| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitDensity |
| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitDisintegration |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace Invariant233.CircleDigitCoupling |
| open CouplingMarginalBounds CouplingMixture OneCoordinateCoupling |
|
|
| lemma baseLaw_le_intervalLaw_of_subset {a b : ℝ} (ha : 0 ≤ a) (hb : b ≤ 1) |
| {s : Set ℝ} (hs : s ⊆ Ico a b) : baseLaw s ≤ intervalLaw a b s := by |
| rw [intervalLaw_eq_base_cond ha hb, ProbabilityTheory.cond_apply measurableSet_Ico, |
| inter_eq_right.mpr hs] |
| have hmass : baseLaw (Ico a b) ≤ 1 := by |
| simpa using (measure_mono (subset_univ (Ico a b)) : baseLaw (Ico a b) ≤ baseLaw univ) |
| simpa only [one_mul, mul_one, mul_comm] using |
| mul_le_mul_right (ENNReal.one_le_inv.mpr hmass) (baseLaw s) |
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| variable {κ m a b t : ℝ} |
| (hκ0 : 0 ≤ κ) (hκ : κ ≤ 1 / 100) (hm : 100 ≤ m) |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) |
| include hκ0 hκ hm ha ha' hb hb' ht0 ht |
|
|
| lemma label_posterior_lower_of_bin (side : Bool) (r : ℕ) [NeZero r] |
| (d : Fin r) (l : Label) |
| (hbin : baseLaw (bin r d) ≤ endpointLaw side m a b l (bin r d)) : |
| labelMass κ m a b t l / 10 ≤ labelsGivenDigit side r κ m a b t d {l} := by |
| rw [labelsGivenDigit_singleton] |
| have h0 : baseLaw (bin r d) ≠ 0 := by rw [baseLaw_bin]; simp |
| have hfin : baseLaw (bin r d) ≠ ∞ := measure_ne_top _ _ |
| calc |
| _ = (labelMass κ m a b t l * baseLaw (bin r d)) / |
| (10 * baseLaw (bin r d)) := (ENNReal.mul_div_mul_right _ _ h0 hfin).symm |
| _ ≤ _ := ENNReal.div_le_div (mul_le_mul_right hbin _) |
| (digitWeight_total_le_ten hκ0 hκ hm ha ha' hb hb' ht0 ht side r d) |
|
|
| lemma top_posterior_lower (r : ℕ) [NeZero r] (d : Fin r) (which : Bool) |
| {z u : ℝ} (hz : z ∈ bin r d) (hzu : z < u) |
| (hu : u ≤ 1 - width a b which) (hsmall : 1 / (r : ℝ) < width a b which / 2) : |
| ENNReal.ofReal (rho κ m (width a b which)) / 10 ≤ |
| labelsGivenDigit false r κ m a b t d {topLabel which} := by |
| have hm0 : 0 < m := by linarith |
| have ha0 := width_pos hm0 ha |
| have hb0 := width_pos hm0 hb |
| have hsubset := bin_subset_top_left d hz hzu hu hsmall |
| have hlike : baseLaw (bin r d) ≤ endpointLaw false m a b (topLabel which) (bin r d) := by |
| cases which <;> simp only [topLabel, endpointLaw, Bool.false_eq_true, ite_false, |
| ite_true, leftLaw] <;> simp only [width, Bool.false_eq_true, ite_false, ite_true] at hsubset <;> apply baseLaw_le_intervalLaw_of_subset |
| all_goals solve | norm_num | linarith | exact hsubset |
| have hpost := label_posterior_lower_of_bin hκ0 hκ hm ha ha' hb hb' ht0 ht false r d (topLabel which) hlike |
| cases which <;> simpa [labelMass, topLabel, componentProbabilities, width] using hpost |
|
|
| lemma bottom_posterior_lower (r : ℕ) [NeZero r] (d : Fin r) (which : Bool) |
| {z u : ℝ} (hz : z ∈ bin r d) (hzu : u ≤ z) |
| (hu : width a b which ≤ u) (hsmall : 1 / (r : ℝ) < width a b which / 2) : |
| ENNReal.ofReal (rho κ m (width a b which)) / 10 ≤ |
| labelsGivenDigit true r κ m a b t d {bottomLabel which} := by |
| have hm0 : 0 < m := by linarith |
| have ha0 := width_pos hm0 ha |
| have hb0 := width_pos hm0 hb |
| have hsubset := bin_subset_bottom_right d hz hzu hu hsmall |
| have hlike : baseLaw (bin r d) ≤ endpointLaw true m a b (bottomLabel which) (bin r d) := by |
| cases which <;> simp only [bottomLabel, endpointLaw, Bool.false_eq_true, ite_false, |
| ite_true, rightLaw] <;> simp only [width, Bool.false_eq_true, ite_false, ite_true] at hsubset <;> apply baseLaw_le_intervalLaw_of_subset |
| all_goals solve | norm_num | linarith | exact hsubset |
| have hpost := label_posterior_lower_of_bin hκ0 hκ hm ha ha' hb hb' ht0 ht true r d (bottomLabel which) hlike |
| cases which <;> simpa [labelMass, bottomLabel, componentProbabilities, width] using hpost |
|
|
| end Invariant233.CircleDigitCoupling |
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|
| end |
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| end OAI |
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