File size: 3,155 Bytes
f61e547 | 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 | import OAI.Probability.FirstPassage.AttachmentGeometry
namespace OAI
namespace PlanarFPP
open Set Filter Metric
open scoped Topology
theorem finite_ball_net (D γ : ℝ) (hγ : 0 < γ) :
∃ S : Finset {v : Plane // ‖v‖ ≤ D},
∀ v : Plane, ‖v‖ ≤ D → ∃ u ∈ S, ‖v-u.val‖ < γ := by
classical
have hc : closedBall (0 : Plane) D ⊆
⋃ u : {v : Plane // ‖v‖ ≤ D}, ball u.val γ := by
intro v hv
have hv' : ‖v‖ ≤ D := by simpa using hv
exact mem_iUnion.mpr ⟨⟨v,hv'⟩,by simpa using hγ⟩
obtain ⟨S,hS⟩ := (isCompact_closedBall (0 : Plane) D).elim_finite_subcover
(fun u : {v : Plane // ‖v‖ ≤ D} => ball u.val γ) (fun _ => isOpen_ball) hc
refine ⟨S,?_⟩
intro v hv
obtain ⟨u,hu,hvu⟩ := mem_iUnion₂.mp (hS (by simpa using hv))
exact ⟨u,hu,hvu⟩
theorem lattice_mem_box_of_norm (p : Vertex) (n : ℕ)
(hp : ‖latticeEmbed p‖ ≤ (n : ℝ)) : p ∈ latticeBox n := by
have h0 := (plane_coordinate_le_norm (latticeEmbed p) 0).trans hp
have h1 := (plane_coordinate_le_norm (latticeEmbed p) 1).trans hp
simp only [latticeEmbed,Matrix.cons_val_zero,Matrix.cons_val_one,
Matrix.cons_val_fin_one] at h0 h1
have hh0 : -(n : ℤ) ≤ p.1 ∧ p.1 ≤ n := by exact_mod_cast abs_le.mp h0
have hh1 : -(n : ℤ) ≤ p.2 ∧ p.2 ≤ n := by exact_mod_cast abs_le.mp h1
simpa only [latticeBox,Finset.mem_product,Finset.mem_Icc] using And.intro hh0 hh1
theorem scaled_net_norm (D : ℝ) (u : {v : Plane // ‖v‖ ≤ D}) (n : ℕ) :
‖latticeEmbed (latticeFloor ((n : ℝ) • u.val))‖ ≤ D*n+2 := by
have ht := norm_add_le
(latticeEmbed (latticeFloor ((n : ℝ) • u.val))-(n : ℝ) • u.val)
((n : ℝ) • u.val)
rw [sub_add_cancel,norm_smul,Real.norm_natCast] at ht
have hu := mul_le_mul_of_nonneg_left u.property (Nat.cast_nonneg n : (0 : ℝ) ≤ n)
linarith [latticeFloor_error ((n : ℝ) • u.val)]
theorem scaled_net_approximates (D γ : ℝ)
(S : Finset {v : Plane // ‖v‖ ≤ D})
(hS : ∀ v : Plane, ‖v‖ ≤ D → ∃ u ∈ S, ‖v-u.val‖ < γ)
(n : ℕ) (hn : 0 < n) (p : Vertex) (hp : ‖latticeEmbed p‖ ≤ D*n) :
∃ u ∈ S, ‖latticeEmbed (latticeFloor ((n : ℝ) • u.val)-p)‖ ≤ γ*n+2 := by
have hn' : (0 : ℝ) < n := by exact_mod_cast hn
let v := (1/(n : ℝ)) • latticeEmbed p
have hv : ‖v‖ ≤ D := by
dsimp [v]
rw [norm_smul,Real.norm_eq_abs,abs_of_pos (one_div_pos.mpr hn')]
rw [one_div_mul_eq_div]
exact (div_le_iff₀ hn').mpr (by nlinarith)
obtain ⟨u,hu,hvu⟩ := hS v hv
refine ⟨u,hu,?_⟩
have he : (n : ℝ) • v = latticeEmbed p := by
dsimp [v]; rw [smul_smul]; simp [hn'.ne']
have hd : ‖(n : ℝ) • u.val-latticeEmbed p‖ ≤ γ*n := by
rw [← he,← smul_sub,norm_smul,Real.norm_natCast,norm_sub_rev]
nlinarith
have ht := norm_add_le
(latticeEmbed (latticeFloor ((n : ℝ) • u.val))-(n : ℝ) • u.val)
((n : ℝ) • u.val-latticeEmbed p)
rw [sub_add_sub_cancel,← latticeEmbed_sub] at ht
linarith [latticeFloor_error ((n : ℝ) • u.val)]
end PlanarFPP
end OAI
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