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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ σ v = σ' v | case a
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ ¬v = x ∧ isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | tauto | case a
x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ ¬v = x ∧ isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | induction F generalizing binders | F : Formula
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders F = replaceFreeAux σ' binders F | case pred_const_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝)
case pred_var_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝)
case eq_
σ σ' : VarName → VarName
a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_
case false_
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_
case not_
σ σ' : VarName → VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_
case imp_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝)
case and_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝)
case or_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝)
case iff_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝)
case forall_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝)
case exists_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝)
case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | all_goals
simp only [replaceFreeAux] | case pred_const_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝)
case pred_var_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝)
case eq_
σ σ' : VarName → VarName
a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_
case false_
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_
case not_
σ σ' : VarName → VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_
case imp_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝)
case and_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝)
case or_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝)
case iff_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝)
case forall_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝)
case exists_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝)
case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝) | case pred_const_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) =
pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)
case pred_var_
σ σ' : VarName → VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) =
pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)
case eq_
σ σ' : VarName → VarName
a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ eq_ (if a✝¹ ∉ binders then σ a✝¹ else a✝¹) (if a✝ ∉ binders then σ a✝ else a✝) =
eq_ (if a✝¹ ∉ binders then σ' a✝¹ else a✝¹) (if a✝ ∉ binders then σ' a✝ else a✝)
case not_
σ σ' : VarName → VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders a✝).not_ = (replaceFreeAux σ' binders a✝).not_
case imp_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders a✝¹).imp_ (replaceFreeAux σ binders a✝) =
(replaceFreeAux σ' binders a✝¹).imp_ (replaceFreeAux σ' binders a✝)
case and_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders a✝¹).and_ (replaceFreeAux σ binders a✝) =
(replaceFreeAux σ' binders a✝¹).and_ (replaceFreeAux σ' binders a✝)
case or_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders a✝¹).or_ (replaceFreeAux σ binders a✝) =
(replaceFreeAux σ' binders a✝¹).or_ (replaceFreeAux σ' binders a✝)
case iff_
σ σ' : VarName → VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders a✝¹).iff_ (replaceFreeAux σ binders a✝) =
(replaceFreeAux σ' binders a✝¹).iff_ (replaceFreeAux σ' binders a✝)
case forall_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ forall_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = forall_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝)
case exists_
σ σ' : VarName → VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = exists_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝)
case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) =
def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
intro x _
split_ifs <;> tauto | σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) =
def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case not_ phi phi_ih =>
simp
tauto | σ σ' : VarName → VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto | σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) =
(replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr! 1
apply phi_ih
simp
tauto | σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp only [replaceFreeAux] | case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝) | case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) =
def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) =
def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs) | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp only [List.map_eq_map_iff] | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | intro x _ | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
x : VarName
a✝ : x ∈ xs
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
x : VarName
a✝ : x ∈ xs
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ eq_ (if x ∉ binders then σ x else x) (if y ∉ binders then σ y else y) =
eq_ (if x ∉ binders then σ' x else x) (if y ∉ binders then σ' y else y) | case h.e'_1
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
case h.e'_2
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_1
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | split_ifs <;> tauto | case h.e'_2
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | σ σ' : VarName → VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_ | σ σ' : VarName → VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | σ σ' : VarName → VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) =
(replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi) | σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧
replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧
replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | congr! 1 | σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi) | case h.e'_2
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | apply phi_ih | case h.e'_2
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi | case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | simp | case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v | case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free | [226, 1] | [260, 10] | tauto | case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
(∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | induction xs | α : Type
f : α → α
xs : List α
⊢ map f xs = xs ↔ ∀ x ∈ xs, f x = x | case nil
α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x
case cons
α : Type
f : α → α
head✝ : α
tail✝ : List α
tail_ih✝ : map f tail✝ = tail✝ ↔ ∀ x ∈ tail✝, f x = x
⊢ map f (head✝ :: tail✝) = head✝ :: tail✝ ↔ ∀ x ∈ head✝ :: tail✝, f x = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | case nil =>
simp | α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | case cons hd tl ih =>
simp
intro _
exact ih | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ map f (hd :: tl) = hd :: tl ↔ ∀ x ∈ hd :: tl, f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | simp | α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | simp | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ map f (hd :: tl) = hd :: tl ↔ ∀ x ∈ hd :: tl, f x = x | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ f hd = hd → (map f tl = tl ↔ ∀ a ∈ tl, f a = a) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | intro _ | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ f hd = hd → (map f tl = tl ↔ ∀ a ∈ tl, f a = a) | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
a✝ : f hd = hd
⊢ map f tl = tl ↔ ∀ a ∈ tl, f a = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | exact ih | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
a✝ : f hd = hd
⊢ map f tl = tl ↔ ∀ a ∈ tl, f a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | induction xs | α : Type
f : α → α
xs : List α
h1 : ∀ x ∈ xs, f x = x
⊢ map f xs = xs | case nil
α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = []
case cons
α : Type
f : α → α
head✝ : α
tail✝ : List α
tail_ih✝ : (∀ x ∈ tail✝, f x = x) → map f tail✝ = tail✝
h1 : ∀ x ∈ head✝ :: tail✝, f x = x
⊢ map f (head✝ :: tail✝) = head✝ :: tail✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | case nil =>
simp | α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp | α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp at h1 | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : ∀ x ∈ hd :: tl, f x = x
⊢ map f (hd :: tl) = hd :: tl | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : f hd = hd ∧ ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | cases h1 | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : f hd = hd ∧ ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl | case intro
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
left✝ : f hd = hd
right✝ : ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd ∧ map f tl = tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | constructor | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd ∧ map f tl = tl | case left
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd
case right
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f tl = tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | exact h1_left | case left
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | exact ih h1_right | case right
