Re: [isabelle] Query towards pseudo constructors



Hi Florian and Andreas,

Many thanks for your previous emails. I now have some further confusions.

Suppose I have a type A:

consts P:: "real ⇒ bool"
typedef A="{x::real. P x}" sorry

Then, I provide a pseudo constructor C for A:

consts C:: "nat ⇒ A"
consts C':: "A ⇒ nat"
lemma [code abstype]:"C (C' x) = x" sorry

After that, I define a function g and provide an code equation:

consts g::"A ⇒ bool"
lemma [code]:  "g (C 0) = True" sorry

Now, I want to check code equations for g and see evaluation of g (C 0):

code_thms g
value "g (C 0)"

but both fail due to

"C" is not a constructor, on left hand side of equation, in theorem:
g (C zero_nat_inst.zero_nat) ≡ True

However, if I specify

code_datatype C

then everything goes as expected. I am a little confused by the behaviour, since I was assuming both "code abstype" and "code_datatype" provide a way to construct a pseudo constructor.

Many thanks for any advice,
Wenda

On 2014-09-05 19:18, Florian Haftmann wrote:
Hi all,

this time I strongly hope that I am more awake than when I wrote the
other two mails ;-).

It's indeed not about surjective vs. injective, but about surjective vs.
total.

Have C :: s => t

If C is a total function, then it is suitable for code_datatype.
If C is surjective (but not necessarily total), then it is suitable for
[code abstype].

Note that in the sense of HOL each function is total.  »C is partial«
here means that in generated code C will only be applied to certain
values of s (»abstract datatype«).

Hope this helps,
	Florian


On 04.09.2014 14:41, Andreas Lochbihler wrote:
Hi Wenda and Florian,

Just my 50 cents: pseudo-constructors with code_datatype can be neither
injective nor surjective. List.set and List.coset are examples of this
case.

Andreas

On 04/09/14 14:13, Florian Haftmann wrote:
Hi Wenda,

On 04.09.2014 14:12, Wenda Li wrote:
Hi Florian,

Many thanks for your clarification, it helps a lot.

A pseudo constructor declared using »code_datatype« must be
surjective. A pseudo constructor declared using »[code abstype]« must
be injective.

Do you mean the other way around? Ratreal::"rat => real" in "Real.thy"
(declared using "code_datatype") is an injective but not surjective
function, while Frct:: "int \times int => rat" (declared using "[code abstype]") is a surjective but not injective function. Sorry if I have
messed with any definitions.

thanks for clarifying, I have made a slip here.

Cheers,
    Florian


Best,
Wenda

On 2014-09-04 08:14, Florian Haftmann wrote:
Hi Wenda,

I am not sure which documentation you have read so far. The »Tutorial on code generation« contains some explanations and examples concerning datatype abstraction. There is also a substantial publication »Data
Refinement in Isabelle/HOL«.

In short, it is helpful to thinkg about pseudo constructors as
morphisms. A pseudo constructor declared using »code_datatype« must be surjective. A pseudo constructor declared using »[code abstype]« must
be injective.

Cheers,
     Florian

On 03.09.2014 21:41, Wenda Li wrote:
Dear data refinement experts,

I am trying to learn something about code generation. Based on what I have seen so far, there are two ways to introduce a pseudo constructor
for an abstract type A:
     1) define constant f:: C => A and g::A => C, and then prove
            lemma [code abstype]: f (g x) = x
,and f becomes a pseudo constructor for the abstract type A
     2) define constant f:: C => A, and then declare
            code_datatype f
Based on my observations, only one of them is used at a time:
     "Int.thy": code_datatype "0::int" Pos Neg
     "Real.thy": code_datatype Ratreal
     "Multiset.thy": code_datatype multiset_of
have adopted the second approach, while
     "Rat.thy": lemma [code abstype]:"Frct (quotient_of q) = q"
"Polynomial.thy": lemma Poly_coeffs [simp, code abstype]: "Poly
(coeffs p) = p"
have adopted the first approach.

It seems that the "code_datatype" approach is more flexible than the
"abstype" approach, as multiple constructors can be introduced (I
am not
sure if it is not case for the "abstype" approach) and the cardinality
of C can be smaller than that of A (e.g. Ratreal in "Real.thy").

My question is: what is the main difference between these two
approaches? When shall I choose one over the other?

Many thanks in advance,
Wenda




--
Wenda Li
PhD Candidate
Computer Laboratory
University of Cambridge




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