Coq unable to unify -- how to change hypothesis?

Question

Coq beginner here.

I have the following silly theorems:

  Theorem plus_same : forall a b c : nat,
      a+b=a+c -> b=c.
  Proof. Admitted.

  Theorem advanced_commutivity:
    forall x y z w : nat, x + y + (z+w) = x + z + (y + w).
  Proof.
    intros x y z w.
    apply (plus_same x (y + (z+w)) (z + (y + w))).

However, when I try to run the apply line, I get an error:

Unable to unify "y + (z + w) = z + (y + w)" with
 "x + y + (z + w) = x + z + (y + w)".

Do I need to change my hypothesis here? How can I apply plus_same here to the arguments in advanced_commutivity proof?

Answer 1

You are misreading your goal: x + y + (z + w) stands for (x + y) + (z + w), because + is registered as left-associative, which is different from x + (y + (z + w)).

So in order to apply your lemma, you should first reassociate your + by rewriting with another Lemma plus_assoc : forall x y z, x + y + z = x + (y + z).