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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)         *)
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(** 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.
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(** * Decidability *)
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Hint Resolve diff_true_false : bool.
Hint Resolve diff_false_true : bool.
Hint Extern 1 (false <> true) => exact diff_false_true : core.
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(** * Order on booleans *)
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Definition leb (b1 b2:bool) :=
match b1 with
| true => b2 = true
| false => True
end.
Hint Unfold leb: bool.
(* Infix "<=" := leb : bool_scope. *)
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(** * Equality *)
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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.