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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *)
(* <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) *)
(************************************************************************)
Require Import Bool ZArith.
Require Import Zify ZifyClasses.
Local Open Scope Z_scope.
(* Instances of [ZifyClasses] for dealing with boolean operators.
Various encodings of boolean are possible. One objective is to
have an encoding that is terse but also lia friendly.
*)
(** [Z_of_bool] is the injection function for boolean *)
Definition Z_of_bool (b : bool) : Z := if b then 1 else 0.
(** [bool_of_Z] is a compatible reverse operation *)
Definition bool_of_Z (z : Z) : bool := negb (Z.eqb z 0).
Lemma Z_of_bool_bound : forall x, 0 <= Z_of_bool x <= 1.
Proof.
destruct x ; simpl; compute; intuition congruence.
Qed.
Instance Inj_bool_Z : InjTyp bool Z :=
{ inj := Z_of_bool ; pred :=(fun x => 0 <= x <= 1) ; cstr := Z_of_bool_bound}.
Add InjTyp Inj_bool_Z.
(** Boolean operators *)
Instance Op_andb : BinOp andb :=
{ TBOp := Z.min ;
TBOpInj := ltac: (destruct n,m; reflexivity)}.
Add BinOp Op_andb.
Instance Op_orb : BinOp orb :=
{ TBOp := Z.max ;
TBOpInj := ltac:(destruct n,m; reflexivity)}.
Add BinOp Op_orb.
Instance Op_implb : BinOp implb :=
{ TBOp := fun x y => Z.max (1 - x) y;
TBOpInj := ltac:(destruct n,m; reflexivity) }.
Add BinOp Op_implb.
Instance Op_xorb : BinOp xorb :=
{ TBOp := fun x y => Z.max (x - y) (y - x);
TBOpInj := ltac:(destruct n,m; reflexivity) }.
Add BinOp Op_xorb.
Instance Op_negb : UnOp negb :=
{ TUOp := fun x => 1 - x ; TUOpInj := ltac:(destruct x; reflexivity)}.
Add UnOp Op_negb.
Instance Op_eq_bool : BinRel (@eq bool) :=
{TR := @eq Z ; TRInj := ltac:(destruct n,m; simpl ; intuition congruence) }.
Add BinRel Op_eq_bool.
Instance Op_true : CstOp true :=
{ TCst := 1 ; TCstInj := eq_refl }.
Add CstOp Op_true.
Instance Op_false : CstOp false :=
{ TCst := 0 ; TCstInj := eq_refl }.
Add CstOp Op_false.
(** Comparisons are encoded using the predicates [isZero] and [isLeZero].*)
Definition isZero (z : Z) := Z_of_bool (Z.eqb z 0).
Definition isLeZero (x : Z) := Z_of_bool (Z.leb x 0).
Instance Op_isZero : UnOp isZero :=
{ TUOp := isZero; TUOpInj := ltac: (reflexivity) }.
Add UnOp Op_isZero.
Instance Op_isLeZero : UnOp isLeZero :=
{ TUOp := isLeZero; TUOpInj := ltac: (reflexivity) }.
Add UnOp Op_isLeZero.
(* Some intermediate lemma *)
Lemma Z_eqb_isZero : forall n m,
Z_of_bool (n =? m) = isZero (n - m).
Proof.
intros ; unfold isZero.
destruct ( n =? m) eqn:EQ.
- simpl. rewrite Z.eqb_eq in EQ.
rewrite EQ. rewrite Z.sub_diag.
reflexivity.
-
destruct (n - m =? 0) eqn:EQ'.
rewrite Z.eqb_neq in EQ.
rewrite Z.eqb_eq in EQ'.
apply Zminus_eq in EQ'.
congruence.
reflexivity.
Qed.
Lemma Z_leb_sub : forall x y, x <=? y = ((x - y) <=? 0).
Proof.
intros.
destruct (x <=?y) eqn:B1 ;
destruct (x - y <=?0) eqn:B2 ; auto.
- rewrite Z.leb_le in B1.
rewrite Z.leb_nle in B2.
rewrite Z.le_sub_0 in B2. tauto.
- rewrite Z.leb_nle in B1.
rewrite Z.leb_le in B2.
rewrite Z.le_sub_0 in B2. tauto.
Qed.
Lemma Z_ltb_leb : forall x y, x <? y = (x +1 <=? y).
Proof.
intros.
destruct (x <?y) eqn:B1 ;
destruct (x + 1 <=?y) eqn:B2 ; auto.
- rewrite Z.ltb_lt in B1.
rewrite Z.leb_nle in B2.
apply Zorder.Zlt_le_succ in B1.
unfold Z.succ in B1.
tauto.
- rewrite Z.ltb_nlt in B1.
rewrite Z.leb_le in B2.
apply Zorder.Zle_lt_succ in B2.
unfold Z.succ in B2.
apply Zorder.Zplus_lt_reg_r in B2.
tauto.
Qed.
