| import OAI.Analysis.IrrationalRotation.Model.MixedRadixProbability |
| import OAI.Analysis.IrrationalRotation.Sampling.MixedRadix |
| import OAI.Analysis.IrrationalRotation.Sampling.FinitePrefixLaw |
| import OAI.Analysis.IrrationalRotation.Analysis.UniformQuantization |
| import OAI.Analysis.IrrationalRotation.Analysis.LimitLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal BigOperators Classical |
|
|
| namespace Invariant233.MixedRadix |
| |
| instance digitLaw_isProbability (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) : |
| IsProbabilityMeasure (digitLaw r hr) := by |
| unfold digitLaw |
| infer_instance |
| |
| theorem measurable_prefix_sum (r : ℕ → ℕ) (N : ℕ) : |
| Measurable (fun d : Digits r => ∑ n ∈ Finset.range N, term d n) := by |
| apply Finset.measurable_sum |
| intro n hn |
| exact ((measurable_of_finite (fun a : Fin (r n) => (a : ℝ))).comp |
| (measurable_pi_apply n)).div_const _ |
| |
| theorem tendsto_prefix_sum {r : ℕ → ℕ} (hr : ∀ n, 0 < r n) (d : Digits r) : |
| Tendsto (fun N => ∑ n ∈ Finset.range N, term d n) atTop (𝓝 (ofDigits d)) := |
| (summable_terms hr d).hasSum.tendsto_sum_nat |
| |
| theorem tendsto_quantized_prefix {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) {x : ℝ} |
| (hx : x ∈ Ico (0 : ℝ) 1) : |
| Tendsto (fun N => (⌊(denominator r N : ℝ) * x⌋₊ : ℝ) / |
| (denominator r N : ℝ)) atTop (𝓝 x) := by |
| let hp : ∀ n, 0 < r n := fun n => lt_of_lt_of_le (by decide : 0 < 2) (hr n) |
| have heq : (fun N => (⌊(denominator r N : ℝ) * x⌋₊ : ℝ) / |
| (denominator r N : ℝ)) = |
| (fun N => ∑ n ∈ Finset.range N, term (canonicalDigits r hp x) n) := by |
| funext N |
| have hq : (denominator r N : ℝ) ≠ 0 := |
| ne_of_gt (by exact_mod_cast denominator_pos hp N) |
| have h := denominator_mul_prefix hp (canonicalDigits r hp x) N |
| rw [prefixRank_canonicalDigits hp hx N] at h |
| exact (eq_div_iff hq).mpr (by simpa [mul_comm] using h) |>.symm |
| rw [heq] |
| have hrec : ofDigits (canonicalDigits r hp x) = x := ofDigits_canonicalDigits hr hx |
| simpa only [hrec] using tendsto_prefix_sum hp (canonicalDigits r hp x) |
| |
| theorem map_prefixRank_uniform (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) (N : ℕ) [NeZero (denominator r N)] : |
| (digitLaw r hr).map |
| (fun d => finitePrefixRankEquiv r hr N (fun i : Fin N => d i)) = |
| (PMF.uniformOfFintype (Fin (denominator r N))).toMeasure := by |
| let : ∀ n, NeZero (r n) := fun n => ⟨ne_of_gt (hr n)⟩ |
| let : NeZero (denominator r N) := ⟨ne_of_gt (denominator_pos hr N)⟩ |
| let E := finitePrefixRankEquiv r hr N |
| have hp : (digitLaw r hr).map (fun d (i : Fin N) => d i) = |
| (PMF.uniformOfFintype ((i : Fin N) → Fin (r i))).toMeasure := by |
| unfold digitLaw |
| exact FiniteUniformLaw.infinitePi_fin_uniform (fun n => Fin (r n)) N |
| have hm : Measurable (fun d : Digits r => fun i : Fin N => d i) := |
| Measurable.of_eval (fun index => measurable_pi_apply (index : ℕ)) |
| change (digitLaw r hr).map (E ∘ (fun d (i : Fin N) => d i)) = _ |
| rw [← Measure.map_map (measurable_of_finite E) hm, hp] |
| exact FiniteUniformLaw.map_equiv E |
| |
| theorem prefix_sum_eq_normalized_rank (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) |
| (N : ℕ) (d : Digits r) : |
| (∑ n ∈ Finset.range N, term d n) = |
| ((finitePrefixRankEquiv r hr N (fun i : Fin N => d i) : ℕ) : ℝ) / |
| (denominator r N : ℝ) := by |
| rw [finitePrefixRankEquiv_val] |
| have heq : prefixRank (extendPrefix r hr N (fun i : Fin N => d i)) N = |
| prefixRank d N := by |
| apply prefixRank_congr |
| intro n hn |
| simp [hn] |
| rw [heq] |
| have hq : (denominator r N : ℝ) ≠ 0 := |
