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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | 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
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