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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationBudgets
namespace OAI
/-! # The resource bound for the actual dependent sampling experiment -/
noncomputable section
namespace LowSpaceSubsetSum
theorem prefix_floor_modulus_lower {a b c u : ℕ} (hu : 256 ≤ u)
(ha : a ≤ u) (hb : b ≤ u) (hc : c ≤ u)
(z : ↥(tripleRange a b c)) (i : Fin 3) :
(2 : ℝ) ^ (prefixExponent a b c i : ℝ) / (u : ℝ) ^ 5 ≤ prefixModulus z.val i := by
obtain ⟨hab, hcz⟩ := Finset.mem_product.mp z.property
obtain ⟨haz, _hbz⟩ := Finset.mem_product.mp hab
fin_cases i
· have h := nested_size_bounds (L := ((a + b + c : ℕ) : ℝ)) hu ha hb hc
le_rfl (by push_cast; linarith) z.property
exact h.1
· have hz : (z.val.1, 1) ∈ tripleRange a b 0 := by
exact Finset.mem_product.mpr ⟨hab, by simp [factorRange]⟩
have h := nested_size_bounds (L := ((a + b : ℕ) : ℝ)) hu ha hb (by omega)
(by simp) (by push_cast; linarith) hz
change (2 : ℝ) ^ ((a + b : ℕ) : ℝ) / (u : ℝ) ^ 5 ≤ (z.val.1.1 * z.val.1.2 : ℕ)
simpa only [tripleModulus, mul_one] using h.1
· have hz : ((z.val.1.1, 1), 1) ∈ tripleRange a 0 0 := by
exact Finset.mem_product.mpr
⟨Finset.mem_product.mpr ⟨haz, by simp [factorRange]⟩, by simp [factorRange]⟩
have h := nested_size_bounds (L := (a : ℝ)) hu ha (by omega) (by omega)
(by simp) (by push_cast; linarith) hz
change (2 : ℝ) ^ (a : ℝ) / (u : ℝ) ^ 5 ≤ z.val.1.1
simpa only [tripleModulus, mul_one] using h.1
namespace RepresentationSetup
variable {n : ℕ} (R : RepresentationSetup n)
theorem actual_modulus_floor_lower (z : Layout n) (p : R.PrimeSpace z)
(hu : 256 ≤ inputScale R.input) (i : Fin 3) :
(2 : ℝ) ^ (⌊R.lowExponent z i⌋₊ : ℝ) / (inputScale R.input : ℝ) ^ 5 ≤
prefixModulus p.val i := by
have h := prefix_floor_modulus_lower hu (R.factorExponents_le_scale z 0)
(R.factorExponents_le_scale z 1) (R.factorExponents_le_scale z 2) p i
rw [R.prefixExponent_eq_floor] at h
exact h
def AllocatedResourceGood (hn : 20000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000)
(h : R.TrialHistory) : Prop :=
let z := h.1.1.1.1
let g := h.1.1.1.2
let f := h.1.1.2
let p := h.1.2
let B := R.allocatedBackground z g (by omega) hlog
RepresentationMoments.GoodResource (prefixModulus_pos p 0) (prefixModulus_pos p 1)
(prefixModulus_pos p 2) (R.signedWeights z) (R.rawLeaf z g f B 0)
(R.rawLeaf z g f B 1) (R.signedTarget z) (inputScale R.input)
(R.resourceGamma z) (R.resourceLambda z g) ((n : ℝ) / 2) h.2
/-- Every actual prime history has the resource tail estimate before observing
its independently uniform low residues. -/
theorem actual_history_resource_failure (hn : 20000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000)
(hu : 256 ≤ inputScale R.input) (h : R.PrimeHistory) :
finiteUniformProbability (fun r : R.Residues h.2 =>
¬R.AllocatedResourceGood hn hlog ⟨h, r⟩) <
(1 / 10 : ℝ) / (inputScale R.input : ℝ) ^ 100 := by
exact R.allocated_resource_failure (prefixModulus_pos h.2 0) (prefixModulus_pos h.2 1)
(prefixModulus_pos h.2 2) h.1.1.1 h.1.1.2 h.1.2 hn hlog
(by exact_mod_cast (show 8 ≤ inputScale R.input by omega)) R.mixerSize_add_one_le_scale
(R.actual_modulus_floor_lower h.1.1.1 h.2 hu 0)
(R.actual_modulus_floor_lower h.1.1.1 h.2 hu 1)
(R.actual_modulus_floor_lower h.1.1.1 h.2 hu 2)
/-- Joint failure with any event fixed before the low residues is small.
In particular this applies to the spread-test event, without conditioning
on the rare surviving representation. -/
theorem actual_joint_resource_failure (hn : 20000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000)
(hu : 256 ≤ inputScale R.input) (passing : R.PrimeHistory → Prop) :
R.trialLaw.probability (fun h => passing h.1 ∧ ¬R.AllocatedResourceGood hn hlog h) ≤
(1 / 10 : ℝ) / (inputScale R.input : ℝ) ^ 100 := by
exact RepresentationMoments.joint_failure_average R.primeLaw (fun h => R.Residues h.2)
passing (fun h r => ¬R.AllocatedResourceGood hn hlog ⟨h, r⟩) (by positivity)
(fun h _ => (R.actual_history_resource_failure hn hlog hu h).le)
end RepresentationSetup
end LowSpaceSubsetSum
end
end OAI
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