| import OAI.Geometry.PeriodicTiling.TilingBasic |
| import OAI.Geometry.PeriodicTiling.ScalarPolynomialPeriodicity |
| import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise |
| import Mathlib.Algebra.Order.BigOperators.Ring.Finset |
| import Mathlib.Tactic.Choose |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| namespace PeriodicTilingThree |
|
|
| theorem coset_configuration_has_period {m : ℕ} |
| (P : Fin m → ℤ → ℝ) (b : Fin m → ℤ) (a : Plane → ℝ) |
| (hzero : ∀ j n, 0 ≤ P j n) |
| (hpair : ∀ i j, i ≠ j → ∀ n r, P i n + P j r ≤ 1) |
| (hsum : ∀ x : Plane, a x = 1 - ∑ j, P j (x.1 + b j * x.2)) |
| (degree : ℕ) (hdegree : 0 < degree) |
| (hpoly : ∀ j, (forwardDiff^[degree]) |
| (fun n => (P j n : AddCircle (1 : ℝ))) = 0) : |
| ∃ v : Plane, v ≠ 0 ∧ Function.Periodic a v := by |
| classical |
| by_cases hbounded : ∀ j, ∃ c : ℝ, c < 1 ∧ ∀ n, P j n ≤ c |
| · have hp : ∀ j, ∃ q : ℕ, 0 < q ∧ Function.Periodic (P j) (q : ℤ) := by |
| intro j |
| obtain ⟨c, hc, hPc⟩ := hbounded j |
| exact real_polynomial_periodic_of_bounded_lt_one degree hdegree |
| (P j) c (hzero j) hPc hc (hpoly j) |
| choose p hp hperiod using hp |
| let Q : ℕ := ∏ j, p j |
| have hQ : 0 < Q := Finset.prod_pos fun j _ => hp j |
| refine ⟨(0, (Q : ℤ)), ?_, ?_⟩ |
| · intro h |
| have hz : (Q : ℤ) = 0 := congrArg Prod.snd h |
| exact (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hQ)) hz |
| · intro x |
| rw [hsum, hsum] |
| apply congrArg (fun y : ℝ => 1 - y) |
| apply Finset.sum_congr rfl |
| intro j _hj |
| have hd : p j ∣ Q := Finset.dvd_prod_of_mem p (Finset.mem_univ j) |
| obtain ⟨k, hk⟩ := hd |
| have hperiodQ : Function.Periodic (P j) (b j * (Q : ℤ)) := by |
| simpa only [hk, Nat.cast_mul, Int.cast_id, mul_assoc, mul_left_comm, mul_comm] using |
| (hperiod j).int_mul (b j * (k : ℤ)) |
| change P j (x.1 + 0 + b j * (x.2 + (Q : ℤ))) = |
| P j (x.1 + b j * x.2) |
| rw [show x.1 + 0 + b j * (x.2 + (Q : ℤ)) = |
| (x.1 + b j * x.2) + b j * (Q : ℤ) by ring] |
| exact hperiodQ _ |
| · obtain ⟨j, hj⟩ := not_forall.mp hbounded |
| have hother : ∀ i, i ≠ j → ∀ n, P i n = 0 := by |
| intro i hij n |
| apply le_antisymm _ (hzero i n) |
| apply le_of_not_gt |
| intro hpos |
| apply hj |
| refine ⟨1 - P i n, by linarith, ?_⟩ |
| intro r |
| have hcross := hpair j i (Ne.symm hij) r n |
| linarith |
| have hone : ∀ x : Plane, a x = 1 - P j (x.1 + b j * x.2) := by |
| intro x |
| rw [hsum] |
| apply congrArg (fun y : ℝ => 1 - y) |
| apply Finset.sum_eq_single j |
| · intro i _hi hij |
| exact hother i hij _ |
| · intro hj' |
| exact (hj' (Finset.mem_univ j)).elim |
| refine ⟨(-b j, 1), ?_, ?_⟩ |
| · intro h |
| have hz : (1 : ℤ) = 0 := congrArg Prod.snd h |
| exact one_ne_zero hz |
| · intro x |
| rw [hone, hone] |
| apply congrArg (fun n : ℤ => 1 - P j n) |
| change x.1 + -b j + b j * (x.2 + 1) = x.1 + b j * x.2 |
| ring |
|
|
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
