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(*
* The Why certification tool
* Copyright (C) 2002 Jean-Christophe FILLIATRE
*
* This software is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public
* License version 2, as published by the Free Software Foundation.
*
* This software is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
*
* See the GNU General Public License version 2 for more details
* (enclosed in the file GPL).
*)
(* $Id: WhyBool.v,v 1.13 2008/03/17 09:30:05 filliatr Exp $ *)
Require Import ZArith.
Require Import Sumbool.
Require Export Bool_nat.
Require Export Zwf.
Definition annot_bool :
forall b:bool, {b' : bool | if b' then b = true else b = false}.
intro b.
exists b.
case b; trivial.
Qed.
Definition if_then_else (A:Set) (a:bool) (b c:A) := if a then b else c.
Implicit Arguments if_then_else.
(* Logical connectives *)
Definition spec_and (A B C D:Prop) (b:bool) :=
if b then A /\ C else B \/ D.
Definition prog_bool_and :
forall Q1 Q2:bool -> Prop,
sig Q1 ->
sig Q2 ->
{b : bool | if b then Q1 true /\ Q2 true else Q1 false \/ Q2 false}.
intros Q1 Q2 H1 H2.
elim H1.
intro b1.
elim H2.
intro b2.
case b1; case b2; intros.
exists true; auto.
exists false; auto.
exists false; auto.
exists false; auto.
Qed.
Definition spec_or (A B C D:Prop) (b:bool) :=
if b then A \/ C else B /\ D.
Definition prog_bool_or :
forall Q1 Q2:bool -> Prop,
sig Q1 ->
sig Q2 ->
{b : bool | if b then Q1 true \/ Q2 true else Q1 false /\ Q2 false}.
intros Q1 Q2 H1 H2.
elim H1.
intro b1.
elim H2.
intro b2.
case b1; case b2; intros.
exists true; auto.
exists true; auto.
exists true; auto.
exists false; auto.
Qed.
Definition spec_not (A B:Prop) (b:bool) := if b then B else A.
Definition prog_bool_not :
forall Q:bool -> Prop,
sig Q -> {b : bool | if b then Q false else Q true}.
intros Q H.
elim H.
intro b.
case b; intro.
exists false; auto.
exists true; auto.
Qed.
(** Conversely, we build a [sumbool] out of a boolean,
for the need of validations *)
Definition btest (q:bool -> Prop) (b:bool) (p:q b) :
{q true} + {q false} :=
match b return q b -> {q true} + {q false} with
| true => fun p => @left _ (q false) p
| false => fun p => @right _ (q false) p
end p.
(** Equality over booleans *)
Definition B_eq_dec : forall x y:bool, {x = y} + {x <> y}.
decide equality.
Qed.
Definition B_eq_bool (x y:bool) := bool_of_sumbool (B_eq_dec x y).
Definition B_noteq_bool (x y:bool) :=
bool_of_sumbool (sumbool_not _ _ (B_eq_dec x y)).
(** Equality over type unit *)
Definition U_eq_dec : forall x y:unit, {x = y} + {x <> y}.
decide equality.
Qed.
Definition U_eq_bool (x y:unit) := bool_of_sumbool (U_eq_dec x y).
Definition U_noteq_bool (x y:unit) :=
bool_of_sumbool (sumbool_not _ _ (U_eq_dec x y)).
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