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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp only [fastReplaceFree] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝ | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ List.map (fun x => if v = x then t else x) xs = xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp only [List.map_eq_self_iff] | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ List.map (fun x => if v = x then t else x) xs = xs | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, (if v = x then t else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, (if v = x then t else x) = x | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, v = x → t = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | intro x a1 a2 | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, v = x → t = x | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
x : VarName
a1 : x ∈ xs
a2 : v = x
⊢ t = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | subst a2 | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
x : VarName
a1 : x ∈ xs
a2 : v = x
⊢ t = x | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
a1 : v ∈ xs
⊢ t = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | contradiction | v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
a1 : v ∈ xs
⊢ t = v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp | v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y | v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ (v = x → t = x) ∧ (v = y → t = y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | tauto | v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ (v = x → t = x) ∧ (v = y → t = y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | tauto | v t : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬isFreeIn v phi
⊢ (fastReplaceFree v t phi).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp | v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi | v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ fastReplaceFree v t phi = phi ∧ fastReplaceFree v t psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | tauto | v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ fastReplaceFree v t phi = phi ∧ fastReplaceFree v t psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | simp | v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ ¬v = x → fastReplaceFree v t phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self | [258, 1] | [290, 10] | tauto | v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ ¬v = x → fastReplaceFree v t phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | induction F | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastReplaceFree t v (fastReplaceFree v t F) = F | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_
case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | any_goals
simp only [occursIn] at h1
simp only [fastReplaceFree] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_
case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝ | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : t ∉ a✝
⊢ pred_const_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) =
pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : t ∉ a✝
⊢ pred_var_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) =
pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
⊢ eq_ (if t = if v = a✝¹ then t else a✝¹ then v else if v = a✝¹ then t else a✝¹)
(if t = if v = a✝ then t else a✝ then v else if v = a✝ then t else a✝) =
eq_ a✝¹ a✝
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝)).not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).imp_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).and_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).or_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).iff_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ fastReplaceFree t v (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ fastReplaceFree t v (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case not_ phi phi_ih =>
congr!
exact phi_ih h1 | v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr! <;> tauto | v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
push_neg at h1
cases h1
case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right
case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [occursIn] at h1 | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝ | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [fastReplaceFree] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝ | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! | v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ def_ X (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs)) = def_ X xs | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs) = xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs) = xs | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) xs = xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [List.map_eq_self_iff] | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) xs = xs | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) x = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) x = x | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, (if t = if v = x then t else x then v else if v = x then t else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | intro x a1 | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, (if t = if v = x then t else x then v else if v = x then t else x) = x | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | by_cases c1 : v = x | case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [if_pos c1] | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if True then v else t) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if True then v else t) = x | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | exact c1 | case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [if_neg c1] | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = x then v else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = x then v else x) = x | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ t = x → v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | intro a2 | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ t = x → v = x | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
a2 : t = x
⊢ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | subst a2 | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
a2 : t = x
⊢ v = x | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
a1 : t ∈ xs
c1 : ¬v = t
⊢ v = t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | contradiction | case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
a1 : t ∈ xs
c1 : ¬v = t
