File size: 4,062 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 88 89 90 91 92 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionCallbacks
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredEncodingCells
namespace OAI
/-! # Live storage of the compression dictionary callbacks
The key is constructed once, the trie inserts only a new path, and the
cached distinct count controls overflow. The lookup callback retains no
new dictionary or list of matches.
-/
namespace LowSpaceSubsetSum.CompressionCallbackCells
variable {n : ℕ}
def weightCode (I : Instance n) (x : ℤ) : (List Bool × ℕ) × ℕ :=
let mag := StoredEncodingCells.bits x.natAbs (2*inputScale I) 0
((decide (x<0)::mag.1.1,mag.1.2+3),mag.2+8)
theorem weightCode_value (I : Instance n) (x : ℤ) :
(weightCode I x).1 = CompressionCallbacks.weightCode I x := by
simp only [weightCode,StoredEncodingCells.bits_value,CompressionCallbacks.weightCode,
CountedWeights.natBits]
theorem weightCode_cells (I : Instance n) (x : ℤ) :
(weightCode I x).2 ≤ 12*inputScale I+12 := by
simp only [weightCode,StoredEncodingCells.bits_cells]
omega
def report (I : Instance n) (C : ℕ) (x y : ℤ) (r : CompressionCallbacks.Dictionary I C) :
((CompressionCallbacks.Dictionary I C × Bool) × ℕ) × ℕ :=
let out := TrieDictionaryCells.safeFeed C (WeightEncoding.weightWidth I)
(CompressionCallbacks.weightCode I) (weightCode I) (weightCode_value I)
(CompressionCallbacks.weightCode_width I) (fun _ => 1) (x+y) r
(((out.1.1,false),out.1.2+6),out.2+16)
theorem report_value (I : Instance n) (C : ℕ) (x y : ℤ)
(r : CompressionCallbacks.Dictionary I C) :
(report I C x y r).1 = CompressionCallbacks.report I C x y r := by
simp only [report,TrieDictionaryCells.safeFeed_value,CompressionCallbacks.report,
CountedDictionaryCallback.callback]
apply Prod.ext
· rfl
· dsimp only; omega
theorem report_cells (I : Instance n) (C : ℕ) (x y : ℤ)
(r : CompressionCallbacks.Dictionary I C) :
(report I C x y r).2 ≤ 26*inputScale I+2*(C+1)+83 := by
have h := TrieDictionaryCells.safeFeed_cells C (WeightEncoding.weightWidth I)
(CompressionCallbacks.weightCode I) (weightCode I) (weightCode_value I)
(CompressionCallbacks.weightCode_width I) (fun _ => 1)
(12*inputScale I+12) 1 (weightCode_cells I) (fun _ => le_rfl) (x+y) r
change _+16 ≤ _
have ha := Nat.add_le_add_right h 16
exact ha.trans (by unfold WeightEncoding.weightWidth; omega)
def probe (I : Instance n) (tree : BinaryTrie.Tree) (target x y : ℤ) (found : Bool) :
((Bool × Bool) × ℕ) × ℕ :=
let code := weightCode I (target-(x+y))
let hit := TrieCells.contains tree code.1.1
(((found || hit.1.1,false),code.1.2+hit.1.2+5),code.2+hit.2+code.1.1.length+16)
theorem probe_value (I : Instance n) (tree : BinaryTrie.Tree) (target x y : ℤ) (found : Bool) :
(probe I tree target x y found).1 = CompressionCallbacks.probe I tree target x y found := by
simp only [probe,weightCode_value,TrieCells.contains_value,CompressionCallbacks.probe]
theorem probe_cells (I : Instance n) (tree : BinaryTrie.Tree) (target x y : ℤ) (found : Bool) :
(probe I tree target x y found).2 ≤ 18*inputScale I+35 := by
have hcode := weightCode_cells I (target-(x+y))
have hwidth : (weightCode I (target-(x+y))).1.1.length = WeightEncoding.weightWidth I := by
rw [weightCode_value,CompressionCallbacks.weightCode_width]
have hh := TrieCells.contains_cells tree (weightCode I (target-(x+y))).1.1
rw [hwidth] at hh
change _+_+_+16 ≤ _
rw [hwidth]
unfold WeightEncoding.weightWidth at hh ⊢
omega
def dictionaryCells {I : Instance n} {C : ℕ} (r : CompressionCallbacks.Dictionary I C) : ℕ :=
2*r.val.records.size+3*BinaryTrie.nodes r.val.keys.tree+8
theorem dictionaryCells_bound (I : Instance n) (C : ℕ) (r : CompressionCallbacks.Dictionary I C) :
dictionaryCells r ≤ (6*inputScale I+8)*(C+1)+8 := by
obtain ⟨hs,ht⟩ := CompressionCallbacks.storage I C r
unfold dictionaryCells
nlinarith only [hs,ht]
end LowSpaceSubsetSum.CompressionCallbackCells
end OAI
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