openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Analysis /BoundedPathPacking.lean
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.BoundedPathGrowth | |
| import OAI.GroupTheory.PolycyclicRecognition.Analysis.MeasurePacking | |
| namespace OAI | |
| /-! | |
| # From polynomial-volume thickenings to bounded-height path packing | |
| The endpoint set need not be measurable. We pack finite separated subsets using | |
| Haar mass and then take a maximal one. Only the enclosing coordinate regions | |
| are measured; this is the precise bridge needed from the Lie-coordinate bounds. | |
| -/ | |
| namespace PolycyclicRecognition | |
| open MeasureTheory MeasureTheory.Measure Metric | |
| open scoped ENNReal | |
| variable {G W : Type*} [Group G] [MetricSpace G] [MeasurableSpace G] | |
| [BorelSpace G] [MeasurableMul G] [IsIsometricSMul G G] [NormedAddCommGroup W] | |
| /-- Endpoints of at most `n` bounded-height steps from a fixed starting point. -/ | |
| def boundedChainEndpoints (π : G → W) (L M : ℝ) (n : ℕ) (x : G) : Set G := | |
| {y | ∃ m ≤ n, BoundedHeightChain π L M m x y} | |
| /-- A polynomial-volume region containing the half-unit thickening of the | |
| bounded-chain endpoints gives the required polynomial covering bound. -/ | |
| theorem boundedChain_cover_of_volume | |
| (μ : Measure G) [IsMulLeftInvariant μ] (π : G → W) | |
| (x : G) (L M : ℝ) (C b : ℕ) (region : ℕ → Set G) | |
| (hq0 : μ (ball (1 : G) (1 / 2)) ≠ 0) | |
| (hqtop : μ (ball (1 : G) (1 / 2)) ≠ ⊤) | |
| (hthick : ∀ n y, y ∈ boundedChainEndpoints π L M n x → | |
| ball y (1 / 2) ⊆ region n) | |
| (hvol : ∀ n, μ (region n) ≤ | |
| ((C * (1 + n) ^ b : ℕ) : ℝ≥0∞) * μ (ball (1 : G) (1 / 2))) : | |
| ∃ centers : ℕ → Finset G, | |
| (∀ n, (centers n).card ≤ C * (1 + n) ^ b) ∧ | |
| ∀ n m, m ≤ n → ∀ y, BoundedHeightChain π L M m x y → | |
| ∃ c ∈ centers n, dist y c ≤ 1 := by | |
| classical | |
| have hn (n : ℕ) := finite_cover_set_of_separated_bound | |
| (boundedChainEndpoints π L M n x) (r := 1) (by norm_num) (C * (1 + n) ^ b) | |
| (fun t ht hs => separated_card_le_polynomial μ t (1 / 2) (region n) C b n hq0 hqtop | |
| (by intro a ha d hd had; have := hs a ha d hd had; linarith) | |
| (fun y hy => hthick n y (ht hy)) (hvol n)) | |
| choose centers _ hcard hcover using hn | |
| exact ⟨centers, hcard, fun n m hm y hy => hcover n y ⟨m, hm, hy⟩⟩ | |
| end PolycyclicRecognition | |
| end OAI | |