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 true_ | false_ => simp only [subAux] 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 (subAux c Ο„ Οƒ 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]
case not_ phi phi_ih => simp only [isFreeIn] at h1 simp only [predVarSet] at h2 simp only [subAux] simp only [Holds] congr! 1 exact phi_ih V V' Οƒ h1 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.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_)
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (pred_const_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_const_ X 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_const_ X 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 [subAux]
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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_const_ X 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (pred_const_ X (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 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_const_ X xs) ↔ Holds D I V E (pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (I' D I V'' E Ο„).pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (I' D I V'' E Ο„).pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (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).pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (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).pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ I.pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ I.pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ I.pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ I.pred_const_ X (List.map V' xs) ↔ I.pred_const_ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x 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 ∈ (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (pred_var_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 ∈ (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 [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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 ∈ {(X, xs.length)}.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ {(X, xs.length)}.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 ∈ predVarFreeVarSet Ο„ (X, xs.length), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 [predVarFreeVarSet] 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ predVarFreeVarSet Ο„ (X, xs.length), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 [subAux]
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 ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (pred_var_ X 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 ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (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 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 ⊒ Holds D (I' D I V'' E Ο„) V' E (pred_var_ X xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else 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 ⊒ (I' D I V'' E Ο„).pred_var_ X (List.map V' xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (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 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 : 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 ⊒ (I' D I V'' E Ο„).pred_var_ X (List.map V' xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else 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 ⊒ (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).pred_var_ X (List.map V' xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (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 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 : 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 ⊒ (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).pred_var_ X (List.map V' xs) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else 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 ⊒ (if h : (Ο„ X (List.map V' xs).length).isSome = true then if (List.map V' xs).length = ((Ο„ X (List.map V' xs).length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X (List.map V' xs).length).get β‹―).1 (List.map V' xs)) E ((Ο„ X (List.map V' xs).length).get β‹―).2 else I.pred_var_ X (List.map V' xs) else I.pred_var_ X (List.map V' xs)) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (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 ⊒ (if h : (Ο„ X (List.map V' xs).length).isSome = true then if (List.map V' xs).length = ((Ο„ X (List.map V' xs).length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X (List.map V' xs).length).get β‹―).1 (List.map V' xs)) E ((Ο„ X (List.map V' xs).length).get β‹―).2 else I.pred_var_ X (List.map V' xs) else I.pred_var_ X (List.map V' xs)) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else 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 ⊒ (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 else I.pred_var_ X (List.map V' xs) else I.pred_var_ X (List.map V' xs)) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (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]
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 : 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 ⊒ (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 else I.pred_var_ X (List.map V' xs) else I.pred_var_ X (List.map V' xs)) ↔ Holds D I V E (if h : (Ο„ X xs.length).isSome = true then if xs.length = ((Ο„ X xs.length).get β‹―).1.length then Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2 else pred_var_ X (List.map Οƒ xs) else pred_var_ X (List.map Οƒ xs))
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 : 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 h✝¹ : (Ο„ X xs.length).isSome = true h✝ : xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2) 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 : 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 h✝¹ : (Ο„ X xs.length).isSome = true h✝ : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ I.pred_var_ X (List.map V' xs) ↔ Holds D I V E (pred_var_ X (List.map Οƒ xs)) 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 : 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 h✝ : 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))
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 c2 => simp only [Holds] simp congr! 1 simp only [List.map_eq_map_iff] intro x a1 simp exact h1 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 : 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 ⊒ I.pred_var_ X (List.map V' xs) ↔ Holds D I V E (pred_var_ X (List.map Οƒ xs))
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 _ c1 => simp only [Holds] simp congr! 1 simp only [List.map_eq_map_iff] intro x a1 simp exact h1 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 : 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))
