url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [Rec.fastReplaceFree] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝)) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ IsSub (def_ a✝¹ a✝) v u (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.pred_const_ | apply IsSub.pred_var_ | v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v ∈ xs → u ∉ binders
⊢ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.pred_const_ | v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v ∈ xs → u ∉ binders
⊢ IsSub (pred_const_ X xs) v u (pred_const_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.pred_var_ | v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v ∈ xs → u ∉ binders
⊢ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.eq_ | v u x y : VarName
binders : Finset VarName
h1 : v = x ∨ v = y → u ∉ binders
⊢ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.true_ | apply IsSub.false_ | v u : VarName
binders : Finset VarName
h1 : True
⊢ IsSub false_ v u false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.true_ | v u : VarName
binders : Finset VarName
h1 : True
⊢ IsSub true_ v u true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.false_ | v u : VarName
binders : Finset VarName
h1 : True
⊢ IsSub false_ v u false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.not_ | v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
⊢ IsSub phi.not_ v u (Rec.fastReplaceFree v u phi).not_ | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact phi_ih binders h1 | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | cases h1 | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi ∧ Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi)) | case intro
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
left✝ : Rec.fastAdmitsAux v u binders phi
right✝ : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.imp_ | apply IsSub.and_ | apply IsSub.or_ | apply IsSub.iff_ | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi)) | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.imp_ | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.imp_ psi) v u ((Rec.fastReplaceFree v u phi).imp_ (Rec.fastReplaceFree v u psi)) | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.and_ | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.and_ psi) v u ((Rec.fastReplaceFree v u phi).and_ (Rec.fastReplaceFree v u psi)) | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.or_ | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.or_ psi) v u ((Rec.fastReplaceFree v u phi).or_ (Rec.fastReplaceFree v u psi)) | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.iff_ | v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi)) | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact phi_ih binders h1_left | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact psi_ih binders h1_right | case a
v u : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi)
binders : Finset VarName
h1_left : Rec.fastAdmitsAux v u binders phi
h1_right : Rec.fastAdmitsAux v u binders psi
⊢ IsSub psi v u (Rec.fastReplaceFree v u psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | cases h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x ∨ Rec.fastAdmitsAux v u (binders ∪ {x}) phi
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) | case inl
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h✝ : v = x
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))
case inr
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h✝ : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case inl h1 =>
split_ifs
case pos c1 | neg c1 =>
first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in
subst h1
simp only [isFreeIn]
simp | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | split_ifs | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 h✝ : v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi)
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
h✝ : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case pos c1 | neg c1 =>
first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in
subst h1
simp only [isFreeIn]
simp | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | subst h1 | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ ¬isFreeIn v (exists_ x phi) | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
c1 : ¬v = v
⊢ ¬isFreeIn v (exists_ v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [isFreeIn] | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
c1 : ¬v = v
⊢ ¬isFreeIn v (exists_ v phi) | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
c1 : ¬v = v
⊢ ¬(¬True ∧ isFreeIn v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp | case a
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
c1 : ¬v = v
⊢ ¬(¬True ∧ isFreeIn v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.forall_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ IsSub (forall_ x phi) v u (forall_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ ¬isFreeIn v (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.exists_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : v = x
c1 : ¬v = x
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | split_ifs | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi)
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case pos c1 =>
first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in
simp only [isFreeIn]
tauto | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [isFreeIn] | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬isFreeIn v (exists_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬(¬v = x ∧ isFreeIn v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | tauto | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬(¬v = x ∧ isFreeIn v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.forall_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ IsSub (forall_ x phi) v u (forall_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬isFreeIn v (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.exists_not_free_in | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | by_cases c2 : isFreeIn v phi | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.forall_free_in | apply IsSub.exists_free_in | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v (exists_ x phi)
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [isFreeIn] | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v (exists_ x phi)
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬v = x ∧ isFreeIn v phi
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | constructor | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬v = x ∧ isFreeIn v phi
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | case pos.a.left
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬v = x
case pos.a.right
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v phi
