| import OAI.Analysis.FiniteFactor.Coupling.Conditioning |
| import OAI.Analysis.FiniteFactor.Coupling.Order |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter Finset |
| open scoped ENNReal |
| noncomputable section |
| namespace CouplingParameters |
|
|
| def eligible (right : Bool) (x u : ℝ) : Prop := if right then u ≤ x else x < u |
| |
| instance eligible_decidable (right : Bool) (x u : ℝ) : Decidable (eligible right x u) := by |
| unfold eligible |
| infer_instance |
| |
| def eligibleIndices {ι : Type*} (right : ι → Bool) (x u : ι → ℝ) (I : Finset ι) : Finset ι := |
| I.filter (fun i => eligible (right i) (x i) (u i)) |
| |
| theorem eligible_measurable (right : Bool) (u : ℝ) : MeasurableSet {x : ℝ | eligible right x u} := by |
| cases right |
| · exact measurableSet_Iio |
| · exact measurableSet_Ici |
| |
| theorem eligible_count_measurable {ι : Type*} (right : ι → Bool) (u : ι → ℝ) (I : Finset ι) : |
| Measurable (fun x : ι → ℝ => (eligibleIndices right x u I).card) := by |
| unfold eligibleIndices |
| simp_rw [Finset.card_filter] |
| apply Finset.measurable_sum |
| intro i _ |
| exact Measurable.ite ((eligible_measurable (right i) (u i)).preimage (measurable_pi_apply i)) |
| measurable_const measurable_const |
| |
| / |
| read from the conditioning vector. Taking right=false or right=true gives |
| the two orientations used in the contradiction. -/ |
| def badHalfEvent {ι : Type*} (right : ι → Bool) (u : ι → ℝ) (I : Finset ι) (q : ℝ) : |
| Set ((ι → ℝ) × (ι → ℝ)) := |
| {z | q/2 ≤ ((eligibleIndices right z.1 u I).card : ℝ) ∧ |
| ∀ i : I, ¬(eligible (right i) (z.1 i) (u i) ∧ z.2 i ∈ crossingPartner (right i) (u i))} |
| |
| theorem badHalfEvent_measurable {ι : Type*} (right : ι → Bool) (u : ι → ℝ) (I : Finset ι) (q : ℝ) : |
| MeasurableSet (badHalfEvent right u I q) := by |
| have hcount : Measurable (fun z : (ι → ℝ) × (ι → ℝ) => |
| ((eligibleIndices right z.1 u I).card : ℝ)) := |
| (measurable_of_countable (fun n : ℕ => (n : ℝ))).comp |
| ((eligible_count_measurable right u I).comp measurable_fst) |
| have hq : MeasurableSet {z : (ι → ℝ) × (ι → ℝ) | q/2 ≤ ((eligibleIndices right z.1 u I).card : ℝ)} := |
| measurableSet_le measurable_const hcount |
| have hi (i : I) : MeasurableSet {z : (ι → ℝ) × (ι → ℝ) | |
| ¬(eligible (right i) (z.1 i) (u i) ∧ z.2 i ∈ crossingPartner (right i) (u i))} := |
| (((eligible_measurable (right i) (u i)).preimage |
| ((measurable_pi_apply (i : ι)).comp measurable_fst)).inter |
| ((crossingPartner_measurable (right i) (u i)).preimage |
| ((measurable_pi_apply (i : ι)).comp measurable_snd))).compl |
| convert hq.inter (MeasurableSet.iInter hi) using 1 |
| ext z |
| simp [badHalfEvent] |
| |
| theorem badHalfEvent_section {ι : Type*} (right : ι → Bool) (x u : ι → ℝ) |
