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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** The type [bool] is defined in the prelude as
[Inductive bool : Set := true : bool | false : bool] *)
(** Most of the lemmas in this file are trivial after breaking all booleans *)
Ltac destr_bool :=
intros; destruct_all bool; simpl in *; trivial; try discriminate.
(** Interpretation of booleans as propositions *)
Definition Is_true (b:bool) :=
match b with
| true => True
| false => False
end.
(*******************)
(** * Decidability *)
(*******************)
Lemma bool_dec : forall b1 b2 : bool, {b1 = b2} + {b1 <> b2}.
Proof.
decide equality.
Defined.
(*********************)
(** * Discrimination *)
(*********************)
Lemma diff_true_false : true <> false.
Proof.
discriminate.
Qed.
Hint Resolve diff_true_false : bool.
Lemma diff_false_true : false <> true.
Proof.
discriminate.
Qed.
Hint Resolve diff_false_true : bool.
Hint Extern 1 (false <> true) => exact diff_false_true : core.
Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False.
Proof.
destr_bool.
Qed.
Lemma not_true_is_false : forall b:bool, b <> true -> b = false.
Proof.
destr_bool; intuition.
Qed.
Lemma not_false_is_true : forall b:bool, b <> false -> b = true.
Proof.
destr_bool; intuition.
Qed.
Lemma not_true_iff_false : forall b, b <> true <-> b = false.
Proof.
destr_bool; intuition.
Qed.
Lemma not_false_iff_true : forall b, b <> false <-> b = true.
Proof.
destr_bool; intuition.
Qed.
(**********************)
(** * Order on booleans *)
(**********************)
Definition leb (b1 b2:bool) :=
match b1 with
| true => b2 = true
| false => True
end.
Hint Unfold leb: bool.
Lemma leb_implb : forall b1 b2, leb b1 b2 <-> implb b1 b2 = true.
Proof.
destr_bool; intuition.
Qed.
(* Infix "<=" := leb : bool_scope. *)
(*************)
(** * Equality *)
(*************)
Definition eqb (b1 b2:bool) : bool :=
match b1, b2 with
| true, true => true
| true, false => false
| false, true => false
| false, false => true
end.
Lemma eqb_subst :
forall (P:bool -> Prop) (b1 b2:bool), eqb b1 b2 = true -> P b1 -> P b2.
Proof.
destr_bool.
Qed.
Lemma eqb_reflx : forall b:bool, eqb b b = true.
Proof.
destr_bool.
Qed.
Lemma eqb_prop : forall a b:bool, eqb a b = true -> a = b.
Proof.
destr_bool.
Qed.
Lemma eqb_true_iff : forall a b:bool, eqb a b = true <-> a = b.
Proof.
destr_bool; intuition.
Qed.
Lemma eqb_false_iff : forall a b:bool, eqb a b = false <-> a <> b.
Proof.
destr_bool; intuition.
Qed.
(************************)
(** * A synonym of [if] on [bool] *)
(************************)
Definition ifb (b1 b2 b3:bool) : bool :=
match b1 with
| true => b2
| false => b3
end.
Open Scope bool_scope.
(****************************)
(** * De Morgan laws *)
(****************************)
Lemma negb_orb : forall b1 b2:bool, negb (b1 || b2) = negb b1 && negb b2.
Proof.
destr_bool.
Qed.
Lemma negb_andb : forall b1 b2:bool, negb (b1 && b2) = negb b1 || negb b2.
Proof.
destr_bool.
Qed.
(********************************)
(** * Properties of [negb] *)
(********************************)
Lemma negb_involutive : forall b:bool, negb (negb b) = b.
Proof.
destr_bool.
Qed.
Lemma negb_involutive_reverse : forall b:bool, b = negb (negb b).
Proof.
destr_bool.
Qed.
Notation negb_elim := negb_involutive (only parsing).
Notation negb_intro := negb_involutive_reverse (only parsing).
Lemma negb_sym : forall b b':bool, b' = negb b -> b = negb b'.
Proof.
destr_bool.
Qed.
Lemma no_fixpoint_negb : forall b:bool, negb b <> b.
Proof.
destr_bool.
Qed.
