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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* 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 *)
(************************************************************************)
(** Binary Integers : results about Zcompare *)
(** Initial author: Pierre Crégut (CNET, Lannion, France *)
(** THIS FILE IS DEPRECATED.
It is now almost entirely made of compatibility formulations
for results already present in BinInt.Z. *)
Require Export BinPos BinInt.
Require Import Lt Gt Plus Mult. (* Useless now, for compatibility only *)
Local Open Scope Z_scope.
(***************************)
(** * Comparison on integers *)
Lemma Zcompare_Gt_Lt_antisym : forall n m:Z, (n ?= m) = Gt <-> (m ?= n) = Lt.
Proof Z.gt_lt_iff.
Lemma Zcompare_antisym n m : CompOpp (n ?= m) = (m ?= n).
Proof eq_sym (Z.compare_antisym n m).
(** * Transitivity of comparison *)
Lemma Zcompare_Lt_trans :
forall n m p:Z, (n ?= m) = Lt -> (m ?= p) = Lt -> (n ?= p) = Lt.
Proof Z.lt_trans.
Lemma Zcompare_Gt_trans :
forall n m p:Z, (n ?= m) = Gt -> (m ?= p) = Gt -> (n ?= p) = Gt.
Proof.
intros n m p. change (n > m -> m > p -> n > p).
Z.swap_greater. intros. now transitivity m.
Qed.
(** * Comparison and opposite *)
Lemma Zcompare_opp n m : (n ?= m) = (- m ?= - n).
Proof.
symmetry. apply Z.compare_opp.
Qed.
(** * Comparison first-order specification *)
Lemma Zcompare_Gt_spec n m : (n ?= m) = Gt -> exists h, n + - m = Zpos h.
Proof.
rewrite Z.compare_sub. unfold Z.sub.
destruct (n+-m) as [|p|p]; try discriminate. now exists p.
Qed.
(** * Comparison and addition *)
Lemma Zcompare_plus_compat n m p : (p + n ?= p + m) = (n ?= m).
Proof.
apply Z.add_compare_mono_l.
Qed.
Lemma Zplus_compare_compat (r:comparison) (n m p q:Z) :
(n ?= m) = r -> (p ?= q) = r -> (n + p ?= m + q) = r.
Proof.
rewrite (Z.compare_sub n), (Z.compare_sub p), (Z.compare_sub (n+p)).
unfold Z.sub. rewrite Z.opp_add_distr. rewrite Z.add_shuffle1.
destruct (n+-m), (p+-q); simpl; intros; now subst.
Qed.
Lemma Zcompare_succ_Gt n : (Z.succ n ?= n) = Gt.
Proof.
apply Z.lt_gt. apply Z.lt_succ_diag_r.
Qed.
Lemma Zcompare_Gt_not_Lt n m : (n ?= m) = Gt <-> (n ?= m+1) <> Lt.
Proof.
change (n > m <-> n >= m+1). Z.swap_greater. symmetry. apply Z.le_succ_l.
Qed.
(** * Successor and comparison *)
Lemma Zcompare_succ_compat n m : (Z.succ n ?= Z.succ m) = (n ?= m).
Proof.
rewrite <- 2 Z.add_1_l. apply Zcompare_plus_compat.
Qed.
(** * Multiplication and comparison *)
Lemma Zcompare_mult_compat :
forall (p:positive) (n m:Z), (Zpos p * n ?= Zpos p * m) = (n ?= m).
Proof.
intros p [|n|n] [|m|m]; simpl; trivial; now rewrite Pos.mul_compare_mono_l.
Qed.
Lemma Zmult_compare_compat_l n m p:
p > 0 -> (n ?= m) = (p * n ?= p * m).
Proof.
intros; destruct p; try discriminate.
symmetry. apply Zcompare_mult_compat.
Qed.
Lemma Zmult_compare_compat_r n m p :
p > 0 -> (n ?= m) = (n * p ?= m * p).
Proof.
intros; rewrite 2 (Zmult_comm _ p); now apply Zmult_compare_compat_l.
Qed.
(** * Relating [x ?= y] to [=], [<=], [<], [>=] or [>] *)
Lemma Zcompare_elim :
forall (c1 c2 c3:Prop) (n m:Z),
(n = m -> c1) ->
(n < m -> c2) ->
(n > m -> c3) -> match n ?= m with
| Eq => c1
| Lt => c2
| Gt => c3
end.
Proof.
intros. case Z.compare_spec; trivial. now Z.swap_greater.
Qed.
Lemma Zcompare_eq_case :
forall (c1 c2 c3:Prop) (n m:Z),
c1 -> n = m -> match n ?= m with
| Eq => c1
| Lt => c2
| Gt => c3
end.
Proof.
intros. subst. now rewrite Z.compare_refl.
Qed.
Lemma Zle_compare :
forall n m:Z,
n <= m -> match n ?= m with
| Eq => True
| Lt => True
| Gt => False
end.
Proof.
intros. case Z.compare_spec; trivial; Z.order.
Qed.
Lemma Zlt_compare :
forall n m:Z,
n < m -> match n ?= m with
| Eq => False
| Lt => True
| Gt => False
end.
Proof.
intros x y H; now rewrite H.
Qed.
Lemma Zge_compare :
forall n m:Z,
n >= m -> match n ?= m with
| Eq => True
| Lt => False
| Gt => True
end.
Proof.
intros. now case Z.compare_spec.
Qed.
Lemma Zgt_compare :
forall n m:Z,
n > m -> match n ?= m with
| Eq => False
| Lt => False
| Gt => True
end.
Proof.
intros x y H; now rewrite H.
Qed.
(** Compatibility notations *)
Notation Zcompare_refl := Z.compare_refl (only parsing).
Notation Zcompare_Eq_eq := Z.compare_eq (only parsing).
Notation Zcompare_Eq_iff_eq := Z.compare_eq_iff (only parsing).
Notation Zcompare_spec := Z.compare_spec (only parsing).
Notation Zmin_l := Z.min_l (only parsing).
Notation Zmin_r := Z.min_r (only parsing).
Notation Zmax_l := Z.max_l (only parsing).
Notation Zmax_r := Z.max_r (only parsing).
Notation Zabs_eq := Z.abs_eq (only parsing).
Notation Zabs_non_eq := Z.abs_neq (only parsing).
Notation Zsgn_0 := Z.sgn_null (only parsing).
Notation Zsgn_1 := Z.sgn_pos (only parsing).
Notation Zsgn_m1 := Z.sgn_neg (only parsing).
(** Not kept: Zcompare_egal_dec *)
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