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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_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/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | induction F generalizing σ | σ : VarName → VarName
c : Char
F : Formula
⊢ (sub σ c F).freeVarSet = Finset.image σ F.freeVarSet | case pred_const_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet
case pred_var_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet
case eq_
c : Char
a✝¹ a✝ : VarName
σ : VarName → VarName
⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet
case true_
c : Char
σ : VarName → VarName
⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet
case false_
c : Char
σ : VarName → VarName
⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet
case not_
c : Char
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet
case imp_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet
case and_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet
case or_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet
case iff_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet
case forall_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet
case exists_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet
case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | all_goals
simp only [sub]
simp only [freeVarSet] | case pred_const_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet
case pred_var_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet
case eq_
c : Char
a✝¹ a✝ : VarName
σ : VarName → VarName
⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet
case true_
c : Char
σ : VarName → VarName
⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet
case false_
c : Char
σ : VarName → VarName
⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet
case not_
c : Char
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet
case imp_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet
case and_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet
case or_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet
case iff_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet
case forall_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet
case exists_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet
case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet | case pred_const_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset
case pred_var_
c : Char
a✝¹ : PredName
a✝ : List VarName
σ : VarName → VarName
⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset
case eq_
c : Char
a✝¹ a✝ : VarName
σ : VarName → VarName
⊢ {σ a✝¹, σ a✝} = Finset.image σ {a✝¹, a✝}
case true_
c : Char
σ : VarName → VarName
⊢ ∅ = Finset.image σ ∅
case false_
c : Char
σ : VarName → VarName
⊢ ∅ = Finset.image σ ∅
case not_
c : Char
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
case imp_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet)
case and_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet)
case or_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet)
case iff_
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet)
case forall_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub
(Function.updateITE σ a✝¹
(if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then
fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet
else a✝¹))
c a✝).freeVarSet \
{if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet
else a✝¹} =
Finset.image σ (a✝.freeVarSet \ {a✝¹})
case exists_
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet
σ : VarName → VarName
⊢ (sub
(Function.updateITE σ a✝¹
(if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then
fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet
else a✝¹))
c a✝).freeVarSet \
{if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet
else a✝¹} =
Finset.image σ (a✝.freeVarSet \ {a✝¹})
case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs =>
apply Finset.ext
intro a
simp | c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case true_ | false_ =>
simp | c : Char
σ : VarName → VarName
⊢ ∅ = Finset.image σ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case not_ phi phi_ih =>
exact phi_ih σ | c : Char
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [phi_ih]
simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ]
split_ifs
case _ c1 =>
obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi))
generalize (
fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at *
have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi)
apply Finset.image_subset_image
simp
have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})
apply Finset.not_mem_mono s2 s1
calc
Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'}
= Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} :=
by
{
apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x')
simp only [Function.updateITE]
simp
}
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} :=
by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x]
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) :=
by exact Finset.sdiff_singleton_eq_self s3
case _ c1 =>
simp at c1
have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x}
apply Finset.image_sdiff_singleton
simp only [Function.updateITE]
simp
simp only [s1]
clear s1
have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})
simp only [Finset.mem_image]
simp
simp only [Function.updateITE]
simp
tauto
simp only [Finset.sdiff_singleton_eq_self s2] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ (sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi).freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} =
Finset.image σ (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [sub] | case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet | case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [freeVarSet] | case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet | case def_
c : Char
a✝¹ : DefName
a✝ : List VarName
σ : VarName → VarName
⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.ext | c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset | case a
c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | intro a | case a
c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset | case a
c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
a : VarName
⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case a
c : Char
X : DefName
xs : List VarName
σ : VarName → VarName
a : VarName
⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | c : Char
σ : VarName → VarName
⊢ ∅ = Finset.image σ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact phi_ih σ | c : Char
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.image_union] | c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ (phi.freeVarSet ∪ psi.freeVarSet) | c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | congr! | c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet | case h.e'_3
c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
case h.e'_4
c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact phi_ih σ | case h.e'_3
c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact psi_ih σ | case h.e'_4
c : Char
phi psi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
σ : VarName → VarName
⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [phi_ih] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ (sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi).freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} =
Finset.image σ (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ Finset.image
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x))
phi.freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x} =
Finset.image σ (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ Finset.image
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x))
phi.freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x} =
Finset.image σ (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ Finset.image
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x))
phi.freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | split_ifs | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
⊢ Finset.image
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x))
phi.freeVarSet \
{if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)
else x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case pos
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
h✝ : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)))
phi.freeVarSet \
{fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
case neg
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
h✝ : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case _ c1 =>
obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi))
generalize (
fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at *
have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi)
apply Finset.image_subset_image
simp
have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})
apply Finset.not_mem_mono s2 s1
calc
Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'}
= Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} :=
by
{
apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x')
simp only [Function.updateITE]
