# Library Coq.Bool.Bool

``` ```
 Booleans
``` ```
 The type `bool` is defined in the prelude as `Inductive bool : Set := true : bool | false : bool`
``` ```
 Interpretation of booleans as Proposition
``` Definition Is_true (b:bool) :=   match b with   | true => True   | false => False   end. Hint Unfold Is_true: bool. Lemma Is_true_eq_left : forall x:bool, x = true -> Is_true x. Proof.   intros; rewrite H; auto with bool. Qed. Lemma Is_true_eq_right : forall x:bool, true = x -> Is_true x. Proof.   intros; rewrite <- H; auto with bool. Qed. Hint Immediate Is_true_eq_right Is_true_eq_left: bool. ```
 Discrimination
``` Lemma diff_true_false : true <> false. Proof. unfold not in |- *; intro contr; change (Is_true false) in |- *. elim contr; simpl in |- *; trivial with bool. Qed. Hint Resolve diff_true_false: bool v62. Lemma diff_false_true : false <> true. Proof. red in |- *; intros H; apply diff_true_false. symmetry in |- *. assumption. Qed. Hint Resolve diff_false_true: bool v62. Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False. intros b H; rewrite H; auto with bool. Qed. Hint Resolve eq_true_false_abs: bool. Lemma not_true_is_false : forall b:bool, b <> true -> b = false. destruct b. intros. red in H; elim H. reflexivity. intros abs. reflexivity. Qed. Lemma not_false_is_true : forall b:bool, b <> false -> b = true. destruct b. intros. reflexivity. intro H; red in H; elim H. reflexivity. Qed. ```
 Order on booleans
``` Definition leb (b1 b2:bool) :=   match b1 with   | true => b2 = true   | false => True   end. Hint Unfold leb: bool v62. ```
 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_refl : forall x:bool, Is_true (eqb x x). destruct x; simpl in |- *; auto with bool. Qed. Lemma eqb_eq : forall x y:bool, Is_true (eqb x y) -> x = y. destruct x; destruct y; simpl in |- *; tauto. Qed. Lemma Is_true_eq_true : forall x:bool, Is_true x -> x = true. destruct x; simpl in |- *; tauto. Qed.   Lemma Is_true_eq_true2 : forall x:bool, x = true -> Is_true x. destruct x; simpl in |- *; auto with bool. Qed. Lemma eqb_subst :  forall (P:bool -> Prop) (b1 b2:bool), eqb b1 b2 = true -> P b1 -> P b2. unfold eqb in |- *. intros P b1. intros b2. case b1. case b2. trivial with bool. intros H. inversion_clear H. case b2. intros H. inversion_clear H. trivial with bool. Qed. Lemma eqb_reflx : forall b:bool, eqb b b = true. intro b. case b. trivial with bool. trivial with bool. Qed. Lemma eqb_prop : forall a b:bool, eqb a b = true -> a = b. destruct a; destruct b; simpl in |- *; intro; discriminate H || reflexivity. Qed. ```
 Logical combinators
```   Definition ifb (b1 b2 b3:bool) : bool :=   match b1 with   | true => b2   | false => b3   end. Definition andb (b1 b2:bool) : bool := ifb b1 b2 false. Definition orb (b1 b2:bool) : bool := ifb b1 true b2. Definition implb (b1 b2:bool) : bool := ifb b1 b2 true. Definition xorb (b1 b2:bool) : bool :=   match b1, b2 with   | true, true => false   | true, false => true   | false, true => true   | false, false => false   end. Definition negb (b:bool) := match b with                             | true => false                             | false => true                             end. Infix "||" := orb (at level 50, left associativity) : bool_scope. Infix "&&" := andb (at level 40, left associativity) : bool_scope. Open Scope bool_scope. Delimit Scope bool_scope with bool. Bind Scope bool_scope with bool. ```
 Lemmas about `negb`
``` Lemma negb_intro : forall b:bool, b = negb (negb b). Proof. destruct b; reflexivity. Qed. Lemma negb_elim : forall b:bool, negb (negb b) = b. Proof. destruct b; reflexivity. Qed.         Lemma negb_orb : forall b1 b2:bool, negb (b1 || b2) = negb b1 && negb b2. Proof.   destruct b1; destruct b2; simpl in |- *; reflexivity. Qed. Lemma negb_andb : forall b1 b2:bool, negb (b1 && b2) = negb b1 || negb b2. Proof.   destruct b1; destruct b2; simpl in |- *; reflexivity. Qed. Lemma negb_sym : forall b b':bool, b' = negb b -> b = negb b'. Proof. destruct b; destruct b'; intros; simpl in |- *; trivial with bool. Qed. Lemma no_fixpoint_negb : forall b:bool, negb b <> b. Proof. destruct b; simpl in |- *; intro; apply diff_true_false;  auto with bool. Qed. Lemma eqb_negb1 : forall b:bool, eqb (negb b) b = false. destruct b. trivial with bool. trivial with bool. Qed.   Lemma eqb_negb2 : forall b:bool, eqb b (negb b) = false. destruct b. trivial with bool. trivial with bool. Qed. Lemma if_negb :  forall (A:Set) (b:bool) (x y:A),    (if negb b then x else y) = (if b then y else x). Proof.   destruct b; trivial. Qed. ```
