openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactOrdinaryProgram.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactBranchChoice | |
| import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingAccuracy | |
| namespace OAI | |
| /-! # The finite ordinary branch of the exact sampler -/ | |
| namespace ContingencyTables.ProfilePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream DensePrograms Polynomial | |
| def exactOrdinaryWord (z : CompletionMargins×List Bool) : MatrixCode := | |
| samplingMatrixAll (exactSamplingParameters z.1,z.2) | |
| def exactOrdinaryWordRealizer : Realizer exactOrdinaryWord := | |
| composition (f := fun z : CompletionMargins×List Bool => (exactSamplingParameters z.1,z.2)) | |
| (g := samplingMatrixAll) | |
| (pair (composition (f := Prod.fst) (g := exactSamplingParameters) | |
| first exactSamplingParametersRealizer) second) samplingMatrixAllRealizer | |
| lemma polynomial_exactOrdinaryWord : PolynomialTime exactOrdinaryWordRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_exactSamplingParameters) polynomial_second) | |
| polynomial_samplingMatrixAll | |
| def exactOrdinary (z : CompletionMargins) : Experiment MatrixCode := | |
| FiniteWordExecution.experiment exactOrdinaryBits exactOrdinaryWord z | |
| noncomputable def exactOrdinaryRealizer : OnlineTree.Realizer exactOrdinary := | |
| FiniteWordExecution.realizer exactOrdinaryBitsRealizer exactOrdinaryWordRealizer | |
| lemma work_exactOrdinary : OnlineTree.WorkBound exactOrdinaryRealizer | |
| (FiniteWordExecution.wordMeasure exactOrdinaryBits) := | |
| FiniteWordExecution.work_realizer polynomial_exactOrdinaryBits polynomial_exactOrdinaryWord | |
| /-- The gate retains the margins for either branch. -/ | |
| def exactGate (z : CompletionMargins) : Experiment (Bool×CompletionMargins) := | |
| (rareBranch z).map (fun rare => (rare,z)) | |
| noncomputable def exactGateRealizer : OnlineTree.Realizer exactGate := | |
| (OnlineTree.pair rareBranchRealizer (OnlineTree.deterministic identity)).congr | |
| (by intro z g; simp only [Experiment.mean_bind,Experiment.mean,Experiment.mean_map,exactGate,id_eq]) | |
| lemma work_exactGate : OnlineTree.WorkBound exactGateRealizer exactParameterMeasure := by | |
| unfold exactGateRealizer | |
| apply OnlineTree.work_congr | |
| apply OnlineTree.work_pair work_rareBranch | |
| (OnlineTree.work_of_polynomial (OnlineTree.polynomial_deterministic polynomial_identity) X | |
| (by intro z; simp only [eval_X,exactParameterMeasure]; omega)) X | |
| intro z | |
| simp only [eval_X,exactParameterMeasure] | |
| omega | |
| end ContingencyTables.ProfilePrograms | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms MatchingFPRAS.PhysicalStream | |
| open scoped BigOperators | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| lemma exactOrdinaryWord_code (htotal : (∑ i,r i)=∑ j,c j) | |
| (bits : Fin (finiteSamplerBits r c (2*correctionPrecision r c)) → Bool) : | |
| exactOrdinaryWord ((List.ofFn r,List.ofFn c),List.ofFn bits)= | |
| matrixCode (scheduledTableDraw r c htotal (2*correctionPrecision r c) bits).val := by | |
| rw [exactOrdinaryWord,exactSamplingParameters_ofFn,samplingMatrixAll_code r c htotal] | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |