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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionCallbackCells
namespace OAI
/-! # Capacity-scaled bounds for the actual compression callbacks -/
namespace LowSpaceSubsetSum.CompressionCallbackBounds
variable {n : ℕ}
private theorem polynomial (I : Instance n) (hn : 10000 ≤ n) :
1000*(inputScale I+1) ≤ inputScale I^4 := by
have hu : 10000 ≤ inputScale I := by unfold inputScale; omega
have h2 : 1000*(inputScale I+1) ≤ inputScale I^2 := by nlinarith only [hu]
exact h2.trans (Nat.pow_le_pow_right (by omega) (by decide : 2 ≤ 4))
theorem dictionary (I : Instance n) (hn : 10000 ≤ n) (C : ℕ)
(r : CompressionCallbacks.Dictionary I C) :
CompressionCallbackCells.dictionaryCells r ≤ inputScale I^4*(C+1) := by
have h := CompressionCallbackCells.dictionaryCells_bound I C r
have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn)
nlinarith only [h,hp]
theorem report (I : Instance n) (hn : 10000 ≤ n) (C : ℕ) (x y : ℤ)
(r : CompressionCallbacks.Dictionary I C) :
(CompressionCallbackCells.report I C x y r).2 ≤ inputScale I^4*(C+1) := by
have h := CompressionCallbackCells.report_cells I C x y r
have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn)
nlinarith only [h,hp]
theorem probe (I : Instance n) (hn : 10000 ≤ n) (C : ℕ)
(tree : BinaryTrie.Tree) (t x y : ℤ) (found : Bool) :
(CompressionCallbackCells.probe I tree t x y found).2 ≤ inputScale I^4*(C+1) := by
have h := CompressionCallbackCells.probe_cells I tree t x y found
have hp := Nat.mul_le_mul_right (C+1) (polynomial I hn)
nlinarith only [h,hp]
end LowSpaceSubsetSum.CompressionCallbackBounds
end OAI
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