| import OAI.Geometry.CylinderCovering.Rational.FieldDefinitions |
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| namespace OAI |
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| open Set MeasureTheory |
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| noncomputable section |
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| namespace SlopeRational |
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| lemma ceilingPolynomial_continuous (d : ℝ) : Continuous (ceilingPolynomial d) := by |
| unfold ceilingPolynomial ceilingCoefficient |
| fun_prop |
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| lemma labelDomain_bounds {d ε q : ℝ} {z : Plane} (hz : z ∈ labelDomain d ε q) : |
| -(1 / 4 : ℝ) ≤ z 1 ∧ z 1 ≤ 1 / 2 ∧ |z 0| ≤ 3 / 8 := by |
| rcases hz with ⟨hw, hfloor, hceil⟩ |
| change 2 * |z 0| ≤ 1 / 2 - z 1 at hw |
| exact ⟨hfloor, by linarith [abs_nonneg (z 0)], by linarith⟩ |
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|
| lemma wedge_isClosed : IsClosed wedge := by |
| unfold wedge radius |
| exact isClosed_le (by fun_prop) (by fun_prop) |
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| lemma labelDomain_isClosed (d ε q : ℝ) : IsClosed (labelDomain d ε q) := by |
| unfold labelDomain |
| exact wedge_isClosed.inter ((isClosed_le |
| (f := fun _ : Plane => -(1 / 4 : ℝ)) (g := fun z => z 1) |
| (by fun_prop) (by fun_prop)).inter |
| (isClosed_le (f := fun z : Plane => -ε ^ 2 * ceilingPolynomial d (z 0) - q) |
| (g := fun z => z 1) |
| (by unfold ceilingPolynomial ceilingCoefficient; fun_prop) (by fun_prop))) |
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| lemma labelDomain_isCompact (d ε q : ℝ) : IsCompact (labelDomain d ε q) := by |
| have hbox : IsCompact ((fun p : ℝ × ℝ => vec2 p.1 p.2) '' |
| (Icc (-(3 / 8 : ℝ)) (3 / 8) ×ˢ Icc (-(1 / 4 : ℝ)) (1 / 2))) := |
| (isCompact_Icc.prod isCompact_Icc).image (by unfold vec2; fun_prop) |
| apply hbox.of_isClosed_subset (labelDomain_isClosed d ε q) |
| intro z hz |
| obtain ⟨hl, hu, hy⟩ := labelDomain_bounds hz |
| refine ⟨(z 0, z 1), ⟨abs_le.mp hy, hl, hu⟩, ?_⟩ |
| ext i |
| fin_cases i <;> simp [vec2] |
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| lemma labelDomain_measurableSet (d ε q : ℝ) : MeasurableSet (labelDomain d ε q) := |
| (labelDomain_isClosed d ε q).measurableSet |
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| lemma labelDomain_mem_iff (d ε q : ℝ) (z : Plane) : |
| z ∈ labelDomain d ε q ↔ |
| z 1 ∈ Icc (max (-(1 / 4 : ℝ)) |
| (-ε ^ 2 * ceilingPolynomial d (z 0) - q)) (1 / 2 - 2 * |z 0|) := by |
| simp only [labelDomain, wedge, radius, mem_ofPred_eq, mem_Icc, max_le_iff] |
| constructor |
| · rintro ⟨hw, hl, hp⟩ |
| exact ⟨⟨hl, hp⟩, by linarith⟩ |
| · rintro ⟨⟨hl, hp⟩, hw⟩ |
| exact ⟨by linarith, hl, hp⟩ |
|
|
| lemma labelDomain_vertical_section (d ε q y : ℝ) : |
| {t : ℝ | vec2 y t ∈ labelDomain d ε q} = |
| Icc (max (-(1 / 4 : ℝ)) (-ε ^ 2 * ceilingPolynomial d y - q)) |
| (1 / 2 - 2 * |y|) := by |
| ext t |
| simpa [vec2] using labelDomain_mem_iff d ε q (vec2 y t) |
|
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| lemma labelDomain_volume_ne_top (d ε q : ℝ) : |
| volume (labelDomain d ε q) ≠ ⊤ := |
| (labelDomain_isCompact d ε q).measure_lt_top.ne |
|
|
| namespace DomainGeometry |
|
|
| def pairEquiv : (ℝ × ℝ) ≃ᵐ Plane := |
| MeasurableEquiv.finTwoArrow.symm.trans (MeasurableEquiv.toLp 2 (Fin 2 → ℝ)) |
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| lemma pairEquiv_measurePreserving : MeasurePreserving pairEquiv volume volume := |
| (PiLp.volume_preserving_toLp (Fin 2)).comp (volume_preserving_finTwoArrow ℝ).symm |
