openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionBoundedTrial.lean
| 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 | |