Library mathcomp.analysis.prodnormedzmodule

From mathcomp Require Import all_ssreflect fingroup ssralg poly ssrnum.
Require Import signed.

This file equips the product of two normedZmodTypes with a canonical normedZmodType structure. It is a short file that has been added here for convenience during the rebase of MathComp-Analysis on top of MathComp 1.1. The contents is likely to be moved elsewhere.

Set Implicit Arguments.

Local Open Scope ring_scope.
Import Order.TTheory GRing.Theory Num.Theory.

Module ProdNormedZmodule.
Section ProdNormedZmodule.
Context {R : numDomainType} {U V : normedZmodType R}.

Definition norm (x : U × V) : R := Num.max `|x .1| `|x .2|.

Lemma normD x y : norm (x + y) ≤ norm x + norm y.

Lemma norm_eq0 x : norm x = 0 → x = 0.

Lemma normMn x n : norm (x *+ n) = (norm x) *+ n.

Lemma normrN x : norm (- x) = norm x.

Definition normedZmodMixin :
@Num.normed_mixin_of R [zmodType of U × V] (Num.NumDomain.class R) :=
@Num.NormedMixin _ _ _ norm normD norm_eq0 normMn normrN.

Canonical normedZmodType := NormedZmodType R (U × V) normedZmodMixin.

Lemma prod_normE (x : normedZmodType) : `|x| = Num.max `|x .1| `|x .2|.

End ProdNormedZmodule.

Module Exports.
Canonical normedZmodType.
Definition prod_normE := @prod_normE.
End Exports.

End ProdNormedZmodule.
Export ProdNormedZmodule.Exports.