In a sprawling legal brief partly unsealed on Wednesday, the special counsel, Jack Smith, laid out his case for why former President Donald J. Trump is not immune from prosecution on federal charges of plotting to overturn the 2020 election.

The redacted brief, made public by Judge Tanya S. Chutkan of the Federal District Court in Washington, adds new details to the already extensive public record of how Mr. Trump lost the race but attempted nonetheless to cling to power.