openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Rotation /CircleRationalGraphCaps.lean
| import OAI.Analysis.IrrationalRotation.Rotation.ParameterCircleReindex | |
| import OAI.Analysis.IrrationalRotation.Analysis.InfiniteRatioCaps | |
| import OAI.Analysis.IrrationalRotation.Analysis.RationalGraphTrimming | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Function | |
| open scoped BigOperators | |
| namespace Invariant233.CircleParameterLaws | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem ae_flattened_floor_support : | |
| ∀ᵐ x ∂circleMeasure, ∀ k : ℤ, | |
| HasFiniteSupport (fun i : ℕ => Int.floor | |
| (CircleWeights.coordinate p.flattenedFrequency i x + | |
| (k : ℝ) * CircleWeights.phase θ p.flattenedFrequency i)) := by | |
| have hq : ∀ i, p.flattenedFrequency i ≠ 0 := by | |
| intro i | |
| change (p.q (i + 1) : ℤ) ≠ 0 | |
| exact_mod_cast (hp.q_pos (i + 1)).ne' | |
| have hs : Summable (CircleWeights.phase θ p.flattenedFrequency) := by | |
| change Summable (fun i : ℕ => Int.fract ((p.q (i + 1) : ℝ) * θ)) | |
| exact hp.summable_phases | |
| exact CircleWeightSupport.ae_all_int_finite_floor_support | |
| p.flattenedFrequency hq (CircleWeights.phase θ p.flattenedFrequency) | |
| (CircleWeights.phase_nonneg θ p.flattenedFrequency) hs | |
| theorem ae_all_rational_graph_floor_support : | |
| ∀ᵐ x ∂circleMeasure, | |
| (∀ k : ℤ, HasFiniteSupport (fun i : ℕ => Int.floor | |
| (CircleWeights.coordinate p.flattenedFrequency i x + | |
| (k : ℝ) * CircleWeights.phase θ p.flattenedFrequency i))) ∧ | |
| ∀ (A : ℕ) (r : ℤ) (k : ℤ), HasFiniteSupport (fun i : ℕ => Int.floor | |
| (CircleWeights.coordinate p.flattenedFrequency i | |
| (x + (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle)) + | |
| (k : ℝ) * CircleWeights.phase θ p.flattenedFrequency i)) := by | |
| refine hp.ae_flattened_floor_support.and ?_ | |
| apply ae_all_iff.mpr | |
| intro A | |
| apply ae_all_iff.mpr | |
| intro r | |
| have hmp : MeasurePreserving | |
| (fun x : Circle => x + (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle)) | |
| circleMeasure circleMeasure := by | |
| change MeasurePreserving _ volume volume | |
| exact measurePreserving_add_right volume _ | |
| let P : Circle → Prop := fun y => ∀ k : ℤ, HasFiniteSupport (fun i : ℕ => | |
| Int.floor (CircleWeights.coordinate p.flattenedFrequency i y + | |
| (k : ℝ) * CircleWeights.phase θ p.flattenedFrequency i)) | |
| have hP : ∀ᵐ y ∂circleMeasure, P y := hp.ae_flattened_floor_support | |
| exact hmp.quasiMeasurePreserving.ae (p := P) hP | |
| theorem ae_rational_graph_ratio_cap : | |
| ∀ᵐ x ∂circleMeasure, ∀ (A : ℕ), 1 ≤ A → ∀ (r k : ℤ), | |
| p.floorCocycle θ k | |
| (x + (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle)) / | |
| p.floorCocycle θ k x ≤ | |
| Real.exp (∑ i ∈ Finset.range (A - 1), p.flattenedCost i) := by | |
| filter_upwards [hp.ae_all_rational_graph_floor_support] with x hx | |
| intro A hA r k | |
| rw [hp.parameter_cocycle_eq_flattened, hp.parameter_cocycle_eq_flattened] | |
| apply CircleWeights.cocycle_ratio_cap_finite_exceptions θ | |
| p.flattenedFrequency p.flattenedCost k _ _ (Finset.range (A - 1)) | |
| (fun i => (hp.flattenedCost_pos i).le) _ (hx.1 k) (hx.2 A r k) | |
| intro i hi | |
| have hAi : A ≤ i + 1 := by | |
| simp only [Finset.mem_range, not_lt] at hi | |
| omega | |
| have he := hp.trimming_coordinate_rational_translation hAi r x | |
| change p.coordinate (i + 1) x ≤ p.coordinate (i + 1) | |
| (x + (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle)) | |
| exact le_of_eq he.symm | |
| end Invariant233.CircleParameterLaws | |
| end | |
| end OAI | |