# Re: [isabelle] Unable to prove easy existential "directly"

```Hallo,

the behaviour of "rule" in the presence of premises is something that I
have not and will probably never understand.

You can replace the ".." with a "by (rule_tac exI)" or a "by (intro
exI)" and that works fine. Personally, I usually prefer the good old "by
blast" in these cases. Another thing that would work is "from exI[OF
this] show "∃b.(P b)" ."

As for the reason that rule fails here but rule_tac works, that is
something from the depths of the internals of Isabelle that are far
outside my area of expertise.

Cheers,
Manuel

On 07/18/2013 05:41 PM, Wilmer RICCIOTTI wrote:
> Hi all,
>
> as a beginner in the use of Isabelle/Isar, I have every day numerous clashes with the Isar way of proving theorems. The strangest one to date is related to proving an existentially quantified formula when you have the same formula with an explicit witness as a hypothesis. That is to say, something similar to this lemma:
>
>   lemma fie : "P a ⟶ (∃b.(P b))"
>   proof
>     assume ha : "P a"
>     thus "∃b.(P b)"..
>   qed
>
> Unsurprisingly, this proof doesn't pose any challenge at all. However I can slightly complicate the formula by means of a definition, and this obvious proof technique won't work any more. Specifically, I define
>
>   definition bijection :: "('a ⇒ 'b) ⇒ bool" where
>     "bijection f = (∀y::'b.∃!x::'a. y = f x)"
>
> and then the same proof as before, with bijection in place of a generic P, fails:
>
>   lemma foo : "bijection (g::'a ⇒ 'b) ⟶ (∃ f.(bijection (f::'a ⇒ 'b)))"
>   proof
>     assume hg : "bijection (g::'a ⇒ 'b)"
>     thus "∃f.(bijection f)"..
>   qed
>
> replacing the implicit ".." with an explicit "proof (rule exI)" fails similarly, leaving me quite puzzled.
> (Un)Interestingly, since "foo" is an instance of "fie", we can easily prove it using "by (rule fie)" and nothing else. However this feels more like a trick to make things work than a solution.
>
> What am I doing wrong?
>
> Best,
> --
> Wilmer Ricciotti
>

```

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