Re: [isabelle] Code setup for Fraction_Field

> 1. I noticed that there are the type classes euclidean_ring and
> euclidean_ring_gcd. Most of the theorems about gcd which I need are in
> euclidean_ring_gcd, but not in euclidean_ring itself. Unfortunately,
> the instantiations for nat, int, etc. are only done for
> euclidean_ring, not for euclidean_ring_gcd. What are the plans for
> these type classes? Should I spend any effort on adding instances for
> euclidean_ring_gcd? Or will this become obsolete with the polishing
> that is still due? 

euclidean_ring_gcd is nothing but euclidean_ring with the additional
assumption that gcd and Gcd (which are introduced in a syntactical
typeclass and therefore have no a priori properties associated with
them) are the same as the euclidean GCD (and the same for LCM).

So, in a sense, there is usually no proof work to be done to make
anything an instance of euclidean_ring_gcd if you already have
euclidean_ring; you can just define "gcd = gcd_eucl", "Gcd = Gcd_eucl"
and so on and you get the instance for free.

If there is already an existing definition of gcd, you have to prove its
equivalence to the Euclidean GCD etc., which you can do by showing that
it fulfils the defining properties of the GCD and that it returns a
normalised GCD, i.e. "1" and not "-1" for integers.

As for whether this will become obsolete: I don't think so. I don't
quite remember what Florian had planned for this theory.



On 04/09/15 11:32, Andreas Lochbihler wrote:
> Hi Florian and Manuel,
> I have experimented a bit with a normalisation function to be used in
> the operations over fraction fields and had a look at
> Euclidean_Algorithm in Number_Theory.
> 2. I am no longer sure that an invariant-based approach is the optimal
> thing for fraction field. In the code equation for the arithmetic
> operation, I don't see any big opportunities to exploit the invariant
> that nominator and denominator are normalised in some unspecified
> form. Only the equality tests become simpler (as normal forms are
> unique and we thus save two multiplications). Instead, it seems much
> simpler to treat cancellation of common factors as an optimisation
> without logical significance. For example, we could just add a call to
> a simplification function before calls to the Fract constructor. The
> simplification function only has to return an element of the same
> equivalence class, so for nat and int, we could use a normalisation
> function based on Euclid's algorithm. For other types, the
> simplification function could also just be the identity. What do you
> think?
> Best,
> Andreas
> On 27/08/15 12:02, Florian Haftmann wrote:
>> Hi Andreas and Prathamesh,
>>> We noticed that the code setup for the theory Fraction_Field in
>>> HOL/Library is broken, at least since Isabelle2013. We would like to
>>> use
>>> code generation in his formalisation of knot theory where the elements
>>> of the fraction fields are polynomials over integers. I had a brief
>>> look
>>> at the theory Fraction_Field and noticed that there is a smart
>>> pseudo-constructor Fract, which is declared as code_datatype. So this
>>> feels as if the code generation was working at some time in the distant
>>> past and got broken somewhen.
>>> Does anyone know about the status of Fraction_Field?
>> I dimly remember that initially Fraction_Field just took over the
>> then-used code setup from the rationals, and maybe it never worked as
>> intended (would need a closer investigation to find out actually).
>> When code generation had been equipped with invariants, I revisited
>> Fraction_Field to make a code setup close to the rationals but soon
>> realized thatâ
>>> I believe that it would be fairly easy to "fix" the problem of code
>>> generator setup by deriving a few code equation from the lemmas, but
>>> this will clearly result in suboptimal code for two reasons. First, we
>>> need tests whether the denominator is 0 all over the place. Second, the
>>> elements of the fraction fields will not be normalised automatically.
>> âI need a generalized gcd for that, and so I let everything stand as
>> it is.
>>> For the application on polynomials, this means that we would need some
>>> Euclidean algorithm for cancelling common factors of polynomials. I
>>> dimly remember that there is some support for the Euclidean
>>> algorithm in
>>> the pipeline. What is the status there?
>> I think you can build on the corresponding theory in Number_Theory, but
>> that needs some further rounds of polishing before I would recommend to
>> turn it into a Âhard dependency.  Instead I suggest to put the code
>> setup into a separate theory ÂExecutable_Fraction_FieldÂ.  This is what
>> we did 8 years ago when code generation was still highly experimentalâ
>> Hope this helps,
>>     Florian
>>> Best,
>>> Andreas and Prathamesh

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