Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
state_before
stringlengths
3
2.09M
state_after
stringlengths
6
2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [Function.updateITE]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c1_left : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet c1_right : x βˆ‰ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) s1 : Β¬v = x ⊒ V'' v = Function.updateITE V x d v
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c1_left : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet c1_right : x βˆ‰ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) s1 : Β¬v = x ⊒ V'' v = if v = x then d else V v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [if_neg s1]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c1_left : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet c1_right : x βˆ‰ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) s1 : Β¬v = x ⊒ V'' v = if v = x then d else V v
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c1_left : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet c1_right : x βˆ‰ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) s1 : Β¬v = x ⊒ V'' v = V 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]
exact h2 v a1
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) x : VarName phi : Formula ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x phi β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E Ο„) V' E phi ↔ Holds D I V E (subAux c Ο„ Οƒ phi)) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x_1 : VarName), Β¬x_1 = x ∧ isFreeIn x_1 phi β†’ V' x_1 = V (Οƒ x_1) h2 : βˆ€ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x d : D v : VarName a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) c1_left : x βˆ‰ Finset.image (Function.updateITE Οƒ x x) phi.freeVarSet c1_right : x βˆ‰ phi.predVarSet.biUnion (predVarFreeVarSet Ο„) s1 : Β¬v = x ⊒ V'' v = V v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
simp only [subAux]
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (def_ X xs) ↔ Holds D I V E (subAux c Ο„ Οƒ (def_ 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 : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (def_ X xs) ↔ Holds D I V E (def_ 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]
induction E generalizing V V' Οƒ
D : Type I : Interpretation D V'' : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' E Ο„) V' E (def_ X xs) ↔ Holds D I V E (def_ X (List.map Οƒ xs))
case nil D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' [] Ο„) V' [] (def_ X xs) ↔ Holds D I V [] (def_ X (List.map Οƒ xs)) case cons D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName head✝ : Definition tail✝ : List Definition tail_ih✝ : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' tail✝ Ο„) V' tail✝ (def_ X xs) ↔ Holds D I V tail✝ (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' (head✝ :: tail✝) Ο„) V' (head✝ :: tail✝) (def_ X xs) ↔ Holds D I V (head✝ :: tail✝) (def_ 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 nil => simp only [Holds]
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' [] Ο„) V' [] (def_ X xs) ↔ Holds D I V [] (def_ 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 [Holds]
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' [] Ο„) V' [] (def_ X xs) ↔ Holds D I V [] (def_ 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 [isFreeIn] at h1
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V (E_hd :: E_tl) (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V (E_hd :: E_tl) (def_ 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 c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V (E_hd :: E_tl) (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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]
have s1 : (List.map V' xs) = (List.map (V ∘ Οƒ) xs)
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ List.map V' xs = List.map (V ∘ Οƒ) xs D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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 [List.map_eq_map_iff]
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ List.map V' xs = List.map (V ∘ Οƒ) xs D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ xs, V' x = (V ∘ Οƒ) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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]
intro x a1
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ βˆ€ x ∈ xs, V' x = (V ∘ Οƒ) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ xs ⊒ V' x = (V ∘ Οƒ) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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]
exact h1 x a1
case s1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x x : VarName a1 : x ∈ xs ⊒ V' x = (V ∘ Οƒ) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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 [s1]
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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]
clear s1
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x s1 : List.map V' xs = List.map (V ∘ Οƒ) xs ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ 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 c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x ⊒ (if X = E_hd.name ∧ xs.length = E_hd.args.length then Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q else Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs)) ↔ if X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length then Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q else Holds D I V E_tl (def_ X (List.map Οƒ xs))
case pos D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x h✝¹ : X = E_hd.name ∧ xs.length = E_hd.args.length h✝ : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q case neg D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x h✝¹ : X = E_hd.name ∧ xs.length = E_hd.args.length h✝ : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) case pos D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x h✝¹ : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h✝ : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q case neg D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x h✝¹ : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) h✝ : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ 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 [List.length_map] at c2 contradiction
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I V E_tl (def_ 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 c2 => simp only [List.length_map] at c2 contradiction
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
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]
have s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
case s2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
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
