openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionBoundedWalkSemantics.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionBoundedWalkProgram | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisWordSemantics | |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteEncodedHistory | |
| namespace OAI | |
| /-! # The bounded physical recorder preserves the complete phase history -/ | |
| namespace ContingencyTables.CompanionBoundedWalkProgram | |
| open MatchingFPRAS CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionArithmeticProposal CompanionOracleDensity CompanionWeightOracle | |
| open CompanionEvaluatedWeightProgram CompanionMultiplierMatrixProgram CommonBasesFPRAS | |
| open CompanionBooleanPhase (State) | |
| open FiniteObservableTransport TwoLiftStates GridBoundary | |
| open scoped Classical | |
| def codeStep (fixed : Fixed) (code : List ℕ) (bits : List Bool) : List ℕ := | |
| next (bits,(fixed,code)) | |
| lemma recordFold_eq (fixed : Fixed) (words : List (List Bool)) (code : List ℕ) | |
| (past : List (List ℕ)) : | |
| words.foldl (fun state bits => recordAdvance (bits,state)) ((fixed,code),past)= | |
| let out := words.foldl (fun state bits => | |
| let d := codeStep fixed state.1 bits; (d,d::state.2)) (code,past) | |
| ((fixed,out.1),out.2) := by | |
| induction words generalizing code past with | |
| | nil => rfl | |
| | cons word words ih => | |
| simpa only [List.foldl_cons,recordAdvance,advance,codeStep] using | |
| ih (next (word,(fixed,code))) (next (word,(fixed,code))::past) | |
| lemma history_eq_trace (fixed : Fixed) (code : List ℕ) (words : List (List Bool)) : | |
| history ((fixed,code),words)=traceStates (codeStep fixed) code words := by | |
| unfold history recordWalk initialRecord | |
| rw [recordFold_eq] | |
| exact traceStates_fold (codeStep fixed) words code | |
| noncomputable section | |
| variable {m n : ℕ} | |
| lemma cap_encode (A : Instance (Fin m) (Fin n)) (S T : Set (Labels A × Bool)) : | |
| ProfilePrograms.capView (((encodeSet (coordinateOrder A) S).length,1), | |
| encodeSet (coordinateOrder A) T)=encodeSet (coordinateOrder A) T := by | |
| apply ProfilePrograms.capView_eq | |
| · simp only [encodeSet,List.length_ofFn] | |
| · intro a ha | |
| obtain ⟨i,rfl⟩ := List.mem_ofFn.mp ha | |
| unfold ProfilePrograms.setProfile | |
| split_ifs <;> omega | |
| /-- The all-input clipping has no effect on a valid encoded transition; | |
| in particular its full row law agrees with the reference phase step. -/ | |
| theorem codeStep_sameLaw (A : Instance (Fin m) (Fin n)) (C K j k : 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) | |
| (xi : ℚ) (hxi : 0 < xi) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) (degree depth : List Bool) | |
| (hdegree : degree.length=PowerOfTwoApproximation.rootDegree (roundingTolerance (innerTolerance (xi/2) j))) | |
| (hdepth : depth.length=PowerOfTwoApproximation.depth (roundingTolerance (innerTolerance (xi/2) j))) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (S₀ S : CompanionBooleanPhase.State A) : | |
| let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth | |
| let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S₀.val) | |
| SameLaw (fun bits : Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool => | |
| codeStep fixed (encodeSet (coordinateOrder A) S.val) (List.ofFn bits)) | |
| (fun bits => encodeSet (coordinateOrder A) | |
| (RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi | |
| (RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w) S bits).val) := by | |
| dsimp only | |
| have h := CompanionMetropolisWordProgram.sameLaw_flat A C K j k hC hj hK hW hB | |
| xi hxi z v degree depth hdegree hdepth w S | |
| have hm := h.map (fun code => ProfilePrograms.capView | |
| (((encodeSet (coordinateOrder A) S₀.val).length,1),code)) | |
| intro phi | |
| simpa only [codeStep,next,bound,candidate,transitionInput,CompanionMetropolisWordProgram.packed, | |
| cap_encode] using hm phi | |
| /-- The physical list recorder and the full reference state history have | |
| the same law, before any first-return scan or phase statistic is applied. -/ | |
| theorem history_sameLaw (A : Instance (Fin m) (Fin n)) (C K j k : 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) | |
| (xi : ℚ) (hxi : 0 < xi) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) (degree depth : List Bool) | |
| (hdegree : degree.length=PowerOfTwoApproximation.rootDegree (roundingTolerance (innerTolerance (xi/2) j))) | |
| (hdepth : depth.length=PowerOfTwoApproximation.depth (roundingTolerance (innerTolerance (xi/2) j))) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (S : CompanionBooleanPhase.State A) (N : ℕ) : | |
| let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth | |
| let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S.val) | |
| SameLaw (fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) => | |
| history ((fixed,encodeSet (coordinateOrder A) S.val),List.ofFn (fun i => List.ofFn (words i)))) | |
| (fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) => (traceStates | |
| (RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi | |
| (RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w)) S (List.ofFn words)).map | |
| (fun T => encodeSet (coordinateOrder A) T.val)) := by | |
| dsimp only | |
| let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth | |
| let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S.val) | |
| have ht := traceStates_sameLaw | |
| (fun code bits => codeStep fixed code (List.ofFn bits)) | |
| (RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi | |
| (RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w)) | |
| (fun T : CompanionBooleanPhase.State A => encodeSet (coordinateOrder A) T.val) | |
| (codeStep_sameLaw A C K j k hC hj hK hW hB xi hxi z v degree depth hdegree hdepth w S) S N | |
| intro phi | |
| rw [show (fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) => | |
| phi (history ((fixed,encodeSet (coordinateOrder A) S.val),List.ofFn (fun i => List.ofFn (words i)))))= | |
| (fun words => phi (traceStates (fun code bits => codeStep fixed code (List.ofFn bits)) | |
| (encodeSet (coordinateOrder A) S.val) (List.ofFn words))) by | |
| funext words | |
| rw [history_eq_trace] | |
| have hm : List.ofFn (fun i => List.ofFn (words i))=(List.ofFn words).map List.ofFn := by | |
| rw [List.map_ofFn] | |
| rfl | |
| rw [hm,traceStates_map]] | |
| exact ht phi | |
| end | |
| end ContingencyTables.CompanionBoundedWalkProgram | |
| end OAI | |