url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
tauto
|
v t : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).not_ = (fastReplaceFree v t phi).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).not_ = (fastReplaceFree v t phi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
simp
|
v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) =
(fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi)
|
v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders phi = fastReplaceFree v t phi ∧ replaceFreeAux v t binders psi = fastReplaceFree v t psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) =
(fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
tauto
|
v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders phi = fastReplaceFree v t phi ∧ replaceFreeAux v t binders psi = fastReplaceFree v t psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders phi = fastReplaceFree v t phi ∧ replaceFreeAux v t binders psi = fastReplaceFree v t psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
split_ifs
|
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) =
if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)
|
case pos
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi
case neg
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : ¬v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) =
if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
case pos c1 =>
congr! 1
apply replaceFreeAux_mem_binders
simp
right
exact c1
|
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
case neg c1 =>
congr! 1
apply phi_ih
simp
tauto
|
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
congr! 1
|
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
apply replaceFreeAux_mem_binders
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∪ {x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
simp
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∪ {x}
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∨ v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
right
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∨ v = x
|
case h.e'_2.h1.h
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v ∈ binders ∨ v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
exact c1
|
case h.e'_2.h1.h
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1.h
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
congr! 1
|
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = fastReplaceFree v t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
apply phi_ih
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = fastReplaceFree v t phi
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ replaceFreeAux v t (binders ∪ {x}) phi = fastReplaceFree v t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
simp
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
|
[163, 1]
|
[216, 12]
|
tauto
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree
|
[219, 1]
|
[226, 7]
|
simp only [replaceFree]
|
F : Formula
v t : VarName
⊢ replaceFree v t F = fastReplaceFree v t F
|
F : Formula
v t : VarName
⊢ replaceFreeAux v t ∅ F = fastReplaceFree v t F
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
⊢ replaceFree v t F = fastReplaceFree v t F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree
|
[219, 1]
|
[226, 7]
|
apply replaceFreeAux_eq_fastReplaceFree
|
F : Formula
v t : VarName
⊢ replaceFreeAux v t ∅ F = fastReplaceFree v t F
|
case h1
F : Formula
v t : VarName
⊢ v ∉ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
⊢ replaceFreeAux v t ∅ F = fastReplaceFree v t F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree
|
[219, 1]
|
[226, 7]
|
simp
|
case h1
F : Formula
v t : VarName
⊢ v ∉ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
F : Formula
v t : VarName
⊢ v ∉ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
induction F
|
F : Formula
v : VarName
⊢ fastReplaceFree v v F = F
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v a✝¹ a✝ : VarName
⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v : VarName
⊢ fastReplaceFree v v true_ = true_
case false_
v : VarName
⊢ fastReplaceFree v v false_ = false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v a✝.not_ = a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v : VarName
⊢ fastReplaceFree v v F = F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
any_goals
simp only [fastReplaceFree]
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v a✝¹ a✝ : VarName
⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v : VarName
⊢ fastReplaceFree v v true_ = true_
case false_
v : VarName
⊢ fastReplaceFree v v false_ = false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v a✝.not_ = a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ pred_const_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = pred_const_ a✝¹ a✝
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ pred_var_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = pred_var_ a✝¹ a✝
case eq_
v a✝¹ a✝ : VarName
⊢ eq_ (if v = a✝¹ then v else a✝¹) (if v = a✝ then v else a✝) = eq_ a✝¹ a✝
case not_
v : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (fastReplaceFree v v a✝).not_ = a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (fastReplaceFree v v a✝¹).imp_ (fastReplaceFree v v a✝) = a✝¹.imp_ a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (fastReplaceFree v v a✝¹).and_ (fastReplaceFree v v a✝) = a✝¹.and_ a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (fastReplaceFree v v a✝¹).or_ (fastReplaceFree v v a✝) = a✝¹.or_ a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (fastReplaceFree v v a✝¹).iff_ (fastReplaceFree v v a✝) = a✝¹.iff_ a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v v a✝)) = forall_ a✝¹ a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v v a✝)) = exists_ a✝¹ a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v a✝¹ a✝ : VarName
⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v : VarName
⊢ fastReplaceFree v v true_ = true_
case false_
v : VarName
⊢ fastReplaceFree v v false_ = false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v a✝.not_ = a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
simp only [List.map_eq_self_iff]
simp
|
v : VarName
X : DefName
xs : List VarName
⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
X : DefName
xs : List VarName
⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
case eq_ x y =>
simp
|
v x y : VarName
⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x y : VarName
⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
case not_ phi phi_ih =>
congr!