⊢ v = t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! | v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ eq_ (if t = if v = x then t else x then v else if v = x then t else x)
(if t = if v = y then t else y then v else if v = y then t else y) =
eq_ x y | case h.e'_1
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
case h.e'_2
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = y then t else y then v else if v = y then t else y) = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | split_ifs <;> tauto | case h.e'_1
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | split_ifs <;> tauto | case h.e'_2
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = y then t else y then v else if v = y then t else y) = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! | v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_ | case h.e'_1
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | exact phi_ih h1 | case h.e'_1
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! <;> tauto | v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | push_neg at h1 | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | cases h1 | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | case intro
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
left✝ : t ≠ x
right✝ : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right
case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | split_ifs | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi | case pos
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
case neg
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [fastReplaceFree] | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ (if t = x then exists_ x phi else exists_ x (fastReplaceFree t v phi)) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [if_neg h1_left] | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ (if t = x then exists_ x phi else exists_ x (fastReplaceFree t v phi)) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ exists_ x (fastReplaceFree t v phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ exists_ x (fastReplaceFree t v phi) = exists_ x phi | case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | apply not_free_in_fastReplaceFree_self | case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v phi = phi | case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ ¬isFreeIn t phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | contrapose! h1_right | case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ ¬isFreeIn t phi | case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
c1 : v = x
h1_right : isFreeIn t phi
⊢ occursIn t phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | exact isFreeIn_imp_occursIn t phi h1_right | case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
c1 : v = x
h1_right : isFreeIn t phi
⊢ occursIn t phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [fastReplaceFree] | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ (if t = x then exists_ x (fastReplaceFree v t phi) else exists_ x (fastReplaceFree t v (fastReplaceFree v t phi))) =
exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | simp only [if_neg h1_left] | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ (if t = x then exists_ x (fastReplaceFree v t phi) else exists_ x (fastReplaceFree t v (fastReplaceFree v t phi))) =
exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ exists_ x (fastReplaceFree t v (fastReplaceFree v t phi)) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | congr! | v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ exists_ x (fastReplaceFree t v (fastReplaceFree v t phi)) = exists_ x phi | case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse | [293, 1] | [349, 30] | exact phi_ih h1_right | case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | induction F | F : Formula
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t F) | case pred_const_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case eq_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (eq_ a✝¹ a✝))
case true_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t true_)
case false_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t false_)
case not_
v t : VarName
h1 : ¬v = t
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t a✝.not_)
case imp_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.imp_ a✝))
case and_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.and_ a✝))
case or_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.or_ a✝))
case iff_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.iff_ a✝))
case forall_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | any_goals
simp only [fastReplaceFree]
simp only [isFreeIn] | case pred_const_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case eq_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (eq_ a✝¹ a✝))
case true_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t true_)
case false_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t false_)
case not_
v t : VarName
h1 : ¬v = t
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t a✝.not_)
case imp_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.imp_ a✝))
case and_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.and_ a✝))
case or_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.or_ a✝))
case iff_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.iff_ a✝))
case forall_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (def_ a✝¹ a✝)) | case pred_const_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝
case pred_var_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝
case eq_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : VarName
⊢ ¬((v = if v = a✝¹ then t else a✝¹) ∨ v = if v = a✝ then t else a✝)
case true_
v t : VarName
h1 : ¬v = t
⊢ ¬False
case false_
v t : VarName
h1 : ¬v = t
⊢ ¬False
case not_
v t : VarName
h1 : ¬v = t
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t a✝)
case imp_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))
case and_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))
case or_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))
case iff_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))
case forall_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
intro x
split_ifs <;> tauto | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
⊢ v ∉ 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/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case eq_ x y =>
split_ifs <;> tauto | v t : VarName
h1 : ¬v = t
x y : VarName
⊢ ¬((v = if v = x then t else x) ∨ v = if v = y then t else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case true_ | false_ =>
