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Diffstat (limited to 'theories/ZArith/Zmax.v')
| -rw-r--r-- | theories/ZArith/Zmax.v | 178 |
1 files changed, 59 insertions, 119 deletions
diff --git a/theories/ZArith/Zmax.v b/theories/ZArith/Zmax.v index 0d6fc94a..53c40ae7 100644 --- a/theories/ZArith/Zmax.v +++ b/theories/ZArith/Zmax.v @@ -5,162 +5,102 @@ (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Zmax.v 10291 2007-11-06 02:18:53Z letouzey $ i*) +(*i $Id$ i*) -Require Import Arith_base. -Require Import BinInt. -Require Import Zcompare. -Require Import Zorder. +(** THIS FILE IS DEPRECATED. Use [Zminmax] instead. *) + +Require Export BinInt Zorder Zminmax. Open Local Scope Z_scope. -(******************************************) -(** Maximum of two binary integer numbers *) +(** [Zmax] is now [Zminmax.Zmax]. Code that do things like + [unfold Zmin.Zmin] will have to be adapted, and neither + a [Definition] or a [Notation] here can help much. *) -Definition Zmax m n := - match m ?= n with - | Eq | Gt => m - | Lt => n - end. (** * Characterization of maximum on binary integer numbers *) -Lemma Zmax_case : forall (n m:Z) (P:Z -> Type), P n -> P m -> P (Zmax n m). -Proof. - intros n m P H1 H2; unfold Zmax in |- *; case (n ?= m); auto with arith. -Qed. - -Lemma Zmax_case_strong : forall (n m:Z) (P:Z -> Type), - (m<=n -> P n) -> (n<=m -> P m) -> P (Zmax n m). -Proof. - intros n m P H1 H2; unfold Zmax, Zle, Zge in *. - rewrite <- (Zcompare_antisym n m) in H1. - destruct (n ?= m); (apply H1|| apply H2); discriminate. -Qed. +Definition Zmax_case := Z.max_case. +Definition Zmax_case_strong := Z.max_case_strong. -Lemma Zmax_spec : forall x y:Z, - x >= y /\ Zmax x y = x \/ - x < y /\ Zmax x y = y. +Lemma Zmax_spec : forall x y, + x >= y /\ Zmax x y = x \/ x < y /\ Zmax x y = y. Proof. - intros; unfold Zmax, Zlt, Zge. - destruct (Zcompare x y); [ left | right | left ]; split; auto; discriminate. + intros x y. rewrite Zge_iff_le. destruct (Z.max_spec x y); auto. Qed. -Lemma Zmax_left : forall n m:Z, n>=m -> Zmax n m = n. -Proof. - intros n m; unfold Zmax, Zge; destruct (n ?= m); auto. - intro H; elim H; auto. -Qed. +Lemma Zmax_left : forall n m, n>=m -> Zmax n m = n. +Proof. intros x y. rewrite Zge_iff_le. apply Zmax_l. Qed. -Lemma Zmax_right : forall n m:Z, n<=m -> Zmax n m = m. -Proof. - intros n m; unfold Zmax, Zle. - generalize (Zcompare_Eq_eq n m). - destruct (n ?= m); auto. - intros _ H; elim H; auto. -Qed. +Definition Zmax_right : forall n m, n<=m -> Zmax n m = m := Zmax_r. (** * Least upper bound properties of max *) -Lemma Zle_max_l : forall n m:Z, n <= Zmax n m. -Proof. - intros; apply Zmax_case_strong; auto with zarith. -Qed. +Definition Zle_max_l : forall n m, n <= Zmax n m := Z.le_max_l. +Definition Zle_max_r : forall n m, m <= Zmax n m := Z.le_max_r. -Notation Zmax1 := Zle_max_l (only parsing). +Definition Zmax_lub : forall n m p, n <= p -> m <= p -> Zmax n m <= p + := Z.max_lub. -Lemma Zle_max_r : forall n m:Z, m <= Zmax n m. -Proof. - intros; apply Zmax_case_strong; auto with zarith. -Qed. +Definition Zmax_lub_lt : forall n m p:Z, n < p -> m < p -> Zmax n m < p + := Z.max_lub_lt. -Notation Zmax2 := Zle_max_r (only parsing). -Lemma Zmax_lub : forall n m p:Z, n <= p -> m <= p -> Zmax n m <= p. -Proof. - intros; apply Zmax_case; assumption. -Qed. +(** * Compatibility with order *) -(** * Semi-lattice properties of max *) +Definition Zle_max_compat_r : forall n m p, n <= m -> Zmax n p <= Zmax m p + := Z.max_le_compat_r. -Lemma Zmax_idempotent : forall n:Z, Zmax n n = n. -Proof. - intros; apply Zmax_case; auto. -Qed. +Definition Zle_max_compat_l : forall