[isabelle-dev] not-exists problem

[Jasmin Blanchette]

> On 13.09.2016, at 09:45, Peter Lammich <lammich at in.tum.de> wrote:
> 
> On Di, 2016-09-13 at 00:46 +0100, Lawrence Paulson wrote:
>> We have a problem with the ∄ operator, when existential quantifiers
>> are nested:
>> 
>> lemma "(∄x. ∃y. P x y) = (~(∃x y. P x y))"
>>   by (rule refl)
> 
> I do not see a particular problem with this,

Like you I find ∄x. ∃y. P x y clear (albeit weird), but ∄x y. P x y is probably a better way to display the same thing.

> lemma "(∄x y. P x y) ⟷ ¬(∃x. ¬(∃y. P x y))"
>     by (rule refl)

That looks like a serious flaw.

> lemma "(∃!x y. P x y) ⟷ (∃!x. (∃!y. P x y))"
>     by (rule refl)

Ditto.

> We should either forbid multiple variables on those quantifiers, or try
> to figure out whether there is a widely-accepted consensus on their
> meaning.

The best would be to forbid multiple variables for ∃!. For ∄, the situation is more complicated. We can forbid them, or we can interpret a single multi-variable ∄ as ∄∃...∃ (and in the other direction for printing).

Jasmin