rewriting hypothesis to false with a contradictory theorem

Question

I want to show that

[seq q x t | x <- iota 0 (t + 1)] != [::]

I decided to destruct iota 0 (t + 1) because I have a lemma that says:

iota 0 (t + 1) != [::]

So the first case of destruct should have iota 0 (t + 1) = [::] which by the theorem mentioned is false, and I can discriminate. How can I rewrite the equation in the first destruct case using the lemma? I cannot figure it out.

Thanks.

Answer 1

You do not need to destruct. Note that iota is defined by recursion on its second variable. Your current goal cannot be simplified because t + 1 does not start with a constructor. However, you can do by rewrite addn1 to put it in a form where it can be solved.

Answer 2

In addition to computation, as Arthur suggests, you can sometimes use contraposition to deal with non-equalities (do Search "contra" for variant versions).

For instance, in your case, you can show, if you add some injectivity constraint:

Lemma foo (q : nat -> nat -> nat) t (injq: injective (q^~ t)) :
  iota 0 (t + 1) != [::] -> [seq q x t | x <- iota 0 (t + 1)] != [::].
Proof.
apply: contra_neq.
rewrite [RHS]( _ : [::] = [seq q x t | x <- [::]]) //.
exact: inj_map. 
Qed.