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.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
exact c1
|
case s1
z : String
t : Var
x : String
c1 : ¬z = x
⊢ ¬z = x
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp only [s1]
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ {free_ x}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
apply Finset.subset_union_right
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ {free_ x}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp only [Var.subst]
|
z : String
t : Var
i : ℕ
⊢ ((free_ z).subst t (bound_ i)).freeVarSet ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
z : String
t : Var
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
conv =>
lhs
simp only [Var.freeVarSet]
|
z : String
t : Var
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
z : String
t : Var
i : ℕ
⊢ ∅ ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp
|
z : String
t : Var
i : ℕ
⊢ ∅ ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
induction F
|
z : String
t : Var
F : Formula
⊢ (Formula.subst (free_ z) t F).freeVarSet ⊆ t.freeVarSet ∪ F.freeVarSet \ {free_ z}
|
case pred_
z : String
t : Var
a✝¹ : String
a✝ : List Var
⊢ (Formula.subst (free_ z) t (pred_ a✝¹ a✝)).freeVarSet ⊆ t.freeVarSet ∪ (pred_ a✝¹ a✝).freeVarSet \ {free_ z}
case not_
z : String
t : Var
a✝ : Formula
a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet ⊆ t.freeVarSet ∪ a✝.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t a✝.not_).freeVarSet ⊆ t.freeVarSet ∪ a✝.not_.freeVarSet \ {free_ z}
case imp_
z : String
t : Var
a✝¹ a✝ : Formula
a_ih✝¹ : (Formula.subst (free_ z) t a✝¹).freeVarSet ⊆ t.freeVarSet ∪ a✝¹.freeVarSet \ {free_ z}
a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet ⊆ t.freeVarSet ∪ a✝.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t (a✝¹.imp_ a✝)).freeVarSet ⊆ t.freeVarSet ∪ (a✝¹.imp_ a✝).freeVarSet \ {free_ z}
case forall_
z : String
t : Var
a✝¹ : String
a✝ : Formula
a_ih✝ : (Formula.subst (free_ z) t a✝).freeVarSet ⊆ t.freeVarSet ∪ a✝.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t (forall_ a✝¹ a✝)).freeVarSet ⊆ t.freeVarSet ∪ (forall_ a✝¹ a✝).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
case not_ phi phi_ih =>
simp only [Formula.subst]
simp only [Formula.freeVarSet]
exact phi_ih
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi.not_).freeVarSet ⊆ t.freeVarSet ∪ phi.not_.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.subst]
simp only [Formula.freeVarSet]
exact phi_ih
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet ⊆ t.freeVarSet ∪ (forall_ x phi).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
X : String
vs : List Var
⊢ (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet ⊆ t.freeVarSet ∪ (pred_ X vs).freeVarSet \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet ⊆ t.freeVarSet ∪ (pred_ X vs).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet ⊆ t.freeVarSet ∪ (pred_ X vs).freeVarSet \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
⊢ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet ⊆
t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp
|
z : String
t : Var
X : String
vs : List Var
⊢ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet ⊆
t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
⊢ ∀ a ∈ vs, ((free_ z).subst t a).freeVarSet ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
intro v a1
|
z : String
t : Var
X : String
vs : List Var
⊢ ∀ a ∈ vs, ((free_ z).subst t a).freeVarSet ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ ((free_ z).subst t v).freeVarSet ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
trans Var.freeVarSet t ∪ Var.freeVarSet v \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ ((free_ z).subst t v).freeVarSet ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ ((free_ z).subst t v).freeVarSet ⊆ t.freeVarSet ∪ v.freeVarSet \ {free_ z}
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ t.freeVarSet ∪ v.freeVarSet \ {free_ z} ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
exact VarSubstFreeVarSet z t v
|
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ ((free_ z).subst t v).freeVarSet ⊆ t.freeVarSet ∪ v.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
apply Finset.union_subset_union_right
|
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ t.freeVarSet ∪ v.freeVarSet \ {free_ z} ⊆ t.freeVarSet ∪ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
case h
z : String
t : Var
X : String
vs : List Var
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.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [← List.mem_toFinset] at a1
|
case h
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
case h
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
apply Finset.sdiff_subset_sdiff
|
case h
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
case h.hst
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
case h.hvu
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ {free_ z} ⊆ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
apply Finset.subset_biUnion_of_mem Var.freeVarSet a1
|
case h.hst
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp
|
case h.hvu
z : String
t : Var
X : String
vs : List Var
v : Var
a1 : v ∈ vs.toFinset
⊢ {free_ z} ⊆ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi.not_).freeVarSet ⊆ t.freeVarSet ∪ phi.not_.freeVarSet \ {free_ z}
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).not_.freeVarSet ⊆ t.freeVarSet ∪ phi.not_.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).not_.freeVarSet ⊆ t.freeVarSet ∪ phi.not_.freeVarSet \ {free_ z}
