[isabelle-dev] (Re-)introducing set as a type constructor rather than as mere abbreviation

[Brian Huffman]

Here's one possible approach to the set-axiomatization/typedef issue:

As a new primitive, we could have something like a "pred_typedef"
operation that uses predicates. This would work just like the
new_type_definition facility of HOL4, etc.

The type "'a set" could be introduced definitionally using "pred_typedef".

After type "'a set" has been defined, we can implement the "typedef"
command, preserving its current behavior, as a thin wrapper on top of
"pred_typedef".

This approach would let us avoid having to axiomatize the 'a set type.
Also, for those of us who like predicates, "pred_typedef" might be
rather useful as a user-level command.

- Brian

On Fri, Aug 26, 2011 at 12:06 AM, Lawrence Paulson <lp15 at cam.ac.uk> wrote:
> indeed yes I'm the person who decided that this primitive should introduce a type as a copy of an existing non-empty set. I have always preferred sets to predicates and the examples I have looked at lately have only strengthened my view. Not to mention numerous occasions when people have re-invented the notion of an image.
> Larry
>
> On 25 Aug 2011, at 23:24, Michael Norrish wrote:
>
>> On 26/08/11 7:26 AM, Florian Haftmann wrote:
>>> Hi Andreas,
>>>
>>>> Let me ask something stupid:
>>>>  why exactly do you guys axiomatize 'a set?
>>>> Sure it's admissable and all, but why not do this definitionally
>>>> via the obvious typedef?
>>>
>>> the answer is rather technical: »typedef« in its current implementation
>>> needs set syntax / set type as prerequisite.  Of course you could change
>>> »typedef« to be based on predicates, but there is some kind of unspoken
>>> agreement not to set one's hand on that ancient and time-honoured Gordon
>>> HOL primitive.
>>
>> HOL88, hol90, hol98 and HOL4 all have new_type_definition.  This
>> principle takes a theorem of the form
>>
>>  ?x. P x
>>
>> HOL Light takes a theorem of the form  P x (removing the dependency on
>> the existential quantifier).
>>
>> To the best of my knowledge, none of these systems ever used sets in
>> this role.
>>
>> Michael
>>
>>
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