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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp at h2_left | v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders | v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | tauto | v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | subst a1 | v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = y
⊢ u ∉ binders | v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp at h2_right | v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders | v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | tauto | v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp at h2 | v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
⊢ fastAdmitsAux v u binders phi | v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
⊢ fastAdmitsAux v u binders phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | exact phi_ih binders h1 h2 | v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
⊢ fastAdmitsAux v u binders phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp at h2 | v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi | v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | tauto | v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | split_ifs at h2 | v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case pos
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
case neg
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | case pos c1 =>
left
exact c1 | v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | left | v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case h
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | exact c1 | case h
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | right | v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case h
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ fastAdmitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | apply phi_ih | case h
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ fastAdmitsAux v u (binders ∪ {x}) phi | case h.h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∪ {x}
case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp | case h.h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∪ {x} | case h.h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∧ ¬v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | tauto | case h.h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∧ ¬v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp only [toIsBoundAux] at h2 | case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) | case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | simp at h2 | case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) | case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux | [677, 1] | [740, 17] | exact h2 | case h.h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_self | [760, 1] | [769, 10] | induction F generalizing binders | F : Formula
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders F | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
⊢ admitsAux v v binders (eq_ a✝¹ a✝)
case true_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders true_
case false_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_self | [760, 1] | [769, 10] | all_goals
simp only [admitsAux] | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
⊢ admitsAux v v binders (eq_ a✝¹ a✝)
case true_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders true_
case false_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (def_ a✝¹ a✝) | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → v ∉ binders
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_self | [760, 1] | [769, 10] | all_goals
tauto | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → v ∉ binders
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_self | [760, 1] | [769, 10] | simp only [admitsAux] | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (def_ a✝¹ a✝) | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_self | [760, 1] | [769, 10] | tauto | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_self | [772, 1] | [778, 23] | simp only [admits] | F : Formula
v : VarName
⊢ admits v v F | F : Formula
v : VarName
⊢ admitsAux v v ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_self | [772, 1] | [778, 23] | apply admitsAux_self | F : Formula
v : VarName
⊢ admitsAux v v ∅ F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝.not_
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | all_goals
simp only [isFreeIn] at h1
simp only [admitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝.not_
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | simp only [isFreeIn] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | simp only [admitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | by_cases c1 : v = x | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ admitsAux v u (binders ∪ {x}) phi | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | apply mem_binders_imp_admitsAux | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi | case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | simp | case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∪ {x} | case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∨ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | tauto | case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∨ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | apply phi_ih | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi | case neg.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ ¬isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | tauto | case neg.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ ¬isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux | [782, 1] | [802, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admits | [805, 1] | [812, 46] | simp only [admits] | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admits v u F | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admits | [805, 1] | [812, 46] | exact not_isFreeIn_imp_admitsAux F v u ∅ h1 | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u ∅ F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u F
h2 : u ∉ binders
⊢ admitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | all_goals
simp only [isBoundIn] at h1
simp only [admitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝
h2 : u ∉ binders
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
push_neg at h1
cases h1
case intro h1_left h1_right =>
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝
h2 : u ∉ binders
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | simp only [isBoundIn] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | simp only [admitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | push_neg at h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {x}) phi | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | cases h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi | case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
left✝ : u ≠ x
right✝ : ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | case intro h1_left h1_right =>
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | apply phi_ih (binders ∪ {x}) h1_right | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | simp | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x} | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | tauto | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux | [816, 1] | [838, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admits | [841, 1] | [849, 7] | simp only [admits] | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ admits v u F | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ admitsAux v u ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admits | [841, 1] | [849, 7] | apply not_isBoundIn_imp_admitsAux F v u ∅ h1 | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ admitsAux v u ∅ F | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admits | [841, 1] | [849, 7] | simp | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | induction F generalizing binders | F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
