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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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
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