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/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.imp_right_
|
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v (phi.imp_ psi)
|
case a
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.and_right_
|
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v (phi.and_ psi)
|
case a
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.or_right_
|
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v (phi.or_ psi)
|
case a
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.iff_right_
|
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v (phi.iff_ psi)
|
case a
v : VarName
phi psi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
psi_ih : isFreeIn v psi → isFreeInInd v psi
c1 : isFreeIn v psi
⊢ isFreeInInd v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
cases h1
|
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1 : ¬v = x ∧ isFreeIn v phi
⊢ isFreeInInd v (exists_ x phi)
|
case intro
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
left✝ : ¬v = x
right✝ : isFreeIn v phi
⊢ isFreeInInd v (exists_ x phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
first | apply isFreeInInd.forall_ | apply isFreeInInd.exists_
|
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v (exists_ x phi)
|
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ ¬v = x
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.forall_
|
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v (forall_ x phi)
|
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ ¬v = x
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
apply isFreeInInd.exists_
|
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v (exists_ x phi)
|
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ ¬v = x
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
exact h1_left
|
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ ¬v = x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_isFreeInInd
|
[382, 1]
|
[413, 30]
|
exact phi_ih h1_right
|
case a
v x : VarName
phi : Formula
phi_ih : isFreeIn v phi → isFreeInInd v phi
h1_left : ¬v = x
h1_right : isFreeIn v phi
⊢ isFreeInInd v phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeInInd_imp_isFreeIn
|
[416, 1]
|
[425, 10]
|
induction h1
|
v : VarName
F : Formula
h1 : isFreeInInd v F
⊢ isFreeIn v F
|
case pred_const_
v : VarName
F : Formula
X✝ : PredName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (pred_const_ X✝ xs✝)
case pred_var_
v : VarName
F : Formula
X✝ : PredName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (pred_var_ X✝ xs✝)
case eq_
v : VarName
F : Formula
x✝ y✝ : VarName
a✝ : v = x✝ ∨ v = y✝
⊢ isFreeIn v (eq_ x✝ y✝)
case not_
v : VarName
F phi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v phi✝.not_
case imp_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.imp_ psi✝)
case imp_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.imp_ psi✝)
case and_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.and_ psi✝)
case and_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.and_ psi✝)
case or_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.or_ psi✝)
case or_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.or_ psi✝)
case iff_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.iff_ psi✝)
case iff_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.iff_ psi✝)
case forall_
v : VarName
F : Formula
x✝ : VarName
phi✝ : Formula
a✝¹ : ¬v = x✝
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (forall_ x✝ phi✝)
case exists_
v : VarName
F : Formula
x✝ : VarName
phi✝ : Formula
a✝¹ : ¬v = x✝
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (exists_ x✝ phi✝)
case def_
v : VarName
F : Formula
X✝ : DefName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (def_ X✝ xs✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeInInd_imp_isFreeIn
|
[416, 1]
|
[425, 10]
|
all_goals
simp only [isFreeIn]
tauto
|
case pred_const_
v : VarName
F : Formula
X✝ : PredName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (pred_const_ X✝ xs✝)
case pred_var_
v : VarName
F : Formula
X✝ : PredName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (pred_var_ X✝ xs✝)
case eq_
v : VarName
F : Formula
x✝ y✝ : VarName
a✝ : v = x✝ ∨ v = y✝
⊢ isFreeIn v (eq_ x✝ y✝)
case not_
v : VarName
F phi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v phi✝.not_
case imp_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.imp_ psi✝)
case imp_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.imp_ psi✝)
case and_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.and_ psi✝)
