| import OAI.Probability.MatroidProphet.Reverse.Hybrid |
| import OAI.Probability.MatroidProphet.RevealComposition |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset |
| variable {α : Type*} [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
|
|
| omit [DecidableEq α] in |
| lemma reversePrefix_flat (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (j : ℕ) : |
| M.IsFlat (reversePrefix M hE κ D C S k j) := by |
| cases j with |
| | zero => |
| by_cases hk : k < 0 |
| · simpa only [reversePrefix, ite_eq_left hk] using Matroid.isFlat_closure (M := M) ∅ |
| · simpa only [reversePrefix, ite_eq_right hk, ← hE] using M.ground_isFlat |
| | succ j => exact Matroid.isFlat_closure (M := M) _ |
|
|
| omit [DecidableEq α] in |
| lemma reverseNominal_flat (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C S : ℕ → Set α) (k : ℤ) (j : ℕ) : |
| M.IsFlat (reverseNominal M hE κ D C S k j) := |
| densityEnabled_flat M hE κ (D j) (activation j) k (reversePrefix_flat M hE κ D C S k j) |
|
|
| omit [DecidableEq α] in |
|
|
| lemma reversePrefix_into_output (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C B S : ℕ → Set α) {l k : ℤ} (hlk : l ≤ k) (j : ℕ) : |
| reversePrefix M hE κ D B (fun i => reverseNominal M hE κ D C S k i) l j ⊆ |
| reversePrefix M hE κ D C S k j := by |
| induction j with |
| | zero => exact guardedPath_mono M hE κ D C 0 hlk |
| | succ j ih => |
| have hn := densityEnabled_mono M hE κ (D j) (activation j) hlk ih |
| have hs : reversePrefix M hE κ D B (fun i => reverseNominal M hE κ D C S k i) l (j+1) ⊆ |
| reverseNominal M hE κ D C S k j := by |
| exact (M.closure_subset_closure (Set.union_subset hn Set.inter_subset_right)).trans_eq |
| (reverseNominal_flat M hE κ D C S k j).closure |
| exact hs.trans (Set.subset_union_left.trans (M.subset_closure _ (by simp [hE]))) |
|
|
| omit [DecidableEq α] in |
| lemma reverseNominal_into_output (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C B S : ℕ → Set α) {l k : ℤ} (hlk : l ≤ k) (j : ℕ) : |
| reverseNominal M hE κ D B (fun i => reverseNominal M hE κ D C S k i) l j ⊆ |
| reverseNominal M hE κ D C S k j := |
| densityEnabled_mono M hE κ (D j) (activation j) hlk |
| (reversePrefix_into_output M hE κ D C B S hlk j) |
|
|
| def ReverseClosed (M : Matroid α) (hE : M.E = Set.univ) (κ : ℕ) |
| (D : ℕ → Set α) (k : ℤ) (S : ℕ → Set α) : Prop := |
| ∀ C j, reverseNominal M hE κ D C S k j ⊆ S j |
|
|
| omit [DecidableEq α] in |
| lemma reverseClosed_univ (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (k : ℤ) : |
| ReverseClosed M hE κ D k (fun _ => Set.univ) := fun _ _ => Set.subset_univ _ |
|
|
| noncomputable def reverseStepTree (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) : |
| RevealTree α (ℕ → Set α) := |
| (sourceReverseTree M hE κ k D S G n).record.map |
| (fun r j => reverseNominal M hE κ D (fun i => (r.positive : Set α) ∩ (G i : Set α)) S k j) |
|
|
| lemma reverseStepTree_run (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (C : Finset α) : |
| (reverseStepTree M hE κ k D S G n).run C = |
| (fun j => reverseNominal M hE κ D |
| (fun i => ((C ∩ (sourceReverseTree M hE κ k D S G n).queried C : Finset α) : Set α) ∩ (G i : Set α)) S k j) := by |
| rw [reverseStepTree, RevealTree.run_map, RevealTree.record_run] |
|
|
