| import OAI.Probability.MatroidProphet.Density.Path |
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| namespace OAI |
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| namespace MatroidProphet |
| open Set |
| variable {α : Type*} [Fintype α] |
|
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| noncomputable def densityEnabled (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) (a k : ℤ) (P : Set α) : Set α := |
| if a ≤ k then densityExpansion M hE κ D P else P |
|
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| lemma densityEnabled_extensive (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) (a k : ℤ) (P : Set α) : |
| P ⊆ densityEnabled M hE κ D a k P := by |
| by_cases hak : a ≤ k |
| · simpa only [densityEnabled, ite_eq_left hak] using densityExpansion_extensive M hE κ D P |
| · simp only [densityEnabled, ite_eq_right hak, Subset.rfl] |
|
|
| lemma densityEnabled_flat (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) (a k : ℤ) {P : Set α} (hP : M.IsFlat P) : |
| M.IsFlat (densityEnabled M hE κ D a k P) := by |
| by_cases hak : a ≤ k |
| · simpa only [densityEnabled, ite_eq_left hak] using densityExpansion_flat M hE κ D P |
| · simpa only [densityEnabled, ite_eq_right hak] using hP |
|
|
| lemma densityEnabled_mono (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) (a : ℤ) {k k' : ℤ} {P P' : Set α} |
| (hkk' : k ≤ k') (hPP' : P ⊆ P') : |
| densityEnabled M hE κ D a k P ⊆ densityEnabled M hE κ D a k' P' := by |
| by_cases hak : a ≤ k |
| · have hak' := hak.trans hkk' |
| simpa only [densityEnabled, ite_eq_left hak, ite_eq_left hak'] using |
| densityExpansion_mono M hE κ D hPP' |
| · by_cases hak' : a ≤ k' |
| · simpa only [densityEnabled, ite_eq_right hak, ite_eq_left hak'] using |
| hPP'.trans (densityExpansion_extensive M hE κ D P') |
| · simpa only [densityEnabled, ite_eq_right hak, ite_eq_right hak'] using hPP' |
|
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| lemma densityEnabled_univ (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D : Set α) (a k : ℤ) : densityEnabled M hE κ D a k univ = univ := |
| Set.eq_univ_of_univ_subset (densityEnabled_extensive M hE κ D a k univ) |
|
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| def activation (h : ℕ) : ℤ := -3 * ((h : ℤ) + 1) |
|
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| noncomputable def guardedPath (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) : ℕ → ℤ → Set α |
| | 0, k => if k < 0 then M.closure ∅ else univ |
| | h + 1, k => |
| M.closure (densityEnabled M hE κ (D h) (activation h) k |
| (guardedPath M hE κ D C h k) ∪ |
| (C h ∩ densityEnabled M hE κ (D h) (activation h) (k + 1) |
| (guardedPath M hE κ D C h (k + 1)))) |
|
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| noncomputable def nominalPath (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : Set α := |
| densityEnabled M hE κ (D h) (activation h) k (guardedPath M hE κ D C h k) |
|
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| lemma guardedPath_succ (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| guardedPath M hE κ D C (h+1) k = |
| M.closure (nominalPath M hE κ D C h k ∪ (C h ∩ nominalPath M hE κ D C h (k+1))) := rfl |
|
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| lemma guardedPath_flat (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| M.IsFlat (guardedPath M hE κ D C h k) := by |
| cases h with |
| | zero => |
| by_cases hk : k < 0 |
| · simpa only [guardedPath, ite_eq_left hk] using Matroid.isFlat_closure (M := M) ∅ |
| · simpa only [guardedPath, ite_eq_right hk, ← hE] using M.ground_isFlat |
| | succ h => exact Matroid.isFlat_closure (M := M) _ |
|
|
| lemma nominalPath_flat (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| M.IsFlat (nominalPath M hE κ D C h k) := |
| densityEnabled_flat M hE κ (D h) (activation h) k (guardedPath_flat M hE κ D C h k) |
|
|
| lemma guardedPath_mono (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : Monotone (guardedPath M hE κ D C h) := by |
| induction h with |
| | zero => |
| intro i j hij |
| by_cases hi : i < 0 |
| · by_cases hj : j < 0 |
