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[helm.git] / helm / matita / library / datatypes / bool.ma

diff --git a/helm/matita/library/datatypes/bool.ma b/helm/matita/library/datatypes/bool.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/datatypes/bool/".
+
+include "logic/equality.ma".
+
+inductive bool : Set \def 
+  | true : bool
+  | false : bool.
+
+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
+| flase \Rightarrow True].
+rewrite > H.simplify.exact I.
+qed.
+
+definition notb : bool \to bool \def
+\lambda b:bool. 
+ match b with 
+ [ true \Rightarrow false
+ | false \Rightarrow true ].
+ 
+theorem notb_elim: \forall b:bool.\forall P:bool \to Prop.
+match b with
+[ true \Rightarrow P false
+| false \Rightarrow P true] \to P (notb b).
+intros 2.elim b.exact H. exact H.
+qed.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "boolean not" 'not x = (cic:/matita/datatypes/bool/notb.con x).
+
+definition andb : bool \to bool \to bool\def
+\lambda b1,b2:bool. 
+ match b1 with 
+ [ 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 (b1 \land b2).
+intros 3.elim b1.exact H. exact H.
+qed.
+
+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. 
+ match b1 with 
+ [ true \Rightarrow true
+ | false \Rightarrow b2].
+
+theorem orb_elim: \forall b1,b2:bool. \forall P:bool \to Prop.
+match b1 with
+[ true \Rightarrow P true
+| false \Rightarrow P b2] \to P (orb b1 b2).
+intros 3.elim b1.exact H. exact H.
+qed.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "boolean or" 'or x y = (cic:/matita/datatypes/bool/orb.con x y).
+
+definition if_then_else : bool \to Prop \to Prop \to Prop \def 
+\lambda b:bool.\lambda P,Q:Prop.
+match b with
+[ true \Rightarrow P
+| 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.