case and_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.and_ psi✝)
case or_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.or_ psi✝)
case or_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.or_ psi✝)
case iff_left_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (phi✝.iff_ psi✝)
case iff_right_
v : VarName
F phi✝ psi✝ : Formula
a✝ : isFreeInInd v psi✝
a_ih✝ : isFreeIn v psi✝
⊢ isFreeIn v (phi✝.iff_ psi✝)
case forall_
v : VarName
F : Formula
x✝ : VarName
phi✝ : Formula
a✝¹ : ¬v = x✝
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (forall_ x✝ phi✝)
case exists_
v : VarName
F : Formula
x✝ : VarName
phi✝ : Formula
a✝¹ : ¬v = x✝
a✝ : isFreeInInd v phi✝
a_ih✝ : isFreeIn v phi✝
⊢ isFreeIn v (exists_ x✝ phi✝)
case def_
v : VarName
F : Formula
X✝ : DefName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (def_ X✝ xs✝)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeInInd_imp_isFreeIn
|
[416, 1]
|
[425, 10]
|
simp only [isFreeIn]
|
case def_
v : VarName
F : Formula
X✝ : DefName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ isFreeIn v (def_ X✝ xs✝)
|
case def_
v : VarName
F : Formula
X✝ : DefName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ v ∈ xs✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeInInd_imp_isFreeIn
|
[416, 1]
|
[425, 10]
|
tauto
|
case def_
v : VarName
F : Formula
X✝ : DefName
xs✝ : List VarName
a✝ : v ∈ xs✝
⊢ v ∈ xs✝
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_iff_isFreeInInd
|
[428, 1]
|
[435, 39]
|
constructor
|
v : VarName
F : Formula
⊢ isFreeIn v F ↔ isFreeInInd v F
|
case mp
v : VarName
F : Formula
⊢ isFreeIn v F → isFreeInInd v F
case mpr
v : VarName
F : Formula
⊢ isFreeInInd v F → isFreeIn v F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_iff_isFreeInInd
|
[428, 1]
|
[435, 39]
|
exact isFreeIn_imp_isFreeInInd v F
|
case mp
v : VarName
F : Formula
⊢ isFreeIn v F → isFreeInInd v F
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_iff_isFreeInInd
|
[428, 1]
|
[435, 39]
|
exact isFreeInInd_imp_isFreeIn v F
|
case mpr
v : VarName
F : Formula
⊢ isFreeInInd v F → isFreeIn v F
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
induction F
|
P : PredName
n : ℕ
F : Formula
⊢ predVarOccursIn P n F ↔ (P, n) ∈ F.predVarSet
|
case pred_const_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ predVarOccursIn P n (pred_const_ a✝¹ a✝) ↔ (P, n) ∈ (pred_const_ a✝¹ a✝).predVarSet
case pred_var_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ predVarOccursIn P n (pred_var_ a✝¹ a✝) ↔ (P, n) ∈ (pred_var_ a✝¹ a✝).predVarSet
case eq_
P : PredName
n : ℕ
a✝¹ a✝ : VarName
⊢ predVarOccursIn P n (eq_ a✝¹ a✝) ↔ (P, n) ∈ (eq_ a✝¹ a✝).predVarSet
case true_
P : PredName
n : ℕ
⊢ predVarOccursIn P n true_ ↔ (P, n) ∈ true_.predVarSet
case false_
P : PredName
n : ℕ
⊢ predVarOccursIn P n false_ ↔ (P, n) ∈ false_.predVarSet
case not_
P : PredName
n : ℕ
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝.not_ ↔ (P, n) ∈ a✝.not_.predVarSet
case imp_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.imp_ a✝) ↔ (P, n) ∈ (a✝¹.imp_ a✝).predVarSet
case and_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.and_ a✝) ↔ (P, n) ∈ (a✝¹.and_ a✝).predVarSet
case or_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.or_ a✝) ↔ (P, n) ∈ (a✝¹.or_ a✝).predVarSet
case iff_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.iff_ a✝) ↔ (P, n) ∈ (a✝¹.iff_ a✝).predVarSet
case forall_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (forall_ a✝¹ a✝) ↔ (P, n) ∈ (forall_ a✝¹ a✝).predVarSet
case exists_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (exists_ a✝¹ a✝) ↔ (P, n) ∈ (exists_ a✝¹ a✝).predVarSet
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ predVarOccursIn P n (def_ a✝¹ a✝) ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
all_goals
simp only [predVarOccursIn]
simp only [Formula.predVarSet]
|
case pred_const_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ predVarOccursIn P n (pred_const_ a✝¹ a✝) ↔ (P, n) ∈ (pred_const_ a✝¹ a✝).predVarSet
case pred_var_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ predVarOccursIn P n (pred_var_ a✝¹ a✝) ↔ (P, n) ∈ (pred_var_ a✝¹ a✝).predVarSet
case eq_
P : PredName
n : ℕ
a✝¹ a✝ : VarName
⊢ predVarOccursIn P n (eq_ a✝¹ a✝) ↔ (P, n) ∈ (eq_ a✝¹ a✝).predVarSet
case true_
P : PredName
n : ℕ
⊢ predVarOccursIn P n true_ ↔ (P, n) ∈ true_.predVarSet
case false_
P : PredName
n : ℕ
⊢ predVarOccursIn P n false_ ↔ (P, n) ∈ false_.predVarSet
case not_
P : PredName
n : ℕ
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝.not_ ↔ (P, n) ∈ a✝.not_.predVarSet
