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 bind... | 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 binde... | 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... |
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 binde... | 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... |
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 : ... | 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 exis... |
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 p... |
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 ... |
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 p... |
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 ... |
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 = fastReplac... |
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... | 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... | 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✝ : VarNam... | 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 ... |
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✝... | 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✝... | 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✝... | 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✝... |
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... |
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... |
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 (fastReplac... | 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 (fastReplac... | 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 ... | 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 : ¬oc... |
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 (fastReplaceFr... | 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 (fastR... |
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_i... | 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 ... | 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 (fastReplaceFr... | 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 (fastR... |
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 ... | 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 ph... |
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 (fastRe... | 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 ph... |
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 (fastR... |
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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