| import OAI.Geometry.CylinderCovering.Caps.TrapezoidGeometryDefinitions | |
| import OAI.Geometry.CylinderCovering.Caps.FiniteShape | |
| import OAI.Geometry.CylinderCovering.Caps.Geometry | |
| import OAI.Geometry.CylinderCovering.Caps.FiniteTheorem | |
| namespace OAI | |
| open Set Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace CylinderCovering.HalfheightFinite | |
| theorem trapezoid_pair {δ : ℝ} (hδ : 0 < δ) {R : ℝ → ℝ} | |
| (hR : ContinuousOn R (Icc (-1 : ℝ) 1)) | |
| (hR0 : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ R r) {N : ℕ} (i : Fin N) : | |
| NondegenerateTrapezoid (firstBase δ R N i) ∧ | |
| NondegenerateTrapezoid (secondBase δ R N i) := by | |
| have hM := RadialRiemann.capMax_nonneg (by norm_num : (-1 : ℝ) < 1) hR hR0 i | |
| obtain ⟨hU, hL, hiU, hiL⟩ := Shape.projected_shape i hδ hM | |
| obtain ⟨hnU, hnL⟩ := Shape.projected_nondegenerate i hδ hM | |
| obtain ⟨ha, hb, vU, vL, hvU, hvL, hU0, hU1, hL0, hL1⟩ := | |
| Shape.projected_parallel_sides i hδ hM | |
| constructor | |
| · exact ⟨Shape.firstVertices δ R N i, hU, hiU, hnU, | |
| 2 * halfwidth N * collar δ N, | |
| 2 * halfwidth N * (maxCap R N i + 3 * collar δ N), | |
| ha, hb, vU, hvU, hU0, hU1⟩ | |
| · exact ⟨Shape.secondVertices δ R N i, hL, hiL, hnL, | |
| 2 * halfwidth N * collar δ N, | |
| 2 * halfwidth N * (maxCap R N i + 3 * collar δ N), | |
| ha, hb, vL, hvL, hL0, hL1⟩ | |
| theorem source_trapezoids (δ : ℝ) (hδ : 0 < δ) (N : ℕ) (i : Fin N) : | |
| NondegenerateTrapezoid (firstBase δ (cubicCap δ) N i) ∧ | |
| NondegenerateTrapezoid (secondBase δ (cubicCap δ) N i) ∧ | |
| NondegenerateTrapezoid (firstBase δ (fractionalCap δ) N i) ∧ | |
| NondegenerateTrapezoid (secondBase δ (fractionalCap δ) N i) := by | |
| obtain ⟨hcU, hcL⟩ := trapezoid_pair hδ (cubicCap_continuous δ).continuousOn | |
| (fun r _ => (cubicCap_positive hδ.le r).le) i | |
| obtain ⟨hfU, hfL⟩ := trapezoid_pair hδ (fractionalCap_continuous δ).continuousOn | |
| (fun r _ => (fractionalCap_positive hδ.le r).le) i | |
| exact ⟨hcU, hcL, hfU, hfL⟩ | |
| theorem eventual_half_minimum : ∀ᶠ δ : ℝ in 𝓝[>] 0, | |
| (∀ᶠ N in atTop, 0 < N ∧ | |
| totalArea δ (cubicCap δ) N < Similarity.minimumArea tetrahedron / 2) ∧ | |
| (∀ᶠ N in atTop, 0 < N ∧ | |
| totalArea δ (fractionalCap δ) N < Similarity.minimumArea tetrahedron / 2) := by | |
| have hthreshold : Similarity.minimumArea tetrahedron / 2 = height / 4 := by | |
| rw [minimum_area] | |
| ring | |
| simpa only [hthreshold] using full_halfheight.2.2.2.2 | |
| theorem full_halfheight_geometry : GeometryStatement := | |
| ⟨unit_edge_geometry, minimum_area, source_trapezoids, eventual_half_minimum⟩ | |
| end CylinderCovering.HalfheightFinite | |
| end | |
| end OAI | |