openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionMetropolisWordSemantics.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisWordProgram | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisLaw | |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordSlices | |
| namespace OAI | |
| /-! # The flat physical transition has the proved phase-step law -/ | |
| namespace ContingencyTables.CompanionMetropolisWordProgram | |
| open MatchingFPRAS CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionArithmeticProposal CompanionOracleDensity CompanionWeightOracle | |
| open CompanionEvaluatedWeightProgram CompanionMultiplierMatrixProgram CommonBasesFPRAS | |
| open CompanionBooleanPhase (State) | |
| open FiniteObservableTransport TwoLiftStates GridBoundary | |
| open scoped Classical | |
| noncomputable section | |
| variable {m n : ℕ} | |
| def packed (frame : CompanionMetropolisProgram.Frame) (k : ℕ) (code : List ℕ) (bits : List Bool) : Input := | |
| (((frame,k),code),bits) | |
| lemma fields_ofFn (frame : CompanionMetropolisProgram.Frame) (k p q : ℕ) (code : List ℕ) | |
| (bits : Fin (1+(p+((q+q)+k))) → Bool) | |
| (hp : proposalWidth (packed frame k code (List.ofFn bits))=p) | |
| (hq : queryWidth (packed frame k code (List.ofFn bits))=q) : | |
| fields (packed frame k code (List.ofFn bits))= | |
| ((FiniteWordSlices.split p q k bits).1 0, | |
| (List.ofFn (FiniteWordSlices.split p q k bits).2.1, | |
| ((List.ofFn (FiniteWordSlices.split p q k bits).2.2.1.1, | |
| List.ofFn (FiniteWordSlices.split p q k bits).2.2.1.2), | |
| List.ofFn (FiniteWordSlices.split p q k bits).2.2.2))) := by | |
| let inp := packed frame k code (List.ofFn bits) | |
| change ((List.ofFn bits).getD 0 false, | |
| (((List.ofFn bits).drop 1).take (proposalWidth inp), | |
| ((((List.ofFn bits).drop (1+proposalWidth inp)).take (queryWidth inp), | |
| ((List.ofFn bits).drop (1+proposalWidth inp+queryWidth inp)).take (queryWidth inp)), | |
| ((List.ofFn bits).drop (1+proposalWidth inp+queryWidth inp+queryWidth inp)).take k)))=_ | |
| change proposalWidth inp=p at hp | |
| change queryWidth inp=q at hq | |
| rw [hp,hq,FiniteWordSlices.holding_bit,FiniteWordSlices.first_slice, | |
| FiniteWordSlices.current_slice,FiniteWordSlices.target_slice,FiniteWordSlices.coin_slice] | |
| lemma proposalWidth_code (A : Instance (Fin m) (Fin n)) | |
| (frame : CompanionMetropolisProgram.Frame) (k : ℕ) (S : Set (Labels A × Bool)) (bits : List Bool) : | |
| proposalWidth (packed frame k (encodeSet (coordinateOrder A) S) bits)= | |
| CompanionBooleanPhase.proposalBits A := by | |
| have hc := Fintype.card_congr (coordinateOrder A) | |
| simp only [Fintype.card_prod,Fintype.card_bool,Fintype.card_fin] at hc | |
| have hd : (CanonicalInitialEncoding.pairOrder (capacities A)).length/2=Fintype.card (Labels A) := by omega | |
| simp only [proposalWidth,packed,encodeSet,List.length_ofFn,hd,CompanionBooleanPhase.proposalBits] | |
| lemma queryWidth_config (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) | |
| (z : View A → ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (degree depth : List Bool) (w : FractionMatrix) (code : List ℕ) (bits : List Bool) : | |
| queryWidth (packed (RootedLabelWeightProgram.config A C K j xi xi z v degree depth,w) k code bits)= | |
| medianEstimatorBits A C K j xi xi := by | |
| exact CompanionEvaluatedWeightProgram.queryWidth A C K j xi z v | |
| lemma output_ofFn (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) | |
| (z : View A → ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (degree depth : List Bool) (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (S : State A) (bits : Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) : | |
| output (packed (RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w) | |
| k (encodeSet (coordinateOrder A) S.val) (List.ofFn bits))= | |
| CompanionMetropolisProgram.output (CompanionMetropolisProgram.programInput A C K j k xi | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth) w S | |
| (CompanionBooleanPhase.stepWordEquiv A C K j k xi bits)) := by | |
| have hf := fields_ofFn | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w) k | |
| (CompanionBooleanPhase.proposalBits A) (medianEstimatorBits A C K j xi xi) | |
| (encodeSet (coordinateOrder A) S.val) bits | |
| (proposalWidth_code A _ k S.val _) (queryWidth_config A C K j k xi z v degree depth _ _ _) | |
| have hi : stepInput (packed | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w) | |
| k (encodeSet (coordinateOrder A) S.val) (List.ofFn bits))= | |
| CompanionMetropolisProgram.programInput A C K j k xi | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth) w S | |
| (CompanionBooleanPhase.stepWordEquiv A C K j k xi bits) := by | |
| exact congrArg (fun fs => | |
| (((RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w), | |
| encodeSet (coordinateOrder A) S.val),fs)) hf | |
| exact congrArg CompanionMetropolisProgram.output hi | |
| /-- The actual polynomial flat-step implementation has exactly the | |
| reference kernel used by the trajectory and return estimates. -/ | |
| theorem sameLaw_flat (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 : State A) : | |
| SameLaw (fun bits : Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool => | |
| output (packed (RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w) | |
| k (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 | |
| let f := fun b : CompanionMetropolisProgram.Fields A C K j k xi => | |
| CompanionMetropolisProgram.output (CompanionMetropolisProgram.programInput A C K j k xi | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth) w S b) | |
| let g := fun b : CompanionMetropolisProgram.Fields A C K j k xi => | |
| 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 | |
| ((CompanionBooleanPhase.stepWordEquiv A C K j k xi).symm b)).val | |
| have hfg : SameLaw f g := CompanionMetropolisProgram.sameLaw_fields A C K j k hC hj hK hW hB | |
| xi hxi z v degree depth hdegree hdepth w S | |
| intro phi | |
| simp_rw [output_ofFn] | |
| have hf := uniformAverage_equiv (CompanionBooleanPhase.stepWordEquiv A C K j k xi) | |
| (fun b => phi (f b)) | |
| have hg := uniformAverage_equiv (CompanionBooleanPhase.stepWordEquiv A C K j k xi) | |
| (fun b => phi (g b)) | |
| refine hf.trans ((hfg phi).trans ?_) | |
| simpa only [g,Equiv.symm_apply_apply] using hg.symm | |
| end | |
| end ContingencyTables.CompanionMetropolisWordProgram | |
| end OAI | |