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | . exact c2 | case pos.a.right
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v phi
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | . have s1 : ¬ u ∈ binders ∪ {x}
exact Rec.fastAdmitsAux_isFreeIn phi v u (binders ∪ {x}) h1 c2
simp at s1
tauto | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.forall_free_in | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub (forall_ x phi) v u (forall_ x (Rec.fastReplaceFree v u phi)) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v (forall_ x phi)
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.exists_free_in | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v (exists_ x phi)
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact c1 | case pos.a.left
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact c2 | case pos.a.right
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | have s1 : ¬ u ∈ binders ∪ {x} | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ ¬x = u | case s1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ u ∉ binders ∪ {x}
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∪ {x}
⊢ ¬x = u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact Rec.fastAdmitsAux_isFreeIn phi v u (binders ∪ {x}) h1 c2 | case s1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ u ∉ binders ∪ {x}
case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∪ {x}
⊢ ¬x = u | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∪ {x}
⊢ ¬x = u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp at s1 | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∪ {x}
⊢ ¬x = u | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∧ ¬u = x
⊢ ¬x = u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | tauto | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
s1 : u ∉ binders ∧ ¬u = x
⊢ ¬x = u | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact phi_ih (binders ∪ {x}) h1 | case pos.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : isFreeIn v phi
⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | have s1 : Rec.fastReplaceFree v u phi = phi | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case s1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
⊢ Rec.fastReplaceFree v u phi = phi
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | exact Rec.not_free_in_fastReplaceFree_self phi v u c2 | case s1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
⊢ Rec.fastReplaceFree v u phi = phi
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [s1] | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | first | apply IsSub.forall_not_free_in | apply IsSub.exists_not_free_in | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [isFreeIn] | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬isFreeIn v (exists_ x phi) | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬(¬v = x ∧ isFreeIn v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | tauto | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬(¬v = x ∧ isFreeIn v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.forall_not_free_in | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (forall_ x phi) v u (forall_ x phi) | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬isFreeIn v (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.exists_not_free_in | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ IsSub (exists_ x phi) v u (exists_ x phi) | case neg.a
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
c2 : ¬isFreeIn v phi
s1 : Rec.fastReplaceFree v u phi = phi
⊢ ¬isFreeIn v (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | apply IsSub.def_ | v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ xs → u ∉ binders
⊢ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | apply Exists.elim h1 | F : Formula
v u : VarName
binders : Finset VarName
h1 : ∃ F', IsSub F v u F'
h2 : u ∉ binders
⊢ Rec.fastAdmitsAux v u binders F | F : Formula
v u : VarName
binders : Finset VarName
h1 : ∃ F', IsSub F v u F'
h2 : u ∉ binders
⊢ ∀ (a : Formula), IsSub F v u a → Rec.fastAdmitsAux v u binders F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | intro F' h1_1 | F : Formula
v u : VarName
binders : Finset VarName
h1 : ∃ F', IsSub F v u F'
h2 : u ∉ binders
⊢ ∀ (a : Formula), IsSub F v u a → Rec.fastAdmitsAux v u binders F | F : Formula
v u : VarName
binders : Finset VarName
h1 : ∃ F', IsSub F v u F'
h2 : u ∉ binders
F' : Formula
h1_1 : IsSub F v u F'
⊢ Rec.fastAdmitsAux v u binders F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | clear h1 | F : Formula
v u : VarName
binders : Finset VarName
h1 : ∃ F', IsSub F v u F'
h2 : u ∉ binders
F' : Formula
h1_1 : IsSub F v u F'
⊢ Rec.fastAdmitsAux v u binders F | F : Formula
v u : VarName
binders : Finset VarName
h2 : u ∉ binders
F' : Formula
h1_1 : IsSub F v u F'
⊢ Rec.fastAdmitsAux v u binders F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | induction h1_1 generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h2 : u ∉ binders
F' : Formula
h1_1 : IsSub F v u F'
⊢ Rec.fastAdmitsAux v u binders F | case pred_const_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_const_ X✝ xs✝)
case pred_var_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_var_ X✝ xs✝)
case eq_
F : Formula
v u : VarName
F' : Formula
x✝ y✝ v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (eq_ x✝ y✝)
case true_
F : Formula
v u : VarName
F' : Formula
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders true_
case false_
F : Formula
v u : VarName
F' : Formula
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders false_
case not_
F : Formula
v u : VarName
F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝.not_
case imp_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.imp_ psi✝)
case and_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.and_ psi✝)
case or_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.or_ psi✝)
case iff_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.iff_ psi✝)
case forall_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝)
case forall_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝)
case exists_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝)
case exists_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝)
case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | all_goals
simp only [Rec.fastAdmitsAux] | case pred_const_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_const_ X✝ xs✝)
case pred_var_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_var_ X✝ xs✝)
case eq_
F : Formula
v u : VarName
F' : Formula
x✝ y✝ v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (eq_ x✝ y✝)
case true_
F : Formula
v u : VarName
F' : Formula
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders true_
case false_
F : Formula
v u : VarName
F' : Formula
