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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | 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
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