openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Sampling /SourceSeedConditionalSupport.lean
| import OAI.Probability.MatroidSecretary.Sampling.SourceSeedSupport | |
| import OAI.Probability.MatroidSecretary.Secretary.Coupling | |
| namespace OAI | |
| /-! | |
| Strict support for the actual finite conditional reconstruction. Every advertised | |
| branch/mask fiber has positive source mass, so the zero-fiber fallback is never | |
| used for these concrete kernels. The conditional law is supported exactly on its | |
| fiber, not on the whole source space; this does not remove the secretary replay's | |
| every-seed mismatch guard. | |
| Source: `sections/secretary.tex`, lines 68--78, SHA256 | |
| `3554fe0f7296782edf63cc7f1305c377e7a7b9d8f8948b9510ea4ad02a702d63`. | |
| -/ | |
| namespace MatroidProphet.SourceSeedSupport | |
| open MeasureTheory Secretary | |
| theorem coupling_fiberMass_pos {n : ℕ} (M : Matroid (Fin n)) | |
| (hE : M.E = Set.univ) (b : SourceObservation n) : | |
| 0 < fiberMass (fun r => (sourceSeedLaw n).real {r}) (couplingObservation M hE) b := by | |
| have hpos : 0 < sourceSeedLaw n {r | couplingObservation M hE r = b} := by | |
| rcases b with ⟨j, P⟩ | |
| simpa only [couplingObservation, Prod.mk.injEq] using | |
| sourceSeedLaw_actual_branch_mask_pos M hE j P | |
| have hr : 0 < ((sourceSeedLaw n).map (couplingObservation M hE)).real {b} := by | |
| rw [measureReal_def, Measure.map_apply (measurable_of_finite _) (measurableSet_singleton _)] | |
| exact ENNReal.toReal_pos hpos.ne' (measure_ne_top _ _) | |
| have hmass : fiberMass (fun r => (sourceSeedLaw n).real {r}) | |
| (couplingObservation M hE) b = | |
| ((sourceSeedLaw n).map (couplingObservation M hE)).real {b} := by | |
| rw [FiniteLaw.map_real_singleton] | |
| unfold fiberMass | |
| apply Finset.sum_congr rfl | |
| intro r _ | |
| by_cases h : couplingObservation M hE r = b <;> simp [h] | |
| rw [hmass] | |
| exact hr | |
| /-- Each complete source seed has positive conditional mass exactly in its | |
| advertised branch/mask fiber, including unused source coordinates. -/ | |
| theorem couplingKernel_singleton_pos_iff {n : ℕ} (M : Matroid (Fin n)) | |
| (hE : M.E = Set.univ) (b : SourceObservation n) (r : Seed (mainSeedBits n)) : | |
| 0 < couplingKernel M hE b {r} ↔ couplingObservation M hE r = b := by | |
| have hb := coupling_fiberMass_pos M hE b | |
| have hr : 0 < (sourceSeedLaw n).real {r} := | |
| ENNReal.toReal_pos (sourceSeedLaw_singleton_pos n r).ne' (measure_ne_top _ _) | |
| rw [couplingKernel, conditionalSourceLaw, FiniteLaw.weightMeasure_singleton, | |
| ENNReal.ofReal_pos] | |
| simp only [fiberKernel, hb.ne', ite_false] | |
| by_cases h : couplingObservation M hE r = b | |
| · simp only [h, ite_true, iff_true] | |
| exact div_pos hr hb | |
| · simp [h] | |
| end MatroidProphet.SourceSeedSupport | |
| end OAI | |