openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /BlockPhaseConstruction.lean
| import OAI.Analysis.IrrationalRotation.Model.PhaseBlocks | |
| import OAI.Analysis.IrrationalRotation.Sampling.LargePhaseAssembly | |
| import OAI.Analysis.IrrationalRotation.Analysis.BlockAngleConstruction | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped BigOperators Topology | |
| namespace Invariant233.ProductPhases | |
| @[simp] theorem card_consecutiveBlock (m : ℕ → ℕ) (j : ℕ) : | |
| (consecutiveBlock m j).card = m j := by | |
| simp [consecutiveBlock] | |
| theorem mem_consecutiveBlock_iff (m : ℕ → ℕ) (j i : ℕ) : | |
| i ∈ consecutiveBlock m j ↔ | |
| (∑ r ∈ Finset.range j, m r) ≤ i ∧ | |
| i < (∑ r ∈ Finset.range j, m r) + m j := by | |
| simp [consecutiveBlock] | |
| theorem enumeration_block_of_mem (m : ℕ → ℕ) | |
| (e : ℕ ≃ Σ j, Fin (m j)) | |
| (he : ∀ a : Σ j, Fin (m j), | |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) | |
| {j i : ℕ} (hi : i ∈ consecutiveBlock m j) : (e i).1 = j := by | |
| obtain ⟨hlo, hhi⟩ := (mem_consecutiveBlock_iff m j i).mp hi | |
| let a : Fin (m j) := ⟨i - (∑ r ∈ Finset.range j, m r), by omega⟩ | |
| have ha : e.symm ⟨j, a⟩ = i := by | |
| rw [he] | |
| dsimp [a] | |
| omega | |
| have heq : e i = ⟨j, a⟩ := by | |
| rw [← ha, e.apply_symm_apply] | |
| exact congrArg Sigma.fst heq | |
| theorem mem_consecutiveBlock_enumeration (m : ℕ → ℕ) | |
| (e : ℕ ≃ Σ j, Fin (m j)) | |
| (he : ∀ a : Σ j, Fin (m j), | |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) | |
| (i : ℕ) : i ∈ consecutiveBlock m (e i).1 := by | |
| have hi : i = (∑ r ∈ Finset.range (e i).1, m r) + (e i).2.val := by | |
| simpa only [e.symm_apply_apply] using he (e i) | |
| apply (mem_consecutiveBlock_iff m (e i).1 i).mpr | |
| have hb := (e i).2.isLt | |
| omega | |
| theorem consecutive_uniformProduct_ae_phaseRanges | |
| (m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j) | |
| (K G : ℕ → ℝ) (hK : ∀ j, 0 < K j) (hG : ∀ j, 0 < G j) | |
| (e : ℕ ≃ Σ j, Fin (m j)) | |
| (he : ∀ a : Σ j, Fin (m j), | |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) : | |
| ∀ᵐ θ ∂Measure.infinitePi (fun i => ProbabilityTheory.cond volume | |
| (Set.Icc (1 / ((m (e i).1 : ℝ) * K (e i).1)) | |
| (2 / ((m (e i).1 : ℝ) * K (e i).1)))), | |
| (∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧ | |
| (∀ᶠ j in atTop, | |
| largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) := by | |
| have hm : ∀ j, 0 < m j := fun j => | |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) | |
| have hN : ∀ i : ℕ, 0 < (m (e i).1 : ℝ) * K (e i).1 := fun i => | |
| mul_pos (Nat.cast_pos.mpr (hm _)) (hK _) | |
| have h := uniformProduct_ae_phaseRanges | |
| (fun i => (m (e i).1 : ℝ) * K (e i).1) hN | |
| (consecutiveBlock m) K G hK hG | |
| (by simpa only [card_consecutiveBlock] using hgrowth) | |
| (by | |
| intro j i hi | |
| rw [enumeration_block_of_mem m e he hi, card_consecutiveBlock]) | |
| simpa only [card_consecutiveBlock] using h | |
| theorem exists_consecutive_phase_angles_and | |
