openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationLayoutPreparation.lean
| 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 | |