case s2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
case s2.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v E_hd.q β†’ Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
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 s2.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v E_hd.q β†’ Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs) v = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) v D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
case s2.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs) x = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
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 s2.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs) x = Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs) x D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
case s2.h1.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ βˆ€ y ∈ xs, (V ∘ Οƒ) y = (V ∘ Οƒ) y case s2.h1.h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ E_hd.args.length = xs.length case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ x ∈ E_hd.args D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
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 c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
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 c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
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 s2
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
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_PredVar
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_const_ = I.pred_const_ case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length E_hd.q β†’ ((I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds)
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.h1.h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ βˆ€ y ∈ xs, (V ∘ Οƒ) 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]
tauto
case s2.h1.h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ E_hd.args.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]
simp only [isFreeIn_iff_mem_freeVarSet] at a1
case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : isFreeIn x E_hd.q ⊒ x ∈ E_hd.args
case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : x ∈ E_hd.q.freeVarSet ⊒ x ∈ E_hd.args
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.mem_toFinset]
case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : x ∈ E_hd.q.freeVarSet ⊒ x ∈ E_hd.args
case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : x ∈ E_hd.q.freeVarSet ⊒ x ∈ E_hd.args.toFinset
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 Finset.mem_of_subset E_hd.h1 a1
case s2.h1.h3 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length x : VarName a1 : x ∈ E_hd.q.freeVarSet ⊒ x ∈ E_hd.args.toFinset
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 [I']
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_const_ = I.pred_const_
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ (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_hd :: E_tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_ = I.pred_const_
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]
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ (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_hd :: E_tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_ = I.pred_const_
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 P ds a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length E_hd.q β†’ ((I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : predVarOccursIn P ds.length E_hd.q ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
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 [predVarOccursIn_iff_mem_predVarSet] at a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : predVarOccursIn P ds.length E_hd.q ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : (P, ds.length) ∈ E_hd.q.predVarSet ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
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 [E_hd.h2] at a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : (P, ds.length) ∈ E_hd.q.predVarSet ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : (P, ds.length) ∈ βˆ… ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
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 a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length P : PredName ds : List D a1 : (P, ds.length) ∈ βˆ… ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ I.pred_var_ P ds
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 [List.length_map] at c2
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I V E_tl (def_ 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]
contradiction
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : X = E_hd.name ∧ xs.length = E_hd.args.length c2 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) (Function.updateListITE V' E_hd.args (List.map (V ∘ Οƒ) xs)) E_tl E_hd.q ↔ Holds D I V E_tl (def_ 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 [List.length_map] at c2
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
contradiction
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : X = E_hd.name ∧ xs.length = E_hd.args.length ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Οƒ xs))) E_tl E_hd.q
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
[123, 1]
[434, 44]
obtain s2 := E_ih V V' Οƒ
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ 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 [isFreeIn] at s2
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : (βˆ€ x ∈ xs, V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ 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]
specialize s2 h1 h2
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : (βˆ€ x ∈ xs, V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ 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 [← s2]
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ 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]
apply Holds_coincide_PredVar
D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ Holds D (I' D I V'' (E_hd :: E_tl) Ο„) V' E_tl (def_ X xs) ↔ Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs)
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_const_ = (I' D I V'' E_tl Ο„).pred_const_ case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ((I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ (I' D I V'' E_tl Ο„).pred_var_ P ds)
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']
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_const_ = (I' D I V'' E_tl Ο„).pred_const_
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (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_hd :: E_tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_ = (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_tl ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_