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f tl = tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | induction xs generalizing ys | α : Type
inst✝ : DecidableEq α
xs ys : List α
h1 : xs.length = ys.length
h2 : xs.Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f xs ∧ map f xs = ys | case nil
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f [] ∧ map f [] = ys
case cons
α : Type
inst✝ : DecidableEq α
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (ys : List α), tail✝.length = ys.length → tail✝.Nodup → ys.Nodup → ∃ f, InjOn f tail✝ ∧ map f tail✝ = ys
ys : List α
h1 : (head✝ :: tail✝).length = ys.length
h2 : (head✝ :: tail✝).Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f (head✝ :: tail✝) ∧ map f (head✝ :: tail✝) = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | have s1 : ys = [] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = []
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | {
apply List.eq_nil_of_length_eq_zero
simp only [← h1]
simp
} | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = []
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [s1] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.intro id | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = [] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] ∧ map id [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | constructor | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] ∧ map id [] = [] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id []
case right
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ map id [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply List.eq_nil_of_length_eq_zero | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = [] | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys.length = 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [← h1] | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys.length = 0 | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ [].length = 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ [].length = 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [List.InjOn] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∀ ⦃x₁ : α⦄, x₁ ∈ [] → ∀ ⦃x₂ : α⦄, x₂ ∈ [] → id x₁ = id x₂ → x₁ = x₂ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∀ ⦃x₁ : α⦄, x₁ ∈ [] → ∀ ⦃x₂ : α⦄, x₂ ∈ [] → id x₁ = id x₂ → x₁ = x₂ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case right
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ map id [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases ys | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys : List α
h1 : (xs_hd :: xs_tl).length = ys.length
h2 : (xs_hd :: xs_tl).Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys | case nil
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = []
case cons
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
head✝ : α
tail✝ : List α
h1 : (xs_hd :: xs_tl).length = (head✝ :: tail✝).length
h3 : (head✝ :: tail✝).Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = head✝ :: tail✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case nil =>
simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h1 : (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length
h3 : (ys_hd :: ys_tl).Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases h2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
left✝ : xs_hd ∉ xs_tl
right✝ : xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases h3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
left✝ : ys_hd ∉ ys_tl
right✝ : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | specialize xs_ih ys_tl h1 h2_right h3_right | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.elim xs_ih | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∀ (a : α → α),
InjOn a xs_tl ∧ map a xs_tl = ys_tl → ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro f a1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∀ (a : α → α),
InjOn a xs_tl ∧ map a xs_tl = ys_tl → ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | clear xs_ih | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [List.InjOn] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases a1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
left✝ : InjOn f xs_tl
right✝ : map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.intro (Function.updateITE f xs_hd ys_hd) | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ (∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂) ∧
Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | constructor | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ (∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂) ∧
Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂
case right
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro x1 x1_mem x2 x2_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂ | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 : α
x1_mem : x1 ∈ xs_hd :: xs_tl
x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at x1_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 : α
x1_mem : x1 ∈ xs_hd :: xs_tl
x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at x2_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [Function.updateITE] | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ ((if x1 = xs_hd then ys_hd else f x1) = if x2 = xs_hd then ys_hd else f x2) → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | split_ifs | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ ((if x1 = xs_hd then ys_hd else f x1) = if x2 = xs_hd then ys_hd else f x2) → x1 = x2 | case pos
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : x1 = xs_hd
h✝ : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2
case neg
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : x1 = xs_hd
h✝ : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2
case pos
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : ¬x1 = xs_hd
h✝ : x2 = xs_hd
⊢ f x1 = ys_hd → x1 = x2
case neg
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : ¬x1 = xs_hd
h✝ : ¬x2 = xs_hd
⊢ f x1 = f x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro _
simp only [c1]
simp only [c2] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x2_mem
case inl c3 =>
contradiction
case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [← a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x1_mem
case inl c3 =>
contradiction
case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
⊢ f x1 = ys_hd → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x1_mem
case inl x1_mem_left =>
contradiction
case inr x1_mem_right =>
cases x2_mem
case inl x2_mem_left =>
contradiction
case inr x2_mem_right =>
simp only [List.InjOn] at a1_left
apply a1_left x1_mem_right x2_mem_right a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ f x1 = f x2 → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro _ | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [c1] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ xs_hd = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [c2] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ xs_hd = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases x2_mem | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
⊢ x1 = x2 | case inl
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
h✝ : x2 = xs_hd
⊢ x1 = x2
case inr
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
h✝ : x2 ∈ xs_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case inl c3 =>
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 = xs_hd
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [← a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 = xs_hd
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | obtain s1 := List.mem_map_of_mem f c3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : f x2 ∈ map f xs_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [a1_right] at s1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : f x2 ∈ map f xs_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : f x2 ∈ ys_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [← a2] at s1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : f x2 ∈ ys_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : ys_hd ∈ ys_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 ∈ xs_tl
s1 : ys_hd ∈ ys_tl
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
⊢ f x1 = ys_hd → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases x1_mem | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
⊢ x1 = x2 | case inl
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
h✝ : x1 = xs_hd
⊢ x1 = x2
case inr
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
h✝ : x1 ∈ xs_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case inl c3 =>
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 = xs_hd
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 = xs_hd
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | obtain s1 := List.mem_map_of_mem f c3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : f x1 ∈ map f xs_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [a1_right] at s1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : f x1 ∈ map f xs_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : f x1 ∈ ys_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [a2] at s1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : f x1 ∈ ys_tl
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : ys_hd ∈ ys_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
a2 : f x1 = ys_hd
c3 : x1 ∈ xs_tl
s1 : ys_hd ∈ ys_tl
⊢ x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ f x1 = f x2 → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : f x1 = f x2
⊢ x1 = x2 |
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