File size: 4,441 Bytes
5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Streams.DictionaryCallback
namespace OAI
/-!
# Provenance of every retained bounded-dictionary key
Capacity failure can freeze a callback, but cannot create an unreported key.
The proof applies even to overflowing complete joins, with no cardinality
hypothesis on a bucket and no knowledge of any solution.
-/
namespace LowSpaceSubsetSum.BinaryTrie
theorem feed_provenance {width : ℕ} (C : ℕ) (r : BuildResult)
(hr : HasWidth r.tree width) (key : List Bool) (hk : key.length = width) :
HasWidth (feed C key r).tree width ∧
keySet (feed C key r).tree ⊆ Insert.insert key (keySet r.tree) := by
by_cases ho : r.overflow = true
· rw [feed_frozen C key r ho]
exact ⟨hr, Finset.subset_insert _ _⟩
have hc : r.overflow = false := Bool.eq_false_iff.mpr ho
by_cases hd : contains r.tree key = true
· simp only [feed, hc, hd, Bool.false_eq_true, ↓reduceIte, tick]
exact ⟨hr, Finset.subset_insert _ _⟩
have hf : contains r.tree key = false := Bool.eq_false_iff.mpr hd
simp only [feed, hc, hf, Bool.false_eq_true, ↓reduceIte]
exact ⟨insert_hasWidth hr hk, by rw [keySet_insert hr hk]⟩
theorem fold_provenance {width : ℕ} (C : ℕ) (keys : List (List Bool)) (r : BuildResult)
(hr : HasWidth r.tree width) (hk : ∀ key ∈ keys, key.length = width) :
keySet (keys.foldl (fun r key => feed C key r) r).tree ⊆ keySet r.tree ∪ keys.toFinset := by
induction keys generalizing r with
| nil => simp
| cons key keys ih =>
have hf := feed_provenance C r hr key (hk key (by simp))
have hi := ih (feed C key r) hf.1 (fun q hq => hk q (by simp [hq]))
intro q hq
rcases Finset.mem_union.mp (hi hq) with h | h
· rcases Finset.mem_insert.mp (hf.2 h) with rfl | h
· simp
· exact Finset.mem_union_left _ h
· exact Finset.mem_union_right _ (by simp [h])
theorem matchFold_provenance {α β : Type*} {C width : ℕ}
(key₁ : α → ℤ) (key₂ : β → ℤ) (encode : α → β → List Bool)
(xs : List α) (ys : List β)
(hwidth : ∀ x ∈ xs, ∀ y ∈ ys, key₁ x = key₂ y → (encode x y).length = width)
(query : List Bool)
(hq : contains (MatchFold.run key₁ key₂ (callback C encode) xs ys
⟨.empty, false, 0⟩).tree query = true) :
∃ x ∈ xs, ∃ y ∈ ys, key₁ x = key₂ y ∧ encode x y = query := by
rw [matchFold_eq_keys] at hq
have hp := fold_provenance C (matchingKeys key₁ key₂ encode xs ys) ⟨.empty, false, 0⟩
(width := width) (by trivial) (by
intro key hk
obtain ⟨x, hx, y, hy, he, rfl⟩ := (mem_matchingKeys _ _ _ _ _ _).mp hk
exact hwidth x hx y hy he)
have hm := hp ((mem_keySet _ _).mpr hq)
simp only [keySet, Finset.empty_union, List.mem_toFinset] at hm
exact (mem_matchingKeys _ _ _ _ _ _).mp hm
theorem join_provenance {α β σ₁ σ₂ : Type*} {C width : ℕ}
(read₁ : σ₁ → Option (α × σ₁)) (key₁ : α → ℤ)
(read₂ : σ₂ → Option (β × σ₂)) (key₂ : β → ℤ) (encode : α → β → List Bool)
(view₁ : σ₁ → List α) (hread₁ : RowMerge.IsReader read₁ view₁)
(view₂ : σ₂ → List β) (hread₂ : RowMerge.IsReader read₂ view₂)
(B fuel₁ fuel₂ controlFuel : ℕ) (s₁ : σ₁) (s₂ : σ₂)
(hsort₁ : (view₁ s₁).Pairwise (fun x y => key₁ x ≤ key₁ y))
(hsort₂ : (view₂ s₂).Pairwise (fun x y => key₂ x ≤ key₂ y))
(hcap₁ : (view₁ s₁).length ≤ fuel₁) (hcap₂ : (view₂ s₂).length ≤ fuel₂)
(hfuel : (view₁ s₁).length + (view₂ s₂).length < controlFuel)
(hwidth : ∀ x ∈ view₁ s₁, ∀ y ∈ view₂ s₂,
key₁ x = key₂ y → (encode x y).length = width)
(query : List Bool)
(hq : contains (StreamJoin.run read₁ key₁ read₂ key₂ (callback C encode)
B fuel₁ fuel₂ controlFuel s₁ s₂ ⟨.empty, false, 0⟩).user.tree query = true) :
∃ x ∈ view₁ s₁, ∃ y ∈ view₂ s₂, key₁ x = key₂ y ∧ encode x y = query := by
rw [StreamJoin.run_complete read₁ key₁ read₂ key₂ (callback C encode)
(fun _ _ _ => rfl) view₁ hread₁ view₂ hread₂ B fuel₁ fuel₂ controlFuel
s₁ s₂ ⟨.empty, false, 0⟩ hsort₁ hsort₂ hcap₁ hcap₂ hfuel] at hq
exact matchFold_provenance key₁ key₂ encode _ _ hwidth query hq
end LowSpaceSubsetSum.BinaryTrie
end OAI
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