| import OAI.Probability.MatroidProphet.Density.Guarded |
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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 reversePrefix (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) : ℕ → Set α |
| | 0 => if k < 0 then M.closure ∅ else univ |
| | h+1 => M.closure (densityEnabled M hE κ (D h) (activation h) k |
| (reversePrefix M hE κ D C S k h) ∪ (C h ∩ S h)) |
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| noncomputable def reverseNominal (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : Set α := |
| densityEnabled M hE κ (D h) (activation h) k (reversePrefix M hE κ D C S k h) |
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| lemma reversePrefix_succ (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reversePrefix M hE κ D C S k (h+1) = |
| M.closure (reverseNominal M hE κ D C S k h ∪ (C h ∩ S h)) := rfl |
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| lemma reversePrefix_agrees (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reversePrefix M hE κ D C (fun j => nominalPath M hE κ D C j (k+1)) k h = |
| guardedPath M hE κ D C h k := by |
| induction h with |
| | zero => rfl |
| | succ h ih => |
| rw [reversePrefix, ih] |
| rfl |
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| lemma reverseNominal_agrees (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reverseNominal M hE κ D C (fun j => nominalPath M hE κ D C j (k+1)) k h = |
| nominalPath M hE κ D C h k := by |
| rw [reverseNominal, reversePrefix_agrees] |
| rfl |
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| lemma reversePrefix_mono (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' S S' : ℕ → Set α) {k k' : ℤ} (hkk' : k ≤ k') |
| (hC : ∀ j, C j ⊆ C' j) (hS : ∀ j, S j ⊆ S' j) (h : ℕ) : |
| reversePrefix M hE κ D C S k h ⊆ reversePrefix M hE κ D C' S' k' h := by |
| induction h with |
| | zero => exact guardedPath_mono M hE κ D C 0 hkk' |
| | succ h ih => |
| exact M.closure_subset_closure |
| (union_subset_union (densityEnabled_mono M hE κ (D h) (activation h) hkk' ih) |
| (inter_subset_inter (hC h) (hS h))) |
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| lemma reverseNominal_mono (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' S S' : ℕ → Set α) {k k' : ℤ} (hkk' : k ≤ k') |
| (hC : ∀ j, C j ⊆ C' j) (hS : ∀ j, S j ⊆ S' j) (h : ℕ) : |
| reverseNominal M hE κ D C S k h ⊆ reverseNominal M hE κ D C' S' k' h := |
| densityEnabled_mono M hE κ (D h) (activation h) hkk' |
| (reversePrefix_mono M hE κ D C C' S S' hkk' hC hS h) |
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| lemma reversePrefix_congr_before (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' S S' : ℕ → Set α) (k : ℤ) (h : ℕ) |
| (hC : ∀ j < h, C j = C' j) (hS : ∀ j < h, S j = S' j) : |
| reversePrefix M hE κ D C S k h = reversePrefix M hE κ D C' S' k h := by |
| induction h with |
| | zero => rfl |
| | succ h ih => |
| rw [reversePrefix, reversePrefix, ih (fun j hj => hC j (by omega)) |
| (fun j hj => hS j (by omega)), hC h (by omega), hS h (by omega)] |
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| lemma reverseNominal_congr_before (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' S S' : ℕ → Set α) (k : ℤ) (h : ℕ) |
| (hC : ∀ j < h, C j = C' j) (hS : ∀ j < h, S j = S' j) : |
| reverseNominal M hE κ D C S k h = reverseNominal M hE κ D C' S' k h := by |
| rw [reverseNominal, reverseNominal, reversePrefix_congr_before M hE κ D C C' S S' k h hC hS] |
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| lemma reversePrefix_subset_later (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reversePrefix M hE κ D C' (fun j => nominalPath M hE κ D C j (k+1)) k h ⊆ |
| guardedPath M hE κ D C h (k+1) := by |
| induction h with |
| | zero => exact guardedPath_mono M hE κ D C 0 (by omega) |
| | succ h ih => |
| have hn := densityEnabled_mono M hE κ (D h) (activation h) (by omega : k ≤ k+1) ih |
| have hs : reversePrefix M hE κ D C' (fun j => nominalPath M hE κ D C j (k+1)) k (h+1) ⊆ |
| nominalPath M hE κ D C h (k+1) := by |
| exact (M.closure_subset_closure (union_subset hn inter_subset_right)).trans_eq |
| (nominalPath_flat M hE κ D C h (k+1)).closure |
| exact hs.trans (nominal_subset_guarded M hE κ D C h (k+1)) |
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| lemma reverseNominal_subset_later (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C C' : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reverseNominal M hE κ D C' (fun j => nominalPath M hE κ D C j (k+1)) k h ⊆ |
| nominalPath M hE κ D C h (k+1) := |
| densityEnabled_mono M hE κ (D h) (activation h) (by omega) |
| (reversePrefix_subset_later M hE κ D C C' k h) |
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| lemma reversePrefix_subset_succ (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reversePrefix M hE κ D C S k h ⊆ reversePrefix M hE κ D C S k (h+1) := |
| (densityEnabled_extensive M hE κ (D h) (activation h) k _).trans |
| (subset_union_left.trans (M.subset_closure _ (by simp [hE]))) |
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| omit [Fintype α] in |
| lemma union_inter_sdiff (N C S : Set α) : N ∪ (C ∩ (S \ N)) = N ∪ (C ∩ S) := by |
| ext e |
| change (e ∈ N ∨ (e ∈ C ∧ e ∈ S ∧ e ∉ N)) ↔ (e ∈ N ∨ (e ∈ C ∧ e ∈ S)) |
