| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| This is the displayed `eq:log-resource` with its conditional product law written |
| as the explicit two-outcome Bernoulli expectation. The finite-tree execution |
| proof justifies applying this identity at every predictably selected fresh query. |
| -/ |
| |
| namespace MatroidProphet |
| |
| open Finset |
| |
| /-- The source's expected one-query decrement, exponential expression, and |
| logarithmic bound, for the actual finite-product failure probabilities. -/ |
| theorem transaction_log_resource |
| {α : Type*} [DecidableEq α] |
| (cert : Finset α → Prop) [DecidablePred cert] (hcert : Monotone cert) |
| (q : α → ℝ) (hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) |
| (V I : Finset α) (e : α) (he : e ∈ V) |
| (hp : 0 < bitsFailure cert q V ∅) : |
| let p := bitsFailure cert q V ∅ |
| let p' := bitsFailure cert q (V.erase e) ∅ |
| let Z := bitsFailure cert q V I / p |
| let next := (1 - q e) * (bitsFailure cert q (V.erase e) I / p') + |
| q e * (bitsFailure cert q (V.erase e) (insert e I) / p') |
| Z - next = Z * (1 - p / p') ∧ |
| Z - next = Z * (1 - Real.exp (-(Real.log p' - Real.log p))) ∧ |
| Z - next ≤ Real.log p' - Real.log p := by |
| dsimp only |
| have hpp' : bitsFailure cert q V ∅ ≤ bitsFailure cert q (V.erase e) ∅ := |
| bitsFailure_antitone cert hcert q hq0 hq1 ∅ (erase_subset e V) |
| have hp' : 0 < bitsFailure cert q (V.erase e) ∅ := hp.trans_le hpp' |
| have hmean : |
| (1 - q e) * (bitsFailure cert q (V.erase e) I / |
| bitsFailure cert q (V.erase e) ∅) + |
| q e * (bitsFailure cert q (V.erase e) (insert e I) / |
| bitsFailure cert q (V.erase e) ∅) = |
| bitsFailure cert q V I / bitsFailure cert q (V.erase e) ∅ := by |
| rw [bitsFailure_split cert q V I e he] |
| ring |
| rw [hmean] |
| have hid : bitsFailure cert q V I / bitsFailure cert q V ∅ - |
| bitsFailure cert q V I / bitsFailure cert q (V.erase e) ∅ = |
| bitsFailure cert q V I / bitsFailure cert q V ∅ * |
| (1 - bitsFailure cert q V ∅ / bitsFailure cert q (V.erase e) ∅) := by |
| field_simp |
| refine ⟨hid, ?_, ?_⟩ |
| · rw [neg_sub, Real.exp_sub, Real.exp_log hp, Real.exp_log hp'] |
| exact hid |
| · exact ratio_resource hp hpp' (bitsFailure_nonneg cert q hq0 hq1 V I) |
| (bitsFailure_forced_antitone cert hcert q hq0 hq1 V (empty_subset I)) |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