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| lemma volume_graph_eq_zero {f : ℝ → ℝ} (hf : Continuous f) : |
| volume {z : Plane | z 1 = f (z 0)} = 0 := by |
| have hs : MeasurableSet {z : Plane | z 1 = f (z 0)} := |
| (isClosed_eq (by fun_prop) (hf.comp (by fun_prop))).measurableSet |
| rw [← pairEquiv_measurePreserving.measure_preimage hs.nullMeasurableSet, Measure.volume_eq_prod] |
| apply Measure.measure_prod_null_of_ae_null (hs.preimage pairEquiv.measurable) |
| filter_upwards with y |
| have heq : Prod.mk y ⁻¹' (pairEquiv ⁻¹' {z : Plane | z 1 = f (z 0)}) = |
| {f y} := by |
| ext t |
| simp [pairEquiv, MeasurableEquiv.finTwoArrow] |
| rw [heq] |
| exact measure_singleton (f y) |
|
|
| end DomainGeometry |
|
|
| lemma frontier_labelDomain_subset (d ε q : ℝ) : |
| frontier (labelDomain d ε q) ⊆ |
| {z : Plane | z 1 = 1 / 2 - 2 * |z 0|} ∪ |
| ({z : Plane | z 1 = -(1 / 4 : ℝ)} ∪ |
| {z : Plane | z 1 = -ε ^ 2 * ceilingPolynomial d (z 0) - q}) := by |
| intro z hz |
| change z ∈ frontier (wedge ∩ |
| ({w : Plane | -(1 / 4 : ℝ) ≤ w 1} ∩ |
| {w : Plane | -ε ^ 2 * ceilingPolynomial d (w 0) - q ≤ w 1})) at hz |
| rcases frontier_inter_subset _ _ hz with hw | hr |
| · left |
| have he := frontier_le_subset_eq |
| (f := fun w : Plane => 2 * |w 0|) |
| (g := fun w : Plane => 1 / 2 - w 1) |
| (by fun_prop) (by fun_prop) hw.1 |
| change 2 * |z 0| = 1 / 2 - z 1 at he |
| change z 1 = 1 / 2 - 2 * |z 0| |
| linarith |
| · right |
| rcases frontier_inter_subset _ _ hr.2 with hf | hp |
| · left |
| exact (frontier_le_subset_eq |
| (f := fun _ : Plane => -(1 / 4 : ℝ)) (g := fun w => w 1) |
| (by fun_prop) (by fun_prop) hf.1).symm |
| · right |
| exact (frontier_le_subset_eq |
| (f := fun w : Plane => -ε ^ 2 * ceilingPolynomial d (w 0) - q) |
| (g := fun w => w 1) |
| (by unfold ceilingPolynomial ceilingCoefficient; fun_prop) |
| (by fun_prop) hp.2).symm |
|
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| lemma volume_frontier_labelDomain (d ε q : ℝ) : |
| volume (frontier (labelDomain d ε q)) = 0 := by |
| apply measure_mono_null (frontier_labelDomain_subset d ε q) |
| apply measure_union_null |
| · exact DomainGeometry.volume_graph_eq_zero (f := fun y => 1 / 2 - 2 * |y|) (by fun_prop) |
| · apply measure_union_null |
| · exact DomainGeometry.volume_graph_eq_zero (f := fun _ => -(1 / 4 : ℝ)) continuous_const |
| · exact DomainGeometry.volume_graph_eq_zero |
| (f := fun y => -ε ^ 2 * ceilingPolynomial d y - q) |
| (by unfold ceilingPolynomial ceilingCoefficient; fun_prop) |
|
|
| lemma labelDomain_volume_le (d ε q : ℝ) : |
| volume (labelDomain d ε q) ≤ ENNReal.ofReal (9 / 16 : ℝ) := by |
| rw [← DomainGeometry.pairEquiv_measurePreserving.measure_preimage |
| (labelDomain_measurableSet d ε q).nullMeasurableSet, Measure.volume_eq_prod] |
| calc |
| (volume.prod volume) (DomainGeometry.pairEquiv ⁻¹' labelDomain d ε q) ≤ |
| (volume.prod volume) |
| (Icc (-(3 / 8 : ℝ)) (3 / 8) ×ˢ Icc (-(1 / 4 : ℝ)) (1 / 2)) := by |
| apply measure_mono |
| intro p hp |
| have hz := labelDomain_bounds hp |
| change -(1 / 4 : ℝ) ≤ p.2 ∧ p.2 ≤ 1 / 2 ∧ |p.1| ≤ 3 / 8 at hz |
| exact ⟨abs_le.mp hz.2.2, hz.1, hz.2.1⟩ |
| _ = ENNReal.ofReal (9 / 16 : ℝ) := by |
| rw [Measure.prod_prod] |
| norm_num [Real.volume_Icc] |
| rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 3 / 4)] |
| norm_num |
| |
| lemma labelDomain_realVolume_le (d ε q : ℝ) : |
| volume.real (labelDomain d ε q) ≤ 9 / 16 := by |
| have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top (labelDomain_volume_le d ε q) |
| simpa only [measureReal_def, ENNReal.toReal_ofReal (by norm_num : (0 : ℝ) ≤ 9 / 16)] using h |
| |
| end SlopeRational |
| |
| end |
| |
| end OAI |
| |
