feat(library/standard): add equivalence inductive predicate

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/equivalence.lean

@@ -0,0 +1,27 @@
+-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Leonardo de Moura
+import logic
+
+namespace equivalence
+section
+  parameter {A : Type}
+  parameter p : A → A → Bool
+  infix `∼`:50 := p
+  definition reflexive  := ∀a, a ∼ a
+  definition symmetric  := ∀a b, a ∼ b → b ∼ a
+  definition transitive := ∀a b c, a ∼ b → b ∼ c → a ∼ c
+end
+
+inductive equivalence {A : Type} (p : A → A → Bool) : Bool :=
+| equivalence_intro : reflexive p → symmetric p → transitive p → equivalence p
+
+theorem equivalence_reflexive [instance] {A : Type} {p : A → A → Bool} (H : equivalence p) : reflexive p
+:= equivalence_rec (λ r s t, r) H
+
+theorem equivalence_symmetric [instance] {A : Type} {p : A → A → Bool} (H : equivalence p) : symmetric p
+:= equivalence_rec (λ r s t, s) H
+
+theorem equivalence_transitive [instance] {A : Type} {p : A → A → Bool} (H : equivalence p) : transitive p
+:= equivalence_rec (λ r s t, t) H
+end