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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/Reals/ROrderedType.v | |
parent | da178a880e3ace820b41d38b191d3785b82991f5 (diff) |
Imported Upstream snapshot 8.3~beta0+13298
Diffstat (limited to 'theories/Reals/ROrderedType.v')
-rw-r--r-- | theories/Reals/ROrderedType.v | 95 |
1 files changed, 95 insertions, 0 deletions
diff --git a/theories/Reals/ROrderedType.v b/theories/Reals/ROrderedType.v new file mode 100644 index 00000000..2b302386 --- /dev/null +++ b/theories/Reals/ROrderedType.v @@ -0,0 +1,95 @@ +(************************************************************************) +(* 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 Rbase Equalities Orders OrdersTac. + +Local Open Scope R_scope. + +(** * DecidableType structure for real numbers *) + +Lemma Req_dec : forall r1 r2:R, {r1 = r2} + {r1 <> r2}. +Proof. + intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse; + intuition eauto 3. +Qed. + +Definition Reqb r1 r2 := if Req_dec r1 r2 then true else false. +Lemma Reqb_eq : forall r1 r2, Reqb r1 r2 = true <-> r1=r2. +Proof. + intros; unfold Reqb; destruct Req_dec as [EQ|NEQ]; auto with *. + split; try discriminate. intro EQ; elim NEQ; auto. +Qed. + +Module R_as_UBE <: UsualBoolEq. + Definition t := R. + Definition eq := @eq R. + Definition eqb := Reqb. + Definition eqb_eq := Reqb_eq. +End R_as_UBE. + +Module R_as_DT <: UsualDecidableTypeFull := Make_UDTF R_as_UBE. + +(** Note that the last module fulfills by subtyping many other + interfaces, such as [DecidableType] or [EqualityType]. *) + + + +(** Note that [R_as_DT] can also be seen as a [DecidableType] + and a [DecidableTypeOrig]. *) + + + +(** * OrderedType structure for binary integers *) + + + +Definition Rcompare x y := + match total_order_T x y with + | inleft (left _) => Lt + | inleft (right _) => Eq + | inright _ => Gt + end. + +Lemma Rcompare_spec : forall x y, CompSpec eq Rlt x y (Rcompare x y). +Proof. + intros. unfold Rcompare. + destruct total_order_T as [[H|H]|H]; auto. +Qed. + +Module R_as_OT <: OrderedTypeFull. + Include R_as_DT. + Definition lt := Rlt. + Definition le := Rle. + Definition compare := Rcompare. + + Instance lt_strorder : StrictOrder Rlt. + Proof. split; [ exact Rlt_irrefl | exact Rlt_trans ]. Qed. + + Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Rlt. + Proof. repeat red; intros; subst; auto. Qed. + + Lemma le_lteq : forall x y, x <= y <-> x < y \/ x = y. + Proof. unfold Rle; auto with *. Qed. + + Definition compare_spec := Rcompare_spec. + +End R_as_OT. + +(** Note that [R_as_OT] can also be seen as a [UsualOrderedType] + and a [OrderedType] (and also as a [DecidableType]). *) + + + +(** * An [order] tactic for real numbers *) + +Module ROrder := OTF_to_OrderTac R_as_OT. +Ltac r_order := ROrder.order. + +(** Note that [r_order] is domain-agnostic: it will not prove + [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *) + |