Email Gateway (Aug 19 2022 at 17:32): Email Gateway (Aug 19 2022 at 17:32):From: "Thiemann, Rene" <Rene.Thiemann@uibk.ac.at>
Dear all,
I wonder whether it is worthwhile to include the notion of a semiring into HOL/Algebra/Ring.
My motivation is an extension of the current AFP/Matrix-entry such that the elements of the matrices
don't have to be class-instances of class semiring, but that they are connected via a locale
semiring. However, currently such a locale does not exist
(only a locale for rings is defined, which rules out the natural numbers), one cannot conveniently study
the semiring of matrices of the natural numbers.
To this end, I would like to modify the locale-structure in HOL-algebra as follows:
An additional locale semiring:
locale semiring = abelian_monoid R + monoid R for R (structure) +
assumes l_distr: "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> (x ⊕ y) ⊗ z = x ⊗ z ⊕ y ⊗ z"
and r_distr: "[| x ∈ carrier R; y ∈ carrier R; z ∈ carrier R |]
==> z ⊗ (x ⊕ y) = z ⊗ x ⊕ z ⊗ y"
and l_null[simp]: "x ∈ carrier R ==> 𝟬 ⊗ x = 𝟬"
and r_null[simp]: "x ∈ carrier R ==> x ⊗ 𝟬 = 𝟬"
Prove the sublocale-property:
context ring
sublocale ring <= semiring
...
Prove several properties like finsum_ldistr already in semiring.
Of course, I can also just copy HOL/Algebra/Ring and modify it locally,
but I believe that semirings should be valuable for other Isabelle users, too.
If desired, I can update a recent repository version and discuss this change further on Isabelle-dev.
Any comments are welcome,
René
Email Gateway (Aug 19 2022 at 17:32):From: Larry Paulson <lp15@cam.ac.uk>
Looks like a good idea to me. Are there any drawbacks?
Larry Paulson
Last updated: Nov 21 2024 at 12:39 UTC