What Kind of Type Definitions are Legal in Local Contexts?

问题

In Isabelle's NEWS file, I found

Command 'typedef' now works within a local theory context -- without introducing dependencies on parameters or assumptions, which is not possible in Isabelle/Pure/HOL. Note that the logical environment may contain multiple interpretations of local typedefs (with different non-emptiness proofs), even in a global theory context.

(which dates back to Isabelle2009-2). Is this the latest news with respect to typedef and local theory contexts? Further, what does the restriction "without introducing dependencies on parameters or assumptions" actually mean?

If it would mean that I cannot use locale parameters in the defining set of a typedef, then I would not consider typedef to be localized at all (since the only allowed instances can easily be moved outside the local context, or am I missing something?).

Is it (or should it, or will it ever be) possible to do something along the lines (where the set used for a typedef depends on the locale parameter V):

datatype ('a, 'b) "term" = Var 'b | Fun 'a "('a, 'b) term list"

locale term_algebra =
  fixes F :: "'a set"
    and V :: "'b set"
begin

definition "domain α = {x : V. α x ~= Var x}"

typedef ('a, 'b) subst =
  "{α :: 'b => ('a, 'b) term. finite (domain α)}"

end

for which I currently obtain:

Locally fixed type arguments "'a", "'b" in type declaration "subst"

回答1:

A few more notes on this:

The local theory infrastructure merely organizes existing module concepts (locale, class etc.) such that definitional mechanisms (definition, theorem, inductive, function etc.) can work uniformly in a variety of contexts. This does not change the logical foundations, so specification elements that cannot depend on term parameters (fixes) or premises (assumes) won't become fundamentally better. They are merely retrofitted into the greater framework, which is already an extra-logical benefit.
The canonical local theory targets are locale and its derivatives like class. These work within the logic according to the principles sketched above: lambda-lifting via some context of fixes and assumes. Other targets with more ambition can be imagined, but need to be implemented by some brave and heroic guys. For example, one could wrap up the AWE theory interpretation mechanism as another local theory target, and then get parametrization of types/consts/axioms --- at the usual cost of going through explicit proof terms to implement admissible inferences within an LCF prover (or at the cost of giving up LCF-ness and do it via some oracle).
Plain typedef as sketched above (and its derivatives like the localized codatatype and datatype of recent HOL-BNF) could be slightly improved in its dependecy type parameters, but this would mean some implementation efforts that don't justify the meagre outcome right now. It would merely allow to write polymorphic type constructions with implicit arguments like this:
```
context fixes type 'a
begin
datatype list = Nil | Cons 'a list
end
```
After export you would get 'a list as usual.

Further complication: fixes type 'a does not exist. Isabelle/Pure handles type parameters implicitly via Hindley-Milner.

回答2:

In the meanwhile, this question was answered on the Isabelle mailing list. The short version is that what I tried to do is simply impossible. What follows is an explanation by @makarius:

[Without introducing dependencies on parameters or assumptions] means you cannot refer to the fixes/assumes of context in the type specification -- this is logically impossible in Isabelle/Pure/HOL. Only the non-emptyness proof lives within the context, and the resulting theorems are local. (Actual dependence of the proof on logical content of the context is hard to get in practice, though.)

'typedef' is formally localized within the range of what is possible. Localization means to work with the local theory infrastructure and the context in the usual ways. For typedef this means extra-logical things like name spaces, syntax, derived declarations etc.

Historically, due to the impossibility to make typedef depend on the logical part of the context, it was not localized at all, and many tool implementations suffer from that until today.

As for my specific example:

You would have to evade the scope clash [...] by using different names for the parameters of type subst. Nonetheless, this does not work from a logical standpoint: the dependency on term parameter V cannot be used in HOL typedef. The local theory concept does not provide magic ways to augment the logic -- it is merely infrastructure for an existing logical framework.

来源：https://stackoverflow.com/questions/16556633/what-kind-of-type-definitions-are-legal-in-local-contexts

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