| lemma reverseStepTree_run_coordinate (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) (C : Finset α) {j : ℕ} (hj : j ≤ n) : |
| (reverseStepTree M hE κ k D S G n).run C j = |
| reverseNominal M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) S k j := by |
| rw [reverseStepTree_run] |
| exact sourceReverseTree_positive_nominal M hE κ k D S G n hG C hj |
|
|
| lemma reverseStepTree_fresh (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (V : Finset α) (hV : ∀ j < n, (G j).filter (fun e => e ∈ S j) ⊆ V) : |
| (reverseStepTree M hE κ k D S G n).Fresh V := by |
| rw [reverseStepTree, RevealTree.fresh_map, RevealTree.record_fresh] |
| exact sourceReverseTree_fresh M hE κ k D S G n hG V hV |
|
|
| lemma reverseStepTree_queried (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (C : Finset α) : |
| (reverseStepTree M hE κ k D S G n).queried C = |
| (sourceReverseTree M hE κ k D S G n).queried C := by |
| rw [reverseStepTree, RevealTree.queried_map, RevealTree.record_queried] |
|
|
| lemma reverseStepTree_closed (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) (C : Finset α) : |
| ReverseClosed M hE κ D (k-1) ((reverseStepTree M hE κ k D S G n).run C) := by |
| rw [reverseStepTree_run] |
| intro B j |
| exact reverseNominal_into_output M hE κ D _ B S (by omega) j |
|
|
| lemma reverseStepTree_remaining (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (k : ℤ) (D S : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (V : Finset α) (hV : ∀ j < n, (G j).filter (fun e => e ∈ S j) ⊆ V) |
| (closed : ReverseClosed M hE κ D k S) (C : Finset α) : |
| ∀ j < n, (G j).filter (fun e => e ∈ (reverseStepTree M hE κ k D S G n).run C j) ⊆ |
| V \ (reverseStepTree M hE κ k D S G n).queried C := by |
| intro j hj e he |
| rcases mem_filter.mp he with ⟨heG, heN⟩ |
| rw [reverseStepTree_run_coordinate M hE κ k D S G n hG C hj.le] at heN |
| refine mem_sdiff.mpr ⟨hV j hj (mem_filter.mpr ⟨heG, closed _ j heN⟩), ?_⟩ |
| rw [reverseStepTree_queried, |
| sourceReverseTree_queried_coordinate_full M hE κ k D S G n hG C hj heG] |
| exact fun h => h.2 heN |
|
|
| noncomputable def reverseTransactions (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (stop : ℤ → (ℕ → Set α) → Prop) : ℕ → ℤ → (ℕ → Set α) → QueryProgram α |
| | 0, _, _ => .stop |
| | m+1, k, S => if stop k S then .stop else |
| (reverseStepTree M hE κ k D S G n).transaction |
| (fun S' => reverseTransactions M hE κ D G n stop m (k-1) S') |
|
|
| lemma reverseTransactions_fresh (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (hG : Pairwise (fun i j => Disjoint (G i) (G j))) |
| (stop : ℤ → (ℕ → Set α) → Prop) (m : ℕ) (k : ℤ) (S : ℕ → Set α) |
| (V : Finset α) (hV : ∀ j < n, (G j).filter (fun e => e ∈ S j) ⊆ V) |
| (closed : ReverseClosed M hE κ D k S) : |
| (reverseTransactions M hE κ D G n stop m k S).Fresh V := by |
| induction m generalizing k S V with |
| | zero => trivial |
| | succ m ih => |
| rw [reverseTransactions] |
| split_ifs with hs |
| · trivial |
| · apply RevealTree.transaction_fresh _ _ V (reverseStepTree_fresh M hE κ k D S G n hG V hV) |
| intro C hC |
| exact ih (k-1) _ _ (reverseStepTree_remaining M hE κ k D S G n hG V hV closed C) |
| (reverseStepTree_closed M hE κ k D S G n C) |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
