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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | tauto | case h.h2
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_t ∉ binders ∧ ¬h1_1_t = h1_1_x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | tauto | case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | induction h1 | F F' : Formula
v u : VarName
h1 : IsSub F v u F'
⊢ Rec.fastReplaceFree v u F = F' | case pred_const_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_const_ X✝ xs✝) = pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case pred_var_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_var_ X✝ xs✝) = pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case eq_
F F' : Formula
v u x✝ y✝ v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (eq_ x✝ y✝) = eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝)
case true_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ true_ = true_
case false_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ false_ = false_
case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ phi✝.not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.imp_ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.and_ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.or_ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.iff_ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi'✝
case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | all_goals
simp only [Rec.fastReplaceFree] | case pred_const_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_const_ X✝ xs✝) = pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case pred_var_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_var_ X✝ xs✝) = pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case eq_
F F' : Formula
v u x✝ y✝ v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (eq_ x✝ y✝) = eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝)
case true_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ true_ = true_
case false_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ false_ = false_
case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ phi✝.not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.imp_ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.and_ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.or_ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.iff_ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi'✝
case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) | case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).imp_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).and_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).or_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).iff_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ (if v✝ = x✝ then forall_ x✝ phi✝ else forall_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (if v✝ = x✝ then forall_ x✝ phi✝ else forall_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ (if v✝ = x✝ then exists_ x✝ phi✝ else exists_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (if v✝ = x✝ then exists_ x✝ phi✝ else exists_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = exists_ x✝ phi'✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case not_ h1_phi h1_v h1_t h1_phi' h1_1 h1_ih =>
tauto | F F' : Formula
v u : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).not_ = h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
imp_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| and_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| or_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| iff_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2 =>
subst h1_ih_1
subst h1_ih_2
rfl | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) = h1_phi'.iff_ h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
forall_not_free_in h1_x h1_phi h1_v h1_t h1_1
| exists_not_free_in h1_x h1_phi h1_v h1_t h1_1 =>
simp only [isFreeIn] at h1_1
simp at h1_1
split_ifs
case pos c1 =>
rfl
case neg c1 =>
congr!
apply Rec.not_free_in_fastReplaceFree_self
exact h1_1 c1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
forall_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 _ _ h1_ih
| exists_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 _ _ h1_ih =>
simp only [isFreeIn] at h1_1
cases h1_1
case intro h1_1_left h1_1_right =>
simp only [if_neg h1_1_left]
subst h1_ih
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [Rec.fastReplaceFree] | case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | tauto | F F' : Formula
v u : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).not_ = h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih_1 | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) = h1_phi'.iff_ h1_psi' | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_psi' : Formula
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih_2 | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_psi' : Formula
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ h1_psi' | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
h1_2 : IsSub h1_psi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_psi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
h1_2 : IsSub h1_psi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_psi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [isFreeIn] at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | split_ifs | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | case pos
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
h✝ : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi
case neg
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
h✝ : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case pos c1 =>
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case neg c1 =>
congr!
apply Rec.not_free_in_fastReplaceFree_self
exact h1_1 c1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | congr! | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi | case h.e'_2
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | apply Rec.not_free_in_fastReplaceFree_self | case h.e'_2
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi | case h.e'_2.h1
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ ¬isFreeIn h1_v h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | exact h1_1 c1 | case h.e'_2.h1
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ ¬isFreeIn h1_v h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [isFreeIn] at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | cases h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | case intro
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
left✝ : ¬h1_v = h1_x
right✝ : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case intro h1_1_left h1_1_right =>
simp only [if_neg h1_1_left]
subst h1_ih
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [if_neg h1_1_left] | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
a✝¹ : ¬h1_x = h1_t
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
a✝ : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
a✝¹ : ¬h1_x = h1_t
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
a✝ : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | induction h1 generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
⊢ Holds D I (Function.updateITE V v (V t)) E F ↔ Holds D I V E F' | case pred_const_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (pred_const_ X✝ xs✝) ↔
Holds D I V E (pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝))
case pred_var_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (pred_var_ X✝ xs✝) ↔
Holds D I V E (pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝))
case eq_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ y✝ v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (eq_ x✝ y✝) ↔
Holds D I V E (eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝))
case true_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝.not_ ↔ Holds D I V E phi'✝.not_
case imp_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.imp_ psi✝) ↔ Holds D I V E (phi'✝.imp_ psi'✝)
case and_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.and_ psi✝) ↔ Holds D I V E (phi'✝.and_ psi'✝)
case or_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.or_ psi✝) ↔ Holds D I V E (phi'✝.or_ psi'✝)
case iff_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.iff_ psi✝) ↔ Holds D I V E (phi'✝.iff_ psi'✝)
case forall_not_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ x✝ phi✝)
case forall_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ x✝ phi'✝)
case exists_not_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ x✝ phi✝)
case exists_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ x✝ phi'✝)
case def_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (def_ X✝ xs✝) ↔
Holds D I V E (def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case pred_const_ h1_X h1_xs h1_v h1_t | pred_var_ h1_X h1_xs h1_v h1_t =>
simp only [Holds]
congr! 1
simp
simp only [List.map_eq_map_iff]
intro x _
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 =>