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
exact phi_ih
|
z : String
t : Var
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t (phi.imp_ psi)).freeVarSet ⊆ t.freeVarSet ∪ (phi.imp_ psi).freeVarSet \ {free_ z}
|
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet ⊆
t.freeVarSet ∪ (phi.imp_ psi).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ ((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet ⊆
t.freeVarSet ∪ (phi.imp_ psi).freeVarSet \ {free_ z}
|
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ∪ (Formula.subst (free_ z) t psi).freeVarSet ⊆
t.freeVarSet ∪ (phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
apply Finset.union_subset_left_right_diff
|
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ∪ (Formula.subst (free_ z) t psi).freeVarSet ⊆
t.freeVarSet ∪ (phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z}
|
case h1
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
case h2
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
exact phi_ih
|
case h1
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
exact psi_ih
|
case h2
z : String
t : Var
phi psi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
psi_ih : (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t psi).freeVarSet ⊆ t.freeVarSet ∪ psi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet ⊆ t.freeVarSet ∪ (forall_ x phi).freeVarSet \ {free_ z}
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet ⊆ t.freeVarSet ∪ (forall_ x phi).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet ⊆ t.freeVarSet ∪ (forall_ x phi).freeVarSet \ {free_ z}
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet
|
[503, 1]
|
[536, 17]
|
exact phi_ih
|
z : String
t : Var
x : String
phi : Formula
phi_ih : (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
⊢ (Formula.subst (free_ z) t phi).freeVarSet ⊆ t.freeVarSet ∪ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
cases v
|
z : String
t v : Var
⊢ v.freeVarSet \ {free_ z} ⊆ ((free_ z).subst t v).freeVarSet
|
case free_
z : String
t : Var
a✝ : String
⊢ (free_ a✝).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (free_ a✝)).freeVarSet
case bound_
z : String
t : Var
a✝ : ℕ
⊢ (bound_ a✝).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (bound_ a✝)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
case free_ x =>
simp only [Var.subst]
split_ifs
case pos c1 =>
simp only [c1]
conv =>
lhs
simp only [Var.freeVarSet]
simp
case neg c1 =>
simp only [Var.freeVarSet]
exact Finset.sdiff_subset {free_ x} {free_ z}
|
z : String
t : Var
x : String
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (free_ x)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
case bound_ i =>
conv =>
lhs
simp only [Var.freeVarSet]
|
z : String
t : Var
i : ℕ
⊢ (bound_ i).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (bound_ i)).freeVarSet
|
z : String
t : Var
i : ℕ
⊢ ∅ \ {free_ z} ⊆ ((free_ z).subst t (bound_ i)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
simp only [Var.subst]
|
z : String
t : Var
x : String
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (free_ x)).freeVarSet
|
z : String
t : Var
x : String
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ (if free_ z = free_ x then t else free_ x).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
split_ifs
|
z : String
t : Var
x : String
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ (if free_ z = free_ x then t else free_ x).freeVarSet
|
case pos
z : String
t : Var
x : String
h✝ : free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ t.freeVarSet
case neg
z : String
t : Var
x : String
h✝ : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ (free_ x).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
case pos c1 =>
simp only [c1]
conv =>
lhs
simp only [Var.freeVarSet]
simp
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ t.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
case neg c1 =>
simp only [Var.freeVarSet]
exact Finset.sdiff_subset {free_ x} {free_ z}
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ (free_ x).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
simp only [c1]
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ t.freeVarSet
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ x} ⊆ t.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
conv =>
lhs
simp only [Var.freeVarSet]
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ x} ⊆ t.freeVarSet
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ {free_ x} \ {free_ x} ⊆ t.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
simp
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ {free_ x} \ {free_ x} ⊆ t.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
simp only [Var.freeVarSet]
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} ⊆ (free_ x).freeVarSet
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ {free_ x} \ {free_ z} ⊆ {free_ x}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
exact Finset.sdiff_subset {free_ x} {free_ z}
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ {free_ x} \ {free_ z} ⊆ {free_ x}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet'
|
[540, 1]
|
[562, 33]
|
conv =>
lhs
simp only [Var.freeVarSet]
|
z : String
t : Var
i : ℕ
⊢ (bound_ i).freeVarSet \ {free_ z} ⊆ ((free_ z).subst t (bound_ i)).freeVarSet
|
z : String
t : Var
i : ℕ
⊢ ∅ \ {free_ z} ⊆ ((free_ z).subst t (bound_ i)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
induction F
|
z : String
t : Var
F : Formula
⊢ F.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t F).freeVarSet
