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Re: [mizar] fraenkel terms over possibly empty types





On Mon, 21 May 2007, Adam Naumowicz wrote:

Hi,

On Mon, 21 May 2007, Jesse Alama wrote:

I can get the second case to go through, but the first case, where A is
assumed to be empty, gives me trouble: I can't finish the proof that X
is empty.

I then gave up on (*) and moved to

 X := (***) { x where x is Element of A : <whatever> & x in A }

With this change my proofs go through, but (***) seems ham-handed to me.
My question for the experts: what other ways are there of dealing with
the problem besides (***)?

To me it's the only solution in this case, albeit an ugly one.

Why was I unable to get the desired typing judgment with (*)?

According to the current MML definition of 'Element of', it's simply not true (in the case X={} ) that

X = {x: where x in Element of X: not contradiction}

because {} is not equal to {{}}. And for the same reason { x where x is Element of A : <whatever>} needn't be a subset of A

another nice example how the holy struggle for non empty dependent types (here Element of A) makes Mizar formalizations close to normal mathematician's expectations

Josef