⊢ admitsAux t v binders (replaceFreeAux v t binders F) | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ admitsAux t v binders (replaceFreeAux v t binders true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ admitsAux t v binders (replaceFreeAux v t binders false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | all_goals
simp only [occursIn] at h1
simp only [replaceFreeAux]
simp only [admitsAux] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ admitsAux t v binders (replaceFreeAux v t binders true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ admitsAux t v binders (replaceFreeAux v t binders false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
⊢ ((t = if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) ∨ t = if v = a✝ ∧ a✝ ∉ binders then t else a✝) ∧ t ∉ binders →
v ∉ binders
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
intro x a1 a2 a3
by_cases c1 : v = x ∧ x ∉ binders
case pos =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case neg =>
simp at c1
specialize a2 c1
subst a2
contradiction | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs ∧ t ∉ binders → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case eq_ x y =>
push_neg at h1
cases h1
case intro h1_left h1_right =>
intro a1
split_ifs at a1
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right
case _ c1 c2 =>
tauto | v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | all_goals
tauto | case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | simp only [occursIn] at h1 | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | simp only [replaceFreeAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ admitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | simp only [admitsAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ admitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | simp | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs ∧ t ∉ binders → v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → t ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | intro x a1 a2 a3 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → t ∉ binders → v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | by_cases c1 : v = x ∧ x ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
⊢ v ∉ binders | case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case pos =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case neg =>
simp at c1
specialize a2 c1
subst a2
contradiction | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | cases c1 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders | case intro
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | subst c1_left | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a3 : t ∉ binders
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | exact c1_right | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a3 : t ∉ binders
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | simp at c1 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : v = x → x ∈ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | specialize a2 c1 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
a3 : t ∉ binders
c1 : v = x → x ∈ binders
⊢ v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a3 : t ∉ binders
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | subst a2 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a3 : t ∉ binders
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders | v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
a3 : x ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | contradiction | v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
a3 : x ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | push_neg at h1 | v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | cases h1 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
left✝ : t ≠ x
right✝ : t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case intro h1_left h1_right =>
intro a1
split_ifs at a1
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right
case _ c1 c2 =>
tauto | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | intro a1 | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
a1 : ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | split_ifs at a1 | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
a1 : ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders
⊢ v ∉ binders | case pos
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders
case neg
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders
case pos
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
h✝¹ : ¬(v = x ∧ x ∉ binders)
h✝ : v = y ∧ y ∉ binders
a1 : (t = x ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders
case neg
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
h✝¹ : ¬(v = x ∧ x ∉ binders)
h✝ : ¬(v = y ∧ y ∉ binders)
a1 : (t = x ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : (t = x ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case _ c1 c2 =>
tauto | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = x ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | cases c1 | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | subst c1_left | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t y : VarName
binders : Finset VarName
h1_right : t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
h1_left : t ≠ v
c1_right : v ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | exact c1_right | v t y : VarName
binders : Finset VarName
h1_right : t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : (t = t ∨ t = t) ∧ t ∉ binders
h1_left : t ≠ v
c1_right : v ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | cases c1 | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | subst c1_left | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t y : VarName
binders : Finset VarName
h1_right : t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
h1_left : t ≠ v
c1_right : v ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | exact c1_right | v t y : VarName
binders : Finset VarName
h1_right : t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = t ∨ t = y) ∧ t ∉ binders
h1_left : t ≠ v
c1_right : v ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | cases c2 | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : (t = x ∨ t = t) ∧ t ∉ binders
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : (t = x ∨ t = t) ∧ t ∉ binders
left✝ : v = y
right✝ : y ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | case intro c2_left c2_right =>
subst c2_left
exact c2_right | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : (t = x ∨ t = t) ∧ t ∉ binders
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | subst c2_left | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : (t = x ∨ t = t) ∧ t ∉ binders
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders | v t x : VarName
binders : Finset VarName
h1_left : t ≠ x
c1 : ¬(v = x ∧ x ∉ binders)
a1 : (t = x ∨ t = t) ∧ t ∉ binders
h1_right : t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | exact c2_right | v t x : VarName
binders : Finset VarName
h1_left : t ≠ x
c1 : ¬(v = x ∧ x ∉ binders)
a1 : (t = x ∨ t = t) ∧ t ∉ binders
h1_right : t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | tauto | v t x y : VarName
binders : Finset VarName
h1_left : t ≠ x
h1_right : t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : (t = x ∨ t = y) ∧ t ∉ binders
⊢ v ∉ binders | no goals |
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