| import OAI.Geometry.PeriodicTiling.EuclideanBasic |
| import Mathlib.Algebra.Order.Floor.Ring |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Lean.Elab.Tactic.Omega |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree |
|
|
| private theorem floor_abs_le_nat {m : ℕ} {x : ℝ} (hx : |x| ≤ (m : ℝ)) : |
| |⌊x⌋| ≤ (m : ℤ) := by |
| apply abs_le.mpr |
| constructor |
| · apply Int.le_floor.mpr |
| exact_mod_cast (abs_le.mp hx).1 |
| · have hh : (⌊x⌋ : ℝ) ≤ (m : ℝ) := (Int.floor_le x).trans (abs_le.mp hx).2 |
| exact_mod_cast hh |
|
|
| private theorem floor_error_lt_one (x : ℝ) : |x - (⌊x⌋ : ℝ)| < 1 := by |
| rw [abs_of_nonneg (sub_nonneg.mpr (Int.floor_le x))] |
| have := Int.lt_floor_add_one x |
| linarith |
|
|
| private theorem floor_successor_error_lt_one {x : ℝ} (hx : x ≠ (⌊x⌋ : ℝ)) : |
| |x - ((⌊x⌋ + 1 : ℤ) : ℝ)| < 1 := by |
| have hlo : (⌊x⌋ : ℝ) < x := lt_of_le_of_ne (Int.floor_le x) hx.symm |
| have hhi := Int.lt_floor_add_one x |
| simp only [Int.cast_add, Int.cast_one] |
| apply abs_lt.mpr |
| constructor <;> linarith |
|
|
| theorem closed_box_rounding_exists {n : ℕ} (m : ℕ) (hm : 2 ≤ m) (δ : Space n) |
| (hδ : ∀ i, |δ i| ≤ (m : ℝ)) |
| (hnot : ¬ ∃ z : Lattice n, δ = (m : ℝ) • castLattice z) : |
| ∃ z : Lattice n, |
| (∀ i, |z i| ≤ (m : ℤ)) ∧ |
| (∃ i, ¬ (m : ℤ) ∣ z i) ∧ |
| (∀ i, |δ i - (z i : ℝ)| < 1) := by |
| classical |
| let a : Lattice n := fun i => ⌊δ i⌋ |
| have habs (i : Fin n) : |a i| ≤ (m : ℤ) := floor_abs_le_nat (hδ i) |
| have herr (i : Fin n) : |δ i - (a i : ℝ)| < 1 := floor_error_lt_one (δ i) |
| by_cases hdiv : ∀ i, (m : ℤ) ∣ a i |
| · have hnonint : ∃ j, δ j ≠ (a j : ℝ) := by |
| by_contra h |
| have heq (i : Fin n) : δ i = (a i : ℝ) := by |
| by_contra hi |
| exact h ⟨i, hi⟩ |
| let z : Lattice n := fun i => Classical.choose (hdiv i) |
| have hz (i : Fin n) : a i = (m : ℤ) * z i := Classical.choose_spec (hdiv i) |
| apply hnot |
| refine ⟨z, ?_⟩ |
| funext i |
| change δ i = (m : ℝ) * (z i : ℝ) |
| rw [heq i] |
| exact_mod_cast hz i |
| obtain ⟨j, hj⟩ := hnonint |
| have hjstrict : (a j : ℝ) < δ j := lt_of_le_of_ne (Int.floor_le (δ j)) hj.symm |
| have hjlt : a j < (m : ℤ) := by |
| have hh : (a j : ℝ) < (m : ℝ) := hjstrict.trans_le (abs_le.mp (hδ j)).2 |
| exact_mod_cast hh |
| let z : Lattice n := Function.update a j (a j + 1) |
| refine ⟨z, ?_, ?_, ?_⟩ |
| · intro i |
| by_cases hij : i = j |
| · subst i |
| simp only [z, Function.update_self] |
| apply abs_le.mpr |
| have ha := (abs_le.mp (habs j)).1 |
| constructor <;> omega |
| · simpa only [z, Function.update_of_ne hij] using habs i |
| · refine ⟨j, ?_⟩ |
| simp only [z, Function.update_self] |
| intro hsucc |
| have hone : (m : ℤ) ∣ 1 := by |
| simpa using dvd_sub hsucc (hdiv j) |
| have hmle : (m : ℤ) ≤ 1 := Int.le_of_dvd (by norm_num : (0 : ℤ) < 1) hone |
| have hmtwo : (2 : ℤ) ≤ (m : ℤ) := by exact_mod_cast hm |
| omega |
| · intro i |
| by_cases hij : i = j |
| · subst i |
| simp only [z, Function.update_self] |
| exact floor_successor_error_lt_one hj |
| · simpa only [z, Function.update_of_ne hij] using herr i |
| · exact ⟨a, habs, not_forall.mp hdiv, herr⟩ |
|
|
| theorem closed_box_rounding (m : ℕ) (hm : 2 ≤ m) (δ : Space 3) |
| (hδ : ∀ i, |δ i| ≤ (m : ℝ)) (hnot : δ ∉ scaledGrid m) : |
| ∃ z : Lattice 3, |
| (∀ i, |z i| ≤ (m : ℤ)) ∧ |
| (∃ i, ¬ (m : ℤ) ∣ z i) ∧ |
| (∀ i, |δ i - (z i : ℝ)| < 1) := by |
| apply closed_box_rounding_exists m hm δ hδ |
| rintro ⟨z, hz⟩ |
| exact hnot (mem_scaledGrid.mpr ⟨z, hz.symm⟩) |
|
|
| theorem closed_box_rounding_norm (m : ℕ) (hm : 2 ≤ m) (δ : Space 3) |
| (hδ : ∀ i, |δ i| ≤ (m : ℝ)) (hnot : δ ∉ scaledGrid m) : |
| ∃ z : Lattice 3, |
| (∀ i, |z i| ≤ (m : ℤ)) ∧ |
| (∃ i, ¬ (m : ℤ) ∣ z i) ∧ |
| ‖δ - castLattice z‖ < 1 := by |
| obtain ⟨z, hz, hdiv, herr⟩ := closed_box_rounding m hm δ hδ hnot |
| refine ⟨z, hz, hdiv, (pi_norm_lt_iff (by norm_num : (0 : ℝ) < 1)).mpr ?_⟩ |
| intro i |
| simpa only [Pi.sub_apply, castLattice_apply, Real.norm_eq_abs] using herr i |
|
|
| end PeriodicTilingThree |
|
|
| end |
|
|
| end OAI |
|
|