| (m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j) | |
| (K G : ℕ → ℝ) (hK : ∀ j, 0 < K j) (hG : ∀ j, 0 < G j) | |
| (hsum : Summable (fun j => 1 / K j)) | |
| (e : ℕ ≃ Σ j, Fin (m j)) | |
| (he : ∀ a : Σ j, Fin (m j), | |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) | |
| {P : (ℕ → ℝ) → Prop} | |
| (hP : ∀ᵐ θ ∂Measure.infinitePi (fun i => ProbabilityTheory.cond volume | |
| (Set.Icc (1 / ((m (e i).1 : ℝ) * K (e i).1)) | |
| (2 / ((m (e i).1 : ℝ) * K (e i).1)))), P θ) : | |
| ∃ θ : ℕ → ℝ, | |
| (∀ j, ∀ i ∈ consecutiveBlock m j, | |
| θ i ∈ Set.Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧ | |
| (∀ i, 0 < θ i) ∧ Summable θ ∧ | |
| (∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧ | |
| LinearIndependent ℚ (fun o : Option ℕ => o.elim 1 θ) ∧ | |
| (∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧ | |
| (∀ᶠ j in atTop, | |
| largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) ∧ P θ := by | |
| have hm : ∀ j, 0 < m j := fun j => | |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) | |
| let N : ℕ → ℝ := fun i => (m (e i).1 : ℝ) * K (e i).1 | |
| have hN : ∀ i, 0 < N i := fun i => | |
| mul_pos (Nat.cast_pos.mpr (hm _)) (hK _) | |
| have hblock := InvariantProjection.IndependentAngles.enumerated_block_reciprocal | |
| m hm K hK hsum e | |
| have hphase := consecutive_uniformProduct_ae_phaseRanges m hgrowth K G hK hG e he | |
| obtain ⟨θ, hθ, hsθ, htθ, hiθ, hph, hp⟩ := | |
| InvariantProjection.IndependentAngles.exists_prescribed_summable_independent_and | |
| N hN hblock.1 (hphase.and hP) | |
| refine ⟨θ, ?_, ?_, hsθ, ?_, hiθ, hph.1, hph.2, hp⟩ | |
| · intro j i hi | |
| simpa only [N, enumeration_block_of_mem m e he hi] using hθ i | |
| · intro i | |
| exact (one_div_pos.mpr (hN i)).trans_le (hθ i).1 | |
| · simpa only [N, hblock.2] using htθ | |
| theorem exists_consecutive_phase_angles | |
| (m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j) | |
| (K G : ℕ → ℝ) (hK : ∀ j, 0 < K j) (hG : ∀ j, 0 < G j) | |
| (hsum : Summable (fun j => 1 / K j)) : | |
| ∃ θ : ℕ → ℝ, | |
| (∀ j, ∀ i ∈ consecutiveBlock m j, | |
| θ i ∈ Set.Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧ | |
| (∀ i, 0 < θ i) ∧ Summable θ ∧ | |
| (∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧ | |
| LinearIndependent ℚ (fun o : Option ℕ => o.elim 1 θ) ∧ | |
| (∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧ | |
| (∀ᶠ j in atTop, | |
| largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) := by | |
| have hm : ∀ j, 0 < m j := fun j => | |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) | |
| obtain ⟨e, he⟩ := InvariantProjection.IndependentAngles.exists_consecutiveBlocks_equiv m hm | |
| obtain ⟨θ, hθ, hpos, hs, hsumθ, hind, hsmall, hlarge, _⟩ := | |
| exists_consecutive_phase_angles_and m hgrowth K G hK hG hsum e he | |
| (P := fun _ => True) (Filter.Eventually.of_forall fun _ => True.intro) | |
| exact ⟨θ, hθ, hpos, hs, hsumθ, hind, hsmall, hlarge⟩ | |
| end Invariant233.ProductPhases | |
| end | |
| end OAI | |