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Lax68.GridsAndWalls

Grids and walls

concepts/Lax68/GridsAndWalls.lean · lax-68

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    Definition

    A nonempty rectangular grid has vertices in rows and columns, with edges between orthogonally consecutive positions. A wall is the brick-wall subgraph obtained by retaining alternating vertical grid edges.

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Hasse
    2
    3/-!
    4---
    5title: Grids and walls
    6type: definition
    7---
    8A nonempty rectangular grid has vertices in rows and columns, with edges
    9between orthogonally consecutive positions. A wall is the brick-wall subgraph
    10obtained by retaining alternating vertical grid edges.
    11-/
    12
    13set_option autoImplicit false
    14
    15namespace Lax68.GridsAndWalls
    16
    17def consecutive (a b : ℕ) : Prop :=
    18 a + 1 = b ∨ b + 1 = a
    19
    20def WallAdjacent {m n : ℕ} (u v : Fin m × Fin n) : Prop :=
    21 (u.1 = v.1 ∧ consecutive u.2.val v.2.val) ∨
    22 (u.2 = v.2 ∧
    23 consecutive u.1.val v.1.val ∧
    24 (Nat.min u.1.val v.1.val + u.2.val) % 2 = 0)
    25
    26def HasGridShape {V : Type*} (G : SimpleGraph V) : Prop :=
    27 ∃ m n : ℕ,
    28 0 < m ∧
    29 0 < n ∧
    30 Nonempty
    31 (G ≃g (SimpleGraph.pathGraph m □ SimpleGraph.pathGraph n))
    32
    33def HasWallShape {V : Type*} (G : SimpleGraph V) : Prop :=
    34 ∃ m n : ℕ,
    35 0 < m ∧
    36 0 < n ∧
    37 ∃ e : Fin m × Fin n ≃ V,
    38 ∀ u v,
    39 G.Adj (e u) (e v) ↔ WallAdjacent u v
    40
    41def IsGrid {V : Type*} (G : SimpleGraph V) : Prop :=
    42 HasGridShape G
    43
    44def IsWall {V : Type*} (G : SimpleGraph V) : Prop :=
    45 HasWallShape G
    46
    47end Lax68.GridsAndWalls
    48

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