openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteCenteredPatternLemmas.lean
| import OAI.Combinatorics.BalancedRyser.Probability.PatternCenteredAssembly | |
| import OAI.Combinatorics.BalancedRyser.Probability.PatternSupportCompatibility | |
| import OAI.Combinatorics.BalancedRyser.Probability.PatternWeightExtensions | |
| namespace OAI | |
| section | |
| namespace Balanced.PatternModel | |
| open scoped _root_.Balanced | |
| noncomputable section | |
| open scoped Classical BigOperators | |
| open MeasureTheory | |
| theorem centered_moment_le_of_exposureBounds (P : PatternModel) | |
| (K₀ : Finset P.Color) (x : P.Sample) (s r : ℕ) (q C : ℝ) | |
| (hq : 1 ≤ q) (hC : 0 ≤ C) | |
| (hsupport : ∀ a, (P.support a).card ≤ r) | |
| (htotal : ∀ K : Finset P.Color, K.card ≤ (2 * s) * r → ∀ y, | |
| P.fixedExpectation (K₀ ∪ K) y P.count ≤ C * q) | |
| (hoverlap : ∀ K : Finset P.Color, K.card ≤ (2 * s) * r → ∀ y, | |
| P.fixedExpectation (K₀ ∪ K) y (P.overlapCount K₀ K) ≤ C) : | |
| P.fixedExpectation K₀ x | |
| (fun ω => |P.count ω - P.fixedExpectation K₀ x P.count| ^ (2 * s)) ≤ | |
| ((2 * s).factorial : ℝ) * 2 ^ (2 * s) * (2 * C) ^ (2 * s) * q ^ s := by | |
| apply PatternModelMoments.centered_moment_bound_of_productWeight_costs | |
| P K₀ x s q (2 * C) hq (by positivity) | |
| · intro pre hpre | |
| have hK : (P.additionalSupport K₀ pre).card ≤ (2 * s) * r := | |
| (P.additionalSupport_card_le K₀ pre r hsupport).trans | |
| (Nat.mul_le_mul_right r (by omega)) | |
| have h := P.productWeight_free_extension_le K₀ x pre (C * q) | |
| (mul_nonneg hC (by linarith)) | |
| (fun y => htotal (P.additionalSupport K₀ pre) hK y) | |
| convert h using 1; ring | |
| · intro pre hpre | |
| have hK : (P.additionalSupport K₀ pre).card ≤ (2 * s) * r := | |
| (P.additionalSupport_card_le K₀ pre r hsupport).trans | |
| (Nat.mul_le_mul_right r (by omega)) | |
| have h := P.productWeight_overlap_extension_le K₀ x pre C hC | |
| (fun y => hoverlap (P.additionalSupport K₀ pre) hK y) | |
| have heq : | |
| (∑ a, if PatternMoments.supportOverlap P.support K₀ pre a then | |
| P.productWeight K₀ x (a :: pre) else 0) = | |
| (∑ a, if Disjoint (P.support a \ K₀) (P.additionalSupport K₀ pre) | |
| then 0 else P.productWeight K₀ x (a :: pre)) := by | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| rw [P.additionalSupport_eq_freshPrefixSupport] | |
| unfold PatternMoments.supportOverlap PatternModel.freshPrefixSupport | |
| split_ifs <;> simp_all | |
| rw [heq] | |
| exact h | |
| theorem overlapCount_le_allExposed (P : PatternModel) | |
| (K₀ K : Finset P.Color) (ω : P.Sample) : | |
| P.overlapCount K₀ K ω ≤ P.overlapCount ∅ (K₀ ∪ K) ω := by | |
| unfold overlapCount | |
| apply Finset.sum_le_sum | |
| intro a _ | |
| by_cases h : Disjoint (P.support a \ K₀) K | |
| · rw [ite_eq_left h] | |
| split_ifs | |
| · exact le_rfl | |
| · exact P.indicator_nonneg a ω | |
| · have hn : ¬ Disjoint (P.support a \ ∅) (K₀ ∪ K) := by | |
| obtain ⟨c, hc, hcK⟩ := Finset.not_disjoint_iff.mp h | |
| exact Finset.not_disjoint_iff.mpr ⟨c, | |
| by simpa using (Finset.mem_sdiff.mp hc).1, | |
| Finset.mem_union_right _ hcK⟩ | |
| rw [ite_eq_right h, ite_eq_right hn] | |
| theorem integrable_fixed_overlapCount (P : PatternModel) | |
| (D : Finset P.Color) (x : P.Sample) (K₀ K : Finset P.Color) : | |
| Integrable (fun ω => P.overlapCount K₀ K (P.fixCoordinates D x ω)) P.measure := by | |
| unfold overlapCount | |
