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Diffstat (limited to 'theories/ZArith/Zmax.v')
| -rw-r--r-- | theories/ZArith/Zmax.v | 144 |
1 files changed, 49 insertions, 95 deletions
diff --git a/theories/ZArith/Zmax.v b/theories/ZArith/Zmax.v index cb2fcf26..31880c17 100644 --- a/theories/ZArith/Zmax.v +++ b/theories/ZArith/Zmax.v @@ -1,106 +1,60 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Zmax.v 14641 2011-11-06 11:59:10Z herbelin $ i*) -(** THIS FILE IS DEPRECATED. Use [Zminmax] instead. *) - -Require Export BinInt Zorder Zminmax. - -Open Local Scope Z_scope. - -(** [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. *) - - -(** * Characterization of maximum on binary integer numbers *) - -Definition Zmax_case := Z.max_case. -Definition Zmax_case_strong := Z.max_case_strong. - -Lemma Zmax_spec : forall x y, - x >= y /\ Zmax x y = x \/ x < y /\ Zmax x y = y. +(** THIS FILE IS DEPRECATED. *) + +Require Export BinInt Zcompare Zorder. + +Local Open Scope Z_scope. + +(** Definition [Z.max] is now [BinInt.Z.max]. *) + +(** Exact compatibility *) + +Notation Zmax_case := Z.max_case (compat "8.3"). +Notation Zmax_case_strong := Z.max_case_strong (compat "8.3"). +Notation Zmax_right := Z.max_r (compat "8.3"). +Notation Zle_max_l := Z.le_max_l (compat "8.3"). +Notation Zle_max_r := Z.le_max_r (compat "8.3"). +Notation Zmax_lub := Z.max_lub (compat "8.3"). +Notation Zmax_lub_lt := Z.max_lub_lt (compat "8.3"). +Notation Zle_max_compat_r := Z.max_le_compat_r (compat "8.3"). +Notation Zle_max_compat_l := Z.max_le_compat_l (compat "8.3"). +Notation Zmax_idempotent := Z.max_id (compat "8.3"). +Notation Zmax_n_n := Z.max_id (compat "8.3"). +Notation Zmax_comm := Z.max_comm (compat "8.3"). +Notation Zmax_assoc := Z.max_assoc (compat "8.3"). +Notation Zmax_irreducible_dec := Z.max_dec (compat "8.3"). +Notation Zmax_le_prime := Z.max_le (compat "8.3"). +Notation Zsucc_max_distr := Z.succ_max_distr (compat "8.3"). +Notation Zmax_SS := Z.succ_max_distr (compat "8.3"). +Notation Zplus_max_distr_l := Z.add_max_distr_l (compat "8.3"). +Notation Zplus_max_distr_r := Z.add_max_distr_r (compat "8.3"). +Notation Zmax_plus := Z.add_max_distr_r (compat "8.3"). +Notation Zmax1 := Z.le_max_l (compat "8.3"). +Notation Zmax2 := Z.le_max_r (compat "8.3"). +Notation Zmax_irreducible_inf := Z.max_dec (compat "8.3"). +Notation Zmax_le_prime_inf := Z.max_le (compat "8.3"). +Notation Zpos_max := Pos2Z.inj_max (compat "8.3"). +Notation Zpos_minus := Pos2Z.inj_sub_max (compat "8.3"). + +(** Slightly different lemmas *) + +Lemma Zmax_spec x y : + x >= y /\ Z.max x y = x \/ x < y /\ Z.max x y = y. Proof. - intros x y. rewrite Zge_iff_le. destruct (Z.max_spec x y); auto. + Z.swap_greater. destruct (Z.max_spec x y); 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. - -Definition Zmax_right : forall n m, n<=m -> Zmax n m = m := Zmax_r. - -(** * Least upper bound properties of max *) - -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. - -Definition Zmax_lub : forall n m p, n <= p -> m <= p -> Zmax n m <= p - := Z.max_lub. - -Definition Zmax_lub_lt : forall n m p:Z, n < p -> m < p -> Zmax n m < p - := Z.max_lub_lt. - - -(** * Compatibility with order *) - -Definition Zle_max_compat_r : forall n m p, n <= m -> Zmax n p <= Zmax m p - := Z.max_le_compat_r. - -Definition Zle_max_compat_l : forall n m p, n <= m -> Zmax p n <= Zmax p m - := Z.max_le_compat_l. - +Lemma Zmax_left n m : n>=m -> Z.max n m = n. +Proof. Z.swap_greater. apply Z.max_l. 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_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. - - -(** * Operations preserving max *) - -Definition Zsucc_max_distr : - forall n m:Z, Zsucc (Zmax n m) = Zmax (Zsucc n) (Zsucc m) - := Z.succ_max_distr. - -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 *) - -Definition Zpos_max : forall p q, Zpos (Pmax p q) = Zmax (Zpos p) (Zpos q) - := Z.pos_max. - -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 *) - -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 *) +Lemma Zpos_max_1 p : Z.max 1 (Z.pos p) = Z.pos p. +Proof. + now destruct p. +Qed. |