Lemma eqb_negb1 : forall b:bool, eqb (negb b) b = false.
Proof.
destr_bool.
Qed.
Lemma eqb_negb2 : forall b:bool, eqb b (negb b) = false.
Proof.
destr_bool.
Qed.
Lemma if_negb :
forall (A:Type) (b:bool) (x y:A),
(if negb b then x else y) = (if b then y else x).
Proof.
destr_bool.
Qed.
Lemma negb_true_iff : forall b, negb b = true <-> b = false.
Proof.
destr_bool; intuition.
Qed.
Lemma negb_false_iff : forall b, negb b = false <-> b = true.
Proof.
destr_bool; intuition.
Qed.
(********************************)
(** * Properties of [orb] *)
(********************************)
Lemma orb_true_iff :
forall b1 b2, b1 || b2 = true <-> b1 = true \/ b2 = true.
Proof.
destr_bool; intuition.
Qed.
Lemma orb_false_iff :
forall b1 b2, b1 || b2 = false <-> b1 = false /\ b2 = false.
Proof.
destr_bool; intuition.
Qed.
Lemma orb_true_elim :
forall b1 b2:bool, b1 || b2 = true -> {b1 = true} + {b2 = true}.
Proof.
destruct b1; simpl; auto.
Defined.
Lemma orb_prop : forall a b:bool, a || b = true -> a = true \/ b = true.
Proof.
intros; apply orb_true_iff; trivial.
Qed.
Lemma orb_true_intro :
forall b1 b2:bool, b1 = true \/ b2 = true -> b1 || b2 = true.
Proof.
intros; apply orb_true_iff; trivial.
Qed.
Hint Resolve orb_true_intro: bool.
Lemma orb_false_intro :
forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false.
Proof.
intros. subst. reflexivity.
Qed.
Hint Resolve orb_false_intro: bool.
Lemma orb_false_elim :
forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false.
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.
Proof.
destr_bool.
Qed.
Hint Resolve orb_true_r: bool.
Lemma orb_true_l : forall b:bool, true || b = true.
Proof.
reflexivity.
Qed.
Notation orb_b_true := orb_true_r (only parsing).
Notation orb_true_b := orb_true_l (only parsing).
(** [false] is neutral for [orb] *)
Lemma orb_false_r : forall b:bool, b || false = b.
Proof.
destr_bool.
Qed.
Hint Resolve orb_false_r: bool.
Lemma orb_false_l : forall b:bool, false || b = b.
Proof.
destr_bool.
Qed.
Hint Resolve orb_false_l: bool.
Notation orb_b_false := orb_false_r (only parsing).
Notation orb_false_b := orb_false_l (only parsing).
(** Complementation *)
Lemma orb_negb_r : forall b:bool, b || negb b = true.
Proof.
destr_bool.
Qed.
Hint Resolve orb_negb_r: bool.
Notation orb_neg_b := orb_negb_r (only parsing).
(** Commutativity *)
Lemma orb_comm : forall b1 b2:bool, b1 || b2 = b2 || b1.
Proof.
destr_bool.
Qed.
(** Associativity *)
Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3.
Proof.
destr_bool.
Qed.
Hint Resolve orb_comm orb_assoc: bool.
(*******************************)
(** * Properties of [andb] *)
(*******************************)
Lemma andb_true_iff :
forall b1 b2:bool, b1 && b2 = true <-> b1 = true /\ b2 = true.
Proof.
destr_bool; intuition.
Qed.
Lemma andb_false_iff :
forall b1 b2:bool, b1 && b2 = false <-> b1 = false \/ b2 = false.
Proof.
destr_bool; intuition.
Qed.
Lemma andb_true_eq :
forall a b:bool, true = a && b -> true = a /\ true = b.
Proof.
destr_bool. auto.
Defined.
Lemma andb_false_intro1 : forall b1 b2:bool, b1 = false -> b1 && b2 = false.
Proof.
intros. apply andb_false_iff. auto.
Qed.
Lemma andb_false_intro2 : forall b1 b2:bool, b2 = false -> b1 && b2 = false.
Proof.
intros. apply andb_false_iff. auto.
Qed.
(** [false] is a zero for [andb] *)
Lemma andb_false_r : forall b:bool, b && false = false.
