[isabelle-dev] locales, groups, metric spaces?
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 :
locale semigroup_add_on_with =
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:
locale vector_space_on_with = ab_group_add_on_with +
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" ...
> isabelle-dev mailing list
> isabelle-dev at in.tum.de
-------------- next part --------------
A non-text attachment was scrubbed...
Size: 5581 bytes
Desc: S/MIME Cryptographic Signature
More information about the isabelle-dev