Re: [isabelle] Defining Union Types
Sum_Type.Projr and Sum_Type.Projl are the destructor view on datatypes, which
are necessarily partial. But most formalisations in Isabelle follow the
constructor view and use case expressions for destruction, for which there is a
reasonable setup (simp rules, split rules, etc.). For example,
> But what is really needed? Haskell does not offer Projl/r at all
but lefts, rights and paritionEithers.
In IsaFoR we use Sum_Type.Projr and Sum_Type.Projl. It would just feel "more
official" if you didn't have to use the cumbersome prefix.
(Sum_Type.Projl x) is (almost) equivalent to (case x of Inl y => y) and
(Sum_Type.Projr x) to (case x of Inr y => y).
I would recommend not to use sum types for such things, but introduce a new type
with an error element. This has the advantage that split rules and the like can
be applied more precisely. If sum types are used for other notions in a
formalisation, too, the general simplification and split rules might slow break
down proof automation because they also apply to the other parts.
A not entirely related idea is the setup of an (executable) error monad using
"+" (which is used heavily in IsaFoR for example).
I suppose that the error messages are irrelevant to the proofs, so they need not
be part of the logic. If the error monad is a type constructor of its own
(rather than a sum type), you can identify all error cases in the logic and
handle the error messages in the code generator only. Here's the idea:
To this end I once tried to setup such a monad to use for partial functions (I
was heading towards a parsec-like parser combinator library; side remark: there
are not many deep properties I wanted to proof about this combinators, but it is
just nice to be able to write also your parser in Isabelle when you use code
generation) but failed to complete since different error cases (i.e., Inl's
containing different error messages) are not equal. Maybe this could be
generalized using some equivalence relation?
datatype 'a err = Error | OK 'a
definition Raise_error :: "String.literal => 'a err"
where "Raise_error msg = Error"
code_datatype Raise_error OK
In the logic, all errors are the same "Error" value, but the generated code uses
Raise_error as constructor which also stores the error message. Hence, the
logical problem with different error messages no longer occurs.
Am 26/07/2011 09:03, schrieb Christian Sternagel:
Talking of the sum type... I think it would be good to have it more
easily accessible. Currently, e.g., I have to write "Sum_Type.Projr" to
get the right projection. As far as I can see it is mainly for internal
use of some packages. But something like Haskell's Either would be
useful for the library (together with a bunch of useful functions and
On 07/26/2011 07:42 AM, Tobias Nipkow wrote:
It is written "+" and defined in theory Sum_Type, which is part of Main.
It is hardly advertised because in most cases it is nicer to define your
own special datatype.
Am 26/07/2011 03:57, schrieb Alfio Martini:
Dear Isabelle Users,
Do we have in Isabelle something like a union (sum) type constructor
the corresponding injections?
I went through the tutorial and did not find use or reference to it. If
there is, can anyone point to an application
or a written example of this?
I assume there must be a simple way to do it.
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