ocaml 3.09 transition

[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
index bdea562cedeff817a299edbe619a69b07084a975..a1b54c80c59cf786d68bc89d881b98de57f61cc7 100644 (file)
--- a/helm/matita/library/higher_order_defs/functions.ma
+++ b/helm/matita/library/higher_order_defs/functions.ma
@@ -15,22 +15,42 @@
 set "baseuri" "cic:/matita/higher_order_defs/functions/".
 
 include "logic/equality.ma".
-include "logic/connectives.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.eq B (f x) (f y) \to (eq A x y).
+  \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.(eq B 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.eq A (f x y) (f y x).
+\def \lambda A.\lambda f.\forall x,y.f x y = f y x.
+
+definition symmetric2: \forall A,B:Type.\forall f:A \to A\to B.Prop
+\def \lambda A,B.\lambda f.\forall x,y.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)).
+\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.
@@ -39,6 +59,9 @@ definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
 
 (* 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)).
+\def \lambda A.\lambda f,g.\forall x,y,z:A. f x (g y z) = g (f x y) (f x z).
 
+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).