openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /ConditionalMultiplicity2.lean
| 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 | |