--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.