simp | v t : VarName
h1 : ¬v = t
⊢ ¬False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case not_ phi phi_ih =>
exact phi_ih | v t : VarName
h1 : ¬v = t
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
tauto | v t : VarName
h1 : ¬v = t
phi psi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
psi_ih : ¬isFreeIn v (fastReplaceFree v t psi)
⊢ ¬(isFreeIn v (fastReplaceFree v t phi) ∨ isFreeIn v (fastReplaceFree v t psi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [fastReplaceFree]
split_ifs
case pos c1 =>
simp only [isFreeIn]
simp
intro a1
contradiction
case neg c1 =>
simp only [isFreeIn]
simp
intro _
exact phi_ih | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp only [fastReplaceFree] | case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (def_ a✝¹ a✝)) | case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp only [isFreeIn] | case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) | case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) xs | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
⊢ ∀ x ∈ xs, ¬(if v = x then t else x) = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | intro x | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
⊢ ∀ x ∈ xs, ¬(if v = x then t else x) = v | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
x : VarName
⊢ x ∈ xs → ¬(if v = x then t else x) = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | split_ifs <;> tauto | v t : VarName
h1 : ¬v = t
X : DefName
xs : List VarName
x : VarName
⊢ x ∈ xs → ¬(if v = x then t else x) = v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | split_ifs <;> tauto | v t : VarName
h1 : ¬v = t
x y : VarName
⊢ ¬((v = if v = x then t else x) ∨ v = if v = y then t else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp | v t : VarName
h1 : ¬v = t
⊢ ¬False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | exact phi_ih | v t : VarName
h1 : ¬v = t
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | tauto | v t : VarName
h1 : ¬v = t
phi psi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
psi_ih : ¬isFreeIn v (fastReplaceFree v t psi)
⊢ ¬(isFreeIn v (fastReplaceFree v t phi) ∨ isFreeIn v (fastReplaceFree v t psi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp only [fastReplaceFree] | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ x phi)) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | split_ifs | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
⊢ ¬isFreeIn v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) | case pos
v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
h✝ : v = x
⊢ ¬isFreeIn v (exists_ x phi)
case neg
v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
h✝ : ¬v = x
⊢ ¬isFreeIn v (exists_ x (fastReplaceFree v t phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case pos c1 =>
simp only [isFreeIn]
simp
intro a1
contradiction | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬isFreeIn v (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | case neg c1 =>
simp only [isFreeIn]
simp
intro _
exact phi_ih | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬isFreeIn v (exists_ x (fastReplaceFree v t phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp only [isFreeIn] | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬isFreeIn v (exists_ x phi) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬(¬v = x ∧ isFreeIn v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬(¬v = x ∧ isFreeIn v phi) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬v = x → ¬isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | intro a1 | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
⊢ ¬v = x → ¬isFreeIn v phi | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
a1 : ¬v = x
⊢ ¬isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | contradiction | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : v = x
a1 : ¬v = x
⊢ ¬isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp only [isFreeIn] | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬isFreeIn v (exists_ x (fastReplaceFree v t phi)) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬(¬v = x ∧ isFreeIn v (fastReplaceFree v t phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | simp | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬(¬v = x ∧ isFreeIn v (fastReplaceFree v t phi)) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬v = x → ¬isFreeIn v (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | intro _ | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 : ¬v = x
⊢ ¬v = x → ¬isFreeIn v (fastReplaceFree v t phi) | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 a✝ : ¬v = x
⊢ ¬isFreeIn v (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree | [352, 1] | [390, 19] | exact phi_ih | v t : VarName
h1 : ¬v = t
x : VarName
phi : Formula
phi_ih : ¬isFreeIn v (fastReplaceFree v t phi)
c1 a✝ : ¬v = x
⊢ ¬isFreeIn v (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | induction E generalizing F binders V V' σ σ' | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
σ σ' : VarName → VarName
binders : Finset VarName
F : Formula
h1 : admitsAux σ binders F
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V E F ↔ Holds D I V' E (fastReplaceFree σ' F) | case nil
D : Type
I : Interpretation D
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
F : Formula
h1 : admitsAux σ binders F
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V [] F ↔ Holds D I V' [] (fastReplaceFree σ' F)
case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
F : Formula
h1 : admitsAux σ binders F
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | induction F generalizing binders V V' σ σ' | case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
F : Formula
h1 : admitsAux σ binders F
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) F ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' F) | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (pred_const_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (pred_var_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (eq_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders true_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' true_)
case cons.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders false_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' false_)
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝.not_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.imp_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.and_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.or_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.iff_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (forall_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (exists_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (def_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | all_goals