|
case pred_
z : String
t : Var
a✝¹ : String
a✝ : List Var
⊢ (pred_ a✝¹ a✝).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (pred_ a✝¹ a✝)).freeVarSet
case not_
z : String
t : Var
a✝ : Formula
a_ih✝ : a✝.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t a✝).freeVarSet
⊢ a✝.not_.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t a✝.not_).freeVarSet
case imp_
z : String
t : Var
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t a✝¹).freeVarSet
a_ih✝ : a✝.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t a✝).freeVarSet
⊢ (a✝¹.imp_ a✝).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (a✝¹.imp_ a✝)).freeVarSet
case forall_
z : String
t : Var
a✝¹ : String
a✝ : Formula
a_ih✝ : a✝.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t a✝).freeVarSet
⊢ (forall_ a✝¹ a✝).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (forall_ a✝¹ a✝)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
case pred_ X vs =>
simp only [Formula.subst]
simp only [Formula.freeVarSet]
induction vs
case nil =>
simp
case cons hd tl ih =>
simp
have s1 : (Var.freeVarSet hd ∪ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) ∪ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z})
exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z}
simp only [s1]
exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X vs).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
case not_ phi phi_ih =>
simp only [Formula.subst]
simp only [Formula.freeVarSet]
exact phi_ih
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.not_.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi.not_).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.subst]
simp only [Formula.freeVarSet]
exact phi_ih
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ (forall_ x phi).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X vs).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (pred_ X vs)).freeVarSet
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X vs).freeVarSet \ {free_ z} ⊆ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
X : String
vs : List Var
⊢ (pred_ X vs).freeVarSet \ {free_ z} ⊆ (pred_ X (List.map ((free_ z).subst t) vs)).freeVarSet
|
z : String
t : Var
X : String
vs : List Var
⊢ vs.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
induction vs
|
z : String
t : Var
X : String
vs : List Var
⊢ vs.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) vs).toFinset.biUnion Var.freeVarSet
|
case nil
z : String
t : Var
X : String
⊢ [].toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet
case cons
z : String
t : Var
X : String
head✝ : Var
tail✝ : List Var
tail_ih✝ :
tail✝.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
(List.map ((free_ z).subst t) tail✝).toFinset.biUnion Var.freeVarSet
⊢ (head✝ :: tail✝).toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
(List.map ((free_ z).subst t) (head✝ :: tail✝)).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
case nil =>
simp
|
z : String
t : Var
X : String
⊢ [].toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
case cons hd tl ih =>
simp
have s1 : (Var.freeVarSet hd ∪ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) ∪ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z})
exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z}
simp only [s1]
exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd :: tl).toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
(List.map ((free_ z).subst t) (hd :: tl)).toFinset.biUnion Var.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp
|
z : String
t : Var
X : String
⊢ [].toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) []).toFinset.biUnion Var.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd :: tl).toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
(List.map ((free_ z).subst t) (hd :: tl)).toFinset.biUnion Var.freeVarSet
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
have s1 : (Var.freeVarSet hd ∪ Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z} = (Var.freeVarSet hd \ {free_ z}) ∪ ((Finset.biUnion (List.toFinset tl) Var.freeVarSet) \ {free_ z})
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
case s1
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
exact Finset.union_sdiff_distrib (Var.freeVarSet hd) (Finset.biUnion (List.toFinset tl) Var.freeVarSet) {free_ z}
|
case s1
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [s1]
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ (hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
exact Finset.union_subset_union (VarSubstFreeVarSet' z t hd) ih
|
z : String
t : Var
X : String
hd : Var
tl : List Var
ih : tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
s1 :
(hd.freeVarSet ∪ tl.toFinset.biUnion Var.freeVarSet) \ {free_ z} =
hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z}
⊢ hd.freeVarSet \ {free_ z} ∪ tl.toFinset.biUnion Var.freeVarSet \ {free_ z} ⊆
((free_ z).subst t hd).freeVarSet ∪ (List.map ((free_ z).subst t) tl).toFinset.biUnion Var.freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.not_.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi.not_).freeVarSet
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.not_.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).not_.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.not_.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).not_.freeVarSet
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
exact phi_ih
|
z : String
t : Var
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ (phi.imp_ psi).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (phi.imp_ psi)).freeVarSet
