openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionMetropolisProgram.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionEvaluatedWeightProgram | |
| import OAI.Combinatorics.ContingencyTables.Machines.FractionMetropolisProgram | |
| namespace OAI | |
| /-! # One physical companion-chain step with independent word fields | |
| The proposed state is computed by the quadratic arithmetic candidate | |
| list. Two disjoint words evaluate the current and candidate states, and | |
| a third field supplies the rational acceptance coin. | |
| -/ | |
| namespace ContingencyTables.CompanionMetropolisProgram | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| abbrev Frame := CompanionEvaluatedWeightProgram.Frame | |
| abbrev Words := Bool × (List Bool × ((List Bool × List Bool) × List Bool)) | |
| abbrev Input := (Frame × List ℕ) × Words | |
| def proposed (z : Input) : Option (List ℕ) := | |
| CompanionComputedProposal.proposalProgram (z.1.2,z.2.2.1) | |
| def proposedRealizer : Realizer proposed := | |
| composition (f := fun z : Input => (z.1.2,z.2.2.1)) | |
| (g := CompanionComputedProposal.proposalProgram) | |
| (pair (composition first second) (composition (composition second second) first)) | |
| CompanionComputedProposal.proposalProgramRealizer | |
| lemma polynomial_proposed : PolynomialTime proposedRealizer := | |
| polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_second) | |
| (polynomial_composition (polynomial_composition polynomial_second polynomial_second) polynomial_first)) | |
| CompanionComputedProposal.polynomial_proposalProgram | |
| private def defaultCode (z : List ℕ × Option (List ℕ)) : List ℕ := z.2.getD z.1 | |
| private def defaultCodeRealizer : Realizer defaultCode := | |
| (optionCases identity second).recode id defaultCode (fun _ => rfl) | |
| (by rintro ⟨d,o⟩; cases o <;> rfl) | |
| private lemma polynomial_defaultCode : PolynomialTime defaultCodeRealizer := by | |
| unfold defaultCodeRealizer | |
| apply polynomial_recode | |
| exact polynomial_optionCases polynomial_identity polynomial_second | |
| def candidate (z : Input) : List ℕ := (proposed z).getD z.1.2 | |
| def candidateRealizer : Realizer candidate := | |
| composition (f := fun z : Input => (z.1.2,proposed z)) | |
| (g := defaultCode) (pair (composition first second) proposedRealizer) defaultCodeRealizer | |
| lemma polynomial_candidate : PolynomialTime candidateRealizer := | |
| polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_proposed) polynomial_defaultCode | |
| attribute [local irreducible] candidate | |
| def currentInput (z : Input) : CompanionEvaluatedWeightProgram.Input := (z.1,z.2.2.2.1.1) | |
| def currentInputRealizer : Realizer currentInput := | |
| pair first (composition (composition (composition (composition second second) second) first) first) | |
| lemma polynomial_currentInput : PolynomialTime currentInputRealizer := | |
| polynomial_pair polynomial_first (polynomial_composition | |
| (polynomial_composition (polynomial_composition (polynomial_composition | |
| polynomial_second polynomial_second) polynomial_second) polynomial_first) polynomial_first) | |
| def targetInput (z : Input) : CompanionEvaluatedWeightProgram.Input := | |
| ((z.1.1,candidate z),z.2.2.2.1.2) | |
| def targetInputRealizer : Realizer targetInput := | |
| pair (pair (composition first first) candidateRealizer) | |
| (composition (composition (composition (composition second second) second) first) second) | |
| lemma polynomial_targetInput : PolynomialTime targetInputRealizer := | |
| polynomial_pair (polynomial_pair (polynomial_composition polynomial_first polynomial_first) polynomial_candidate) | |
| (polynomial_composition (polynomial_composition (polynomial_composition | |
| (polynomial_composition polynomial_second polynomial_second) polynomial_second) polynomial_first) polynomial_second) | |
| attribute [local irreducible] currentInput targetInput CompanionEvaluatedWeightProgram.output | |
| def currentValue (z : Input) : TreeFraction.Fraction := CompanionEvaluatedWeightProgram.output (currentInput z) | |
| def currentValueRealizer : Realizer currentValue := | |
| composition (f := currentInput) (g := CompanionEvaluatedWeightProgram.output) | |
| currentInputRealizer CompanionEvaluatedWeightProgram.outputRealizer | |
| lemma polynomial_currentValue : PolynomialTime currentValueRealizer := | |
| polynomial_composition polynomial_currentInput CompanionEvaluatedWeightProgram.polynomial_output | |
| def targetValue (z : Input) : TreeFraction.Fraction := CompanionEvaluatedWeightProgram.output (targetInput z) | |
| def targetValueRealizer : Realizer targetValue := | |
| composition (f := targetInput) (g := CompanionEvaluatedWeightProgram.output) | |
| targetInputRealizer CompanionEvaluatedWeightProgram.outputRealizer | |
| lemma polynomial_targetValue : PolynomialTime targetValueRealizer := | |
| polynomial_composition polynomial_targetInput CompanionEvaluatedWeightProgram.polynomial_output | |
| attribute [local irreducible] currentValue targetValue | |
| def acceptanceInput (z : Input) : FractionMetropolisProgram.Input := | |
| ((currentValue z,targetValue z),z.2.2.2.2) | |
| def acceptanceInputRealizer : Realizer acceptanceInput := | |
| pair (pair currentValueRealizer targetValueRealizer) | |
| (composition (composition (composition second second) second) second) | |
| lemma polynomial_acceptanceInput : PolynomialTime acceptanceInputRealizer := | |
| polynomial_pair (polynomial_pair polynomial_currentValue polynomial_targetValue) | |
| (polynomial_composition (polynomial_composition (polynomial_composition | |
| polynomial_second polynomial_second) polynomial_second) polynomial_second) | |
| def accepted (z : Input) : Bool := FractionMetropolisProgram.accept (acceptanceInput z) | |
| def acceptedRealizer : Realizer accepted := | |
| composition (f := acceptanceInput) (g := FractionMetropolisProgram.accept) | |
| acceptanceInputRealizer FractionMetropolisProgram.acceptRealizer | |
| lemma polynomial_accepted : PolynomialTime acceptedRealizer := | |
| polynomial_composition polynomial_acceptanceInput FractionMetropolisProgram.polynomial_accept | |
| attribute [local irreducible] accepted | |
| def output (z : Input) : List ℕ := if z.2.1 then z.1.2 else if accepted z then candidate z else z.1.2 | |
| def outputRealizer : Realizer output := | |
| choose (composition second first) (composition first second) | |
| (choose acceptedRealizer candidateRealizer (composition first second)) | |
| theorem polynomial_output : PolynomialTime outputRealizer := | |
| polynomial_choose (polynomial_composition polynomial_second polynomial_first) | |
| (polynomial_composition polynomial_first polynomial_second) | |
| (polynomial_choose polynomial_accepted polynomial_candidate | |
| (polynomial_composition polynomial_first polynomial_second)) | |
| lemma output_hold (z : Input) (h : z.2.1=true) : output z=z.1.2 := by | |
| simp only [output,h,ite_true] | |
| end ContingencyTables.CompanionMetropolisProgram | |
| end OAI | |