The police decided not to prosecute.
Is it exactly the same to say: the police didn't decide to prosecute.
the police didn't prosecute.
At school you learn how to negate with "didn't". Somehow, nobody at school teaches the negation with "not" (my own experience; don't feel addressed).
When do we use or prefer "not" over "didn't"?
The police decided, "We won't prosecute".
The police did not decide, "We will prosecute".
The first is a decision; the second isn't.
I'm sure that when you learnt to negate with 'didn't', they were talking about a sentence with one verb in it:
"John saw the dog."; "John didn't see the dog."
You're presenting a different structure.