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

Yes, it is true for any function (even if it isn't injective, although
then it's not an "if-and-only-if").

The point is that if you actually want it to use this to prove that two
lists (or data type values in general) are not equal, you need to plug
in a concrete function f. And "size" happens to be one that is good for
this particular case.

Sometimes less general theorems can achieve more because they guide the
automation more (and us humans, too!). That is particularly true for
induction rules.


On 01/05/2020 22:59, Miguel Pagano wrote:
> On Fri, 1 May 2020 at 17:58, Richard Waldinger <waldinger at
> <mailto:waldinger at>> wrote:
>     isn’t the “size” theorem true for any function, not just size?  or
>     am i missing something?
> It shouldn't be true for non-injective functions.
>     > On May 1, 2020, at 1:51 PM, Manuel Eberl <eberlm at
>     <mailto: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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