| ne_of_gt (by exact_mod_cast denominator_pos hr N) |
| exact (eq_div_iff hq).mpr (by simpa [mul_comm] using denominator_mul_prefix hr d N) |
| |
| theorem map_prefix_sum_eq_quantized (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) (N : ℕ) : |
| (digitLaw r hr).map (fun d => ∑ n ∈ Finset.range N, term d n) = |
| (volume.restrict (Ico (0 : ℝ) 1)).map |
| (fun x => (⌊(denominator r N : ℝ) * x⌋₊ : ℝ) / (denominator r N : ℝ)) := by |
| let : ∀ n, NeZero (r n) := fun n => ⟨ne_of_gt (hr n)⟩ |
| let : NeZero (denominator r N) := ⟨ne_of_gt (denominator_pos hr N)⟩ |
| let R : Digits r → Fin (denominator r N) := |
| fun d => finitePrefixRankEquiv r hr N (fun i : Fin N => d i) |
| let g : Fin (denominator r N) → ℝ := fun a => (a : ℝ) / (denominator r N : ℝ) |
| have hR : Measurable R := (measurable_of_finite (finitePrefixRankEquiv r hr N)).comp |
| (Measurable.of_eval (fun index : Fin N => measurable_pi_apply (index : ℕ))) |
| have hg : Measurable g := measurable_of_finite _ |
| have heq : (fun d => ∑ n ∈ Finset.range N, term d n) = g ∘ R := by |
| funext d |
| exact prefix_sum_eq_normalized_rank r hr N d |
| rw [heq, ← Measure.map_map hg hR] |
| have hRlaw : (digitLaw r hr).map R = |
| (PMF.uniformOfFintype (Fin (denominator r N))).toMeasure := |
| map_prefixRank_uniform r hr N |
| rw [hRlaw, ← UniformQuantization.map_quantize_uniform (denominator r N), |
| Measure.map_map hg (UniformQuantization.measurable_quantize _)] |
| apply Measure.map_congr |
| filter_upwards [ae_restrict_mem (μ := (volume : Measure ℝ)) measurableSet_Ico] with x hx |
| have hq : (0 : ℝ) < denominator r N := by exact_mod_cast denominator_pos hr N |
| have hfloor : ⌊(denominator r N : ℝ) * x⌋₊ < denominator r N := |
| (Nat.floor_lt (mul_nonneg hq.le hx.1)).mpr (by nlinarith [hx.2]) |
| simp [g, UniformQuantization.quantize, Nat.mod_eq_of_lt hfloor] |
| |
| theorem map_ofDigits_eq_volume (r : ℕ → ℕ) (hr : ∀ n, 2 ≤ r n) : |
| (digitLaw r (fun n => lt_of_lt_of_le (by decide : 0 < 2) (hr n))).map ofDigits = |
| volume.restrict (Ico (0 : ℝ) 1) := by |
| let hp : ∀ n, 0 < r n := fun n => lt_of_lt_of_le (by decide : 0 < 2) (hr n) |
| let μ := digitLaw r hp |
| let ν := (volume : Measure ℝ).restrict (Ico (0 : ℝ) 1) |
| have : IsProbabilityMeasure ν := ⟨by simp [ν]⟩ |
| have : IsProbabilityMeasure μ := digitLaw_isProbability r hp |
| have hX : ∀ N, AEMeasurable (fun d : Digits r => ∑ n ∈ Finset.range N, term d n) μ := |
| fun N => (measurable_prefix_sum r N).aemeasurable |
| have hY : ∀ N, AEMeasurable |
| (fun x : ℝ => (⌊(denominator r N : ℝ) * x⌋₊ : ℝ) / (denominator r N : ℝ)) ν := by |
| intro N |
| exact (((measurable_of_countable (fun n : ℕ => (n : ℝ))).comp |
| (measurable_const.mul measurable_id).nat_floor).div_const _).aemeasurable |
| have hZ : AEMeasurable (ofDigits : Digits r → ℝ) μ := |
| (continuous_ofDigits hr).measurable.aemeasurable |
| have hW : AEMeasurable (id : ℝ → ℝ) ν := measurable_id.aemeasurable |
| have hXZ : ∀ᵐ d ∂μ, Tendsto (fun N => ∑ n ∈ Finset.range N, term d n) |
| atTop (𝓝 (ofDigits d)) := Filter.Eventually.of_forall (tendsto_prefix_sum hp) |
| have hYW : ∀ᵐ x ∂ν, Tendsto |
| (fun N => (⌊(denominator r N : ℝ) * x⌋₊ : ℝ) / (denominator r N : ℝ)) |
| atTop (𝓝 (id x)) := by |
| filter_upwards [ae_restrict_mem (μ := (volume : Measure ℝ)) measurableSet_Ico] with x hx |
| exact tendsto_quantized_prefix hr hx |
| have h := Invariant233.map_eq_of_ae_limits_of_map_eq hX hY hZ hW hXZ hYW |
| (map_prefix_sum_eq_quantized r hp) |
| simpa only [Measure.map_id] using h |
| |
| end Invariant233.MixedRadix |
| |
| end |
| |
| end OAI |
| |
