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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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.
Require Import ZifyInst.
Local Open Scope Z_scope.
(* Instances of [ZifyClasses] for dealing with boolean operators. *)
Instance Inj_bool_bool : InjTyp bool bool :=
{ inj := (fun x => x) ; pred := (fun b => b = true \/ b = false) ;
cstr := (ltac:(intro b; destruct b; tauto))}.
Add Zify InjTyp Inj_bool_bool.
(** Boolean operators *)
Instance Op_andb : BinOp andb :=
{ TBOp := andb ; TBOpInj := fun _ _ => eq_refl}.
Add Zify BinOp Op_andb.
Instance Op_orb : BinOp orb :=
{ TBOp := orb ; TBOpInj := fun _ _ => eq_refl}.
Add Zify BinOp Op_orb.
Instance Op_implb : BinOp implb :=
{ TBOp := implb; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_implb.
Definition xorb_eq : forall b1 b2,
xorb b1 b2 = andb (orb b1 b2) (negb (eqb b1 b2)).
Proof.
destruct b1,b2 ; reflexivity.
Qed.
Instance Op_xorb : BinOp xorb :=
{ TBOp := fun x y => andb (orb x y) (negb (eqb x y)); TBOpInj := xorb_eq }.
Add Zify BinOp Op_xorb.
Instance Op_eqb : BinOp eqb :=
{ TBOp := eqb; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_eqb.
Instance Op_negb : UnOp negb :=
{ TUOp := negb ; TUOpInj := fun _ => eq_refl}.
Add Zify UnOp Op_negb.
Instance Op_eq_bool : BinRel (@eq bool) :=
{TR := @eq bool ; TRInj := ltac:(reflexivity) }.
Add Zify BinRel Op_eq_bool.
Instance Op_true : CstOp true :=
{ TCst := true ; TCstInj := eq_refl }.
Add Zify CstOp Op_true.
Instance Op_false : CstOp false :=
{ TCst := false ; TCstInj := eq_refl }.
Add Zify CstOp Op_false.
(** Comparison over Z *)
Instance Op_Zeqb : BinOp Z.eqb :=
{ TBOp := Z.eqb ; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_Zeqb.
Instance Op_Zleb : BinOp Z.leb :=
{ TBOp := Z.leb; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_Zleb.
Instance Op_Zgeb : BinOp Z.geb :=
{ TBOp := Z.geb; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_Zgeb.
Instance Op_Zltb : BinOp Z.ltb :=
{ TBOp := Z.ltb ; TBOpInj := fun _ _ => eq_refl }.
Add Zify BinOp Op_Zltb.
Instance Op_Zgtb : BinOp Z.gtb :=
{ TBOp := Z.gtb; TBOpInj := fun _ _ => eq_refl }.
Add Zify 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 := Z.eqb; TBOpInj := Z_of_nat_eqb_iff }.
Add Zify BinOp Op_nat_eqb.
Instance Op_nat_leb : BinOp Nat.leb :=
{ TBOp := Z.leb; TBOpInj := Z_of_nat_leb_iff }.
Add Zify BinOp Op_nat_leb.
Instance Op_nat_ltb : BinOp Nat.ltb :=
{ TBOp := Z.ltb; TBOpInj := Z_of_nat_ltb_iff }.
Add Zify BinOp Op_nat_ltb.
Lemma b2n_b2z : forall x, Z.of_nat (Nat.b2n x) = Z.b2z x.
Proof.
intro. destruct x ; reflexivity.
Qed.
Instance Op_b2n : UnOp Nat.b2n :=
{ TUOp := Z.b2z; TUOpInj := b2n_b2z }.
Add Zify UnOp Op_b2n.
Instance Op_b2z : UnOp Z.b2z :=
{ TUOp := Z.b2z; TUOpInj := fun _ => eq_refl }.
Add Zify UnOp Op_b2z.
Lemma b2z_spec : forall b, (b = true /\ Z.b2z b = 1) \/ (b = false /\ Z.b2z b = 0).
Proof.
destruct b ; simpl; intuition congruence.
Qed.
Instance b2zSpec : UnOpSpec Z.b2z :=
{ UPred := fun b r => (b = true /\ r = 1) \/ (b = false /\ r = 0);
USpec := b2z_spec
}.
Add Zify UnOpSpec b2zSpec.
Ltac elim_bool_cstr :=
repeat match goal with
| C : ?B = true \/ ?B = false |- _ =>
destruct C as [C|C]; rewrite C in *
end.
Ltac Zify.zify_post_hook ::= elim_bool_cstr.
|
