# Re: [isabelle] Weird nesting of Quantifiers

```
You can pull the existentials up so that there are only foralls using
the symmetric versions of the all_simps rewrites.

Try out
apply (simp only: all_simps[symmetric] cong: imp_cong)

I'm not sure how consistently that will work, but that followed by
regular simplification got me the expected result in all the simple
test cases I typed.

BTW, I have a dim recollection that I used to see goals like this more
often. The "a = .." and "b = .." looks like it came from "(a, b)" being
in the image of some projection, maybe. I think I managed to get more
```
image/map type constructions to simplify a bit before quantifiers appeared.
```
Best regards,
Thomas.

On 2021-02-14 17:58, Tobias Nipkow wrote:
```
```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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