To forgive is maybe right but to forget only divine

I have been pondering over this for quite some time now whenever I hear the two opposing views each held equally strongly. One says "I don't believe in forgetting even when I forgive" and another says "Do not forgive if you can't forget".
Which do I believe? Ideally it would be right to forget if you forgive and for a long time I disagreed with the former train of thoughts. But what about learning from your mistakes? When someone wrongs you, you try to learn to guard yourself not just against similar situations but also similar people sometimes. Forgiveness is releasing yourself and the other person from any hard feelings. But trust broken is not always reparable. After the hurt comes the anger and after that again hurt , finally culminating in peace when you don't feel anything much. At that stage it would be criminal to expect a person to forget that their trust had been broken and go back to the same situation. Trust is like the ship of a relationship. Once broken, its drowned forever. You can retrieve mementos to remember it by but you can't mend it. But does that mean that forgiveness itself is not possible sometimes or that one can't forget even after forgiving? I wonder still.