case imp_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.imp_ a✝) ↔ (P, n) ∈ (a✝¹.imp_ a✝).predVarSet
case and_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.and_ a✝) ↔ (P, n) ∈ (a✝¹.and_ a✝).predVarSet
case or_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.or_ a✝) ↔ (P, n) ∈ (a✝¹.or_ a✝).predVarSet
case iff_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (a✝¹.iff_ a✝) ↔ (P, n) ∈ (a✝¹.iff_ a✝).predVarSet
case forall_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (forall_ a✝¹ a✝) ↔ (P, n) ∈ (forall_ a✝¹ a✝).predVarSet
case exists_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n (exists_ a✝¹ a✝) ↔ (P, n) ∈ (exists_ a✝¹ a✝).predVarSet
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ predVarOccursIn P n (def_ a✝¹ a✝) ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet
|
case pred_const_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ ∅
case pred_var_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ P = a✝¹ ∧ n = a✝.length ↔ (P, n) ∈ {(a✝¹, a✝.length)}
case eq_
P : PredName
n : ℕ
a✝¹ a✝ : VarName
⊢ False ↔ (P, n) ∈ ∅
case true_
P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅
case false_
P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅
case not_
P : PredName
n : ℕ
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
case imp_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝¹ ∨ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝¹.predVarSet ∪ a✝.predVarSet
case and_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝¹ ∨ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝¹.predVarSet ∪ a✝.predVarSet
case or_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝¹ ∨ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝¹.predVarSet ∪ a✝.predVarSet
case iff_
P : PredName
n : ℕ
a✝¹ a✝ : Formula
a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝¹ ∨ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝¹.predVarSet ∪ a✝.predVarSet
case forall_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
case exists_
P : PredName
n : ℕ
a✝¹ : VarName
a✝ : Formula
a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
⊢ predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ ∅
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
|
P : PredName
n : ℕ
X : DefName
xs : List VarName
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case eq_ x y =>
simp
|
P : PredName
n : ℕ
x y : VarName
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case true_ | false_ =>
tauto
|
P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case not_ phi phi_ih =>
tauto
|
P : PredName
n : ℕ
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto
|
P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∪ psi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
tauto
|
P : PredName
n : ℕ
x : VarName
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
simp only [predVarOccursIn]
|
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ predVarOccursIn P n (def_ a✝¹ a✝) ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet
|
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
simp only [Formula.predVarSet]
|
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet
|
case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ ∅
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
simp
|
P : PredName
n : ℕ
X : DefName
xs : List VarName
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
simp
|
P : PredName
n : ℕ
x y : VarName
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
tauto
|
P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
tauto
|
P : PredName
n : ℕ
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
simp
|
P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∪ psi.predVarSet
|
P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∨ (P, n) ∈ psi.predVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
tauto
|
P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∨ (P, n) ∈ psi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.predVarOccursIn_iff_mem_predVarSet
|
[438, 1]
|
[464, 10]
|
tauto
|
P : PredName
n : ℕ
x : VarName
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
induction F
|
v : VarName
F : Formula
h1 : isBoundIn v F
⊢ occursIn v F
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isBoundIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)
case true_
v : VarName
h1 : isBoundIn v true_
⊢ occursIn v true_
case false_
v : VarName
h1 : isBoundIn v false_
⊢ occursIn v false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝.not_
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isBoundIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