simp
simp only [if_pos c1]
case _ c1 =>
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (pred_var_ h1_X h1_xs) ↔
Holds D I V E (pred_var_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case eq_ h1_x h1_y h1_v h1_t =>
simp only [Holds]
simp only [Function.updateITE]
simp only [eq_comm]
congr! 1 <;> { split_ifs <;> rfl } | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (eq_ h1_x h1_y) ↔
Holds D I V E (eq_ (if h1_v = h1_x then h1_t else h1_x) (if h1_v = h1_y then h1_t else h1_y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case true_ _ _ | false_ _ _ =>
simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case not_ h1_phi h1_v h1_t h1_phi' _ h1_ih =>
simp only [Holds]
congr! 1
apply h1_ih | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case
forall_not_free_in h1_x h1_phi h1_v h1_t h1_1
| exists_not_free_in h1_x h1_phi h1_v h1_t h1_1 =>
simp only [isFreeIn] at h1_1
simp at h1_1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply Holds_coincide_Var
intro x a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
rfl
case _ c1 c2 =>
subst c2
tauto
case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case
forall_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 h1_2 _ h1_ih
| exists_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 h1_2 _ h1_ih =>
simp only [isFreeIn] at h1_1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
specialize h1_ih (Function.updateITE V h1_x d)
simp only [← h1_ih]
apply Holds_coincide_Var
intro x _
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 c2 c3 =>
subst c2
cases h1_1
case intro h1_1_left h1_1_right =>
contradiction
case _ | _ | _ =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (pred_var_ h1_X h1_xs) ↔
Holds D I V E (pred_var_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ I.pred_var_ h1_X (List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) ↔
I.pred_var_ h1_X (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ I.pred_var_ h1_X (List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) ↔
I.pred_var_ h1_X (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs =
List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs =
List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs) | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x _ | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
h✝ : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
h✝ : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
simp
simp only [if_pos c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = V (if h1_v = x then h1_t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [if_pos c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = V (if h1_v = x then h1_t else x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = V (if h1_v = x then h1_t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = V (if h1_v = x then h1_t else x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (eq_ h1_x h1_y) ↔
Holds D I V E (eq_ (if h1_v = h1_x then h1_t else h1_x) (if h1_v = h1_y then h1_t else h1_y)) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Function.updateITE V h1_v (V h1_t) h1_x = Function.updateITE V h1_v (V h1_t) h1_y ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Function.updateITE V h1_v (V h1_t) h1_x = Function.updateITE V h1_v (V h1_t) h1_y ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_x = h1_v then h1_t else h1_x) = V (if h1_y = h1_v then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 <;> { split_ifs <;> rfl } | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_x = h1_v then h1_t else h1_x) = V (if h1_y = h1_v then h1_t else h1_y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs <;> rfl | case a.h.e'_3
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ (if h1_y = h1_v then V h1_t else V h1_y) = V (if h1_y = h1_v then h1_t else h1_y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ ¬Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ ¬Holds D I V E h1_phi' | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (h1_phi.iff_ h1_psi) ↔ Holds D I V E (h1_phi'.iff_ h1_psi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi) ↔
(Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi) ↔
(Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih_1 | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih_2 | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [isFreeIn] at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro d | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Holds_coincide_Var | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ ∀ (v : VarName),
isFreeIn v h1_phi → Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v = Function.updateITE V h1_x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x a1 | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ ∀ (v : VarName),
isFreeIn v h1_phi → Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v = Function.updateITE V h1_x d v | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x = Function.updateITE V h1_x d x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x = Function.updateITE V h1_x d x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) = if x = h1_x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) = if x = h1_x then d else V x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝ : x = h1_x
⊢ d = d
case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝¹ : ¬x = h1_x
h✝ : x = h1_v
⊢ V h1_t = V x
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝¹ : ¬x = h1_x
h✝ : ¬x = h1_v
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : x = h1_x
⊢ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
subst c2
tauto | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : x = h1_v
⊢ V h1_t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : ¬x = h1_v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : x = h1_x
⊢ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | subst c2 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : x = h1_v
⊢ V h1_t = V x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_t : VarName
V : VarAssignment D
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
h1_1 : ¬x = h1_x → ¬isFreeIn x h1_phi
⊢ V h1_t = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | tauto | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_t : VarName
V : VarAssignment D
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
h1_1 : ¬x = h1_x → ¬isFreeIn x h1_phi
⊢ V h1_t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : ¬x = h1_v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [isFreeIn] at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro d | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | specialize h1_ih (Function.updateITE V h1_x d) | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [← h1_ih] | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Holds_coincide_Var | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ ∀ (v : VarName),
isFreeIn v h1_phi →
Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x _ | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ ∀ (v : VarName),
isFreeIn v h1_phi →
Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) v | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) =
if x = h1_v then if h1_t = h1_x then d else V h1_t else if x = h1_x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) =
if x = h1_v then if h1_t = h1_x then d else V h1_t else if x = h1_x then d else V x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if h1_x = x then d else if h1_v = x then V h1_t else V x) =
if h1_v = x then if h1_x = h1_t then d else V h1_t else if h1_x = x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if h1_x = x then d else if h1_v = x then V h1_t else V x) =
if h1_v = x then if h1_x = h1_t then d else V h1_t else if h1_x = x then d else V x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : h1_x = x
h✝ : h1_v = x
⊢ d = V h1_t
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : h1_x = x
h✝ : ¬h1_v = x
⊢ d = d
case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : h1_v = x
⊢ V h1_t = V h1_t
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 c3 =>
subst c2
cases h1_1
case intro h1_1_left h1_1_right =>
contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
c1 : isFreeIn x h1_phi
c2 : h1_x = x
c3 : h1_v = x
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ | _ | _ =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | subst c2 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
c1 : isFreeIn x h1_phi
c2 : h1_x = x
c3 : h1_v = x
⊢ d = V h1_t | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
⊢ d = V h1_t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | cases h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
⊢ d = V h1_t | case intro
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
left✝ : ¬h1_v = h1_x
right✝ : isFreeIn h1_v h1_phi
⊢ d = V h1_t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case intro h1_1_left h1_1_right =>
contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x | no goals |
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