| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Topology ENNReal |
| open Classical Filter |
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| namespace TriangularUniversal |
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| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
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| def LocallyFinite (C : Set Plane) : Prop := |
| ∀ R : ℝ, (C ∩ Metric.closedBall (0 : Plane) R).Finite |
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| def diskPoints (C : Set Plane) (R : ℝ) : Finset Plane := |
| if h : (C ∩ Metric.closedBall (0 : Plane) R).Finite then h.toFinset else ∅ |
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| def diskCount (C : Set Plane) (R : ℝ) : ℕ := (diskPoints C R).card |
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| def DensityOne (C : Set Plane) : Prop := |
| Tendsto (fun R : ℝ => (diskCount C R : ℝ) / (Real.pi * R ^ 2)) |
| atTop (𝓝 1) |
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| def AdmissiblePotential (g : ℝ → ℝ) : Prop := |
| ContDiffOn ℝ (⊤ : ℕ∞) g (Set.Ioi 0) ∧ |
| (∀ t : ℝ, 0 < t → 0 ≤ g t) ∧ |
| ∀ (r : ℕ) (t : ℝ), 0 < t → 0 ≤ (-1 : ℝ) ^ r * iteratedDeriv r g t |
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| def diskEnergy (g : ℝ → ℝ) (C : Set Plane) (R : ℝ) : ℝ≥0∞ := |
| (diskCount C R : ℝ≥0∞)⁻¹ * |
| ∑ x ∈ diskPoints C R, ∑ y ∈ (diskPoints C R).erase x, |
| ENNReal.ofReal (g (‖x - y‖ ^ 2)) |
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| def energy (g : ℝ → ℝ) (C : Set Plane) : ℝ≥0∞ := |
| Filter.liminf (diskEnergy g C) atTop |
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| end TriangularUniversal |
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| namespace AtomicTriangular |
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| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
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| def b : ℝ := Real.sqrt 3 / 2 |
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| def triangularPoint (jk : ℤ × ℤ) : Plane := |
| (Real.sqrt b)⁻¹ • |
| (EuclideanSpace.single 0 ((jk.1 : ℝ) + (jk.2 : ℝ) / 2) + |
| EuclideanSpace.single 1 ((jk.2 : ℝ) * b)) |
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| def A : Set Plane := Set.range triangularPoint |
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| abbrev LocallyFinite := TriangularUniversal.LocallyFinite |
| abbrev diskPoints := TriangularUniversal.diskPoints |
| abbrev diskCount := TriangularUniversal.diskCount |
| abbrev DensityOne := TriangularUniversal.DensityOne |
| abbrev AdmissiblePotential := TriangularUniversal.AdmissiblePotential |
| abbrev diskEnergy := TriangularUniversal.diskEnergy |
| abbrev energy := TriangularUniversal.energy |
|
|
| / |
| def latticeEnergy (g : ℝ → ℝ) : ℝ≥0∞ := |
| ∑' a : {x : Plane // x ∈ A ∧ x ≠ 0}, ENNReal.ofReal (g (‖a.val‖ ^ 2)) |
|
|
| / |
| theorem universal_energy_minimum (g : ℝ → ℝ) (C : Set Plane) |
| (hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) : |
| latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := by |
| sorry |
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| end AtomicTriangular |
| end |
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| end OAI |
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|
