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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLayoutExecution
import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationParameterPreparation
namespace OAI
/-! # Preparing stored layout data and prime widths after the bounded draw -/
noncomputable section
namespace LowSpaceSubsetSum.RepresentationLayoutPreparation
variable {n : ℕ}
structure Data (n : ℕ) where
permutation : Vector (Fin n) n
signs : IndexedMask n
domains : Vector (List (Fin n)) 3
parameters : MixerParameterExecution.Parameters
def run (D : RepresentationData n) (s : D.zeroSetup.layoutCoupling.Seed) :
Option (Data n) × ℕ :=
let layout := RepresentationLayoutExecution.run D.zeroSetup s
match layout.1 with
| none => (none, layout.2 + 1)
| some (v, signs) =>
let prepared := RepresentationParameterPreparation.run D v
(some ⟨v, signs, prepared.1.1, prepared.1.2⟩, layout.2 + prepared.2 + 1)
theorem prepared (D : RepresentationData n) (s : D.zeroSetup.layoutCoupling.Seed)
(data : Data n) (h : (run D s).1 = some data) :
(data.domains, data.parameters) =
(RepresentationParameterPreparation.run D data.permutation).1 := by
unfold run at h
cases hl : (RepresentationLayoutExecution.run D.zeroSetup s).1 with
| none => simp only [hl, reduceCtorEq] at h
| some pair =>
rcases pair with ⟨v, signs⟩
simp only [hl, Option.some.injEq] at h
subst data
rfl
theorem value (D : RepresentationData n) (s : D.zeroSetup.layoutCoupling.Seed)
(data : Data n) (h : (run D s).1 = some data) :
∃ e, D.zeroSetup.layoutCoupling.bounded s = some (e, data.signs.semantic) ∧
(∀ i : Fin n, data.permutation[i.val] = e i) ∧
(∀ j : Fin 3, data.domains[j.val] = (permuteSubset e (D.mixers j)).sort (· ≤ ·)) ∧
(∀ j : Fin 4, data.parameters.widths[j.val] =
D.zeroSetup.samplerPrimeWidths (e, data.signs.semantic) j) := by
unfold run at h
cases hl : (RepresentationLayoutExecution.run D.zeroSetup s).1 with
| none => simp only [hl, reduceCtorEq] at h
| some pair =>
rcases pair with ⟨v, signs⟩
simp only [hl, Option.some.injEq] at h
subst data
obtain ⟨e, he, hv⟩ := RepresentationLayoutExecution.value D.zeroSetup s v signs hl
refine ⟨e, he, hv, ?_, ?_⟩
· intro j
exact RepresentationMixerPreparation.domains_value D v e hv j
· intro j
exact RepresentationParameterPreparation.widths_value D v e hv signs.semantic j
theorem complete (D : RepresentationData n) (s : D.zeroSetup.layoutCoupling.Seed)
(e : Equiv.Perm (Fin n)) (signs : Finset (Fin n))
(h : D.zeroSetup.layoutCoupling.bounded s = some (e, signs)) :
∃ data, (run D s).1 = some data ∧ data.signs.semantic = signs := by
obtain ⟨v, mask, hv, hm, _⟩ := RepresentationLayoutExecution.complete D.zeroSetup s e signs h
refine ⟨⟨v, mask, (RepresentationParameterPreparation.run D v).1.1,
(RepresentationParameterPreparation.run D v).1.2⟩, ?_, hm⟩
simp only [run, hv]
theorem work (D : RepresentationData n) (hn : 10000 ≤ n)
(s : D.zeroSetup.layoutCoupling.Seed) :
(run D s).2 ≤ n * n ^ 3 + 10 * (n + 1) ^ 2 + 4 * n + 6 +
inputScale D.input ^ 6 * 2 ^ mixerSize n := by
have hw := RepresentationLayoutExecution.work D.zeroSetup s
unfold run
cases hl : (RepresentationLayoutExecution.run D.zeroSetup s).1 with
| none => simp only [hl]; omega
| some pair =>
rcases pair with ⟨v, signs⟩
obtain ⟨e, _, hv⟩ := RepresentationLayoutExecution.value D.zeroSetup s v signs hl
have hp := RepresentationParameterPreparation.work D hn v e hv
simp only [hl]
omega
end LowSpaceSubsetSum.RepresentationLayoutPreparation
end
end OAI
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