openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactResidualInput.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingParameters | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixTrial | |
| import OAI.Combinatorics.ContingencyTables.Machines.ScheduledResidualLaw | |
| namespace OAI | |
| /-! # Computing the complete input to the exact matrix rejection loop -/ | |
| namespace ContingencyTables.ProfilePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms Polynomial | |
| attribute [local irreducible] exactResidualInput scheduledResidualCounts originalCandidates | |
| def exactRareMeasure (z : CompletionMargins) : ℕ := | |
| exactParameterMeasure z+scheduledResidualMeasure (exactResidualInput z)+ | |
| originalEnumerationMeasure z+weight z | |
| def exactRareWeights (z : CompletionMargins) : List ℕ := | |
| scheduledResidualCounts (exactResidualInput z) | |
| def exactRareWeightsRealizer : Realizer exactRareWeights := | |
| composition (f := exactResidualInput) (g := scheduledResidualCounts) | |
| exactResidualInputRealizer scheduledResidualCountsRealizer | |
| lemma work_exactRareWeights : WorkBound exactRareWeightsRealizer exactRareMeasure := by | |
| unfold exactRareWeightsRealizer | |
| have hp : WorkBound exactResidualInputRealizer exactRareMeasure := | |
| work_remeasure work_exactResidualInput X (by | |
| intro z | |
| simp only [eval_X,exactRareMeasure] | |
| omega) | |
| apply work_composition hp work_scheduledResidualCounts X | |
| intro z | |
| simp only [eval_X,exactRareMeasure] | |
| omega | |
| attribute [local irreducible] exactRareWeights | |
| def exactMatrixInput (z : CompletionMargins) : ExactMatrixTrial.Input := | |
| ((originalCandidates z,GreedyFeasibleTable.listMatrix z.1 z.2),exactRareWeights z) | |
| def exactMatrixInputRealizer : Realizer exactMatrixInput := | |
| pair (pair originalCandidatesRealizer GreedyFeasibleTable.tableRealizer) exactRareWeightsRealizer | |
| lemma work_exactMatrixInput : WorkBound exactMatrixInputRealizer exactRareMeasure := by | |
| have hi (z : CompletionMargins) : weight z ≤ X.eval (exactRareMeasure z) := by | |
| simp only [eval_X,exactRareMeasure] | |
| omega | |
| have he : WorkBound originalCandidatesRealizer exactRareMeasure := | |
| work_remeasure work_originalCandidates X (by | |
| intro z | |
| simp only [eval_X,exactRareMeasure] | |
| omega) | |
| exact work_pair (work_pair he (work_of_polynomial GreedyFeasibleTable.polynomial_table X hi) X hi) | |
| work_exactRareWeights X hi | |
| end ContingencyTables.ProfilePrograms | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms OriginalTableBox ResidualMixture FirstSuccess CompletionCounts | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n) | |
| local notation "D" => correctionPrecision r c | |
| local notation "q" => largeTailCount r (d^20)*largeTailCount c (d^20) | |
| lemma exactRareWeights_pos | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8))) | |
| (htotal : (∑ i,r i)=∑ j,c j) : | |
| 0<(exactRareWeights (List.ofFn r,List.ofFn c)).sum := by | |
| rw [exactRareWeights,exactResidualInput_ofFn] | |
| exact scheduledResidualCounts_sum r c hPL hCheeger htotal | |
| lemma exactMatrixInput_code (htotal : (∑ i,r i)=∑ j,c j) : | |
| exactMatrixInput (List.ofFn r,List.ofFn c)= | |
| ((originalCandidates (List.ofFn r,List.ofFn c),matrixCode (GreedyFeasibleTable.table r c htotal).val), | |
| scheduledResidualCounts (((List.ofFn r,List.ofFn c),2*D),2^D)) := by | |
| unfold exactMatrixInput exactRareWeights | |
| rw [exactResidualInput_ofFn,GreedyFeasibleTable.listMatrix_ofFn] | |
| rfl | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |