/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 433 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 3 ms] (6) TRS for Loop Detection (7) InfiniteLowerBoundProof [FINISHED, 0 ms] (8) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt) -> U22(isList) U22(tt) -> tt U31(tt) -> tt U41(tt) -> U42(isNeList) U42(tt) -> tt U51(tt) -> U52(isList) U52(tt) -> tt U61(tt) -> tt U71(tt) -> U72(isPal) U72(tt) -> tt U81(tt) -> tt isList -> U11(isNeList) isList -> tt isList -> U21(isList) isNeList -> U31(isQid) isNeList -> U41(isList) isNeList -> U51(isNeList) isNePal -> U61(isQid) isNePal -> U71(isQid) isPal -> U81(isNePal) isPal -> tt isQid -> tt S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(nil) -> nil encArg(tt) -> tt encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1)) -> U11(encArg(x_1)) encArg(cons_U21(x_1)) -> U21(encArg(x_1)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1)) -> U31(encArg(x_1)) encArg(cons_U41(x_1)) -> U41(encArg(x_1)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(encArg(x_1)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1)) -> U61(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_isList) -> isList encArg(cons_isNeList) -> isNeList encArg(cons_isNePal) -> isNePal encArg(cons_isPal) -> isPal encArg(cons_isQid) -> isQid encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U11(x_1) -> U11(encArg(x_1)) encode_tt -> tt encode_U21(x_1) -> U21(encArg(x_1)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isList -> isList encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1) -> U41(encArg(x_1)) encode_U42(x_1) -> U42(encArg(x_1)) encode_isNeList -> isNeList encode_U51(x_1) -> U51(encArg(x_1)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1) -> U61(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isPal -> isPal encode_U81(x_1) -> U81(encArg(x_1)) encode_isQid -> isQid encode_isNePal -> isNePal ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt) -> U22(isList) U22(tt) -> tt U31(tt) -> tt U41(tt) -> U42(isNeList) U42(tt) -> tt U51(tt) -> U52(isList) U52(tt) -> tt U61(tt) -> tt U71(tt) -> U72(isPal) U72(tt) -> tt U81(tt) -> tt isList -> U11(isNeList) isList -> tt isList -> U21(isList) isNeList -> U31(isQid) isNeList -> U41(isList) isNeList -> U51(isNeList) isNePal -> U61(isQid) isNePal -> U71(isQid) isPal -> U81(isNePal) isPal -> tt isQid -> tt The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(tt) -> tt encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1)) -> U11(encArg(x_1)) encArg(cons_U21(x_1)) -> U21(encArg(x_1)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1)) -> U31(encArg(x_1)) encArg(cons_U41(x_1)) -> U41(encArg(x_1)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(encArg(x_1)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1)) -> U61(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_isList) -> isList encArg(cons_isNeList) -> isNeList encArg(cons_isNePal) -> isNePal encArg(cons_isPal) -> isPal encArg(cons_isQid) -> isQid encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U11(x_1) -> U11(encArg(x_1)) encode_tt -> tt encode_U21(x_1) -> U21(encArg(x_1)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isList -> isList encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1) -> U41(encArg(x_1)) encode_U42(x_1) -> U42(encArg(x_1)) encode_isNeList -> isNeList encode_U51(x_1) -> U51(encArg(x_1)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1) -> U61(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isPal -> isPal encode_U81(x_1) -> U81(encArg(x_1)) encode_isQid -> isQid encode_isNePal -> isNePal Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt) -> U22(isList) U22(tt) -> tt U31(tt) -> tt U41(tt) -> U42(isNeList) U42(tt) -> tt