openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactPrefixFeasibility.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPrefixLaw | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixLookup | |
| namespace OAI | |
| /-! # Feasibility is preserved on both branches and every finite retry prefix -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms MatchingFPRAS.PhysicalStream | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| attribute [local irreducible] exactOrdinary correctionPrefix exactMatrixInput | |
| ExactCoreProgram.prefixExperiment scheduledTableDraw ExactIntegerOnline.trial | |
| scheduledResidualCounts originalCandidates | |
| def FeasibleCode (X : MatrixCode) : Prop := ∃ T : Table r c,X=matrixCode T.val | |
| def FeasibleOutcome (o : Option MatrixCode) : Prop := o.elim True (FeasibleCode r c) | |
| lemma exactOrdinary_feasible (htotal : (∑ i,r i)=∑ j,c j) : | |
| (exactOrdinary (List.ofFn r,List.ofFn c)).All (FeasibleCode r c) := by | |
| rw [Experiment.all_indicator,exactOrdinary_mean r c htotal] | |
| have he (bits : Bits (finiteSamplerBits r c (2*correctionPrecision r c))) : | |
| (if FeasibleCode r c (matrixCode (scheduledTableDraw r c htotal | |
| (2*correctionPrecision r c) bits).val) then (0:ℚ) else 1)=0 := | |
| ite_eq_left ⟨_,rfl⟩ | |
| simp_rw [he] | |
| exact average_const _ _ | |
| lemma exactTrial_feasible (htotal : (∑ i,r i)=∑ j,c j) : | |
| (ExactMatrixTrial.trial (exactMatrixInput (List.ofFn r,List.ofFn c))).All | |
| (FeasibleOutcome r c) := by | |
| rw [exactMatrixInput_code r c htotal,ExactMatrixTrial.trial,Experiment.all_map] | |
| apply (Experiment.all_true _).mono | |
| intro o _ | |
| cases o with | |
| | none => trivial | |
| | some i => | |
| exact ExactMatrixTrial.lookup_feasible r c (GreedyFeasibleTable.table r c htotal) i | |
| lemma correctionPrefix_feasible (htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) : | |
| (correctionPrefix (List.ofFn r,List.ofFn c) N).All (FeasibleOutcome r c) := by | |
| unfold correctionPrefix | |
| induction N with | |
| | zero => trivial | |
| | succ N ih => | |
| rw [OnlineIntegerRetry.prefixExperiment,Experiment.all_bind] | |
| apply (exactTrial_feasible r c htotal).mono | |
| intro o ho | |
| cases o with | |
| | none => exact ih | |
| | some X => exact ho | |
| lemma exactPrefix_feasible (htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) : | |
| (ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).All (FeasibleOutcome r c) := by | |
| rw [ExactCoreProgram.prefixExperiment,Experiment.all_bind] | |
| apply (Experiment.all_true _).mono | |
| intro rare _ | |
| cases rare with | |
| | false => | |
| rw [ite_eq_right Bool.false_ne_true,Experiment.all_map] | |
| exact exactOrdinary_feasible r c htotal | |
| | true => exact correctionPrefix_feasible r c htotal N | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |