| 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 | |