openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Pivots /RandomizedMarkedTransactions.lean
| import OAI.Probability.MatroidSecretary.Pivots.MarkedTransactions | |
| import OAI.Probability.MatroidSecretary.Pivots.CertificateAuxSeed | |
| import Mathlib.MeasureTheory.Integral.Bochner.Basic | |
| import Mathlib.MeasureTheory.Integral.IntegrableOn | |
| namespace OAI | |
| /-! | |
| # The complete disjoint-query certificate contract | |
| This is the independent-auxiliary-seed form of `lem:transactions`, with explicit | |
| success declarations rather than counting every certifying transaction. The | |
| inner finite product expectation has the original Bernoulli law for every seed. | |
| Soundness is required only on bit vectors with positive product mass; this keeps | |
| deterministic Bernoulli coordinates within scope without restricting impossible | |
| executions. Integrability follows from the proved logarithmic budget. | |
| -/ | |
| namespace MatroidProphet.MarkedQueryProgram | |
| open Finset MeasureTheory | |
| /-- On every sound execution with positive failure probability, successes are | |
| bounded by the number of coordinates, even if the chosen finite tree contains | |
| arbitrarily many empty or unsuccessful transactions. -/ | |
| lemma run_le_card_of_positive_failure | |
| {α : Type*} [Fintype α] [DecidableEq α] | |
| (cert : Finset α → Prop) [DecidablePred cert] (hcert : Monotone cert) | |
| (q : α → ℝ) (hδ : 0 < bitsFailure cert q univ ∅) | |
| (prog : MarkedQueryProgram α) (fresh : prog.Fresh univ) | |
| (S : Finset α) (sound : prog.SoundAt cert S ∅) : | |
| prog.run S ≤ (Fintype.card α : ℝ) := by | |
| have hempty : ¬ cert ∅ := by | |
| intro hc | |
| have hz := bitsFailure_certifies cert hcert q univ ∅ hc | |
| rw [hz] at hδ | |
| exact lt_irrefl 0 hδ | |
| have hu := prog.erase.run_le_card cert hempty S univ (prog.erase_fresh univ fresh) | |
| simpa using (prog.run_le_certifying_run cert S ∅ sound).trans hu | |
| /-- Arbitrary independent auxiliary randomization preserves the certificate | |
| budget for explicitly declared, sound successes. The source's uniform finite | |
| transaction bound is not needed: each seed fixes a finite adaptive tree. -/ | |
| theorem disjoint_query_certificates_auxiliary | |
| {α : Type*} [Fintype α] [DecidableEq α] | |
| (cert : Finset α → Prop) [DecidablePred cert] (hcert : Monotone cert) | |
| (q : α → ℝ) (hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) | |
| (hδ : 0 < bitsFailure cert q univ ∅) | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (prog : Ω → MarkedQueryProgram α) | |
| (fresh : ∀ ω, (prog ω).Fresh univ) | |
| (sound : ∀ ω S, 0 < bitsWeight q univ S → (prog ω).SoundAt cert S ∅) | |
| (meas : ∀ S, Measurable (fun ω => (prog ω).run S)) : | |
| Integrable (fun ω => bitsExpectation q univ (prog ω).run) μ ∧ | |
| (∫ ω, bitsExpectation q univ (prog ω).run ∂μ) ≤ | |
| Real.log (1 / bitsFailure cert q univ ∅) := by | |
| let f : Ω → ℝ := fun ω => bitsExpectation q univ (prog ω).run | |
| have hf0 (ω : Ω) : 0 ≤ f ω := | |
| bitsExpectation_nonneg q hq0 hq1 univ | |
| (fun S _ => (prog ω).run_nonneg S) | |
| have hf1 (ω : Ω) : f ω ≤ Real.log (1 / bitsFailure cert q univ ∅) := | |
| disjoint_query_certificates_support cert hcert q hq0 hq1 hδ | |
| (prog ω) (fresh ω) (sound ω) | |
| have hfm : Measurable f := by | |
| unfold f bitsExpectation | |
| exact Finset.measurable_sum _ fun S _ => measurable_const.mul (meas S) | |
| have hfi : Integrable f μ := by | |
| apply Integrable.of_bound hfm.aestronglyMeasurable | |
| (Real.log (1 / bitsFailure cert q univ ∅)) | |
| exact Filter.Eventually.of_forall fun ω => by | |
| rw [Real.norm_eq_abs, abs_of_nonneg (hf0 ω)] | |
| exact hf1 ω | |
| refine ⟨hfi, ?_⟩ | |
| calc | |
| (∫ ω, f ω ∂μ) ≤ ∫ _ : Ω, Real.log (1 / bitsFailure cert q univ ∅) ∂μ := | |
| integral_mono hfi (integrable_const _) hf1 | |
| _ = Real.log (1 / bitsFailure cert q univ ∅) := by simp | |
| end MatroidProphet.MarkedQueryProgram | |
| end OAI | |