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 : P... | case pred_const_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ ∅
case pred_var_
P : PredName
n : ℕ
a✝¹ : PredName
a✝ : List VarName
⊢ P = a✝¹ ∧ n = a✝.length ↔ (P, n) ∈ {(a✝¹, a✝.length)}
case eq_
P : PredName
n : ℕ
a✝¹ a✝ : VarName
⊢ False ↔ (P, n) ∈ ∅
case true_
P : PredName
n : ℕ
⊢ False ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp | P : PredName
n : ℕ
X : DefName
xs : List VarName
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case eq_ x y =>
simp | P : PredName
n : ℕ
x y : VarName
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case true_ | false_ =>
tauto | P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case not_ phi phi_ih =>
tauto | P : PredName
n : ℕ
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto | P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∪ psi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
tauto | P : PredName
n : ℕ
x : VarName
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | simp only [predVarOccursIn] | case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ predVarOccursIn P n (def_ a✝¹ a✝) ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet | case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | simp only [Formula.predVarSet] | case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet | case def_
P : PredName
n : ℕ
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ (P, n) ∈ ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | simp | P : PredName
n : ℕ
X : DefName
xs : List VarName
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | simp | P : PredName
n : ℕ
x y : VarName
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | tauto | P : PredName
n : ℕ
⊢ False ↔ (P, n) ∈ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | tauto | P : PredName
n : ℕ
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | simp | P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∪ psi.predVarSet | P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∨ (P, n) ∈ psi.predVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | tauto | P : PredName
n : ℕ
phi psi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
psi_ih : predVarOccursIn P n psi ↔ (P, n) ∈ psi.predVarSet
⊢ predVarOccursIn P n phi ∨ predVarOccursIn P n psi ↔ (P, n) ∈ phi.predVarSet ∨ (P, n) ∈ psi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.predVarOccursIn_iff_mem_predVarSet | [438, 1] | [464, 10] | tauto | P : PredName
n : ℕ
x : VarName
phi : Formula
phi_ih : predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet
⊢ predVarOccursIn P n phi ↔ (P, n) ∈ phi.predVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | induction F | v : VarName
F : Formula
h1 : isBoundIn v F
⊢ occursIn v F | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isBoundIn v (eq_ a✝¹ a✝... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | all_goals
simp only [isBoundIn] at h1 | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isBoundIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isBoundIn v (eq_ a✝¹ a✝... | case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarN... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | all_goals
simp only [occursIn]
tauto | case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝
⊢ occursIn v a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : isBoundIn v a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)
case and_
v : VarN... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | simp only [isBoundIn] at h1 | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isBoundIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | simp only [occursIn] | case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝) | case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ v = a✝¹ ∨ occursIn v a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isBoundIn_imp_occursIn | [467, 1] | [478, 10] | tauto | case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ → occursIn v a✝
h1 : v = a✝¹ ∨ isBoundIn v a✝
⊢ v = a✝¹ ∨ occursIn v a✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | induction F | v : VarName
F : Formula
h1 : isFreeIn v F
⊢ occursIn v F | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | all_goals
simp only [isFreeIn] at h1 | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢... | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)
case not_
v : VarName
a✝ : Fo... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | all_goals
simp only [occursIn]
tauto | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)
case not_
v : VarName
a✝ : Fo... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | simp only [isFreeIn] at h1 | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝) | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | simp only [occursIn] | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝) | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Binders.lean | FOL.NV.isFreeIn_imp_occursIn | [481, 1] | [492, 10] | tauto | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/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 = (... | 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_ :=... |
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_,
... | 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 =... | 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... | 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_,
... | 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 a... | 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.p... | 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_,
... | 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_,
... |
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_,
... | 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_,
... | 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)... |
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)... | 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 ∈ binder... |
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 ∈ binder... | 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✝ ... |
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 ... | 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✝... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... |
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 : ... | 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 : ... | 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... |
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... | 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... |
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... | 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_righ... |
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_righ... | 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_righ... |
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_righ... | 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_righ... |
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_righ... | 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_righ... |
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_righ... | 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_righ... | 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_righ... |
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_righ... | 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_righ... |
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_righ... | 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_righ... | 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, isFreeI... |
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, isFreeI... | 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, isFr... |
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, isFr... | 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 =... | 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 =... |
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 =... | 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... | 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... |
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... | 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_,
... | 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_,
... |
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_,
... | 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_,
... |
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_,
... | 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_,
... |
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_,
... | 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_ :=... |
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_ :=... | 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_cons... | 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_cons... |
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_cons... | 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_cons... |
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_cons... | 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_cons... |
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_cons... | 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_ := ... |
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_cons... | 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... |
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... | 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... | 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.p... | 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.p... |
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.p... | 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.p... |
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.p... | 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.p... |
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.p... | 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_... |
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_... | 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_... |
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_... | 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_... |
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_... | 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_... |
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_... | 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_... |
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_... | 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_... |
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_... | 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_... |
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_... | 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_... |
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.p... | 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.p... | 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_... |
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.p... | 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_... |
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.p... | 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.p... |
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.p... | 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_... | 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... |
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_ := ... | 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_ := ... | 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_ := ... |
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_ := ... | 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_ :=... | 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_ :=... |
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