n m p, n <= m -> Zmax p n <= Zmax p m + := Z.max_le_compat_l. -Lemma Zmax_comm : forall n m:Z, Zmax n m = Zmax m n. -Proof. - intros; do 2 apply Zmax_case_strong; intros; - apply Zle_antisym; auto with zarith. -Qed. -Lemma Zmax_assoc : forall n m p:Z, Zmax n (Zmax m p) = Zmax (Zmax n m) p. -Proof. - intros n m p; repeat apply Zmax_case_strong; intros; - reflexivity || (try apply Zle_antisym); eauto with zarith. -Qed. +(** * Semi-lattice properties of max *) + +Definition Zmax_idempotent : forall n, Zmax n n = n := Z.max_id. +Definition Zmax_comm : forall n m, Zmax n m = Zmax m n := Z.max_comm. +Definition Zmax_assoc : forall n m p, Zmax n (Zmax m p) = Zmax (Zmax n m) p + := Z.max_assoc. (** * Additional properties of max *) -Lemma Zmax_irreducible_inf : forall n m:Z, Zmax n m = n \/ Zmax n m = m. -Proof. - intros; apply Zmax_case; auto. -Qed. +Lemma Zmax_irreducible_dec : forall n m, {Zmax n m = n} + {Zmax n m = m}. +Proof. exact Z.max_dec. Qed. + +Definition Zmax_le_prime : forall n m p, p <= Zmax n m -> p <= n \/ p <= m + := Z.max_le. -Lemma Zmax_le_prime_inf : forall n m p:Z, p <= Zmax n m -> p <= n \/ p <= m. -Proof. - intros n m p; apply Zmax_case; auto. -Qed. (** * Operations preserving max *) -Lemma Zsucc_max_distr : - forall n m:Z, Zsucc (Zmax n m) = Zmax (Zsucc n) (Zsucc m). -Proof. - intros n m; unfold Zmax in |- *; rewrite (Zcompare_succ_compat n m); - elim_compare n m; intros E; rewrite E; auto with arith. -Qed. +Definition Zsucc_max_distr : + forall n m:Z, Zsucc (Zmax n m) = Zmax (Zsucc n) (Zsucc m) + := Z.succ_max_distr. -Lemma Zplus_max_distr_r : forall n m p:Z, Zmax (n + p) (m + p) = Zmax n m + p. -Proof. - intros x y n; unfold Zmax in |- *. - rewrite (Zplus_comm x n); rewrite (Zplus_comm y n); - rewrite (Zcompare_plus_compat x y n). - case (x ?= y); apply Zplus_comm. -Qed. +Definition Zplus_max_distr_l : forall n m p:Z, Zmax (p + n) (p + m) = p + Zmax n m + := Z.plus_max_distr_l. + +Definition Zplus_max_distr_r : forall n m p:Z, Zmax (n + p) (m + p) = Zmax n m + p + := Z.plus_max_distr_r. (** * Maximum and Zpos *) -Lemma Zpos_max : forall p q, Zpos (Pmax p q) = Zmax (Zpos p) (Zpos q). -Proof. - intros; unfold Zmax, Pmax; simpl; generalize (Pcompare_Eq_eq p q). - destruct Pcompare; auto. - intro H; rewrite H; auto. -Qed. +Definition Zpos_max : forall p q, Zpos (Pmax p q) = Zmax (Zpos p) (Zpos q) + := Z.pos_max. -Lemma Zpos_max_1 : forall p, Zmax 1 (Zpos p) = Zpos p. -Proof. - intros; unfold Zmax; simpl; destruct p; simpl; auto. -Qed. +Definition Zpos_max_1 : forall p, Zmax 1 (Zpos p) = Zpos p + := Z.pos_max_1. (** * Characterization of Pminus in term of Zminus and Zmax *) -Lemma Zpos_minus : forall p q, Zpos (Pminus p q) = Zmax 1 (Zpos p - Zpos q). -Proof. - intros. - case_eq (Pcompare p q Eq). - intros H; rewrite (Pcompare_Eq_eq _ _ H). - rewrite Zminus_diag. - unfold Zmax; simpl. - unfold Pminus; rewrite Pminus_mask_diag; auto. - intros; rewrite Pminus_Lt; auto. - destruct (Zmax_spec 1 (Zpos p - Zpos q)) as [(H1,H2)|(H1,H2)]; auto. - elimtype False; clear H2. - assert (H1':=Zlt_trans 0 1 _ Zlt_0_1 H1). - generalize (Zlt_0_minus_lt _ _ H1'). - unfold Zlt; simpl. - rewrite (ZC2 _ _ H); intro; discriminate. - intros; simpl; rewrite H. - symmetry; apply Zpos_max_1. -Qed. +Definition Zpos_minus : + forall p q, Zpos (Pminus p q) = Zmax 1 (Zpos p - Zpos q) + := Zpos_minus. +(* begin hide *) +(* Compatibility *) +Notation Zmax1 := Zle_max_l (only parsing). +Notation Zmax2 := Zle_max_r (only parsing). +Notation Zmax_irreducible_inf := Zmax_irreducible_dec (only parsing). +Notation Zmax_le_prime_inf := Zmax_le_prime (only parsing). +(* end hide *) |