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101
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.elim a2_right | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∀ (a_1 : List (List α)),
(∀ r ∈ a_1, r ∈ RegExp.languageOf α e) ∧ xs ++ a_1.join = w →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro ys a3 | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∀ (a_1 : List (List α)),
(∀ r ∈ a_1, r ∈ RegExp.languageOf α e) ∧ xs ++ a_1.join = w →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | clear a2_right | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | cases a3 | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case intro
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
left✝ : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
right✝ : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | case _ a3_left a3_right =>
apply Exists.intro [(a :: w)]
simp
simp only [← ih]
sorry | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.intro [(a :: w)] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ (∀ r ∈ [a :: w], r ∈ RegExp.languageOf α e) ∧ [a :: w].join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ (∀ r ∈ [a :: w], r ∈ RegExp.languageOf α e) ∧ [a :: w].join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ a :: w ∈ RegExp.languageOf α e |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp only [← ih] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ a :: w ∈ RegExp.languageOf α e | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | induction F | P : PredName
zs : List VarName
H F : Formula
h1 : F.predVarSet = ∅
⊢ replace P zs H F = F | case pred_const_
P : PredName
zs : List VarName
H : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
P : PredName
zs : List VarName
H : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
P : PredName
zs : List VarName
H : Formula
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
P : PredName
zs : List VarName
H : Formula
h1 : true_.predVarSet = ∅
⊢ replace P zs H true_ = true_
case false_
P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_
case not_
P : PredName
zs : List VarName
H a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : a✝.not_.predVarSet = ∅
⊢ replace P zs H a✝.not_ = a✝.not_
case imp_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.imp_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.and_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.or_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.iff_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
P : PredName
zs : List VarName
H : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (forall_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
P : PredName
zs : List VarName
H : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (exists_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
P : PredName
zs : List VarName
H : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case pred_const_ X xs =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case pred_var_ X xs =>
simp only [predVarSet] at h1
simp at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case eq_ x y =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ replace P zs H (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case true_ | false_ =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case not_ phi phi_ih =>
simp only [predVarSet] at h1
simp only [replace]
congr!
exact phi_ih h1 | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [predVarSet] at h1
simp only [replace]
congr!
exact phi_ih h1 | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case def_ X xs =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ replace P zs H (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ replace P zs H (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ (replace P zs H phi).not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ (replace P zs H phi).not_ = phi.not_ | case h.e'_1
P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1 | case h.e'_1
P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : (phi.iff_ psi).predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [Finset.union_eq_empty] at h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | cases h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | case intro
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
left✝ : phi.predVarSet = ∅
right✝ : psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (replace P zs H phi).iff_ (replace P zs H psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (replace P zs H phi).iff_ (replace P zs H psi) = phi.iff_ psi | case h.e'_1
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H phi = phi
case h.e'_2
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1_left | case h.e'_1
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact psi_ih h1_right | case h.e'_2
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (replace P zs H phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (replace P zs H phi) = exists_ x phi | case h.e'_2
P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1 | case h.e'_2
P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ replace P zs H (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | set E_ref := E | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E P zs H) V E F ↔ Holds D I V E (replace P zs H F) | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := E
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | induction E generalizing F binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := E
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) | case nil
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
V : VarAssignment D
F : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := []
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
case cons
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
V : VarAssignment D
F : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := head✝ :: tail✝
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case nil.def_ X xs =>
simp only [replace]
simp only [E_ref]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (replace P zs H (def_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | induction F generalizing binders V | case cons
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
V : VarAssignment D
F : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := head✝ :: tail✝
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) | case cons.pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_const_ a✝¹ a✝) ↔
