Re: [isabelle] Non-idempotence of datatype constructors



On 03/05/2020 11:46, Traytel Dmitriy wrote:
> while your code works for the examples you show, it does not seem to handle nested recursion.

Yes, I am aware of that. Nested datatypes make things considerably more
complicated, so I'd rather not support that for now. I'd have to
1. somehow figure out /all/ the other datatypes involved in the
definition of my datatype and
2. figure out a robust way to expand their size functions.

I have no idea how to do 1 with the information provided by the BNF
package (other than inspecting the types of the constructors to figure
out the nesting).

I also have no idea how to do 2 properly – the "size" plugin provides
some information with BNF_LFP_Size.size_of, but oddly enough it only
provides the equations for the "regular" size function "size", but not
for the generalised one (e.g. "size_list"), which appear in the size
functions of nested datatypes.

Note that I really do want to do 2, because relying on the simplifier
itself to solve the precondition "size lhs ≠ size rhs" is fragile and
breaks at least in some cases in the AFP.


> I am sympathetic to the proposal of having a 'proper
> subexpression' ordering defined for
> all datatypes (e.g., via a plugin similarly to size).
> Its usefulness goes beyond the
> acyclicity rules which this thread is about.

Absolutely! It's probably no surprise to you when I say that I won't
implement this. But if you (or one of our other datatype experts) ever
does it, do let me know!

> But one would like to have some reasonable simp rules for subexp_x
> (which may be as hard as the original problem that you are trying to solve). 
> In particular, if F is itself a datatype that belongs to the subexp type class, 
> its notion of subexp should be linked to the one of x.

Already for mutual and nested datatypes, you would probably need an
entire family of relations for each combination of types (e.g. if you
have mutually recursive datatypes A and B, you would probably need
subexpr_A_A, subexpr_A_B, subexpr_B_A, subexpr_B_B).

