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".
19 definition injective: \forall A,B:Type.\forall f:A \to B.Prop
20 \def \lambda A,B.\lambda f.
21 \forall x,y:A.f x = f y \to x=y.
23 (* we have still to attach exists *)
24 definition surjective: \forall A,B:Type.\forall f:A \to B.Prop
25 \def \lambda A,B.\lambda f.
26 \forall z:B.ex A (\lambda x:A.z=f x).
28 definition symmetric: \forall A:Type.\forall f:A \to A\to A.Prop
29 \def \lambda A.\lambda f.\forall x,y.f x y = f y x.
31 definition symmetric2: \forall A,B:Type.\forall f:A \to A\to B.Prop
32 \def \lambda A,B.\lambda f.\forall x,y.f x y = f y x.
34 definition associative: \forall A:Type.\forall f:A \to A\to A.Prop
35 \def \lambda A.\lambda f.\forall x,y,z.f (f x y) z = f x (f y z).
37 (* functions and relations *)
38 definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
39 \forall f:A \to A.Prop \def
40 \lambda A. \lambda R. \lambda f. \forall x,y:A.R x y \to R (f x) (f y).
42 (* functions and functions *)
43 definition distributive: \forall A:Type.\forall f,g:A \to A \to A.Prop
44 \def \lambda A.\lambda f,g.\forall x,y,z:A. f x (g y z) = g (f x y) (f x z).
46 definition distributive2: \forall A,B:Type.\forall f:A \to B \to B.
47 \forall g: B\to B\to B. Prop
48 \def \lambda A,B.\lambda f,g.\forall x:A.\forall y,z:B. f x (g y z) = g (f x y) (f x z).