Library Stdlib.Bool.Bool


From Stdlib Require Import DecidableClass HLevelsBase.

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 by case analysis

Local Ltac Tauto.intuition_solver ::= auto with bool.

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.

A proof of the resulting propositions can be wrapped to make it reduce whenever the boolean is true, even if the original proof is sealed.

Definition transparent_Is_true' (b : bool) : (True -> Is_true b) -> Is_true b :=
  match b with
  | true => fun _ => I
  | false => fun H => False_rect _ (H I)
  end.

Notation transparent_Is_true b pf := (transparent_Is_true' b (fun _ => pf)).

Example transparent_Is_true_true pf : transparent_Is_true' true pf = I
  := eq_refl.
This is useful for efficiently implementing subset types in cases where the subset-membership check is cheap enough to be performed after every operation, instead of keeping around (a closure for) the validity proof. Thunking (True ->) is to avoid cbv eagerly computing the proof.
For Example: 
#[projections(primitive)]
Record bounded_nat (m : nat) : Set := of_nat_value {
  to_nat : nat ;
  to_nat_range' : Is_true (Nat.ltb to_nat m) }.

Axiom three_less_than_seven : Is_true (Nat.ltb 3 7).
Example three_two_ways :
  of_nat_value 7 3 (transparent_Is_true _ three_less_than_seven) =
  of_nat_value 7 3 I.
Proof. compute. reflexivity. Qed.
For advanced dependently-typed use, equalities between these proofs can in turn be proven with a transparent and easy-to-normalize lemma:

Definition Is_true_hprop b : IsHProp (Is_true b) :=
  match b with
  | true => true_hprop
  | false => false_hprop
  end.

The previous proof is based on the encode-decode method of HoTT; it is a special case of the more general result that decidable equality implies IsHSet (originally Hedberg's theorem).

Decidability


Lemma bool_dec : forall b1 b2 : bool, {b1 = b2} + {b1 <> b2}.

Discrimination


Lemma diff_true_false : true <> false.
#[global]
Hint Resolve diff_true_false : bool.

Lemma diff_false_true : false <> true.
#[global]
Hint Resolve diff_false_true : bool.
#[global]
Hint Extern 1 (false <> true) => exact diff_false_true : core.

Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False.

Lemma not_true_is_false : forall b:bool, b <> true -> b = false.

Lemma not_false_is_true : forall b:bool, b <> false -> b = true.

Lemma not_true_iff_false : forall b, b <> true <-> b = false.

Lemma not_false_iff_true : forall b, b <> false <-> b = true.

Order on booleans


#[ local ] Definition le (b1 b2:bool) :=
  match b1 with
    | true => b2 = true
    | false => True
  end.
#[global]
Hint Unfold le: bool.

Lemma le_implb : forall b1 b2, le b1 b2 <-> implb b1 b2 = true.

#[ local ] Definition lt (b1 b2:bool) :=
  match b1 with
    | true => False
    | false => b2 = true
  end.
#[global]
Hint Unfold lt: bool.

#[ local ] Definition compare (b1 b2 : bool) :=
  match b1, b2 with
   | false, true => Lt
   | true, false => Gt
   | _, _ => Eq
  end.

Lemma compare_spec : forall b1 b2,
  CompareSpec (b1 = b2) (lt b1 b2) (lt b2 b1) (compare b1 b2).

Equality


Definition eqb (b1 b2:bool) : bool :=
  match b1, b2 with
    | true, true => true
    | true, false => false
    | false, true => false
    | false, false => true
  end.

Register eqb as core.bool.eqb.

Lemma eqb_subst :
  forall (P:bool -> Prop) (b1 b2:bool), eqb b1 b2 = true -> P b1 -> P b2.

Lemma eqb_reflx : forall b:bool, eqb b b = true.

Lemma eqb_prop : forall a b:bool, eqb a b = true -> a = b.

Lemma eqb_true_iff : forall a b:bool, eqb a b = true <-> a = b.

#[global]
Instance Decidable_eq_bool : forall (x y : bool), Decidable (eq x y) := {
  Decidable_spec := eqb_true_iff x y
}.

