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

Fabian Immler immler at in.tum.de
Mon Apr 15 17:31:02 CEST 2019

```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 :

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 :
"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:

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


http://isabelle.in.tum.de/repos/isabelle/file/538919322852/src/HOL/Types_To_Sets/Examples/Group_On_With.thy#l12

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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