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stringlengths 6
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply PrefixAppend | case star_repetition
V_N V_T : Type
R : V_N → PE V_N V_T
e✝ : PE V_N V_T
xs_1✝ xs_2✝ ys✝ : List V_T
n1✝ n2✝ : ℕ
a✝¹ : Interpretation V_N V_T R (e✝, xs_1✝ ++ xs_2✝ ++ ys✝) (n1✝, some xs_1✝)
a✝ : Interpretation V_N V_T R (e✝.star, xs_2✝ ++ ys✝) (n2✝, some xs_2✝)
ih :
∀ m < n1✝ + n2✝ + 1,
∀ (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1✝ ++ xs_2✝ ++ ys✝) (m, some (xs_1✝ ++ xs_2✝)) →
(xs_1✝ ++ xs_2✝).IsPrefix (xs_1✝ ++ xs_2✝ ++ ys✝)
⊢ (xs_1✝ ++ xs_2✝).IsPrefix (xs_1✝ ++ xs_2✝ ++ ys✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n _ (R A) | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
A : V_N
n : ℕ
ih : ∀ m < n + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
⊢ ys.IsPrefix xs | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
A : V_N
n : ℕ
ih : ∀ m < n + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
⊢ n < n + 1
V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
A : V_N
n : ℕ
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) → ys.IsPrefix xs
⊢ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
A : V_N
n : ℕ
ih : ∀ m < n + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
⊢ n < n + 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_1 | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
A : V_N
n : ℕ
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) → ys.IsPrefix xs
⊢ ys.IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n2 _ e2 | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : ℕ
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : ∀ m < n1 + n2 + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
⊢ ys.IsPrefix xs | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : ℕ
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : ∀ m < n1 + n2 + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
⊢ n2 < n1 + n2 + 1
V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : ℕ
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) → ys.IsPrefix xs
⊢ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : ℕ
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : ∀ m < n1 + n2 + 1, ∀ (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) → ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
⊢ n2 < n1 + n2 + 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_2 | V_N V_T : Type
R : V_N → PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : ℕ
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) → ys.IsPrefix xs
⊢ ys.IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | induction F | F : Formula
τ : PredName → PredName
h1 : F.predVarSet = ∅
⊢ sub τ F = F | case pred_const_
τ : PredName → PredName
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
τ : PredName → PredName
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
τ : PredName → PredName
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
τ : PredName → PredName
h1 : true_.predVarSet = ∅
⊢ sub τ true_ = true_
case false_
τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_
case not_
τ : PredName → PredName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : a✝.not_.predVarSet = ∅
⊢ sub τ a✝.not_ = a✝.not_
case imp_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.imp_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.and_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.or_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.iff_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
τ : PredName → PredName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (forall_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
τ : PredName → PredName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (exists_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
τ : PredName → PredName
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_const_ X xs =>
simp only [sub] | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ sub τ (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_var_ X xs =>
simp only [predVarSet] at h1
simp at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case eq_ x y =>
simp only [sub] | τ : PredName → PredName
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ sub τ (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case true_ | false_ =>
simp only [sub] | τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case not_ phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case def_ X xs =>
simp only [sub] | τ : PredName → PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ sub τ (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ sub τ (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ sub τ (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ (sub τ phi).not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ (sub τ phi).not_ = phi.not_ | case h.e'_1
τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_1
τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : (phi.iff_ psi).predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [Finset.union_eq_empty] at h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | cases h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | case intro
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
left✝ : phi.predVarSet = ∅
right✝ : psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (sub τ phi).iff_ (sub τ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (sub τ phi).iff_ (sub τ psi) = phi.iff_ psi | case h.e'_1
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ phi = phi
case h.e'_2
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1_left | case h.e'_1
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact psi_ih h1_right | case h.e'_2
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (sub τ phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (sub τ phi) = exists_ x phi | case h.e'_2
τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_2
τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ sub τ (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction E generalizing F V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
τ : PredName → PredName
F : Formula
⊢ Holds D I V E (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
V E F | case nil
D : Type
I : Interpretation D
τ : PredName → PredName
V : VarAssignment D
F : Formula
⊢ Holds D I V [] (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] F
case cons
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
F : Formula
⊢ Holds D I V (head✝ :: tail✝) (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case nil.def_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case cons.def_ hd tl ih X xs =>
simp only [Holds] at ih
simp at ih
simp only [sub]
simp only [Holds]
split_ifs
case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q τ hd.h2] at ih
apply ih
case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tl F
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction F generalizing V | case cons
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
F : Formula
⊢ Holds D I V (head✝ :: tail✝) (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) F | case cons.pred_const_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ true_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) true_
case cons.false_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_
case cons.not_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ a✝.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.imp_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.and_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.or_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.iff_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (forall_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (def_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_const_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_var_ X xs =>
simp only [sub]
split_ifs
case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1]
case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case eq_ x y =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ x y)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case true_ | false_ =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case not_ phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
congr! 1
apply phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intros d
apply phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (if xs = [] then pred_var_ (τ X) [] else pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (if xs = [] then pred_var_ (τ X) [] else pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | case pos
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
h✝ : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs)
case neg
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
h✝ : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) (List.map V []) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) (List.map V []) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) [] ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_pos c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) [] ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ x y)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (eq_ x y) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (eq_ x y) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) false_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) false_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi).not_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi).not_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ¬Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ¬Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at psi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) ((sub τ phi).iff_ (sub τ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) ((sub τ phi).iff_ (sub τ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ (Holds D I V (head✝ :: tail✝) (sub τ phi) ↔ Holds D I V (head✝ :: tail✝) (sub τ psi)) ↔
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ (Holds D I V (head✝ :: tail✝) (sub τ phi) ↔ Holds D I V (head✝ :: tail✝) (sub τ psi)) ↔
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply psi_ih | case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (exists_ x (sub τ phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (exists_ x (sub τ phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | intros d | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply forall_congr' | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∀ (d : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tl F
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | case pos
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
h✝ : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q τ hd.h2] at ih
apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub_no_predVar hd.q τ hd.h2] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih V (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] at h1 | F : Formula
τ : PredName → PredName
h1 : F.IsValid
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | intro D I V E | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F) |
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