Lemma eqb_false_iff : forall a b:bool, eqb a b = false <-> a <> b.

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.

Lemma negb_andb : forall b1 b2:bool, negb (b1 && b2) = negb b1 || negb b2.

Properties of negb


Lemma negb_involutive : forall b:bool, negb (negb b) = b.

Lemma negb_involutive_reverse : forall b:bool, b = negb (negb b).

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'.

Lemma no_fixpoint_negb : forall b:bool, negb b <> b.

Lemma eqb_negb1 : forall b:bool, eqb (negb b) b = false.

Lemma eqb_negb2 : forall b:bool, eqb b (negb b) = false.

Lemma if_negb :
  forall (A:Type) (b:bool) (x y:A),
    (if negb b then x else y) = (if b then y else x).

Lemma negb_true_iff : forall b, negb b = true <-> b = false.

Lemma negb_false_iff : forall b, negb b = false <-> b = true.

Properties of orb


Lemma orb_true_iff :
  forall b1 b2, b1 || b2 = true <-> b1 = true \/ b2 = true.

Lemma orb_false_iff :
  forall b1 b2, b1 || b2 = false <-> b1 = false /\ b2 = false.

Lemma orb_true_elim :
  forall b1 b2:bool, b1 || b2 = true -> {b1 = true} + {b2 = true}.

Lemma orb_prop : forall a b:bool, a || b = true -> a = true \/ b = true.

Lemma orb_true_intro :
  forall b1 b2:bool, b1 = true \/ b2 = true -> b1 || b2 = true.
#[global]
Hint Resolve orb_true_intro: bool.

Lemma orb_false_intro :
  forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false.
#[global]
Hint Resolve orb_false_intro: bool.

Lemma orb_false_elim :
  forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false.

Lemma orb_diag : forall b, b || b = b.

true is a zero for orb

Lemma orb_true_r : forall b:bool, b || true = true.
#[global]
Hint Resolve orb_true_r: bool.

Lemma orb_true_l : forall b:bool, true || b = true.

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.
#[global]
Hint Resolve orb_false_r: bool.

Lemma orb_false_l : forall b:bool, false || b = b.
#[global]
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.
#[global]
Hint Resolve orb_negb_r: bool.

Lemma orb_negb_l : forall b:bool, negb b || b = true.

Notation orb_neg_b := orb_negb_r (only parsing).

Commutativity

Lemma orb_comm : forall b1 b2:bool, b1 || b2 = b2 || b1.

Associativity

Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3.
#[global]
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.

Lemma andb_false_iff :
  forall b1 b2:bool, b1 && b2 = false <-> b1 = false \/ b2 = false.

Lemma andb_true_eq :
  forall a b:bool, true = a && b -> true = a /\ true = b.

Lemma andb_false_intro1 : forall b1 b2:bool, b1 = false -> b1 && b2 = false.

Lemma andb_false_intro2 : forall b1 b2:bool, b2 = false -> b1 && b2 = false.

false is a zero for andb

Lemma andb_false_r : forall b:bool, b && false = false.

Lemma andb_false_l : forall b:bool, false && b = false.

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.

true is neutral for andb

Lemma andb_true_r : forall b:bool, b && true = b.

Lemma andb_true_l : forall b:bool, true && b = b.

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}.
#[global]
Hint Resolve andb_false_elim: bool.

Complementation

Lemma andb_negb_r : forall b:bool, b && negb b = false.
#[global]
Hint Resolve andb_negb_r: bool.

Lemma andb_negb_l : forall b:bool, negb b && b = false.

Notation andb_neg_b := andb_negb_r (only parsing).

Commutativity

Lemma andb_comm : forall b1 b2:bool, b1 && b2 = b2 && b1.

Associativity

Lemma andb_assoc : forall b1 b2 b3:bool, b1 && (b2 && b3) = b1 && b2 && b3.

#[global]
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.

Lemma andb_orb_distrib_l :
 forall b1 b2 b3:bool, (b1 || b2) && b3 = b1 && b3 || b2 && b3.

Lemma orb_andb_distrib_r :
  forall b1 b2 b3:bool, b1 || b2 && b3 = (b1 || b2) && (b1 || b3).

