diff options
| -rw-r--r-- | doc/changelog/10-standard-library/12018-master+implb-characterization.rst | 19 | ||||
| -rw-r--r-- | theories/Bool/Bool.v | 158 |
2 files changed, 144 insertions, 33 deletions
diff --git a/doc/changelog/10-standard-library/12018-master+implb-characterization.rst b/doc/changelog/10-standard-library/12018-master+implb-characterization.rst new file mode 100644 index 0000000000..4b0abdfa3b --- /dev/null +++ b/doc/changelog/10-standard-library/12018-master+implb-characterization.rst @@ -0,0 +1,19 @@ +- **Added:** + Added lemmas + :g:`orb_negb_l`, + :g:`andb_negb_l`, + :g:`implb_true_iff`, + :g:`implb_false_iff`, + :g:`implb_true_r`, + :g:`implb_false_r`, + :g:`implb_true_l`, + :g:`implb_false_l`, + :g:`implb_same`, + :g:`implb_contrapositive`, + :g:`implb_negb`, + :g:`implb_curry`, + :g:`implb_andb_distrib_r`, + :g:`implb_orb_distrib_r`, + :g:`implb_orb_distrib_l` in library :g:`Bool` + (`#12018 <https://github.com/coq/coq/pull/12018>`_,` + by Hugo Herbelin).` diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 1d5e3e54ff..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. @@ -305,6 +308,11 @@ Proof. Qed. Hint Resolve orb_negb_r: bool. +Lemma orb_negb_l : forall b:bool, negb b || b = true. +Proof. + destr_bool. +Qed. + Notation orb_neg_b := orb_negb_r (only parsing). (** Commutativity *) @@ -322,9 +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. @@ -404,6 +412,11 @@ Proof. Qed. Hint Resolve andb_negb_r: bool. +Lemma andb_negb_l : forall b:bool, negb b && b = false. +Proof. + destr_bool. +Qed. + Notation andb_neg_b := andb_negb_r (only parsing). (** Commutativity *) @@ -422,9 +435,9 @@ Qed. Hint Resolve andb_comm andb_assoc: bool. -(*******************************************) +(*****************************************) (** * Properties mixing [andb] and [orb] *) -(*******************************************) +(*****************************************) (** Distributivity *) @@ -476,9 +489,88 @@ Notation absoption_andb := absorption_andb (only parsing). Notation absoption_orb := absorption_orb (only parsing). (* end hide *) -(*********************************) -(** * Properties of [xorb] *) -(*********************************) +(****************************) +(** * Properties of [implb] *) +(****************************) + +Lemma implb_true_iff : forall b1 b2:bool, implb b1 b2 = true <-> (b1 = true -> b2 = true). +Proof. + destr_bool; intuition. +Qed. + +Lemma implb_false_iff : forall b1 b2:bool, implb b1 b2 = false <-> (b1 = true /\ b2 = false). +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. +Qed. + +Lemma implb_false_r : forall b:bool, implb b false = negb b. +Proof. + destr_bool. +Qed. + +Lemma implb_true_l : forall b:bool, implb true b = b. +Proof. + destr_bool. +Qed. + +Lemma implb_false_l : forall b:bool, implb false b = true. +Proof. + destr_bool. +Qed. + +Lemma implb_same : forall b:bool, implb b b = true. +Proof. + destr_bool. +Qed. + +Lemma implb_contrapositive : forall b1 b2:bool, implb (negb b1) (negb b2) = implb b2 b1. +Proof. + destr_bool. +Qed. + +Lemma implb_negb : forall b1 b2:bool, implb (negb b1) b2 = implb (negb b2) b1. +Proof. + destr_bool. +Qed. + +Lemma implb_curry : forall b1 b2 b3:bool, implb (b1 && b2) b3 = implb b1 (implb b2 b3). +Proof. + destr_bool. +Qed. + +Lemma implb_andb_distrib_r : forall b1 b2 b3:bool, implb b1 (b2 && b3) = implb b1 b2 && implb b1 b3. +Proof. + destr_bool. +Qed. + +Lemma implb_orb_distrib_r : forall b1 b2 b3:bool, implb b1 (b2 || b3) = implb b1 b2 || implb b1 b3. +Proof. + destr_bool. +Qed. + +Lemma implb_orb_distrib_l : forall b1 b2 b3:bool, implb (b1 || b2) b3 = implb b1 b3 && implb b2 b3. +Proof. + destr_bool. +Qed. + +(***************************) +(** * Properties of [xorb] *) +(***************************) (** [false] is neutral for [xorb] *) @@ -632,9 +724,9 @@ Proof. Qed. Hint Resolve trans_eq_bool : core. -(*****************************************) +(***************************************) (** * Reflection of [bool] into [Prop] *) -(*****************************************) +(***************************************) (** [Is_true] and equality *) @@ -752,10 +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. @@ -776,11 +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 |