1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(*i $Id$ i*)
(** THIS FILE IS DEPRECATED. Use [Zminmax] instead. *)
Require Import BinInt Zorder Zminmax.
Open Local Scope Z_scope.
(** [Zmin] is now [Zminmax.Zmin]. 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 the minimum on binary integer numbers *)
Definition Zmin_case := Z.min_case.
Definition Zmin_case_strong := Z.min_case_strong.
Lemma Zmin_spec : forall x y,
x <= y /\ Zmin x y = x \/ x > y /\ Zmin x y = y.
Proof.
intros x y. rewrite Zgt_iff_lt, Z.min_comm. destruct (Z.min_spec y x); auto.
Qed.
(** * Greatest lower bound properties of min *)
Definition Zle_min_l : forall n m, Zmin n m <= n := Z.le_min_l.
Definition Zle_min_r : forall n m, Zmin n m <= m := Z.le_min_r.
Definition Zmin_glb : forall n m p, p <= n -> p <= m -> p <= Zmin n m
:= Z.min_glb.
Definition Zmin_glb_lt : forall n m p, p < n -> p < m -> p < Zmin n m
:= Z.min_glb_lt.
(** * Compatibility with order *)
Definition Zle_min_compat_r : forall n m p, n <= m -> Zmin n p <= Zmin m p
:= Z.min_le_compat_r.
Definition Zle_min_compat_l : forall n m p, n <= m -> Zmin p n <= Zmin p m
:= Z.min_le_compat_l.
(** * Semi-lattice properties of min *)
Definition Zmin_idempotent : forall n, Zmin n n = n := Z.min_id.
Notation Zmin_n_n := Zmin_idempotent (only parsing).
Definition Zmin_comm : forall n m, Zmin n m = Zmin m n := Z.min_comm.
Definition Zmin_assoc : forall n m p, Zmin n (Zmin m p) = Zmin (Zmin n m) p
:= Z.min_assoc.
(** * Additional properties of min *)
Lemma Zmin_irreducible_inf : forall n m, {Zmin n m = n} + {Zmin n m = m}.
Proof. exact Z.min_dec. Qed.
Lemma Zmin_irreducible : forall n m, Zmin n m = n \/ Zmin n m = m.
Proof. intros; destruct (Z.min_dec n m); auto. Qed.
Notation Zmin_or := Zmin_irreducible (only parsing).
Lemma Zmin_le_prime_inf : forall n m p, Zmin n m <= p -> {n <= p} + {m <= p}.
Proof. intros n m p; apply Zmin_case; auto. Qed.
(** * Operations preserving min *)
Definition Zsucc_min_distr :
forall n m, Zsucc (Zmin n m) = Zmin (Zsucc n) (Zsucc m)
:= Z.succ_min_distr.
Notation Zmin_SS := Z.succ_min_distr (only parsing).
Definition Zplus_min_distr_r :
forall n m p, Zmin (n + p) (m + p) = Zmin n m + p
:= Z.plus_min_distr_r.
Notation Zmin_plus := Z.plus_min_distr_r (only parsing).
(** * Minimum and Zpos *)
Definition Zpos_min : forall p q, Zpos (Pmin p q) = Zmin (Zpos p) (Zpos q)
:= Z.pos_min.
|
