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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import ZAxioms ZProperties BinInt Zcompare Zorder ZArith_dec
Zbool Zeven Zsqrt_def Zpow_def Zlog_def Zgcd_def Zdiv_def.
Local Open Scope Z_scope.
(** Bi-directional induction for Z. *)
Theorem Z_bi_induction :
forall A : Z -> Prop, Proper (eq ==> iff) A ->
A 0 -> (forall n : Z, A n <-> A (Zsucc n)) -> forall n : Z, A n.
Proof.
intros A A_wd A0 AS n; apply Zind; clear n.
assumption.
intros; rewrite <- Zsucc_succ'. now apply -> AS.
intros n H. rewrite <- Zpred_pred'. rewrite Zsucc_pred in H. now apply <- AS.
Qed.
(** * Implementation of [ZAxiomsMiniSig] by [BinInt.Z] *)
Module Z
<: ZAxiomsSig <: UsualOrderedTypeFull <: TotalOrder
<: UsualDecidableTypeFull.
Definition t := Z.
Definition eqb := Zeq_bool.
Definition zero := 0.
Definition one := 1.
Definition two := 2.
Definition succ := Zsucc.
Definition pred := Zpred.
Definition add := Zplus.
Definition sub := Zminus.
Definition mul := Zmult.
Definition lt := Zlt.
Definition le := Zle.
Definition compare := Zcompare.
Definition min := Zmin.
Definition max := Zmax.
Definition opp := Zopp.
Definition abs := Zabs.
Definition sgn := Zsgn.
Definition eq_dec := Z_eq_dec.
Definition bi_induction := Z_bi_induction.
(** Basic operations. *)
Definition eq_equiv : Equivalence (@eq Z) := eq_equivalence.
Local Obligation Tactic := simpl_relation.
Program Instance succ_wd : Proper (eq==>eq) Zsucc.
Program Instance pred_wd : Proper (eq==>eq) Zpred.
Program Instance add_wd : Proper (eq==>eq==>eq) Zplus.
Program Instance sub_wd : Proper (eq==>eq==>eq) Zminus.
Program Instance mul_wd : Proper (eq==>eq==>eq) Zmult.
Definition pred_succ n := eq_sym (Zpred_succ n).
Definition add_0_l := Zplus_0_l.
Definition add_succ_l := Zplus_succ_l.
Definition sub_0_r := Zminus_0_r.
Definition sub_succ_r := Zminus_succ_r.
Definition mul_0_l := Zmult_0_l.
Definition mul_succ_l := Zmult_succ_l.
Lemma one_succ : 1 = Zsucc 0.
Proof.
reflexivity.
Qed.
Lemma two_succ : 2 = Zsucc 1.
Proof.
reflexivity.
Qed.
(** Boolean Equality *)
Definition eqb_eq x y := iff_sym (Zeq_is_eq_bool x y).
(** Order *)
Program Instance lt_wd : Proper (eq==>eq==>iff) Zlt.
Definition lt_eq_cases := Zle_lt_or_eq_iff.
Definition lt_irrefl := Zlt_irrefl.
Definition lt_succ_r := Zlt_succ_r.
(** Comparison *)
Definition compare_spec := Zcompare_spec.
(** Minimum and maximum *)
Definition min_l := Zmin_l.
Definition min_r := Zmin_r.
Definition max_l := Zmax_l.
Definition max_r := Zmax_r.
(** Properties specific to integers, not natural numbers. *)
Program Instance opp_wd : Proper (eq==>eq) Zopp.
Definition succ_pred n := eq_sym (Zsucc_pred n).
Definition opp_0 := Zopp_0.
Definition opp_succ := Zopp_succ.
(** Absolute value and sign *)
Definition abs_eq := Zabs_eq.
Definition abs_neq := Zabs_non_eq.
Definition sgn_null := Zsgn_0.
Definition sgn_pos := Zsgn_1.
Definition sgn_neg := Zsgn_m1.
(** Even and Odd *)
Definition Even n := exists m, n=2*m.
Definition Odd n := exists m, n=2*m+1.
Lemma even_spec n : Zeven_bool n = true <-> Even n.
Proof. rewrite Zeven_bool_iff. exact (Zeven_ex_iff n). Qed.
Lemma odd_spec n : Zodd_bool n = true <-> Odd n.
Proof. rewrite Zodd_bool_iff. exact (Zodd_ex_iff n). Qed.
Definition even := Zeven_bool.
Definition odd := Zodd_bool.
(** Power *)
Program Instance pow_wd : Proper (eq==>eq==>eq) Zpower.
Definition pow_0_r := Zpower_0_r.
Definition pow_succ_r := Zpower_succ_r.
Definition pow_neg_r := Zpower_neg_r.
Definition pow := Zpower.
(** Sqrt *)
(** NB : we use a new Zsqrt defined in Zsqrt_def, the previous
module Zsqrt.v is now Zsqrt_compat.v *)
Definition sqrt_spec := Zsqrt_spec.
Definition sqrt_neg := Zsqrt_neg.
Definition sqrt := Zsqrt.
(** Log2 *)
Definition log2_spec := Zlog2_spec.
Definition log2_nonpos := Zlog2_nonpos.
Definition log2 := Zlog2.
(** Gcd *)
Definition gcd_divide_l := Zgcd_divide_l.
Definition gcd_divide_r := Zgcd_divide_r.
Definition gcd_greatest := Zgcd_greatest.
Definition gcd_nonneg := Zgcd_nonneg.
Definition divide := Zdivide'.
Definition gcd := Zgcd.
(** Div / Mod / Quot / Rem *)
Program Instance div_wd : Proper (eq==>eq==>eq) Zdiv.
Program Instance mod_wd : Proper (eq==>eq==>eq) Zmod.
Program Instance quot_wd : Proper (eq==>eq==>eq) Zquot.
Program Instance rem_wd : Proper (eq==>eq==>eq) Zrem.
Definition div_mod := Z_div_mod_eq_full.
Definition mod_pos_bound := Zmod_pos_bound.
Definition mod_neg_bound := Zmod_neg_bound.
Definition mod_bound_pos := fun a b (_:0<=a) => Zmod_pos_bound a b.
Definition div := Zdiv.
Definition modulo := Zmod.
Definition quot_rem := fun a b (_:b<>0) => Z_quot_rem_eq a b.
Definition rem_bound_pos := Zrem_bound.
Definition rem_opp_l := fun a b (_:b<>0) => Zrem_opp_l a b.
Definition rem_opp_r := fun a b (_:b<>0) => Zrem_opp_r a b.
Definition quot := Zquot.
Definition remainder := Zrem.
(** We define [eq] only here to avoid refering to this [eq] above. *)
Definition eq := (@eq Z).
(** Now the generic properties. *)
Include ZProp
<+ UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.
End Z.
(** * An [order] tactic for integers *)
Ltac z_order := Z.order.
(** Note that [z_order] is domain-agnostic: it will not prove
[1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
Section TestOrder.
Let test : forall x y, x<=y -> y<=x -> x=y.
Proof.
z_order.
Qed.
End TestOrder.
(** Z forms a ring *)
(*Lemma Zring : ring_theory 0 1 NZadd NZmul NZsub Zopp NZeq.
Proof.
constructor.
exact Zadd_0_l.
exact Zadd_comm.
exact Zadd_assoc.
exact Zmul_1_l.
exact Zmul_comm.
exact Zmul_assoc.
exact Zmul_add_distr_r.
intros; now rewrite Zadd_opp_minus.
exact Zadd_opp_r.
Qed.
Add Ring ZR : Zring.*)
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