openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationExtraPreparation.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationResiduePreparation | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationExtraDraw | |
| namespace OAI | |
| /-! # The complete stored representation prefix through the extra prime -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.RepresentationExtraPreparation | |
| variable {n : ℕ} | |
| structure Data (n : ℕ) where | |
| residues : RepresentationResiduePreparation.Data n | |
| q : ℕ | |
| theorem width_ideal (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (s : (canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.Seed) | |
| (data : RepresentationResiduePreparation.Data n) | |
| (h : (RepresentationResiduePreparation.run I hn hlog s).1 = some data) : | |
| data.primes.stored.stored.layout.parameters.widths[3] = | |
| (canonicalRepresentationData I).zeroSetup.extraPrimeParameter | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s).1.1.1.1) := by | |
| obtain ⟨_, _, hp, _⟩ := RepresentationResiduePreparation.source I hn hlog s data h | |
| obtain ⟨_, _, hs, _, _⟩ := RepresentationPrimePreparation.source I hn hlog s.1 data.primes hp | |
| exact RepresentationPrimePreparation.widths_ideal I hn hlog s.1.1 data.primes.stored hs 3 | |
| def run (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| Option (Data n) × ℕ := | |
| let prepared := RepresentationResiduePreparation.run I hn hlog s.1 | |
| match h : prepared.1 with | |
| | none => (none, prepared.2 + 1) | |
| | some data => | |
| let out := RepresentationExtraDraw.draw (canonicalRepresentationData I).zeroSetup | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1) | |
| (by omega) data.primes.stored.stored.layout.parameters.widths[3] | |
| (width_ideal I hn hlog s.1 data h) s.2 | |
| (out.1.map (fun q => ⟨data, q⟩), prepared.2 + out.2 + 1) | |
| theorem work (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| (run I hn hlog s).2 ≤ (RepresentationResiduePreparation.run I hn hlog s.1).2 + | |
| n ^ 3 + SamplerExecution.primeTestBudget n + 2 := by | |
| simp only [run] | |
| split | |
| · dsimp only | |
| omega | |
| next data h => | |
| have hw := RepresentationExtraDraw.work (canonicalRepresentationData I).zeroSetup | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1) | |
| (by omega) data.primes.stored.stored.layout.parameters.widths[3] | |
| (width_ideal I hn hlog s.1 data h) s.2 | |
| change _ + _ + 1 ≤ _ | |
| omega | |
| theorem complete (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) | |
| (history : (canonicalRepresentationData I).zeroSetup.ExtraTrialHistory) | |
| (h : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).bounded s = some history) : | |
| ∃ data, (run I hn hlog s).1 = some data ∧ data.q = history.2.val := by | |
| let R := (canonicalRepresentationData I).zeroSetup | |
| have hp : R.trialHistoryCoupling.bounded s.1 ≠ none := by | |
| intro hf | |
| dsimp only [R] at hf | |
| simp only [RepresentationSetup.extraHistoryCoupling, SamplerCoupling.bind, hf, | |
| ↓reduceIte, reduceCtorEq] at h | |
| have hb : R.trialHistoryCoupling.bounded s.1 = some (R.trialHistoryCoupling.ideal s.1) := | |
| (R.trialHistoryCoupling.agrees s.1).resolve_left hp | |
| dsimp only [R] at hb | |
| simp only [RepresentationSetup.extraHistoryCoupling, SamplerCoupling.bind, hb, | |
| reduceCtorEq, ↓reduceIte] at h | |
| obtain ⟨q, hq, he⟩ := Option.map_eq_some_iff.mp h | |
| cases he | |
| obtain ⟨data, hd, _⟩ := RepresentationResiduePreparation.complete I hn hlog s.1 | |
| (R.trialHistoryCoupling.ideal s.1) hb | |
| have hv := RepresentationExtraDraw.value R (R.trialHistoryCoupling.ideal s.1).1.1.1.1 | |
| (by omega) data.primes.stored.stored.layout.parameters.widths[3] | |
| (width_ideal I hn hlog s.1 data hd) s.2 | |
| rw [hq] at hv | |
| refine ⟨⟨data, q.val⟩, ?_, rfl⟩ | |
| unfold run | |
| dsimp only | |
| split | |
| next hf => | |
| rw [hd] at hf | |
| contradiction | |
| next data' hd' => | |
| have he : data' = data := Option.some.inj (hd'.symm.trans hd) | |
| subst data' | |
| dsimp only [R] at hv | |
| simpa only [Option.map_some] using | |
| congrArg (Option.map (fun q => (⟨data, q⟩ : Data n))) hv | |
| theorem source (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) | |
| (data : Data n) (h : (run I hn hlog s).1 = some data) : | |
| ∃ history : (canonicalRepresentationData I).zeroSetup.ExtraTrialHistory, | |
| ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).bounded s = some history ∧ | |
| (RepresentationResiduePreparation.run I hn hlog s.1).1 = some data.residues ∧ | |
| data.q = history.2.val := by | |
| unfold run at h | |
| dsimp only at h | |
| split at h | |
| · contradiction | |
| next residues hr => | |
| obtain ⟨q, hq, hd⟩ := Option.map_eq_some_iff.mp h | |
| cases hd | |
| have hv := RepresentationExtraDraw.value (canonicalRepresentationData I).zeroSetup | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1) | |
| (by omega) residues.primes.stored.stored.layout.parameters.widths[3] | |
| (width_ideal I hn hlog s.1 residues hr) s.2 | |
| obtain ⟨p, hp, he⟩ := Option.map_eq_some_iff.mp (hv.symm.trans hq) | |
| obtain ⟨history, hh, _, _⟩ := RepresentationResiduePreparation.source I hn hlog s.1 residues hr | |
| have hi := (canonicalRepresentationData I).zeroSetup.trialHistoryExecution.ideal_eq_of_some | |
| s.1 history (((canonicalRepresentationData I).zeroSetup.trialHistoryExecution.value_eq s.1).trans hh) | |
| have hb : (canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.bounded s.1 = | |
| some ((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1) := by | |
| rw [hi] | |
| exact hh | |
| refine ⟨⟨(canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1, p⟩, | |
| ?_, hr, he.symm⟩ | |
| simp only [RepresentationSetup.extraHistoryCoupling, SamplerCoupling.bind, hb, | |
| reduceCtorEq, ↓reduceIte, hp, Option.map_some] | |
| end LowSpaceSubsetSum.RepresentationExtraPreparation | |
| end | |
| end OAI | |