You introduced r
before using induction on y
. In general you'll want to use induction before introducing anything so the induction hypothesis is as general as possible.
Conjecture injectivity : forall n m, s n = s m -> n = m.
Theorem T1: forall x y r, (plus x y) = r -> (Add x y r).
Proof.
intros x y. induction y.
simpl. intros r H. rewrite H. apply add_z.
intro r. case r.
simpl. intro H. discriminate H.
simpl. intros n H. apply add_s. apply IHy. apply injectivity. apply H.
Qed.