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a97875e | 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 | import OAI.Analysis.CircleDomains.Topology.EqualScaleCellSeparation
namespace OAI
/-! Equal-scale three-dimensional cells, colored by sixteen residues
in each coordinate. Enlarged balls in one color are disjoint. -/
open Set Function Metric
noncomputable section
namespace CircleDomainRigidity
abbrev GraphGridColor := (Fin 16 × Fin 16) × Fin 16
abbrev GraphGridIndex := (ℤ × ℤ) × ℤ
abbrev GraphSpace := (ℝ × ℝ) × ℝ
def graphGridInterval (r : Fin 16) (k : ℤ) (s : ℝ) : Set ℝ :=
Icc ((16*(k:ℝ)+(r:ℕ))*s) ((16*(k:ℝ)+(r:ℕ)+1)*s)
def graphGridCell (r : GraphGridColor) (k : GraphGridIndex) (s : ℝ) : Set GraphSpace :=
(graphGridInterval r.1.1 k.1.1 s ×ˢ graphGridInterval r.1.2 k.1.2 s) ×ˢ
graphGridInterval r.2 k.2 s
lemma graphGridCell_dist_gt {s : ℝ} (hs : 0 < s) (r : GraphGridColor)
{k l : GraphGridIndex} (hkl : k ≠ l) {x y : GraphSpace}
(hx : x ∈ graphGridCell r k s) (hy : y ∈ graphGridCell r l s) :
8*s < dist x y := by
by_cases h₀ : k.1.1 = l.1.1
· by_cases h₁ : k.1.2 = l.1.2
· have h₂ : k.2 ≠ l.2 := by
intro h
exact hkl (Prod.ext (Prod.ext h₀ h₁) h)
have hh := equal_scale_cell_separation hs h₂ (r.2:ℕ) hx.2 hy.2
exact hh.trans_le (by
rw [Prod.dist_eq,Real.dist_eq]
exact le_max_right _ _)
· have hh := equal_scale_cell_separation hs h₁ (r.1.2:ℕ) hx.1.2 hy.1.2
exact hh.trans_le (by
rw [Prod.dist_eq,Prod.dist_eq,Real.dist_eq]
exact (le_max_right _ _).trans (le_max_left _ _))
· have hh := equal_scale_cell_separation hs h₀ (r.1.1:ℕ) hx.1.1 hy.1.1
exact hh.trans_le (by
rw [Prod.dist_eq,Prod.dist_eq,Real.dist_eq]
exact (le_max_left _ _).trans (le_max_left _ _))
lemma graphGridCell_enlarged_disjoint {s : ℝ} (hs : 0 < s) (r : GraphGridColor)
{k l : GraphGridIndex} (hkl : k ≠ l) {x y : GraphSpace}
(hx : x ∈ graphGridCell r k s) (hy : y ∈ graphGridCell r l s) :
Disjoint (closedBall x (4*s)) (closedBall y (4*s)) := by
apply Set.disjoint_left.mpr
intro z hzx hzy
have ht := dist_triangle x z y
rw [dist_comm x z] at ht
have hle : dist x y ≤ 8*s := by
linarith [mem_closedBall.mp hzx, mem_closedBall.mp hzy]
exact (not_lt_of_ge hle) (graphGridCell_dist_gt hs r hkl hx hy)
lemma exists_graphGridInterval {s : ℝ} (hs : 0 < s) (x : ℝ) :
∃ (r : Fin 16) (k : ℤ), x ∈ graphGridInterval r k s := by
let n : ℤ := Int.floor (x/s)
have hn0 : 0 ≤ n % 16 := Int.emod_nonneg _ (by norm_num)
have hn16 : n % 16 < 16 := Int.emod_lt_of_pos _ (by norm_num)
let r : Fin 16 := ⟨(n%16).toNat,by omega⟩
refine ⟨r,n/16,?_⟩
have he : (16:ℝ)*(n/16:ℤ)+(r:ℕ) = (n:ℝ) := by
have hr : ((r:ℕ):ℝ) = ((n%16:ℤ):ℝ) := by
dsimp only [r]
exact_mod_cast (Int.natCast_toNat_eq_self.mpr hn0)
rw [hr]
have hh : (16:ℤ)*(n/16)+n%16 = n := by simpa only [add_comm] using Int.emod_add_mul_ediv n 16
exact_mod_cast hh
change (16*(n/16:ℤ)+(r:ℕ))*s ≤ x ∧ x ≤ (16*(n/16:ℤ)+(r:ℕ)+1)*s
rw [he]
have hlo := (le_div_iff₀ hs).mp (Int.floor_le (x/s))
have hhi := (div_lt_iff₀ hs).mp (Int.lt_floor_add_one (x/s))
exact ⟨hlo,hhi.le⟩
lemma exists_graphGridCell {s : ℝ} (hs : 0 < s) (x : GraphSpace) :
∃ (r : GraphGridColor) (k : GraphGridIndex), x ∈ graphGridCell r k s := by
obtain ⟨r₀,k₀,h₀⟩ := exists_graphGridInterval hs x.1.1
obtain ⟨r₁,k₁,h₁⟩ := exists_graphGridInterval hs x.1.2
obtain ⟨r₂,k₂,h₂⟩ := exists_graphGridInterval hs x.2
exact ⟨((r₀,r₁),r₂),((k₀,k₁),k₂),⟨⟨h₀,h₁⟩,h₂⟩⟩
lemma graphGridCell_dist_le {s : ℝ} (r : GraphGridColor) (k : GraphGridIndex)
{x y : GraphSpace} (hx : x ∈ graphGridCell r k s) (hy : y ∈ graphGridCell r k s) :
dist x y ≤ s := by
have hcoord {a x y : ℝ} (hx : x ∈ Icc a (a+s)) (hy : y ∈ Icc a (a+s)) :
|x-y| ≤ s := abs_le.mpr ⟨by linarith [hx.1,hy.2],by linarith [hx.2,hy.1]⟩
have h₀ := hcoord (a := (16*(k.1.1:ℝ)+(r.1.1:ℕ))*s)
(by simpa only [graphGridInterval,add_mul,one_mul] using hx.1.1)
(by simpa only [graphGridInterval,add_mul,one_mul] using hy.1.1)
have h₁ := hcoord (a := (16*(k.1.2:ℝ)+(r.1.2:ℕ))*s)
(by simpa only [graphGridInterval,add_mul,one_mul] using hx.1.2)
(by simpa only [graphGridInterval,add_mul,one_mul] using hy.1.2)
have h₂ := hcoord (a := (16*(k.2:ℝ)+(r.2:ℕ))*s)
(by simpa only [graphGridInterval,add_mul,one_mul] using hx.2)
(by simpa only [graphGridInterval,add_mul,one_mul] using hy.2)
simpa only [Prod.dist_eq,Real.dist_eq,max_le_iff] using And.intro (And.intro h₀ h₁) h₂
end CircleDomainRigidity
end
end OAI
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