Lemma orb_andb_distrib_l :
  forall b1 b2 b3:bool, b1 && b2 || b3 = (b1 || b3) && (b2 || b3).

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.

Lemma absorption_orb : forall b1 b2:bool, b1 || b1 && b2 = b1.


Properties of implb


Lemma implb_true_iff : forall b1 b2:bool, implb b1 b2 = true <-> (b1 = true -> b2 = true).

Lemma implb_false_iff : forall b1 b2:bool, implb b1 b2 = false <-> (b1 = true /\ b2 = false).

Lemma implb_orb : forall b1 b2:bool, implb b1 b2 = negb b1 || b2.

Lemma implb_negb_orb : forall b1 b2:bool, implb (negb b1) b2 = b1 || b2.

Lemma implb_true_r : forall b:bool, implb b true = true.

Lemma implb_false_r : forall b:bool, implb b false = negb b.

Lemma implb_true_l : forall b:bool, implb true b = b.

Lemma implb_false_l : forall b:bool, implb false b = true.

Lemma implb_same : forall b:bool, implb b b = true.

Lemma implb_contrapositive : forall b1 b2:bool, implb (negb b1) (negb b2) = implb b2 b1.

Lemma implb_negb : forall b1 b2:bool, implb (negb b1) b2 = implb (negb b2) b1.

Lemma implb_curry : forall b1 b2 b3:bool, implb (b1 && b2) b3 = implb b1 (implb b2 b3).

Lemma implb_andb_distrib_r : forall b1 b2 b3:bool, implb b1 (b2 && b3) = implb b1 b2 && implb b1 b3.

Lemma implb_orb_distrib_r : forall b1 b2 b3:bool, implb b1 (b2 || b3) = implb b1 b2 || implb b1 b3.

Lemma implb_orb_distrib_l : forall b1 b2 b3:bool, implb (b1 || b2) b3 = implb b1 b3 && implb b2 b3.

Properties of xorb

false is neutral for xorb

Lemma xorb_false_r : forall b:bool, xorb b false = b.

Lemma xorb_false_l : forall b:bool, xorb false b = b.

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.

Lemma xorb_true_l : forall b:bool, xorb true b = negb b.

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.

Commutativity

Lemma xorb_comm : forall b b':bool, xorb b b' = xorb b' b.

Associativity

Lemma xorb_assoc_reverse :
  forall b b' b'':bool, xorb (xorb b b') b'' = xorb b (xorb b' b'').

Notation xorb_assoc := xorb_assoc_reverse (only parsing).
Lemma xorb_eq : forall b b':bool, xorb b b' = false -> b = b'.

Lemma xorb_move_l_r_1 :
  forall b b' b'':bool, xorb b b' = b'' -> b' = xorb b b''.

Lemma xorb_move_l_r_2 :
  forall b b' b'':bool, xorb b b' = b'' -> b = xorb b'' b'.

Lemma xorb_move_r_l_1 :
  forall b b' b'':bool, b = xorb b' b'' -> xorb b' b = b''.

Lemma xorb_move_r_l_2 :
  forall b b' b'':bool, b = xorb b' b'' -> xorb b b'' = b'.

Lemma negb_xorb a b : negb (xorb a b) = Bool.eqb a b.

Lemma negb_xorb_l : forall b b', negb (xorb b b') = xorb (negb b) b'.

Lemma negb_xorb_r : forall b b', negb (xorb b b') = xorb b (negb b').

Lemma xorb_negb_negb : forall b b', xorb (negb b) (negb b') = xorb b b'.

Lemmas about the b = true embedding of bool to Prop

Definition not_eqtrue_false : not (eq_true false) :=
  fun H => match H with end.

Definition transparent_eq_true' b : (True -> eq_true b) -> eq_true b :=
  match b with
  | true => fun _ => is_eq_true
  | _ => fun H => False_rect _ (not_eqtrue_false (H I))
  end.

Notation transparent_eq_true b pf := (transparent_eq_true' b (fun _ => pf)).

Definition eq_any_transparent_eq_true b pf' : forall pf, transparent_eq_true b pf = pf'
  := match pf' with is_eq_true => fun _ => eq_refl end.

