| import OAI.Probability.MatroidProphet.BranchMix | |
| namespace OAI | |
| namespace MatroidProphet | |
| open MeasureTheory Finset | |
| noncomputable def maximumRounded {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) : ℝ := by | |
| classical | |
| exact Finset.univ.sup' Finset.univ_nonempty (fun o : Option (Fin n) => | |
| match o with | |
| | none => 0 | |
| | some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) | |
| lemma maximumRounded_nonneg {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) : | |
| 0 ≤ maximumRounded M w := by | |
| classical | |
| exact Finset.le_sup' (s := Finset.univ) | |
| (f := fun o : Option (Fin n) => match o with | |
| | none => 0 | |
| | some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) | |
| (Finset.mem_univ none) | |
| lemma roundedWeight_le_maximumRounded {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) | |
| (e : Fin n) (he : M.Indep ({e} : Set (Fin n))) : | |
| roundedWeight weightBase (w e) ≤ maximumRounded M w := by | |
| classical | |
| have h := Finset.le_sup' (s := Finset.univ) | |
| (f := fun o : Option (Fin n) => match o with | |
| | none => 0 | |
| | some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) | |
| (Finset.mem_univ (some e)) | |
| change (if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) ≤ | |
| maximumRounded M w at h | |
| rwa [ite_eq_left he] at h | |
| noncomputable def maximumPayoff {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) : ℝ := | |
| ∫ r, hiddenWorstReward (roundedMaximumHidden M (seedSet (bits := n))) w r | |
| ∂bernoulliSeedLaw (fun _ => (1 / 2 : ℝ)) (fun _ => by norm_num) (fun _ => by norm_num) | |
| lemma maximumPayoff_nonneg {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e) : | |
| 0 ≤ maximumPayoff M w := | |
| integral_nonneg (fun r => hiddenWorstReward_nonneg _ w hw r) | |
| lemma roundedWeight_le_four_maximumPayoff {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) | |
| (hw : ∀ e, 0 ≤ w e) (e : Fin n) (he : M.Indep ({e} : Set (Fin n))) : | |
| roundedWeight weightBase (w e) ≤ 4 * maximumPayoff M w := by | |
| classical | |
| let a : Fin n → WithBot ℤ := fun f => Priority.asLevel (roundedLevel weightBase (w f)) | |
| by_cases hea : a e = ⊥ | |
| · have hu : roundedWeight weightBase (w e) = 0 := by | |
| change levelWeight weightBase (a e) = 0 | |
| rw [hea] | |
| rfl | |
| rw [hu] | |
| exact mul_nonneg (by norm_num) (maximumPayoff_nonneg M w hw) | |
| · let P : Finset (Fin n) := Finset.univ.filter fun f => M.Indep ({f} : Set (Fin n)) ∧ a f ≠ ⊥ | |
| have heP : e ∈ P := Finset.mem_filter.mpr ⟨Finset.mem_univ e, he, hea⟩ | |
| obtain ⟨g, hgP, htop⟩ := Maximum.exists_top a P ⟨e, heP⟩ | |
| have hg := (Finset.mem_filter.mp hgP).2 | |
| have hwin : w g / 4 ≤ maximumPayoff M w := | |
| Maximum.maximum_integral_top M (fun x => Priority.asLevel (roundedLevel weightBase x)) | |
| ((measurable_of_countable Priority.asLevel).comp (measurable_roundedLevel weightBase)) | |
| (fun x => x ≠ ⊥) w hw g hg.1 hg.2 | |
| (fun f hf haf hfg => htop f (Finset.mem_filter.mpr ⟨Finset.mem_univ f, hf, haf⟩) hfg) | |
| have hkey : a e ≤ a g := by | |
| by_cases heg : e = g | |
| · exact heg ▸ le_rfl | |
| · rcases htop e heP heg with hlt | ⟨heq, _⟩ | |
| · exact hlt.le | |
| · exact heq.symm.le | |
| have hu : roundedWeight weightBase (w e) ≤ roundedWeight weightBase (w g) := | |
| levelWeight_mono (by norm_num [weightBase]) hkey | |
| have hraw := roundedWeight_le (by norm_num [weightBase] : 1 < weightBase) (hw g) | |
| nlinarith | |
| theorem maximum_branch_guarantee {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) | |
| (hw : ∀ e, 0 ≤ w e) : maximumRounded M w / 4 ≤ maximumPayoff M w := by | |
| classical | |
| have hnonneg := maximumPayoff_nonneg M w hw | |
| have hfour : maximumRounded M w ≤ 4 * maximumPayoff M w := by | |
| apply Finset.sup'_le | |
| intro o _ | |
| cases o with | |
| | none => exact mul_nonneg (by norm_num) hnonneg | |
| | some e => | |
| change (if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) ≤ _ | |
| split_ifs with he | |
| · exact roundedWeight_le_four_maximumPayoff M w hw e he | |
| · exact mul_nonneg (by norm_num) hnonneg | |
| nlinarith | |
| end MatroidProphet | |
| end OAI | |