set "baseuri" "cic:/matita/datatypes/bool/".
include "logic/equality.ma".
+include "higher_order_defs/functions.ma".
inductive bool : Set \def
| true : bool
change with
match true with
[ true \Rightarrow False
-| flase \Rightarrow True].
+| false \Rightarrow True].
rewrite > H.simplify.exact I.
qed.
intros 2.elim b.exact H. exact H.
qed.
+theorem notb_notb: \forall b:bool. notb (notb b) = b.
+intros.
+elim b;reflexivity.
+qed.
+
+theorem injective_notb: injective bool bool notb.
+unfold injective.
+intros.
+rewrite < notb_notb.
+rewrite < (notb_notb y).
+apply eq_f.
+assumption.
+qed.
+
(*CSC: the URI must disappear: there is a bug now *)
interpretation "boolean not" 'not x = (cic:/matita/datatypes/bool/notb.con x).
intros 3.elim b1.exact H. exact H.
qed.
+theorem and_true: \forall a,b:bool.
+andb a b =true \to a =true \land b= true.
+intro.elim a
+ [split
+ [reflexivity|assumption]
+ |apply False_ind.
+ apply not_eq_true_false.
+ apply sym_eq.
+ assumption
+ ]
+qed.
+
theorem andb_true_true: \forall b1,b2. (b1 \land b2) = true \to b1 = true.
intro. elim b1.
reflexivity.
assumption.
qed.
+theorem andb_true_true_r: \forall b1,b2. (b1 \land b2) = true \to b2 = true.
+intro. elim b1
+ [assumption
+ |apply False_ind.apply not_eq_true_false.
+ apply sym_eq.assumption
+ ]
+qed.
+
definition orb : bool \to bool \to bool\def
\lambda b1,b2:bool.
match b1 with
intros.
apply eq_to_eq_to_eq_p_q.
exact bool_to_decidable_eq.
-qed.
+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.
\ No newline at end of file