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[helm.git] / helm / software / matita / dama / constructive_higher_order_relations.ma
1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 set "baseuri" "cic:/matita/constructive_higher_order_relations".
16
17 include "constructive_connectives.ma".
18
19 definition cotransitive ≝
20  λC:Type.λlt:C→C→Type.∀x,y,z:C. lt x y → lt x z ∨ lt z y. 
21
22 definition coreflexive ≝ λC:Type.λlt:C→C→Type. ∀x:C. ¬ (lt x x).
23
24 definition antisymmetric ≝
25  λC:Type.λle:C→C→Type.λeq:C→C→Type.∀x,y:C.le x y → le y x → eq x y.
26
27 definition symmetric ≝
28  λC:Type.λle:C→C→Type.∀x,y:C.le x y → le y x.
29
30 definition transitive ≝
31  λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
32
33 definition associative ≝
34  λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z).
35
36 definition commutative ≝
37  λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).