| import OAI.Analysis.IrrationalRotation.Sampling.MixedRadix |
|
|
| namespace OAI |
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|
| open Filter |
| open scoped Topology |
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|
| namespace Invariant233.MixedRadix |
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|
| theorem denominator_radices {q : ℕ → ℕ} (hq0 : q 0 = 1) |
| (hdiv : ∀ n, q n ∣ q (n + 1)) (N : ℕ) : denominator (radices q) N = q N := by |
| induction N with |
| | zero => simpa using hq0.symm |
| | succ N ih => |
| rw [denominator_succ, ih, radices] |
| exact Nat.mul_div_cancel' (hdiv N) |
|
|
| theorem radices_ge_two {q : ℕ → ℕ} (hq : ∀ n, 0 < q n) |
| (hstrict : StrictMono q) (hdiv : ∀ n, q n ∣ q (n + 1)) (n : ℕ) : |
| 2 ≤ radices q n := by |
| have heq : q n * radices q n = q (n + 1) := Nat.mul_div_cancel' (hdiv n) |
| have hi := hstrict (Nat.lt_succ_self n) |
| have hp := hq n |
| by_contra hnot |
| have hle : radices q n ≤ 1 := by omega |
| nlinarith |
|
|
| theorem radices_pos {q : ℕ → ℕ} (hq : ∀ n, 0 < q n) |
| (hstrict : StrictMono q) (hdiv : ∀ n, q n ∣ q (n + 1)) (n : ℕ) : |
| 0 < radices q n := lt_of_lt_of_le (by decide) (radices_ge_two hq hstrict hdiv n) |
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| theorem coordinate_eq_tailValue {q : ℕ → ℕ} (hq0 : q 0 = 1) |
| (hq : ∀ n, 0 < q n) (hstrict : StrictMono q) (hdiv : ∀ n, q n ∣ q (n + 1)) |
| (d : Digits (radices q)) (hnt : Nonterminal d) (i : ℕ) : |
| Int.fract ((q i : ℝ) * ofDigits d) = tailValue d i := by |
| rw [← denominator_radices hq0 hdiv i] |
| exact fract_denominator_mul_ofDigits (radices_pos hq hstrict hdiv) d hnt i |
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|
| theorem coordinates_ordered {q : ℕ → ℕ} (hq0 : q 0 = 1) |
| (hq : ∀ n, 0 < q n) (hstrict : StrictMono q) (hdiv : ∀ n, q n ∣ q (n + 1)) |
| {d e : Digits (radices q)} (hd : Nonterminal d) (he : Nonterminal e) |
| (hde : ∀ n, d n ≤ e n) (i : ℕ) : |
| Int.fract ((q i : ℝ) * ofDigits d) ≤ Int.fract ((q i : ℝ) * ofDigits e) := by |
| rw [coordinate_eq_tailValue hq0 hq hstrict hdiv d hd i, |
| coordinate_eq_tailValue hq0 hq hstrict hdiv e he i] |
| exact tailValue_mono (radices_pos hq hstrict hdiv) hde i |
|
|
| theorem abs_coordinate_sub_raw_le {q : ℕ → ℕ} (hq0 : q 0 = 1) |
| (hq : ∀ n, 0 < q n) (hstrict : StrictMono q) (hdiv : ∀ n, q n ∣ q (n + 1)) |
| (d : Digits (radices q)) (hnt : Nonterminal d) (i : ℕ) {X : ℝ} (hX : 0 ≤ X) |
| (hfloor : ⌊(radices q i : ℝ) * X⌋₊ = (d i : ℕ)) : |
| |Int.fract ((q i : ℝ) * ofDigits d) - X| ≤ 1 / (radices q i : ℝ) := by |
| rw [coordinate_eq_tailValue hq0 hq hstrict hdiv d hnt i] |
| exact abs_tailValue_sub_raw_le_of_floor (radices_pos hq hstrict hdiv) d i hX hfloor |
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| @[simp] theorem hybrid_apply_lt {r : ℕ → ℕ} (d e : Digits r) {N n : ℕ} (hn : n < N) : |
| hybrid d e N n = e n := ite_eq_left hn |
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|
| @[simp] theorem hybrid_apply_ge {r : ℕ → ℕ} (d e : Digits r) {N n : ℕ} (hn : N ≤ n) : |
| hybrid d e N n = d n := ite_eq_right (not_lt.mpr hn) |
|
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| theorem nonterminal_hybrid {r : ℕ → ℕ} (d e : Digits r) (hd : Nonterminal d) (N : ℕ) : |
| Nonterminal (hybrid d e N) := by |
| intro j |
| obtain ⟨n, hn, hdigit⟩ := hd (max j N) |
| have hjn : j ≤ n := (le_max_left j N).trans hn |
| have hNn : N ≤ n := (le_max_right j N).trans hn |
| exact ⟨n, hjn, by simpa [hybrid_apply_ge d e hNn] using hdigit⟩ |
|
|
| theorem hybrid_ordered {r : ℕ → ℕ} {d e : Digits r} (hde : ∀ n, d n ≤ e n) (N n : ℕ) : |
| d n ≤ hybrid d e N n ∧ hybrid d e N n ≤ e n := by |
| by_cases hn : n < N |
| · simpa [hybrid, hn] using hde n |
| · simpa [hybrid, hn] using hde n |
|
|
| theorem tailValue_hybrid {r : ℕ → ℕ} (d e : Digits r) (N : ℕ) : |
| tailValue (hybrid d e N) N = tailValue d N := by |
| unfold tailValue |
| congr 1 |
| apply tsum_congr |
| intro n |
| simp [term] |
|
|
| theorem denominator_mul_hybrid_sub {r : ℕ → ℕ} (hr : ∀ n, 0 < r n) |
| (d e : Digits r) (N : ℕ) : |
| (denominator r N : ℝ) * (ofDigits (hybrid d e N) - ofDigits d) = |
| ((prefixRank (hybrid d e N) N : ℤ) - (prefixRank d N : ℤ) : ℤ) := by |
| rw [mul_sub, denominator_mul_ofDigits hr (hybrid d e N) N, |
| denominator_mul_ofDigits hr d N, tailValue_hybrid] |
| push_cast |
| ring |
|
|
| theorem hybrid_sub_eq_int_div {r : ℕ → ℕ} (hr : ∀ n, 0 < r n) |
| (d e : Digits r) (N : ℕ) : |
| ∃ z : ℤ, ofDigits (hybrid d e N) - ofDigits d = (z : ℝ) / (denominator r N : ℝ) := by |
| refine ⟨(prefixRank (hybrid d e N) N : ℤ) - (prefixRank d N : ℤ), ?_⟩ |
| have hq : (denominator r N : ℝ) ≠ 0 := ne_of_gt (by exact_mod_cast denominator_pos hr N) |
| apply (eq_div_iff hq).mpr |
| simpa only [mul_comm] using denominator_mul_hybrid_sub hr d e N |
|
|
| theorem tendsto_ofDigits_hybrid {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) (d e : Digits r) : |
| Tendsto (fun N => ofDigits (hybrid d e N)) atTop (𝓝 (ofDigits e)) := by |
| apply tendsto_ofDigits_of_agree_prefix hr e (hybrid d e) |
| intro N n hn |
| exact hybrid_apply_lt d e hn |
|
|
| end Invariant233.MixedRadix |
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| end OAI |
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