46 lines
1.8 KiB
Text
46 lines
1.8 KiB
Text
|
namespace sigma
|
||
|
|
open lift
|
||
|
|
open sigma.ops
|
||
|
|
variables {A : Type} {B : A → Type}
|
||
|
|
|
||
|
|
variables {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂}
|
||
|
|
|
||
|
|
definition dpair.inj (H : ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) : Σ (e₁ : a₁ = a₂), eq.rec b₁ e₁ = b₂ :=
|
||
|
|
down (no_confusion H (λ e₁ e₂, ⟨e₁, e₂⟩))
|
||
|
|
|
||
|
|
definition dpair.inj₁ (H : ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) : a₁ = a₂ :=
|
||
|
|
(dpair.inj H).1
|
||
|
|
|
||
|
|
definition dpair.inj₂ (H : ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) : eq.rec b₁ (dpair.inj₁ H) = b₂ :=
|
||
|
|
(dpair.inj H).2
|
||
|
|
|
||
|
|
end sigma
|
||
|
|
|
||
|
|
structure foo :=
|
||
|
|
mk :: (A : Type) (B : A → Type) (a : A) (b : B a)
|
||
|
|
|
||
|
|
set_option pp.implicit true
|
||
|
|
|
||
|
|
namespace foo
|
||
|
|
open lift sigma.ops
|
||
|
|
universe variables l₁ l₂
|
||
|
|
variables {A₁ : Type.{l₁}} {B₁ : A₁ → Type.{l₂}} {a₁ : A₁} {b₁ : B₁ a₁}
|
||
|
|
variables {A₂ : Type.{l₁}} {B₂ : A₂ → Type.{l₂}} {a₂ : A₂} {b₂ : B₂ a₂}
|
||
|
|
|
||
|
|
definition foo.inj (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) :
|
||
|
|
Σ (e₁ : A₁ = A₂) (e₂ : eq.rec B₁ e₁ = B₂) (e₃ : eq.rec a₁ e₁ = a₂), eq.rec (eq.rec (eq.rec b₁ e₁) e₂) e₃ = b₂ :=
|
||
|
|
down (no_confusion H (λ e₁ e₂ e₃ e₄, ⟨e₁, e₂, e₃, e₄⟩))
|
||
|
|
|
||
|
|
definition foo.inj₁ (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) : A₁ = A₂ :=
|
||
|
|
(foo.inj H).1
|
||
|
|
|
||
|
|
definition foo.inj₂ (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) : eq.rec B₁ (foo.inj₁ H) = B₂ :=
|
||
|
|
(foo.inj H).2.1
|
||
|
|
|
||
|
|
definition foo.inj₃ (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) : eq.rec a₁ (foo.inj₁ H) = a₂ :=
|
||
|
|
(foo.inj H).2.2.1
|
||
|
|
|
||
|
|
definition foo.inj₄ (H : mk A₁ B₁ a₁ b₁ = mk A₂ B₂ a₂ b₂) : eq.rec (eq.rec (eq.rec b₁ (foo.inj₁ H)) (foo.inj₂ H)) (foo.inj₃ H) = b₂ :=
|
||
|
|
(foo.inj H).2.2.2
|
||
|
|
end foo
|