| · simp only [guardedPath, ite_eq_left hi, ite_eq_left hj, Subset.rfl] |
| · simp only [guardedPath, ite_eq_left hi, ite_eq_right hj, subset_univ] |
| · have hj : ¬ j < 0 := by omega |
| simp only [guardedPath, ite_eq_right hi, ite_eq_right hj, Subset.rfl] |
| | succ h ih => |
| intro i j hij |
| apply M.closure_subset_closure |
| exact union_subset_union (densityEnabled_mono M hE κ (D h) (activation h) hij (ih hij)) |
| (inter_subset_inter_right (C h) |
| (densityEnabled_mono M hE κ (D h) (activation h) (by omega) (ih (by omega)))) |
|
|
| lemma nominalPath_mono (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : Monotone (nominalPath M hE κ D C h) := by |
| intro i j hij |
| exact densityEnabled_mono M hE κ (D h) (activation h) hij |
| (guardedPath_mono M hE κ D C h hij) |
|
|
| lemma guarded_subset_nominal (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| guardedPath M hE κ D C h k ⊆ nominalPath M hE κ D C h k := |
| densityEnabled_extensive M hE κ (D h) (activation h) k _ |
|
|
| lemma nominal_subset_guarded (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| nominalPath M hE κ D C h k ⊆ guardedPath M hE κ D C (h+1) k := |
| subset_union_left.trans (M.subset_closure _ (by simp [hE])) |
|
|
| lemma guarded_subset_next_nominal (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : |
| guardedPath M hE κ D C (h+1) k ⊆ nominalPath M hE κ D C h (k+1) := by |
| have hs : nominalPath M hE κ D C h k ∪ (C h ∩ nominalPath M hE κ D C h (k+1)) ⊆ |
| nominalPath M hE κ D C h (k+1) := |
| union_subset (nominalPath_mono M hE κ D C h (by omega)) inter_subset_right |
| exact (M.closure_subset_closure hs).trans_eq (nominalPath_flat M hE κ D C h (k+1)).closure |
|
|
| lemma guardedPath_nonnegative (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : 0 ≤ k) : |
| guardedPath M hE κ D C h k = univ := by |
| induction h generalizing k with |
| | zero => simp [guardedPath, not_lt_of_ge hk] |
| | succ h ih => |
| rw [guardedPath, ih hk, ih (by omega), densityEnabled_univ, densityEnabled_univ] |
| simp [hE] |
|
|
| lemma nominalPath_nonnegative (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : 0 ≤ k) : |
| nominalPath M hE κ D C h k = univ := by |
| rw [nominalPath, guardedPath_nonnegative M hE κ D C h hk, densityEnabled_univ] |
|
|
| lemma guardedPath_early (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : k < -3 * (h : ℤ) - 1) : |
| guardedPath M hE κ D C h k = M.closure ∅ := by |
| induction h generalizing k with |
| | zero => |
| have hk' : k < 0 := by omega |
| simp [guardedPath, hk'] |
| | succ h ih => |
| have hka : ¬ activation h ≤ k := by dsimp [activation]; push_cast at hk; omega |
| have hk'a : ¬ activation h ≤ k+1 := by dsimp [activation]; push_cast at hk; omega |
| have hkold : k < -3 * (h : ℤ) - 1 := by push_cast at hk; omega |
| have hk'old : k+1 < -3 * (h : ℤ) - 1 := by omega |
| rw [guardedPath, densityEnabled, densityEnabled, ite_eq_right hka, ite_eq_right hk'a, |
| ih hkold, ih hk'old] |
| rw [union_eq_self_of_subset_right inter_subset_right, M.closure_closure] |
|
|
| lemma nominalPath_early (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : k < activation h) : |
| nominalPath M hE κ D C h k = M.closure ∅ := by |
| rw [nominalPath, densityEnabled, ite_eq_right (not_le_of_gt hk)] |
| apply guardedPath_early |
| dsimp [activation] at hk |
| omega |
|
|
| lemma nominalPath_baseline (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : |
| nominalPath M hE κ D C h (activation h) = |
| densityExpansion M hE κ (D h) (M.closure ∅) := by |
| rw [nominalPath, densityEnabled, ite_eq_left le_rfl, guardedPath_early] |
| dsimp [activation] |
| omega |
|
|
| lemma nominalPath_baseline_gap (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : |
| nominalPath M hE κ D C h (activation h + 1) = |
| nominalPath M hE κ D C h (activation h) := by |
| rw [nominalPath_baseline, nominalPath, densityEnabled, ite_eq_left (by omega), guardedPath_early] |
| dsimp [activation] |
| omega |
|
|
| end MatroidProphet |
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| end OAI |
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