Hi Rob,
Many thanks for the reply and clarification.
All below makes sense. I can see how effectively excluding terms such as
|- n = 1 /\ n = {1}
would in any case impose a big overhead on run-time, requiring some
dynamic type-checking when rules are applied, etc. To exclude such cases
nonetheless I suppose it will be cheapest to do manual checks in one's
own code where this duplicity can arise, and maybe raise custom error
messages or carry out renaming if feasible.
> ... finding a good name for it seems the hardest problem just now :-)
Maybe something like "match_inst_rule" ?
> Yes. I have been meaning to get round to this for a long time. It
> isn't quite as simple as you might think because the Z tactics are
> designed to keep in Z, but the back-chaining tactics often need to
> produce an existentially quantified goal and that would need to be
> converted from HOL back into Z.
Okay, I see. This happens I suppose when the theorem "asms | ant => suc"
used for backward-chaining contains free variables in the antecedent
which do not occur anywhere else, i.e. the assumptions or conclusion of
the implication used for backward-chaining. As far as I can see the
internal bc_rule wraps such free variables into an existential
quantification after the theorem used for backward-chaining has been
matched and suitably instantiated. So, basically, one needs to
reimplement bc_rule as well to ensure that the subgoal generated
(existential quantifiers) remains in the Z language. I might have a
crack at it next week and send a personal email if any success ;-).
Cheers, Rob, for taking the suggestions below into considerations with
the next release of ProofPower, and thanks for the feedback in general,
Frank
Rob Arthan wrote:
Frank,
On 7 Apr 2009, at 14:39, Frank Zeyda wrote:
Dear Roger,
Many thanks for the reply.
....
The additional feature of better error handling is easily supported
with another line of code handling possible exceptions.
The idea behind the caller parameter in many of these internal functions
was to make the reporting of the function that was the real detector of
the error more precise. In this case, apply_matches_rule isn't doing
anything very subtle so you are right that the calling function could do
it with a simple handler.
Maybe apply_matches_rule could be a nice function to have in the
general interface, what about exposing it?
In many instances in the original development of ProofPower, there were
instances of general purpose functions like this that were nearly
general enough and useful enough to "productise" but not quite (e.g.,
because of the effort involved in getting the error handling completely
general or in documenting exactly what the function does, or even just
thinking up the right name for the function). I will certainly consider
exposing apply_matches_rule - finding a good name for it seems the
hardest problem just now :-)
>> A second case is when y occurs free in both thm and term,
>> is not substituted but nonetheless introduced through the
substitution.
>> I presume this is okay
>> as long as the types of y are identical in thm and term.
>
> Its still OK if the types are not the same, they will be
> logically distinct variables and its a confusing situation
> you shold seek to avoid.
This sounds a little bit curious, is there a document that explains
more about this situation?
There are at least two accounts of the semantics of HOL and they agree
on this point as do the classical references on type theory and on
many-sorted first-order logic. For HOL, I have in mind the account by
Andy Pitts in the Cambridge HOL documentation and my account in HOL in
spc00{1,2,3,4}.doc supplied with ProofPower. It isn't really curious if
you think about it the other way round: how could the semantics possibly
consider two variables with different types to be the same? In Church's
simple type theory and its polymorphic variant HOL, the types are disjoint.
So although the variables have the same name they are actually treated
as logical distinct by the fact that they have different types? So
instantiation would have be sensitive to variable types (not just
names) and might result in one variables n to be substitution, while
another n is left unaffected e.g. if it has a different type?
For example, assume we have a theorem thm
n = 1 |- n = {1}
Then (asm_inst_term_rule [(2, n)] thm) would yield
2 = 1 |- n = {1} ?
Absolutely! Instantiation works just like that (it even has to rename
bound variables if the instantiation would cause a capture problem). But
this is a tiny price to pay: the alternative approaches are very
unattractive: e.g., the abstract data type of terms could ban terms that
used the same variable name with different types but that would impose a
big runtime overhead on the constructors for applications and
lambda-abstraction.
Thanks for pointing this out, it clarified one or two behaviours of
ProofPower for me which I could not explain before.
> I think the main problem you will have is in matching against terms
> containing bound variables.
I haven't given considerations to this, but as far as I can see such a
situation may not arise in the particular application. Thanks for
pointing this out!
Cheers once more,
Frank
PS: What about a function that supports backward-chaining with Z
theorems, something like z_bc_tac and z_bc_thm_tac? I suppose this
could be quite useful too. Is there anything more that needs to be
done other than rewriting the Z universal quantifier into a HOL one,
and using the HOL backward-chaining tactics?
Yes. I have been meaning to get round to this for a long time. It isn't
quite as simple as you might think because the Z tactics are designed to
keep in Z, but the back-chaining tactics often need to produce an
existentially quantified goal and that would need to be converted from
HOL back into Z.
PPS: A final comment. To implement some custom error handling I
noticed that there was no functions that could be used to infer the id
of an error message (of type MESSAGE). Since the MESSAGE datatype is
not exposed, we cannot take the message apart; the only solution seems
to be to dissect the string of the error message. If I'm overlooking
something please let me know, otherwise it would be beneficial to have
some function get_id in the general interface of BasicError to extract
the id of an error message ;-).
You aren't overlooking anything. I have thought that a function like you
get_id would be useful in some circumstances. I think I will add a
function that will just let you take the MESSAGE type apart, but with a
warning that it should be used with caution since ti makes your code
very sensitive to changes in the code you are calling.
By the way, it is very good to see that you are boxing very clever with
ProofPower just now!
Regards,
Rob.
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