simp only [admitsAux] at h1
simp only [fastReplaceFree]
simp only [Holds] | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (pred_const_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (pred_var_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (eq_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders true_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' true_)
case cons.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders false_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' false_)
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝.not_
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.imp_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.and_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.or_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (a✝¹.iff_ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (forall_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (exists_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (def_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (def_ a✝¹ a✝)) | case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ I.pred_const_ a✝¹ (List.map V a✝) ↔ I.pred_const_ a✝¹ (List.map V' (List.map σ' a✝))
case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ I.pred_var_ a✝¹ (List.map V a✝) ↔ I.pred_var_ a✝¹ (List.map V' (List.map σ' a✝))
case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : (a✝¹ ∉ binders → σ a✝¹ ∉ binders) ∧ (a✝ ∉ binders → σ a✝ ∉ binders)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ V a✝¹ = V a✝ ↔ V' (σ' a✝¹) = V' (σ' a✝)
case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ ¬Holds D I V (head✝ :: tail✝) a✝ ↔ ¬Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝)
case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝¹ ∧ admitsAux σ binders a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ → Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹) → Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝)
case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝¹ ∧ admitsAux σ binders a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∧ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹) ∧ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝)
case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝¹ ∧ admitsAux σ binders a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∨ Holds D I V (head✝ :: tail✝) a✝ ↔
Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹) ∨ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝)
case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝¹ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹))
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders a✝¹ ∧ admitsAux σ binders a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V (head✝ :: tail✝) a✝) ↔
(Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝¹) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ (binders ∪ {a✝¹}) a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∀ (d : D),
Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders a✝ →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' a✝))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ (binders ∪ {a✝¹}) a✝
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
∃ d, Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ (List.map σ' a✝).length = head✝.args.length then
Holds D I (Function.updateListITE V' head✝.args (List.map V' (List.map σ' a✝))) tail✝ head✝.q
else Holds D I V' tail✝ (def_ a✝¹ (List.map σ' a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V V' σ σ' binders h1 h2 h2' h3 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName),
admitsAux σ binders phi →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi))
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders phi
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [admitsAux] at h1 | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : admitsAux σ binders (def_ a✝¹ a✝)
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (def_ a✝¹ a✝)) | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [fastReplaceFree] | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' (def_ a✝¹ a✝)) | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ (List.map σ' a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [Holds] | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ (List.map σ' a✝)) | case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ a✝, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ (List.map σ' a✝).length = head✝.args.length then
Holds D I (Function.updateListITE V' head✝.args (List.map V' (List.map σ' a✝))) tail✝ head✝.q
else Holds D I V' tail✝ (def_ a✝¹ (List.map σ' a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V' (List.map σ' xs)) | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ List.map V xs = List.map V' (List.map σ' xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ List.map V xs = List.map V' (List.map σ' xs) | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ List.map V xs = List.map (V' ∘ σ') xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ List.map V xs = List.map (V' ∘ σ') xs | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ ∀ x ∈ xs, V x = (V' ∘ σ') x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | intro v a1 | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ ∀ x ∈ xs, V x = (V' ∘ σ') x | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
⊢ V v = (V' ∘ σ') v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | apply h2 | case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
⊢ V v = (V' ∘ σ') v | case a.h.e'_4.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
⊢ v ∈ binders ∨ σ' v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | by_cases c1 : v ∈ binders | case a.h.e'_4.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
⊢ v ∈ binders ∨ σ' v ∉ binders | case pos
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
c1 : v ∈ binders
⊢ v ∈ binders ∨ σ' v ∉ binders
case neg
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
c1 : v ∉ binders
⊢ v ∈ binders ∨ σ' v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | left | case pos
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
c1 : v ∈ binders
⊢ v ∈ binders ∨ σ' v ∉ binders | case pos.h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
admitsAux σ binders F →
(∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
(∀ v ∈ binders, v = σ' v) →
(∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
v : VarName
a1 : v ∈ xs
c1 : v ∈ binders
⊢ v ∈ binders |
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