| 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 |
|
|