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 [c1] 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 : 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 : xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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) c1 : (Ο„ X xs.length).isSome = true c2 : xs.length = ((Ο„ X xs.length).get β‹―).1.length h2 : βˆ€ x ∈ if h : True then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true c2 : xs.length = ((Ο„ X xs.length).get β‹―).1.length h2 : βˆ€ x ∈ if h : True then ((Ο„ X xs.length).get β‹―).2.freeVarSet \ ((Ο„ X xs.length).get β‹―).1.toFinset else βˆ…, V'' x = V x ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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) c1 : (Ο„ X xs.length).isSome = true c2 : xs.length = ((Ο„ X xs.length).get β‹―).1.length h2 : βˆ€ x ∈ ((Ο„ X xs.length).get β‹―).2.freeVarSet, x βˆ‰ ((Ο„ X xs.length).get β‹―).1 β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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]
set zs := (Option.get (Ο„ X (List.length xs)) (_ : Option.isSome (Ο„ X (List.length xs)) = true)).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) c1 : (Ο„ X xs.length).isSome = true c2 : xs.length = ((Ο„ X xs.length).get β‹―).1.length h2 : βˆ€ x ∈ ((Ο„ X xs.length).get β‹―).2.freeVarSet, x βˆ‰ ((Ο„ X xs.length).get β‹―).1 β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' ((Ο„ X xs.length).get β‹―).1 (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο„ X xs.length).get β‹―).1 (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length h2 : βˆ€ x ∈ ((Ο„ X xs.length).get β‹―).2.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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]
set H := (Option.get (Ο„ X (List.length xs)) (_ : Option.isSome (Ο„ X (List.length xs)) = true)).2
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length h2 : βˆ€ x ∈ ((Ο„ X xs.length).get β‹―).2.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E ((Ο„ X xs.length).get β‹―).2 ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c ((Ο„ X xs.length).get β‹―).2)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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 := Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateListITE id zs (xs.map Οƒ)) c H
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (V ∘ Function.updateListITE id zs (List.map Οƒ xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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.updateListITE_comp] 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (V ∘ Function.updateListITE id zs (List.map Οƒ xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE (V ∘ id) zs (List.map V (List.map Οƒ xs))) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE (V ∘ id) zs (List.map V (List.map Οƒ xs))) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H)
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H
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]
clear s1
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x s1 : Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Οƒ xs)) c H) ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H
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 Holds_coincide_Var
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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H ↔ Holds D I (Function.updateListITE V zs (List.map (V ∘ Οƒ) xs)) E H
case h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ βˆ€ (v : VarName), isFreeIn v H β†’ Function.updateListITE V'' zs (List.map V' xs) v = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) 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]
intro x a1
case h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x ⊒ βˆ€ (v : VarName), isFreeIn v H β†’ Function.updateListITE V'' zs (List.map V' xs) v = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) v
case h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) 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]
by_cases c3 : x ∈ zs
case h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) x
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) x 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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) 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 Function.updateListITE_map_mem_ext
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) x
case pos.h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ βˆ€ y ∈ xs, V' y = (V ∘ Οƒ) y case pos.h2 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ zs.length = xs.length case pos.h3 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ x ∈ zs
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.h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ βˆ€ y ∈ xs, V' y = (V ∘ Οƒ) y
case pos.h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ βˆ€ y ∈ xs, V' y = 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]
exact h1
case pos.h1 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ βˆ€ y ∈ xs, V' y = 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 [← c2]
case pos.h2 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ zs.length = xs.length
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]
exact c3
case pos.h3 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x ∈ zs ⊒ x ∈ zs
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.updateListITE_not_mem V'' x zs (List.map V' xs) c3]
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) x
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ V'' x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) 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 only [Function.updateListITE_not_mem V x zs (List.map (V ∘ Οƒ ) xs) c3]
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ V'' x = Function.updateListITE V zs (List.map (V ∘ Οƒ) xs) x
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 : PredName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ 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 neg 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ V'' x = V x
case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ x ∈ H.freeVarSet case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ x βˆ‰ zs
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
case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ x ∈ H.freeVarSet
case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName c3 : x βˆ‰ zs a1 : x ∈ H.freeVarSet ⊒ x ∈ H.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 a1
case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName c3 : x βˆ‰ zs a1 : x ∈ H.freeVarSet ⊒ x ∈ H.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]
exact c3
case neg.a 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) c1 : (Ο„ X xs.length).isSome = true zs : List VarName := ((Ο„ X xs.length).get β‹―).1 c2 : xs.length = zs.length H : Formula := ((Ο„ X xs.length).get β‹―).2 h2 : βˆ€ x ∈ H.freeVarSet, x βˆ‰ zs β†’ V'' x = V x x : VarName a1 : isFreeIn x H c3 : x βˆ‰ zs ⊒ x βˆ‰ zs
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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (Ο„ X xs.length).isSome = true c2 : Β¬xs.length = ((Ο„ X xs.length).get β‹―).1.length ⊒ 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 : (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
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 : (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
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)
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„)