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Diffstat (limited to 'theories/NArith/NOrderedType.v')
-rw-r--r-- | theories/NArith/NOrderedType.v | 60 |
1 files changed, 60 insertions, 0 deletions
diff --git a/theories/NArith/NOrderedType.v b/theories/NArith/NOrderedType.v new file mode 100644 index 00000000..c5dd395b --- /dev/null +++ b/theories/NArith/NOrderedType.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 BinNat Equalities Orders OrdersTac. + +Local Open Scope N_scope. + +(** * DecidableType structure for [N] binary natural numbers *) + +Module N_as_UBE <: UsualBoolEq. + Definition t := N. + Definition eq := @eq N. + Definition eqb := Neqb. + Definition eqb_eq := Neqb_eq. +End N_as_UBE. + +Module N_as_DT <: UsualDecidableTypeFull := Make_UDTF N_as_UBE. + +(** Note that the last module fulfills by subtyping many other + interfaces, such as [DecidableType] or [EqualityType]. *) + + + +(** * OrderedType structure for [N] numbers *) + +Module N_as_OT <: OrderedTypeFull. + Include N_as_DT. + Definition lt := Nlt. + Definition le := Nle. + Definition compare := Ncompare. + + Instance lt_strorder : StrictOrder Nlt. + Proof. split; [ exact Nlt_irrefl | exact Nlt_trans ]. Qed. + + Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Nlt. + Proof. repeat red; intros; subst; auto. Qed. + + Definition le_lteq := Nle_lteq. + Definition compare_spec := Ncompare_spec. + +End N_as_OT. + +(** Note that [N_as_OT] can also be seen as a [UsualOrderedType] + and a [OrderedType] (and also as a [DecidableType]). *) + + + +(** * An [order] tactic for [N] numbers *) + +Module NOrder := OTF_to_OrderTac N_as_OT. +Ltac n_order := NOrder.order. + +(** Note that [n_order] is domain-agnostic: it will not prove + [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *) + |