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695bd70 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | import OAI.Geometry.RuledCovering.ConeVolume
import OAI.Geometry.RuledCovering.SingularOverlap
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
lemma positivepart_increment {a h t : ℝ} (hh : 0 ≤ h) :
0 ≤ max (a+h-t) 0-max (a-t) 0 ∧ max (a+h-t) 0-max (a-t) 0 ≤ h := by
by_cases ht : t ≤ a
· rw [max_eq_left (by linarith),max_eq_left (by linarith)]
constructor <;> linarith
· rw [show max (a-t) 0 = 0 from max_eq_right (by linarith), sub_zero]
exact ⟨le_max_right _ _, max_le (by linarith) hh⟩
lemma positivepart_floor_increment {a d h t : ℝ} (hd : d ≤ a) (hh : 0 ≤ h) :
max (a+h-max d t) 0-max (a-max d t) 0 = max (a+h-t) 0-max (a-t) 0 := by
by_cases ht : t ≤ d
· rw [max_eq_left ht,max_eq_left (by linarith),max_eq_left (by linarith),
max_eq_left (by linarith),max_eq_left (by linarith)]
ring
· rw [max_eq_right (le_of_not_ge ht)]
lemma rpow_five_halves_div_square {c : ℝ} (hc : 0 < c) :
c^(5/2:ℝ)/c^2=c^(1/2:ℝ) := by
rw [← Real.rpow_natCast,← Real.rpow_sub hc]
norm_num
def capIncrementQuotient (p c b : ℝ) : ℝ :=
(max ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-|b|) 0 -
max ((1/2:ℝ)-|b|) 0)/c^2
lemma capIncrementQuotient_nonneg_bound {p c b : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (hc : 0 < c) :
0 ≤ capIncrementQuotient p c b ∧
capIncrementQuotient p c b ≤ ceilingPolynomial p b+c^(1/2:ℝ) := by
have hd := ceiling_nonneg hp hp2 b
have hi := positivepart_increment (a := (1/2:ℝ)) (t := |b|)
(h := c^2*ceilingPolynomial p b+c^(5/2:ℝ)) (by positivity)
have he : (c^2*ceilingPolynomial p b+c^(5/2:ℝ))/c^2 = ceilingPolynomial p b+c^(1/2:ℝ) := by
rw [add_div,mul_div_cancel_left₀ _ (pow_ne_zero _ hc.ne'),rpow_five_halves_div_square hc]
unfold capIncrementQuotient
rw [← add_assoc] at hi
exact ⟨div_nonneg hi.1 (sq_nonneg c),(div_le_div_of_nonneg_right hi.2 (sq_nonneg c)).trans_eq he⟩
lemma capIncrementQuotient_inside {p c b : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2)
(hc : 0 < c) (hb : |b| ≤ 1/2) :
capIncrementQuotient p c b = ceilingPolynomial p b+c^(1/2:ℝ) := by
have hd := ceiling_nonneg hp hp2 b
have hn : 0 ≤ c^2*ceilingPolynomial p b+c^(5/2:ℝ) := by positivity
unfold capIncrementQuotient
rw [max_eq_left (by linarith),max_eq_left (by linarith)]
have he : ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-|b|-(1/2-|b|)) =
c^2*ceilingPolynomial p b+c^(5/2:ℝ) := by ring
rw [he,add_div,mul_div_cancel_left₀ _ (pow_ne_zero _ hc.ne'),rpow_five_halves_div_square hc]
lemma ceilingPolynomial_continuous (p : ℝ) : Continuous (ceilingPolynomial p) := by
unfold ceilingPolynomial
fun_prop
lemma singular_ceiling_eventually (p : ℝ) :
∀ᶠ c : ℝ in 𝓝[>] 0, ∀ b : ℝ, |b| ≤ 2 →
(1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ) ≤ 2 := by
obtain ⟨C,hC⟩ := (isCompact_Icc (a := (-2:ℝ)) (b := 2)).exists_bound_of_continuousOn
(ceilingPolynomial_continuous p).continuousOn
have ht : Tendsto (fun c : ℝ => c^2*C+c^(5/2:ℝ)) (𝓝[>] 0) (𝓝 0) := by
have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds
simpa using ((h0.pow 2).mul_const C).add (right_rpow_tendsto_zero (q := (5/2:ℝ)) (by norm_num))
filter_upwards [ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<3/2))] with c hc b hb
have h := (le_abs_self (ceilingPolynomial p b)).trans (by simpa only [Real.norm_eq_abs] using hC b (abs_le.mp hb))
have hh := mul_le_mul_of_nonneg_left h (sq_nonneg c)
linarith
lemma capIncrementQuotient_tendsto {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (b : ℝ) :
Tendsto (fun c : ℝ => capIncrementQuotient p c b) (𝓝[>] 0)
(𝓝 ((Icc (-(1/2:ℝ)) (1/2)).indicator (ceilingPolynomial p) b)) := by
by_cases hb : |b| ≤ 1/2
· rw [Set.indicator_of_mem (show b ∈ Icc (-(1/2:ℝ)) (1/2) from abs_le.mp hb)]
have h : Tendsto (fun c : ℝ => ceilingPolynomial p b+c^(1/2:ℝ)) (𝓝[>] 0)
(𝓝 (ceilingPolynomial p b)) := by
simpa using (right_rpow_tendsto_zero (q := (1/2:ℝ)) (by norm_num)).const_add (ceilingPolynomial p b)
apply h.congr'
filter_upwards [self_mem_nhdsWithin] with c hc
exact (capIncrementQuotient_inside hp hp2 hc hb).symm
· rw [Set.indicator_of_notMem (by simpa only [mem_Icc,← abs_le] using hb)]
have ht : Tendsto (fun c : ℝ => (1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ))
(𝓝[>] 0) (𝓝 (1/2:ℝ)) := by
have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds
simpa using (tendsto_const_nhds.add ((h0.pow 2).mul_const (ceilingPolynomial p b))).add
