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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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationSamplerExecution
import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionSamplerCoupling
namespace OAI
/-! # Executed bounded permutation and prime draws for compression -/
noncomputable section
namespace LowSpaceSubsetSum.CompressionData
variable {n : ℕ} (D : CompressionData n)
def samplerExecution (hn : 168000000 ≤ n) : SamplerExecution.Execution (D.sampler hn) :=
SamplerExecution.bind (SamplerExecution.permutation n n) _ _
(fun _ => SamplerExecution.prime (by unfold compressionPrimeExponent; omega) n)
theorem samplerExecution_bounded (hn : 168000000 ≤ n) :
(D.samplerExecution hn).Bounded (n * n ^ 3 + n ^ 3) (SamplerExecution.primeTestBudget n) := by
have hp := SamplerExecution.permutation_bounded n n
have hq := SamplerExecution.prime_bounded
(show 4 ≤ compressionPrimeExponent n by unfold compressionPrimeExponent; omega) n
have hq' := hq.mono le_rfl (Nat.mul_le_mul_left (n ^ 3)
(Nat.pow_le_pow_right (by decide) (show
(compressionPrimeExponent n + 1) / 2 ≤ (n + 1) / 2 by
unfold compressionPrimeExponent
omega)))
intro s
have hb := SamplerExecution.bind_bounded (SamplerExecution.permutation n n) _ _
(fun _ => SamplerExecution.prime (by unfold compressionPrimeExponent; omega) n)
hp (fun _ => hq')
simpa only [samplerExecution, SamplerExecution.primeTestBudget, Nat.zero_add] using hb s
theorem samplerExecution_operations (hn : 168000000 ≤ n) (s : (D.sampler hn).Seed) :
((D.samplerExecution hn).run s).reads + ((D.samplerExecution hn).run s).tests ≤
3 * inputScale D.input ^ 4 * 2 ^ ((n + 1) / 2) := by
have h := D.samplerExecution_bounded hn s
have hu : n ≤ inputScale D.input := by unfold inputScale; omega
have hu1 : 1 ≤ inputScale D.input := by unfold inputScale; omega
have h3 := Nat.pow_le_pow_left hu 3
have h34 : inputScale D.input ^ 3 ≤ inputScale D.input ^ 4 :=
pow_le_pow_right' hu1 (by decide)
have h4 : n * n ^ 3 ≤ inputScale D.input ^ 4 := by
have hm := Nat.mul_le_mul hu h3
simpa only [← pow_succ'] using hm
have hread : n * n ^ 3 + n ^ 3 ≤ 2 * inputScale D.input ^ 4 := by omega
have hT : 1 ≤ 2 ^ ((n + 1) / 2) := Nat.one_le_two_pow
have hreads : n * n ^ 3 + n ^ 3 ≤ 2 * inputScale D.input ^ 4 * 2 ^ ((n + 1) / 2) :=
hread.trans (Nat.le_mul_of_pos_right _ (by omega))
have htests := Nat.mul_le_mul_right (2 ^ ((n + 1) / 2)) (h3.trans h34)
unfold SamplerExecution.primeTestBudget at h
nlinarith only [h.1, h.2, hreads, htests]
end LowSpaceSubsetSum.CompressionData
end
end OAI
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