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.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/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
induction F generalizing binders V
|
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
binders : Finset VarName
F : Formula
h1 : admitsAux τ binders F
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E F ↔
Holds D I V E (replace τ F)
|
case pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_const_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_const_ a✝¹ a✝) ↔
Holds D I V E (replace τ (pred_const_ a✝¹ a✝))
case pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_var_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ a✝¹ a✝) ↔
Holds D I V E (replace τ (pred_var_ a✝¹ a✝))
case eq_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (eq_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (eq_ a✝¹ a✝) ↔
Holds D I V E (replace τ (eq_ a✝¹ a✝))
case true_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders true_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E true_ ↔
Holds D I V E (replace τ true_)
case false_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E false_ ↔
Holds D I V E (replace τ false_)
case not_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders a✝.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝.not_ ↔
Holds D I V E (replace τ a✝.not_)
case imp_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝¹ ↔
Holds D I V E (replace τ a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (a✝¹.imp_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (a✝¹.imp_ a✝) ↔
Holds D I V E (replace τ (a✝¹.imp_ a✝))
case and_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝¹ ↔
Holds D I V E (replace τ a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (a✝¹.and_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (a✝¹.and_ a✝) ↔
Holds D I V E (replace τ (a✝¹.and_ a✝))
case or_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝¹ ↔
Holds D I V E (replace τ a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (a✝¹.or_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (a✝¹.or_ a✝) ↔
Holds D I V E (replace τ (a✝¹.or_ a✝))
case iff_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝¹ ↔
Holds D I V E (replace τ a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (a✝¹.iff_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (a✝¹.iff_ a✝) ↔
Holds D I V E (replace τ (a✝¹.iff_ a✝))
case forall_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (forall_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (forall_ a✝¹ a✝) ↔
Holds D I V E (replace τ (forall_ a✝¹ a✝))
case exists_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E a✝ ↔
Holds D I V E (replace τ a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (exists_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ a✝¹ a✝) ↔
Holds D I V E (replace τ (exists_ a✝¹ a✝))
case def_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (def_ a✝¹ a✝) ↔
Holds D I V E (replace τ (def_ a✝¹ a✝))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case pred_const_ X xs =>
simp only [replace]
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_const_ X xs) ↔
Holds D I V E (replace τ (pred_const_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case eq_ x y =>
simp only [replace]
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (eq_ x y) ↔
Holds D I V E (replace τ (eq_ x y))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case true_ | false_ =>
simp only [replace]
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E false_ ↔
Holds D I V E (replace τ false_)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case not_ phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
congr! 1
exact phi_ih V binders h1 h2
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi.not_)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
intro v a1
simp only [Function.updateITE]
simp at a1
push_neg at a1
cases a1
case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (exists_ x phi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (replace τ (exists_ x phi))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_const_ X xs) ↔
Holds D I V E (replace τ (pred_const_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_const_ X xs) ↔
Holds D I V E (pred_const_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_const_ X xs) ↔
Holds D I V E (pred_const_ X xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [admitsAux] at h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (pred_var_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2 ∧
(∀ x ∈ binders, ¬(isFreeIn x (τ X xs.length).2 ∧ x ∉ (τ X xs.length).1)) ∧ xs.length = (τ X xs.length).1.length
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp at h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2 ∧
(∀ x ∈ binders, ¬(isFreeIn x (τ X xs.length).2 ∧ x ∉ (τ X xs.length).1)) ∧ xs.length = (τ X xs.length).1.length
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1 :
Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2 ∧
(∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1) ∧ xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
cases h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1 :
Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2 ∧
(∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1) ∧ xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
left✝ : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
right✝ : (∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1) ∧ xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
cases h1_right
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right : (∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1) ∧ xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
left✝ : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
right✝ : xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
obtain s1 :=
Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id (τ X xs.length).fst xs)
(τ X xs.length).snd h1_left
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (V ∘ Function.updateListITE id (τ X xs.length).1 xs) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Function.updateListITE_comp] at s1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (V ∘ Function.updateListITE id (τ X xs.length).1 xs) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE (V ∘ id) (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp at s1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE (V ∘ id) (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [s2] at s1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
s2 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s2 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
clear s2
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s2 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (pred_var_ X xs) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if (List.map V xs).length = (τ X (List.map V xs).length).1.length then
Holds D I (Function.updateListITE V' (τ X (List.map V xs).length).1 (List.map V xs)) E
