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I thought you need to add "is/has (been)" in between? e.g. "Mission is accomplished", "Mission has been accomplished"

So as "All done", shouldn't it be "All are done", "All have been completed", etc.?

Further example, "Have you finished your homework?"

"Yes, all my homework completed", shouldn't it be "Yes, all my homework has been completed"?

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It is arguably idiomatic (https://www.dictionary.com/browse/idiom). In a few phrases, following no particular pattern (of which I am aware), the word "is" is omitted. That being said, the omission of "to be" in short utterances could also be argued as a general feature of English, and thus these phrases might not be particularly idiomatic. For example, one could even say "Done." When an objection is raised in court, a judge may say "Overruled" or "Objection sustained," or something similar, again implicitly including "the" and "is." As for the second example, one would not say: "Yes, all my homework completed." In longer sentences, verbs are not optional. In very short ones, they may be implied. It sounds perfectly natural to say, "Homework completed." It does not sound natural to say, "Yes, all my homework completed."

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    I'd add only one thing: this elliptical construction is used, even in highly formal contexts, to emphasize a very short, sharp, strong statement — no argument possible. (This is probably the main reason for it not to be used: if the statement really isn't that strong, it makes no sense to use such a powerful construction.) – Nathan Tuggy Jul 22 '15 at 17:24
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"Mission accomplished" is typical of short military style and stands for "The mission has been accomplished". The leader of a group of soldiers may return and report to his officer "Mission accomplished".

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