openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /AllSamplingProgram.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.UnitTrialProgram | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ProfilePublicProgram | |
| namespace OAI | |
| /-! # A single computed sampler for large and all-small rectangles -/ | |
| namespace ContingencyTables.ProfilePrograms | |
| open MatchingFPRAS.TreeTyped DensePrograms | |
| def parameterUnitWords (z : SamplingParameters) : ℕ := | |
| parameterTrials z*(parameterSteps z*parameterProposalWidth z) | |
| def parameterUnitWordsRealizer : Realizer parameterUnitWords := | |
| composition (f := fun z : SamplingParameters => | |
| (parameterTrials z,parameterSteps z*parameterProposalWidth z)) | |
| (g := fun p : ℕ×ℕ => p.1*p.2) (pair parameterTrialsRealizer | |
| (composition (f := fun z : SamplingParameters => (parameterSteps z,parameterProposalWidth z)) | |
| (g := fun p : ℕ×ℕ => p.1*p.2) (pair parameterStepsRealizer parameterProposalWidthRealizer) multiply)) multiply | |
| theorem polynomial_parameterUnitWords : PolynomialTime parameterUnitWordsRealizer := | |
| polynomial_composition (polynomial_pair polynomial_parameterTrials | |
| (polynomial_composition (polynomial_pair polynomial_parameterSteps polynomial_parameterProposalWidth) | |
| polynomial_multiply)) polynomial_multiply | |
| attribute [local irreducible] parameterUnitWords | |
| abbrev SamplingWordInput := SamplingParameters×List Bool | |
| def unitSamplingBits (z : SamplingWordInput) : List Bool := | |
| z.2.take (parameterUnitWords z.1) | |
| def unitSamplingBitsRealizer : Realizer unitSamplingBits := | |
| composition (f := fun z : SamplingWordInput => (z.2,parameterUnitWords z.1)) | |
| (g := fun z : List Bool×ℕ => z.1.take z.2) | |
| (pair (f := fun z : SamplingWordInput => z.2) | |
| (g := fun z : SamplingWordInput => parameterUnitWords z.1) second | |
| (composition (f := fun z : SamplingWordInput => z.1) | |
| (g := parameterUnitWords) first parameterUnitWordsRealizer)) takeFast | |
| theorem polynomial_unitSamplingBits : PolynomialTime unitSamplingBitsRealizer := | |
| polynomial_composition (polynomial_pair polynomial_second | |
| (polynomial_composition polynomial_first polynomial_parameterUnitWords)) polynomial_takeFast | |
| def unitSamplingInput (z : SamplingWordInput) : ProfileRetryData := | |
| (samplingRetryParameters z.1,unitSamplingBits z) | |
| def unitSamplingInputRealizer : Realizer unitSamplingInput := | |
| pair (composition (f := fun z : SamplingWordInput => z.1) (g := samplingRetryParameters) | |
| first samplingRetryParametersRealizer) unitSamplingBitsRealizer | |
| theorem polynomial_unitSamplingInput : PolynomialTime unitSamplingInputRealizer := | |
| polynomial_pair (polynomial_composition polynomial_first polynomial_samplingRetryParameters) | |
| polynomial_unitSamplingBits | |
| def unitSamplingMatrix (z : SamplingWordInput) : MatrixCode := unitProfileRetry (unitSamplingInput z) | |
| def unitSamplingMatrixRealizer : Realizer unitSamplingMatrix := | |
| composition (f := unitSamplingInput) (g := unitProfileRetry) | |
| unitSamplingInputRealizer unitProfileRetryRealizer | |
| theorem polynomial_unitSamplingMatrix : PolynomialTime unitSamplingMatrixRealizer := | |
| polynomial_composition polynomial_unitSamplingInput polynomial_unitProfileRetry | |
| def parameterLargeIndices (row : Bool) (z : SamplingParameters) : List ℕ := | |
| largeIndices (marginThreshold z.1,if row then z.1.1 else z.1.2) | |
| def parameterLargeIndicesRealizer (row : Bool) : Realizer (parameterLargeIndices row) := by | |
| cases row with | |
| | false => | |
| exact composition | |
| (f := fun z : SamplingParameters => (marginThreshold z.1,z.1.2)) (g := largeIndices) | |
| (pair (composition (f := fun z : SamplingParameters => z.1) (g := marginThreshold) | |
| first marginThresholdRealizer) (composition first second)) largeIndicesRealizer | |
| | true => | |
| exact composition | |
| (f := fun z : SamplingParameters => (marginThreshold z.1,z.1.1)) (g := largeIndices) | |