 A few lemmas about `or`
``` Lemma orb_prop : forall a b:bool, a || b = true -> a = true \/ b = true. destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);  auto with bool. Qed. Lemma orb_prop2 : forall a b:bool, Is_true (a || b) -> Is_true a \/ Is_true b. destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);  auto with bool. Qed. Lemma orb_true_intro :  forall b1 b2:bool, b1 = true \/ b2 = true -> b1 || b2 = true. destruct b1; auto with bool. destruct 1; intros. elim diff_true_false; auto with bool. rewrite H; trivial with bool. Qed. Hint Resolve orb_true_intro: bool v62. Lemma orb_b_true : forall b:bool, b || true = true. auto with bool. Qed. Hint Resolve orb_b_true: bool v62. Lemma orb_true_b : forall b:bool, true || b = true. trivial with bool. Qed. Definition orb_true_elim :   forall b1 b2:bool, b1 || b2 = true -> {b1 = true} + {b2 = true}. destruct b1; simpl in |- *; auto with bool. Defined. Lemma orb_false_intro :  forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false. intros b1 b2 H1 H2; rewrite H1; rewrite H2; trivial with bool. Qed. Hint Resolve orb_false_intro: bool v62. Lemma orb_b_false : forall b:bool, b || false = b. Proof.   destruct b; trivial with bool. Qed. Hint Resolve orb_b_false: bool v62. Lemma orb_false_b : forall b:bool, false || b = b. Proof.   destruct b; trivial with bool. Qed. Hint Resolve orb_false_b: bool v62. Lemma orb_false_elim :  forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false. Proof.   destruct b1.   intros; elim diff_true_false; auto with bool.   destruct b2.   intros; elim diff_true_false; auto with bool.   auto with bool. Qed. Lemma orb_neg_b : forall b:bool, b || negb b = true. Proof.   destruct b; reflexivity. Qed. Hint Resolve orb_neg_b: bool v62. Lemma orb_comm : forall b1 b2:bool, b1 || b2 = b2 || b1. destruct b1; destruct b2; reflexivity. Qed. Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3. Proof.   destruct b1; destruct b2; destruct b3; reflexivity. Qed. Hint Resolve orb_comm orb_assoc orb_b_false orb_false_b: bool v62. ```
 A few lemmas about `and`
``` Lemma andb_prop : forall a b:bool, a && b = true -> a = true /\ b = true. Proof.   destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);    auto with bool. Qed. Hint Resolve andb_prop: bool v62. Definition andb_true_eq :   forall a b:bool, true = a && b -> true = a /\ true = b. Proof.   destruct a; destruct b; auto. Defined. Lemma andb_prop2 :  forall a b:bool, Is_true (a && b) -> Is_true a /\ Is_true b. Proof.   destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);    auto with bool. Qed. Hint Resolve andb_prop2: bool v62. Lemma andb_true_intro :  forall b1 b2:bool, b1 = true /\ b2 = true -> b1 && b2 = true. Proof.   destruct b1; destruct b2; simpl in |- *; tauto || auto with bool. Qed. Hint Resolve andb_true_intro: bool v62. Lemma andb_true_intro2 :  forall b1 b2:bool, Is_true b1 -> Is_true b2 -> Is_true (b1 && b2). Proof.   