File size: 7,203 Bytes
785c727 | 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 | import Mathlib
namespace OAI
open Set MeasureTheory
noncomputable section
namespace CylinderCovering.SimplexLayerCake
theorem integral_shadow_profile (L : ℝ) (hL : 0 < L) (n : ℕ) :
(∫ s in (0 : ℝ)..L, (1 - s / L) ^ n) = L / (n + 1 : ℝ) := by
have hLne : L ≠ 0 := ne_of_gt hL
have h := intervalIntegral.integral_comp_sub_mul (a := (0 : ℝ)) (b := L)
(c := L⁻¹) (fun x : ℝ => x ^ n) (inv_ne_zero hLne) 1
simp only [inv_inv, mul_zero, sub_zero, inv_mul_cancel₀ hLne, sub_self,
smul_eq_mul, integral_pow, one_pow, zero_pow (Nat.succ_ne_zero n),
sub_zero] at h
simpa only [div_eq_mul_inv, mul_comm, one_mul] using h
theorem integral_eq_area_mul_length_div
{α : Type*} [MeasurableSpace α] (μ : Measure α) (ℓ : α → ℝ)
(A L : ℝ) (hL : 0 < L) (n : ℕ)
(hℓ : Integrable ℓ μ) (hℓnonneg : 0 ≤ᵐ[μ] ℓ)
(hℓbound : ℓ ≤ᵐ[μ] fun _ => L)
(hprofile : ∀ s ∈ Ioo (0 : ℝ) L,
μ.real {x | s ≤ ℓ x} = A * (1 - s / L) ^ n) :
(∫ x, ℓ x ∂μ) = A * L / (n + 1 : ℝ) := by
rw [hℓ.integral_eq_integral_Ioc_meas_le hℓnonneg hℓbound,
integral_Ioc_eq_integral_Ioo]
calc
(∫ s in Ioo (0 : ℝ) L, μ.real {x | s ≤ ℓ x}) =
∫ s in Ioo (0 : ℝ) L, A * (1 - s / L) ^ n :=
setIntegral_congr_fun measurableSet_Ioo hprofile
_ = ∫ s in (0 : ℝ)..L, A * (1 - s / L) ^ n := by
rw [intervalIntegral.integral_of_le hL.le, integral_Ioc_eq_integral_Ioo]
_ = A * L / (n + 1 : ℝ) := by
rw [intervalIntegral.integral_const_mul, integral_shadow_profile L hL n]
ring
theorem prod_real_eq_integral_fiber
{α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(μ : Measure α) (ν : Measure β) [SFinite ν]
{S : Set (α × β)} (hS : MeasurableSet S) (hSfinite : (μ.prod ν) S ≠ ⊤) :
(μ.prod ν).real S = ∫ x, ν.real (Prod.mk x ⁻¹' S) ∂μ := by
have hm := measurable_measure_prodMk_left (ν := ν) hS
have hfinite : (∫⁻ x, ν (Prod.mk x ⁻¹' S) ∂μ) ≠ ⊤ := by
rwa [← Measure.prod_apply hS]
rw [Measure.real, Measure.prod_apply hS]
exact (integral_toReal hm.aemeasurable (ae_lt_top hm hfinite)).symm
theorem integrable_fiber_measure
{α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(μ : Measure α) (ν : Measure β) [SFinite ν]
{S : Set (α × β)} (hS : MeasurableSet S) (hSfinite : (μ.prod ν) S ≠ ⊤) :
Integrable (fun x => ν.real (Prod.mk x ⁻¹' S)) μ := by
have hm := measurable_measure_prodMk_left (ν := ν) hS
have hfinite : (∫⁻ x, ν (Prod.mk x ⁻¹' S) ∂μ) ≠ ⊤ := by
rwa [← Measure.prod_apply hS]
exact (integrable_toReal_iff hm.aemeasurable
((ae_lt_top hm hfinite).mono fun _ h => h.ne)).mpr hfinite
theorem prod_real_eq_area_mul_length_div
{α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(μ : Measure α) (ν : Measure β) [SFinite ν]
{S : Set (α × β)} (hS : MeasurableSet S) (hSfinite : (μ.prod ν) S ≠ ⊤)
(A L : ℝ) (hL : 0 < L) (n : ℕ)
(hbound : (fun x => ν.real (Prod.mk x ⁻¹' S)) ≤ᵐ[μ] fun _ => L)
(hprofile : ∀ s ∈ Ioo (0 : ℝ) L,
μ.real {x | s ≤ ν.real (Prod.mk x ⁻¹' S)} = A * (1 - s / L) ^ n) :
(μ.prod ν).real S = A * L / (n + 1 : ℝ) := by
rw [prod_real_eq_integral_fiber μ ν hS hSfinite]
exact integral_eq_area_mul_length_div μ _ A L hL n
(integrable_fiber_measure μ ν hS hSfinite)