I guess everything gets complicated once you involve nested datatypes…

Manuel



 See code below, with your simproc enabled. Also I would register it
only for types that belong to the size class, i.e.,
> 
> simproc_setup datatype_no_proper_subterm ("(x :: 'a :: size) = y") = ‹K datatype_no_proper_subterm_simproc›
> 
> Your retrieval of mutual types looks reasonable to me. As usual with Isabelle/ML, the most reliable documentation is the code.
> 
> I am sympathetic to the proposal of having a 'proper subexpression' ordering defined for all datatypes (e.g., via a plugin similarly to size). Its usefulness goes beyond the acyclicity rules which this thread is about. In particular, the 'proper subexpression' ordering can be used for 'strong induction' or to prove termination of functions in cases when size does not exists. (Provided that we also have the fact that this ordering is wellfounded proved.) Something along the following lines:
> 
> class subexp =
>   fixes subexp :: "'a ⇒ 'a ⇒ bool" (infixl "⊏" 65)
>   assumes wf: "wfP subexp"
> 
> declare [[typedef_overloaded]]
> 
> bnf_axiomatization 'a F [wits: "'a F"]
> 
> datatype x = Ctor "x F"
> 
> instantiation x :: subexp begin
> 
> definition subexp_x :: "x ⇒ x ⇒ bool" where
>   "subexp_x = tranclp (λx y. case y of Ctor f ⇒ x ∈ set_F f)"
> 
> instance
>   apply intro_classes
>   apply (unfold subexp_x_def)
>   apply (rule wfP_trancl)
>   apply (rule wfPUNIVI)
>   subgoal premises prems for P x
>     apply (induct x)
>     apply (rule prems[rule_format])
>     apply (simp only: x.case)
>     done
>   done
> 
> end
> 
> But one would like to have some reasonable simp rules for subexp_x (which may be as hard as the original problem that you are trying to solve). In particular, if F is itself a datatype that belongs to the subexp type class, its notion of subexp should be linked to the one of x.
> 
> Dmitriy
> 
> datatype 'a rtree = Leaf | Node 'a "'a rtree list"
> 
> lemma "Node x (a # xs) ≠ a"
>   apply simp? ―‹no_change›
>   apply (rule size_neq_size_imp_neq)
>   apply simp
>   done
> 
> lemma "Node x [c,a,b] ≠ a"
>   apply simp? ―‹no_change›
>   apply (rule size_neq_size_imp_neq)
>   apply simp
>   done
> 
> 
> 
> On 2 May 2020, at 18:04, Manuel Eberl <eberlm at in.tum.de<mailto:eberlm at in.tum.de>> wrote:
> 
> I attached a proof of concept (works with Isabelle 2020) using the
> simple size-based approach, including some example applications.
> 
> It works well, although I'm not sure what the proper way to get the
> datatype information is (e.g. the list of all the constructors of the
> datatype and the associated other datatypes in case of mutual recursion).
> 
> Is the ML interface of the BNF package documented anywhere (in
> particular this aspect)?
> 
> Manuel
> 
> 
> On 02/05/2020 16:19, Manuel Eberl wrote:
> True, but after your suggestion, I realised that the solution with the
> "proper subexpression" relation (or, alternatively, the size function)
> combined with a simproc that produces these theorems on the spot is
> actually the superior approach in every respect. It's simpler, more
> general, and probably more performant.
> 
> I can try to come up with a proof-of-concept implementation using the
> size function approach, since that needs no additional new features from
> the BNF package.
> 
> Manuel
> 
> 
> On 02/05/2020 16:16, Tobias Nipkow wrote:
> A first version which only deals with depth 1 would already cover most
> of the practical cases.
> 
> Tobias
> 
> On 02/05/2020 15:50, Manuel Eberl wrote:
> That sounds like a good idea.
> 
> However, if such a simproc is to work for any nesting of
> constructors,having pre-proven theorems for every constructor will not
> be enough.Instead, I suppose one would have to introduce a
> "proper-subexpression"relation for datatypes (e.g. xs < Cons x xs) along
> with a proof thatthis relation has the obvious properties (irreflexive,
> asymmetric,transitive).
> 
> I guess that is something that only a datatype package plugin similar
> tothe one for the "size" function could provide. Having looked at the
> codebriefly, I think only the people who wrote the BNF package could (or
> atleast should) implement that.
> 
> Alternatively, one could just use the size function (as someone
> alreadysuggested in this thread) to get pretty much the same thing,
> except thatit won't work for all datatypes (e.g. infinitely branching
> ones).
> 
> Manuel
> 
> 
> On 02/05/2020 15:36, Tobias Nipkow wrote:
> I do think such rules are useful, esp if they are there by default. I
> suggest they are best handled by a simproc that is triggered by any
> "(=)" but that checks immediately if the two argumenst are of the
> appropriate type and form. That can be done very quickly (there are
> similar simprocs already). The simproc should work for any datatype and
> any degree of nesting of the constructors.
> 
> Tobias
> 
> 
> On 01/05/2020 22:51, Manuel Eberl wrote:
> Firstly, I don't think these theorem is especially useful. You might
> have planned to add this to the simplifier, but its term net
> doesn't do
> any magic here. It will end up checking every term that matches
> "Cons x
> xs = ys" for whether "xs" can match "ys". I'm not sure if that
> matching
> is equality, alpha-equivalent or unifiable.
> 
> I honestly never think /that/ much about the performance
> implications of
> simp rules (as long as they're unconditional). At least for lists, by
> the way, this is already a simp rule by default though, and lists are
> probably by far the most prevalent data type in the Isabelle universe.
> 
> But you're certainly right that it would make sense to keep a look at
> this performance impact if one wanted to add these to the simp set for
> all datatypes by default, and I agree that the rules are probably not
> helpful /that/ often. Still, it might be nice to have them available
> nonetheless.
> 
> Secondly, you can prove these theorems by using this handy trivial
> theorem : "size x ~= size y ==> x ~= y". Apparently that theorem
> has the
> name  Sledgehammer.size_ne_size_imp_ne - presumably the sledgehammer
> uses it to prove such inequalities.
> 
> It's even easier to prove it by induction. Plus, in fact, the "size"
> thing only works if the data type even has a sensible size function.
> That is not always the case, e.g. when you nest the datatype through a
> function.
> 
> My main point however is that when you have a datatype with a dozen
> binary constructors, there's quite a bit of copying & pasting involved
> before you've proven all those theorems. Since it can (probably) be
> automated very easily, why not do that? Whether or not these should be
> simp lemmas by default is another question.
> 
> Manuel
> 
> 
> 
> 
> 
> <Foo.thy>
> 




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