openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationConcentration.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionConcentration | |
| namespace OAI | |
| /-! | |
| # The low-mean layout event | |
| These estimates concern one uniform permutation of a fixed layout. All six | |
| half blocks and all three mixers have the required marginals without any | |
| independence assumption. The spread test retains the integer floor used by | |
| the program. The separate balance law connects this layout law to the | |
| conditional signed-solution experiment. | |
| -/ | |
| namespace LowSpaceSubsetSum | |
| open scoped BigOperators | |
| theorem finiteUniformProbability_union_le {Ω : Type*} [Fintype Ω] [Nonempty Ω] | |
| (P Q : Ω → Prop) : | |
| finiteUniformProbability (fun ω => P ω ∨ Q ω) ≤ | |
| finiteUniformProbability P + finiteUniformProbability Q := by | |
| have h := finiteUniformProbability_exists_le_sum | |
| (fun (b : Bool) (ω : Ω) => if b then P ω else Q ω) | |
| simpa [Bool.exists_bool, Fintype.sum_bool, or_comm, add_comm] using h | |
| /-- The exact pairwise integer spread test; it also bounds the largest list | |
| by the smallest list times the displayed factor. -/ | |
| def mixerSpreadTest {ι : Type*} [Fintype ι] [DecidableEq ι] (a : ι → ℤ) | |
| (M : Fin 3 → Finset ι) (n : ℕ) : Prop := | |
| ∀ i j, distinctSumCount a (M i) ≤ distinctSumCount a (M j) * 2 ^ (n / 2000) | |
| theorem spread_floor_margin {n : ℕ} (hn : 10000 ≤ n) : | |
| (n : ℝ) / 2500 ≤ (n / 2000 : ℕ) := by | |
| have hrem := Nat.mod_lt n (by norm_num : 0 < 2000) | |
| have hdiv := Nat.mod_add_div n 2000 | |
| have hnR : (10000 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hfloor : (n : ℝ) < 2000 * ((n / 2000 : ℕ) + 1 : ℝ) := by | |
| exact_mod_cast (show n < 2000 * (n / 2000 + 1) by omega) | |
| linarith | |
| theorem mixerSpreadTest_of_centered {n m : ℕ} (hn : 10000 ≤ n) | |
| {ι : Type*} [Fintype ι] [DecidableEq ι] (a : ι → ℤ) (M : Fin 3 → Finset ι) | |
| (hM : ∀ i, (M i).card = m) (ν : ℝ) | |
| (hcenter : ∀ i, |deficiency a (M i) - ν| ≤ (n : ℝ) / 5000) : | |
| mixerSpreadTest a M n := by | |
| intro i j | |
| have hi := (abs_le.mp (hcenter i)).1 | |
| have hj := (abs_le.mp (hcenter j)).2 | |
| have hfloor := spread_floor_margin hn | |
| have hexp : ((M i).card : ℝ) - deficiency a (M i) ≤ | |
| ((M j).card : ℝ) - deficiency a (M j) + (n / 2000 : ℕ) := by | |
| rw [hM i, hM j] | |
| linarith | |
| have hpow := Real.rpow_le_rpow_of_exponent_le (by norm_num : (1 : ℝ) ≤ 2) hexp | |
| rw [Real.rpow_add (by norm_num : (0 : ℝ) < 2), | |
| two_pow_card_sub_deficiency, two_pow_card_sub_deficiency, | |
| Real.rpow_natCast] at hpow | |
| exact_mod_cast hpow | |
| /-- Simultaneous concentration for the six halves and three pre-order | |
| mixers. This is a marginal union bound, with explicit absolute threshold. -/ | |
| theorem representation_layout_good_probability {n s m : ℕ} (hn : 0 < n) | |
| (hs : (s : ℝ) ≤ (3 / 25 : ℝ) * n) | |
| (hm : (m : ℝ) ≤ (6 / 25 : ℝ) * n) | |
| (a : Fin n → ℤ) (H : Fin 6 → Finset (Fin n)) | |
| (M : Fin 3 → Finset (Fin n)) | |
| (hH : ∀ i, (H i).card = s) (hM : ∀ i, (M i).card = m) | |
| (hmean : meanDeficiency a Finset.univ s < (153 / 20000 : ℝ) * n) : | |
| 1 - (90000000 : ℝ) / n ≤ finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| (∀ i, deficiency a (permuteSubset e (H i)) ≤ (39 / 5000 : ℝ) * n) ∧ | |
| (∀ i, |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m| ≤ (n : ℝ) / 5000)) := by | |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn | |
| have hhalf (i : Fin 6) : finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| (39 / 5000 : ℝ) * n < deficiency a (permuteSubset e (H i))) ≤ | |
| (12000000 : ℝ) / n := by | |
| have hinc : ∀ e : Equiv.Perm (Fin n), | |
| (39 / 5000 : ℝ) * n < deficiency a (permuteSubset e (H i)) → | |
| (n : ℝ) / 10000 ≤ |deficiency a (permuteSubset e (H i)) - | |