all_goals
simp only [isBoundIn] at h1
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isBoundIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)
case true_
v : VarName
h1 : isBoundIn v true_
⊢ occursIn v true_
case false_
v : VarName
h1 : isBoundIn v false_
⊢ occursIn v false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝.not_
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isBoundIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
all_goals
simp only [occursIn]
tauto
|
case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
simp only [isBoundIn] at h1
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isBoundIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
simp only [occursIn]
|
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)
|
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ v = a✝¹ ∨ occursIn v a✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isBoundIn_imp_occursIn
|
[467, 1]
|
[478, 10]
|
tauto
|
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ v = a✝¹ ∨ occursIn v a✝
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
induction F
|
v : VarName
F : Formula
h1 : isFreeIn v F
⊢ occursIn v F
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)
case true_
v : VarName
h1 : isFreeIn v true_
⊢ occursIn v true_
case false_
v : VarName
h1 : isFreeIn v false_
⊢ occursIn v false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝.not_
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
all_goals
simp only [isFreeIn] at h1
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)
case true_
v : VarName
h1 : isFreeIn v true_
⊢ occursIn v true_
case false_
v : VarName
h1 : isFreeIn v false_
⊢ occursIn v false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝.not_
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)
case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
all_goals
simp only [occursIn]
tauto
|
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)
case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
simp only [isFreeIn] at h1
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
simp only [occursIn]
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Binders.lean
|
FOL.NV.isFreeIn_imp_occursIn
|
[481, 1]
|
[492, 10]
|
tauto
|
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
induction F
|
F : Formula
⊢ fastReplaceFree id F = F
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ fastReplaceFree id (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
⊢ fastReplaceFree id true_ = true_
case false_
⊢ fastReplaceFree id false_ = false_
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id a✝.not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
all_goals
simp only [fastReplaceFree]
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ fastReplaceFree id (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
⊢ fastReplaceFree id true_ = true_
case false_
⊢ fastReplaceFree id false_ = false_
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id a✝.not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ pred_const_ a✝¹ (List.map id a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ pred_var_ a✝¹ (List.map id a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ eq_ (id a✝¹) (id a✝) = eq_ a✝¹ a✝
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝).not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).imp_ (fastReplaceFree id a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).and_ (fastReplaceFree id a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).or_ (fastReplaceFree id a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).iff_ (fastReplaceFree id a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ forall_ a✝¹ (fastReplaceFree (Function.updateITE id a✝¹ a✝¹) a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ exists_ a✝¹ (fastReplaceFree (Function.updateITE id a✝¹ a✝¹) a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map id a✝) = def_ a✝¹ a✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr!
simp
|
X : DefName
xs : List VarName
⊢ def_ X (List.map id xs) = def_ X xs
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case eq_ x y =>
congr!
|
x y : VarName
⊢ eq_ (id x) (id y) = eq_ x y
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case not_ phi phi_ih =>
congr!
|
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ (fastReplaceFree id phi).not_ = phi.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr!
|
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
⊢ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr!
