Integrations

[helm.git] / weblib / basics / bool.ma

diff --git a/weblib/basics/bool.ma b/weblib/basics/bool.ma
index adbfd9cab304d0455ac9626ed3203602384b2734..6bc18e055f2131d16bb1e7ccc758fbcd65892a79 100644 (file)
--- a/weblib/basics/bool.ma
+++ b/weblib/basics/bool.ma
@@ -38,6 +38,14 @@ theorem notb_notb: ∀b:bool. notb (notb b) = b.
 theorem injective_notb: injective bool bool notb.
 #b1 #b2 #H // qed.
 
+theorem noteq_to_eqnot: ∀b1,b2. b1 ≠ b2 → b1 = notb b2.
+* * // #H @False_ind /2/
+qed.
+
+theorem eqnot_to_noteq: ∀b1,b2. b1 = notb b2 → b1 ≠ b2.
+* * normalize // #H @(not_to_not … not_eq_true_false) //
+qed.
+
 definition andb : bool → bool → bool ≝
 λb1,b2:bool. match b1 with [ true ⇒ b2 | false ⇒ false ].
 
@@ -53,6 +61,9 @@ theorem andb_true_l: ∀ b1,b2. (b1 ∧ b2) = true → b1 = true.
 theorem andb_true_r: ∀b1,b2. (b1 ∧ b2) = true → b2 = true.
 #b1 #b2 (cases b1) normalize // (cases b2) // qed.
 
+theorem andb_true: ∀b1,b2. (b1 ∧ b2) = true → b1 = true ∧ b2 = true.
+/3/ qed.
+
 definition orb : bool → bool → bool ≝
 λb1,b2:bool.match b1 with [ true ⇒ true | false ⇒ b2].
  
@@ -62,8 +73,28 @@ theorem orb_elim: ∀ b1,b2:bool. ∀ P:bool → Prop.
 match b1 with [ true ⇒ P true | false ⇒ P b2] → P (orb b1 b2).
 #b1 #b2 #P (elim b1) normalize // qed.
 
-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 with [ true ⇒ $t | false ⇒ $f]  }.
 
 theorem bool_to_decidable_eq: 
   ∀b1,b2:bool. decidable (b1=b2).