X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fdatatypes%2Fbool.ma;h=3292e6789610f6114c590be58d18c729a3bf92fe;hb=97c2d258a5c524eb5c4b85208899d80751a2c82f;hp=d99456dc89e1b02ee4522c26ee90cc5ab6930ded;hpb=03fcee16d9c262aad38a47d0a409b684a965cc3f;p=helm.git

diff --git a/helm/matita/library/datatypes/bool.ma b/helm/matita/library/datatypes/bool.ma
index d99456dc8..3292e6789 100644
--- a/helm/matita/library/datatypes/bool.ma
+++ b/helm/matita/library/datatypes/bool.ma
@@ -19,9 +19,20 @@ include "logic/equality.ma".
 inductive bool : Set \def 
   | true : bool
   | false : bool.
-  
-theorem not_eq_true_false : \lnot true = false.
-simplify.intro.
+
+theorem bool_elim: \forall P:bool \to Prop. \forall b:bool.
+  (b = true \to P true)
+  \to (b = false \to P false)
+  \to P b.
+  intros 2 (P b).
+  elim b;
+  [ apply H; reflexivity
+  | apply H1; reflexivity
+  ]
+qed.
+
+theorem not_eq_true_false : true \neq false.
+unfold Not.intro.
 change with 
 match true with
 [ true \Rightarrow False
@@ -51,15 +62,21 @@ definition andb : bool \to bool \to bool\def
  [ true \Rightarrow b2
  | false \Rightarrow false ].
 
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "boolean and" 'and x y = (cic:/matita/datatypes/bool/andb.con x y).
+
 theorem andb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
 match b1 with
 [ true \Rightarrow P b2
-| false \Rightarrow P false] \to P (andb b1 b2).
+| false \Rightarrow P false] \to P (b1 \land b2).
 intros 3.elim b1.exact H. exact H.
 qed.
 
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "boolean and" 'and x y = (cic:/matita/datatypes/bool/andb.con x y).
+theorem andb_true_true: \forall b1,b2. (b1 \land b2) = true \to b1 = true.
+intro. elim b1.
+reflexivity.
+assumption.
+qed.
 
 definition orb : bool \to bool \to bool\def
 \lambda b1,b2:bool. 
@@ -84,3 +101,26 @@ match b with
 | false  \Rightarrow Q].
 
 (*CSC: missing notation for if_then_else *)
+
+theorem bool_to_decidable_eq:
+ \forall b1,b2:bool. decidable (b1=b2).
+ intros.
+ unfold decidable.
+ elim b1.
+  elim b2.
+   left. reflexivity.
+   right. exact not_eq_true_false.
+  elim b2.
+   right. unfold Not. intro.
+   apply not_eq_true_false.
+   symmetry. exact H.
+   left. reflexivity.
+qed.
+
+theorem P_x_to_P_x_to_eq:
+ \forall A:Set. \forall P: A \to bool.
+  \forall x:A. \forall p1,p2:P x = true. p1 = p2.
+ intros.
+ apply eq_to_eq_to_eq_p_q.
+ exact bool_to_decidable_eq.
+qed.