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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 53 54 55 | import OAI.Combinatorics.BalancedRyser.Probability.ConditionalMultiplicity
namespace OAI
noncomputable section
open scoped Classical
open _root_.Balanced _root_.OAI.Balanced Balanced.CandidateExposure
universe u
variable {F : Type u} [Field F] [Fintype F] [DecidableEq F]
example (M : MergedPairs F) (p : Point F) :
exposedMultiplicity ∅ M p = 0 := by
simp [exposedMultiplicity]
example (M : MergedPairs F) (p : Point F) :
freshMultiplicity Finset.univ M p = 0 := by
let : IsEmpty {d : Direction F // d ∉ (Finset.univ : Finset (Direction F))} :=
⟨fun d => d.2 (Finset.mem_univ _)⟩
simp [freshMultiplicity]
example (M : MergedPairs F) (p : Point F) :
freshMultiplicity ∅ M p = sampledMultiplicity M p := by
have h := sampledMultiplicity_eq_exposed_add_fresh ∅ M p
simpa [exposedMultiplicity] using h.symm
example (a : MergedPairs F) (p : Point F) :
conditionalProbability Finset.univ a
{M | sampledMultiplicity M p = sampledMultiplicity a p} = 1 := by
let : IsEmpty {d : Direction F // d ∉ (Finset.univ : Finset (Direction F))} :=
⟨fun d => d.2 (Finset.mem_univ _)⟩
have hsplit := sampledMultiplicity_eq_exposed_add_fresh Finset.univ a p
have hfixed : exposedMultiplicity Finset.univ a p = sampledMultiplicity a p := by
simpa [freshMultiplicity] using hsplit.symm
simpa [Direction, hfixed] using
sampledMultiplicity_conditional_atom Finset.univ a p 0
example (D : Finset (Direction F)) (a : MergedPairs F) (p : Point F)
(n : ℕ) (hn : n < exposedMultiplicity D a p) :
conditionalProbability D a {M | sampledMultiplicity M p = n} = 0 := by
unfold conditionalProbability
have hz : (exposedLaw D a).toOuterMeasure
{M | sampledMultiplicity M p = n} = 0 := by
apply (PMF.toOuterMeasure_apply_eq_zero_iff _ _).mpr
apply Set.disjoint_left.mpr
intro M hM he
have hsplit := sampledMultiplicity_eq_fixed_add_fresh D a M p hM
change sampledMultiplicity M p = n at he
omega
rw [hz, ENNReal.toReal_zero]
end
end OAI
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