Definition eq_true_hprop b : IsHProp (eq_true b) :=
  match b return IsHProp (eq_true b) with
  | true => fun p q =>
    eq_trans
      (eq_sym (eq_any_transparent_eq_true _ p is_eq_true))
      (eq_any_transparent_eq_true _ q is_eq_true)
  | false => fun H => False_rect _ (not_eqtrue_false H)
  end.

Lemma eq_iff_eq_true : forall b1 b2, b1 = b2 <-> (b1 = true <-> b2 = true).

Lemma eq_true_iff_eq : forall b1 b2, (b1 = true <-> b2 = true) -> b1 = b2.

Notation bool_1 := eq_true_iff_eq (only parsing).
Lemma eq_true_negb_classical : forall b:bool, negb b <> true -> b = true.

Notation bool_3 := eq_true_negb_classical (only parsing).
Lemma eq_true_negb_classical_iff : forall b:bool, negb b <> true <-> b = true.

Lemma eq_true_not_negb : forall b:bool, b <> true -> negb b = true.

Lemma eq_true_not_negb_iff : forall b:bool, b <> true <-> negb b = true.

Notation bool_6 := eq_true_not_negb (only parsing).
#[global]
Hint Resolve eq_true_not_negb : bool.


Lemma absurd_eq_bool : forall b b':bool, False -> b = b'.


Lemma absurd_eq_true : forall b, False -> b = true.
#[global]
Hint Resolve absurd_eq_true : core.


Lemma trans_eq_bool : forall x y z:bool, x = y -> y = z -> x = z.
#[global]
Hint Resolve trans_eq_bool : core.

Reflection of bool into Prop

Is_true and equality

#[global]
Hint Unfold Is_true: bool.

Lemma Is_true_eq_true : forall x:bool, Is_true x -> x = true.

Lemma Is_true_eq_left : forall x:bool, x = true -> Is_true x.

Lemma Is_true_eq_right : forall x:bool, true = x -> Is_true x.

Notation Is_true_eq_true2 := Is_true_eq_right (only parsing).

#[global]
Hint Immediate Is_true_eq_right Is_true_eq_left: bool.

Lemma eqb_refl : forall x:bool, Is_true (eqb x x).

Lemma eqb_eq : forall x y:bool, Is_true (eqb x y) -> x = y.

Is_true and connectives

Lemma orb_prop_elim :
  forall a b:bool, Is_true (a || b) -> Is_true a \/ Is_true b.

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).

Lemma andb_prop_intro :
  forall b1 b2:bool, Is_true b1 /\ Is_true b2 -> Is_true (b1 && b2).
#[global]
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.
#[global]
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.

Lemma eq_bool_prop_elim : forall b1 b2, b1 = b2 -> (Is_true b1 <-> Is_true b2).

Lemma negb_prop_elim : forall b, Is_true (negb b) -> ~ Is_true b.

Lemma negb_prop_intro : forall b, ~ Is_true b -> Is_true (negb b).

Lemma negb_prop_classical : forall b, ~ Is_true (negb b) -> Is_true b.

Lemma negb_prop_involutive : forall b, Is_true b -> ~ Is_true (negb b).

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').

Lemma negb_if : forall (A:Type)(a a':A)(b:bool),
 (if negb b then a else a') =
 (if b then a' else a).

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.

Lemma orb_lazy_alt : forall a b : bool, a || b = a ||| b.

Reflect: a specialized inductive type for

relating propositions and booleans, as popularized by the Ssreflect library.

Notation reflect := Datatypes.reflect (only parsing).
Notation ReflectT := Datatypes.ReflectT (only parsing).
Notation ReflectF := Datatypes.ReflectF (only parsing).

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).

Lemma iff_reflect : forall P b, (P<->b=true) -> reflect P b.

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}.

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').

Notations
Module BoolNotations.
Infix "<=" := le : bool_scope.
Infix "<" := lt : bool_scope.
Infix "?=" := compare (at level 70) : bool_scope.
Infix "=?" := eqb (at level 70) : bool_scope.
End BoolNotations.