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders false_
case not_
F : Formula
v u : VarName
F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝.not_
case imp_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.imp_ psi✝)
case and_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.and_ psi✝)
case or_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.or_ psi✝)
case iff_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.iff_ psi✝)
case forall_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝)
case forall_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝)
case exists_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝)
case exists_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝)
case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝) | case pred_const_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders
case pred_var_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders
case eq_
F : Formula
v u : VarName
F' : Formula
x✝ y✝ v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ v✝ = y✝ → t✝ ∉ binders
case not_
F : Formula
v u : VarName
F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝
case imp_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case and_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case or_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case iff_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case forall_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝
case forall_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝
case exists_not_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝
case exists_free_in
F : Formula
v u : VarName
F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝
case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | case
forall_free_in h1_1_x h1_1_phi h1_1_v h1_1_t _ _ h1_1_2 _ h1_1_ih
| exists_free_in h1_1_x h1_1_phi h1_1_v h1_1_t _ _ h1_1_2 _ h1_1_ih =>
right
apply h1_1_ih
simp
tauto | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | all_goals
tauto | case pred_const_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders
case pred_var_
F : Formula
v u : VarName
F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders
case eq_
F : Formula
v u : VarName
F' : Formula
x✝ y✝ v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ = x✝ ∨ v✝ = y✝ → t✝ ∉ binders
case not_
F : Formula
v u : VarName
F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝
case imp_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case and_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case or_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case iff_
F : Formula
v u : VarName
F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝
a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝
case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | simp only [Rec.fastAdmitsAux] | case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝) | case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | simp only [isFreeIn] at h1_1_1 | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
h1_1_1 : ¬isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
h1_1_1 : ¬(¬h1_1_v = h1_1_x ∧ isFreeIn h1_1_v h1_1_phi)
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | simp at h1_1_1 | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
h1_1_1 : ¬(¬h1_1_v = h1_1_x ∧ isFreeIn h1_1_v h1_1_phi)
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | by_cases c1 : h1_1_v = h1_1_x | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case pos
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
case neg
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | left | case pos
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case pos.h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | exact c1 | case pos.h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | right | case neg
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case neg.h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | apply Rec.not_isFreeIn_imp_fastAdmitsAux | case neg.h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case neg.h.h1
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ ¬isFreeIn h1_1_v h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | exact h1_1_1 c1 | case neg.h.h1
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
binders : Finset VarName
h2 : h1_1_t ∉ binders
h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi
c1 : ¬h1_1_v = h1_1_x
⊢ ¬isFreeIn h1_1_v h1_1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | right | F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | apply h1_1_ih | case h
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi | case h.h2
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_t ∉ binders ∪ {h1_1_x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | simp | case h.h2
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_t ∉ binders ∪ {h1_1_x} | case h.h2
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_t ∉ binders ∧ ¬h1_1_t = h1_1_x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | tauto | case h.h2
F : Formula
v u : VarName
F' : Formula
h1_1_x : VarName
h1_1_phi : Formula
h1_1_v h1_1_t : VarName
phi'✝ : Formula
a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi)
h1_1_2 : ¬h1_1_x = h1_1_t
a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝
h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi
binders : Finset VarName
h2 : h1_1_t ∉ binders
⊢ h1_1_t ∉ binders ∧ ¬h1_1_t = h1_1_x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux | [197, 1] | [231, 10] | tauto | case def_
F : Formula
v u : VarName
F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
binders : Finset VarName
h2 : t✝ ∉ binders
⊢ v✝ ∈ xs✝ → t✝ ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | induction h1 | F F' : Formula
v u : VarName
h1 : IsSub F v u F'
⊢ Rec.fastReplaceFree v u F = F' | case pred_const_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_const_ X✝ xs✝) = pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case pred_var_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_var_ X✝ xs✝) = pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case eq_
F F' : Formula
v u x✝ y✝ v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (eq_ x✝ y✝) = eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝)
case true_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ true_ = true_
case false_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ false_ = false_
case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ phi✝.not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.imp_ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.and_ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.or_ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.iff_ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi'✝
