diff options
Diffstat (limited to 'theories/Bool/Bool.v')
| -rw-r--r-- | theories/Bool/Bool.v | 33 |
1 files changed, 28 insertions, 5 deletions
diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 9509d9fd..d5d11cea 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Bool.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** The type [bool] is defined in the prelude as [Inductive bool : Set := true : bool | false : bool] *) @@ -259,6 +257,11 @@ Proof. intros. apply orb_false_iff; trivial. Qed. +Lemma orb_diag : forall b, b || b = b. +Proof. + destr_bool. +Qed. + (** [true] is a zero for [orb] *) Lemma orb_true_r : forall b:bool, b || true = true. @@ -364,6 +367,11 @@ Qed. Notation andb_b_false := andb_false_r (only parsing). Notation andb_false_b := andb_false_l (only parsing). +Lemma andb_diag : forall b, b && b = b. +Proof. + destr_bool. +Qed. + (** [true] is neutral for [andb] *) Lemma andb_true_r : forall b:bool, b && true = b. @@ -547,6 +555,21 @@ Proof. destr_bool. Qed. +Lemma negb_xorb_l : forall b b', negb (xorb b b') = xorb (negb b) b'. +Proof. + destruct b,b'; trivial. +Qed. + +Lemma negb_xorb_r : forall b b', negb (xorb b b') = xorb b (negb b'). +Proof. + destruct b,b'; trivial. +Qed. + +Lemma xorb_negb_negb : forall b b', xorb (negb b) (negb b') = xorb b b'. +Proof. + destruct b,b'; trivial. +Qed. + (** Lemmas about the [b = true] embedding of [bool] to [Prop] *) Lemma eq_iff_eq_true : forall b1 b2, b1 = b2 <-> (b1 = true <-> b2 = true). @@ -768,7 +791,7 @@ Qed. Lemma iff_reflect : forall P b, (P<->b=true) -> reflect P b. Proof. destr_bool; intuition. -Qed. +Defined. (** It would be nice to join [reflect_iff] and [iff_reflect] in a unique [iff] statement, but this isn't allowed since @@ -779,7 +802,7 @@ Qed. Lemma reflect_dec : forall P b, reflect P b -> {P}+{~P}. Proof. destruct 1; auto. -Qed. +Defined. (** Reciprocally, from a decidability, we could state a [reflect] as soon as we have a [bool_of_sumbool]. *) |