simp only [Function.updateITE_id]
exact phi_ih
|
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp only [fastReplaceFree]
|
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝
|
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map id a✝) = def_ a✝¹ a✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
X : DefName
xs : List VarName
⊢ def_ X (List.map id xs) = def_ X xs
|
case h.e'_2
X : DefName
xs : List VarName
⊢ List.map id xs = xs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp
|
case h.e'_2
X : DefName
xs : List VarName
⊢ List.map id xs = xs
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
x y : VarName
⊢ eq_ (id x) (id y) = eq_ x y
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ (fastReplaceFree id phi).not_ = phi.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
⊢ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree (Function.updateITE id x x) phi = phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp only [Function.updateITE_id]
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree (Function.updateITE id x x) phi = phi
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree id phi = phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
exact phi_ih
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree id phi = phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
induction F generalizing σ σ'
|
F : Formula
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v F → σ v = σ' v
⊢ fastReplaceFree σ F = fastReplaceFree σ' F
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_const_ a✝¹ a✝) = fastReplaceFree σ' (pred_const_ a✝¹ a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_var_ a✝¹ a✝) = fastReplaceFree σ' (pred_var_ a✝¹ a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (eq_ a✝¹ a✝) = fastReplaceFree σ' (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v true_ → σ v = σ' v
⊢ fastReplaceFree σ true_ = fastReplaceFree σ' true_
case false_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v false_ → σ v = σ' v
⊢ fastReplaceFree σ false_ = fastReplaceFree σ' false_
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → σ v = σ' v
⊢ fastReplaceFree σ a✝.not_ = fastReplaceFree σ' a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.imp_ a✝) = fastReplaceFree σ' (a✝¹.imp_ a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.and_ a✝) = fastReplaceFree σ' (a✝¹.and_ a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.or_ a✝) = fastReplaceFree σ' (a✝¹.or_ a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.iff_ a✝) = fastReplaceFree σ' (a✝¹.iff_ a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (forall_ a✝¹ a✝) = fastReplaceFree σ' (forall_ a✝¹ a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (exists_ a✝¹ a✝) = fastReplaceFree σ' (exists_ a✝¹ a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
all_goals
simp only [isFreeIn] at h1
simp only [fastReplaceFree]
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_const_ a✝¹ a✝) = fastReplaceFree σ' (pred_const_ a✝¹ a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_var_ a✝¹ a✝) = fastReplaceFree σ' (pred_var_ a✝¹ a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (eq_ a✝¹ a✝) = fastReplaceFree σ' (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v true_ → σ v = σ' v
⊢ fastReplaceFree σ true_ = fastReplaceFree σ' true_
case false_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v false_ → σ v = σ' v
⊢ fastReplaceFree σ false_ = fastReplaceFree σ' false_
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → σ v = σ' v
⊢ fastReplaceFree σ a✝.not_ = fastReplaceFree σ' a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.imp_ a✝) = fastReplaceFree σ' (a✝¹.imp_ a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.and_ a✝) = fastReplaceFree σ' (a✝¹.and_ a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.or_ a✝) = fastReplaceFree σ' (a✝¹.or_ a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.iff_ a✝) = fastReplaceFree σ' (a✝¹.iff_ a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (forall_ a✝¹ a✝) = fastReplaceFree σ' (forall_ a✝¹ a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (exists_ a✝¹ a✝) = fastReplaceFree σ' (exists_ a✝¹ a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝)
|
case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ pred_const_ a✝¹ (List.map σ a✝) = pred_const_ a✝¹ (List.map σ' a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ pred_var_ a✝¹ (List.map σ a✝) = pred_var_ a✝¹ (List.map σ' a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = a✝¹ ∨ v = a✝ → σ v = σ' v
⊢ eq_ (σ a✝¹) (σ a✝) = eq_ (σ' a✝¹) (σ' a✝)
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝).not_ = (fastReplaceFree σ' a✝).not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).imp_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).imp_ (fastReplaceFree σ' a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).and_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).and_ (fastReplaceFree σ' a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).or_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).or_ (fastReplaceFree σ' a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).iff_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).iff_ (fastReplaceFree σ' a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → σ v = σ' v
⊢ forall_ a✝¹ (fastReplaceFree (Function.updateITE σ a✝¹ a✝¹) a✝) =
forall_ a✝¹ (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → σ v = σ' v
⊢ exists_ a✝¹ (fastReplaceFree (Function.updateITE σ a✝¹ a✝¹) a✝) =
exists_ a✝¹ (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ def_ a✝¹ (List.map σ a✝) = def_ a✝¹ (List.map σ' a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
exact h1
|
