BigQ: starting to create and use an interface QSig

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11028 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 84 insertions, 0 deletions

diff --git a/theories/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v
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+(************************************************************************)
+(*  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        *)
+(************************************************************************)
+
+(*i $Id$ i*)
+
+Require Import QArith Qpower.
+
+Open Scope Q_scope.
+
+(** * QSig *)
+
+(** Interface of a rich structure about rational numbers.
+    Specifications are written via translation to Q.
+*)
+
+Module Type QType.
+
+ Parameter t : Type.
+
+ Parameter to_Q : t -> Q.
+ Notation "[ x ]" := (to_Q x).
+
+ Definition eq x y := [x] == [y].
+
+ Parameter of_Q : Q -> t.
+ Parameter spec_of_Q: forall x, to_Q (of_Q x) == x.
+
+ Parameter zero : t.
+ Parameter one : t.
+ Parameter minus_one : t.
+
+ Parameter spec_0: [zero] == 0.
+ Parameter spec_1: [one] == 1.
+ Parameter spec_m1: [minus_one] == -(1).
+
+ Parameter compare : t -> t -> comparison.
+
+ Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]).
+
+ Definition lt n m := compare n m = Lt.
+ Definition le n m := compare n m <> Gt.
+ Definition min n m := match compare n m with Gt => m | _ => n end.
+ Definition max n m := match compare n m with Lt => m | _ => n end.
+
+ Parameter add  : t -> t -> t.
+
+ Parameter spec_add: forall x y, [add x y] == [x] + [y].
+
+ Parameter sub : t -> t -> t.
+
+ Parameter spec_sub: forall x y, [sub x y] == [x] - [y].
+
+ Parameter opp : t -> t.
+
+ Parameter spec_opp: forall x, [opp x] == - [x].
+
+ Parameter mul : t -> t -> t.
+
+ Parameter spec_mul: forall x y, [mul x y] == [x] * [y].
+
+ Parameter square : t -> t.
+
+ Parameter spec_square: forall x, [square x] == [x] ^ 2.
+
+ Parameter inv : t -> t.
+
+ Parameter spec_inv : forall x, [inv x] == / [x].
+
+ Parameter div : t -> t -> t.
+
+ Parameter spec_div: forall x y, [div x y] == [x] / [y].
+
+ Parameter power_pos : t -> positive -> t.
+
+ Parameter spec_power_pos: forall x n, [power_pos x n] == [x] ^ Zpos n.
+
+End QType.
+
+(* TODO: add norm function and variants, add eq_bool, what about Qc ? *)
\ No newline at end of file