Proof.
destr_bool.
Qed.
Lemma andb_false_l : forall b:bool, false && b = false.
Proof.
reflexivity.
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.
Proof.
destr_bool.
Qed.
Lemma andb_true_l : forall b:bool, true && b = b.
Proof.
reflexivity.
Qed.
Notation andb_b_true := andb_true_r (only parsing).
Notation andb_true_b := andb_true_l (only parsing).
Lemma andb_false_elim :
forall b1 b2:bool, b1 && b2 = false -> {b1 = false} + {b2 = false}.
Proof.
destruct b1; simpl; auto.
Defined.
Hint Resolve andb_false_elim: bool.
(** Complementation *)
Lemma andb_negb_r : forall b:bool, b && negb b = false.
Proof.
destr_bool.
Qed.
Hint Resolve andb_negb_r: bool.
Notation andb_neg_b := andb_negb_r (only parsing).
(** Commutativity *)
Lemma andb_comm : forall b1 b2:bool, b1 && b2 = b2 && b1.
Proof.
destr_bool.
Qed.
(** Associativity *)
Lemma andb_assoc : forall b1 b2 b3:bool, b1 && (b2 && b3) = b1 && b2 && b3.
Proof.
destr_bool.
Qed.
Hint Resolve andb_comm andb_assoc: bool.
(*******************************************)
(** * Properties mixing [andb] and [orb] *)
(*******************************************)
(** Distributivity *)
Lemma andb_orb_distrib_r :
forall b1 b2 b3:bool, b1 && (b2 || b3) = b1 && b2 || b1 && b3.
Proof.
destr_bool.
Qed.
Lemma andb_orb_distrib_l :
forall b1 b2 b3:bool, (b1 || b2) && b3 = b1 && b3 || b2 && b3.
Proof.
destr_bool.
Qed.
Lemma orb_andb_distrib_r :
forall b1 b2 b3:bool, b1 || b2 && b3 = (b1 || b2) && (b1 || b3).
Proof.
destr_bool.
Qed.
Lemma orb_andb_distrib_l :
forall b1 b2 b3:bool, b1 && b2 || b3 = (b1 || b3) && (b2 || b3).
Proof.
destr_bool.
Qed.
(* Compatibility *)
Notation demorgan1 := andb_orb_distrib_r (only parsing).
Notation demorgan2 := andb_orb_distrib_l (only parsing).
Notation demorgan3 := orb_andb_distrib_r (only parsing).
Notation demorgan4 := orb_andb_distrib_l (only parsing).
(** Absorption *)
Lemma absorption_andb : forall b1 b2:bool, b1 && (b1 || b2) = b1.
Proof.
destr_bool.
Qed.
Lemma absorption_orb : forall b1 b2:bool, b1 || b1 && b2 = b1.
Proof.
destr_bool.
Qed.
(* begin hide *)
(* Compatibility *)
Notation absoption_andb := absorption_andb (only parsing).
Notation absoption_orb := absorption_orb (only parsing).
(* end hide *)
(*********************************)
(** * Properties of [xorb] *)
(*********************************)
(** [false] is neutral for [xorb] *)
Lemma xorb_false_r : forall b:bool, xorb b false = b.
Proof.
destr_bool.
Qed.
Lemma xorb_false_l : forall b:bool, xorb false b = b.
Proof.
destr_bool.
Qed.
Notation xorb_false := xorb_false_r (only parsing).
Notation false_xorb := xorb_false_l (only parsing).
(** [true] is "complementing" for [xorb] *)
Lemma xorb_true_r : forall b:bool, xorb b true = negb b.
Proof.
reflexivity.
Qed.
Lemma xorb_true_l : forall b:bool, xorb true b = negb b.
Proof.
reflexivity.
Qed.
Notation xorb_true := xorb_true_r (only parsing).
Notation true_xorb := xorb_true_l (only parsing).
(** Nilpotency (alternatively: identity is a inverse for [xorb]) *)
Lemma xorb_nilpotent : forall b:bool, xorb b b = false.
Proof.
destr_bool.
Qed.
(** Commutativity *)
Lemma xorb_comm : forall b b':bool, xorb b b' = xorb b' b.
