I want to change the goal from S x = S y to x = y. It's like inversion, but for the goal instead of a hypothesis.
Such a tactic seems legit, because when we have x = y, we can simply use rewrite and reflexivity to prove the goal.
Currently I always find myself using assert (x = y) to introduce a new subgoal, but it's tedious to write when x and y are complex expression.
The tactic apply f_equal. will do what you want, for any constructor or function.
The lema f_equal shows that for any function f, you always have x = y -> f x = f y. This allows you to reduce the goal from f x = f y to x = y:
Proposition myprop (x y: nat) (H : x = y) : S x = S y.
Proof.
apply f_equal. assumption.
Qed.
(The injection tactic implements the converse implication — that for some functions, and in particular for constructors, f x = f y -> x = y.)