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Diffstat (limited to 'theories/Arith/NatOrderedType.v')
-rw-r--r-- | theories/Arith/NatOrderedType.v | 64 |
1 files changed, 0 insertions, 64 deletions
diff --git a/theories/Arith/NatOrderedType.v b/theories/Arith/NatOrderedType.v deleted file mode 100644 index fb4bf233..00000000 --- a/theories/Arith/NatOrderedType.v +++ /dev/null @@ -1,64 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -Require Import Lt Peano_dec Compare_dec EqNat - Equalities Orders OrdersTac. - - -(** * DecidableType structure for Peano numbers *) - -Module Nat_as_UBE <: UsualBoolEq. - Definition t := nat. - Definition eq := @eq nat. - Definition eqb := beq_nat. - Definition eqb_eq := beq_nat_true_iff. -End Nat_as_UBE. - -Module Nat_as_DT <: UsualDecidableTypeFull := Make_UDTF Nat_as_UBE. - -(** Note that the last module fulfills by subtyping many other - interfaces, such as [DecidableType] or [EqualityType]. *) - - - -(** * 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_irrefl | exact lt_trans ]. Qed. - - Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt. - Proof. repeat red; intros; subst; auto. Qed. - - Definition le_lteq := le_lt_or_eq_iff. - Definition compare_spec := nat_compare_spec. - -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 := 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]. *) - -Section Test. -Let test : forall x y : nat, x<=y -> y<=x -> x=y. -Proof. nat_order. Qed. -End Test. |