openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Singular /EdgeOneResultsDefinitions.lean
| import OAI.Geometry.CylinderCovering.Singular.EdgeOneDifferentialDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped Topology | |
| namespace SingularEdgeOne | |
| open RuledApproximation | |
| structure DictionaryStatement : Prop where | |
| field_components : ∀ (p w l : ℝ), 0 < l → | |
| edgeU p w l = fieldP p (2 * w) l / 4 ∧ | |
| edgeV p w l = fieldQ p (2 * w) l / 2 | |
| hamiltonian : ∀ (p w l : ℝ), 0 < l → | |
| edgeField p (vec2 w l) = | |
| vec2 (-deriv (fun r : ℝ => edgeHamiltonian p w r) l) | |
| (deriv (fun v : ℝ => edgeHamiltonian p v l) w) | |
| ceiling_formula : ∀ (p w c : ℝ), | |
| edgeCeiling p w = w ^ 2 * edgeU p w (1 / 2) ∧ | |
| c ^ 2 * ceiling p (2 * w) = (4 * c) ^ 2 * edgeCeiling p w | |
| physical_energy : ∀ (p w c l : ℝ), 0 < l → | |
| (4 * c) ^ 2 * | |
| (edgeU p w l ^ 2 + (1 / Real.sqrt 2 : ℝ) ^ 2 * edgeV p w l ^ 2) = | |
| c ^ 2 * (fieldP p (2 * w) l ^ 2 + 2 * fieldQ p (2 * w) l ^ 2) | |
| tetrahedron_image : physicalSimilarity '' tetrahedron = edgeTetrahedron | |
| lower_image : ∀ (p c : ℝ) (D : Set Plane), (∀ b ∈ D, 0 < b 1) → | |
| physicalSimilarity '' physicalRuled (Real.sqrt 2) D | |
| (fun b => c • singularField p b) 1 = edgeLower p (4 * c) (labelMap '' D) | |
| upper_image : ∀ (p c : ℝ) (D : Set Plane), (∀ b ∈ D, 0 < b 1) → | |
| physicalSimilarity '' upperSegments p c D = edgeUpper p (4 * c) (labelMap '' D) | |
| physical_cost : ∀ (p c : ℝ) (D : Set Plane), MeasurableSet D → (∀ z ∈ D, 0 < z 1) → | |
| (1 / Real.sqrt 2) * | |
| (∫ z in labelMap '' D, physicalWeight (1 / Real.sqrt 2) ((4 * c) • edgeField p z)) = | |
| (Real.sqrt 2 * (∫ z in D, physicalWeight (Real.sqrt 2) (c • singularField p z))) / 4 | |
| derivative_conjugacy : ∀ (p c : ℝ) (z : Plane), 0 < z 1 → | |
| ((4 * c) / 2) • fderiv ℝ (edgeField p) (labelMap z) = | |
| c • labelContinuous.toContinuousLinearMap.comp | |
| ((fderiv ℝ (singularField p) z).comp labelContinuous.symm.toContinuousLinearMap) | |
| spectral_conjugacy : ∀ (p c mu : ℝ) (z : Plane), 0 < z 1 → | |
| (Module.End.HasEigenvalue | |
| (((4 * c) / 2) • fderiv ℝ (edgeField p) (labelMap z)).toLinearMap mu ↔ | |
| Module.End.HasEigenvalue ((c • fderiv ℝ (singularField p) z).toLinearMap) mu) | |
| exact_parameters : ∀ (k gamma : ℝ), 0 ≤ k → | |
| (k / 4) ^ gamma = (4 : ℝ) ^ (-gamma) * k ^ gamma ∧ | |
| (k / 4) ^ (5 / 2 : ℝ) = k ^ (5 / 2 : ℝ) / 32 | |
| boundary_integral : ∀ p : ℝ, | |
| (∫ w in (-1 / 4 : ℝ)..(1 / 4 : ℝ), edgeCeiling p w) = edgeA p | |
| energy_integral : ∀ p : ℝ, 1 < p → | |
| (∫ z in edgeDomain0, edgeEnergy p (z 0) (z 1)) = edgeG p | |
| coefficient_formula : ∀ p : ℝ, p ≠ 0 → | |
| edgeA p - edgeG p = (2 * p ^ 2 + 10 * p - 13) / (15360 * p) ∧ | |
| edgeA p - edgeG p = coefficient p / 32 | |
| coefficient_negative : ∀ p : ℝ, 1 < p → p < (Real.sqrt 51 - 5) / 2 → | |
| edgeA p - edgeG p < 0 | |
| coefficient_limits : | |
| Tendsto edgeA (𝓝 1) (𝓝 (16 / 15360)) ∧ | |
| Tendsto edgeG (𝓝 1) (𝓝 (17 / 15360)) | |
| end SingularEdgeOne | |
| end | |
| end OAI | |