| import OAI.Geometry.CylinderCovering.Caps.FiniteGeometry | |
| import OAI.Geometry.CylinderCovering.Caps.FiniteProperties | |
| import OAI.Geometry.CylinderCovering.Caps.FiniteLimit | |
| import OAI.Geometry.CylinderCovering.Caps.FiniteAsymptotics | |
| import OAI.Geometry.CylinderCovering.Caps.CapRegularity | |
| namespace OAI | |
| open Set Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace CylinderCovering.HalfheightFinite | |
| theorem family_conclusion_of_covered {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} | |
| (hR : ContinuousOn R (Icc (-1:ℝ) 1)) | |
| (hR0 : ∀ r ∈ Icc (-1:ℝ) 1, 0 ≤ R r) | |
| (hcover : HalfHeightCaps.Covered δ R) : FamilyConclusion δ R := by | |
| refine ⟨?_, totalArea_tendsto hδ hR hR0⟩ | |
| intro N hN | |
| refine ⟨?_, finite_coverage hδ hR hcover hN⟩ | |
| intro i | |
| exact ⟨firstBase_properties δ R N i, secondBase_properties δ R N i, | |
| firstBase_area hδ hR hR0 N i, secondBase_area hδ hR hR0 N i⟩ | |
| theorem cubic_family {δ : ℝ} (hδ : 0 < δ) : | |
| FamilyConclusion δ (cubicCap δ) := by | |
| apply family_conclusion_of_covered hδ.le (cubicCap_continuous δ).continuousOn | |
| (fun r _ => (cubicCap_positive hδ.le r).le) | |
| exact HalfHeightCaps.cubic_covers hδ | |
| theorem fractional_family {δ : ℝ} (hδ : 0 < δ) (hsmall : δ ≤ 1) : | |
| FamilyConclusion δ (fractionalCap δ) := by | |
| apply family_conclusion_of_covered hδ.le (fractionalCap_continuous δ).continuousOn | |
| (fun r _ => (fractionalCap_positive hδ.le r).le) | |
| exact HalfHeightCaps.fractional_covers hδ hsmall | |
| theorem eventually_cubic_family : | |
| ∀ᶠ δ : ℝ in 𝓝[>] 0, FamilyConclusion δ (cubicCap δ) := by | |
| filter_upwards [self_mem_nhdsWithin] with δ hδ | |
| exact cubic_family hδ | |
| theorem eventually_fractional_family : | |
| ∀ᶠ δ : ℝ in 𝓝[>] 0, FamilyConclusion δ (fractionalCap δ) := by | |
| have hs : ∀ᶠ δ : ℝ in 𝓝[>] 0, δ < 1 := | |
| Filter.Eventually.filter_mono nhdsWithin_le_nhds | |
| (gt_mem_nhds (by norm_num : (0:ℝ) < 1)) | |
| filter_upwards [self_mem_nhdsWithin, hs] with δ hδ hsmall | |
| exact fractional_family hδ hsmall.le | |
| theorem full_halfheight : Statement := by | |
| exact ⟨fun δ hδ _ => cubic_family hδ, eventually_fractional_family, | |
| cubic_cost_bigO, fractional_cost_bigO, | |
| eventual_finite_saving eventually_cubic_family eventually_fractional_family⟩ | |
| end CylinderCovering.HalfheightFinite | |
| end | |
| end OAI | |