openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleRationalSamplingIndex.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRationalGraphCaps | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| private theorem exists_nat_rational_representative (d : ℕ) (hd : 0 < d) (z : ℤ) : | |
| ∃ r : ℕ, r < d ∧ | |
| (((z : ℝ) / (d : ℝ) : ℝ) : Circle) = | |
| (((r : ℝ) / (d : ℝ) : ℝ) : Circle) := by | |
| have hdZ : (0 : ℤ) < d := by exact_mod_cast hd | |
| have hn : 0 ≤ z % (d : ℤ) := Int.emod_nonneg z hdZ.ne' | |
| refine ⟨(z % (d : ℤ)).toNat, (Int.toNat_lt hn).mpr (Int.emod_lt_of_pos z hdZ), ?_⟩ | |
| apply sub_eq_zero.mp | |
| rw [← AddCircle.coe_sub] | |
| apply (AddCircle.coe_eq_zero_iff (1 : ℝ)).mpr | |
| refine ⟨z / (d : ℤ), ?_⟩ | |
| have hcast : (((z % (d : ℤ)).toNat : ℕ) : ℝ) = (z % (d : ℤ) : ℤ) := by | |
| exact_mod_cast Int.toNat_of_nonneg hn | |
| have hdiv := congrArg (fun n : ℤ => (n : ℝ)) | |
| (Int.emod_add_mul_ediv z (d : ℤ)) | |
| push_cast at hdiv | |
| rw [hcast, zsmul_eq_mul, mul_one] | |
| have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast hd.ne' | |
| field_simp | |
| nlinarith | |
| namespace CircleParameterLaws | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem rational_shift_later_denominator {A a : ℕ} (hAa : A ≤ a) (r : ℤ) : | |
| ∃ s : ℕ, s < p.q a ∧ | |
| (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle) = | |
| (((s : ℝ) / (p.q a : ℝ) : ℝ) : Circle) := by | |
| obtain ⟨d, hd⟩ := hp.trimming_frequency_dvd_of_le hAa | |
| have hqa : (p.q A : ℝ) ≠ 0 := by exact_mod_cast (hp.q_pos A).ne' | |
| have hqb : (p.q a : ℝ) ≠ 0 := by exact_mod_cast (hp.q_pos a).ne' | |
| have he : ((r * (d : ℤ) : ℤ) : ℝ) / (p.q a : ℝ) = | |
| (r : ℝ) / (p.q A : ℝ) := by | |
| have hdR : (p.q a : ℝ) = (p.q A : ℝ) * d := by exact_mod_cast hd | |
| push_cast | |
| field_simp | |
| rw [hdR] | |
| ring | |
| obtain ⟨s, hs, heq⟩ := exists_nat_rational_representative (p.q a) (hp.q_pos a) | |
| (r * (d : ℤ)) | |
| refine ⟨s, hs, ?_⟩ | |
| rwa [he] at heq | |
| theorem exists_rational_sampling_index {A j : ℕ} | |
| (hAj : A ≤ p.blockEnd (j - 1)) (r : ℤ) : | |
| ∃ s : ℕ, s < p.q (p.blockEnd (j - 1)) ∧ | |
| (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle) = (p.shift j s : Circle) := by | |
| exact hp.rational_shift_later_denominator hAj r | |
| theorem exists_rational_sampling_index_of_late {A j : ℕ} | |
| (hAj : A + 1 ≤ j) (r : ℤ) : | |
| ∃ s : ℕ, s < p.q (p.blockEnd (j - 1)) ∧ | |
| (((r : ℝ) / (p.q A : ℝ) : ℝ) : Circle) = (p.shift j s : Circle) := by | |
| apply hp.exists_rational_sampling_index | |
| exact (show A ≤ j - 1 by omega).trans (p.blockEnd_ge_index hp.m_pos (j - 1)) | |
| end CircleParameterLaws | |
| end Invariant233 | |
| end | |
| end OAI | |