X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fhigher_order_defs%2Ffunctions.ma;h=a1b54c80c59cf786d68bc89d881b98de57f61cc7;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=da90d421177817ff8561794d66250f97c6ce4fda;hpb=5a702cea95883f7095c16b450e065ccb1714fc5a;p=helm.git

diff --git a/helm/matita/library/higher_order_defs/functions.ma b/helm/matita/library/higher_order_defs/functions.ma
index da90d4211..a1b54c80c 100644
--- a/helm/matita/library/higher_order_defs/functions.ma
+++ b/helm/matita/library/higher_order_defs/functions.ma
@@ -16,14 +16,24 @@ set "baseuri" "cic:/matita/higher_order_defs/functions/".
 
 include "logic/equality.ma".
 
+definition compose \def
+  \lambda A,B,C:Type.\lambda f:(B\to C).\lambda g:(A\to B).\lambda x:A.
+  f (g x).
+
+notation "hvbox(a break \circ b)" 
+  left associative with precedence 70
+for @{ 'compose $a $b }.
+
+interpretation "function composition" 'compose f g =
+  (cic:/matita/higher_order_defs/functions/compose.con _ _ _ f g).
+
 definition injective: \forall A,B:Type.\forall f:A \to B.Prop
 \def \lambda A,B.\lambda f.
   \forall x,y:A.f x = f y \to 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.z=f x).
+  \forall z:B. \exists x:A.z=f x.
 
 definition symmetric: \forall A:Type.\forall f:A \to A\to A.Prop
 \def \lambda A.\lambda f.\forall x,y.f x y = f y x.
@@ -34,6 +44,14 @@ definition symmetric2: \forall A,B:Type.\forall f:A \to A\to B.Prop
 definition associative: \forall A:Type.\forall f:A \to A\to A.Prop
 \def \lambda A.\lambda f.\forall x,y,z.f (f x y) z = f x (f y z).
 
+theorem eq_f_g_h:
+  \forall A,B,C,D:Type.
+  \forall f:C \to D.\forall g:B \to C.\forall h:A \to B.
+  f \circ (g \circ h) = (f \circ g) \circ h.
+  intros.
+  reflexivity.
+qed.
+
 (* functions and relations *)
 definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
 \forall f:A \to A.Prop \def
@@ -46,3 +64,4 @@ definition distributive: \forall A:Type.\forall f,g:A \to A \to A.Prop
 definition distributive2: \forall A,B:Type.\forall f:A \to B \to B.
 \forall g: B\to B\to B. Prop
 \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).
+