File size: 2,588 Bytes
dd14c26 | 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 56 57 58 59 60 61 | 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
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