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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.Trace
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
attribute [local instance] Classical.propDecidable
noncomputable def reversePrefixTree (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) :
ℕ → ℤ → (ℕ → Set α) → RevealTree α (ℕ → Set α)
| 0, _, S => .leaf S
| m+1, k, S => (reverseStepTree M hE κ k D S G n).bind
(fun S' => reversePrefixTree M hE κ D G n m (k-1) S')
lemma reversePrefixTree_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)))
(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) :
(reversePrefixTree M hE κ D G n m k S).Fresh V := by
induction m generalizing k S V with
| zero => trivial
| succ m ih =>
apply RevealTree.fresh_bind _ _ 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)
lemma reversePrefixTree_remaining (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)))
(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) (C : Finset α) :
∀ j < n, (G j).filter (fun e => e ∈ (reversePrefixTree M hE κ D G n m k S).run C j) ⊆
V \ (reversePrefixTree M hE κ D G n m k S).queried C := by
induction m generalizing k S V C with
| zero => simpa only [reversePrefixTree, RevealTree.run, RevealTree.queried, sdiff_empty] using hV
| succ m ih =>
rw [reversePrefixTree, RevealTree.run_bind, RevealTree.queried_bind]
have hr := 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)
(C \ (reverseStepTree M hE κ k D S G n).queried C)
intro j hj e he
have hx := mem_sdiff.mp (hr j hj he)
have hy := mem_sdiff.mp hx.1
exact mem_sdiff.mpr ⟨hy.1, fun hm => (mem_union.mp hm).elim hy.2 hx.2⟩
lemma reversePrefixTree_run (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)))
(m : ℕ) (k : ℤ) (S : ℕ → Set α) (closed : ReverseClosed M hE κ D k S) (C : Finset α) :
(reversePrefixTree M hE κ D G n m k S).run C = reverseFinal M hE κ D G n m k S C := by
induction m generalizing k S C with
| zero => rfl
| succ m ih =>
rw [reversePrefixTree, RevealTree.run_bind, reverseFinal]
let W := univ \ (reverseStepTree M hE κ k D S G n).queried C
have hW := reverseStepTree_remaining M hE κ k D S G n hG univ
(fun _ _ => subset_univ _) closed C
have hc := reverseStepTree_closed M hE κ k D S G n C
have hcongr := (reversePrefixTree M hE κ D G n m (k-1)
((reverseStepTree M hE κ k D S G n).run C)).run_congr W
(C \ (reverseStepTree M hE κ k D S G n).queried C) C
(reversePrefixTree_fresh M hE κ D G n hG m (k-1) _ W hW hc)
(fun e he => by
have hne := (mem_sdiff.mp he).2
simp only [mem_sdiff, hne, not_false_eq_true, and_true])
rw [hcongr, ih (k-1) _ hc C]
theorem reversePrefixTree_tower (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)))
(m : ℕ) (k : ℤ) (S : ℕ → Set α) (closed : ReverseClosed M hE κ D k S) (q : α → ℝ)
(f : (ℕ → Set α) → Finset α → ℝ)
(hf : ∀ R C C', (∀ j < n, ∀ e ∈ G j, e ∈ R j → (e ∈ C ↔ e ∈ C')) → f R C = f R C') :
bitsExpectation q univ (fun C => f ((reversePrefixTree M hE κ D G n m k S).run C) C) =
bitsExpectation q univ (fun C => bitsExpectation q univ
(f ((reversePrefixTree M hE κ D G n m k S).run C))) := by
let tree := reversePrefixTree M hE κ D G n m k S
have hrem := reversePrefixTree_remaining M hE κ D G n hG m k S univ
(fun _ _ => subset_univ _) closed
have he : bitsExpectation q univ (fun C => f (tree.run C) C) =
bitsExpectation q univ (fun C => f (tree.run C) (C \ tree.queried C)) := by
apply bitsExpectation_congr
intro C hC
apply hf
intro j hj e heG heR
have hx := (mem_sdiff.mp (hrem C j hj (mem_filter.mpr ⟨heG, heR⟩))).2
change e ∉ tree.queried C at hx
simp only [mem_sdiff, hx, not_false_eq_true, and_true]
rw [he, tree.continuation_expectation q univ
(reversePrefixTree_fresh M hE κ D G n hG m k S univ (fun _ _ => subset_univ _) closed)]
apply bitsExpectation_congr
intro C hC
symm
apply bitsExpectation_restrict q univ _ (subset_univ _) (f (tree.run C))
intro T T' hTT'
apply hf
intro j hj e heG heR
exact hTT' e (hrem C j hj (mem_filter.mpr ⟨heG, heR⟩))
lemma reverseNominal_mask_congr (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D S : ℕ → Set α) (G : ℕ → Finset α) (k : ℤ) (n : ℕ)
(C C' : Finset α)
(hC : ∀ j < n, ∀ e ∈ G j, e ∈ S j → (e ∈ C ↔ e ∈ C')) :
reverseNominal M hE κ D (fun j => (C : Set α) ∩ (G j : Set α)) S k n =
reverseNominal M hE κ D (fun j => (C' : Set α) ∩ (G j : Set α)) S k n := by
unfold reverseNominal
congr 1
apply reversePrefix_congr_guard_inter
intro j hj
ext e
by_cases heG : e ∈ G j
· by_cases heS : e ∈ S j
· simp only [Set.mem_inter_iff, mem_coe, heG, heS, and_true, hC j hj e heG heS]
· simp [heS]
· simp [heG]
lemma reverseHybrid_mask_congr (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D S : ℕ → Set α) (G : ℕ → Finset α) (k : ℤ) (n : ℕ)
(C C' T : Finset α)
(hC : ∀ j < n, ∀ e ∈ G j, e ∈ S j → (e ∈ C ↔ e ∈ C')) :
reverseNominal M hE κ D (transitionHybrid M hE κ k D S G C T) S k n =
reverseNominal M hE κ D (transitionHybrid M hE κ k D S G C' T) S k n := by
unfold reverseNominal
congr 1
apply reversePrefix_congr_guard_inter
intro j hj
have hn := reverseNominal_mask_congr M hE κ D S G k j C C'
(fun i hi => hC i (by omega))
ext e
by_cases heG : e ∈ G j
· by_cases heS : e ∈ S j
· simp only [transitionHybrid, hn, Set.mem_inter_iff, Set.mem_union, Set.mem_sdiff,
mem_coe, heG, heS, true_and, and_true, hC j hj e heG heS]
· simp [heS]
· simp [transitionHybrid, heG]
end MatroidProphet
end OAI
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