Holds D I V E_ref (replace P zs H (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_var_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ a✝¹ a✝) ↔ Holds D I V E_ref (replace P zs H (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (eq_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (eq_ a✝¹ a✝) ↔ Holds D I V E_ref (replace P zs H (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders true_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref true_ ↔ Holds D I V E_ref (replace P zs H true_)
case cons.false_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref false_ ↔ Holds D I V E_ref (replace P zs H false_)
case cons.not_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders a✝.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref a✝.not_ ↔ Holds D I V E_ref (replace P zs H a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝¹ ↔ Holds D I V E_ref (replace P zs H a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (a✝¹.imp_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (a✝¹.imp_ a✝) ↔ Holds D I V E_ref (replace P zs H (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝¹ ↔ Holds D I V E_ref (replace P zs H a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (a✝¹.and_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (a✝¹.and_ a✝) ↔ Holds D I V E_ref (replace P zs H (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝¹ ↔ Holds D I V E_ref (replace P zs H a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (a✝¹.or_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (a✝¹.or_ a✝) ↔ Holds D I V E_ref (replace P zs H (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝¹ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝¹ ↔ Holds D I V E_ref (replace P zs H a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (a✝¹.iff_ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (a✝¹.iff_ a✝) ↔ Holds D I V E_ref (replace P zs H (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (forall_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (forall_ a✝¹ a✝) ↔ Holds D I V E_ref (replace P zs H (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders a✝ →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref a✝ ↔ Holds D I V E_ref (replace P zs H a✝))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (exists_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ a✝¹ a✝) ↔ Holds D I V E_ref (replace P zs H (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ a✝¹ a✝)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ a✝¹ a✝) ↔ Holds D I V E_ref (replace P zs H (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pred_const_ X xs =>
simp only [replace]
simp only [Holds]
simp only [I']
simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_const_ X xs) ↔ Holds D I V E_ref (replace P zs H (pred_const_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case eq_ x y =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (eq_ x y) ↔ Holds D I V E_ref (replace P zs H (eq_ x y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case true_ | false_ =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref false_ ↔ Holds D I V E_ref (replace P zs H false_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | 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
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | 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
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (exists_ x phi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_const_ X xs) ↔ Holds D I V E_ref (replace P zs H (pred_const_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_const_ X xs) ↔ Holds D I V E_ref (pred_const_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_const_ X xs) ↔ Holds D I V E_ref (pred_const_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (I' D I V' E_ref P zs H).pred_const_ X (List.map V xs) ↔ I.pred_const_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (I' D I V' E_ref P zs H).pred_const_ X (List.map V xs) ↔ I.pred_const_ X (List.map V xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E_ref H
else I.pred_var_ Q ds).pred_const_
X (List.map V xs) ↔
I.pred_const_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_const_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E_ref H
else I.pred_var_ Q ds).pred_const_
X (List.map V xs) ↔
I.pred_const_ X (List.map V xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (pred_var_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ X xs) ↔ Holds D I V E_ref (replace P zs H (pred_var_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ X xs) ↔ Holds D I V E_ref (replace P zs H (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ X xs) ↔ Holds D I V E_ref (replace P zs H (pred_var_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ X xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (pred_var_ X xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (I' D I V' E_ref P zs H).pred_var_ X (List.map V xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (I' D I V' E_ref P zs H).pred_var_ X (List.map V xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E_ref H
else I.pred_var_ Q ds).pred_var_
X (List.map V xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E_ref H
else I.pred_var_ Q ds).pred_var_
X (List.map V xs) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = P ∧ (List.map V xs).length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = P ∧ (List.map V xs).length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
if X = P ∧ xs.length = zs.length then
Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h✝ : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h✝ : ¬(X = P ∧ xs.length = zs.length)
h1 : True
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg c1 =>
split_ifs
case pos c2 =>
contradiction
case neg c2 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Sub.Var.All.Rec.admits] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H ∧ ∀ x ∈ binders, isFreeIn x H → x ∈ zs
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1 : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H ∧ ∀ x ∈ binders, isFreeIn x H → x ∈ zs
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
left✝ : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
right✝ : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | have s1 :
Holds D I (V ∘ Function.updateListITE id zs xs) E_ref H ↔
Holds D I V E_ref (Sub.Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) :=
by
exact Sub.Var.All.Rec.substitution_theorem D I V E_ref (Function.updateListITE id zs xs) H h1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (V ∘ Function.updateListITE id zs xs) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Function.updateListITE_comp] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (V ∘ Function.updateListITE id zs xs) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE (V ∘ id) zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE (V ∘ id) zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [s2] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
