37 lines
902 B
Text
37 lines
902 B
Text
/-
|
|
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Authors: Leonardo de Moura
|
|
-/
|
|
prelude
|
|
import init.relation
|
|
set_option new_elaborator true
|
|
structure [class] setoid (A : Type) :=
|
|
(r : A → A → Prop) (iseqv : equivalence r)
|
|
|
|
namespace setoid
|
|
infix ` ≈ ` := setoid.r
|
|
|
|
variable {A : Type}
|
|
variable [s : setoid A]
|
|
include s
|
|
|
|
attribute [refl]
|
|
theorem refl (a : A) : a ≈ a :=
|
|
match setoid.iseqv A with
|
|
| ⟨H_refl, H_symm, H_trans⟩ := H_refl a
|
|
end
|
|
|
|
attribute [symm]
|
|
theorem symm {a b : A} (Hab : a ≈ b) : b ≈ a :=
|
|
match setoid.iseqv A with
|
|
| ⟨H_refl, H_symm, H_trans⟩ := H_symm Hab
|
|
end
|
|
|
|
attribute [trans]
|
|
theorem trans {a b c : A} (Hab : a ≈ b) (Hbc : b ≈ c) : a ≈ c :=
|
|
match setoid.iseqv A with
|
|
| ⟨H_refl, H_symm, H_trans⟩ := H_trans Hab Hbc
|
|
end
|
|
end setoid
|