[isabelle] Semirings in HOL/Algebra/Ring



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 â carrier R; y â carrier R; z â carrier R |]
      ==> (x â y) â z = x â z â y â z"
    and r_distr: "[| x â carrier R; y â carrier R; z â carrier R |]
      ==> z â (x â y) = z â x â z â y"
    and l_null[simp]: "x â carrier R ==> ð â x = ð"
    and r_null[simp]: "x â carrier R ==> x â ð = ð"

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,
RenÃ





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