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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 | import OAI.Geometry.RuledCovering.PlanarIntegration
import OAI.Geometry.RuledCovering.SingularBounds
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
def coneTriangle (a : ℝ) : Set Plane :=
{z | 0 ≤ z 1 ∧ z 1 ≤ a ∧ |z 0| ≤ z 1}
lemma coneTriangle_isCompact (a : ℝ) : IsCompact (coneTriangle a) := by
have hc : IsClosed (coneTriangle a) := by
unfold coneTriangle
exact (isClosed_le (f := fun _ : Plane => (0:ℝ)) (g := fun z => z 1) (by fun_prop) (by fun_prop)).inter
((isClosed_le (f := fun z : Plane => z 1) (g := fun _ => a) (by fun_prop) (by fun_prop)).inter
(isClosed_le (f := fun z : Plane => |z 0|) (g := fun z => z 1) (by fun_prop) (by fun_prop)))
exact (isCompact_closedBall (0:Plane) (2*a)).of_isClosed_subset hc (by
intro z hz
rw [Metric.mem_closedBall,dist_zero_right]
have he := plane_norm_le_sum z
rw [abs_of_nonneg hz.1] at he
exact he.trans (by linarith [hz.2.1,hz.2.2]))
lemma coneTriangle_volume {a : ℝ} (ha : 0 ≤ a) : volume.real (coneTriangle a) = a^2 := by
have heq : ∀ z : Plane, z ∈ coneTriangle a ↔ z 1 ∈ Icc 0 a ∧ z 0 ∈ Icc (-z 1) (z 1) := by
intro z
simp only [coneTriangle,mem_ofPred_eq,mem_Icc,← abs_le]
tauto
rw [plane_volume_horizontal_interval (coneTriangle_isCompact a).measurableSet
(coneTriangle_isCompact a).measure_lt_top.ne (s := Icc 0 a) (l := fun r => -r) (u := id) measurableSet_Icc heq]
have he : (∫ r in Icc (0:ℝ) a, max (r- -r) 0) = ∫ r in (0:ℝ)..a, 2*r := by
rw [intervalIntegral.integral_of_le ha,← integral_Icc_eq_integral_Ioc]
apply setIntegral_congr_fun measurableSet_Icc
intro r hr
change max (r- -r) 0 = 2*r
rw [max_eq_left (by linarith [hr.1])]
ring
change (∫ r in Icc (0:ℝ) a, max (r- -r) 0) = a^2
rw [he,intervalIntegral.integral_const_mul]
have hh := integral_pow (a := (0:ℝ)) (b := a) 1
simp only [pow_one] at hh
rw [hh]
norm_num
ring
def coneBand (d a : ℝ) : Set Plane :=
{z | d ≤ z 1 ∧ z 1 ≤ a ∧ |z 0| ≤ z 1}
lemma coneBand_isCompact (d a : ℝ) : IsCompact (coneBand d a) := by
apply (coneTriangle_isCompact a).of_isClosed_subset
· unfold coneBand
exact (isClosed_le (f := fun _ : Plane => d) (g := fun z => z 1) (by fun_prop) (by fun_prop)).inter
((isClosed_le (f := fun z : Plane => z 1) (g := fun _ => a) (by fun_prop) (by fun_prop)).inter
(isClosed_le (f := fun z : Plane => |z 0|) (g := fun z => z 1) (by fun_prop) (by fun_prop)))
· intro z hz
exact ⟨(abs_nonneg _).trans hz.2.2,hz.2⟩
lemma coneBand_volume {d a : ℝ} (hd : 0 ≤ d) (hda : d ≤ a) :
volume.real (coneBand d a) = a^2-d^2 := by
have heq : ∀ z : Plane, z ∈ coneBand d a ↔ z 1 ∈ Icc d a ∧ z 0 ∈ Icc (-z 1) (z 1) := by
intro z
simp only [coneBand,mem_ofPred_eq,mem_Icc,← abs_le]
tauto
rw [plane_volume_horizontal_interval (coneBand_isCompact d a).measurableSet
(coneBand_isCompact d a).measure_lt_top.ne (s := Icc d a) (l := fun r => -r) (u := id) measurableSet_Icc heq]
have he : (∫ r in Icc d a, max (r- -r) 0) = ∫ r in d..a, 2*r := by
rw [intervalIntegral.integral_of_le hda,← integral_Icc_eq_integral_Ioc]
apply setIntegral_congr_fun measurableSet_Icc
intro r hr
change max (r- -r) 0 = 2*r
rw [max_eq_left (by linarith [hr.1])]
ring
change (∫ r in Icc d a, max (r- -r) 0) = a^2-d^2
rw [he,intervalIntegral.integral_const_mul]
have hh := integral_pow (a := d) (b := a) 1
simp only [pow_one] at hh
rw [hh]
norm_num
ring
lemma coneBand_volume_vertical {d a : ℝ} (ha : a ≤ 2) :
volume.real (coneBand d a) = ∫ b in Icc (-2:ℝ) 2, max (a-max d |b|) 0 := by
apply plane_volume_vertical_interval (coneBand_isCompact d a).measurableSet
(coneBand_isCompact d a).measure_lt_top.ne measurableSet_Icc
intro z
change (d ≤ z 1 ∧ z 1 ≤ a ∧ |z 0| ≤ z 1) ↔
z 0 ∈ Icc (-2:ℝ) 2 ∧ z 1 ∈ Icc (max d |z 0|) a
simp only [mem_Icc,max_le_iff]
constructor
· intro hz
have hb := abs_le.mp (hz.2.2.trans (hz.2.1.trans ha))
exact ⟨hb,⟨⟨hz.1,hz.2.2⟩,hz.2.1⟩⟩
· rintro ⟨_,⟨⟨hd,hb⟩,ha⟩⟩
exact ⟨hd,ha,hb⟩
lemma singularDomain_volume_vertical (p gamma c : 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) = ∫ b in Icc (-2:ℝ) 2,
max ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-max (c^gamma) |b|) 0 := by
apply plane_volume_vertical_interval (singularDomain_isCompact p gamma c).measurableSet
(singularDomain_isCompact p gamma c).measure_lt_top.ne measurableSet_Icc
intro z
unfold singularDomain
simp only [mem_ofPred_eq,mem_Icc,max_le_iff]
constructor
· intro hz
have hb := abs_le.mp (hz.2.2.1.trans hz.2.1)
exact ⟨hb,⟨⟨hz.1,hz.2.2.1⟩,hz.2.2.2⟩⟩
· rintro ⟨hb,⟨⟨hd,hb'⟩,hceil⟩⟩
exact ⟨hd,hceil.trans (hcap (z 0) (abs_le.mpr hb)),hb',hceil⟩
end RuledApproximation
end
end OAI
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