How to leave a goal unfinished in Coq

Question

In Isabelle, you can leave a goal unfinished in two ways:

• sorry: will leave your proof and the fact can be used in later proofs.
• oops: will leave the proof but the fact cannot be used in later proofs.

Is there similar functionality in Coq that would allow me to leave a goal unfinished to come back later to it? This is useful to sketch the structure of the proof while not filling in all the details. Note that the approach in How to switch the current goal in Coq? modifies the proof structure. That's not what I'm looking for.

Answer 1

You have several ways to terminate a proof in Coq. You probably know Qed which asserts the proof is completed. There is also Defined for when you want the proof to have computational content.

The things you are looking for are:

• Admitted which admits the proof, so it can be used later;
• Abort which gives up on proving the lemma.

They can be used that way:

Lemma foo : forall n, n = 0.
Proof.
  intro n. destruct n.
  - reflexivity.
  -
Abort.

Lemma bar : forall n, n = n.
Admitted.

In both cases you can have a partial proof script before the Admitted / Abort or none at all.

As @HTNW points out, you can also use the equivalent tactics admit and give_up inside subgoals of the proof. Using those, the proof however has to be concluded using either Admitted or Abort .

Lemma lem : forall A, A + A -> A.
Proof.
  intros A h.
  destruct h.
  - admit.
  - give_up.
Admitted.

The partial proof is in any case thrown away with this solution. If for some reason you want to keep using a partial proof (for instance if you want to compute with it), a common trick is to use an axiom to close the goals that you want to leave for later.

Axiom cheating : forall A, A.

Tactic Notation "cheat" := apply cheating.

Lemma lem : forall A, A + A -> A.
Proof.
  intros A h.
  destruct h.
  - cheat.
  - cheat.
Defined. (* This is now ok *)

You have to be careful using that trick though. You can check that a lemma has been proven without axioms using Print Assumptions lem . If it says "closed under context" you're good, otherwise it will lists the axioms it depends on and if cheating appears you know it's not entirely proven.