| import OAI.Combinatorics.BalancedRyser.Local.InjectionExposure |
| import OAI.Combinatorics.BalancedRyser.Probability.FiniteUniformConditioning |
|
|
| namespace OAI |
|
|
| namespace Balanced.InjectionExposure |
|
|
| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
| [Nonempty (I ↪ J)] |
|
|
| theorem original_uniform_conditioned (K : Finset I) (base : I ↪ J) |
| (h : ∃ e ∈ (↑(completions K base) : Set (I ↪ J)), |
| e ∈ (PMF.uniformOfFintype (I ↪ J)).support) : |
| (PMF.uniformOfFintype (I ↪ J)).filter (↑(completions K base)) h = |
| PMF.uniformOfFinset (completions K base) (completions_nonempty K base) := by |
| classical |
| let S := completions K base |
| have : Nonempty (↑S : Set (I ↪ J)) := ⟨⟨base, by simp [S]⟩⟩ |
| rw [uniform_filter_eq_subtype_map] |
| ext e |
| by_cases he : e ∈ completions K base |
| · rw [PMF.map_apply, tsum_eq_single (⟨e, he⟩ : (↑S : Set (I ↪ J)))] |
| · simp only [PMF.uniformOfFintype_apply, PMF.uniformOfFinset_apply, |
| he, ite_true] |
| exact congrArg (fun n : ℕ => (n : ENNReal)⁻¹) |
| (Fintype.card_coe (completions K base)) |
| · intro f hf |
| rw [ite_eq_right] |
| intro hef |
| exact hf (Subtype.ext hef.symm) |
| · rw [PMF.uniformOfFinset_apply_of_notMem (completions_nonempty K base) he, |
| PMF.map_apply] |
| apply ENNReal.tsum_eq_zero.mpr |
| intro f |
| rw [ite_eq_right] |
| intro hef |
| exact he (hef ▸ f.property) |
|
|
| theorem original_conditioned_next_event (K : Finset I) (base : I ↪ J) |
| (h : ∃ e ∈ (↑(completions K base) : Set (I ↪ J)), |
| e ∈ (PMF.uniformOfFintype (I ↪ J)).support) |
| (i : I) (hi : i ∉ K) (P : J → Prop) [DecidablePred P] : |
| (((PMF.uniformOfFintype (I ↪ J)).filter (↑(completions K base)) h).toOuterMeasure |
| {e | P (e i)}).toReal = |
| (((remaining K base).filter P).card : ℝ) / ((remaining K base).card : ℝ) := by |
| rw [original_uniform_conditioned K base h] |
| exact next_event_probability K base i hi P |
|
|
| end Balanced.InjectionExposure |
|
|
| end OAI |
|
|