| (pair (composition (f := fun z : SamplingParameters => z.1) (g := marginThreshold) | |
| first marginThresholdRealizer) (composition first first)) largeIndicesRealizer | |
| theorem polynomial_parameterLargeIndices (row : Bool) : PolynomialTime (parameterLargeIndicesRealizer row) := by | |
| cases row with | |
| | false => exact polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_marginThreshold) | |
| (polynomial_composition polynomial_first polynomial_second)) polynomial_largeIndices | |
| | true => exact polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_marginThreshold) | |
| (polynomial_composition polynomial_first polynomial_first)) polynomial_largeIndices | |
| def hasLargeRectangle (z : SamplingParameters) : Bool := | |
| decide (0<(parameterLargeIndices true z).length) && decide (0<(parameterLargeIndices false z).length) | |
| def hasLargeRectangleRealizer : Realizer hasLargeRectangle := | |
| composition (f := fun z : SamplingParameters => | |
| (decide (0<(parameterLargeIndices true z).length),decide (0<(parameterLargeIndices false z).length))) | |
| (g := fun p : Bool×Bool => p.1 && p.2) | |
| (pair | |
| (composition (f := fun z : SamplingParameters => (0,(parameterLargeIndices true z).length)) | |
| (g := fun p : ℕ×ℕ => decide (p.1<p.2)) | |
| (pair (constant 0) (composition (f := parameterLargeIndices true) (g := List.length) | |
| (parameterLargeIndicesRealizer true) listLength)) less) | |
| (composition (f := fun z : SamplingParameters => (0,(parameterLargeIndices false z).length)) | |
| (g := fun p : ℕ×ℕ => decide (p.1<p.2)) | |
| (pair (constant 0) (composition (f := parameterLargeIndices false) (g := List.length) | |
| (parameterLargeIndicesRealizer false) listLength)) less)) boolAnd | |
| theorem polynomial_hasLargeRectangle : PolynomialTime hasLargeRectangleRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition (polynomial_pair (polynomial_constant 0) | |
| (polynomial_composition (polynomial_parameterLargeIndices true) polynomial_listLength)) polynomial_less) | |
| (polynomial_composition (polynomial_pair (polynomial_constant 0) | |
| (polynomial_composition (polynomial_parameterLargeIndices false) polynomial_listLength)) polynomial_less)) | |
| polynomial_boolAnd | |
| def samplingMatrixAll (z : SamplingParameters×List Bool) : MatrixCode := | |
| if hasLargeRectangle z.1 then samplingMatrix z else unitSamplingMatrix z | |
| def samplingMatrixAllRealizer : Realizer samplingMatrixAll := | |
| choose (composition (f := fun z : SamplingParameters×List Bool => z.1) (g := hasLargeRectangle) | |
| first hasLargeRectangleRealizer) samplingMatrixRealizer unitSamplingMatrixRealizer | |
| theorem polynomial_samplingMatrixAll : PolynomialTime samplingMatrixAllRealizer := | |
| polynomial_choose (polynomial_composition polynomial_first polynomial_hasLargeRectangle) | |
| polynomial_samplingMatrix polynomial_unitSamplingMatrix | |
| def publicSamplingAll (z : List MatchingFPRAS.Symbol×List Bool) : List MatchingFPRAS.Symbol := | |
| GreedyFeasibleTable.printMatrix (samplingMatrixAll (parsedSampling z.1,z.2)) | |
| def publicSamplingAllRealizer : Realizer publicSamplingAll := | |
| composition (f := fun z : List MatchingFPRAS.Symbol×List Bool => | |
| samplingMatrixAll (parsedSampling z.1,z.2)) (g := GreedyFeasibleTable.printMatrix) | |
| (composition (f := fun z : List MatchingFPRAS.Symbol×List Bool => (parsedSampling z.1,z.2)) | |
| (g := samplingMatrixAll) (pair (composition (f := fun z : List MatchingFPRAS.Symbol×List Bool => z.1) | |
| (g := parsedSampling) first parsedSamplingRealizer) second) samplingMatrixAllRealizer) | |
| GreedyFeasibleTable.printMatrixRealizer | |
| theorem polynomial_publicSamplingAll : PolynomialTime publicSamplingAllRealizer := | |
| polynomial_composition (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_parsedSampling) polynomial_second) | |
| polynomial_samplingMatrixAll) GreedyFeasibleTable.polynomial_printMatrix | |
| end ContingencyTables.ProfilePrograms | |
| end OAI | |