| import OAI.Probability.FirstPassage.EndpointCoarse |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set Finset |
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| lemma gridMesh_scale_bound (m J : ℕ) : |
| (2 : ℝ)^m ≤ (2 : ℝ)^(4*J+1)*(gridMesh m J : ℝ) := |
| (div_le_iff₀ (by exact_mod_cast gridMesh_pos m J)).mp (gridMesh_count_ratio m J) |
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| noncomputable def coarseColumnIndices (A : ℝ) (J : ℕ) : Finset ℤ := |
| gridIndexRange ((A+2)*(2 : ℝ)^(4*J+1)) |
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| lemma coarse_column_index (A : ℝ) (hA : 0 ≤ A) (m J : ℕ) (X : ℤ) |
| (hX : gridMesh m J ∣ X) (hbound : |(X : ℝ)| ≤ (A+2)*(2 : ℝ)^m) : |
| ∃ i ∈ coarseColumnIndices A J, X=gridMesh m J*i := by |
| obtain ⟨i,hi⟩ := hX |
| refine ⟨i,?_,hi⟩ |
| apply mem_gridIndexRange_of_abs_le |
| have hd : (0 : ℝ) < gridMesh m J := by exact_mod_cast gridMesh_pos m J |
| have hprod := mul_le_mul_of_nonneg_left (gridMesh_scale_bound m J) (by linarith : 0 ≤ A+2) |
| rw [hi,Int.cast_mul,abs_mul,abs_of_pos hd] at hbound |
| nlinarith |
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| noncomputable def coarseRegionFactor (A : ℝ) (J : ℕ) : ℝ := (2*A+8)*(2 : ℝ)^(4*J+1) |
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| lemma endpointBox_into_rareFastRegion (b A R : ℝ) (hA : 0 ≤ A) (k m J : ℕ) (X Y : ℤ) |
| (hX : |(X : ℝ)| ≤ (A+2)*(2 : ℝ)^m) |
| (hspan : (gridMesh m J : ℝ) ≤ ((Y-X : ℤ) : ℝ)) : |
| endpointBox b (A+4) R k m ⊆ |
| rareFastRegion b (coarseRegionFactor A J) (R+1) ((2 : ℝ)^k) (Y-X) (endpointVertex b 0 X) := by |
| have hd : (0 : ℝ) < gridMesh m J := by exact_mod_cast gridMesh_pos m J |
| have hY : (0 : ℝ) ≤ ((Y-X : ℤ) : ℝ) := hd.le.trans hspan |
| have hscale : (2 : ℝ)^m ≤ (2 : ℝ)^(4*J+1)*((Y-X : ℤ) : ℝ) := |
| (gridMesh_scale_bound m J).trans (mul_le_mul_of_nonneg_left hspan (by positivity)) |
| intro z hz |
| have hzabs : |(z.1 : ℝ)-(X : ℝ)| ≤ (2*A+6)*(2 : ℝ)^m := |
| (abs_sub _ _).trans (by linarith [hz.1]) |
| have habs : |(z.1 : ℝ)-(X : ℝ)| ≤ coarseRegionFactor A J*((Y-X : ℤ) : ℝ) := by |
| have hc := mul_le_mul_of_nonneg_left hscale (by linarith : 0 ≤ 2*A+8) |
| unfold coarseRegionFactor |
| have hp : (0 : ℝ) < 2^m := by positivity |
| nlinarith |
| have hcoord : ((z-endpointVertex b 0 X).1 : ℝ)=(z.1 : ℝ)-(X : ℝ) := by simp [endpointVertex] |
| have htrans : transverse b (z-endpointVertex b 0 X)=transverse b z-transverse b (endpointVertex b 0 X) := by |
| simp [transverse]; ring |
| have ht : |transverse b (endpointVertex b 0 X)| ≤ 1 := by simpa using (endpointVertex_close b 0 X).le |
| have hs : (1 : ℝ) ≤ 2^k := one_le_pow₀ (by norm_num) |
| refine ⟨?_,?_,?_⟩ |
| · rw [hcoord] |
| have hh := (abs_le.mp habs).1 |
| linarith |
| · rw [hcoord] |
| have hh := (abs_le.mp habs).2 |
| nlinarith |
| · rw [htrans] |
| exact (abs_sub _ _).trans (by linarith [hz.2]) |
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| end PlanarFPP |
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| end OAI |
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