File size: 3,359 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionSamplerCoupling
namespace OAI
/-! # Actual compression acceptance after bounded fresh-word sampling -/
noncomputable section
namespace LowSpaceSubsetSum.CompressionData
variable {n : ℕ} (D : CompressionData n)
def AcceptedWithCompactTests (B : ℕ) (h : CompressionSetup.TrialHistory n) : Prop :=
D.compactTests h.1 = true ∧ D.compressionTrial B h.1 h.2 = true
/-- Keep the actual seven compact-list tests in the successful event; later
resource bounds can use them without an independence premise. -/
theorem compact_acceptance_probability (B : ℕ) (hn : 168000000 ≤ n)
(hsolve : Solvable D.input)
(hmean : (153 / 20000 : ℝ) * n ≤
meanDeficiency D.weights Finset.univ (compressionHalfSize n))
(hlog : Real.logb 2 (inputScale D.input) ≤ (n : ℝ) / 1000000000) :
(1 / 4 : ℝ) ≤ (D.trialLaw hn).probability (D.AcceptedWithCompactTests B) := by
obtain ⟨S, hS⟩ := hsolve
let R : CompressionSetup n := ⟨D.input, S, hS⟩
apply (R.successful_probability hn hmean hlog).trans
apply FiniteLaw.probability_mono
intro h hh
exact ⟨(R.compactTests_iff h.1).mpr hh.1, R.compressionTrial_true B h.1 h.2 hh.1 hh.2⟩
theorem bounded_compact_acceptance_probability (B : ℕ) (hn : 168000000 ≤ n)
(hsolve : Solvable D.input)
(hmean : (153 / 20000 : ℝ) * n ≤
meanDeficiency D.weights Finset.univ (compressionHalfSize n))
(hlog : Real.logb 2 (inputScale D.input) ≤ (n : ℝ) / 1000000000) :
(1 / 8 : ℝ) ≤ (D.sampler hn).boundedLaw.probability
(fun o => ∃ h, o = some h ∧ D.AcceptedWithCompactTests B h) := by
have hp := D.compact_acceptance_probability B hn hsolve hmean hlog
have hl := D.sampler_event hn (D.AcceptedWithCompactTests B)
have he : 1 / (2 : ℝ) ^ n ≤ (1 / 8 : ℝ) := by
have hh := pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (by omega : 3 ≤ n)
norm_num at hh
exact one_div_le_one_div_of_le (by norm_num) hh
linarith
theorem bounded_compression_acceptance_probability (B : ℕ) (hn : 168000000 ≤ n)
(hsolve : Solvable D.input)
(hmean : (153 / 20000 : ℝ) * n ≤
meanDeficiency D.weights Finset.univ (compressionHalfSize n))
(hlog : Real.logb 2 (inputScale D.input) ≤ (n : ℝ) / 1000000000) :
1 / (inputScale D.input : ℝ) ^ 110 ≤ (D.sampler hn).boundedLaw.probability
(fun o => ∃ h, o = some h ∧ D.compressionTrial B h.1 h.2 = true) := by
have hu : (2 : ℝ) ≤ inputScale D.input := by
exact_mod_cast (show 2 ≤ inputScale D.input by unfold inputScale; omega)
have hp : (8 : ℝ) ≤ (inputScale D.input : ℝ) ^ 110 := by
calc
_ ≤ (2 : ℝ) ^ 110 := by norm_num
_ ≤ _ := pow_le_pow_left₀ (by norm_num) hu _
apply (one_div_le_one_div_of_le (by norm_num) hp).trans
apply (D.bounded_compact_acceptance_probability B hn hsolve hmean hlog).trans
apply FiniteLaw.probability_mono
rintro o ⟨h, ho, _, hh⟩
exact ⟨h, ho, hh⟩
theorem sampled_acceptance_sound (B : ℕ) (_hn : 168000000 ≤ n)
(o : Option (CompressionSetup.TrialHistory n))
(h : ∃ h, o = some h ∧ D.compressionTrial B h.1 h.2 = true) : Solvable D.input := by
obtain ⟨h, _, hh⟩ := h
exact D.compressionTrial_sound B h.1 h.2 hh
end LowSpaceSubsetSum.CompressionData
end
end OAI
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