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"have crossed" is correct and usable in written English.
You can use it when talking about something that happened in the past or to indicate a change of status. For example: "I have crossed the finish line and won the race!".
Exact(59)
He could have crossed somebody.
You have crossed it.
Blacks seem to have crossed that line".
The 'Guignols' have crossed the line.
"I may have crossed societal boundaries.
"We have crossed that river," he said.
Andrew Mason appears to have crossed them.
They have crossed a line.
Honor and virtue have crossed the sea.
But sports, too, have crossed oceans.
"Then I would have crossed the street.
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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