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author | Stephane Glondu <steph@glondu.net> | 2010-07-21 09:46:51 +0200 |
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committer | Stephane Glondu <steph@glondu.net> | 2010-07-21 09:46:51 +0200 |
commit | 5b7eafd0f00a16d78f99a27f5c7d5a0de77dc7e6 (patch) | |
tree | 631ad791a7685edafeb1fb2e8faeedc8379318ae /theories/NArith/POrderedType.v | |
parent | da178a880e3ace820b41d38b191d3785b82991f5 (diff) |
Imported Upstream snapshot 8.3~beta0+13298
Diffstat (limited to 'theories/NArith/POrderedType.v')
-rw-r--r-- | theories/NArith/POrderedType.v | 60 |
1 files changed, 60 insertions, 0 deletions
diff --git a/theories/NArith/POrderedType.v b/theories/NArith/POrderedType.v new file mode 100644 index 00000000..9c0f8261 --- /dev/null +++ b/theories/NArith/POrderedType.v @@ -0,0 +1,60 @@ +(************************************************************************) +(* 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 BinPos Equalities Orders OrdersTac. + +Local Open Scope positive_scope. + +(** * DecidableType structure for [positive] numbers *) + +Module Positive_as_UBE <: UsualBoolEq. + Definition t := positive. + Definition eq := @eq positive. + Definition eqb := Peqb. + Definition eqb_eq := Peqb_eq. +End Positive_as_UBE. + +Module Positive_as_DT <: UsualDecidableTypeFull + := Make_UDTF Positive_as_UBE. + +(** Note that the last module fulfills by subtyping many other + interfaces, such as [DecidableType] or [EqualityType]. *) + + + +(** * OrderedType structure for [positive] numbers *) + +Module Positive_as_OT <: OrderedTypeFull. + Include Positive_as_DT. + Definition lt := Plt. + Definition le := Ple. + Definition compare p q := Pcompare p q Eq. + + Instance lt_strorder : StrictOrder Plt. + Proof. split; [ exact Plt_irrefl | exact Plt_trans ]. Qed. + + Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Plt. + Proof. repeat red; intros; subst; auto. Qed. + + Definition le_lteq := Ple_lteq. + Definition compare_spec := Pcompare_spec. + +End Positive_as_OT. + +(** Note that [Positive_as_OT] can also be seen as a [UsualOrderedType] + and a [OrderedType] (and also as a [DecidableType]). *) + + + +(** * An [order] tactic for positive numbers *) + +Module PositiveOrder := OTF_to_OrderTac Positive_as_OT. +Ltac p_order := PositiveOrder.order. + +(** Note that [p_order] is domain-agnostic: it will not prove + [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *) |