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]
case h1 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ (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_hd :: E_tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_ = (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_tl ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds).pred_const_
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 P ds a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ((I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ (I' D I V'' E_tl Ο„).pred_var_ P ds)
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) P : PredName ds : List D a1 : predVarOccursIn P ds.length (def_ X xs) ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ (I' D I V'' E_tl Ο„).pred_var_ P ds
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 [predVarOccursIn] at a1
case h2 D : Type I : Interpretation D V'' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName E_hd : Definition E_tl : List Definition E_ih : βˆ€ (V V' : VarAssignment D) (Οƒ : VarName β†’ VarName), (βˆ€ (x : VarName), isFreeIn x (def_ X xs) β†’ V' x = V (Οƒ x)) β†’ (βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x) β†’ (Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs))) V V' : VarAssignment D Οƒ : VarName β†’ VarName h1 : βˆ€ x ∈ xs, V' x = V (Οƒ x) h2 : βˆ€ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο„), V'' x = V x c1 : Β¬(X = E_hd.name ∧ xs.length = E_hd.args.length) c2 : Β¬(X = E_hd.name ∧ (List.map Οƒ xs).length = E_hd.args.length) s2 : Holds D (I' D I V'' E_tl Ο„) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map Οƒ xs)) P : PredName ds : List D a1 : predVarOccursIn P ds.length (def_ X xs) ⊒ (I' D I V'' (E_hd :: E_tl) Ο„).pred_var_ P ds ↔ (I' D I V'' E_tl Ο„).pred_var_ P ds
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
[437, 1]
[449, 9]
apply substitution_theorem_aux
D : Type I : Interpretation D V : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula ⊒ Holds D (I' D I V E Ο„) V E F ↔ Holds D I V E (sub c Ο„ F)
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula ⊒ βˆ€ (x : VarName), isFreeIn x F β†’ V x = V (id x) case h2 D : Type I : Interpretation D V : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula ⊒ βˆ€ x ∈ F.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
[437, 1]
[449, 9]
simp
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula ⊒ βˆ€ (x : VarName), isFreeIn x F β†’ V x = V (id 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
[437, 1]
[449, 9]
simp
case h2 D : Type I : Interpretation D V : VarAssignment D E : Env c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula ⊒ βˆ€ x ∈ F.predVarSet.biUnion (predVarFreeVarSet Ο„), V x = V x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
[452, 1]
[464, 11]
simp only [IsValid] at h1
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : F.IsValid ⊒ (sub c Ο„ F).IsValid
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (sub c Ο„ F).IsValid
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_is_valid
[452, 1]
[464, 11]
simp only [IsValid]
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (sub c Ο„ F).IsValid
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο„ F)
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_is_valid
[452, 1]
[464, 11]
intro D I V E
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο„ F)
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : 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 ⊒ Holds D I V E (sub c Ο„ F)
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_is_valid
[452, 1]
[464, 11]
simp only [← substitution_theorem]
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : 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 ⊒ Holds D I V E (sub c Ο„ F)
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : 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 ⊒ Holds D (I' D I V E Ο„) V E F
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_is_valid
[452, 1]
[464, 11]
apply h1
c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) F : Formula h1 : 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 ⊒ Holds D (I' D I V E Ο„) V E F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
induction h1 generalizing V
D : Type I J : Interpretation D V : VarAssignment D E : Env A : Formula P : PredName zs : List VarName H B : Formula h1 : IsSub P zs H A B h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) ⊒ Holds D I V E B ↔ Holds D J V E A
case pred_const_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName xs✝ : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ X✝ xs✝) ↔ Holds D J V E (pred_const_ X✝ xs✝) case pred_not_occurs_in D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName xs✝ : List VarName a✝ : Β¬(X✝ = P ∧ xs✝.length = zs.length) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_var_ X✝ xs✝) ↔ Holds D J V E (pred_var_ X✝ xs✝) case pred_occurs_in D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName ts✝ : List VarName a✝¹ : X✝ = P ∧ ts✝.length = zs.length a✝ : Var.All.Rec.admits (Function.updateListITE id zs ts✝) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs ts✝) H) ↔ Holds D J V E (pred_var_ P ts✝) case eq_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ y✝ : VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (eq_ x✝ y✝) ↔ Holds D J V E (eq_ x✝ y✝) case true_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E true_ ↔ Holds D J V E true_ case false_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E false_ ↔ Holds D J V E false_ case not_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ phi'✝ : Formula a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E phi'✝.not_ ↔ Holds D J V E phi✝.not_ case imp_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.imp_ psi'✝) ↔ Holds D J V E (phi✝.imp_ psi✝) case and_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.and_ psi'✝) ↔ Holds D J V E (phi✝.and_ psi✝) case or_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.or_ psi'✝) ↔ Holds D J V E (phi✝.or_ psi✝) case iff_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.iff_ psi'✝) ↔ Holds D J V E (phi✝.iff_ psi✝) case forall_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ : VarName phi✝ phi'✝ : Formula a✝¹ : Β¬isFreeIn x✝ H a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (forall_ x✝ phi'✝) ↔ Holds D J V E (forall_ x✝ phi✝) case exists_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ : VarName phi✝ phi'✝ : Formula a✝¹ : Β¬isFreeIn x✝ H a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (exists_ x✝ phi'✝) ↔ Holds D J V E (exists_ x✝ phi✝) case def_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : DefName xs✝ : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (def_ X✝ xs✝) ↔ Holds D J V E (def_ X✝ xs✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case