Booleans

diff --git a/helm/software/matita/nlibrary/basics/bool.ma b/helm/software/matita/nlibrary/basics/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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/eq.ma".
+include "basics/functions.ma".
+
+ninductive bool: Type ≝ 
+  | true : bool
+  | false : bool.
+
+(*
+ntheorem 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.*)
+
+(* ndestrcut does not work *)
+ntheorem not_eq_true_false : true \neq false.
+#Heq; nchange with match true with [true ⇒ False|false ⇒ True];
+nrewrite > Heq; //; nqed.
+
+ndefinition notb : bool → bool ≝
+\lambda b:bool. match b with 
+ [true ⇒ false
+ |false ⇒ true ].
+
+interpretation "boolean not" 'not x = (notb x).
+
+ntheorem notb_elim: ∀ b:bool.∀ P:bool → Prop.
+match b with
+[ true ⇒ P false
+| false ⇒ P true] → P (notb b).
+#b; #P; nelim b; nnormalize; //; nqed.
+
+ntheorem notb_notb: ∀b:bool. notb (notb b) = b.
+#b; nelim b; //; nqed.
+
+ntheorem injective_notb: injective bool bool notb.
+#b1; #b2; #H; //; nqed.
+
+ndefinition andb : bool → bool → bool ≝
+\lambda b1,b2:bool. match b1 with 
+ [ true ⇒ b2
+ | false ⇒ false ].
+
+interpretation "boolean and" 'and x y = (andb x y).
+
+ntheorem andb_elim: ∀ b1,b2:bool. ∀ P:bool → Prop.
+match b1 with
+ [ true ⇒ P b2
+ | false ⇒ P false] → P (b1 ∧ b2).
+#b1; #b2; #P; nelim b1; nnormalize; //; nqed.
+
+(*
+ntheorem and_true: ∀ a,b:bool. 
+andb a b =true → a =true ∧ b= true.
+#a; #b; ncases a; nnormalize;#H;napply conj;//;
+  [split
+    [reflexivity|assumption]
+  |apply False_ind.
+   apply not_eq_true_false.
+   apply sym_eq.
+   assumption
+  ]
+qed. *)
+
+ntheorem andb_true_l: ∀ b1,b2. (b1 ∧ b2) = true → b1 = true.
+#b1; ncases b1; nnormalize; //; nqed.
+
+ntheorem andb_true_r: \forall b1,b2. (b1 ∧ b2) = true → b2 = true.
+#b1; ncases b1; nnormalize; //; 
+#b2; ncases b2; //; nqed.
+
+ndefinition orb : bool → bool → bool ≝
+λ b1,b2:bool. 
+ match b1 with 
+ [ true ⇒ true
+ | false ⇒ b2].
+ 
+interpretation "boolean or" 'or x y = (orb x y).
+
+ntheorem orb_elim: ∀ b1,b2:bool. ∀ P:bool → Prop.
+match b1 with
+ [ true ⇒ P true
+ | false ⇒ P b2] → P (orb b1 b2).
+#b1; #b2; #P; nelim b1; nnormalize; //; nqed.
+
+ndefinition if_then_else: ∀A:Type. bool → A → A → A ≝ 
+λA:Type.λb:bool.λ P,Q:A. match b with
+ [ true ⇒ P
+ | false  ⇒ Q].
+
+ntheorem bool_to_decidable_eq:
+ ∀b1,b2:bool. decidable (b1=b2).
+#b1; #b2; ncases b1; ncases b2; /2/;
+@2;/2/; nqed.
+
+ntheorem true_or_false:
+∀b:bool. b = true ∨ b = false.
+#b; ncases b; /2/; nqed.
+
+
+(*
+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.
+
+
+(* some basic properties of and - or*)
+theorem andb_sym: \forall A,B:bool.
+(A \land B) = (B \land A).
+intros.
+elim A;
+  elim B;
+    simplify;
+    reflexivity.
+qed.
+
+theorem andb_assoc: \forall A,B,C:bool.
+(A \land (B \land C)) = ((A \land B) \land C).
+intros.
+elim A;
+  elim B;
+    elim C;
+      simplify;
+      reflexivity.
+qed.
+
+theorem orb_sym: \forall A,B:bool.
+(A \lor B) = (B \lor A).
+intros.
+elim A;
+  elim B;
+    simplify;
+    reflexivity.
+qed.
+
+theorem true_to_true_to_andb_true: \forall A,B:bool.
+A = true \to B = true \to (A \land B) = true.
+intros.
+rewrite > H.
+rewrite > H1.
+reflexivity.
+qed. *)