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/LN/Paper.lean
|
LN.VarOpenFreeVarSet
|
[216, 1]
|
[238, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
induction F generalizing j
|
j : ℕ
z : String
F : Formula
⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}
|
case pred_
z a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ {free_ z}
case not_
z : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ {free_ z}
case imp_
z : String
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ {free_ z}
a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ {free_ z}
case forall_
z a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
F : Formula
⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
case not_ phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.open]
|
z X : String
vs : List Var
j : ℕ
⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.freeVarSet]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp
|
z X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
intro v a1
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
trans Var.freeVarSet v ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
exact VarOpenFreeVarSet j z v
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
apply Finset.union_subset_union_left
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
|
case h
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
apply Finset.subset_biUnion_of_mem
|
case h
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp
|
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
exact a1
|
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.open]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.open]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ {free_ z}
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
apply Finset.union_subset_union_left_right
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
exact phi_ih j
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
exact psi_ih j
|
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.open]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
simp only [Formula.freeVarSet]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet
|
[241, 1]
|
[273, 17]
|
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
cases v
|
j : ℕ
zs : List String
v : Var
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
|
case free_
j : ℕ
zs : List String
a✝ : String
⊢ (Var.openList j (List.map free_ zs) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
case bound_
j : ℕ
zs : List String
a✝ : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
v : Var
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
case free_ x =>
simp only [Var.openList]
simp only [Var.freeVarSet]
simp
|
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
case bound_ i =>
simp only [Var.openList]
split_ifs
case pos c1 =>
simp only [Var.freeVarSet]
simp
case pos c1 c2 =>
simp
simp only [Var.freeVarSet]
simp
apply List.get_mem
case neg c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.openList]
|
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
zs : List String
x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
x : String
⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp
|
j : ℕ
zs : List String
x : String
⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
x : String
⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.openList]
|
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
⊢ (if i < j then bound_ i
else
if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆
(bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
split_ifs
|
j : ℕ
zs : List String
i : ℕ
⊢ (if i < j then bound_ i
else
if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆
(bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
case pos
j : ℕ
zs : List String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
case pos
j : ℕ
zs : List String
i : ℕ
h✝¹ : ¬i < j
h✝ : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
case neg
j : ℕ
zs : List String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
⊢ (if i < j then bound_ i
else
if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆
(bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
case pos c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
case pos c1 c2 =>
simp
simp only [Var.freeVarSet]
simp
apply List.get_mem
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
case neg c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
apply List.get_mem
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenListFreeVarSet
|
[279, 1]
|
[303, 11]
|
simp
|
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
induction F generalizing j
|
j : ℕ
zs : List String
F : Formula
⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset
|
case pred_
zs : List String
a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)).freeVarSet ⊆
(pred_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
case not_
zs : List String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ (List.map free_ zs).toFinset
case imp_
zs : List String
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ (List.map free_ zs).toFinset
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)).freeVarSet ⊆
(a✝¹.imp_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
case forall_
zs : List String
a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)).freeVarSet ⊆
(forall_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
zs : List String
F : Formula
⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
case not_ phi phi_ih =>
simp only [Formula.openList]
simp only [Formula.freeVarSet]
apply phi_ih
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.openList]
simp only [Formula.freeVarSet]
apply phi_ih
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.openList]
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (pred_ X vs)).freeVarSet ⊆
(pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆
(pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (pred_ X vs)).freeVarSet ⊆
(pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.freeVarSet]
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆
(pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆
(pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs,
(Var.openList j (List.map free_ zs) a).freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
intro v a1
|
zs : List String
X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs,
(Var.openList j (List.map free_ zs) a).freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs,
(Var.openList j (List.map free_ zs) a).freeVarSet ⊆
vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
trans v.freeVarSet ∪ (zs.map free_).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
exact VarOpenListFreeVarSet j zs v
|
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply Finset.union_subset_union_left
|
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
|
case h
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply Finset.subset_biUnion_of_mem
|
case h
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp
|
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
exact a1
|
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.xs
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.openList]
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.freeVarSet]
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply phi_ih
|
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.openList]
|
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (phi.imp_ psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (phi.imp_ psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.freeVarSet]
|
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply Finset.union_subset_union_left_right
|
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
|
case h1
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
case h2
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆
phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
exact phi_ih j
|
case h1
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
exact psi_ih j
|
case h2
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
zs : List String
phi psi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
psi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.openList]
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.freeVarSet]
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList (j + 1) (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply phi_ih
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList (j + 1) (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList (j + 1) (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
cases v
|
j : ℕ
z : String
v : Var
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
|
case free_
j : ℕ
z a✝ : String
⊢ (free_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (free_ a✝)).freeVarSet
case bound_
j : ℕ
z : String
a✝ : ℕ
⊢ (bound_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (bound_ a✝)).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
v : Var
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case free_ x =>
simp only [Var.open]
simp only [Var.freeVarSet]
simp
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case bound_ i =>
simp only [Var.open]
split_ifs
case _ c1 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.open]
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet
|
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.open]
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
split_ifs
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet
|
case pos
j : ℕ
z : String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
case pos
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
case neg
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ ∅ ⊆ {free_ z}
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ ∅ ⊆ {free_ z}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ ∅ ⊆ {free_ z}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
induction F generalizing j
|
j : ℕ
z : String
F : Formula
⊢ F.freeVarSet ⊆ (Formula.open j (free_ z) F).freeVarSet
|
case pred_
z a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (pred_ a✝¹ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet
case not_
z : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ a✝.not_.freeVarSet ⊆ (Formula.open j (free_ z) a✝.not_).freeVarSet
case imp_
z : String
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (j : ℕ), a✝¹.freeVarSet ⊆ (Formula.open j (free_ z) a✝¹).freeVarSet
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ (a✝¹.imp_ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet
case forall_
z a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ (forall_ a✝¹ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
j : ℕ
z : String
F : Formula
⊢ F.freeVarSet ⊆ (Formula.open j (free_ z) F).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
case not_ phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ vs.toFinset.biUnion Var.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp
|
z X : String
vs : List Var
j : ℕ
⊢ vs.toFinset.biUnion Var.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ x ∈ vs, x.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ vs.toFinset.biUnion Var.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
intro v a1
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ x ∈ vs, x.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
⊢ ∀ x ∈ vs, x.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
trans Var.freeVarSet (Var.open j (free_ z) v)
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply VarOpenFreeVarSet'
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply Finset.subset_biUnion_of_mem Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ (List.map (Var.open j (free_ z)) vs).toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply List.mem_toFinset.mpr
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ (List.map (Var.open j (free_ z)) vs).toFinset
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ List.map (Var.open j (free_ z)) vs
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ (List.map (Var.open j (free_ z)) vs).toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
exact List.mem_map_of_mem (Var.open j (free_ z)) a1
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ List.map (Var.open j (free_ z)) vs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ List.map (Var.open j (free_ z)) vs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi).not_.freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi).not_.freeVarSet
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi).not_.freeVarSet
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
TACTIC:
|
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