| import OAI.Combinatorics.ContingencyTables.Sampling.ExactCoreProgram |
|
|
| namespace OAI |
|
|
| /-! # The actual finite prefixes of the fixed exact-sampling program -/ |
|
|
| namespace ContingencyTables.ExactCoreProgram |
| open MatchingFPRAS.PhysicalStream ProfilePrograms |
| attribute [local irreducible] rareBranch exactOrdinary correctionPrefix exactPrecision |
|
|
| lemma bool_mean (p : Experiment Bool) (a b : ℚ) : |
| p.mean (fun x => if x then b else a)= |
| (1-p.mean (fun x => if x then 1 else 0))*a+ |
| p.mean (fun x => if x then 1 else 0)*b := by |
| induction p with |
| | done x => cases x <;> simp [Experiment.mean] |
| | delay n p ih => exact ih |
| | read p ih => |
| simp only [Experiment.mean,ih] |
| ring |
|
|
| lemma prefix_mean (z : DensePrograms.CompletionMargins) (N : ℕ) |
| (g : Option DensePrograms.MatrixCode → ℚ) : |
| (prefixExperiment z N).mean g= |
| (1-1/(2:ℚ)^(exactPrecision z))*((exactOrdinary z).map some).mean g+ |
| (1/(2:ℚ)^(exactPrecision z))*(correctionPrefix z N).mean g := by |
| rw [prefixExperiment,Experiment.mean_bind] |
| have he (b : Bool) : |
| (if b then correctionPrefix z N else (exactOrdinary z).map some).mean g= |
| if b then (correctionPrefix z N).mean g else ((exactOrdinary z).map some).mean g := by |
| cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] |
| simp_rw [he] |
| rw [bool_mean,rareBranch_probability] |
|
|
| end ContingencyTables.ExactCoreProgram |
|
|
| namespace ContingencyTables.SmallGraphProfiles |
| open ProfilePrograms DensePrograms MatchingFPRAS MatchingFPRAS.PhysicalStream |
| open ResidualMixture FirstSuccess CompletionCounts |
| attribute [local irreducible] scheduledTableDraw exactOrdinary exactOrdinaryWord |
| exactSamplingParameters ExactCoreProgram.prefixExperiment correctionPrefix |
| 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 exactOrdinary_mean (htotal : (∑ i,r i)=∑ j,c j) (g : MatrixCode → ℚ) : |
| (exactOrdinary (List.ofFn r,List.ofFn c)).mean g= |
| average (fun bits : Bits (finiteSamplerBits r c (2*D)) => |
| g (matrixCode (scheduledTableDraw r c htotal (2*D) bits).val)) := by |
| rw [exactOrdinary,FiniteWordExecution.experiment_mean,exactOrdinaryBits_ofFn] |
| simp_rw [exactOrdinaryWord_code r c htotal] |
|
|
| lemma table_event_average {L : ℕ} (draw : Bits L → Table r c) (X : Table r c) : |
| average (fun bits => if matrixCode (draw bits).val=matrixCode X.val then (1:ℚ) else 0)= |
| (mapLaw (uniformLaw (α := Bits L)) draw).mass X := by |
| have he (Y : Table r c) : matrixCode Y.val=matrixCode X.val ↔ Y=X := by |
| constructor |
| · intro h |
| apply Subtype.ext |
| change List.ofFn (fun i => List.ofFn (Y.val i))=List.ofFn (fun i => List.ofFn (X.val i)) at h |
| funext i |
| exact List.ofFn_injective (congrFun (List.ofFn_injective h) i) |
| · rintro rfl; rfl |
| simp_rw [he] |
| simp only [average,mapLaw,uniformLaw,uniformMass,Fintype.card_fun,Fintype.card_bool, |
| Fintype.card_fin,Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro bits _ |
| split_ifs <;> simp |
|
|
| lemma exactOrdinary_mass (htotal : (∑ i,r i)=∑ j,c j) (X : Table r c) : |
| (exactOrdinary (List.ofFn r,List.ofFn c)).mean |
| (fun Y => if Y=matrixCode X.val then 1 else 0)= |
| (mapLaw (uniformLaw (α := Bits (finiteSamplerBits r c (2*D)))) |
| (scheduledTableDraw r c htotal (2*D))).mass X := |
| (exactOrdinary_mean r c htotal _).trans (table_event_average r c _ X) |
|
|
| lemma exactPrefix_failure |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) |
| (hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8))) |
| (htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) : |
| (ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).mean |
| (fun o => if o=none then 1 else 0)≤(1/(2:ℚ)^D)*(1/2:ℚ)^N := by |
| rw [ExactCoreProgram.prefix_mean,exactPrecision_ofFn,Experiment.mean_map] |
| have hz : (exactOrdinary (List.ofFn r,List.ofFn c)).mean |
| (fun a => if some a=none then (1:ℚ) else 0)=0 := by |
| rw [exactOrdinary_mean r c htotal] |
| simp only [reduceCtorEq,ite_false,average_const] |
| rw [hz,mul_zero,zero_add] |
| exact mul_le_mul_of_nonneg_left (correctionPrefix_failure r c hPL hCheeger htotal N) |
| (by positivity) |
|
|
| lemma exactPrefix_limit |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) |
| (hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8))) |
| (htotal : (∑ i,r i)=∑ j,c j) (X : Table r c) : |
| Filter.Tendsto (fun N => ((ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).mean |
| (fun o => if o=some (matrixCode X.val) then 1 else 0):ℝ)) |
| Filter.atTop (nhds ((Fintype.card (Table r c):ℝ)⁻¹)) := by |
| have ht := corrected_prefix_limit r c hPL hCheeger htotal X |
| convert ht using 1 |
| funext N |
| rw [ExactCoreProgram.prefix_mean,exactPrecision_ofFn,Experiment.mean_map] |
| simp only [Option.some.injEq,exactOrdinary_mass r c htotal,Rat.cast_add,Rat.cast_mul, |
| Rat.cast_sub,Rat.cast_div,Rat.cast_pow,Rat.cast_one,Rat.cast_ofNat] |
|
|
| end |
| end ContingencyTables.SmallGraphProfiles |
|
|
| end OAI |
|
|
