1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/higher_order_defs/functions/".
17 include "logic/equality.ma".
18 include "logic/connectives.ma".
20 definition injective: \forall A,B:Type.\forall f:A \to B.Prop
21 \def \lambda A,B.\lambda f.
22 \forall x,y:A.eq B (f x) (f y) \to (eq A x y).
24 (* we have still to attach exists *)
25 definition surjective: \forall A,B:Type.\forall f:A \to B.Prop
26 \def \lambda A,B.\lambda f.
27 \forall z:B.ex A (\lambda x:A.(eq B z (f x))).
29 definition symmetric: \forall A:Type.\forall f:A \to A\to A.Prop
30 \def \lambda A.\lambda f.\forall x,y.eq A (f x y) (f y x).
32 definition associative: \forall A:Type.\forall f:A \to A\to A.Prop
33 \def \lambda A.\lambda f.\forall x,y,z.eq A (f (f x y) z) (f x (f y z)).
35 (* functions and relations *)
36 definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
37 \forall f:A \to A.Prop \def
38 \lambda A. \lambda R. \lambda f. \forall x,y:A.R x y \to R (f x) (f y).
40 (* functions and functions *)
41 definition distributive: \forall A:Type.\forall f,g:A \to A \to A.Prop
42 \def \lambda A.\lambda f,g.\forall x,y,z:A.eq A (f x (g y z)) (g (f x y) (f x z)).