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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Streams.MeasuredMixerCount
namespace OAI
/-! # Executed splitting of a stored mixer domain -/
namespace LowSpaceSubsetSum.MixerHalfLists
variable {n : ℕ}
def split (xs : List (Fin n)) : (List (Fin n) × List (Fin n)) × ℕ :=
((xs.take (xs.length / 2), xs.drop (xs.length / 2)), 4 * xs.length + 3)
theorem first_source (V : Finset (Fin n)) :
(split (V.sort (· ≤ ·))).1.1 = (MixerParameterCount.firstHalf V).sort (· ≤ ·) := by
simp only [split, Finset.length_sort, MixerParameterCount.firstHalf]
symm
exact (List.toFinset_sort (r := (· ≤ ·)) ((Finset.sort_nodup V (· ≤ ·)).take)).mpr
((Finset.pairwise_sort V (· ≤ ·)).sublist (List.take_sublist _ _))
theorem drop_set (V : Finset (Fin n)) :
((V.sort (· ≤ ·)).drop (V.card / 2)).toFinset = MixerParameterCount.secondHalf V := by
let xs := V.sort (· ≤ ·)
have hnd : (xs.take (V.card / 2) ++ xs.drop (V.card / 2)).Nodup := by
rw [List.take_append_drop]
exact Finset.sort_nodup V (· ≤ ·)
have hd := (List.nodup_append.mp hnd).2.2
ext i
simp only [MixerParameterCount.secondHalf, Finset.mem_sdiff,
MixerParameterCount.firstHalf, List.mem_toFinset]
change i ∈ xs.drop (V.card / 2) ↔ i ∈ V ∧ i ∉ xs.take (V.card / 2)
have hmem : i ∈ xs ↔ i ∈ V := Finset.mem_sort (· ≤ ·)
have happ : i ∈ xs ↔ i ∈ xs.take (V.card / 2) ∨ i ∈ xs.drop (V.card / 2) := by
rw [← List.mem_append, List.take_append_drop]
constructor
· intro hi
exact ⟨hmem.mp (happ.mpr (Or.inr hi)), fun hj => hd i hj i hi rfl⟩
· rintro ⟨hi, hn⟩
exact (happ.mp (hmem.mpr hi)).resolve_left hn
theorem second_source (V : Finset (Fin n)) :
(split (V.sort (· ≤ ·))).1.2 = (MixerParameterCount.secondHalf V).sort (· ≤ ·) := by
simp only [split, Finset.length_sort]
rw [← drop_set]
symm
exact (List.toFinset_sort (r := (· ≤ ·)) ((Finset.sort_nodup V (· ≤ ·)).drop)).mpr
((Finset.pairwise_sort V (· ≤ ·)).sublist (List.drop_sublist _ _))
def count (a : Fin n → ℤ) (xs : List (Fin n)) : ℕ × ℕ :=
let halves := split xs
let out := MeasuredMixerCount.run a halves.1.1 halves.1.2
(out.1, halves.2 + out.2 + 1)
theorem count_source (a : Fin n → ℤ) (V : Finset (Fin n)) :
(count a (V.sort (· ≤ ·))).1 = distinctSumCount a V := by
simp only [count, first_source, second_source]
exact MeasuredMixerCount.mixerCount_value a V
theorem count_work (a : Fin n → ℤ) (xs : List (Fin n)) (hx : xs.length ≤ n) :
(count a xs).2 ≤
MeasuredMixerCount.coefficient n * 2 ^ (xs.length / 2) +
MeasuredMixerCount.coefficient n * 2 ^ (xs.length - xs.length / 2) +
(4 * (xs.length / 2 + 1) ^ 2 + 32) * 2 ^ (xs.length / 2) +
(10 * (xs.length / 2) + 21) * 2 ^ xs.length + 4 * xs.length + 10 := by
have hl : (xs.take (xs.length / 2)).length = xs.length / 2 := by
rw [List.length_take, Nat.min_eq_left (Nat.div_le_self _ _)]
have hr : (xs.drop (xs.length / 2)).length = xs.length - xs.length / 2 :=
List.length_drop
have h := MeasuredMixerCount.run_work a (xs.take (xs.length / 2))
(xs.drop (xs.length / 2)) (by rw [hl]; omega) (by rw [hr]; omega)
rw [hl, hr, Nat.add_sub_of_le (Nat.div_le_self _ _)] at h
change 4 * xs.length + 3 +
(MeasuredMixerCount.run a (xs.take (xs.length / 2)) (xs.drop (xs.length / 2))).2 + 1 ≤ _
omega
theorem count_scale (I : Instance n) (hn : 10000 ≤ n) (a : Fin n → ℤ)
(xs : List (Fin n)) (hx : xs.length ≤ n) :
(count a xs).2 ≤ inputScale I ^ 4 * 2 ^ xs.length := by
let u := inputScale I
have hu : n + 1 ≤ u := by dsimp [u, inputScale]; omega
have hu0 : 10000 ≤ u := by omega
have hnu : n ≤ u := by omega
have hxu : xs.length / 2 + 1 ≤ u := by omega
have hn2 := Nat.pow_le_pow_left hnu 2
have hx2 := Nat.pow_le_pow_left hxu 2
have hu2 : 1 ≤ u ^ 2 := Nat.one_le_pow _ _ (by omega)
have hc : MeasuredMixerCount.coefficient n ≤ 40 * u ^ 2 := by
unfold MeasuredMixerCount.coefficient
nlinarith only [hn2, hu, hu2]
have hbig : 40 * u ^ 2 ≤ u ^ 3 := by
calc
_ ≤ u * u ^ 2 := Nat.mul_le_mul_right _ (by omega)
_ = _ := (pow_succ' _ _).symm
have hp : 4 * (xs.length / 2 + 1) ^ 2 + 32 ≤ u ^ 3 := by omega
have hlinear : 10 * (xs.length / 2) + 21 ≤ u ^ 3 := by
nlinarith only [hxu, hu2, hbig]
have hend : 4 * xs.length + 10 ≤ u ^ 3 := by
nlinarith only [hx, hu, hu2, hbig]
have hleft : 2 ^ (xs.length / 2) ≤ 2 ^ xs.length :=
Nat.pow_le_pow_right (by decide) (Nat.div_le_self _ _)
have hright : 2 ^ (xs.length - xs.length / 2) ≤ 2 ^ xs.length :=
Nat.pow_le_pow_right (by decide) (Nat.sub_le _ _)
have hT : 1 ≤ 2 ^ xs.length := Nat.one_le_two_pow
have hcL := Nat.mul_le_mul (hc.trans hbig) hleft
have hcR := Nat.mul_le_mul (hc.trans hbig) hright
have hpL := Nat.mul_le_mul hp hleft
have hl := Nat.mul_le_mul_right (2 ^ xs.length) hlinear
have he := hend.trans (Nat.le_mul_of_pos_right _ hT)
have hw := count_work a xs hx
have hf : 5 * u ^ 3 ≤ u ^ 4 := by
calc
_ ≤ u * u ^ 3 := Nat.mul_le_mul_right _ (by omega)
_ = _ := (pow_succ' _ _).symm
have hfinal := Nat.mul_le_mul_right (2 ^ xs.length) hf
change (count a xs).2 ≤ u ^ 4 * 2 ^ xs.length
nlinarith only [hw, hcL, hcR, hpL, hl, he, hfinal]
end LowSpaceSubsetSum.MixerHalfLists
end OAI
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