|
v : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (fastReplaceFree v v phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (fastReplaceFree v v phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr!
|
v : VarName
phi psi : Formula
phi_ih : fastReplaceFree v v phi = phi
psi_ih : fastReplaceFree v v psi = psi
⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
phi psi : Formula
phi_ih : fastReplaceFree v v phi = phi
psi_ih : fastReplaceFree v v psi = psi
⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp
intro _
exact phi_ih
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp only [fastReplaceFree]
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp
|
v : VarName
X : DefName
xs : List VarName
⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs
|
v : VarName
X : DefName
xs : List VarName
⊢ List.map (fun x => if v = x then v else x) xs = xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
X : DefName
xs : List VarName
⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp only [List.map_eq_self_iff]
|
v : VarName
X : DefName
xs : List VarName
⊢ List.map (fun x => if v = x then v else x) xs = xs
|
v : VarName
X : DefName
xs : List VarName
⊢ ∀ x ∈ xs, (if v = x then v else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
X : DefName
xs : List VarName
⊢ List.map (fun x => if v = x then v else x) xs = xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp
|
v : VarName
X : DefName
xs : List VarName
⊢ ∀ x ∈ xs, (if v = x then v else x) = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
X : DefName
xs : List VarName
⊢ ∀ x ∈ xs, (if v = x then v else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp
|
v x y : VarName
⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x y : VarName
⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
congr!
|
v : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (fastReplaceFree v v phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (fastReplaceFree v v phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
congr!
|
v : VarName
phi psi : Formula
phi_ih : fastReplaceFree v v phi = phi
psi_ih : fastReplaceFree v v psi = psi
⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v : VarName
phi psi : Formula
phi_ih : fastReplaceFree v v phi = phi
psi_ih : fastReplaceFree v v psi = psi
⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
simp
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ ¬v = x → fastReplaceFree v v phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
intro _
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ ¬v = x → fastReplaceFree v v phi = phi
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
a✝ : ¬v = x
⊢ fastReplaceFree v v phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
⊢ ¬v = x → fastReplaceFree v v phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self
|
[230, 1]
|
[255, 17]
|
exact phi_ih
|
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
a✝ : ¬v = x
⊢ fastReplaceFree v v phi = phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v x : VarName
phi : Formula
phi_ih : fastReplaceFree v v phi = phi
a✝ : ¬v = x
⊢ fastReplaceFree v v phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
induction F
|
F : Formula
v t : VarName
h1 : ¬isFreeIn v F
⊢ fastReplaceFree v t F = F
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ fastReplaceFree v t (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬isFreeIn v true_
⊢ fastReplaceFree v t true_ = true_
case false_
v t : VarName
h1 : ¬isFreeIn v false_
⊢ fastReplaceFree v t false_ = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v a✝.not_
⊢ fastReplaceFree v t a✝.not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ fastReplaceFree v t (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ fastReplaceFree v t (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ fastReplaceFree v t (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ fastReplaceFree v t (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ fastReplaceFree v t (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ fastReplaceFree v t (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬isFreeIn v F
⊢ fastReplaceFree v t F = F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
any_goals
simp only [isFreeIn] at h1
simp only [fastReplaceFree]
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ fastReplaceFree v t (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬isFreeIn v true_
⊢ fastReplaceFree v t true_ = true_
case false_
v t : VarName
h1 : ¬isFreeIn v false_
⊢ fastReplaceFree v t false_ = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v a✝.not_
⊢ fastReplaceFree v t a✝.not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ fastReplaceFree v t (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ fastReplaceFree v t (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ fastReplaceFree v t (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ fastReplaceFree v t (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ fastReplaceFree v t (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ fastReplaceFree v t (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∉ a✝
⊢ pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∉ a✝
⊢ pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝) = eq_ a✝¹ a✝
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v a✝
⊢ (fastReplaceFree v t a✝).not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ (fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ (fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ (fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ (fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree v t (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ fastReplaceFree v t (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬isFreeIn v true_
⊢ fastReplaceFree v t true_ = true_
case false_
v t : VarName
h1 : ¬isFreeIn v false_
⊢ fastReplaceFree v t false_ = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v a✝.not_
⊢ fastReplaceFree v t a✝.not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ fastReplaceFree v t (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ fastReplaceFree v t (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ fastReplaceFree v t (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ fastReplaceFree v t (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ fastReplaceFree v t (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ fastReplaceFree v t (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
simp only [List.map_eq_self_iff]
simp
intro x a1 a2
subst a2
contradiction
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
case eq_ x y =>
simp
tauto
|
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
case not_ phi phi_ih =>
tauto
|
v t : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬isFreeIn v phi
⊢ (fastReplaceFree v t phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬isFreeIn v phi
⊢ (fastReplaceFree v t phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto
|
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp
tauto
|
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp only [isFreeIn] at h1
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp only [fastReplaceFree]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∉ a✝
⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ List.map (fun x => if v = x then t else x) xs = xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp only [List.map_eq_self_iff]
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ List.map (fun x => if v = x then t else x) xs = xs
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, (if v = x then t else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ List.map (fun x => if v = x then t else x) xs = xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, (if v = x then t else x) = x
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, v = x → t = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, (if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