| (I : Finset ι) (q : ℝ) (hq : q/2 ≤ ((eligibleIndices right x u I).card : ℝ)) : |
| Prod.mk x ⁻¹' badHalfEvent right u I q = |
| Set.pi (eligibleIndices right x u I : Set ι) (fun i => (crossingPartner (right i) (u i))ᶜ) := by |
| ext y |
| simp only [Set.mem_preimage, badHalfEvent, Set.mem_ofPred_eq, Set.mem_pi, |
| Finset.mem_coe, eligibleIndices, Finset.mem_filter, Set.mem_compl_iff] |
| constructor |
| · intro h i hi |
| exact fun hy => h.2 ⟨i, hi.1⟩ ⟨hi.2, hy⟩ |
| · intro h |
| refine ⟨hq, fun i hi => ?_⟩ |
| exact h i ⟨i.2, hi.1⟩ hi.2 |
| |
| theorem vectorPosterior_badHalf_bound {ι : Type*} |
| (p : ι → CouplingParameters) (right : ι → Bool) (w : ι → Fin 2) |
| (x u : ι → ℝ) (I : Finset ι) (q m ρ L : ℝ) |
| (hx : ∀ i ∈ I, x i ∈ Set.Ico 0 1) |
| (hu : ∀ i ∈ I, (p i).width (w i) ≤ u i ∧ u i ≤ 1-(p i).width (w i)) |
| (hρ : ∀ i ∈ I, (p i).rho (w i) = ρ) |
| (hρ0 : 0 ≤ ρ) (hq : m/20 ≤ q) (hL : L ≤ m*ρ) : |
| vectorPosterior p right x (Prod.mk x ⁻¹' badHalfEvent right u I q) ≤ |
| ENNReal.ofReal (Real.exp (-L/400)) := by |
| by_cases hn : q/2 ≤ ((eligibleIndices right x u I).card : ℝ) |
| · rw [badHalfEvent_section right x u I q hn] |
| apply vectorPosterior_noncrossing_source_bound p right w x u (eligibleIndices right x u I) m ρ L |
| · intro i hi |
| exact hx i (Finset.mem_filter.mp hi).1 |
| · intro i hi |
| exact hu i (Finset.mem_filter.mp hi).1 |
| · intro i hi |
| exact (Finset.mem_filter.mp hi).2 |
| · intro i hi |
| exact hρ i (Finset.mem_filter.mp hi).1 |
| · exact hρ0 |
| · linarith |
| · exact hL |
| · have he : Prod.mk x ⁻¹' badHalfEvent right u I q = ∅ := by |
| ext y |
| simp [badHalfEvent, hn] |
| rw [he, measure_empty] |
| exact zero_le |
| |
| theorem vectorEndpoint_unitCube {ι : Type*} [Countable ι] |
| (p : ι → CouplingParameters) (right : ι → Bool) : |
| ∀ᵐ x ∂(vectorCoupling p).map (vectorEndpoint right), ∀ i, x i ∈ Set.Ico 0 1 := by |
| rw [vectorEndpoint_law, ae_all_iff] |
| intro i |
| have hi : ∀ᵐ t ∂(p i).selectedLaw true (right i), t ∈ Set.Ico 0 1 := |
| (withDensity_absolutelyContinuous _ _).ae_le (ae_restrict_mem measurableSet_Ico) |
| have hh := Measure.infinitePi_map_eval (fun i => (p i).selectedLaw true (right i)) i |
| rw [← hh] at hi |
| exact ae_of_ae_map (measurable_pi_apply i).aemeasurable hi |
| |
| / |
| countably infinite masked-sum theorem. No conditional-independence assumption |
| or posterior bound is an input. -/ |
| theorem badHalf_condExp_bound {ι : Type*} [Countable ι] |
| (p : ι → CouplingParameters) (right : ι → Bool) (w : ι → Fin 2) |
| (u : ι → ℝ) (I : Finset ι) (q m ρ L : ℝ) |
| (hu : ∀ i ∈ I, (p i).width (w i) ≤ u i ∧ u i ≤ 1-(p i).width (w i)) |
| (hρ : ∀ i ∈ I, (p i).rho (w i) = ρ) |
| (hρ0 : 0 ≤ ρ) (hq : m/20 ≤ q) (hL : L ≤ m*ρ) : |
| ∀ᵐ ω ∂vectorCoupling p, |
| (vectorCoupling p)[ |
| (fun ω => (badHalfEvent right u I q).indicator (fun _ => (1 : ℝ)) |
