CODEWITHSUNDEEP
coq
thor
thor

Reputation: 22460

How to express "there exists a unique X" in Coq?

I was wondering if there is a succinct way of writing that there exists a unique something (i.e. write unique existential quantifier) in Coq?

For example, to say that there exists an x s.t. 2 + x = 4:

Goal exists x, 2 + x = 4.

How can I write that there exists a unique x with the same property?

I know I can replicate the predicate in the s.t. part like this:

Goal exists x, 2 + x = 4 /\ forall y, 2 + y = 4 -> y = x.

But this is a lot of repetition in general, and is there a way to somehow encode a new quantifier, and write:

Goal exists1, 2 + x = 4.

to express the same goal?

Upvotes: 9

Views: 1822

Answers (1)

Daniel Schepler
Daniel Schepler

Reputation: 3103

Coq already provides an exists! notation. For example:

Goal exists! x, 2 + x = 4.
Proof.
exists 2. split.
+ reflexivity.
+ intros. injection H; intro.
  symmetry; assumption.
Qed.

Upvotes: 14

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