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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Certification of Imperative Programs / Jean-Christophe Fillitre *)
(* $Id: ProgBool.v,v 1.4.2.1 2004/07/16 19:30:00 herbelin Exp $ *)
Require Import ZArith.
Require Export Bool_nat.
Require Export Sumbool.
Definition annot_bool :
forall b:bool, {b' : bool | if b' then b = true else b = false}.
Proof.
intro b.
exists b. case b; trivial.
Qed.
(* 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}.
Proof.
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}.
Proof.
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}.
Proof.
intros Q H.
elim H. intro b.
case b; intro.
exists false; auto. exists true; auto.
Qed.
|
