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.
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
On 2021-02-14 17:58, Tobias Nipkow wrote:
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
On 14/02/2021 16:44, Peter Lammich wrote:
in my current formalization, I frequently end up with goals that I
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
"∀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
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
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