case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | all_goals
simp only [Rec.fastReplaceFree] | case pred_const_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_const_ X✝ xs✝) = pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case pred_var_
F F' : Formula
v u : VarName
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (pred_var_ X✝ xs✝) = pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)
case eq_
F F' : Formula
v u x✝ y✝ v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (eq_ x✝ y✝) = eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝)
case true_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ true_ = true_
case false_
F F' : Formula
v u v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ false_ = false_
case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ phi✝.not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.imp_ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.and_ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.or_ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (phi✝.iff_ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (forall_ x✝ phi✝) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ Rec.fastReplaceFree v✝ t✝ (exists_ x✝ phi✝) = exists_ x✝ phi'✝
case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) | case not_
F F' : Formula
v u : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).not_ = phi'✝.not_
case imp_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).imp_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.imp_ psi'✝
case and_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).and_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.and_ psi'✝
case or_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).or_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.or_ psi'✝
case iff_
F F' : Formula
v u : VarName
phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ psi✝ = psi'✝
⊢ (Rec.fastReplaceFree v✝ t✝ phi✝).iff_ (Rec.fastReplaceFree v✝ t✝ psi✝) = phi'✝.iff_ psi'✝
case forall_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
⊢ (if v✝ = x✝ then forall_ x✝ phi✝ else forall_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = forall_ x✝ phi✝
case forall_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (if v✝ = x✝ then forall_ x✝ phi✝ else forall_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = forall_ x✝ phi'✝
case exists_not_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
⊢ (if v✝ = x✝ then exists_ x✝ phi✝ else exists_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = exists_ x✝ phi✝
case exists_free_in
F F' : Formula
v u x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : Rec.fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ (if v✝ = x✝ then exists_ x✝ phi✝ else exists_ x✝ (Rec.fastReplaceFree v✝ t✝ phi✝)) = exists_ x✝ phi'✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case not_ h1_phi h1_v h1_t h1_phi' h1_1 h1_ih =>
tauto | F F' : Formula
v u : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).not_ = h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
imp_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| and_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| or_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2
| iff_ h1_phi h1_psi h1_v h1_t h1_phi' h1_psi' h1_1 h1_2 h1_ih_1 h1_ih_2 =>
subst h1_ih_1
subst h1_ih_2
rfl | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) = h1_phi'.iff_ h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
forall_not_free_in h1_x h1_phi h1_v h1_t h1_1
| exists_not_free_in h1_x h1_phi h1_v h1_t h1_1 =>
simp only [isFreeIn] at h1_1
simp at h1_1
split_ifs
case pos c1 =>
rfl
case neg c1 =>
congr!
apply Rec.not_free_in_fastReplaceFree_self
exact h1_1 c1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case
forall_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 _ _ h1_ih
| exists_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 _ _ h1_ih =>
simp only [isFreeIn] at h1_1
cases h1_1
case intro h1_1_left h1_1_right =>
simp only [if_neg h1_1_left]
subst h1_ih
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [Rec.fastReplaceFree] | case def_
F F' : Formula
v u : VarName
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
⊢ Rec.fastReplaceFree v✝ t✝ (def_ X✝ xs✝) = def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | tauto | F F' : Formula
v u : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).not_ = h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih_1 | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
h1_1 : IsSub h1_phi h1_v h1_t h1_phi'
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) = h1_phi'.iff_ h1_psi' | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_psi' : Formula
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih_2 | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_psi' : Formula
h1_2 : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_2 : Rec.fastReplaceFree h1_v h1_t h1_psi = h1_psi'
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ h1_psi' | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
h1_2 : IsSub h1_psi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_psi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u : VarName
h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_1 : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
h1_2 : IsSub h1_psi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_psi)
⊢ (Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) =
(Rec.fastReplaceFree h1_v h1_t h1_phi).iff_ (Rec.fastReplaceFree h1_v h1_t h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [isFreeIn] at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | split_ifs | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi | case pos
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
h✝ : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi
case neg
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
h✝ : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case pos c1 =>
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case neg c1 =>
congr!
apply Rec.not_free_in_fastReplaceFree_self
exact h1_1 c1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : h1_v = h1_x
⊢ exists_ h1_x h1_phi = exists_ h1_x h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | congr! | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi | case h.e'_2
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | apply Rec.not_free_in_fastReplaceFree_self | case h.e'_2
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi | case h.e'_2.h1
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ ¬isFreeIn h1_v h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | exact h1_1 c1 | case h.e'_2.h1
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
c1 : ¬h1_v = h1_x
⊢ ¬isFreeIn h1_v h1_phi | no goals |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.