-definition if_then_else: ∀A:Type[0]. bool → A → A → A ≝
-λA.λb.λ P,Q:A. match b with [ true ⇒ P | false ⇒ Q].
+lemma orb_true_r1: ∀b1,b2:bool.
+ b1 = true → (b1 ∨ b2) = true.
+#b1 #b2 #H >H // qed.
+
+lemma orb_true_r2: ∀b1,b2:bool.
+ b2 = true → (b1 ∨ b2) = true.
+#b1 #b2 #H >H cases b1 // qed.
+
+lemma orb_true_l: ∀b1,b2:bool.
+ (b1 ∨ b2) = true → (b1 = true) ∨ (b2 = true).
+* normalize /2/ qed.
+
+definition xorb : bool → bool → bool ≝
+λb1,b2:bool.
+ match b1 with
+ [ true ⇒ match b2 with [ true ⇒ false | false ⇒ true ]
+ | false ⇒ match b2 with [ true ⇒ true | false ⇒ false ]].
+
+notation > "'if' term 46 e 'then' term 46 t 'else' term 46 f" non associative with precedence 46
+ for @{ match $e in bool with [ true ⇒ $t | false ⇒ $f] }.
+notation < "hvbox('if' \nbsp term 46 e \nbsp break 'then' \nbsp term 46 t \nbsp break 'else' \nbsp term 49 f \nbsp)" non associative with precedence 46
+ for @{ match $e return $T with [ true ⇒ $t | false ⇒ $f] }.