U51(tt) -> U52(isList) U52(tt) -> tt U61(tt) -> tt U71(tt) -> U72(isPal) U72(tt) -> tt U81(tt) -> tt isList -> U11(isNeList) isList -> tt isList -> U21(isList) isNeList -> U31(isQid) isNeList -> U41(isList) isNeList -> U51(isNeList) isNePal -> U61(isQid) isNePal -> U71(isQid) isPal -> U81(isNePal) isPal -> tt isQid -> tt The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(tt) -> tt encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1)) -> U11(encArg(x_1)) encArg(cons_U21(x_1)) -> U21(encArg(x_1)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1)) -> U31(encArg(x_1)) encArg(cons_U41(x_1)) -> U41(encArg(x_1)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(encArg(x_1)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1)) -> U61(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_isList) -> isList encArg(cons_isNeList) -> isNeList encArg(cons_isNePal) -> isNePal encArg(cons_isPal) -> isPal encArg(cons_isQid) -> isQid encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U11(x_1) -> U11(encArg(x_1)) encode_tt -> tt encode_U21(x_1) -> U21(encArg(x_1)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isList -> isList encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1) -> U41(encArg(x_1)) encode_U42(x_1) -> U42(encArg(x_1)) encode_isNeList -> isNeList encode_U51(x_1) -> U51(encArg(x_1)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1) -> U61(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isPal -> isPal encode_U81(x_1) -> U81(encArg(x_1)) encode_isQid -> isQid encode_isNePal -> isNePal Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt) -> U22(isList) U22(tt) -> tt U31(tt) -> tt U41(tt) -> U42(isNeList) U42(tt) -> tt U51(tt) -> U52(isList) U52(tt) -> tt U61(tt) -> tt U71(tt) -> U72(isPal) U72(tt) -> tt U81(tt) -> tt isList -> U11(isNeList) isList -> tt isList -> U21(isList) isNeList -> U31(isQid) isNeList -> U41(isList) isNeList -> U51(isNeList) isNePal -> U61(isQid) isNePal -> U71(isQid) isPal -> U81(isNePal) isPal -> tt isQid -> tt The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(tt) -> tt encArg(cons___(x_1, x_2)) -> __(encArg(x_1), encArg(x_2)) encArg(cons_U11(x_1)) -> U11(encArg(x_1)) encArg(cons_U21(x_1)) -> U21(encArg(x_1)) encArg(cons_U22(x_1)) -> U22(encArg(x_1)) encArg(cons_U31(x_1)) -> U31(encArg(x_1)) encArg(cons_U41(x_1)) -> U41(encArg(x_1)) encArg(cons_U42(x_1)) -> U42(encArg(x_1)) encArg(cons_U51(x_1)) -> U51(encArg(x_1)) encArg(cons_U52(x_1)) -> U52(encArg(x_1)) encArg(cons_U61(x_1)) -> U61(encArg(x_1)) encArg(cons_U71(x_1)) -> U71(encArg(x_1)) encArg(cons_U72(x_1)) -> U72(encArg(x_1)) encArg(cons_U81(x_1)) -> U81(encArg(x_1)) encArg(cons_isList) -> isList encArg(cons_isNeList) -> isNeList encArg(cons_isNePal) -> isNePal encArg(cons_isPal) -> isPal encArg(cons_isQid) -> isQid encode___(x_1, x_2) -> __(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_U11(x_1) -> U11(encArg(x_1)) encode_tt -> tt encode_U21(x_1) -> U21(encArg(x_1)) encode_U22(x_1) -> U22(encArg(x_1)) encode_isList -> isList encode_U31(x_1) -> U31(encArg(x_1)) encode_U41(x_1) -> U41(encArg(x_1)) encode_U42(x_1) -> U42(encArg(x_1)) encode_isNeList -> isNeList encode_U51(x_1) -> U51(encArg(x_1)) encode_U52(x_1) -> U52(encArg(x_1)) encode_U61(x_1) -> U61(encArg(x_1)) encode_U71(x_1) -> U71(encArg(x_1)) encode_U72(x_1) -> U72(encArg(x_1)) encode_isPal -> isPal encode_U81(x_1) -> U81(encArg(x_1)) encode_isQid -> isQid encode_isNePal -> isNePal Rewrite Strategy: FULL ---------------------------------------- (7) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence isList ->^+ U21(isList) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [ ]. The result substitution is [ ]. ---------------------------------------- (8) BOUNDS(INF, INF)