(right_rpow_tendsto_zero (q := (5/2:ℝ)) (by norm_num))
apply tendsto_const_nhds.congr'
filter_upwards [ht.eventually (gt_mem_nhds (lt_of_not_ge hb))] with c hc
unfold capIncrementQuotient
rw [max_eq_right (by linarith),max_eq_right (by linarith)]
simp
lemma capIncrementQuotient_integral_tendsto {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) :
Tendsto (fun c : ℝ => ∫ b in Icc (-2:ℝ) 2, capIncrementQuotient p c b) (𝓝[>] 0)
(𝓝 (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b)) := by
have hm : ∀ c : ℝ, Continuous (capIncrementQuotient p c) := by
intro c
unfold capIncrementQuotient
exact (((continuous_const.add (continuous_const.mul (ceilingPolynomial_continuous p))).add continuous_const).sub continuous_abs).max continuous_const |>.sub
((continuous_const.sub continuous_abs).max continuous_const) |>.div_const _
have h := tendsto_integral_filter_of_dominated_convergence
(μ := volume.restrict (Icc (-2:ℝ) 2)) (fun b => ceilingPolynomial p b+1)
(Filter.Eventually.of_forall (fun c => (hm c).aestronglyMeasurable))
(show ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ᵐ b ∂volume.restrict (Icc (-2:ℝ) 2),
‖capIncrementQuotient p c b‖ ≤ ceilingPolynomial p b+1 from by
have he : ∀ᶠ c : ℝ in 𝓝[>] 0, c ≤ 1 :=
(eventually_lt_nhds (by norm_num : (0:ℝ)<1)).filter_mono nhdsWithin_le_nhds |>.mono (fun _ h => h.le)
filter_upwards [self_mem_nhdsWithin,he] with c hc hc1
exact Filter.Eventually.of_forall (fun b => by
obtain ⟨h0,h1⟩ := capIncrementQuotient_nonneg_bound hp hp2 hc (b := b)
rw [Real.norm_eq_abs,abs_of_nonneg h0]
exact h1.trans (by linarith [Real.rpow_le_one (z := (1/2:ℝ)) hc.le hc1 (by norm_num)])))
(((ceilingPolynomial_continuous p).add continuous_const).continuousOn.integrableOn_Icc)
(Filter.Eventually.of_forall (capIncrementQuotient_tendsto hp hp2))
have he : (∫ b in Icc (-2:ℝ) 2, (Icc (-(1/2:ℝ)) (1/2)).indicator (ceilingPolynomial p) b) =
∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b := by
rw [integral_indicator measurableSet_Icc,Measure.restrict_restrict measurableSet_Icc]
congr 2
apply Set.inter_eq_left.mpr
intro b hb
constructor <;> linarith [hb.1,hb.2]
exact he ▸ h
lemma singularDomain_volume_difference {p gamma c : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2)
(hc : 0 < c) (hd : c^gamma ≤ (1/2:ℝ))
(hcap : ∀ b : ℝ, |b| ≤ 2 → (1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ) ≤ 2) :
(volume.real (singularDomain p gamma c)-(1/4-(c^gamma)^2))/c^2 =
∫ b in Icc (-2:ℝ) 2, capIncrementQuotient p c b := by
have hv := coneBand_volume (d := c^gamma) (a := (1/2:ℝ)) (by positivity) hd
norm_num at hv
rw [← hv,singularDomain_volume_vertical p gamma c hcap,coneBand_volume_vertical (by norm_num)]
have hI₁ : IntegrableOn (fun b : ℝ => max ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-max (c^gamma) |b|) 0)
(Icc (-2:ℝ) 2) := by
apply ContinuousOn.integrableOn_Icc
apply Continuous.continuousOn
exact (((continuous_const.add (continuous_const.mul (ceilingPolynomial_continuous p))).add continuous_const).sub
(continuous_const.max continuous_abs)).max continuous_const
have hI₂ : IntegrableOn (fun b : ℝ => max ((1/2:ℝ)-max (c^gamma) |b|) 0) (Icc (-2:ℝ) 2) := by
apply ContinuousOn.integrableOn_Icc
exact ((continuous_const.sub (continuous_const.max continuous_abs)).max continuous_const).continuousOn
rw [← integral_sub hI₁ hI₂,← integral_div]
apply setIntegral_congr_fun measurableSet_Icc
intro b _
unfold capIncrementQuotient
apply congrArg (fun x : ℝ => x/c^2)
have he := positivepart_floor_increment (a := (1/2:ℝ)) (d := c^gamma)
(h := c^2*ceilingPolynomial p b+c^(5/2:ℝ)) (t := |b|) hd
(by have := ceiling_nonneg hp hp2 b; positivity)
simpa only [← add_assoc] using he
lemma singularDomain_volume_tendsto {p gamma : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (hg : 1 < gamma) :
Tendsto (fun c : ℝ => (volume.real (singularDomain p gamma c)-1/4)/c^2) (𝓝[>] 0)
(𝓝 (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b)) := by
have h := (capIncrementQuotient_integral_tendsto hp hp2).sub (singular_tip_small hg).tendsto_div_nhds_zero
simp only [sub_zero] at h
apply h.congr'
filter_upwards [self_mem_nhdsWithin,singular_ceiling_eventually p,
(right_rpow_tendsto_zero (lt_trans (by norm_num) hg)).eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))] with c hc hcap hd
rw [← singularDomain_volume_difference hp hp2 hc hd.le hcap]
ring
end RuledApproximation
end
end OAI
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