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
|
|---|---|---|---|---|---|---|---|---|
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 |
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} |
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 |
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 |
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} |
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} |
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} |
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} |
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} |
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 |
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 |
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 |
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 |
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 |
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} |
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} |
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 |
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} |
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} |
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} |
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 |
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 |
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} |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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
⊢ ∅ ⊆ ∅ |
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 |
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} |
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 |
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
⊢ ∅ ⊆ ∅ |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.