File size: 2,650 Bytes
f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | 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
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