openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionCanonicalPhaseSupport.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionCanonicalPhaseLaw | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartSupport | |
| namespace OAI | |
| /-! # Representation preservation for every executed counting phase | |
| The restart support statement is obtained from its actual finite-word law. | |
| The remaining trajectory and observation words may be unused padding for | |
| that law; their presence still supplies the required precision markers. | |
| -/ | |
| namespace ContingencyTables.CompanionCanonicalPhaseSupport | |
| open MatchingFPRAS CommonBasesFPRAS FiniteObservableTransport | |
| open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionEvaluatedWeightProgram CompanionStoredRepresentation CompanionHistoryReadback | |
| open CompanionCanonicalPhaseLaw CompanionOracleDensity GridBoundary | |
| open scoped Classical | |
| noncomputable section | |
| variable {m n : ℕ} | |
| lemma restart_prefix_le (A : Instance (Fin m) (Fin n)) (C K H N R L W j : ℕ) | |
| (epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (hj : j < H) | |
| (hwidth : RootedPhaseHistory.totalWidth A (rawSettings A C K H N R L epsilon hK hW v) ≤ W) : | |
| CompanionComputedInitial.wordLength A | |
| (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+ | |
| CompanionRestartProgram.sumBits A C K j | |
| (CompanionInitializedRestartProgram.parameters A H | |
| (CompanionTargetMixing.restartTime A H L) N R L epsilon) ≤ W := by | |
| let s := rawSettings A C K H N R L epsilon hK hW v | |
| calc | |
| _ = RootedRestartWords.initialStageBits A C K H L s.k R s.h j s.xi := | |
| CompanionOuterWidths.initialWidth_eq A C K H N R L j epsilon | |
| _ ≤ RootedPhaseHistory.width A s j := Nat.le_add_right _ _ | |
| _ ≤ RootedPhaseHistory.totalWidth A s := RootedPhaseHistory.width_le A s j hj | |
| _ ≤ W := hwidth | |
| theorem output_rep (A : Instance (Fin m) (Fin n)) (C K H N R L W : ℕ) | |
| (epsilon : TreeFraction.Fraction) (hC : 0 < C) (hK : 1 ≤ K) | |
| (hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=binWidth A C) | |
| (hxi : 0 < TreeFraction.val (CountingPhaseParametersProgram.noise | |
| (CompanionInitializedRestartProgram.parameters A H | |
| (CompanionTargetMixing.restartTime A H L) N R L epsilon))) | |
| (z : View A→ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (initial : CompanionPhaseFinishProgram.Stored) (history : List CompanionPhaseFinishProgram.Stored) | |
| (hj : history.length < H) (hphase : history.length+1 ≤ rationalHeight A C) | |
| (hwidth : RootedPhaseHistory.totalWidth A (rawSettings A C K H N R L epsilon hK hW v) ≤ W) | |
| (hdegree : ∀ i,i ≤ history.length+1 → PowerOfTwoApproximation.rootDegree | |
| (roundingTolerance (innerTolerance | |
| ((rawSettings A C K H N R L epsilon hK hW v).xi/2) i)) ≤ W) | |
| (hdepth : ∀ i,i ≤ history.length+1 → PowerOfTwoApproximation.depth | |
| (roundingTolerance (innerTolerance | |
| ((rawSettings A C K H N R L epsilon hK hW v).xi/2) i)) ≤ W) | |
| (hinit : StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length initial) | |
| (hh : ∀ q∈history,StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length q) | |
| (bits : Fin W→Bool) : | |
| StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length | |
| (CompanionOuterPhaseProgram.output | |
| (((CompanionRestartInitializer.data A C K H | |
| (CompanionTargetMixing.restartTime A H L) N R L epsilon z v,initial),history),List.ofFn bits)) := by | |
| let t := CompanionTargetMixing.restartTime A H L | |
| let j := history.length | |
| let p := CompanionInitializedRestartProgram.parameters A H t N R L epsilon | |
| let a := wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)) | |
| let b := CompanionRestartProgram.sumBits A C K j p | |
| let padding := W-(a+b) | |
| let w := matrixAt (CanonicalInitialEncoding.copyOrder (capacities A)).length initial history | |
| let E := estimateAt initial history | |
| have hprefix : a+b ≤ W := restart_prefix_le A C K H N R L W j epsilon hK hW v hj hwidth | |
| have hlen : a+(b+padding)=W := by | |
| rw [←Nat.add_assoc] | |
| exact Nat.add_sub_of_le hprefix | |
| have hl := CompanionInitializedRestartProgram.output_padded_sameLaw A C K H t N R L j padding | |
| hC (by omega) hK hW hB epsilon hxi z v w (fun i => E (i.val+1)) | |
| (fun i => by | |
| unfold CompanionRestartStageProgram.oracleData | |
| rw [RootedOracleConfigProgram.rootDegree_data] | |
| exact (hdegree (i.val+1) (by omega)).trans_eq hlen.symm) | |
| (fun i => by | |
| unfold CompanionRestartStageProgram.oracleData | |
| rw [RootedOracleConfigProgram.depth_data] | |
| exact (hdepth (i.val+1) (by omega)).trans_eq hlen.symm) | |
| have hl' := SameLaw.word_length hlen | |
| (fun bs => CompanionInitializedRestartProgram.output | |
| (CompanionInitializedRestartProgram.data A C K H t N R L j epsilon z v w | |
| (fun i => E (i.val+1)) bs)) _ hl | |
| obtain ⟨sourceBits,hs⟩ := hl'.exists_preimage bits | |
| have hd := outer_data_readback A C K H t N R L epsilon z v initial history hinit hh (List.ofFn bits) | |
| apply CompanionInitializedRestartProgram.outer_output_rep_of_supported A _ hinit hh | |
| refine ⟨CompanionInitializedRestartProgram.model A C K H t N R L j epsilon hK hW v w sourceBits,?_⟩ | |
| rw [←hd] | |
| change CompanionInitializedRestartProgram.output | |
| (CompanionOuterPhaseProgram.restartInput | |
| (CompanionOuterPhaseProgram.data A C K H t N R L j epsilon z v w E (List.ofFn bits)))=_ | |
| rw [CompanionOuterPhaseProgram.restartInput_data] | |
| exact hs.symm | |
| end | |
| end ContingencyTables.CompanionCanonicalPhaseSupport | |
| end OAI | |