openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingRealization.lean
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingTests | |
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterPhaseRealization | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology BigOperators | |
| namespace Invariant233.CharacterSampling | |
| theorem exists_realized_sampling_block {θ : ℝ} (hθ : Irrational θ) | |
| {m : ℕ} (hm : 200 ≤ m) (target : Fin m → ℝ) | |
| (htarget0 : ∀ i, 0 < target i) (htarget1 : ∀ i, target i < 1) | |
| {q₀ : ℕ} (hq₀ : 0 < q₀) (R : ℝ) | |
| (U : Set (Fin m → ℝ)) (hU : U ∈ 𝓝 target) | |
| (r : ℕ) (powers : Fin r → Finset ℕ) (δ : Fin r → Bool → ℝ) | |
| (hcut : ∀ k, ∀ n ∈ powers k, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0) | |
| (hbound : ∀ k endpoint, | |
| (∫ y : UnitAddTorus (Fin m), squaredError m (powers k) endpoint (target,y)) < δ k endpoint) : | |
| ∃ q : ℕ → ℕ, | |
| q 0 = q₀ ∧ (∀ i, 0 < q i) ∧ StrictMono q ∧ | |
| (∀ i, q i ∣ q (i + 1)) ∧ | |
| (∀ i < m, R ≤ (q (i + 1) : ℝ) / (q i : ℝ)) ∧ | |
| (fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)) ∈ U ∧ | |
| ∀ k endpoint, | |
| (∫ x : UnitAddCircle, squaredError m (powers k) endpoint | |
| ((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)), | |
| fun i : Fin m => (q (i.val + 1) : ℤ) • x)) < δ k endpoint := by | |
| classical | |
| let target' : ℕ → ℝ := fun i => if h : i < m then target ⟨i,h⟩ else 1/2 | |
| have ht0 : ∀ i, 0 < target' i := by | |
| intro i | |
| dsimp only [target'] | |
| split_ifs with hi | |
| · exact htarget0 ⟨i,hi⟩ | |
| · norm_num | |
| have ht1 : ∀ i, target' i < 1 := by | |
| intro i | |
| dsimp only [target'] | |
| split_ifs with hi | |
| · exact htarget1 ⟨i,hi⟩ | |
| · norm_num | |
| obtain ⟨Q, hzero, hpos, hstrict, hdiv, hgap, hphase'⟩ := | |
| PrescribedPhaseRealization.exists_phase_approximation_family hθ target' ht0 ht1 hq₀ | |
| let phase : ℕ → Fin m → ℝ := fun j i => Int.fract ((Q j (i.val + 1) : ℝ) * θ) | |
| have hphase : Tendsto phase atTop (𝓝 target) := by | |
| apply tendsto_pi_nhds.mpr | |
| intro i | |
| simpa only [phase, target', dite_eq_left i.isLt] using hphase' i.val | |
| have hratios : ∀ᶠ j in atTop, ∀ i < m, | |
| R ≤ (Q j (i + 1) : ℝ) / (Q j i : ℝ) := by | |
| have hfinite : ∀ᶠ j in atTop, ∀ i ∈ Finset.range m, | |
| R ≤ (Q j (i + 1) : ℝ) / (Q j i : ℝ) := by | |
| rw [Finset.eventually_all] | |
| intro i hi | |
| exact (hgap i).eventually_ge_atTop R | |
| simpa only [Finset.mem_range] using hfinite | |
| have htests : ∀ᶠ j in atTop, ∀ k endpoint, | |
| (∫ x : UnitAddCircle, squaredError m (powers k) endpoint | |
| (phase j, fun i : Fin m => (Q j (i.val + 1) : ℤ) • x)) < δ k endpoint := by | |
| apply Filter.eventually_all.mpr | |
| intro k | |
| apply Filter.eventually_all.mpr | |
| intro endpoint | |
| have hlim := tendsto_integral_squaredError hm (powers k) endpoint | |
| (fun j i => Q j (i + 1)) (fun j i => hpos j (i + 1)) | |
| (fun j i l hil => (hstrict j).monotone (Nat.add_le_add_right hil 1)) | |
| (fun i _ => hgap (i + 1)) hphase (hcut k) | |
| exact hlim.eventually_lt_const (hbound k endpoint) | |
| obtain ⟨j, ⟨⟨hjratio, hjU⟩, hjtests⟩⟩ := | |
| ((hratios.and (hphase.eventually hU)).and htests).exists | |
| exact ⟨Q j, hzero j, hpos j, hstrict j, hdiv j, hjratio, hjU, hjtests⟩ | |
| end Invariant233.CharacterSampling | |
| end | |
| end OAI | |