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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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | 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
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