openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterPhaseRealization.lean
| import OAI.Analysis.IrrationalRotation.Sampling.PrescribedPhaseRealization | |
| import OAI.Analysis.IrrationalRotation.Sampling.CharacterLawConvergence | |
| import Mathlib.Analysis.SpecificLimits.Basic | |
| namespace OAI | |
| namespace Invariant233 | |
| namespace PrescribedPhaseRealization | |
| open Filter | |
| open scoped Topology | |
| theorem exists_phase_approximation_family {θ : ℝ} (hθ : Irrational θ) | |
| (target : ℕ → ℝ) (htarget₀ : ∀ i, 0 < target i) | |
| (htarget₁ : ∀ i, target i < 1) {q₀ : ℕ} (hq₀ : 0 < q₀) : | |
| ∃ q : ℕ → ℕ → ℕ, | |
| (∀ j, q j 0 = q₀) ∧ (∀ j i, 0 < q j i) ∧ | |
| (∀ j, StrictMono (q j)) ∧ (∀ j i, q j i ∣ q j (i + 1)) ∧ | |
| (∀ i, Tendsto (fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) ∧ | |
| ∀ i, Tendsto (fun j => Int.fract ((q j (i + 1) : ℝ) * θ)) atTop | |
| (𝓝 (target i)) := by | |
| classical | |
| have hex : ∀ j : ℕ, ∃ q : ℕ → ℕ, | |
| q 0 = q₀ ∧ (∀ i, 0 < q i) ∧ | |
| ∀ i, q i ∣ q (i + 1) ∧ q i < q (i + 1) ∧ | |
| (j + 1) * q i ≤ q (i + 1) ∧ | |
| |Int.fract ((q (i + 1) : ℝ) * θ) - target i| < 1 / ((j : ℝ) + 1) := by | |
| intro j | |
| exact exists_divisible_phase_approximation hθ target | |
| (fun _ => 1 / ((j : ℝ) + 1)) (fun _ => j + 1) | |
| htarget₀ htarget₁ (fun _ => by positivity) hq₀ | |
| choose q hzero hpos hstep using hex | |
| refine ⟨q, hzero, hpos, ?_, ?_, ?_, ?_⟩ | |
| · intro j | |
| exact strictMono_nat_of_lt_succ (fun i => (hstep j i).2.1) | |
| · intro j i | |
| exact (hstep j i).1 | |
| · intro i | |
| apply tendsto_atTop_mono (f := fun j : ℕ => (j : ℝ) + 1) | |
| · intro j | |
| apply (le_div_iff₀ (by exact_mod_cast hpos j i)).mpr | |
| exact_mod_cast (hstep j i).2.2.1 | |
| · exact tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop | |
| · intro i | |
| apply tendsto_iff_dist_tendsto_zero.mpr | |
| apply squeeze_zero (fun _ => dist_nonneg) | |
| (fun j => ?_) tendsto_one_div_add_atTop_nhds_zero_nat | |
| simpa only [Real.dist_eq] using (hstep j i).2.2.2.le | |
| theorem exists_joint_phase_character_limit {θ : ℝ} (hθ : Irrational θ) | |
| (target : ℕ → ℝ) (htarget₀ : ∀ i, 0 < target i) | |
| (htarget₁ : ∀ i, target i < 1) {q₀ : ℕ} (hq₀ : 0 < q₀) : | |
| ∃ q : ℕ → ℕ → ℕ, | |
| (∀ j, q j 0 = q₀) ∧ (∀ j i, 0 < q j i) ∧ | |
| (∀ j, StrictMono (q j)) ∧ (∀ j i, q j i ∣ q j (i + 1)) ∧ | |
| (∀ i, Tendsto (fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) ∧ | |
| (∀ i, Tendsto (fun j => Int.fract ((q j (i + 1) : ℝ) * θ)) atTop | |
| (𝓝 (target i))) ∧ | |
| ∀ m : ℕ, Tendsto (fun j => CharacterLaws.characterPhaseLaw | |
| (fun i : Fin m => q j (i.val + 1)) | |
| (fun i : Fin m => Int.fract ((q j (i.val + 1) : ℝ) * θ))) atTop | |
| (𝓝 (CharacterLaws.haarPhaseLaw m (fun i : Fin m => target i.val))) := by | |
| obtain ⟨q, hzero, hpos, hstrict, hdiv, hgap, hphase⟩ := | |
| exists_phase_approximation_family hθ target htarget₀ htarget₁ hq₀ | |
| refine ⟨q, hzero, hpos, hstrict, hdiv, hgap, hphase, ?_⟩ | |
| intro m | |
| apply CharacterLaws.tendsto_characterPhaseLaw (fun j i => q j (i + 1)) m | |
| · intro j i | |
| exact hpos j (i + 1) | |
| · intro j i k hik | |
| exact (hstrict j).monotone (Nat.add_le_add_right hik 1) | |
| · intro i _hi | |
| exact hgap (i + 1) | |
| · exact tendsto_pi_nhds.mpr (fun i : Fin m => hphase i.val) | |
| end PrescribedPhaseRealization | |
| end Invariant233 | |
| end OAI | |