is_closed

formal source

 344
 345/-- A path is **closed** if its total displacement is zero along every axis.
 346    Closed paths on ℤ³ are the analogues of loops in topology. -/
 347def is_closed {D : ℕ} (p : LatticePath D) : Prop :=
 348  ∀ k : Fin D, winding_number p k = 0
 349
 350/-- The empty path is closed. -/
 351theorem empty_is_closed {D : ℕ} : is_closed ([] : LatticePath D) :=
 352  fun k => winding_empty k
 353
 354/-- A cancelling pair is a closed path. -/
 355theorem cancelling_pair_closed {D : ℕ} (a : Fin D) :
 356    is_closed ([LatticeStep.plus a, LatticeStep.minus a] : LatticePath D) := by
 357  intro k
 358  simp [winding_number, step_displacement]
 359  split <;> simp
 360
 361/-
 362**THEOREM (Closed Paths Are Topologically Trivial in D ≤ 2)**:
 363    In D ≤ 2, every closed path can be reduced to the empty path by
 364    removing cancelling pairs. The path is homotopically trivial.
 365
 366    In D = 3, this is NOT true — closed paths can be knotted.
 367    Knotted paths have non-trivial linking with other paths.
 368
 369    This asymmetry is WHY D = 3 supports topological conservation
 370    while D ≤ 2 does not. -/
 371
 372/-- A square loop on the D-lattice: go +k, +j, −k, −j for two axes j ≠ k.
 373    This is a closed non-trivial loop that exists iff D ≥ 2. -/
 374def square_loop {D : ℕ} (j k : Fin D) : LatticePath D :=
 375  [.plus j, .plus k, .minus j, .minus k]
 376
 377theorem square_loop_closed {D : ℕ} (j k : Fin D) :