(τ X (List.map V xs).length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if (List.map V xs).length = (τ X (List.map V xs).length).1.length then
Holds D I (Function.updateListITE V' (τ X (List.map V xs).length).1 (List.map V xs)) E
(τ X (List.map V xs).length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E (replace τ (pred_var_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if (List.map V xs).length = (τ X (List.map V xs).length).1.length then
Holds D I (Function.updateListITE V' (τ X (List.map V xs).length).1 (List.map V xs)) E
(τ X (List.map V xs).length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E
(if xs.length = (τ X xs.length).1.length then
Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
else pred_var_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if (List.map V xs).length = (τ X (List.map V xs).length).1.length then
Holds D I (Function.updateListITE V' (τ X (List.map V xs).length).1 (List.map V xs)) E
(τ X (List.map V xs).length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E
(if xs.length = (τ X xs.length).1.length then
Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
else pred_var_ X xs)
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if xs.length = (τ X xs.length).1.length then
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E
(if xs.length = (τ X xs.length).1.length then
Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
else pred_var_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [if_pos h1_right_right]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ (if xs.length = (τ X xs.length).1.length then
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E
(if xs.length = (τ X xs.length).1.length then
Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
else pred_var_ X xs)
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact s1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply Holds_coincide_Var
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I (Function.updateListITE V' (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2
|
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ ∀ (v : VarName),
isFreeIn v (τ X xs.length).2 →
Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
intro v a1
|
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
⊢ ∀ (v : VarName),
isFreeIn v (τ X xs.length).2 →
Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
by_cases c1 : v ∈ (τ X xs.length).fst
|
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply Function.updateListITE_mem_eq_len V V' v (τ X xs.length).fst (List.map V xs) c1
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ (τ X xs.length).1.length = (List.map V xs).length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ (τ X xs.length).1.length = (List.map V xs).length
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ (τ X xs.length).1.length = xs.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
symm
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ (τ X xs.length).1.length = xs.length
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ xs.length = (τ X xs.length).1.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact h1_right_right
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∈ (τ X xs.length).1
⊢ xs.length = (τ X xs.length).1.length
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
by_cases c2 : v ∈ binders
|
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∈ binders
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
specialize h1_right_left v c2 a1
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∈ binders
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∈ binders
h1_right_left : v ∈ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
contradiction
|
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∈ binders
h1_right_left : v ∈ (τ X xs.length).1
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
specialize h2 v c2
|
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
h2 : V v = V' v
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply Function.updateListITE_mem'
|
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
h2 : V v = V' v
⊢ Function.updateListITE V (τ X xs.length).1 (List.map V xs) v =
Function.updateListITE V' (τ X xs.length).1 (List.map V xs) v
|
case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
h2 : V v = V' v
⊢ V v = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact h2
|
case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2
h1_right_left : ∀ x ∈ binders, isFreeIn x (τ X xs.length).2 → x ∈ (τ X xs.length).1
h1_right_right : xs.length = (τ X xs.length).1.length
s1 :
Holds D I (Function.updateListITE V (τ X xs.length).1 (List.map V xs)) E (τ X xs.length).2 ↔
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (τ X xs.length).1 xs) (τ X xs.length).2)
v : VarName
a1 : isFreeIn v (τ X xs.length).2
c1 : v ∉ (τ X xs.length).1
c2 : v ∉ binders
h2 : V v = V' v
⊢ V v = V' v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (eq_ x y) ↔
Holds D I V E (replace τ (eq_ x y))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (eq_ x y) ↔
Holds D I V E (eq_ x y)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (eq_ x y) ↔
Holds D I V E (eq_ x y)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E false_ ↔
Holds D I V E (replace τ false_)
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E false_ ↔
Holds D I V E false_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E false_ ↔
Holds D I V E false_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [admitsAux] at h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi.not_)
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi.not_)
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi.not_ ↔
Holds D I V E (replace τ phi).not_
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
¬Holds D I V E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
congr! 1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
¬Holds D I V E (replace τ phi)
|
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact phi_ih V binders h1 h2
|
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [admitsAux] at h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (phi.iff_ psi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (phi.iff_ psi) ↔
Holds D I V E (replace τ (phi.iff_ psi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (phi.iff_ psi) ↔
Holds D I V E (replace τ (phi.iff_ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (phi.iff_ psi) ↔
Holds D I V E (replace τ (phi.iff_ psi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (phi.iff_ psi) ↔
Holds D I V E ((replace τ phi).iff_ (replace τ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (phi.iff_ psi) ↔
Holds D I V E ((replace τ phi).iff_ (replace τ psi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi) ↔
(Holds D I V E (replace τ phi) ↔ Holds D I V E (replace τ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
cases h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders phi ∧ admitsAux τ binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi) ↔
(Holds D I V E (replace τ phi) ↔ Holds D I V E (replace τ psi))
|