| apply integrable_finsetSum | |
| intro a _ | |
| by_cases h : Disjoint (P.support a \ K₀) K | |
| · simp only [ite_eq_left h] | |
| exact integrable_const 0 | |
| · simp only [ite_eq_right h] | |
| exact P.integrable_fixed_indicator D x a | |
| theorem centered_moment_le_of_allExposedBounds (P : PatternModel) | |
| (K₀ : Finset P.Color) (x : P.Sample) (s r : ℕ) (q C : ℝ) | |
| (hq : 1 ≤ q) (hC : 0 ≤ C) | |
| (hsupport : ∀ a, (P.support a).card ≤ r) | |
| (htotal : ∀ D : Finset P.Color, D.card ≤ K₀.card + (2 * s) * r → ∀ y, | |
| P.fixedExpectation D y P.count ≤ C * q) | |
| (hoverlap : ∀ D : Finset P.Color, D.card ≤ K₀.card + (2 * s) * r → ∀ y, | |
| P.fixedExpectation D y (P.overlapCount ∅ D) ≤ C) : | |
| P.fixedExpectation K₀ x | |
| (fun ω => |P.count ω - P.fixedExpectation K₀ x P.count| ^ (2 * s)) ≤ | |
| ((2 * s).factorial : ℝ) * 2 ^ (2 * s) * (2 * C) ^ (2 * s) * q ^ s := by | |
| apply P.centered_moment_le_of_exposureBounds K₀ x s r q C hq hC hsupport | |
| · intro K hK y | |
| exact htotal (K₀ ∪ K) ((Finset.card_union_le _ _).trans (Nat.add_le_add_left hK _)) y | |
| · intro K hK y | |
| apply le_trans ?_ (hoverlap (K₀ ∪ K) | |
| ((Finset.card_union_le _ _).trans (Nat.add_le_add_left hK _)) y) | |
| apply integral_mono (P.integrable_fixed_overlapCount (K₀ ∪ K) y K₀ K) | |
| (P.integrable_fixed_overlapCount (K₀ ∪ K) y ∅ (K₀ ∪ K)) | |
| intro ω | |
| exact P.overlapCount_le_allExposed K₀ K _ | |
| end | |
| end Balanced.PatternModel | |
| end | |
| section | |
| open scoped BigOperators | |
| namespace Balanced | |
| open scoped _root_.Balanced | |
| universe u | |
| theorem uniformCenteredPatternMoments_of_exposureBounds | |
| (P : ℕ → PatternModel.{u}) (K₀ : ∀ q, Finset (P q).Color) | |
| (hsupport : UniformPatternSupports P) | |
| (htotal : UniformPatternTotalBound P K₀ (fun q => (q : ℝ))) | |
| (hoverlap : UniformPatternOverlapBound P K₀) : | |
| UniformCenteredPatternMoments P K₀ := by | |
| classical | |
| intro s _hs | |
| obtain ⟨C, hC, q₀, hcost⟩ := | |
| uniform_productWeight_extensions P K₀ hsupport htotal hoverlap (2 * s) | |
| refine ⟨((2 * s).factorial : ℝ) * 2 ^ (2 * s) * C ^ (2 * s), | |
| by positivity, max q₀ 1, ?_⟩ | |
| intro q hq x | |
| have hq₀ : q₀ ≤ q := (le_max_left q₀ 1).trans hq | |
| have hq₁ : (1 : ℝ) ≤ q := by | |
| exact_mod_cast (le_max_right q₀ 1).trans hq | |
| apply PatternModelMoments.centered_moment_bound_of_productWeight_costs | |
| (P q) (K₀ q) x s q C hq₁ hC.le | |
| · intro pre hpre | |
| exact (hcost q hq₀ x pre hpre).1 | |
| · intro pre hpre | |
| have heq : | |
| (∑ a, if PatternMoments.supportOverlap (P q).support (K₀ q) pre a then | |
| (P q).productWeight (K₀ q) x (a :: pre) else 0) = | |
| (∑ a, if Disjoint ((P q).support a \ K₀ q) | |
| ((P q).additionalSupport (K₀ q) pre) then 0 else | |
| (P q).productWeight (K₀ q) x (a :: pre)) := by | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| rw [(P q).additionalSupport_eq_freshPrefixSupport] | |
| unfold PatternMoments.supportOverlap PatternModel.freshPrefixSupport | |
| split_ifs <;> simp_all | |
| rw [heq] | |
| exact (hcost q hq₀ x pre hpre).2 | |
| theorem patternMoments_proof : PatternMomentsResult.{u} := by | |
| intro P K₀ hsupport _hinitial | |
| constructor | |
| · exact uniformPatternMoments_of_totalBound P K₀ hsupport | |
| · intro htotal hoverlap | |
| exact uniformCenteredPatternMoments_of_exposureBounds P K₀ hsupport htotal hoverlap | |
| end Balanced | |
| end | |
| end OAI | |