openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactSamplingParameters.lean
| import OAI.Combinatorics.ContingencyTables.Machines.ScheduledResidualProgram | |
| import OAI.Combinatorics.ContingencyTables.Sampling.CompiledResidualCorrection | |
| namespace OAI | |
| /-! # Computing the exact sampler's precision from its margin-only input -/ | |
| namespace ContingencyTables.ProfilePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms Polynomial | |
| attribute [local irreducible] marginDimension marginBinaryLength parameterWordCount | |
| def sixthPower : Realizer (fun n : ℕ => n^6) := | |
| (composition (f := fun n : ℕ => n^3) (g := fun n : ℕ => n^2) | |
| cubicPower squarePower).congr (by intro n; simp only [Function.comp_apply,←pow_mul]) | |
| lemma polynomial_sixthPower : PolynomialTime sixthPower := | |
| polynomial_congr _ (polynomial_composition polynomial_cubicPower polynomial_squarePower) | |
| def exactPrecision (z : CompletionMargins) : ℕ := (marginDimension z)^6*(marginBinaryLength z)^2 | |
| def exactPrecisionRealizer : Realizer exactPrecision := | |
| composition (f := fun z : CompletionMargins => ((marginDimension z)^6,(marginBinaryLength z)^2)) | |
| (g := fun p : ℕ×ℕ => p.1*p.2) | |
| (pair (composition (f := marginDimension) (g := fun n : ℕ => n^6) marginDimensionRealizer sixthPower) | |
| (composition (f := marginBinaryLength) (g := fun n : ℕ => n^2) marginBinaryLengthRealizer squarePower)) | |
| multiply | |
| lemma polynomial_exactPrecision : PolynomialTime exactPrecisionRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_marginDimension polynomial_sixthPower) | |
| (polynomial_composition polynomial_marginBinaryLength polynomial_squarePower)) polynomial_multiply | |
| attribute [local irreducible] exactPrecision | |
| def exactSamplingParameters (z : CompletionMargins) : SamplingParameters := (z,2*exactPrecision z) | |
| def exactSamplingParametersRealizer : Realizer exactSamplingParameters := | |
| pair identity (composition (f := fun z : CompletionMargins => (2,exactPrecision z)) | |
| (g := fun p : ℕ×ℕ => p.1*p.2) (pair (constant 2) exactPrecisionRealizer) multiply) | |
| lemma polynomial_exactSamplingParameters : PolynomialTime exactSamplingParametersRealizer := | |
| polynomial_pair polynomial_identity (polynomial_composition | |
| (polynomial_pair (polynomial_constant 2) polynomial_exactPrecision) polynomial_multiply) | |
| def exactBranchDenominator (z : CompletionMargins) : ℕ := BinaryPowerPrograms.power (2,exactPrecision z) | |
| def exactBranchDenominatorRealizer : Realizer exactBranchDenominator := | |
| composition (f := fun z : CompletionMargins => (2,exactPrecision z)) (g := BinaryPowerPrograms.power) | |
| (pair (constant 2) exactPrecisionRealizer) BinaryPowerPrograms.powerRealizer | |
| def exactParameterMeasure (z : CompletionMargins) : ℕ := exactPrecision z+weight z | |
| lemma work_exactBranchDenominator : WorkBound exactBranchDenominatorRealizer exactParameterMeasure := by | |
| unfold exactBranchDenominatorRealizer | |
| apply work_composition (work_of_polynomial (polynomial_pair (polynomial_constant 2) | |
| polynomial_exactPrecision) X (by intro z; simp only [eval_X,exactParameterMeasure]; omega)) | |
| BinaryPowerPrograms.work_power (X+C (weight (2:ℕ))) | |
| intro z | |
| simp only [eval_add,eval_X,eval_C,exactParameterMeasure] | |
| omega | |
| def exactResidualInput (z : CompletionMargins) : SamplingParameters×ℕ := | |
