From f1ef0a9e283af00cace679efd5775062c2a8f05c Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Wed, 22 Feb 2012 12:31:42 +0000
Subject: [PATCH] Integrations

---
 weblib/basics/bool.ma  | 35 +++++++++++++++++++++++++++++++++--
 weblib/basics/logic.ma | 22 +++++++++++++++++++++-
 2 files changed, 54 insertions(+), 3 deletions(-)

diff --git a/weblib/basics/bool.ma b/weblib/basics/bool.ma
index adbfd9cab..6bc18e055 100644
--- 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).
diff --git a/weblib/basics/logic.ma b/weblib/basics/logic.ma
index c64f887c6..aeee17a73 100644
--- a/weblib/basics/logic.ma
+++ b/weblib/basics/logic.ma
@@ -137,8 +137,28 @@ inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
 definition iff :=
  λ A,B. (A → B) a title="logical and" href="cic:/fakeuri.def(1)"∧/a (B → A).
 
-interpretation "iff" 'iff a b = (iff a b).  
+interpretation "iff" 'iff a b = (iff a b).
+  
+lemma iff_sym: ∀A,B. A ↔ B → B ↔ A.
+#A #B * /3/ qed.
+
+lemma iff_trans:∀A,B,C. A ↔ B → B ↔ C → A ↔ C.
+#A #B #C * #H1 #H2 * #H3 #H4 % /3/ qed.
+
+lemma iff_not: ∀A,B. A ↔ B → ¬A ↔ ¬B.
+#A #B * #H1 #H2 % /3/ qed.
+
+lemma iff_and_l: ∀A,B,C. A ↔ B → C ∧ A ↔ C ∧ B.
+#A #B #C * #H1 #H2 % * /3/ qed.  
+
+lemma iff_and_r: ∀A,B,C. A ↔ B → A ∧ C ↔ B ∧ C.
+#A #B #C * #H1 #H2 % * /3/ qed.  
+
+lemma iff_or_l: ∀A,B,C. A ↔ B → C ∨ A ↔ C ∨ B.
+#A #B #C * #H1 #H2 % * /3/ qed.  
 
+lemma iff_or_r: ∀A,B,C. A ↔ B → A ∨ C ↔ B ∨ C.
+#A #B #C * #H1 #H2 % * /3/ qed.  
 (* cose per destruct: da rivedere *) 
 
 definition R0 ≝ λT:Type[0].λt:T.t.
-- 
2.39.2