|
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ (phi.imp_ psi).freeVarSet \ {free_ z} ⊆
((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ (phi.imp_ psi).freeVarSet \ {free_ z} ⊆
((Formula.subst (free_ z) t phi).imp_ (Formula.subst (free_ z) t psi)).freeVarSet
|
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ (phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z} ⊆
(Formula.subst (free_ z) t phi).freeVarSet ∪ (Formula.subst (free_ z) t psi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
apply Finset.diff_union_subset
|
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ (phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z} ⊆
(Formula.subst (free_ z) t phi).freeVarSet ∪ (Formula.subst (free_ z) t psi).freeVarSet
|
case h1
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
case h2
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
apply phi_ih
|
case h1
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
apply psi_ih
|
case h2
z : String
t : Var
phi psi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
psi_ih : psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
⊢ psi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t psi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.subst]
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ (forall_ x phi).freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t (forall_ x phi)).freeVarSet
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ (forall_ x phi).freeVarSet \ {free_ z} ⊆ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
simp only [Formula.freeVarSet]
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ (forall_ x phi).freeVarSet \ {free_ z} ⊆ (forall_ x (Formula.subst (free_ z) t phi)).freeVarSet
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaSubstFreeVarSet'
|
[565, 1]
|
[600, 17]
|
exact phi_ih
|
z : String
t : Var
x : String
phi : Formula
phi_ih : phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
⊢ phi.freeVarSet \ {free_ z} ⊆ (Formula.subst (free_ z) t phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
cases v
|
v : Var
σ : String → String
j : ℕ
z : String
h1 : σ z = z
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) v) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) v)
|
case free_
σ : String → String
j : ℕ
z : String
h1 : σ z = z
a✝ : String
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (free_ a✝)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (free_ a✝))
case bound_
σ : String → String
j : ℕ
z : String
h1 : σ z = z
a✝ : ℕ
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (bound_ a✝)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ a✝))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
case free_ x =>
conv =>
lhs
simp only [Var.open]
simp only [Var.substFun]
simp only [str_fun_to_var_fun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
x : String
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (free_ x)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (free_ x))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
x : String
⊢ free_ (σ x) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (free_ x))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
case bound_ i =>
conv =>
lhs
simp only [Var.open]
split_ifs
case pos c1 =>
simp only [Var.substFun]
simp only [Var.open]
simp only [if_pos c1]
case pos c1 c2 =>
simp only [Var.substFun]
simp only [str_fun_to_var_fun]
simp only [Var.open]
simp only [if_neg c1]
simp only [if_pos c2]
simp only [h1]
case neg c1 c2 =>
simp only [Var.substFun]
simp only [Var.open]
simp only [if_neg c1]
simp only [if_neg c2]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (bound_ i)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
conv =>
lhs
simp only [Var.open]
simp only [Var.substFun]
simp only [str_fun_to_var_fun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
x : String
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (free_ x)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (free_ x))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
x : String
⊢ free_ (σ x) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (free_ x))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
conv =>
lhs
simp only [Var.open]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
⊢ Var.substFun (str_fun_to_var_fun σ) (Var.open j (free_ z) (bound_ i)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
⊢ Var.substFun (str_fun_to_var_fun σ) (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
split_ifs
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
⊢ Var.substFun (str_fun_to_var_fun σ) (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
case pos
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
h✝ : i < j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
case pos
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
h✝¹ : ¬i < j
h✝ : i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
case neg
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ (i - 1)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
case pos c1 =>
simp only [Var.substFun]
simp only [Var.open]
simp only [if_pos c1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
case pos c1 c2 =>
simp only [Var.substFun]
simp only [str_fun_to_var_fun]
simp only [Var.open]
simp only [if_neg c1]
simp only [if_pos c2]
simp only [h1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
case neg c1 c2 =>
simp only [Var.substFun]
simp only [Var.open]
simp only [if_neg c1]
