| import OAI.Probability.RadialTransition.KernelError | |
| namespace OAI | |
| noncomputable section | |
| open Function MeasureTheory Set Filter | |
| open scoped BigOperators ENNReal NNReal ContDiff | |
| namespace RadialTransition | |
| lemma grid_inner_coords {m D : ℕ} {i : GridIndex m} (hi : i ∉ boundaryGrid m D) (d : Fin 3) : | |
| D ≤ (i d:ℕ) ∧ (i d:ℕ)+D < m := by | |
| have hn : ¬ ((i d:ℕ) < D ∨ m-D ≤ (i d:ℕ)) := fun h => hi (mem_boundaryGrid.mpr ⟨d,h⟩) | |
| omega | |
| lemma grid_inner_reach {m : ℕ} {T : ℝ} (hT : 0 ≤ T) (i : GridIndex m) | |
| (hi : i ∉ boundaryGrid m (cellMargin T).toNat) {j : Cell} (hj : CellReach T (gridEmbed i) j) : | |
| ∃ k : GridIndex m, gridEmbed k = j := by | |
| have hD : 0 ≤ cellMargin T := by unfold cellMargin; have := Int.ceil_nonneg (by positivity : 0 ≤ 4*T); omega | |
| have hd (d : Fin 3) : 0 ≤ j d ∧ j d < (m:ℤ) := by | |
| have hc := grid_inner_coords hi d | |
| have ha := abs_le.mp (hj d) | |
| have he : ((cellMargin T).toNat:ℤ) = cellMargin T := Int.toNat_of_nonneg hD | |
| change -cellMargin T ≤ j d-(i d:ℤ) ∧ j d-(i d:ℤ) ≤ cellMargin T at ha | |
| omega | |
| refine ⟨fun d => ⟨(j d).toNat, by have := hd d; omega⟩, ?_⟩ | |
| funext d | |
| exact Int.toNat_of_nonneg (hd d).1 | |
| lemma macroKernel_support_inside {q m : ℕ} (hq : 0 < q) {R : ℝ} (hR : 0 ≤ R) (i : GridIndex m) | |
| (hi : i ∉ boundaryGrid m (cellMargin (R/q+1)).toNat) : | |
| ∀ y ∉ halfCube ((m:ℝ)*q/4), scaledBump R (macroOrigin q (gridEmbed i)-y) = 0 := by | |
| intro y hy | |
| by_cases hR0 : R = 0 | |
| · simp [hR0, scaledBump] | |
| have hRpos : 0 < R := lt_of_le_of_ne hR (Ne.symm hR0) | |
| apply scaledBump_zero hRpos | |
| rw [← dist_eq_norm] | |
| apply le_of_not_gt | |
| intro hd | |
| have hm : y ∈ macroRegion q (macroOf q (cellOf y)) := rfl | |
| have hr := macroReach_of_dist hq (macroOrigin_mem hq (gridEmbed i)) hm hd.le | |
| have hT : 0 ≤ R/(q:ℝ)+1 := by positivity | |
| obtain ⟨j, hj⟩ := grid_inner_reach hT i hi hr | |
| rw [← hj] at hm | |
| exact hy (macroRegion_subset_halfCube hq j hm) | |
| lemma macroKernelLower_diagonal {q : ℕ} {R : ℝ} (hR : 0 < R) (i : Cell) : | |
| macroKernelLower q R i i ≤ (R^3)⁻¹*bumpIntegral⁻¹ := by | |
| apply max_le | |
| · have := bumpIntegral_pos; positivity | |
| · have hb := scaledBump_bound hR (macroOrigin q i-macroOrigin q i) | |
| have hn := macroKernelError_nonneg q R i i | |
| have ha := le_abs_self (scaledBump R (macroOrigin q i-macroOrigin q i)) | |
| change macroKernel q R i i ≤ _ at ha | |
| linarith | |
| def kernelRowError (q : ℕ) (R : ℝ) : ℝ := | |
| (10000*(bumpLip:ℝ)+bumpIntegral⁻¹)*(q:ℝ)/R | |
| lemma kernelRowError_nonneg (q : ℕ) {R : ℝ} (hR : 0 ≤ R) : 0 ≤ kernelRowError q R := by | |
| unfold kernelRowError | |
| have := bumpIntegral_pos | |
| positivity | |
| lemma macroKernelLower_row {q m : ℕ} (hq : 0 < q) {R : ℝ} (hR : (q:ℝ)/4 ≤ R) (i : GridIndex m) | |
| (hi : i ∉ boundaryGrid m (cellMargin (R/q+1)).toNat) : | |
| 1-kernelRowError q R ≤ ((q:ℝ)/4)^3*∑ j : GridIndex m, | |
| (if i = j then 0 else macroKernelLower q R (gridEmbed i) (gridEmbed j)) := by | |
| classical | |
| have hq0 : (0:ℝ) < q := by exact_mod_cast hq | |
| have hR0 : 0 < R := lt_of_lt_of_le (by positivity) hR | |
| have hall := macroKernelLower_row_all hq hR (gridEmbed i) (macroKernel_support_inside hq hR0.le i hi) | |