(.of_forall fun _ => ENNReal.toReal_nonneg) hbound hprofile
theorem compact_fiber {E : Type*} [NormedAddCommGroup E]
{S : Set (E × ℝ)} (hS : IsCompact S) (x : E) :
IsCompact (Prod.mk x ⁻¹' S) := by
have hi : Isometry (Prod.mk x : ℝ → E × ℝ) := by
apply isometry_iff_dist_eq.mpr
intro a b
simp
exact hi.isClosedEmbedding.isCompact_preimage hS
theorem convex_fiber {E : Type*} [AddCommGroup E] [Module ℝ E]
{S : Set (E × ℝ)} (hS : Convex ℝ S) (x : E) :
Convex ℝ (Prod.mk x ⁻¹' S) :=
hS.affine_preimage ((AffineMap.const ℝ ℝ x).prod (AffineMap.id ℝ ℝ))
theorem le_fiber_measure_iff_exists_segment
{T : Set ℝ} (hT : IsCompact T) (hconv : Convex ℝ T)
{s : ℝ} (hs : 0 < s) :
s ≤ volume.real T ↔ ∃ t : ℝ, t ∈ T ∧ t + s ∈ T := by
by_cases hne : T.Nonempty
· let a := sInf T
let b := sSup T
have hI : T = Icc a b := eq_Icc_of_connected_compact (hconv.isConnected hne) hT
have hab : a ≤ b := by
have ha : a ∈ T := hT.sInf_mem hne
rw [hI] at ha
exact ha.2
rw [hI, Measure.real, Real.volume_Icc, ENNReal.toReal_ofReal (sub_nonneg.mpr hab)]
constructor
· intro h
refine ⟨a, ⟨le_rfl, hab⟩, ?_⟩
exact ⟨by linarith, by linarith⟩
· rintro ⟨t, ht, hts⟩
linarith [ht.1, hts.2]
· have hzero : T = ∅ := Set.not_nonempty_iff_eq_empty.mp hne
simp [hzero, not_le.mpr hs]
theorem fiber_superlevel_eq_projected_overlap
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{S : Set (E × ℝ)} (hS : IsCompact S) (hconv : Convex ℝ S)
{s : ℝ} (hs : 0 < s) :
{x | s ≤ volume.real (Prod.mk x ⁻¹' S)} =
Prod.fst '' {z : E × ℝ | z ∈ S ∧ (z.1, z.2 + s) ∈ S} := by
ext x
change (s ≤ volume.real (Prod.mk x ⁻¹' S)) ↔ _
rw [le_fiber_measure_iff_exists_segment (compact_fiber hS x) (convex_fiber hconv x) hs]
constructor
· rintro ⟨t, ht, hts⟩
exact ⟨(x,t), ⟨ht,hts⟩, rfl⟩
· rintro ⟨⟨y,t⟩, ⟨ht,hts⟩, hxy⟩
change y = x at hxy
subst y
exact ⟨t,ht,hts⟩
theorem fiber_measure_le_of_chord_bound
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{S : Set (E × ℝ)} (hS : IsCompact S) (hconv : Convex ℝ S)
{L : ℝ} (hL : 0 ≤ L)
(hchord : ∀ x : E, ∀ t r : ℝ, (x,t) ∈ S → (x,r) ∈ S → r - t ≤ L)
(x : E) : volume.real (Prod.mk x ⁻¹' S) ≤ L := by
by_cases hpos : 0 < volume.real (Prod.mk x ⁻¹' S)
· obtain ⟨t, ht, htr⟩ :=
(le_fiber_measure_iff_exists_segment (compact_fiber hS x)
(convex_fiber hconv x) hpos).mp le_rfl
have h := hchord x t (t + volume.real (Prod.mk x ⁻¹' S)) ht htr
linarith
· exact (le_of_not_gt hpos).trans hL
theorem compact_convex_prod_volume_of_overlap_profile
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E]
(μ : Measure E) {S : Set (E × ℝ)}
(hS : IsCompact S) (hconv : Convex ℝ S)
(hSfinite : (μ.prod volume) S ≠ ⊤)
(A L : ℝ) (hL : 0 < L) (n : ℕ)
(hchord : ∀ x : E, ∀ t r : ℝ, (x,t) ∈ S → (x,r) ∈ S → r - t ≤ L)
(hprofile : ∀ s ∈ Ioo (0 : ℝ) L,
μ.real (Prod.fst '' {z : E × ℝ | z ∈ S ∧ (z.1, z.2 + s) ∈ S}) =
A * (1 - s / L) ^ n) :
(μ.prod volume).real S = A * L / (n + 1 : ℝ) := by
apply prod_real_eq_area_mul_length_div μ volume hS.measurableSet hSfinite A L hL n
· exact .of_forall (fiber_measure_le_of_chord_bound hS hconv hL.le hchord)
· intro s hs
rw [fiber_superlevel_eq_projected_overlap hS hconv hs.1]
exact hprofile s hs
end CylinderCovering.SimplexLayerCake
end
end OAI
|