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ def_ X (List.map σ xs) = def_ X (List.map σ' xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case not_ phi phi_ih =>
congr! 1
exact phi_ih σ σ' h1
|
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ (fastReplaceFree σ phi).not_ = (fastReplaceFree σ' phi).not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr! 1
apply phi_ih
intro v a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
rfl
case _ c1 =>
apply h1
tauto
|
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) =
exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [isFreeIn] at h1
|
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝)
|
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [fastReplaceFree]
|
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝)
|
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ def_ a✝¹ (List.map σ a✝) = def_ a✝¹ (List.map σ' a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ def_ X (List.map σ xs) = def_ X (List.map σ' xs)
|
case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ List.map σ xs = List.map σ' xs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [List.map_eq_map_iff]
|
case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ List.map σ xs = List.map σ' xs
|
case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ ∀ x ∈ xs, σ x = σ' x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact h1
|
case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ ∀ x ∈ xs, σ x = σ' x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ eq_ (σ x) (σ y) = eq_ (σ' x) (σ' y)
|
case h.e'_1
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ x = σ' x
case h.e'_2
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ y = σ' y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_1
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ x = σ' x
|
case h.e'_1.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x ∨ x = y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
left
|
case h.e'_1.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x ∨ x = y
|
case h.e'_1.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
rfl
|
case h.e'_1.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_2
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ y = σ' y
|
case h.e'_2.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = x ∨ y = y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
right
|
case h.e'_2.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = x ∨ y = y
|
case h.e'_2.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
rfl
|
case h.e'_2.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = y
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ (fastReplaceFree σ phi).not_ = (fastReplaceFree σ' phi).not_
|
case h.e'_1
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact phi_ih σ σ' h1
|
case h.e'_1
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ (fastReplaceFree σ phi).iff_ (fastReplaceFree σ psi) = (fastReplaceFree σ' phi).iff_ (fastReplaceFree σ' psi)
|
case h.e'_1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi
case h.e'_2
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply phi_ih
|
case h.e'_1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
intro v a1
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ σ v = σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ σ v = σ' v
|
case h.e'_1.h1.a
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi ∨ isFreeIn v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
left
|
case h.e'_1.h1.a
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi ∨ isFreeIn v psi
|
case h.e'_1.h1.a.h
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact a1
|
case h.e'_1.h1.a.h
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ isFreeIn v phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply psi_ih
|
case h.e'_2
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi
|
case h.e'_2.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
intro v a1
|
case h.e'_2.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v
|
case h.e'_2.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ σ v = σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_2.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ σ v = σ' v
|
case h.e'_2.h1.a
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
right
|
case h.e'_2.h1.a
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi
|
case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact a1
|
case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) =
exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)
|
case h.e'_2
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply phi_ih
|
case h.e'_2
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
intro v a1
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [Function.updateITE]
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then x else σ v) = if v = x then x else σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
split_ifs
|
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then x else σ v) = if v = x then x else σ' v
|
case pos
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
h✝ : v = x
⊢ x = x
case neg
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
h✝ : ¬v = x
⊢ σ v = σ' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case _ c1 =>
rfl
|
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
⊢ x = x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case _ c1 =>
apply h1
tauto
|
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ σ v = σ' v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
rfl
|
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
⊢ x = x
|
no goals
|
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