A usual forced mate position is where the winning player can play in a way that ensures checkmate, no matter what (legal) moves the opponent makes (unless the game ends due to a non-play reason, i.e. resignation, draw accepted, time ran out.)
What I'm asking about here is what I'll call a double-forced mate: A position where checkmate is ensured no matter what (legal) moves either player makes.
With this rule alone we can find a position such as this, where white has only one legal move, resulting in checkmate. Let's call this type 1.
[FEN "K1k5/1q6/P7/B3N3/8/8/8/8 w - - 0 1"] 1. axb7#
The only option is
axb7#. However, this isn't too satisfying as a "double" forced mate, as there was only a single possible move to continue. We can explore positions that have multiple possibilities, but still ensure checkmate absolutely.
[FEN "k7/2QQQQQQ/KQ1QQQQQ/QQQ1QQQQ/QQQQ1QQQ/QQQQQ1QQ/QQQQQQ1Q/QQQQQQQ1 w - - 0 1"]
White has many legal moves, but every one of them results in an immediate checkmate. Still, I feel this question isn't quite answered, as in a true double-forced checkmate, both players should play a role. (Also, this position clearly isn't possible from a standard board start. And even if it were, it doesn't seem that it could be achieved without first creating a stalemate.) Let's finally look to type 3.
The true double-forced mate: A position where checkmate is ensured no matter what (legal) moves either player makes, however there is at least some line where both players make a move. Meaning not every legal ply in the start position is an immediate checkmating move.
Do you know of (or can you find) any such chess positions of (type 3) double-forced mate? Or can you prove that this is not possible?
Also, share if you have any interesting or more plausible examples of type 1 and 2.