Proof.
destr_bool.
Qed.
(** Associativity *)
Lemma xorb_assoc_reverse :
forall b b' b'':bool, xorb (xorb b b') b'' = xorb b (xorb b' b'').
Proof.
destr_bool.
Qed.
Notation xorb_assoc := xorb_assoc_reverse (only parsing). (* Compatibility *)
Lemma xorb_eq : forall b b':bool, xorb b b' = false -> b = b'.
Proof.
destr_bool.
Qed.
Lemma xorb_move_l_r_1 :
forall b b' b'':bool, xorb b b' = b'' -> b' = xorb b b''.
Proof.
destr_bool.
Qed.
Lemma xorb_move_l_r_2 :
forall b b' b'':bool, xorb b b' = b'' -> b = xorb b'' b'.
Proof.
destr_bool.
Qed.
Lemma xorb_move_r_l_1 :
forall b b' b'':bool, b = xorb b' b'' -> xorb b' b = b''.
Proof.
destr_bool.
Qed.
Lemma xorb_move_r_l_2 :
forall b b' b'':bool, b = xorb b' b'' -> xorb b b'' = b'.
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).
Proof.
destr_bool; intuition.
Qed.
Lemma eq_true_iff_eq : forall b1 b2, (b1 = true <-> b2 = true) -> b1 = b2.
Proof.
apply eq_iff_eq_true.
Qed.
Notation bool_1 := eq_true_iff_eq (only parsing). (* Compatibility *)
Lemma eq_true_negb_classical : forall b:bool, negb b <> true -> b = true.
Proof.
destr_bool; intuition.
Qed.
Notation bool_3 := eq_true_negb_classical (only parsing). (* Compatibility *)
Lemma eq_true_not_negb : forall b:bool, b <> true -> negb b = true.
Proof.
destr_bool; intuition.
Qed.
Notation bool_6 := eq_true_not_negb (only parsing). (* Compatibility *)
Hint Resolve eq_true_not_negb : bool.
(* An interesting lemma for auto but too strong to keep compatibility *)
Lemma absurd_eq_bool : forall b b':bool, False -> b = b'.
Proof.
contradiction.
Qed.
(* A more specific one that preserves compatibility with old hint bool_3 *)
Lemma absurd_eq_true : forall b, False -> b = true.
Proof.
contradiction.
Qed.
Hint Resolve absurd_eq_true : core.
(* A specific instance of eq_trans that preserves compatibility with
old hint bool_2 *)
Lemma trans_eq_bool : forall x y z:bool, x = y -> y = z -> x = z.
Proof.
apply eq_trans.
Qed.
Hint Resolve trans_eq_bool : core.
(*****************************************)
(** * Reflection of [bool] into [Prop] *)
(*****************************************)
(** [Is_true] and equality *)
Hint Unfold Is_true: bool.
Lemma Is_true_eq_true : forall x:bool, Is_true x -> x = true.
Proof.
destr_bool; tauto.
Qed.
Lemma Is_true_eq_left : forall x:bool, x = true -> Is_true x.
Proof.
intros; subst; auto with bool.
Qed.
Lemma Is_true_eq_right : forall x:bool, true = x -> Is_true x.
Proof.
intros; subst; auto with bool.
Qed.
Notation Is_true_eq_true2 := Is_true_eq_right (only parsing).
Hint Immediate Is_true_eq_right Is_true_eq_left: bool.
Lemma eqb_refl : forall x:bool, Is_true (eqb x x).
Proof.
destr_bool.
Qed.
Lemma eqb_eq : forall x y:bool, Is_true (eqb x y) -> x = y.
Proof.
destr_bool; tauto.
Qed.
(** [Is_true] and connectives *)
Lemma orb_prop_elim :
forall a b:bool, Is_true (a || b) -> Is_true a \/ Is_true b.
Proof.
destr_bool; tauto.
Qed.
Notation orb_prop2 := orb_prop_elim (only parsing).
Lemma orb_prop_intro :
forall a b:bool, Is_true a \/ Is_true b -> Is_true (a || b).
Proof.
destr_bool; tauto.
Qed.
Lemma andb_prop_intro :
forall b1 b2:bool, Is_true b1 /\ Is_true b2 -> Is_true (b1 && b2).
