Monday Mutant 77: Proof of Quilt / Battleships

Ten ships (as indicated below the grid: one four-cell ship, two three-cell ships, three two-cell ships, and four one-cell ships) are hidden in the grid. The ships may be rotated from the orientations shown (without changing the numbers), but may not overlap or occupy cells which share a corner or an edge. A number to the right of a row or below a column indicates how many cells in that row or column are occupied by ships. These ships also function as black cells in a Proof of Quilt puzzle, and the numbers on some of the ship segments function as they normally would in such a puzzle. None of the black cells already present in the grid are part of the ships. Locate the ships and solve the Proof of Quilt puzzle.

This could be my strangest Monday Mutant in a long time.