# 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.

@[simp]
theorem SetTheory.PGame.Domineering.shiftUp_apply :
∀ (a : ), SetTheory.PGame.Domineering.shiftUp a = Prod.map id (fun (x : ) => x + 1) a
@[simp]
theorem SetTheory.PGame.Domineering.shiftUp_symm_apply :
∀ (a : ), = Prod.map id (fun (x : ) => x + -1) a

The equivalence (x, y) ↦ (x, y+1).

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@[simp]
theorem SetTheory.PGame.Domineering.shiftRight_symm_apply :
∀ (a : ), = Prod.map (fun (x : ) => x + -1) id a
@[simp]
theorem SetTheory.PGame.Domineering.shiftRight_apply :
∀ (a : ), SetTheory.PGame.Domineering.shiftRight a = Prod.map (fun (x : ) => x + 1) id a

The equivalence (x, y) ↦ (x+1, y).

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@[reducible, inline]

A Domineering board is an arbitrary finite subset of ℤ × ℤ.

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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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theorem SetTheory.PGame.Domineering.mem_left (x : ) :
x b (x.1, x.2 - 1) b
theorem SetTheory.PGame.Domineering.mem_right (x : ) :
x b (x.1 - 1, x.2) b

After Left moves, two vertically adjacent squares are removed from the board.

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After Left moves, two horizontally adjacent squares are removed from the board.

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The instance describing allowed moves on a Domineering board.

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• One or more equations did not get rendered due to their size.

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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• = id inferInstance

The Domineering board with two squares arranged vertically, in which Left has the only move.

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The L shaped Domineering board, in which Left is exactly half a move ahead.

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