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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 | 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
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