# [isabelle-dev] locales, groups, metric spaces?

Lawrence Paulson lp15 at cam.ac.uk
Tue Apr 16 13:29:08 CEST 2019

```And what about metric spaces themselves? (Not that we could include very much in the next release, but still)
Larry

> On 15 Apr 2019, at 16:31, Fabian Immler <immler at in.tum.de> wrote:
>
>
> On 4/15/2019 5:57 AM, Lawrence Paulson wrote:
>> In the context of the recent discussions about Algebra, we could revisit these issues in the context of metric spaces, which we still don’t have (except as type classes). A metric space has a carrier and a binary relation, so syntactically it’s similar to a monoid, except that we don’t expect to extend one with additional fields. So, at least, we should be able to avoid records.
>> But what about the path from metric spaces to normed vector spaces, etc.?
> Do you mean with explicit carrier sets and without records?
>
> The algebraic part could look like this [1]:
>
>  fixes S::"'a set" and pls::"'a⇒'a⇒'a"
>  assumes add_assoc: "a ∈ S ⟹ b ∈ S ⟹ c ∈ S ⟹ pls (pls a b) c = pls a (pls b c)"
>  assumes add_mem: "a ∈ S ⟹ b ∈ S ⟹ pls a b ∈ S"
>
> Which leads up to the notion of vector space [2]:
> "vector_space_on_with S pls mns um zero (scl::'a::field\<Rightarrow>_)"
> which looks horrible here because of the explicit mention of all of the parameters (I don't recall why). Written as a locale with named parameters, it would look much nicer:
>
> locale vector_space_on_with = ab_group_add_on_with +
>  fixes scl::"'f::comm_ring_1⇒_"
>  assumes "x ∈ S ⟹ y ∈ S ⟹ scl a (pls x y) = pls (scl a x) (scl a y)"  …
>
> For a normed vector space, I guess one would write something like this (assuming that the parameters for the carrier set have the same name in metric_space and vector_space_on_with)
>
> locale normed_vector_space = metric_space + vector_space_on_with +
>  fixes norm::"'a => real"
>  assumes "dist x y = norm (x - y)"
>  assumes "norm (x + y) <= norm x + norm y" ...
>
> Fabian
>
>
> [1] http://isabelle.in.tum.de/repos/isabelle/file/538919322852/src/HOL/Types_To_Sets/Examples/Group_On_With.thy#l12
> [2] http://isabelle.in.tum.de/repos/isabelle/file/538919322852/src/HOL/Types_To_Sets/Examples/Linear_Algebra_On_With.thy#l90
>
>
>> Larry
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>

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