openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionMetropolisLaw.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisProgram | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionEvaluatedWeightLaw | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteMetropolisTransport | |
| import OAI.Combinatorics.ContingencyTables.Rooted.RootedBooleanPhase | |
| namespace OAI | |
| /-! # The actual computed proposal and two oracle calls have the phase law -/ | |
| namespace ContingencyTables.CompanionMetropolisProgram | |
| open MatchingFPRAS CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionBooleanPhase (State) | |
| open CompanionArithmeticProposal CompanionOrderedProposal CompanionOracleDensity CompanionWeightOracle | |
| open CompanionEvaluatedWeightProgram CompanionMultiplierMatrixProgram CommonBasesFPRAS | |
| open FiniteObservableTransport TwoLiftStates GridBoundary | |
| open scoped Classical | |
| noncomputable section | |
| variable {m n : ℕ} | |
| abbrev Fields (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) := | |
| (Fin 1 → Bool) × ((Fin (CompanionBooleanPhase.proposalBits A) → Bool) × | |
| ((WeightWord A C K j xi × WeightWord A C K j xi) × (Fin k → Bool))) | |
| def programInput (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) | |
| (cfg : RootedLabelWeightProgram.Config) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (S : State A) (b : Fields A C K j k xi) : Input := | |
| (((cfg,matrixCode w),encodeSet (coordinateOrder A) S.val), | |
| (b.1 0,(List.ofFn b.2.1,((List.ofFn b.2.2.1.1,List.ofFn b.2.2.1.2),List.ofFn b.2.2.2)))) | |
| def codeAccept (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) | |
| (cfg : RootedLabelWeightProgram.Config) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (S T : State A) | |
| (b : (WeightWord A C K j xi × WeightWord A C K j xi) × (Fin k → Bool)) : Bool := | |
| FractionMetropolisProgram.accept | |
| ((CompanionEvaluatedWeightProgram.output (((cfg,matrixCode w),encodeSet (coordinateOrder A) S.val),List.ofFn b.1.1), | |
| CompanionEvaluatedWeightProgram.output (((cfg,matrixCode w),encodeSet (coordinateOrder A) T.val),List.ofFn b.1.2)), | |
| List.ofFn b.2) | |
| lemma output_programInput (A : Instance (Fin m) (Fin n)) (C K j k : ℕ) (xi : ℚ) | |
| (cfg : RootedLabelWeightProgram.Config) | |
| (w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (S : State A) (b : Fields A C K j k xi) : | |
| output (programInput A C K j k xi cfg w S b)= | |
| encodeSet (coordinateOrder A) | |
| (if b.1 0 then S else proposedMove S (orderedProposal A (coordinateOrder A) S b.2.1) | |
| (codeAccept A C K j k xi cfg w S) b.2.2).val := by | |
| cases hb : b.1 0 with | |
| | true => simp only [output,programInput,hb,ite_true] | |
| | false => | |
| cases hp : orderedProposal A (coordinateOrder A) S b.2.1 with | |
| | none => | |
| simp only [output,programInput,hb,Bool.false_eq_true,ite_false,accepted,acceptanceInput, | |
| currentValue,targetValue,currentInput,targetInput,candidate,proposed, | |
| CompanionComputedProposal.proposalProgram_eq,hp,Option.map_none,Option.getD_none,proposedMove] | |
| split_ifs <;> simp_all | |
| | some T => | |
| simp only [output,programInput,hb,Bool.false_eq_true,ite_false,accepted,acceptanceInput, | |
| currentValue,targetValue,currentInput,targetInput,candidate,proposed, | |
| CompanionComputedProposal.proposalProgram_eq,hp,Option.map_some,Option.getD_some,proposedMove, | |
| codeAccept] | |
| split_ifs <;> simp_all | |
| /-- Every computed step has the complete reference phase-step law. The | |
| proposal and both query outcomes are transported, including failures. -/ | |
| theorem sameLaw_fields (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 b : Fields A C K j k xi => output (programInput A C K j k xi | |
| (RootedLabelWeightProgram.config A C K j xi xi z v degree depth) w S b)) | |
| (fun b : 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) := by | |
| let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth | |
| let refAccept := fun T : State A => fun b : | |
| (WeightWord A C K j xi × WeightWord A C K j xi) × (Fin k → Bool) => | |
| decide (FairBits.rationalThresholdAccept k | |
| (RootedOracleAcceptance.acceptance A (RootedCompletion.rowColumn A.T) C K j hK hW xi | |
| (RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w) S.val T.val b.1) b.2) | |
| have hp : SameLaw (orderedProposal A (coordinateOrder A) S) (CompanionBooleanPhase.proposal A S) := | |
| SameLaw.of_mapLaw_eq _ _ (orderedProposal_law A (coordinateOrder A) S) | |
| have ha (T : State A) : SameLaw (codeAccept A C K j k xi cfg w S T) (refAccept T) := | |
| acceptance_sameLaw A C K j k hC hj hK hW hB xi hxi z v degree depth hdegree hdepth w S.val T.val | |
| have hm := proposedMove_sameLaw S _ _ _ _ hp ha | |
| have hh := (holding_sameLaw S _ _ hm).map (fun T : State A => encodeSet (coordinateOrder A) T.val) | |
| intro phi | |
| have hleft : (fun b : Fields A C K j k xi => phi (output (programInput A C K j k xi cfg w S b)))= | |
| (fun b : Fields A C K j k xi => phi (encodeSet (coordinateOrder A) | |
| (if b.1 0 then S else proposedMove S (orderedProposal A (coordinateOrder A) S b.2.1) | |
| (codeAccept A C K j k xi cfg w S) b.2.2).val)) := by | |
| funext b | |
| rw [output_programInput] | |
| rw [hleft] | |
| have he := hh phi | |
| refine he.trans ?_ | |
| apply congrArg uniformAverage | |
| funext b | |
| apply congrArg phi | |
| unfold RootedBooleanPhase.step | |
| dsimp only | |
| simp only [Equiv.apply_symm_apply] | |
| cases hb : b.1 0 <;> cases hp : CompanionBooleanPhase.proposal A S b.2.1 | |
| all_goals simp only [proposedMove,Bool.false_eq_true,ite_false,ite_true,refAccept, | |
| decide_eq_true_eq] | |
| end | |
| end ContingencyTables.CompanionMetropolisProgram | |
| end OAI | |