Proof.
destr_bool; tauto.
Qed.
Hint Resolve andb_prop_intro: bool.
Notation andb_true_intro2 :=
(fun b1 b2 H1 H2 => andb_prop_intro b1 b2 (conj H1 H2))
(only parsing).
Lemma andb_prop_elim :
forall a b:bool, Is_true (a && b) -> Is_true a /\ Is_true b.
Proof.
destr_bool; auto.
Qed.
Hint Resolve andb_prop_elim: bool.
Notation andb_prop2 := andb_prop_elim (only parsing).
Lemma eq_bool_prop_intro :
forall b1 b2, (Is_true b1 <-> Is_true b2) -> b1 = b2.
Proof.
destr_bool; tauto.
Qed.
Lemma eq_bool_prop_elim : forall b1 b2, b1 = b2 -> (Is_true b1 <-> Is_true b2).
Proof.
destr_bool; tauto.
Qed.
Lemma negb_prop_elim : forall b, Is_true (negb b) -> ~ Is_true b.
Proof.
destr_bool; tauto.
Qed.
Lemma negb_prop_intro : forall b, ~ Is_true b -> Is_true (negb b).
Proof.
destr_bool; tauto.
Qed.
Lemma negb_prop_classical : forall b, ~ Is_true (negb b) -> Is_true b.
Proof.
destr_bool; tauto.
Qed.
Lemma negb_prop_involutive : forall b, Is_true b -> ~ Is_true (negb b).
Proof.
destr_bool; tauto.
Qed.
(** Rewrite rules about andb, orb and if (used in romega) *)
Lemma andb_if : forall (A:Type)(a a':A)(b b' : bool),
(if b && b' then a else a') =
(if b then if b' then a else a' else a').
Proof.
destr_bool.
Qed.
Lemma negb_if : forall (A:Type)(a a':A)(b:bool),
(if negb b then a else a') =
(if b then a' else a).
Proof.
destr_bool.
Qed.
(*****************************************)
(** * Alternative versions of [andb] and [orb]
with lazy behavior (for vm_compute) *)
(*****************************************)
Declare Scope lazy_bool_scope.
Notation "a &&& b" := (if a then b else false)
(at level 40, left associativity) : lazy_bool_scope.
Notation "a ||| b" := (if a then true else b)
(at level 50, left associativity) : lazy_bool_scope.
Local Open Scope lazy_bool_scope.
Lemma andb_lazy_alt : forall a b : bool, a && b = a &&& b.
Proof.
reflexivity.
Qed.
Lemma orb_lazy_alt : forall a b : bool, a || b = a ||| b.
Proof.
reflexivity.
Qed.
(*****************************************)
(** * Reflect: a specialized inductive type for
relating propositions and booleans,
as popularized by the Ssreflect library. *)
(*****************************************)
Inductive reflect (P : Prop) : bool -> Set :=
| ReflectT : P -> reflect P true
| ReflectF : ~ P -> reflect P false.
Hint Constructors reflect : bool.
(** Interest: a case on a reflect lemma or hyp performs clever
unification, and leave the goal in a convenient shape
(a bit like case_eq). *)
(** Relation with iff : *)
Lemma reflect_iff : forall P b, reflect P b -> (P<->b=true).
Proof.
destruct 1; intuition; discriminate.
Qed.
Lemma iff_reflect : forall P b, (P<->b=true) -> reflect P b.
Proof.
destr_bool; intuition.
Defined.
(** It would be nice to join [reflect_iff] and [iff_reflect]
in a unique [iff] statement, but this isn't allowed since
[iff] is in Prop. *)
(** Reflect implies decidability of the proposition *)
Lemma reflect_dec : forall P b, reflect P b -> {P}+{~P}.
Proof.
destruct 1; auto.
Defined.
(** Reciprocally, from a decidability, we could state a
[reflect] as soon as we have a [bool_of_sumbool]. *)
(** For instance, we could state the correctness of [Bool.eqb] via [reflect]: *)
Lemma eqb_spec (b b' : bool) : reflect (b = b') (eqb b b').
Proof.
destruct b, b'; now constructor.
Defined.
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