Re: [isabelle] [EXTERNAL] Non-idempotence of datatype constructors

isn’t the “size” theorem true for any function, not just size?  or am i missing something?

> On May 1, 2020, at 1:51 PM, Manuel Eberl <eberlm at> 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

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