intro x a1 a2
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, v = x → t = x
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
x : VarName
a1 : x ∈ xs
a2 : v = x
⊢ t = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
⊢ ∀ x ∈ xs, v = x → t = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
subst a2
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
x : VarName
a1 : x ∈ xs
a2 : v = x
⊢ t = x
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
a1 : v ∈ xs
⊢ t = v
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
x : VarName
a1 : x ∈ xs
a2 : v = x
⊢ t = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
contradiction
|
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
a1 : v ∈ xs
⊢ t = v
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : v ∉ xs
a1 : v ∈ xs
⊢ t = v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp
|
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y
|
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ (v = x → t = x) ∧ (v = y → t = y)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
tauto
|
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ (v = x → t = x) ∧ (v = y → t = y)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
h1 : ¬(v = x ∨ v = y)
⊢ (v = x → t = x) ∧ (v = y → t = y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
tauto
|
v t : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬isFreeIn v phi
⊢ (fastReplaceFree v t phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬isFreeIn v phi
⊢ (fastReplaceFree v t phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp
|
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi
|
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ fastReplaceFree v t phi = phi ∧ fastReplaceFree v t psi = psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
tauto
|
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ fastReplaceFree v t phi = phi ∧ fastReplaceFree v t psi = psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
⊢ fastReplaceFree v t phi = phi ∧ fastReplaceFree v t psi = psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
simp
|
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ ¬v = x → fastReplaceFree v t phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self
|
[258, 1]
|
[290, 10]
|
tauto
|
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ ¬v = x → fastReplaceFree v t phi = phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ ¬v = x → fastReplaceFree v t phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
induction F
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastReplaceFree t v (fastReplaceFree v t F) = F
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_
case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastReplaceFree t v (fastReplaceFree v t F) = F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
any_goals
simp only [occursIn] at h1
simp only [fastReplaceFree]
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_
case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : t ∉ a✝
⊢ pred_const_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) =
pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : t ∉ a✝
⊢ pred_var_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) =
pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
⊢ eq_ (if t = if v = a✝¹ then t else a✝¹ then v else if v = a✝¹ then t else a✝¹)
(if t = if v = a✝ then t else a✝ then v else if v = a✝ then t else a✝) =
eq_ a✝¹ a✝
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝)).not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).imp_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).and_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).or_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ (fastReplaceFree t v (fastReplaceFree v t a✝¹)).iff_ (fastReplaceFree t v (fastReplaceFree v t a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ fastReplaceFree t v (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ fastReplaceFree t v (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝
case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_
case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case not_ phi phi_ih =>
congr!
exact phi_ih h1
|
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr! <;> tauto
|
v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
push_neg at h1
cases h1
case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right
case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [occursIn] at h1
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [fastReplaceFree]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ def_ a✝¹ (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) a✝)) = def_ a✝¹ a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : t ∉ a✝
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr!
|
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ def_ X (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs)) = def_ X xs
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs) = xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ def_ X (List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs)) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs) = xs
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) xs = xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map (fun x => if t = x then v else x) (List.map (fun x => if v = x then t else x) xs) = xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [List.map_eq_self_iff]
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) xs = xs
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) x = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ List.map ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) xs = xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) x = x
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, (if t = if v = x then t else x then v else if v = x then t else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((fun x => if t = x then v else x) ∘ fun x => if v = x then t else x) x = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
intro x a1
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, (if t = if v = x then t else x then v else if v = x then t else x) = x
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, (if t = if v = x then t else x then v else if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
by_cases c1 : v = x
|
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [if_pos c1]
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if True then v else t) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if True then v else t) = x
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ (if True then v else t) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
exact c1
|
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ v = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [if_neg c1]
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = x then v else x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = x then v else x) = x
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ t = x → v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ (if t = x then v else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
intro a2
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ t = x → v = x
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
a2 : t = x
⊢ v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ t = x → v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
subst a2
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
a2 : t = x
⊢ v = x
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
a1 : t ∈ xs
c1 : ¬v = t
⊢ v = t
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
a2 : t = x
⊢ v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
contradiction
|
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
a1 : t ∈ xs
c1 : ¬v = t
⊢ v = t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
h1 : t ∉ xs
a1 : t ∈ xs
c1 : ¬v = t
⊢ v = t
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr!
