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/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | all_goals
simp only [isBoundIn] at h1
simp only [fastAdmitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h... | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
bi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | simp only [isBoundIn] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | simp only [fastAdmitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | push_neg at h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | cases h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
left✝ : u ≠ x
right✝ : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | right | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | apply phi_ih (binders ∪ {x}) h1_right | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | simp | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x} | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | tauto | case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits | [388, 1] | [396, 7] | simp only [fastAdmits] | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmits v u F | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmits v u F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits | [388, 1] | [396, 7] | apply not_isBoundIn_imp_fastAdmitsAux F v u ∅ h1 | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits | [388, 1] | [396, 7] | simp | F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | induction F generalizing binders | F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t F) | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t ... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | all_goals
simp only [occursIn] at h1
simp only [fastReplaceFree] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t ... | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | any_goals
simp only [fastAdmitsAux] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉... | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | 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 [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursI... | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | all_goals
tauto | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Fi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp only [occursIn] at h1 | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (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
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp only [fastReplaceFree] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp only [fastAdmitsAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | push_neg at h1 | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t... | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t p... | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | cases h1 | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t p... | case intro
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
left✝ : t ≠ x
right✝ : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (... | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursIn
exact contra
case neg c1 =>
simp only [fastAdmitsAux]
right
apply phi_ih (binder... | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastRep... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | split_ifs | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastRep... | case pos
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
case neg
v t x ... | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | case pos c1 =>
simp only [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursIn
exact contra | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (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), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | case neg c1 =>
simp only [fastAdmitsAux]
right
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (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), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp only [fastAdmitsAux] | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi) | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | subst c1 | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi | v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | right | v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi | case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAux t v (binders ∪ {v}) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | apply not_isFreeIn_imp_fastAdmitsAux | case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAux t v (binders ∪ {v}) phi | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn t phi | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | intro contra | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn t phi | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | apply h1_right | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ False | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ occursIn t phi | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : is... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | apply isFreeIn_imp_occursIn | case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ occursIn t phi | case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ isFreeIn t phi | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : is... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | exact contra | case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ isFreeIn t phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp only [fastAdmitsAux] | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi)) | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {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), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | right | v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi... | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v (binders ∪ {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), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | apply phi_ih (binders ∪ {x}) h1_right | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi) | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | simp | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x} | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | tauto | case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux | [400, 1] | [437, 10] | tauto | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits | [440, 1] | [448, 7] | simp only [fastAdmits] | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (fastReplaceFree v t F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits | [440, 1] | [448, 7] | apply fastReplaceFree_aux_fastAdmitsAux F v t ∅ h1 | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits | [440, 1] | [448, 7] | simp | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | induction F generalizing binders | F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders F) | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | all_goals
simp only [occursIn] at h1
simp only [replaceFreeAux]
simp only [fastAdmitsAux] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_... | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case eq_ x y =>
push_neg at h1
intros a1
split_ifs at a1
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c2
case int... | v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | all_goals
tauto | case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : F... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
cas... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | simp only [occursIn] at h1 | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (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
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | simp only [replaceFreeAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝)) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | simp only [fastAdmitsAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝)) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | simp | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs → v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | intro x a1 a2 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | by_cases c1 : v = x ∧ x ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders | case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | cases c1 | case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders | case pos.intro
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case _ c1_left c1_right =>
subst c1_left
exact c1_right | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | subst c1_left | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | exact c1_right | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | push_neg at c1 | case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders | case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | specialize a2 c1 | case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders | case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | subst a2 | case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders | case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | contradiction | case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | push_neg at h1 | v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | intros a1 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | split_ifs at a1 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders | case pos
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
case neg
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
case ... | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case _ c1 c2 =>
tauto | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | cases c1 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | subst c1_left | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | exact c1_right | v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | cases c1 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case intro c1_left c1_right =>
subst c1_left
exact c1_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | subst c1_left | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders | v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | exact c1_right | v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | cases c2 | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders | case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
left✝ : v = y
right✝ : y ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | case intro c2_left c2_right =>
subst c2_left
exact c2_right | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | subst c2_left | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders | v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | exact c2_right | v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | tauto | v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux | [452, 1] | [500, 10] | tauto | case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits | [503, 1] | [511, 48] | simp only [replaceFree] | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFree v t F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits | [503, 1] | [511, 48] | simp only [fastAdmits] | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits | [503, 1] | [511, 48] | exact replaceFreeAux_fastAdmitsAux F v t ∅ h1 | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | induction F generalizing S | F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u S F
h2 : u ∉ T
⊢ fastAdmitsAux v u (S ∪ T) F | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u S F
h2 : u ∉ T
⊢ fastAdmitsAux v u (S ∪ T) F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | all_goals
simp only [fastAdmitsAux] at h1
simp only [fastAdmitsAux] | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
... | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case eq_
v u... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp
cases h1
case inl h1 =>
tauto
case inr h1 =>
specialize phi_ih (S ∪ {x}) h1
right
simp only [Finset.union_right_comm_assoc]
exact phi_ih | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | any_goals
simp only [Finset.mem_union] | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case eq_
v u... | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | all_goals
tauto | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : F... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | simp only [fastAdmitsAux] at h1 | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | simp only [fastAdmitsAux] | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | simp | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | cases h1 | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | case inl
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h✝ : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
case inr
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Fo... | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | case inl h1 =>
tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | case inr h1 =>
specialize phi_ih (S ∪ {x}) h1
right
simp only [Finset.union_right_comm_assoc]
exact phi_ih | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | specialize phi_ih (S ∪ {x}) h1 | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) ... |
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