Re: [isabelle] Weird nesting of Quantifiers



Hi Peter,

I don't think there is any automation for this. I automated "(∀x. x = t --> P x) = P t" and variations on this, but without nested quantifiers. I am sure the latter could be added and it would be worth it, but somebody would need to do it. I am happy to provide pointers...

Tobias

On 14/02/2021 16:44, Peter Lammich wrote:
Hi List,

in my current formalization, I frequently end up with goals that I feel
should be solvable by auto or blast, but they get stuck due to
containing a precondition similar to this:

(∀a. (∃x y. a = f x y ∧ P x y) ⟶ Q a)

The exact precondition can vary in the number of universal and
existential quantified variables, and the position and number of the
determining a = ... conjuncts, e.g.

(∀a b. (∃x y. a = f x y ∧ P x y ∧ b=g x) ⟶ Q a b)

Anyway, the above preconditions are, obviously, equal to the following
simpler ones:

"∀x y. P x y ⟶ Q (f x y)"
"∀x y. P x y ⟶ Q (f x y) (g x)"

currently, I have to manually prove these equivalences, for every
instance of quantified variables, etc, and then can solve the goal
easily by rewriting and auto.

Is there any way, e.g. a simproc or so, to automate this process, or is
my only solution to bloat up the otherwise fully automatic proofs by
those weird auxiliary lemmas (of which I could, of course, prove
instances for the most common cases globally and add them to the
simpset)

--
   Peter



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