Showing a process is a martingale from Ito's lemma

Question

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing $$\mathbb{E}[t-W^2(t)|\mathcal{F}_s] = s - W^2(s)$$ for any $s \leq t$. It was suggested to me that you can also show this is a martingale by computing $$d(t-W^2(t))$$ using Ito's lemma. I am unclear on how to do this and answers to related questions on this sight are not clear to me.

Answer 1

Apply Ito's lemma to $f(x,t) = t - x^2$

$$ \mathrm{d} f(W_t,t) = \mathrm{d}t - 2 W_t \mathrm{d}W_t - \frac{1}{2} 2 \mathrm{d}t = 2 W_t \mathrm{d}W_t$$

It is a property of an Ito integral $I_t = \int_0^t a(s, \omega) \mathrm{d}X_s$ where $X_t$ is a martingale and $a(t,\omega)$ is adapted that $I_t$ is a martingale. QED

EDIT: you need a few more technical conditions on $a(s,\omega)$ regarding measurability and square intetgability that are certainly satisfied in this case.