h✝ : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
h✝ : ¬(X = P ∧ xs.length = zs.length)
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos c2 =>
exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
c2 : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg _ =>
exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
h✝ : ¬(X = P ∧ xs.length = zs.length)
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact Sub.Var.All.Rec.substitution_theorem D I V E_ref (Function.updateListITE id zs xs) H h1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
⊢ Holds D I (V ∘ Function.updateListITE id zs xs) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ ∀ (v : VarName),
isFreeIn v H → Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro v a1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
⊢ ∀ (v : VarName),
isFreeIn v H → Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | by_cases c2 : v ∈ zs | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Function.updateListITE_mem_eq_len V V' v zs (List.map V xs) c2 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
⊢ zs.length = (List.map V xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases c1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
⊢ zs.length = (List.map V xs).length | case pos.intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
left✝ : X = P
right✝ : xs.length = zs.length
⊢ zs.length = (List.map V xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos.intro c1_left c1_right =>
simp
symm
exact c1_right | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ zs.length = (List.map V xs).length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ zs.length = (List.map V xs).length | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ zs.length = xs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | symm | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ zs.length = xs.length | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ xs.length = zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact c1_right | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∈ zs
c1_left : X = P
c1_right : xs.length = zs.length
⊢ xs.length = zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | by_cases c3 : v ∈ binders | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∈ binders
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∉ binders
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | specialize h1_right v c3 a1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∈ binders
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∈ binders
h1_right : v ∈ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | contradiction | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∈ binders
h1_right : v ∈ zs
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Function.updateListITE_mem' | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∉ binders
⊢ Function.updateListITE V zs (List.map V xs) v = Function.updateListITE V' zs (List.map V xs) v | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∉ binders
⊢ V v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact h2 v c3 | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s1 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
v : VarName
a1 : isFreeIn v H
c2 : v ∉ zs
c3 : v ∉ binders
⊢ V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
c2 : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = P ∧ xs.length = zs.length
h1_left : Var.All.Rec.admitsAux (Function.updateListITE id zs xs) ∅ H
h1_right : ∀ x ∈ binders, isFreeIn x H → x ∈ zs
s2 :
Holds D I (Function.updateListITE V zs (List.map V xs)) E_ref H ↔
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
s1 :
Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
h✝ : ¬(X = P ∧ xs.length = zs.length)
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
⊢ (if X = P ∧ xs.length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
else I.pred_var_ X (List.map V xs)) ↔
Holds D I V E_ref
(if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
else pred_var_ X xs) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
h✝ : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
h✝ : ¬(X = P ∧ xs.length = zs.length)
⊢ I.pred_var_ X (List.map V xs) ↔ Holds D I V E_ref (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos c2 =>
contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
c2 : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg c2 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
c2 : ¬(X = P ∧ xs.length = zs.length)
⊢ I.pred_var_ X (List.map V xs) ↔ Holds D I V E_ref (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
c2 : X = P ∧ xs.length = zs.length
⊢ Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H ↔
Holds D I V E_ref (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = P ∧ xs.length = zs.length)
h1 : True
c2 : ¬(X = P ∧ xs.length = zs.length)
⊢ I.pred_var_ X (List.map V xs) ↔ Holds D I V E_ref (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (eq_ x y) ↔ Holds D I V E_ref (replace P zs H (eq_ x y)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (eq_ x y) ↔ Holds D I V E_ref (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (eq_ x y)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (eq_ x y) ↔ Holds D I V E_ref (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref false_ ↔ Holds D I V E_ref (replace P zs H false_) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref false_ ↔ Holds D I V E_ref false_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders false_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref false_ ↔ Holds D I V E_ref false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi.not_
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi.not_) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi.not_) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi.not_ ↔ Holds D I V E_ref (replace P zs H phi).not_ | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ¬Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ ¬Holds D I V E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ¬Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ ¬Holds D I V E_ref (replace P zs H phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact phi_ih V binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi) | no goals |
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