| (exactSamplingParameters z,exactBranchDenominator z) | |
| def exactResidualInputRealizer : Realizer exactResidualInput := | |
| pair exactSamplingParametersRealizer exactBranchDenominatorRealizer | |
| lemma work_exactResidualInput : WorkBound exactResidualInputRealizer exactParameterMeasure := | |
| work_pair (work_of_polynomial polynomial_exactSamplingParameters X | |
| (by intro z; simp only [eval_X,exactParameterMeasure]; omega)) | |
| work_exactBranchDenominator X (by intro z; simp only [eval_X,exactParameterMeasure]; omega) | |
| /-- The ordinary branch needs precisely the existing public sampler's word. -/ | |
| def exactOrdinaryBits (z : CompletionMargins) : ℕ := parameterWordCount (exactSamplingParameters z) | |
| def exactOrdinaryBitsRealizer : Realizer exactOrdinaryBits := | |
| composition (f := exactSamplingParameters) (g := parameterWordCount) | |
| exactSamplingParametersRealizer parameterWordCountRealizer | |
| lemma polynomial_exactOrdinaryBits : PolynomialTime exactOrdinaryBitsRealizer := | |
| polynomial_composition polynomial_exactSamplingParameters polynomial_parameterWordCount | |
| def exactParsedMargins (as : List Symbol) : CompletionMargins := (parsedSampling as).1 | |
| def exactParsedMarginsRealizer : Realizer exactParsedMargins := | |
| composition (f := parsedSampling) (g := fun p : SamplingParameters => p.1) parsedSamplingRealizer first | |
| lemma polynomial_exactParsedMargins : PolynomialTime exactParsedMarginsRealizer := | |
| polynomial_composition polynomial_parsedSampling polynomial_first | |
| end ContingencyTables.ProfilePrograms | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms MatchingFPRAS.LiteralParse | |
| open scoped BigOperators | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| lemma exactPrecision_ofFn : exactPrecision (List.ofFn r,List.ofFn c)=correctionPrecision r c := by | |
| rw [exactPrecision,marginDimension_ofFn,marginBinaryLength_ofFn] | |
| rfl | |
| lemma exactSamplingParameters_ofFn : | |
| exactSamplingParameters (List.ofFn r,List.ofFn c)= | |
| ((List.ofFn r,List.ofFn c),2*correctionPrecision r c) := by | |
| rw [exactSamplingParameters,exactPrecision_ofFn] | |
| lemma exactBranchDenominator_ofFn : | |
| exactBranchDenominator (List.ofFn r,List.ofFn c)=2^(correctionPrecision r c) := by | |
| rw [exactBranchDenominator,BinaryPowerPrograms.power_eq,exactPrecision_ofFn] | |
| lemma exactResidualInput_ofFn : | |
| exactResidualInput (List.ofFn r,List.ofFn c)= | |
| (((List.ofFn r,List.ofFn c),2*correctionPrecision r c),2^(correctionPrecision r c)) := by | |
| rw [exactResidualInput,exactSamplingParameters_ofFn,exactBranchDenominator_ofFn] | |
| lemma exactOrdinaryBits_ofFn : | |
| exactOrdinaryBits (List.ofFn r,List.ofFn c)=finiteSamplerBits r c (2*correctionPrecision r c) := by | |
| rw [exactOrdinaryBits,exactSamplingParameters_ofFn,parameterWordCount_ofFn] | |
| lemma exactParsedMargins_spec : | |
| exactParsedMargins (Algorithms.encodeMargins r c)=(List.ofFn r,List.ofFn c) := by | |
| unfold exactParsedMargins parsedSampling samplingInput | |
| rw [GreedyFeasibleTable.encodeMargins_natWords,GreedyFeasibleTable.parse_natWords] | |
| have h := GreedyFeasibleTable.marginLists_spec m n (List.ofFn r) (List.ofFn c) [] | |
| List.length_ofFn List.length_ofFn | |
| simpa only [List.append_nil] using h | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |