File size: 3,666 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 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 | import OAI.Probability.FirstPassage.GridMeshes
namespace OAI
namespace PlanarFPP
noncomputable def meshFloor (x d : ℤ) : ℤ := x / d * d
lemma meshFloor_bounds (x d : ℤ) (hd : 0 < d) :
meshFloor x d ≤ x ∧ x < meshFloor x d+d := by
have h0 := Int.emod_nonneg x (ne_of_gt hd)
have h1 := Int.emod_lt_of_pos x hd
have he := Int.emod_add_ediv_mul x d
dsimp [meshFloor]
omega
lemma meshFloor_dvd (x d : ℤ) : d ∣ meshFloor x d := by
exact ⟨x/d,by simp [meshFloor,mul_comm]⟩
lemma meshFloor_of_dvd (x d : ℤ) (h : d ∣ x) : meshFloor x d=x := by
exact Int.ediv_mul_cancel h
lemma meshFloor_greatest (x d y : ℤ) (hd : 0 < d) (hy : y ≤ x) (hdy : d ∣ y) :
y ≤ meshFloor x d := by
obtain ⟨z,rfl⟩ := hdy
have hz : z ≤ x/d := (Int.le_ediv_iff_mul_le hd).mpr (by simpa [mul_comm] using hy)
exact (show d*z ≤ x/d*d by nlinarith)
lemma meshFloor_refinement (x d D : ℤ) (hd : 0 < d) (hD : 0 < D) (hdD : d ∣ D) :
meshFloor x D ≤ meshFloor x d ∧ meshFloor x d-meshFloor x D < D := by
have hb := meshFloor_bounds x D hD
have hb' := meshFloor_bounds x d hd
exact ⟨meshFloor_greatest x d _ hd hb.1 (dvd_trans hdD (meshFloor_dvd x D)),by omega⟩
theorem meshFloor_refinement_count (x d D : ℤ) (hd : 0 < d) (hD : 0 < D)
(hdD : d ∣ D) (hratio : D ≤ 16*d) :
∃ n : ℕ, n < 16 ∧ meshFloor x D+(n : ℤ)*d=meshFloor x d := by
have hb := meshFloor_refinement x d D hd hD hdD
have hdiv : d ∣ meshFloor x d-meshFloor x D :=
dvd_sub (meshFloor_dvd x d) (dvd_trans hdD (meshFloor_dvd x D))
obtain ⟨z,hz⟩ := hdiv
have hz0 : 0 ≤ z := by nlinarith [hb.1]
have hz16 : z < 16 := by nlinarith [hb.2]
refine ⟨z.toNat,by omega,?_⟩
rw [Int.toNat_of_nonneg hz0]
nlinarith
noncomputable def endpointColumn (x : ℤ) (m j : ℕ) : ℤ := meshFloor x (gridMesh m j)
lemma endpointColumn_last (x : ℤ) (m : ℕ) :
endpointColumn x m (gridLastLevel m)=x := by
simp [endpointColumn,gridLastLevel_unit,meshFloor]
lemma endpointColumn_step (x : ℤ) (m j : ℕ) :
∃ n : ℕ, n < 16 ∧ endpointColumn x m j+(n : ℤ)*gridMesh m (j+1)=
endpointColumn x m (j+1) :=
meshFloor_refinement_count x _ _ (gridMesh_pos m (j+1)) (gridMesh_pos m j)
(gridMesh_divides m j) (gridMesh_ratio_le m j)
lemma gridMesh_dvd_of_le (m i j : ℕ) (hij : i ≤ j) : gridMesh m j ∣ gridMesh m i := by
apply pow_dvd_pow
unfold gridMeshExponent
omega
lemma endpointColumn_mono (x : ℤ) (m : ℕ) : Monotone (endpointColumn x m) := by
intro i j hij
exact meshFloor_greatest x _ _ (gridMesh_pos m j)
(meshFloor_bounds x _ (gridMesh_pos m i)).1
(dvd_trans (gridMesh_dvd_of_le m i j hij) (meshFloor_dvd x _))
noncomputable def entranceColumn (x : ℤ) (m J : ℕ) : ℤ :=
endpointColumn x m J-gridMesh m J
noncomputable def exitColumn (x : ℤ) (m J : ℕ) : ℤ := -entranceColumn (-x) m J
lemma entranceColumn_span (x : ℤ) (m J : ℕ) :
gridMesh m J ≤ x-entranceColumn x m J ∧
x-entranceColumn x m J < 2*gridMesh m J := by
have hb := meshFloor_bounds x (gridMesh m J) (gridMesh_pos m J)
dsimp [entranceColumn,endpointColumn]
omega
lemma exitColumn_span (x : ℤ) (m J : ℕ) :
gridMesh m J ≤ exitColumn x m J-x ∧
exitColumn x m J-x < 2*gridMesh m J := by
have hb := entranceColumn_span (-x) m J
dsimp [exitColumn]
omega
lemma entranceColumn_dvd (x : ℤ) (m J : ℕ) : gridMesh m J ∣ entranceColumn x m J :=
dvd_sub (meshFloor_dvd x _) (dvd_refl _)
lemma exitColumn_dvd (x : ℤ) (m J : ℕ) : gridMesh m J ∣ exitColumn x m J :=
dvd_neg.mpr (entranceColumn_dvd (-x) m J)
end PlanarFPP
end OAI
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