Domineering as a combinatorial game.
We define the game of Domineering, played on a chessboard of arbitrary shape (possibly even disconnected). Left moves by placing a domino vertically, while Right moves by placing a domino horizontally.
This is only a fragment of a full development; in order to successfully analyse positions we would need some more theorems. Most importantly, we need a general statement that allows us to discard irrelevant moves. Specifically to domineering, we need the fact that disjoint parts of the chessboard give sums of games.
A Domineering board is an arbitrary finite subset of ℤ × ℤ.
Left can play anywhere that a square and the square below it are open.
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Right can play anywhere that a square and the square to the left are open.
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After Left moves, two vertically adjacent squares are removed from the board.
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- pgame.domineering.move_left b m = (finset.erase b m).erase (m.fst, m.snd - 1)
After Left moves, two horizontally adjacent squares are removed from the board.
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- pgame.domineering.move_right b m = (finset.erase b m).erase (m.fst - 1, m.snd)
The instance describing allowed moves on a Domineering board.
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- pgame.domineering.state = {turn_bound := λ (s : pgame.domineering.board), finset.card s / 2, L := λ (s : pgame.domineering.board), finset.image (pgame.domineering.move_left s) (pgame.domineering.left s), R := λ (s : pgame.domineering.board), finset.image (pgame.domineering.move_right s) (pgame.domineering.right s), left_bound := pgame.domineering.state._proof_1, right_bound := pgame.domineering.state._proof_2}
Construct a pre-game from a Domineering board.
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All games of Domineering are short, because each move removes two squares.
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The Domineering board with two squares arranged vertically, in which Left has the only move.
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- pgame.domineering.one = pgame.domineering [(0, 0), (0, 1)].to_finset
The L shaped Domineering board, in which Left is exactly half a move ahead.
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- pgame.domineering.L = pgame.domineering [(0, 2), (0, 1), (0, 0), (1, 0)].to_finset
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- pgame.short_one = id (pgame.short_domineering [(0, 0), (0, 1)].to_finset)
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- pgame.short_L = id (pgame.short_domineering [(0, 2), (0, 1), (0, 0), (1, 0)].to_finset)