openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteConditionalRatios.lean
| import OAI.Combinatorics.BalancedRyser.Probability.FiniteUniformFibers | |
| namespace OAI | |
| namespace Balanced.AlignedListProbability | |
| noncomputable section | |
| attribute [local instance] Classical.propDecidable | |
| theorem filter_outer_ratio {Ω : Type*} (p : PMF Ω) (C E : Set Ω) | |
| (h : ∃ x ∈ C, x ∈ p.support) : | |
| (p.filter C h).toOuterMeasure E = | |
| p.toOuterMeasure (C ∩ E) / p.toOuterMeasure C := by | |
| simp only [PMF.toOuterMeasure_apply, div_eq_mul_inv] | |
| rw [← ENNReal.tsum_mul_right] | |
| apply tsum_congr | |
| intro x | |
| by_cases hc : x ∈ C <;> by_cases he : x ∈ E <;> | |
| simp [Set.indicator, hc, he] | |
| theorem supported_real_ne_zero {Ω : Type*} (p : PMF Ω) (C : Set Ω) | |
| (h : ∃ x ∈ C, x ∈ p.support) : (p.toOuterMeasure C).toReal ≠ 0 := by | |
| apply ENNReal.toReal_ne_zero.mpr | |
| constructor | |
| · intro hz | |
| have hd := (p.toOuterMeasure_apply_eq_zero_iff C).mp hz | |
| obtain ⟨x, hx, hs⟩ := h | |
| exact Set.disjoint_left.mp hd hs hx | |
| · rw [PMF.toOuterMeasure_apply] | |
| exact p.tsum_coe_indicator_ne_top C | |
| theorem marginal_conditional_ratio {Ω Y : Type*} (p : PMF Ω) (q : PMF Y) | |
| (C : Set Ω) (hC : ∃ x ∈ C, x ∈ p.support) (f : Ω → Y) | |
| (hm : (p.filter C hC).map f = q) (B E : Set Y) : | |
| (p.toOuterMeasure (C ∩ f ⁻¹' (B ∩ E))).toReal / | |
| (p.toOuterMeasure (C ∩ f ⁻¹' B)).toReal = | |
| (q.toOuterMeasure (B ∩ E)).toReal / (q.toOuterMeasure B).toReal := by | |
| have ht (K : Set Y) : | |
| (p.toOuterMeasure (C ∩ f ⁻¹' K)).toReal / (p.toOuterMeasure C).toReal = | |
| (q.toOuterMeasure K).toReal := by | |
| have h := congrArg (fun law => (law.toOuterMeasure K).toReal) hm | |
| rw [PMF.toOuterMeasure_map_apply, filter_outer_ratio, ENNReal.toReal_div] at h | |
| exact h | |
| calc | |
| _ = ((p.toOuterMeasure (C ∩ f ⁻¹' (B ∩ E))).toReal / | |
| (p.toOuterMeasure C).toReal) / | |
| ((p.toOuterMeasure (C ∩ f ⁻¹' B)).toReal / | |
| (p.toOuterMeasure C).toReal) := | |
| (div_div_div_cancel_right₀ (supported_real_ne_zero p C hC) _ _).symm | |
| _ = _ := by rw [ht, ht] | |
| end | |
| end Balanced.AlignedListProbability | |
| end OAI | |