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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
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