openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionEvaluatedWeightProgram.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionStateMultiplierProgram | |
| import OAI.Combinatorics.ContingencyTables.Rooted.RootedLabelWeightProgram | |
| import OAI.Combinatorics.ContingencyTables.Rooted.RootedOracleAcceptance | |
| namespace OAI | |
| /-! # The executed multiplier times labelled weight | |
| Both factors are computed from the supplied current state. The matrix | |
| lookup and the oracle consume no hidden finite-state enumeration. | |
| -/ | |
| namespace ContingencyTables.CompanionEvaluatedWeightProgram | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| open CompanionMultiplierMatrixProgram | |
| abbrev Frame := RootedLabelWeightProgram.Config × FractionMatrix | |
| abbrev Input := (Frame × List ℕ) × List Bool | |
| def multiplierInput (z : Input) : CompanionStateMultiplierProgram.Input := | |
| (z.1.1.2,(PairedCatalogProgram.catalog z.1.2,z.1.2)) | |
| def multiplierInputRealizer : Realizer multiplierInput := | |
| pair (composition (composition first first) second) | |
| (pair (composition (f := fun z : Input => z.1.2) (g := PairedCatalogProgram.catalog) | |
| (composition first second) PairedCatalogProgram.catalogRealizer) (composition first second)) | |
| lemma polynomial_multiplierInput : PolynomialTime multiplierInputRealizer := | |
| polynomial_pair (polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_second) | |
| (polynomial_pair (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| PairedCatalogProgram.polynomial_catalog) (polynomial_composition polynomial_first polynomial_second)) | |
| def multiplier (z : Input) : TreeFraction.Fraction := | |
| CompanionStateMultiplierProgram.output (multiplierInput z) | |
| def multiplierRealizer : Realizer multiplier := | |
| composition (f := multiplierInput) (g := CompanionStateMultiplierProgram.output) | |
| multiplierInputRealizer CompanionStateMultiplierProgram.outputRealizer | |
| lemma polynomial_multiplier : PolynomialTime multiplierRealizer := | |
| polynomial_composition polynomial_multiplierInput CompanionStateMultiplierProgram.polynomial_output | |
| def weightInput (z : Input) : RootedLabelWeightProgram.Input := (z.1.1.1,(z.1.2,z.2)) | |
| def weightInputRealizer : Realizer weightInput := | |
| pair (composition (composition first first) first) (pair (composition first second) second) | |
| lemma polynomial_weightInput : PolynomialTime weightInputRealizer := | |
| polynomial_pair (polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_first) | |
| (polynomial_pair (polynomial_composition polynomial_first polynomial_second) polynomial_second) | |
| attribute [local irreducible] weightInput RootedLabelWeightProgram.output | |
| def weight (z : Input) : TreeFraction.Fraction := RootedLabelWeightProgram.output (weightInput z) | |
| def weightRealizer : Realizer weight := | |
| composition (f := weightInput) (g := RootedLabelWeightProgram.output) | |
| weightInputRealizer RootedLabelWeightProgram.outputRealizer | |
| lemma polynomial_weight : PolynomialTime weightRealizer := | |
| polynomial_composition polynomial_weightInput RootedLabelWeightProgram.polynomial_output | |
| attribute [local irreducible] multiplier weight | |
| def output (z : Input) : TreeFraction.Fraction := TreeFraction.mul (multiplier z,weight z) | |
| def outputRealizer : Realizer output := | |
| composition (f := fun z : Input => (multiplier z,weight z)) (g := TreeFraction.mul) | |
| (pair multiplierRealizer weightRealizer) TreeFraction.mulRealizer | |
| lemma polynomial_output : PolynomialTime outputRealizer := | |
| polynomial_composition (polynomial_pair polynomial_multiplier polynomial_weight) TreeFraction.polynomial_mul | |
| attribute [local irreducible] output | |
| noncomputable section | |
| open CompanionRationalUpdates | |
| open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionArithmeticProposal GridBoundary CompanionOracleDensity CommonBasesFPRAS TwoLiftStates | |
| open FiniteObservableTransport | |
| open scoped Classical | |
| variable {m n : ℕ} | |
| abbrev LabelIndex (A : Instance (Fin m) (Fin n)) := | |
| Fin (CanonicalInitialEncoding.copyOrder (capacities A)).length | |
| def sourceWeights (A : Instance (Fin m) (Fin n)) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (a b : Pair A) : ℚ := | |
| TreeFraction.val (w ((pairLabels A).symm a.1.1) ((pairLabels A).symm b.1.1)) | |
| lemma multiplier_code (A : Instance (Fin m) (Fin n)) | |
| (cfg : RootedLabelWeightProgram.Config) (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (S : Set (Labels A × Bool)) (bits : List Bool) : | |
| TreeFraction.val (multiplier (((cfg,matrixCode w),encodeSet (coordinateOrder A) S),bits))= | |
| CommonBasesFPRAS.rationalMultiplier pairs (sourceWeights A w) S := by | |
| unfold multiplier multiplierInput | |
| rw [CompanionComputedProposal.generated_catalog] | |
| exact CompanionStateMultiplierProgram.output_code A (pairLabels A) (coordinateOrder A) S w | |
| /-- Arbitrary observables of the executed weighted query agree with the | |
| proved mathematical oracle, including outcomes on its failure event. -/ | |
| theorem sameLaw_evaluated (A : Instance (Fin m) (Fin n)) (C K j : 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 : Set (Labels A × Bool)) : | |
| SameLaw (fun bits : Fin (repetitionCount xi*RootedEstimatorLayoutProgram.wordWidth | |
| (RootedEstimatorMedianProgram.context A C K j xi (CompanionWeightOracle.profile A S) v)) → Bool => | |
| TreeFraction.val (output (((RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w), | |
| encodeSet (coordinateOrder A) S),List.ofFn bits))) | |
| (RootedOracleAcceptance.evaluatedWeight A (RootedCompletion.rowColumn A.T) C K j hK hW xi | |
| (RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w) S) := by | |
| have hw := RootedLabelWeightProgram.sameLaw_weight A C K j hC hj hK hW hB xi hxi z v | |
| degree depth hdegree hdepth S | |
| intro phi | |
| have he (bits : Fin (repetitionCount xi*RootedEstimatorLayoutProgram.wordWidth | |
| (RootedEstimatorMedianProgram.context A C K j xi (CompanionWeightOracle.profile A S) v)) → Bool) : | |
| TreeFraction.val (output (((RootedLabelWeightProgram.config A C K j xi xi z v degree depth,matrixCode w), | |
| encodeSet (coordinateOrder A) S),List.ofFn bits))= | |
| CommonBasesFPRAS.rationalMultiplier pairs (sourceWeights A w) S * | |
| TreeFraction.val (RootedLabelWeightProgram.output | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth, | |
| (encodeSet (coordinateOrder A) S,List.ofFn bits))) := by | |
| rw [output,TreeFraction.mul_val,multiplier_code] | |
| unfold weight weightInput | |
| rfl | |
| simp_rw [he] | |
| exact hw (fun q => phi (CommonBasesFPRAS.rationalMultiplier pairs (sourceWeights A w) S*q)) | |
| end | |
| end ContingencyTables.CompanionEvaluatedWeightProgram | |
| end OAI | |