| have hs : (∑ j : GridIndex m, if i = j then (0:ℝ) else macroKernelLower q R (gridEmbed i) (gridEmbed j)) = | |
| (∑ j : GridIndex m, macroKernelLower q R (gridEmbed i) (gridEmbed j))-macroKernelLower q R (gridEmbed i) (gridEmbed i) := by | |
| have hh := Finset.sum_erase_add (Finset.univ : Finset (GridIndex m)) | |
| (fun j => macroKernelLower q R (gridEmbed i) (gridEmbed j)) (Finset.mem_univ i) | |
| have he : (∑ j : GridIndex m, if i = j then (0:ℝ) else macroKernelLower q R (gridEmbed i) (gridEmbed j)) = | |
| ∑ j ∈ Finset.univ.erase i, macroKernelLower q R (gridEmbed i) (gridEmbed j) := by | |
| simp only [Finset.sum_ite, Finset.sum_const_zero, zero_add] | |
| congr 1 | |
| ext j | |
| simp [ne_comm] | |
| rw [he] | |
| linarith | |
| rw [hs, mul_sub] | |
| have hd := mul_le_mul_of_nonneg_left (macroKernelLower_diagonal (q := q) hR0 (gridEmbed i)) (show 0 ≤ ((q:ℝ)/4)^3 by positivity) | |
| have hsq : ((q:ℝ)/4)^2 ≤ R^2 := pow_le_pow_left₀ (by positivity) hR 2 | |
| have hd' : ((q:ℝ)/4)^3*((R^3)⁻¹*bumpIntegral⁻¹) ≤ bumpIntegral⁻¹*q/R := by | |
| apply (le_div_iff₀ hR0).mpr | |
| have hb := bumpIntegral_pos | |
| have he : ((q:ℝ)/4)^3*((R^3)⁻¹*bumpIntegral⁻¹)*R = | |
| bumpIntegral⁻¹*((q:ℝ)/4)*(((q:ℝ)/4)^2/R^2) := by field_simp | |
| rw [he] | |
| have hh : ((q:ℝ)/4)^2/R^2 ≤ 1 := (div_le_one (by positivity)).mpr hsq | |
| calc | |
| _ ≤ bumpIntegral⁻¹*((q:ℝ)/4)*1 := mul_le_mul_of_nonneg_left hh (by positivity) | |
| _ ≤ _ := by nlinarith [inv_nonneg.mpr hb.le] | |
| unfold kernelRowError | |
| linear_combination hall + hd + hd' | |
| lemma kernelInterior_ratio (D : ℕ) : | |
| Tendsto (fun m : ℕ => (((Finset.univ : Finset (GridIndex m)) \ boundaryGrid m D).card:ℝ)/(m:ℝ)^3) | |
| atTop (nhds 1) := by | |
| classical | |
| have hcard (m : ℕ) : (((Finset.univ : Finset (GridIndex m)) \ boundaryGrid m D).card:ℝ) = | |
| (m:ℝ)^3-((boundaryGrid m D).card:ℝ) := by | |
| have hh := Finset.card_sdiff_add_card_inter (Finset.univ : Finset (GridIndex m)) (boundaryGrid m D) | |
| simp only [Finset.univ_inter, Finset.card_univ, Fintype.card_fun, Fintype.card_fin] at hh | |
| have hr : (((Finset.univ : Finset (GridIndex m)) \ boundaryGrid m D).card:ℝ)+((boundaryGrid m D).card:ℝ) = (m:ℝ)^3 := by exact_mod_cast hh | |
| linarith | |
| have hz : Tendsto (fun m : ℕ => (6:ℝ)*D/(m:ℝ)) atTop (nhds (0:ℝ)) := | |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop | |
| have hsmall : Tendsto (fun m : ℕ => ((boundaryGrid m D).card:ℝ)/(m:ℝ)^3) atTop (nhds (0:ℝ)) := by | |
| apply squeeze_zero' (Eventually.of_forall (fun _ => by positivity)) ?_ hz | |
| filter_upwards [eventually_gt_atTop (0:ℕ)] with m hm | |
| have hm0 : (0:ℝ) < m := by exact_mod_cast hm | |
| have hb : ((boundaryGrid m D).card:ℝ) ≤ 6*D*(m:ℝ)^2 := by exact_mod_cast card_boundaryGrid m D | |
| apply (div_le_iff₀ (by positivity : 0 < (m:ℝ)^3)).mpr | |
| calc | |
| _ ≤ 6*D*(m:ℝ)^2 := hb | |
| _ = _ := by field_simp | |
| have ht' : Tendsto (fun m : ℕ => 1-((boundaryGrid m D).card:ℝ)/(m:ℝ)^3) atTop (nhds (1:ℝ)) := by | |
| simpa using (tendsto_const_nhds.sub hsmall : Tendsto (fun m : ℕ => (1:ℝ)-((boundaryGrid m D).card:ℝ)/(m:ℝ)^3) atTop (nhds (1-0))) | |
| apply ht'.congr' | |
| filter_upwards [eventually_gt_atTop (0:ℕ)] with m hm | |
| have hm0 : (0:ℝ) < m := by exact_mod_cast hm | |
| rw [hcard] | |
| field_simp | |
| end RadialTransition | |
| end | |
| end OAI | |