# [isabelle-dev] cardinality primitives in Isabelle/HOL?

Traytel Dmitriy traytel at inf.ethz.ch
Sun Dec 30 14:26:34 CET 2018

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> On 28 Dec 2018, at 19:36, Lawrence Paulson <lp15 at cam.ac.uk> wrote:
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> Tons of useful stuff here.
>
> Some syntactic ambiguities, particularly around the =o relation, which is also defined as Set_Algebras.elt_set_eq.

True. The same holds for +o and *o (which are less widely used in the ordinals library).

I'm not sure what a good naming is in these cases. "=o" makes sense for me as "equality on ordinals". It would be ok for me to play with capitalization or other abbreviations of ordinals, e.g., "=O" or "=ord". Maybe Andrei can comment, as he picked the current notation.

I don't know what "=o" stands for in the case of Set_Algebras.elt_set_eq (is this a reference to the Oh-notation?) and why there is need for an alternative notation for ∈. I believe the better textbooks are those that write "f ∈ O(g)" instead of "f = O(g)".

Along similar lines, I find the notation "f <o g" in Set_Algebras extremely confusing, because it is a function and not a relation. Thus, we get  terms like "k <o g =o O(h)", which for me are hard to parse. Maybe something with a minus (e.g., "-o"?) would be better here?

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> I don’t suppose there’s any chance of using quotients to define actual cardinals and use ordinary equality?

Not really. "(=o) :: 'a rel ⇒ 'b rel ⇒ bool" is too polymorphic for this to work properly.  (The relation arguments are ordinals represented by well-orders.) One can quotient modulo a restricted relation "(=o) :: 'a rel ⇒ 'a rel ⇒ bool", but then the equality on the quotient type won't allow us to compare "|UNIV :: nat set| :: nat rel" to "|UNIV :: real set| :: real rel".

> And it still makes sense to introduce the actual notion of equipollence and similar relations.

This is the usual trade-off between many similar definitions with dedicated reasoning support (transitivity, congruence, and monotonicity rules and such) or a few core ones. Possibly an abbreviation is sufficient in this case.

Dmitriy
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