pred_const_ h1_X h1_ts => simp only [Holds] simp only [h3_const]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ h1_X h1_ts) ↔ Holds D J V E (pred_const_ h1_X h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case pred_occurs_in h1_X h1_ts h1_1 h1_2 => obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2 obtain s2 := Function.updateListITE_comp id V zs h1_ts simp only [s2] at s1 simp at s1 specialize h2 h1_X (List.map V h1_ts) simp only [s1] at h2 simp only [Holds] apply h2 simp exact h1_1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case eq_ h1_x h1_y => simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x h1_y : VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (eq_ h1_x h1_y) ↔ Holds D J V E (eq_ h1_x h1_y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case true_ | false_ => simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E false_ ↔ Holds D J V E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case not_ h1_phi h1_phi' _ h1_ih => simp only [Holds] congr! 1 exact h1_ih V h2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E h1_phi'.not_ ↔ Holds D J V E h1_phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case forall_ h1_x h1_phi h1_phi' h1_1 _ h1_ih | exists_ h1_x h1_phi h1_phi' h1_1 _ h1_ih => simp only [Holds] first | apply forall_congr' | apply exists_congr intro d apply h1_ih intro Q ds a1 specialize h2 Q ds a1 have s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H := by apply Holds_coincide_Var intro v a1 apply Function.updateListITE_updateIte intro contra subst contra contradiction simp only [h2] at s1 exact s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : Β¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (exists_ h1_x h1_phi') ↔ Holds D J V E (exists_ h1_x h1_phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ h1_X h1_ts) ↔ Holds D J V E (pred_const_ h1_X h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ I.pred_const_ h1_X (List.map V h1_ts) ↔ J.pred_const_ h1_X (List.map V h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [h3_const]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ I.pred_const_ h1_X (List.map V h1_ts) ↔ J.pred_const_ h1_X (List.map V h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at h1_1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : Β¬(h1_X = P ∧ h1_ts.length = zs.length) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply Holds_coincide_PredVar
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_const_ = J.pred_const_ case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β†’ (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact h3_const
case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_const_ = J.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro X ds a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β†’ (I.pred_var_ P ds ↔ J.pred_var_ P ds)
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts) ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [predVarOccursIn] at a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts) ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : X = h1_X ∧ ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
cases a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : X = h1_X ∧ ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case h2.intro D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D left✝ : X = h1_X right✝ : ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
subst a1_left
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1_left : X = h1_X a1_right : ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply h3_var
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ Β¬(X = P ∧ ds.length = zs.length)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ Β¬(X = P ∧ ds.length = zs.length)
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ X = P β†’ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro a2
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ X = P β†’ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length a2 : X = P ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
subst a2
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length a2 : X = P ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : X = X β†’ Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at h1_1
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : X = X β†’ Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro contra
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply h1_1
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ False
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
trans List.length ds
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = zs.length
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = ds.length D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [eq_comm]
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = ds.length
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = h1_ts.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact a1_right
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = h1_ts.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact contra
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = zs.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
obtain s2 := Function.updateListITE_comp id V zs h1_ts
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [s2] at s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE (V ∘ id) zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE (V ∘ id) zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
specialize h2 h1_X (List.map V h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [s1] at h2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply h2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ h1_X = P ∧ (List.map V h1_ts).length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ h1_X = P ∧ (List.map V h1_ts).length = zs.length
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ h1_X = P ∧ h1_ts.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact h1_1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ h1_X = P ∧ h1_ts.length = zs.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x h1_y : VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (eq_ h1_x h1_y) ↔ Holds D J V E (eq_ h1_x h1_y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E false_ ↔ Holds D J V E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E h1_phi'.not_ ↔ Holds D J V E h1_phi.not_
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Β¬Holds D I V E h1_phi' ↔ Β¬Holds D J V E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
congr! 1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Β¬Holds D I V E h1_phi' ↔ Β¬Holds D J V E h1_phi
case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact h1_ih V h2
case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi
no goals