openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionConcentration.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.PermutationSampling | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.DeficiencyVariance | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.Representatives | |
| namespace OAI | |
| /-! | |
| # The compressed-block trial's high-deficiency partition event | |
| The sample space is the actual uniform permutation of input indices. Each of | |
| the seven blocks is marginally uniform; independence between blocks is never | |
| assumed. The floor in the retained block-list capacity is included. | |
| -/ | |
| namespace LowSpaceSubsetSum | |
| open scoped BigOperators | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| theorem permutation_deficiency_tail (a : ι → ℤ) (K : Finset ι) | |
| {t : ℝ} (ht : 0 < t) : | |
| finiteUniformProbability (fun e : Equiv.Perm ι => | |
| t ≤ |deficiency a (permuteSubset e K) - meanDeficiency a Finset.univ K.card|) ≤ | |
| (K.card : ℝ) / t ^ 2 := by | |
| classical | |
| rw [permutationBlock_uniform_ratio K | |
| (fun V => t ≤ |deficiency a V - meanDeficiency a Finset.univ K.card|)] | |
| exact deficiency_tail_fraction_le a Finset.univ K.card (Finset.card_le_univ K) ht | |
| /-- Exact seven-block union bound in the high-mean case. -/ | |
| theorem compression_partition_good_probability {n s : ℕ} (hn : 0 < n) | |
| (hs : (s : ℝ) ≤ (3 / 25 : ℝ) * n) | |
| (a : Fin n → ℤ) (K : Fin 7 → Finset (Fin n)) (hK : ∀ i, (K i).card = s) | |
| (hmean : (153 / 20000 : ℝ) * n ≤ meanDeficiency a Finset.univ s) : | |
| 1 - (84000000 : ℝ) / n ≤ finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| ∀ i, (151 / 20000 : ℝ) * n ≤ deficiency a (permuteSubset e (K i))) := by | |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn | |
| have ht : (0 : ℝ) < (n : ℝ) / 10000 := by positivity | |
| have hbad (i : Fin 7) : finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| deficiency a (permuteSubset e (K i)) < (151 / 20000 : ℝ) * n) ≤ | |
| (12000000 : ℝ) / n := by | |
| have hinc : ∀ e : Equiv.Perm (Fin n), | |
| deficiency a (permuteSubset e (K i)) < (151 / 20000 : ℝ) * n → | |
| (n : ℝ) / 10000 ≤ | |
| |deficiency a (permuteSubset e (K i)) - meanDeficiency a Finset.univ s| := by | |
| intro e he | |
| have habs := neg_le_abs | |
| (deficiency a (permuteSubset e (K i)) - meanDeficiency a Finset.univ s) | |
| linarith | |
| apply (finiteUniformProbability_mono hinc).trans | |
| have htail := permutation_deficiency_tail a (K i) ht | |
| rw [hK i] at htail | |
| apply htail.trans | |
| calc | |
| (s : ℝ) / ((n : ℝ) / 10000) ^ 2 ≤ | |
| ((3 / 25 : ℝ) * n) / ((n : ℝ) / 10000) ^ 2 := | |
| div_le_div_of_nonneg_right hs (sq_nonneg _) | |
| _ = (12000000 : ℝ) / n := by field_simp; ring | |
| have hunion := (finiteUniformProbability_exists_le_sum | |
| (fun (i : Fin 7) (e : Equiv.Perm (Fin n)) => | |
| deficiency a (permuteSubset e (K i)) < (151 / 20000 : ℝ) * n)).trans | |
| (Finset.sum_le_sum (fun i _ => hbad i)) | |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hunion | |
| have hunion' : finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| ∃ i, deficiency a (permuteSubset e (K i)) < (151 / 20000 : ℝ) * n) ≤ | |
| (84000000 : ℝ) / n := by | |
| convert hunion using 1 | |
| norm_num | |
| ring | |
| have hcompl : (fun e : Equiv.Perm (Fin n) => | |
| ∀ i, (151 / 20000 : ℝ) * n ≤ deficiency a (permuteSubset e (K i))) = | |
| (fun e => ¬∃ i, deficiency a (permuteSubset e (K i)) < (151 / 20000 : ℝ) * n) := by | |
| ext e | |
| simp | |
| rw [hcompl, finiteUniformProbability_compl] | |
| linarith | |
| noncomputable def compressionCapExponent (n : ℕ) : ℕ := | |
| ⌊((mixerSize n / 2 : ℕ) : ℝ) - (3 / 400 : ℝ) * n⌋₊ | |
| theorem compression_block_size_test {n : ℕ} (hn : 20000 ≤ n) | |
| (a : Fin n → ℤ) (V : Finset (Fin n)) (hV : V.card = mixerSize n / 2) | |
| (hd : (151 / 20000 : ℝ) * n ≤ deficiency a V) : | |
| distinctSumCount a V ≤ 2 ^ compressionCapExponent n := by | |
| have hnR : (20000 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hfloor := Nat.lt_floor_add_one | |
| (((mixerSize n / 2 : ℕ) : ℝ) - (3 / 400 : ℝ) * n) | |
| have hexp : (V.card : ℝ) - deficiency a V ≤ (compressionCapExponent n : ℝ) := by | |
| simp only [compressionCapExponent, hV] | |
| linarith | |
| have hcount : (distinctSumCount a V : ℝ) ≤ (2 : ℝ) ^ compressionCapExponent n := by | |
| rw [← two_pow_card_sub_deficiency a V] | |
| simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le | |
| (by norm_num : (1 : ℝ) ≤ 2) hexp | |
| exact_mod_cast hcount | |
| /-- The actual seven size tests pass with probability at least one half for | |
| an absolute explicit threshold. This is only the partition event, before | |
| prime sampling and the solution-bucket argument. -/ | |
| theorem compression_size_tests_probability {n : ℕ} (hn : 168000000 ≤ n) | |
| (a : Fin n → ℤ) (K : Fin 7 → Finset (Fin n)) | |
| (hK : ∀ i, (K i).card = mixerSize n / 2) | |
| (hmean : (153 / 20000 : ℝ) * n ≤ | |
| meanDeficiency a Finset.univ (mixerSize n / 2)) : | |
| (1 / 2 : ℝ) ≤ finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| ∀ i, distinctSumCount a (permuteSubset e (K i)) ≤ 2 ^ compressionCapExponent n) := by | |
| have hs : ((mixerSize n / 2 : ℕ) : ℝ) ≤ (3 / 25 : ℝ) * n := by | |
| have hdiv : mixerSize n = 2 * (mixerSize n / 2) := by | |
| simp only [mixerSize] | |
| omega | |
| have hu := (mixerSize_bounds n).2 | |
| rw [hdiv, Nat.cast_mul, Nat.cast_ofNat] at hu | |
| linarith | |
| have hgood := compression_partition_good_probability (by omega : 0 < n) | |
| hs a K hK hmean | |
| have hnR : (168000000 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hhalf : (1 / 2 : ℝ) ≤ 1 - (84000000 : ℝ) / n := by | |
| have hd : (84000000 : ℝ) / n ≤ 1 / 2 := by | |
| apply (div_le_iff₀ (by linarith : (0 : ℝ) < n)).mpr | |
| linarith | |
| linarith | |
| apply (hhalf.trans hgood).trans | |
| apply finiteUniformProbability_mono | |
| intro e he i | |
| exact compression_block_size_test (by omega) a (permuteSubset e (K i)) | |
| (by rw [permuteSubset_card, hK i]) (he i) | |
| end LowSpaceSubsetSum | |
| end OAI | |