case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
left✝ : admitsAux τ binders phi
right✝ : admitsAux τ binders psi
⊢ (Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi) ↔
(Holds D I V E (replace τ phi) ↔ Holds D I V E (replace τ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
congr! 1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux τ binders phi
h1_right : admitsAux τ binders psi
⊢ (Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi) ↔
(Holds D I V E (replace τ phi) ↔ Holds D I V E (replace τ psi))
|
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux τ binders phi
h1_right : admitsAux τ binders psi
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux τ binders phi
h1_right : admitsAux τ binders psi
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact phi_ih V binders h1_left h2
|
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux τ binders phi
h1_right : admitsAux τ binders psi
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact psi_ih V binders h1_right h2
|
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux τ binders phi
h1_right : admitsAux τ binders psi
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E psi ↔
Holds D I V E (replace τ psi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [admitsAux] at h1
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (exists_ x phi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (replace τ (exists_ x phi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (replace τ (exists_ x phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (replace τ (exists_ x phi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (exists_ x (replace τ phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (exists_ x phi) ↔
Holds D I V E (exists_ x (replace τ phi))
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
first | apply forall_congr' | apply exists_congr
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E (replace τ phi)
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x a) E phi ↔
Holds D I (Function.updateITE V x a) E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
intro d
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x a) E phi ↔
Holds D I (Function.updateITE V x a) E (replace τ phi)
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi ↔
Holds D I (Function.updateITE V x d) E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi ↔
Holds D I (Function.updateITE V x d) E (replace τ phi)
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
intro v a1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ Function.updateITE V x d v = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Function.updateITE]
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ Function.updateITE V x d v = V' v
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ (if v = x then d else V v) = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp at a1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ (if v = x then d else V v) = V' v
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ ¬v = x
⊢ (if v = x then d else V v) = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
push_neg at a1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ ¬v = x
⊢ (if v = x then d else V v) = V' v
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ v ≠ x
⊢ (if v = x then d else V v) = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
cases a1
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ v ≠ x
⊢ (if v = x then d else V v) = V' v
|
case h.intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
left✝ : v ∉ binders
right✝ : v ≠ x
⊢ (if v = x then d else V v) = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ (if v = x then d else V v) = V' v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply forall_congr'
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∀ (d : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi) ↔
∀ (d : D), Holds D I (Function.updateITE V x d) E (replace τ phi)
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x a) E phi ↔
Holds D I (Function.updateITE V x a) E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
apply exists_congr
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x d) E phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E (replace τ phi)
|
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
(Function.updateITE V x a) E phi ↔
Holds D I (Function.updateITE V x a) E (replace τ phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [if_neg a1_right]
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ (if v = x then d else V v) = V' v
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ V v = V' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
exact h2 v a1_left
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux τ binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E phi ↔
Holds D I V E (replace τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ V v = V' v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
cases E
|
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) E (τ X ds.length).2
else I.pred_var_ X ds }
V E (def_ X xs) ↔
Holds D I V E (replace τ (def_ X xs))
|
case nil
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) [] (τ X ds.length).2
else I.pred_var_ X ds }
V [] (def_ X xs) ↔
Holds D I V [] (replace τ (def_ X xs))
case cons
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
head✝ : Definition
tail✝ : List Definition
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (head✝ :: tail✝) (τ X ds.length).2
else I.pred_var_ X ds }
V (head✝ :: tail✝) (def_ X xs) ↔
Holds D I V (head✝ :: tail✝) (replace τ (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
case nil =>
simp only [replace]
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) [] (τ X ds.length).2
else I.pred_var_ X ds }
V [] (def_ X xs) ↔
Holds D I V [] (replace τ (def_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) [] (τ X ds.length).2
else I.pred_var_ X ds }
V [] (def_ X xs) ↔
Holds D I V [] (replace τ (def_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) [] (τ X ds.length).2
else I.pred_var_ X ds }
V [] (def_ X xs) ↔
Holds D I V [] (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) [] (τ X ds.length).2
else I.pred_var_ X ds }
V [] (def_ X xs) ↔
Holds D I V [] (def_ X xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
|
[109, 1]
|
[238, 15]
|
simp only [replace]
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
hd : Definition
tl : List Definition
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ X ds.length).2
else I.pred_var_ X ds }
V (hd :: tl) (def_ X xs) ↔
Holds D I V (hd :: tl) (replace τ (def_ X xs))
|
D : Type
I : Interpretation D
V' : VarAssignment D
τ : PredName → ℕ → List VarName × Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
hd : Definition
tl : List Definition
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (τ X ds.length).1.length then
Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ X ds.length).2
else I.pred_var_ X ds }
V (hd :: tl) (def_ X xs) ↔
Holds D I V (hd :: tl) (def_ X xs)
|
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