34 lines
812 B
Text
34 lines
812 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.logic
|
||
universes u
|
||
class setoid (α : Type u) :=
|
||
(r : α → α → Prop) (iseqv : equivalence r)
|
||
|
||
namespace setoid
|
||
infix ` ≈ ` := setoid.r
|
||
|
||
variable {α : Type u}
|
||
variable [s : setoid α]
|
||
include s
|
||
|
||
@[refl] lemma refl (a : α) : a ≈ a :=
|
||
match setoid.iseqv α with
|
||
| ⟨h_refl, h_symm, h_trans⟩ := h_refl a
|
||
end
|
||
|
||
@[symm] lemma symm {a b : α} (hab : a ≈ b) : b ≈ a :=
|
||
match setoid.iseqv α with
|
||
| ⟨h_refl, h_symm, h_trans⟩ := h_symm hab
|
||
end
|
||
|
||
@[trans] lemma trans {a b c : α} (hab : a ≈ b) (hbc : b ≈ c) : a ≈ c :=
|
||
match setoid.iseqv α with
|
||
| ⟨h_refl, h_symm, h_trans⟩ := h_trans hab hbc
|
||
end
|
||
end setoid
|