/- Copyright (c) 2014 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 -- TODO(Leo): remove duplication between this file and algebra/relation.lean -- We need some of the following definitions asap when "initializing" Lean. variables {A B : Type} (R : B → B → Prop) infix [local] `≺`:50 := R definition reflexive := ∀x, x ≺ x definition symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x definition transitive := ∀⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z definition irreflexive := ∀x, ¬ x ≺ x definition anti_symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x → x = y definition empty_relation := λa₁ a₂ : A, false definition subrelation (Q R : B → B → Prop) := ∀⦃x y⦄, Q x y → R x y definition inv_image (f : A → B) : A → A → Prop := λa₁ a₂, f a₁ ≺ f a₂ theorem inv_image.trans (f : A → B) (H : transitive R) : transitive (inv_image R f) := λ (a₁ a₂ a₃ : A) (H₁ : inv_image R f a₁ a₂) (H₂ : inv_image R f a₂ a₃), H H₁ H₂ theorem inv_image.irreflexive (f : A → B) (H : irreflexive R) : irreflexive (inv_image R f) := λ (a : A) (H₁ : inv_image R f a a), H (f a) H₁ inductive tc {A : Type} (R : A → A → Prop) : A → A → Prop := base : ∀a b, R a b → tc R a b, trans : ∀a b c, tc R a b → tc R b c → tc R a c