Documentation

Mathlib.SetTheory.Game.Domineering

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

The equivalence (x, y) ↦ (x, y+1).

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

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

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