openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionInitializedRestartJoint.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitializedRestartLaw | |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordJointPadding | |
| namespace OAI | |
| /-! # Joint restart law with the unused trajectory and query suffix -/ | |
| namespace ContingencyTables.CompanionInitializedRestartProgram | |
| open MatchingFPRAS CommonBasesFPRAS FirstSuccess FairBits FiniteObservableTransport | |
| open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionArithmeticProposal CompanionEvaluatedWeightProgram | |
| open CompanionOracleDensity TwoLiftStates GridBoundary | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n : ℕ} | |
| def jointModel (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ) | |
| (epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (bits : Fin (CompanionInitialDraw.initialBits A | |
| (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) → Bool) : | |
| PlainIndex (pairs : List (CompanionRationalUpdates.Pair A))×(Fin padding → Bool) := | |
| let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H) | |
| let widths := CompanionRestartProgram.sourceWidth A C K (parameters A H T N R L epsilon) | |
| let blocks := splitWordEquiv (CompanionInitialDraw.initialBits A B) | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding) bits | |
| let tail := splitWordEquiv (CompanionRestartProgram.sumBits A C K j | |
| (parameters A H T N R L epsilon)) padding blocks.2 | |
| (varyingWalk widths | |
| (CompanionRestartProgram.sourceStep A C K (parameters A H T N R L epsilon) hK hW v w) | |
| (CompanionInitialDraw.draw A B blocks.1) j (varyingWordEquiv widths j tail.1),tail.2) | |
| theorem output_joint_sameLaw (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ) | |
| (hC : 0 < C) (hj : j ≤ rationalHeight A C) (hK : 1 ≤ K) | |
| (hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=CompanionOracleDensity.binWidth A C) | |
| (epsilon : TreeFraction.Fraction) | |
| (hxi : 0 < TreeFraction.val (CountingPhaseParametersProgram.noise (parameters A H T N R L epsilon))) | |
| (z : View A → ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (estimates : Fin j → TreeFraction.Fraction) | |
| (hdegree : ∀ i : Fin j,RootedOracleConfigProgram.rootDegree | |
| (CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v | |
| (List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤ | |
| wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) | |
| (hdepth : ∀ i : Fin j,RootedOracleConfigProgram.depth | |
| (CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v | |
| (List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤ | |
| wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) : | |
| SameLaw (fun bits : Fin (wordLength A | |
| (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) → Bool => | |
| (output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)), | |
| (splitWordEquiv (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) | |
| padding (splitWordEquiv | |
| (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))) | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding) bits).2).2)) | |
| (fun bits => | |
| (encodeSet (coordinateOrder A) (jointModel A C K H T N R L j padding epsilon hK hW v w bits).1.val.val, | |
| (jointModel A C K H T N R L j padding epsilon hK hW v w bits).2)) := by | |
| let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H) | |
| let p := parameters A H T N R L epsilon | |
| let n := CompanionRestartProgram.sumBits A C K j p | |
| let reserve := List.replicate (wordLength A B+(n+padding)) false | |
| let stageList := CompanionRestartProgram.stagesData A C K j p z v reserve w | |
| let init := fun bits : Fin (wordLength A B) → Bool => CompanionRestartInitializer.output | |
| (CompanionRestartInitializer.data A C K H T N R L epsilon z v,List.ofFn bits) | |
| let realInit := CompanionInitialDraw.draw A B | |
| let encode := fun S : PlainIndex (pairs : List (CompanionRationalUpdates.Pair A)) => | |
| encodeSet (coordinateOrder A) S.val.val | |
| let run := fun code (bits : Fin (n+padding) → Bool) => | |
| (CompanionRestartProgram.output ((stageList,code),List.ofFn bits),(splitWordEquiv n padding bits).2) | |
| let sourceRun := fun S (bits : Fin n → Bool) => encode (varyingWalk | |
| (CompanionRestartProgram.sourceWidth A C K p) | |
| (CompanionRestartProgram.sourceStep A C K p hK hW v w) S j | |
| (varyingWordEquiv (CompanionRestartProgram.sourceWidth A C K p) j bits)) | |
| let realRun := fun S (bits : Fin (n+padding) → Bool) => | |
| (sourceRun S (splitWordEquiv n padding bits).1,(splitWordEquiv n padding bits).2) | |
| have hi : SameLaw init (fun bits => encode (realInit bits)) := | |
| CompanionRestartInitializer.output_sameLaw A C K H T N R L epsilon z v | |
| have hs : ∀ S,SameLaw (run (encode S)) (realRun S) := by | |
| intro S | |
| have hn := CompanionRestartProgram.widthSum_data A C K j p z v reserve w S.val.val | |
| have hh := CompanionRestartProgram.output_native_sameLaw A C K j hC hj hK hW hB p hxi z v reserve | |
| (fun i => by simpa only [reserve,List.length_replicate] using hdegree i) | |
| (fun i => by simpa only [reserve,List.length_replicate] using hdepth i) w S | |
| exact restart_append_joint_sameLaw (CompanionRestartProgram.baseData A C K j p z v reserve w S.val.val) | |
| n n hn (sourceRun S) hh padding | |
| have hf := flat_encoded_bind (wordLength A B) (n+padding) (CompanionInitialDraw.initialBits A B) (n+padding) | |
| init realInit encode run realRun hi hs | |
| intro phi | |
| have h := hf phi | |
| have he : (fun bits : Fin (wordLength A B+(n+padding)) → Bool => | |
| (output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)), | |
| (splitWordEquiv n padding (splitWordEquiv (wordLength A B) (n+padding) bits).2).2))= | |
| (fun bits => run (init (splitWordEquiv (wordLength A B) (n+padding) bits).1) | |
| (splitWordEquiv (wordLength A B) (n+padding) bits).2) := by | |
| funext bits | |
| apply Prod.ext | |
| · exact output_split A C K H T N R L j (n+padding) epsilon z v w estimates bits | |
| · rfl | |
| rw [he] | |
| exact h | |
| theorem jointModel_sameLaw (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ) | |
| (epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) : | |
| SameLaw (fun bits => | |
| (encodeSet (coordinateOrder A) (jointModel A C K H T N R L j padding epsilon hK hW v w bits).1.val.val, | |
| (jointModel A C K H T N R L j padding epsilon hK hW v w bits).2)) | |
| (fun bits : | |
| (Fin (CompanionInitialDraw.initialBits A | |
| (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) → Bool)× | |
| (Fin padding → Bool) => | |
| (encodeSet (coordinateOrder A) (model A C K H T N R L j epsilon hK hW v w bits.1).val.val,bits.2)) := by | |
| let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H) | |
| let p := parameters A H T N R L epsilon | |
| let widths := CompanionRestartProgram.sourceWidth A C K p | |
| exact nested_word_joint (CompanionInitialDraw.initialBits A B) | |
| (CompanionRestartProgram.sumBits A C K j p) padding | |
| (fun initBits stageBits => encodeSet (coordinateOrder A) | |
| (varyingWalk widths (CompanionRestartProgram.sourceStep A C K p hK hW v w) | |
| (CompanionInitialDraw.draw A B initBits) j (varyingWordEquiv widths j stageBits)).val.val) | |
| end | |
| end ContingencyTables.CompanionInitializedRestartProgram | |
| end OAI | |