lean2/library/standard/struc/equivalence.lean
2014-08-07 16:59:08 -07:00

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----------------------------------------------------------------------------------------------------
-- 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.connectives.prop
namespace equivalence
section
parameter {A : Type}
parameter p : A → A → Prop
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 → Prop) : Prop :=
| equivalence_intro : reflexive p → symmetric p → transitive p → equivalence p
theorem equivalence_reflexive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : reflexive p :=
equivalence_rec (λ r s t, r) H
theorem equivalence_symmetric [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : symmetric p :=
equivalence_rec (λ r s t, s) H
theorem equivalence_transitive [instance] {A : Type} {p : A → A → Prop} (H : equivalence p) : transitive p :=
equivalence_rec (λ r s t, t) H
end equivalence