| meanDeficiency a Finset.univ s| := by | |
| intro e he | |
| have habs := le_abs_self | |
| (deficiency a (permuteSubset e (H i)) - meanDeficiency a Finset.univ s) | |
| linarith | |
| apply (finiteUniformProbability_mono hinc).trans | |
| have htail := permutation_deficiency_tail a (H i) | |
| (show 0 < (n : ℝ) / 10000 by positivity) | |
| rw [hH 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 hmixer (i : Fin 3) : finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| (n : ℝ) / 5000 < |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m|) ≤ (6000000 : ℝ) / n := by | |
| have hinc : ∀ e : Equiv.Perm (Fin n), | |
| (n : ℝ) / 5000 < |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m| → | |
| (n : ℝ) / 5000 ≤ |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m| := fun _ he => he.le | |
| apply (finiteUniformProbability_mono hinc).trans | |
| have htail := permutation_deficiency_tail a (M i) | |
| (show 0 < (n : ℝ) / 5000 by positivity) | |
| rw [hM i] at htail | |
| apply htail.trans | |
| calc | |
| (m : ℝ) / ((n : ℝ) / 5000) ^ 2 ≤ | |
| ((6 / 25 : ℝ) * n) / ((n : ℝ) / 5000) ^ 2 := | |
| div_le_div_of_nonneg_right hm (sq_nonneg _) | |
| _ = (6000000 : ℝ) / n := by field_simp; ring | |
| have hhalves := (finiteUniformProbability_exists_le_sum | |
| (fun (i : Fin 6) (e : Equiv.Perm (Fin n)) => | |
| (39 / 5000 : ℝ) * n < deficiency a (permuteSubset e (H i)))).trans | |
| (Finset.sum_le_sum (fun i _ => hhalf i)) | |
| have hmixers := (finiteUniformProbability_exists_le_sum | |
| (fun (i : Fin 3) (e : Equiv.Perm (Fin n)) => | |
| (n : ℝ) / 5000 < |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m|)).trans | |
| (Finset.sum_le_sum (fun i _ => hmixer i)) | |
| have hbad := (finiteUniformProbability_union_le | |
| (fun e : Equiv.Perm (Fin n) => ∃ i, | |
| (39 / 5000 : ℝ) * n < deficiency a (permuteSubset e (H i))) | |
| (fun e : Equiv.Perm (Fin n) => ∃ i, (n : ℝ) / 5000 < | |
| |deficiency a (permuteSubset e (M i)) - meanDeficiency a Finset.univ m|)).trans | |
| (add_le_add hhalves hmixers) | |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hbad | |
| norm_num at hbad | |
| have hsum : (6 : ℝ) * (12000000 / (n : ℝ)) + 3 * (6000000 / (n : ℝ)) = | |
| 90000000 / (n : ℝ) := by ring | |
| rw [hsum] at hbad | |
| have hcompl : (fun e : Equiv.Perm (Fin n) => | |
| (∀ i, deficiency a (permuteSubset e (H i)) ≤ (39 / 5000 : ℝ) * n) ∧ | |
| (∀ i, |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m| ≤ (n : ℝ) / 5000)) = | |
| (fun e => ¬((∃ i, (39 / 5000 : ℝ) * n < deficiency a (permuteSubset e (H i))) ∨ | |
| (∃ i, (n : ℝ) / 5000 < |deficiency a (permuteSubset e (M i)) - | |
| meanDeficiency a Finset.univ m|))) := by | |
| ext e | |
| simp only [not_or, not_exists, not_lt] | |
| rw [hcompl, finiteUniformProbability_compl] | |
| linarith | |
| theorem representation_layout_tests_probability {n s m : ℕ} (hn : 180000000 ≤ n) | |
| (hs : (s : ℝ) ≤ (3 / 25 : ℝ) * n) | |
| (hm : (m : ℝ) ≤ (6 / 25 : ℝ) * n) | |
| (a : Fin n → ℤ) (H : Fin 6 → Finset (Fin n)) | |
| (M : Fin 3 → Finset (Fin n)) | |
| (hH : ∀ i, (H i).card = s) (hM : ∀ i, (M i).card = m) | |
| (hmean : meanDeficiency a Finset.univ s < (153 / 20000 : ℝ) * n) : | |
| (1 / 2 : ℝ) ≤ finiteUniformProbability (fun e : Equiv.Perm (Fin n) => | |
| (∀ i, deficiency a (permuteSubset e (H i)) ≤ (39 / 5000 : ℝ) * n) ∧ | |
| mixerSpreadTest a (fun i => permuteSubset e (M i)) n) := by | |
| have hgood := representation_layout_good_probability (by omega : 0 < n) | |
| hs hm a H M hH hM hmean | |
| have hnR : (180000000 : ℝ) ≤ n := by exact_mod_cast hn | |
| have hhalf : (1 / 2 : ℝ) ≤ 1 - (90000000 : ℝ) / n := by | |
| have hd : (90000000 : 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 | |
| exact ⟨he.1, mixerSpreadTest_of_centered (by omega) a _ | |
| (fun i => by rw [permuteSubset_card, hM i]) _ he.2⟩ | |
| end LowSpaceSubsetSum | |
| end OAI | |