| 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 | |