blob: c4e71632cf9f9157119abebf140cd788846290f4 (
plain)
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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Peano_dec Compare_dec
DecidableType2 OrderedType2 OrderedType2Facts.
(** * DecidableType structure for Peano numbers *)
Module Nat_as_MiniDT <: MiniDecidableType.
Definition t := nat.
Definition eq_dec := eq_nat_dec.
End Nat_as_MiniDT.
Module Nat_as_DT <: UsualDecidableType := Make_UDT Nat_as_MiniDT.
(** Note that [Nat_as_DT] can also be seen as a [DecidableType]
and a [DecidableTypeOrig]. *)
(** * OrderedType structure for Peano numbers *)
Module Nat_as_OT <: OrderedTypeFull.
Include Nat_as_DT.
Definition lt := lt.
Definition le := le.
Definition compare := nat_compare.
Instance lt_strorder : StrictOrder lt.
Proof. split; [ exact Lt.lt_irrefl | exact Lt.lt_trans ]. Qed.
Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt.
Proof. repeat red; intros; subst; auto. Qed.
Lemma le_lteq : forall x y, x <= y <-> x < y \/ x=y.
Proof. intuition; subst; auto using Lt.le_lt_or_eq. Qed.
Lemma compare_spec : forall x y, Cmp eq lt x y (compare x y).
Proof.
intros; unfold compare.
destruct (nat_compare x y) as [ ]_eqn; constructor.
apply nat_compare_eq; auto.
apply nat_compare_Lt_lt; auto.
apply nat_compare_Gt_gt; auto.
Qed.
End Nat_as_OT.
(* Note that [Nat_as_OT] can also be seen as a [UsualOrderedType]
and a [OrderedType] (and also as a [DecidableType]). *)
(** * An [order] tactic for Peano numbers *)
Module NatOrder := OTF_to_OrderTac Nat_as_OT.
Ltac nat_order :=
change (@eq nat) with NatOrder.OrderElts.eq in *;
NatOrder.order.
(** Note that [nat_order] is domain-agnostic: it will not prove
[1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
|