destruct b1; destruct b2; simpl in |- *; tauto. Qed. Hint Resolve andb_true_intro2: bool v62. Lemma andb_false_intro1 : forall b1 b2:bool, b1 = false -> b1 && b2 = false. destruct b1; destruct b2; simpl in |- *; tauto || auto with bool. Qed. Lemma andb_false_intro2 : forall b1 b2:bool, b2 = false -> b1 && b2 = false. destruct b1; destruct b2; simpl in |- *; tauto || auto with bool. Qed. Lemma andb_b_false : forall b:bool, b && false = false. destruct b; auto with bool. Qed. Lemma andb_false_b : forall b:bool, false && b = false. trivial with bool. Qed. Lemma andb_b_true : forall b:bool, b && true = b. destruct b; auto with bool. Qed. Lemma andb_true_b : forall b:bool, true && b = b. trivial with bool. Qed. Definition andb_false_elim :   forall b1 b2:bool, b1 && b2 = false -> {b1 = false} + {b2 = false}. destruct b1; simpl in |- *; auto with bool. Defined. Hint Resolve andb_false_elim: bool v62. Lemma andb_neg_b : forall b:bool, b && negb b = false. destruct b; reflexivity. Qed. Hint Resolve andb_neg_b: bool v62. Lemma andb_comm : forall b1 b2:bool, b1 && b2 = b2 && b1. destruct b1; destruct b2; reflexivity. Qed. Lemma andb_assoc : forall b1 b2 b3:bool, b1 && (b2 && b3) = b1 && b2 && b3. destruct b1; destruct b2; destruct b3; reflexivity. Qed. Hint Resolve andb_comm andb_assoc: bool v62. ```
 Properties of `xorb`
``` Lemma xorb_false : forall b:bool, xorb b false = b. Proof.   destruct b; trivial. Qed. Lemma false_xorb : forall b:bool, xorb false b = b. Proof.   destruct b; trivial. Qed. Lemma xorb_true : forall b:bool, xorb b true = negb b. Proof.   trivial. Qed. Lemma true_xorb : forall b:bool, xorb true b = negb b. Proof.   destruct b; trivial. Qed. Lemma xorb_nilpotent : forall b:bool, xorb b b = false. Proof.   destruct b; trivial. Qed. Lemma xorb_comm : forall b b':bool, xorb b b' = xorb b' b. Proof.   destruct b; destruct b'; trivial. Qed. Lemma xorb_assoc :  forall b b' b'':bool, xorb (xorb b b') b'' = xorb b (xorb b' b''). Proof.   destruct b; destruct b'; destruct b''; trivial. Qed. Lemma xorb_eq : forall b b':bool, xorb b b' = false -> b = b'. Proof.   destruct b; destruct b'; trivial.   unfold xorb in |- *. intros. rewrite H. reflexivity. Qed. Lemma xorb_move_l_r_1 :  forall b b' b'':bool, xorb b b' = b'' -> b' = xorb b b''. Proof.   intros. rewrite <- (false_xorb b'). rewrite <- (xorb_nilpotent b). rewrite xorb_assoc.   rewrite H. reflexivity. Qed. Lemma xorb_move_l_r_2 :  forall b b' b'':bool, xorb b b' = b'' -> b = xorb b'' b'. Proof.   intros. rewrite xorb_comm in H. rewrite (xorb_move_l_r_1 b' b b'' H). apply xorb_comm. Qed. Lemma xorb_move_r_l_1 :  forall b b' b'':bool, b = xorb b' b'' -> xorb b' b = b''. Proof.   intros. rewrite H. rewrite <- xorb_assoc. rewrite xorb_nilpotent. apply false_xorb. Qed. Lemma xorb_move_r_l_2 :  forall b b' b'':bool, b = xorb b' b'' -> xorb b b'' = b'. Proof.   intros. rewrite H. rewrite xorb_assoc. rewrite xorb_nilpotent. apply xorb_false. Qed. ```
 De Morgan's law
``` Lemma demorgan1 :  forall b1 b2 b3:bool, b1 && (b2 || b3) = b1 && b2 || b1 && b3. destruct b1; destruct b2; destruct b3; reflexivity. Qed. Lemma demorgan2 :  forall b1 b2 b3:bool, (b1 || b2) && b3 = b1 && b3 || b2 && b3. destruct b1; destruct b2; destruct b3; reflexivity. Qed. Lemma demorgan3 :  forall b1 b2 b3:bool, b1 || b2 && b3 = (b1 || b2) && (b1 || b3). destruct b1; destruct b2; destruct b3; reflexivity. Qed. Lemma demorgan4 :  forall b1 b2 b3:bool, b1 && b2 || b3 = (b1 || b3) && (b2 || b3). destruct b1; destruct b2; destruct b3; reflexivity. Qed. Lemma absoption_andb : forall b1 b2:bool, b1 && (b1 || b2) = b1. Proof.   destruct b1; destruct b2; simpl in |- *; reflexivity. Qed. Lemma absoption_orb : forall b1 b2:bool, b1 || b1 && b2 = b1. Proof.   destruct b1; destruct b2; simpl in |- *; reflexivity. Qed. ```
 Misc. equalities between booleans (to be used by Auto)
``` Lemma bool_1 : forall b1 b2:bool, (b1 = true <-> b2 = true) -> b1 = b2. Proof.  intros b1 b2; case b1; case b2; intuition. Qed. Lemma bool_2 : forall b1 b2:bool, b1 = b2 -> b1 = true -> b2 = true. Proof.  intros b1 b2; case b1; case b2; intuition. Qed. Lemma bool_3 : forall b:bool, negb b <> true -> b = true. Proof.   destruct b; intuition. Qed. Lemma bool_4 : forall b:bool, b = true -> negb b <> true. Proof.   destruct b; intuition. Qed. Lemma bool_5 : forall b:bool, negb b = true -> b <> true. Proof.   destruct b; intuition. Qed. Lemma bool_6 : forall b:bool, b <> true -> negb b = true. Proof.   destruct b; intuition. Qed. Hint Resolve bool_1 bool_2 bool_3 bool_4 bool_5 bool_6. ```
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