| (vectorEndpoint right ω, vectorEndpoint (fun i => !right i) ω)) | |
| (inferInstance : MeasurableSpace (ι → ℝ)).comap (vectorEndpoint right)] ω ≤ Real.exp (-L/400) := by |
| apply vector_condExp_event_le p right (badHalfEvent_measurable right u I q) |
| filter_upwards [vectorEndpoint_unitCube p right] with x hx |
| have hh := vectorPosterior_badHalf_bound p right w x u I q m ρ L (fun i _ => hx i) hu hρ hρ0 hq hL |
| have hh' := ENNReal.toReal_mono ENNReal.ofReal_ne_top hh |
| simpa only [measureReal_def, ENNReal.toReal_ofReal (Real.exp_pos _).le] using hh' |
| |
| / |
| Thus the strict second half of the source bad event lies in the reverse |
| badHalfEvent; using a weak inequality there only enlarges that event. -/ |
| theorem noncrossing_eligible_complement {ι : Type*} (x y u : ι → ℝ) (I : Finset ι) |
| (hxy : ∀ i ∈ I, x i ≤ y i) (hn : ∀ i ∈ I, ¬(x i < u i ∧ u i ≤ y i)) : |
| (eligibleIndices (fun _ => false) x u I).card + |
| (eligibleIndices (fun _ => true) y u I).card = I.card := by |
| have he : eligibleIndices (fun _ => true) y u I = I.filter (fun i => ¬(x i < u i)) := by |
| ext i |
| simp only [eligibleIndices, Finset.mem_filter, eligible, ↓reduceIte] |
| constructor |
| · rintro ⟨hi, hy⟩ |
| exact ⟨hi, fun hx => hn i hi ⟨hx, hy⟩⟩ |
| · rintro ⟨hi, hx⟩ |
| exact ⟨hi, le_trans (le_of_not_gt hx) (hxy i hi)⟩ |
| rw [he] |
| exact Finset.card_filter_add_card_filter_not (fun i => x i < u i) |
| |
| / |
| conditional estimate was proved above. All coordinates in I are accounted for. -/ |
| theorem second_bad_half_in_reverse {ι : Type*} (x y u : ι → ℝ) (I : Finset ι) |
| (hxy : ∀ i ∈ I, x i ≤ y i) |
| (hn : ∀ i ∈ I, ¬(x i < u i ∧ u i ≤ y i)) |
| (hcount : ((eligibleIndices (fun _ => false) x u I).card : ℝ) < (I.card : ℝ)/2) : |
| (y,x) ∈ badHalfEvent (fun _ => true) u I I.card := by |
| constructor |
| · have he := noncrossing_eligible_complement x y u I hxy hn |
| have her : ((eligibleIndices (fun _ => false) x u I).card : ℝ) + |
| ((eligibleIndices (fun _ => true) y u I).card : ℝ) = (I.card : ℝ) := by exact_mod_cast he |
| linarith |
| · intro i hi |
| exact hn i i.2 ⟨hi.2, hi.1⟩ |
| |
| / |
| itself a theorem about the actual full augmented law. -/ |
| theorem vectorCoupling_ordered {ι : Type*} [Countable ι] (p : ι → CouplingParameters) : |
| vectorCoupling p {ω | ∀ i, (ω i).2.1 ≤ (ω i).2.2} = 1 := by |
| have he : {ω : ι → Fin 6 × (ℝ × ℝ) | ∀ i, (ω i).2.1 ≤ (ω i).2.2} = |
| Set.pi Set.univ (fun _ : ι => {z : Fin 6 × (ℝ × ℝ) | z.2.1 ≤ z.2.2}) := by ext ω; simp |
| rw [he, vectorCoupling, Measure.infinitePi_pi_univ] |
| · simp [labelledCoupling_ordered] |
| · intro i |
| exact orderedPairs_measurable.preimage measurable_snd |
| |
| end CouplingParameters |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