(** Comparison over Z *)
Instance Op_Zeqb : BinOp Z.eqb :=
{ TBOp := fun x y => isZero (Z.sub x y) ; TBOpInj := Z_eqb_isZero}.
Instance Op_Zleb : BinOp Z.leb :=
{ TBOp := fun x y => isLeZero (x-y) ;
TBOpInj :=
ltac: (intros;unfold isLeZero;
rewrite Z_leb_sub;
auto) }.
Add BinOp Op_Zleb.
Instance Op_Zgeb : BinOp Z.geb :=
{ TBOp := fun x y => isLeZero (y-x) ;
TBOpInj := ltac:(
intros;
unfold isLeZero;
rewrite Z.geb_leb;
rewrite Z_leb_sub;
auto) }.
Add BinOp Op_Zgeb.
Instance Op_Zltb : BinOp Z.ltb :=
{ TBOp := fun x y => isLeZero (x+1-y) ;
TBOpInj := ltac:(
intros;
unfold isLeZero;
rewrite Z_ltb_leb;
rewrite <- Z_leb_sub;
reflexivity) }.
Instance Op_Zgtb : BinOp Z.gtb :=
{ TBOp := fun x y => isLeZero (y-x+1) ;
TBOpInj := ltac:(
intros;
unfold isLeZero;
rewrite Z.gtb_ltb;
rewrite Z_ltb_leb;
rewrite Z_leb_sub;
rewrite Z.add_sub_swap;
reflexivity) }.
Add BinOp Op_Zgtb.
(** Comparison over nat *)
Lemma Z_of_nat_eqb_iff : forall n m,
(n =? m)%nat = (Z.of_nat n =? Z.of_nat m).
Proof.
intros.
rewrite Nat.eqb_compare.
rewrite Z.eqb_compare.
rewrite Nat2Z.inj_compare.
reflexivity.
Qed.
Lemma Z_of_nat_leb_iff : forall n m,
(n <=? m)%nat = (Z.of_nat n <=? Z.of_nat m).
Proof.
intros.
rewrite Nat.leb_compare.
rewrite Z.leb_compare.
rewrite Nat2Z.inj_compare.
reflexivity.
Qed.
Lemma Z_of_nat_ltb_iff : forall n m,
(n <? m)%nat = (Z.of_nat n <? Z.of_nat m).
Proof.
intros.
rewrite Nat.ltb_compare.
rewrite Z.ltb_compare.
rewrite Nat2Z.inj_compare.
reflexivity.
Qed.
Instance Op_nat_eqb : BinOp Nat.eqb :=
{ TBOp := fun x y => isZero (Z.sub x y) ;
TBOpInj := ltac:(
intros; simpl;
rewrite <- Z_eqb_isZero;
f_equal; apply Z_of_nat_eqb_iff) }.
Add BinOp Op_nat_eqb.
Instance Op_nat_leb : BinOp Nat.leb :=
{ TBOp := fun x y => isLeZero (x-y) ;
TBOpInj := ltac:(
intros;
rewrite Z_of_nat_leb_iff;
unfold isLeZero;
rewrite Z_leb_sub;
auto) }.
Add BinOp Op_nat_leb.
Instance Op_nat_ltb : BinOp Nat.ltb :=
{ TBOp := fun x y => isLeZero (x+1-y) ;
TBOpInj := ltac:(
intros;
rewrite Z_of_nat_ltb_iff;
unfold isLeZero;
rewrite Z_ltb_leb;
rewrite <- Z_leb_sub;
reflexivity) }.
Add BinOp Op_nat_ltb.
(** Injected boolean operators *)
Lemma Z_eqb_ZSpec_ok : forall x, 0 <= isZero x <= 1 /\
(x = 0 <-> isZero x = 1).
Proof.
intros.
unfold isZero.
destruct (x =? 0) eqn:EQ.
- apply Z.eqb_eq in EQ.
simpl. intuition try congruence;
compute ; congruence.
- apply Z.eqb_neq in EQ.
simpl. intuition try congruence;
compute ; congruence.
Qed.
Instance Z_eqb_ZSpec : UnOpSpec isZero :=
{| UPred := fun n r => 0 <= r <= 1 /\ (n = 0 <-> isZero n = 1) ; USpec := Z_eqb_ZSpec_ok |}.
Add Spec Z_eqb_ZSpec.
Lemma leZeroSpec_ok : forall x, x <= 0 /\ isLeZero x = 1 \/ x > 0 /\ isLeZero x = 0.
Proof.
intros.
unfold isLeZero.
destruct (x <=? 0) eqn:EQ.
- apply Z.leb_le in EQ.
simpl. intuition congruence.
- simpl.
apply Z.leb_nle in EQ.
apply Zorder.Znot_le_gt in EQ.
tauto.
Qed.
Instance leZeroSpec : UnOpSpec isLeZero :=
{| UPred := fun n r => (n<=0 /\ r = 1) \/ (n > 0 /\ r = 0) ; USpec := leZeroSpec_ok|}.
Add Spec leZeroSpec.
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