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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 | import OAI.Probability.FirstPassage.EndpointCoarse
namespace OAI
namespace PlanarFPP
open Set Finset
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)
noncomputable def coarseColumnIndices (A : ℝ) (J : ℕ) : Finset ℤ :=
gridIndexRange ((A+2)*(2 : ℝ)^(4*J+1))
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
noncomputable def coarseRegionFactor (A : ℝ) (J : ℕ) : ℝ := (2*A+8)*(2 : ℝ)^(4*J+1)
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])
end PlanarFPP
end OAI
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