openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionInitializedRestartLaw.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitializedRestartSemantics | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartNativeLaw | |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteEncodedBind | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartPadding | |
| namespace OAI | |
| /-! # Exact law of the complete executed initializer and restart prefix -/ | |
| 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 parameters (A : Instance (Fin m) (Fin n)) (H T N R L : ℕ) | |
| (epsilon : TreeFraction.Fraction) : CountingPhaseParametersProgram.Input := | |
| ([Fintype.card (Labels A),A.W,H,T,N,R,L],epsilon) | |
| def model (A : Instance (Fin m) (Fin n)) (C K H T N R L j : 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)) → Bool) : | |
| PlainIndex (pairs : List (CompanionRationalUpdates.Pair A)) := | |
| let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H) | |
| let widths := CompanionRestartProgram.sourceWidth A C K (parameters A H T N R L epsilon) | |
| let split := splitWordEquiv (CompanionInitialDraw.initialBits A B) | |
| (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) bits | |
| varyingWalk widths | |
| (CompanionRestartProgram.sourceStep A C K (parameters A H T N R L epsilon) hK hW v w) | |
| (CompanionInitialDraw.draw A B split.1) j (varyingWordEquiv widths j split.2) | |
| /-- The actual list program has the complete finite-word law, including | |
| every initialization bias, rejected return and fallback. -/ | |
| theorem output_padded_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))) | |
| (fun bits => encodeSet (coordinateOrder A) (model A C K H T N R L j epsilon hK hW v w bits).val.val) := 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) | |
| let realRun := 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)) | |
| 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_sameLaw (CompanionRestartProgram.baseData A C K j p z v reserve w S.val.val) | |
| n hn (realRun S) hh padding | |
| have hf := flat_encoded_bind (wordLength A B) (n+padding) (CompanionInitialDraw.initialBits A B) n | |
| 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)))= | |
| (fun bits => run (init (splitWordEquiv (wordLength A B) (n+padding) bits).1) | |
| (splitWordEquiv (wordLength A B) (n+padding) bits).2) := by | |
| funext bits | |
| exact output_split A C K H T N R L j (n+padding) epsilon z v w estimates bits | |
| rw [he] | |
| exact h | |
| /-- When the supplied restart time is the computed paper schedule, | |
| the reference experiment is precisely initializedStages. -/ | |
| theorem model_eq_initializedStages (A : Instance (Fin m) (Fin n)) (C K H N R L j : 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 (RootedRestartWords.initialStageBits A C K H L | |
| (CountingPhaseParametersProgram.precision | |
| (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon)) | |
| R (CountingPhaseSchedule.initialPrecision H) j | |
| (TreeFraction.val (CountingPhaseParametersProgram.noise | |
| (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon)))) → Bool) : | |
| model A C K H (CompanionTargetMixing.restartTime A H L) N R L j epsilon hK hW v w bits= | |
| RootedRestartWords.initializedStages A (RootedCompletion.rowColumn A.T) C K H L | |
| (CountingPhaseParametersProgram.precision | |
| (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon)) | |
| R (CountingPhaseSchedule.initialPrecision H) j hK hW | |
| (TreeFraction.val (CountingPhaseParametersProgram.noise | |
| (parameters A H (CompanionTargetMixing.restartTime A H L) N R L epsilon))) | |
| (RootedCatalogGrid.gridEquiv A K v) (fun i => sourceWeights A (w i)) bits := rfl | |
| end | |
| end ContingencyTables.CompanionInitializedRestartProgram | |
| end OAI | |