# [isabelle-dev] Problem with factorial-ring in combination with containers

Manuel Eberl eberlm at in.tum.de
Mon Oct 3 15:29:48 CEST 2016

```This is a problem that I have given quite some thought in the past. The
problem is the following: You have a theory A providing certain
operations on sets (in this case: Gcd) and a theory B providing
implementations for sets (in this case: Containers).

The problem is that the code equations for the operations from A depend
on the implementation that is chosen for sets. A cannot give code
equations for every possible implementation of sets, while B cannot
possibly import every theory that has operations involving sets and give
code equations for it.

The best possible solution would be to imitate the way it is currently
done for setsum, setprod, etc: Define a sufficiently general combinator
that iterates over the set and give the code equations in A in terms of
this combinator. Then B only has to reimplement this generic combinator.

That would be the cleanest solution, but I'm not sure how such a
combinator would look like. The folding operation would probably have to
satisfy some associativity/commutativity laws and have that information
available at the type level (similar to the cfc type in Containers).

By the way, my current workaround for this problem is to declare all
problematic constants as "code_abort".

Cheers,

Manuel

On 03/10/16 15:21, Thiemann, Rene wrote:
> Dear all,
>
> in the following theory, the export-code fails:
>
> (Isabelle 957ba35d1338, AFP 618f04bf906f)
>
> theory Problem
>   imports
>   "~~/src/HOL/Library/Polynomial_Factorial"
>   "\$AFP/thys/Containers/Set_Impl"
> begin
>
> definition foo :: "'a :: factorial_semiring_gcd ⇒ 'a ⇒ 'a" where
>   "foo x y = gcd y x"
>
> definition bar :: "int ⇒ int" where
>   "bar x = foo x (x - 1)"
>
> end
>
>
> The problem arises from two issues:
> - factorial_semiring_gcd requires code for
>   Gcd :: “Set ‘a => ‘a”, not only for the binary “gcd :: ‘a => ‘a => ‘a”!
> - the code-equation for Gcd is Gcd_eucl_set: "Gcd_eucl (set xs) = foldl gcd_eucl 0 xs”
>   where “set” is only a constructor if one does not load the container-library.
>
> It would be nice, if one can either alter factorial_semiring_gcd so that it does not
> require “Gcd” anymore, or if the code-equation is modified in a way that permits
> co-existence with containers. (Of course, similarly for Lcm).
>
>
> With best regards,
> Akihisa, Sebastiaan, and René
>
> PS: We currently solve the problem by disabling Gcd and Lcm as follows:
>
> lemma [code]: "(Gcd_eucl :: rat set ⇒ rat) = Code.abort (STR ''no Gcd on sets'') (λ _. Gcd_eucl)"  by simp
> lemma [code]: "(Lcm_eucl :: rat set ⇒ rat) = Code.abort (STR ''no Lcm on sets'') (λ _. Lcm_eucl)"  by simp
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```