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The phrase "having cooperated" is correct and usable in written English.
It is a present participle phrase, which means it describes an action that occurred before the main verb in the sentence. It can be used to show a cause and effect relationship or to indicate that something was done before something else. Example: "Having cooperated with the team, Sarah was chosen as the leader for the project." In this sentence, the phrase "having cooperated" shows that Sarah's cooperation with the team led to her being chosen as the project leader.
Exact(14)
Witnesses, she said, might not always be forthcoming about having cooperated with the F.B.I.
Two weeks later she telephoned and threatened to have her friends retaliate against them for having cooperated with the prosecution.
Mr. Scajola said a fourth individual was detained by the police in Munich, suspected of having cooperated with the group in Milan.
Mr. Smiley's lawyer, Richard A. Reeve, asked the judge to give his client credit for having cooperated with the authorities once he was arrested.
Knauf also expects to be fined, while BPB is hoping for a reduced fine in return for having cooperated with the inquiry.
Mr. Khan has been described as having cooperated with Pakistani and American interrogators, and some American officials said that the information he himself provided, as distinct from the computer records, may also have pointed to the prospect of a current threat of terrorism in New York and Washington.
Similar(45)
Not everyone has cooperated".
"We have cooperated fully".
"I have cooperated fully.
He said the members had cooperated fully.
But this year, Mother Nature has cooperated.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com