simp
}
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} :=
by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x]
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) :=
by exact Finset.sdiff_singleton_eq_self s3 | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)))
phi.freeVarSet \
{fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | case _ c1 =>
simp at c1
have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x}
apply Finset.image_sdiff_singleton
simp only [Function.updateITE]
simp
simp only [s1]
clear s1
have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})
simp only [Finset.mem_image]
simp
simp only [Function.updateITE]
simp
tauto
simp only [Finset.sdiff_singleton_eq_self s2] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)))
phi.freeVarSet \
{fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
s1 :
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)))
phi.freeVarSet \
{fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | generalize (
fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at * | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
s1 :
fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)))
phi.freeVarSet \
{fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_subset_image | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2.h
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2.h
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s3
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.not_mem_mono s2 s1 | case s3
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | calc
Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'}
= Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} :=
by
{
apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x')
simp only [Function.updateITE]
simp
}
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} :=
by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x]
_ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) :=
by exact Finset.sdiff_singleton_eq_self s3 | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} =
Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'} | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Function.updateITE σ x x' x = x' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Function.updateITE σ x x' x = x' | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ (if True then x' else σ x) = x' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ (if True then x' else σ x) = x' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | exact Finset.sdiff_singleton_eq_self s3 | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x
x' : VarName
s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
s2 :
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆
Finset.image (Function.updateITE σ x x) phi.freeVarSet
s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp at c1 | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x} | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | apply Finset.image_sdiff_singleton | case s1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Function.updateITE σ x x x = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Function.updateITE σ x x x = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ (if True then x else σ x) = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s1.h1
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ (if True then x else σ x) = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [s1] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | clear s1 | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s1 :
Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x}
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.mem_image] | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Function.updateITE] | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | tauto | case s2
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image | [48, 1] | [125, 52] | simp only [Finset.sdiff_singleton_eq_self s2] | c : Char
x : VarName
phi : Formula
phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet
σ : VarName → VarName
c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x
s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} =
Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | induction F generalizing σ V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
σ : VarName → VarName
c : Char
F : Formula
⊢ Holds D I V E (sub σ c F) ↔ Holds D I (V ∘ σ) E F | case pred_const_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (pred_const_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (pred_var_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ a✝ : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (eq_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c true_) ↔ Holds D I (V ∘ σ) E true_
case false_
D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_
case not_
D : Type
I : Interpretation D
E : Env
c : Char
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c a✝.not_) ↔ Holds D I (V ∘ σ) E a✝.not_
case imp_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (a✝¹.imp_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (a✝¹.and_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (a✝¹.or_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (a✝¹.iff_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (forall_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (exists_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E : Env
c : Char
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (def_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case pred_const_ X xs | pred_var_ X xs | eq_ x y =>
simp only [sub]
simp only [Holds]
simp | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case true_ | false_ =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case not_ phi phi_ih =>
simp only [sub]
simp only [Holds]
congr! 1
exact phi_ih V σ | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y) | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y) | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | D : Type
I : Interpretation D
E : Env
c : Char
x y : VarName
V : VarAssignment D
σ : VarName → VarName
⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_ | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_ | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_ | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ ¬Holds D I V E (sub σ c phi) ↔ ¬Holds D I (V ∘ σ) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ ¬Holds D I V E (sub σ c phi) ↔ ¬Holds D I (V ∘ σ) E phi | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V σ | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (phi.iff_ psi)) ↔ Holds D I (V ∘ σ) E (phi.iff_ psi) | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E ((sub σ c phi).iff_ (sub σ c psi)) ↔ Holds D I (V ∘ σ) E (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E ((sub σ c phi).iff_ (sub σ c psi)) ↔ Holds D I (V ∘ σ) E (phi.iff_ psi) | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ (Holds D I V E (sub σ c phi) ↔ Holds D I V E (sub σ c psi)) ↔ (Holds D I (V ∘ σ) E phi ↔ Holds D I (V ∘ σ) E psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ (Holds D I V E (sub σ c phi) ↔ Holds D I V E (sub σ c psi)) ↔ (Holds D I (V ∘ σ) E phi ↔ Holds D I (V ∘ σ) E psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V σ | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact psi_ih V σ | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
psi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c psi) ↔ Holds D I (V ∘ σ) E psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E (sub σ c (exists_ x phi)) ↔ Holds D I (V ∘ σ) E (exists_ x phi) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E
(exists_
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi)) ↔
Holds D I (V ∘ σ) E (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ Holds D I V E
(exists_
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi)) ↔
Holds D I (V ∘ σ) E (exists_ x phi) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ (∃ d,
Holds D I
(Function.updateITE V
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi)) ↔
∃ d, Holds D I (Function.updateITE (V ∘ σ) x d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ (∃ d,
Holds D I
(Function.updateITE V
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi)) ↔
∃ d, Holds D I (Function.updateITE (V ∘ σ) x d) E phi | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ ∀ (a : D),
Holds D I
(Function.updateITE V
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi) ↔
Holds D I (Function.updateITE (V ∘ σ) x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro d | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
⊢ ∀ (a : D),
Holds D I
(Function.updateITE V
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi) ↔
Holds D I (Function.updateITE (V ∘ σ) x a) E phi | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi
V : VarAssignment D
σ : VarName → VarName
d : D
⊢ Holds D I
(Function.updateITE V
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)
d)
E
(sub
(Function.updateITE σ x
(if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet
else x))
c phi) ↔
Holds D I (Function.updateITE (V ∘ σ) x d) E phi |
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