29 lines
1.2 KiB
Text
29 lines
1.2 KiB
Text
|
import data.sigma tools.tactic
|
||
|
|
|
||
|
|
namespace sigma
|
||
|
|
definition no_confusion_type {A : Type} {B : A → Type} (P : Type) (v₁ v₂ : sigma B) : Type :=
|
||
|
|
cases_on v₁
|
||
|
|
(λ (a₁ : A) (b₁ : B a₁), cases_on v₂
|
||
|
|
(λ (a₂ : A) (b₂ : B a₂),
|
||
|
|
(Π (eq₁ : a₁ = a₂), eq.rec_on eq₁ b₁ = b₂ → P) → P))
|
||
|
|
|
||
|
|
definition no_confusion {A : Type} {B : A → Type} {P : Type} {v₁ v₂ : sigma B} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
|
||
|
|
assume H₁₂ : v₁ = v₂,
|
||
|
|
have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
|
||
|
|
assume H₁₁, cases_on v₁
|
||
|
|
(λ (a₁ : A) (b₁ : B a₁) (h : Π (eq₁ : a₁ = a₁), eq.rec_on eq₁ b₁ = b₁ → P),
|
||
|
|
h rfl rfl),
|
||
|
|
eq.rec_on H₁₂ aux H₁₂
|
||
|
|
|
||
|
|
theorem sigma.mk.inj_1 {A : Type} {B : A → Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (Heq : dpair a₁ b₁ = dpair a₂ b₂) : a₁ = a₂ :=
|
||
|
|
begin
|
||
|
|
apply (no_confusion Heq), intros, assumption
|
||
|
|
end
|
||
|
|
|
||
|
|
theorem sigma.mk.inj_2 {A : Type} {B : A → Type} (a₁ a₂ : A) (b₁ : B a₁) (b₂ : B a₂) (Heq : dpair a₁ b₁ = dpair a₂ b₂) :
|
||
|
|
eq.rec_on (sigma.mk.inj_1 Heq) b₁ = b₂ :=
|
||
|
|
begin
|
||
|
|
apply (no_confusion Heq), intros, eassumption
|
||
|
|
end
|
||
|
|
end sigma
|