|
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ eq_ (if t = if v = x then t else x then v else if v = x then t else x)
(if t = if v = y then t else y then v else if v = y then t else y) =
eq_ x y
|
case h.e'_1
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
case h.e'_2
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = y then t else y then v else if v = y then t else y) = y
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ eq_ (if t = if v = x then t else x then v else if v = x then t else x)
(if t = if v = y then t else y then v else if v = y then t else y) =
eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
split_ifs <;> tauto
|
case h.e'_1
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = x then t else x then v else if v = x then t else x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
split_ifs <;> tauto
|
case h.e'_2
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = y then t else y then v else if v = y then t else y) = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t x y : VarName
h1 : ¬(t = x ∨ t = y)
⊢ (if t = if v = y then t else y then v else if v = y then t else y) = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr!
|
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_
|
case h.e'_1
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
exact phi_ih h1
|
case h.e'_1
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
v t : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬occursIn t phi
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr! <;> tauto
|
v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi psi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
psi_ih : ¬occursIn t psi → fastReplaceFree t v (fastReplaceFree v t psi) = psi
h1 : ¬(occursIn t phi ∨ occursIn t psi)
⊢ (fastReplaceFree t v (fastReplaceFree v t phi)).iff_ (fastReplaceFree t v (fastReplaceFree v t psi)) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
push_neg at h1
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : ¬(t = x ∨ occursIn t phi)
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
cases h1
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
case intro
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
left✝ : t ≠ x
right✝ : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right
case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
split_ifs
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
case pos
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
case neg
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastReplaceFree t v (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case pos c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
apply not_free_in_fastReplaceFree_self
contrapose! h1_right
exact isFreeIn_imp_occursIn t phi h1_right
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
case neg c1 =>
simp only [fastReplaceFree]
simp only [if_neg h1_left]
congr!
exact phi_ih h1_right
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [fastReplaceFree]
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ (if t = x then exists_ x phi else exists_ x (fastReplaceFree t v phi)) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v (exists_ x phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [if_neg h1_left]
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ (if t = x then exists_ x phi else exists_ x (fastReplaceFree t v phi)) = exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ exists_ x (fastReplaceFree t v phi) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ (if t = x then exists_ x phi else exists_ x (fastReplaceFree t v phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr!
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ exists_ x (fastReplaceFree t v phi) = exists_ x phi
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ exists_ x (fastReplaceFree t v phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
apply not_free_in_fastReplaceFree_self
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v phi = phi
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ ¬isFreeIn t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastReplaceFree t v phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
contrapose! h1_right
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ ¬isFreeIn t phi
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
c1 : v = x
h1_right : isFreeIn t phi
⊢ occursIn t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ ¬isFreeIn t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
exact isFreeIn_imp_occursIn t phi h1_right
|
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
c1 : v = x
h1_right : isFreeIn t phi
⊢ occursIn t phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h1
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
c1 : v = x
h1_right : isFreeIn t phi
⊢ occursIn t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [fastReplaceFree]
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ (if t = x then exists_ x (fastReplaceFree v t phi) else exists_ x (fastReplaceFree t v (fastReplaceFree v t phi))) =
exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (exists_ x (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
simp only [if_neg h1_left]
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ (if t = x then exists_ x (fastReplaceFree v t phi) else exists_ x (fastReplaceFree t v (fastReplaceFree v t phi))) =
exists_ x phi
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ exists_ x (fastReplaceFree t v (fastReplaceFree v t phi)) = exists_ x phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ (if t = x then exists_ x (fastReplaceFree v t phi) else exists_ x (fastReplaceFree t v (fastReplaceFree v t phi))) =
exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
congr!
|
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ exists_ x (fastReplaceFree t v (fastReplaceFree v t phi)) = exists_ x phi
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ exists_ x (fastReplaceFree t v (fastReplaceFree v t phi)) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_inverse
|
[293, 1]
|
[349, 30]
|
exact phi_ih h1_right
|
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
v t x : VarName
phi : Formula
phi_ih : ¬occursIn t phi → fastReplaceFree t v (fastReplaceFree v t phi) = phi
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastReplaceFree t v (fastReplaceFree v t phi) = phi
TACTIC:
|
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