| tauto |
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| noncomputable def departingGuards (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : Set α := |
| C h ∩ (S h \ reverseNominal M hE κ D C S k h) |
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| lemma reversePrefix_departing_only (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reversePrefix M hE κ D (departingGuards M hE κ D C S k) S k h = |
| reversePrefix M hE κ D C S k h := by |
| induction h with |
| | zero => rfl |
| | succ h ih => |
| rw [reversePrefix, reversePrefix, ih] |
| change M.closure (reverseNominal M hE κ D C S k h ∪ |
| ((C h ∩ (S h \ reverseNominal M hE κ D C S k h)) ∩ S h)) = _ |
| have hi : (C h ∩ (S h \ reverseNominal M hE κ D C S k h)) ∩ S h = |
| C h ∩ (S h \ reverseNominal M hE κ D C S k h) := inter_eq_self_of_subset_left (fun _ he => he.2.1) |
| rw [hi, union_inter_sdiff] |
| rfl |
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| lemma reverseNominal_departing_only (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (h : ℕ) : |
| reverseNominal M hE κ D (departingGuards M hE κ D C S k) S k h = |
| reverseNominal M hE κ D C S k h := by |
| rw [reverseNominal, reverseNominal, reversePrefix_departing_only] |
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| noncomputable def departureBatch (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C G : ℕ → Set α) (h : ℕ) (k : ℤ) : Set α := |
| G h ∩ (nominalPath M hE κ D C h (k+1) \ nominalPath M hE κ D C h k) |
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| lemma departureBatch_disjoint_time (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C G : ℕ → Set α) (h : ℕ) {k l : ℤ} (hkl : k ≠ l) : |
| Disjoint (departureBatch M hE κ D C G h k) (departureBatch M hE κ D C G h l) := by |
| apply Set.disjoint_left.mpr |
| intro e he hf |
| rcases lt_or_gt_of_ne hkl with hlt | hgt |
| · exact hf.2.2 (nominalPath_mono M hE κ D C h (by omega : k+1 ≤ l) he.2.1) |
| · exact he.2.2 (nominalPath_mono M hE κ D C h (by omega : l+1 ≤ k) hf.2.1) |
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| lemma departureBatch_disjoint_group (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C G : ℕ → Set α) {h j : ℕ} (hG : Disjoint (G h) (G j)) (k l : ℤ) : |
| Disjoint (departureBatch M hE κ D C G h k) (departureBatch M hE κ D C G j l) := |
| hG.mono inter_subset_left inter_subset_left |
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| noncomputable def hybridGuards (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) (h : ℕ) : Set α := |
| (C h ∩ nominalPath M hE κ D C h (b-1)) ∪ |
| (T h ∩ (nominalPath M hE κ D C h b \ nominalPath M hE κ D C h (b-1))) |
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| noncomputable def lowerCompetition (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) : ℕ → Set α |
| | 0 => ∅ |
| | h+1 => lowerCompetition M hE κ D C T b h ∪ |
| (T h ∩ (nominalPath M hE κ D C h b \ nominalPath M hE κ D C h (b-1))) |
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| lemma hybridPrefix_contains (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) (h : ℕ) : |
| guardedPath M hE κ D C h (b-2) ∪ lowerCompetition M hE κ D C T b h ⊆ |
| reversePrefix M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) h := by |
| induction h with |
| | zero => |
| exact union_subset (guardedPath_mono M hE κ D C 0 (by omega)) (empty_subset _) |
| | succ h ih => |
| let A := reversePrefix M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) h |
| let N := reverseNominal M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) h |
| have hFn : nominalPath M hE κ D C h (b-2) ⊆ N := |
| densityEnabled_mono M hE κ (D h) (activation h) (by omega) |
| (subset_union_left.trans ih) |
| have hguards : C h ∩ nominalPath M hE κ D C h (b-1) ⊆ |
| hybridGuards M hE κ D C T b h ∩ nominalPath M hE κ D C h b := by |
| intro e he |
| exact ⟨Or.inl he, nominalPath_mono M hE κ D C h (by omega) he.2⟩ |
| have hF : guardedPath M hE κ D C (h+1) (b-2) ⊆ |
| reversePrefix M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) (h+1) := by |
| rw [guardedPath_succ, reversePrefix_succ] |
| have ht : b-2+1 = b-1 := by omega |
| rw [ht] |
| exact M.closure_subset_closure (union_subset_union hFn hguards) |
| apply union_subset hF |
| change lowerCompetition M hE κ D C T b h ∪ _ ⊆ _ |
| apply union_subset |
| · exact (subset_union_right.trans ih).trans |
| (reversePrefix_subset_succ M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) h) |
| · intro e he |
| apply M.subset_closure _ (by simp [hE]) |
| exact Or.inr ⟨Or.inr he, he.2.1⟩ |
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| lemma hybrid_containment (M : Matroid α) (hE : M.E = univ) |
| (κ : ℕ) (D C T : ℕ → Set α) (b : ℤ) (h : ℕ) (henabled : activation h ≤ b-1) : |
| densityExpansion M hE κ (D h) |
| (guardedPath M hE κ D C h (b-2) ∪ lowerCompetition M hE κ D C T b h) ⊆ |
| reverseNominal M hE κ D (hybridGuards M hE κ D C T b) |
| (fun j => nominalPath M hE κ D C j b) (b-1) h := by |
| rw [reverseNominal, densityEnabled, ite_eq_left henabled] |
| exact densityExpansion_mono M hE κ (D h) (hybridPrefix_contains M hE κ D C T b h) |
|
|
| end MatroidProphet |
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| end OAI |
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