openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteExposureTower.lean
| import OAI.Combinatorics.BalancedRyser.Probability.FiniteFixedLaw | |
| import OAI.Combinatorics.BalancedRyser.Probability.PatternMomentModel | |
| namespace OAI | |
| namespace Balanced.ConditionalRestriction | |
| noncomputable section | |
| open scoped Classical | |
| open MeasureTheory | |
| universe u | |
| variable {D A : Type u} [Fintype D] [Fintype A] [DecidableEq D] [Nonempty A] | |
| [MeasurableSpace A] [MeasurableSingletonClass A] | |
| private def uniformArrayModel : PatternModel.{u} where | |
| Color := D | |
| Value := fun _ => A | |
| law := fun _ => (PMF.uniformOfFintype A).toMeasure | |
| Label := PUnit | |
| support := fun _ => ∅ | |
| event := fun _ => ∅ | |
| eventMeasurable := fun _ => MeasurableSet.empty | |
| dependsOn := by intros; rfl | |
| theorem filtered_integral_mul_le (old extra : Finset D) (a : D → A) | |
| (f z : (D → A) → ℝ) (C Bf Bz : ℝ) (hC : 0 ≤ C) | |
| (hfn : ∀ ω, 0 ≤ f ω) (hzn : ∀ ω, 0 ≤ z ω) | |
| (hfb : ∀ ω, f ω ≤ Bf) (hzb : ∀ ω, z ω ≤ Bz) | |
| (hdepends : ∀ ω ω', (∀ d ∈ old ∪ extra, ω d = ω' d) → f ω = f ω') | |
| (hbound : ∀ y : D → A, | |
| (∫ ω, z ω ∂((PMF.uniformOfFintype (D → A)).filter | |
| {ω | ∀ d, d ∈ old ∪ extra → ω d = y d} | |
| (fixed_supported (fun d => d ∈ old ∪ extra) y)).toMeasure) ≤ C) : | |
| (∫ ω, f ω * z ω ∂((PMF.uniformOfFintype (D → A)).filter | |
| {ω | ∀ d, d ∈ old → ω d = a d} | |
| (fixed_supported (fun d => d ∈ old) a)).toMeasure) ≤ | |
| C * ∫ ω, f ω ∂((PMF.uniformOfFintype (D → A)).filter | |
| {ω | ∀ d, d ∈ old → ω d = a d} | |
| (fixed_supported (fun d => d ∈ old) a)).toMeasure := by | |
| let P : PatternModel := uniformArrayModel (D := D) (A := A) | |
| have hconvert (J : Finset D) (y : D → A) (G : (D → A) → ℝ) : | |
| P.fixedExpectation J y G = | |
| ∫ ω, G ω ∂((PMF.uniformOfFintype (D → A)).filter | |
| {ω | ∀ d, d ∈ J → ω d = y d} | |
| (fixed_supported (fun d => d ∈ J) y)).toMeasure := by | |
| rw [← integral_fixArray_eq_filtered (fun d => d ∈ J) y G] | |
| apply integral_congr_ae | |
| filter_upwards [] with ω | |
| congr 1 | |
| funext d | |
| dsimp [P, uniformArrayModel, PatternModel.fixCoordinates, fixArray] | |
| exact @ite_cond_congr A (d ∈ J) (d ∈ J) | |
| (Finset.decidableMem d J) (Classical.propDecidable _) (y d) (ω d) rfl | |
| rw [← hconvert old a (fun ω => f ω * z ω), ← hconvert old a f] | |
| apply P.fixedExpectation_mul_le_of_fixedBounds old extra a f z C Bf Bz hC | |
| (show Measurable f from measurable_of_countable f) | |
| (show Measurable z from measurable_of_countable z) hfn hzn hfb hzb hdepends | |
| intro y | |
| exact (hconvert (old ∪ extra) y z).trans_le (hbound y) | |
| theorem integral_eq_of_eq_on_support {Ω : Type*} [Fintype Ω] | |
| [MeasurableSpace Ω] [MeasurableSingletonClass Ω] | |
| (p : PMF Ω) (f g : Ω → ℝ) (h : ∀ ω ∈ p.support, f ω = g ω) : | |
| (∫ ω, f ω ∂p.toMeasure) = ∫ ω, g ω ∂p.toMeasure := by | |
| simp only [PMF.integral_eq_sum, smul_eq_mul] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| by_cases hω : ω ∈ p.support | |
| · rw [h ω hω] | |
| · have hp : p ω = 0 := by simpa [PMF.mem_support_iff] using hω | |
| simp [hp] | |
| end | |
| end Balanced.ConditionalRestriction | |
| end OAI | |