Definition by existence

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Victor Porton

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Feb 24, 2009, 10:41:44 AM2/24/09
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I'm an Isabelle novice.

Let there is therorem "exists X such that P(X)". I want to define Y as
arbitrarily chosen (fixed) X confirming to this theorem.

What is the syntax for this?

hen...@topoi.pooq.com

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Feb 24, 2009, 10:06:32 AM2/24/09
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On Tue, Feb 24, 2009 at 07:41:44AM -0800, Victor Porton wrote:
>
> I'm an Isabelle novice.
>
> Let there is therorem "exists X such that P(X)". I want to define Y as
> arbitrarily chosen (fixed) X confirming to this theorem.

Would being able to use this notation be equivalent to the axiom of
choice?

-- hendrik

slawekk

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Feb 24, 2009, 12:33:43 PM2/24/09
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> Let there is therorem "exists X such that P(X)". I want to define Y as
> arbitrarily chosen (fixed) X confirming to this theorem.
>
> What is the syntax for this?

Let's say the theorem "exists X such that P(X)" is called my_theorem.
Then you can say in some proof:

obtain Y where "P(Y)" using my_theorem by auto

From that point on you can refer to the set Y and its property P(Y) in
this proof. This type of reasoning does not require the axiom of
choice.

Victor Porton

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Feb 24, 2009, 12:49:34 PM2/24/09
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24.02.09, 20:33, "slawekk" <skok...@yahoo.com>:

As far as I understand, "obtain" can be used only in proofs.

I want something similar to "obtain" but in definitions (outside of proofs).

slawekk

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Feb 24, 2009, 1:48:43 PM2/24/09
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>
> I want something similar to "obtain" but in definitions (outside of proofs).

If you had uniqueness you could use the "THE" keyword. Something like:

definition
"Q \<equiv> THE Y. P(Y)"

I used it a couple of times, although I prefer different ways of
getting the only set satisfying a predicate.

Since you don't have uniqueness you may check the similar SOME
construct.

definition
"Q \<equiv> SOME Y. P(Y)"

It might be what you want but I have never used it so I am not sure.

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