New version of the library, a bit more structured.

[helm.git] / helm / matita / library / higher_order_defs / functions.ma

diff --git a/helm/matita/library/higher_order_defs/functions.ma b/helm/matita/library/higher_order_defs/functions.ma
new file mode 100644 (file)
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+++ b/helm/matita/library/higher_order_defs/functions.ma
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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/higher_order_defs/functions/".
+
+include "logic/equality.ma".
+include "logic/connectives.ma".
+
+definition injective: \forall A,B:Type.\forall f:A \to B.Prop
+\def \lambda A,B.\lambda f.
+  \forall x,y:A.eq B (f x) (f y) \to (eq A x y).
+
+(* we have still to attach exists *)
+definition surjective: \forall A,B:Type.\forall f:A \to B.Prop
+\def \lambda A,B.\lambda f.
+  \forall z:B.ex A (\lambda x:A.(eq B z (f x))).
+
+definition symmetric: \forall A:Type.\forall f:A \to A\to A.Prop
+\def \lambda A.\lambda f.\forall x,y.eq A (f x y) (f y x).
+
+definition associative: \forall A:Type.\forall f:A \to A\to A.Prop
+\def \lambda A.\lambda f.\forall x,y,z.eq A (f (f x y) z) (f x (f y z)).
+
+(* functions and relations *)
+definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
+\forall f:A \to A.Prop \def
+\lambda A. \lambda R. \lambda f. \forall x,y:A.R x y \to R (f x) (f y).
+
+(* functions and functions *)
+definition distributive: \forall A:Type.\forall f,g:A \to A \to A.Prop
+\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)).
+
+