Coq - How to prove eqb_neq?

Question

I'm trying to prove eqb_neq:

Theorem eqb_neq : forall x y : nat,
  x =? y = false <-> x <> y.

This is my current proof status:

During the proof I reached a final step where I just need to prove the additional helper theorem:

Theorem eqb_false_helper : forall n m : nat,
    n <> m -> S n <> S m.

I've tried multiple strategies but now I'm not even sure it's possible to prove this helper theorem.

I'm not sure how to prove the base case using induction:

What else can I try? Any tips for eqb_neq or the helper theorem?

Thanks

I have to kindly ask you to refrain from posting (partial) solutions to SF exercises on public forums. If you can repost your question to remove your proof, we'd be happy to help you. — Arthur Azevedo De Amorim, May 27 '20 at 13:09

score 1 · Accepted Answer · answered May 27 '20 at 13:13

It is actually quite simple for your helper theorem if you just unfold not :

Theorem eqb_false_helper : forall n m : nat,
    n <> m -> S n <> S m.
Proof.
unfold not; intros.
apply H; injection H0; intro; assumption.
Qed.

You actually just need to prove that S n = S m -> False, you assume that n = m -> False, thus you can prove that S n = S m -> n = m, which is done injecting hypothesis S n = S m.

Coq - How to prove eqb_neq?

1 Answers1