simp only [if_neg c2]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ (i - 1)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.substFun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ i) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ bound_ i = Var.open j (free_ z) (bound_ i)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.open]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ bound_ i = Var.open j (free_ z) (bound_ i)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ bound_ i = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [if_pos c1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : i < j
⊢ bound_ i = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.substFun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (free_ z) = Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ str_fun_to_var_fun σ (free_ z) = Var.open j (free_ z) (bound_ i)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [str_fun_to_var_fun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ str_fun_to_var_fun σ (free_ z) = Var.open j (free_ z) (bound_ i)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = Var.open j (free_ z) (bound_ i)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.open]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = Var.open j (free_ z) (bound_ i)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [if_neg c1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = if i = j then free_ z else bound_ (i - 1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [if_pos c2]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = if i = j then free_ z else bound_ (i - 1)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = free_ z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [h1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ free_ (σ z) = free_ z
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.substFun]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ Var.substFun (str_fun_to_var_fun σ) (bound_ (i - 1)) =
Var.open j (free_ z) (Var.substFun (str_fun_to_var_fun σ) (bound_ i))
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = Var.open j (free_ z) (bound_ i)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [Var.open]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = Var.open j (free_ z) (bound_ i)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [if_neg c1]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = if i = j then free_ z else bound_ (i - 1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.SubOpenVar
|
[611, 1]
|
[647, 28]
|
simp only [if_neg c2]
|
σ : String → String
j : ℕ
z : String
h1 : σ z = z
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ bound_ (i - 1) = if i = j then free_ z else bound_ (i - 1)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
funext v
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d = shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)
|
case h
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
v : Var
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
simp
|
case h
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
v : Var
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) v
|
case h
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
v : Var
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = shift D V d (Var.openList (j + 1) (List.map free_ zs) v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
cases v
|
case h
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
v : Var
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d v = shift D V d (Var.openList (j + 1) (List.map free_ zs) v)
|
case h.free_
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
a✝ : String
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ a✝) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ a✝))
case h.bound_
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
a✝ : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ a✝) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ a✝))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
case _ x =>
simp only [Var.openList]
simp only [shift]
simp
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ x))
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ V (Var.openList j (List.map free_ zs) (free_ x)) = V (free_ x)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
case _ i =>
cases i
case zero =>
simp only [shift]
simp only [Var.openList]
simp
case succ i =>
simp only [shift]
simp only [Var.openList]
simp
split
case _ c1 =>
have s1 : i + 1 < j + 1
linarith
simp only [if_pos s1]
case _ c1 =>
have s1 : ¬ i + 1 < j + 1
linarith
simp only [if_neg s1]
split
case _ c2 =>
simp
case _ c2 =>
simp
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ i) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ i))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (free_ x))
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (free_ x)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
simp only [shift]
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (free_ x) = shift D V d (free_ x)
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ (V ∘ Var.openList j (List.map free_ zs)) (free_ x) = V (free_ x)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftVarOpenList
|
[815, 1]
|
[854, 17]
|
simp
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ (V ∘ Var.openList j (List.map free_ zs)) (free_ x) = V (free_ x)
|
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
x : String
⊢ V (Var.openList j (List.map free_ zs) (free_ x)) = V (free_ x)
|
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