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... |
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... | 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 : P... |
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 : P... | 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 ... |
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✝... |
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✝... | 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 : VarN... |
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 : VarN... | 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✝)
⊢... |
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✝)
⊢... | 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✝ : Fo... |
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✝ : Fo... | 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_
⊢ fastReplac... |
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_
⊢ fastReplac... | 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... |
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),... |
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),... | 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 (L... |
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) =
ex... | 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 → Var... | 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' : 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 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' : 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' ... |
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' ... | 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' ... |
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' ... | 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... |
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... | 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... |
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... | 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' : 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' ... |
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' ... | 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' ... |
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' ... | 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... |
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... | 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... |
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... | 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) =
ex... | 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 = fastRe... |
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 = fastRe... | 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... |
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... | 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... |
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... | 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 els... |
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 els... | 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... |
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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