diff options
Diffstat (limited to 'theories')
| -rw-r--r-- | theories/Bool/Bool.v | 85 |
1 files changed, 49 insertions, 36 deletions
diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 756089907c..8708f5a2bf 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.v @@ -9,9 +9,12 @@ (************************************************************************) (** The type [bool] is defined in the prelude as - [Inductive bool : Set := true : bool | false : bool] *) +[[ +Inductive bool : Set := true : bool | false : bool +]] + *) -(** Most of the lemmas in this file are trivial after breaking all booleans *) +(** Most of the lemmas in this file are trivial by case analysis *) Ltac destr_bool := intros; destruct_all bool; simpl in *; trivial; try discriminate. @@ -75,9 +78,9 @@ Proof. destr_bool; intuition. Qed. -(**********************) +(************************) (** * Order on booleans *) -(**********************) +(************************) Definition leb (b1 b2:bool) := match b1 with @@ -93,9 +96,9 @@ Qed. (* Infix "<=" := leb : bool_scope. *) -(*************) +(***************) (** * Equality *) -(*************) +(***************) Definition eqb (b1 b2:bool) : bool := match b1, b2 with @@ -131,9 +134,9 @@ Proof. destr_bool; intuition. Qed. -(************************) +(**********************************) (** * A synonym of [if] on [bool] *) -(************************) +(**********************************) Definition ifb (b1 b2 b3:bool) : bool := match b1 with @@ -143,9 +146,9 @@ Definition ifb (b1 b2 b3:bool) : bool := Open Scope bool_scope. -(****************************) -(** * De Morgan laws *) -(****************************) +(*********************) +(** * De Morgan laws *) +(*********************) Lemma negb_orb : forall b1 b2:bool, negb (b1 || b2) = negb b1 && negb b2. Proof. @@ -157,9 +160,9 @@ Proof. destr_bool. Qed. -(********************************) -(** * Properties of [negb] *) -(********************************) +(***************************) +(** * Properties of [negb] *) +(***************************) Lemma negb_involutive : forall b:bool, negb (negb b) = b. Proof. @@ -212,9 +215,9 @@ Proof. Qed. -(********************************) -(** * Properties of [orb] *) -(********************************) +(**************************) +(** * Properties of [orb] *) +(**************************) Lemma orb_true_iff : forall b1 b2, b1 || b2 = true <-> b1 = true \/ b2 = true. @@ -327,9 +330,9 @@ Proof. Qed. Hint Resolve orb_comm orb_assoc: bool. -(*******************************) -(** * Properties of [andb] *) -(*******************************) +(***************************) +(** * Properties of [andb] *) +(***************************) Lemma andb_true_iff : forall b1 b2:bool, b1 && b2 = true <-> b1 = true /\ b2 = true. @@ -432,9 +435,9 @@ Qed. Hint Resolve andb_comm andb_assoc: bool. -(*******************************************) +(*****************************************) (** * Properties mixing [andb] and [orb] *) -(*******************************************) +(*****************************************) (** Distributivity *) @@ -486,9 +489,9 @@ Notation absoption_andb := absorption_andb (only parsing). Notation absoption_orb := absorption_orb (only parsing). (* end hide *) -(*********************************) -(** * Properties of [implb] *) -(*********************************) +(****************************) +(** * Properties of [implb] *) +(****************************) Lemma implb_true_iff : forall b1 b2:bool, implb b1 b2 = true <-> (b1 = true -> b2 = true). Proof. @@ -500,6 +503,16 @@ Proof. destr_bool; intuition. Qed. +Lemma implb_orb : forall b1 b2:bool, implb b1 b2 = negb b1 || b2. +Proof. + destr_bool. +Qed. + +Lemma implb_negb_orb : forall b1 b2:bool, implb (negb b1) b2 = b1 || b2. +Proof. + destr_bool. +Qed. + Lemma implb_true_r : forall b:bool, implb b true = true. Proof. destr_bool. @@ -555,9 +568,9 @@ Proof. destr_bool. Qed. -(*********************************) -(** * Properties of [xorb] *) -(*********************************) +(***************************) +(** * Properties of [xorb] *) +(***************************) (** [false] is neutral for [xorb] *) @@ -711,9 +724,9 @@ Proof. Qed. Hint Resolve trans_eq_bool : core. -(*****************************************) +(***************************************) (** * Reflection of [bool] into [Prop] *) -(*****************************************) +(***************************************) (** [Is_true] and equality *) @@ -831,10 +844,10 @@ Proof. destr_bool. Qed. -(*****************************************) +(***********************************************) (** * Alternative versions of [andb] and [orb] - with lazy behavior (for vm_compute) *) -(*****************************************) + with lazy behavior (for vm_compute) *) +(***********************************************) Declare Scope lazy_bool_scope. @@ -855,11 +868,11 @@ Proof. reflexivity. Qed. -(*****************************************) +(************************************************) (** * Reflect: a specialized inductive type for relating propositions and booleans, - as popularized by the Ssreflect library. *) -(*****************************************) + as popularized by the Ssreflect library. *) +(************************************************) Inductive reflect (P : Prop) : bool -> Set := | ReflectT : P -> reflect P true |