/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 224 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [FINISHED, 0 ms] (8) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) lowers(x, nil) -> nil lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) greaters(x, nil) -> nil greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_lowers(x_1, x_2)) -> lowers(encArg(x_1), encArg(x_2)) encArg(cons_greaters(x_1, x_2)) -> greaters(encArg(x_1), encArg(x_2)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) encode_lowers(x_1, x_2) -> lowers(encArg(x_1), encArg(x_2)) encode_greaters(x_1, x_2) -> greaters(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) lowers(x, nil) -> nil lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) greaters(x, nil) -> nil greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_lowers(x_1, x_2)) -> lowers(encArg(x_1), encArg(x_2)) encArg(cons_greaters(x_1, x_2)) -> greaters(encArg(x_1), encArg(x_2)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) encode_lowers(x_1, x_2) -> lowers(encArg(x_1), encArg(x_2)) encode_greaters(x_1, x_2) -> greaters(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) lowers(x, nil) -> nil lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) greaters(x, nil) -> nil greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_lowers(x_1, x_2)) -> lowers(encArg(x_1), encArg(x_2)) encArg(cons_greaters(x_1, x_2)) -> greaters(encArg(x_1), encArg(x_2)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) encode_lowers(x_1, x_2) -> lowers(encArg(x_1), encArg(x_2)) encode_greaters(x_1, x_2) -> greaters(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: qsort(nil) -> nil qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) lowers(x, nil) -> nil lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) greaters(x, nil) -> nil greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(<=(x_1, x_2)) -> <=(encArg(x_1), encArg(x_2)) encArg(cons_qsort(x_1)) -> qsort(encArg(x_1)) encArg(cons_lowers(x_1, x_2)) -> lowers(encArg(x_1), encArg(x_2)) encArg(cons_greaters(x_1, x_2)) -> greaters(encArg(x_1), encArg(x_2)) encode_qsort(x_1) -> qsort(encArg(x_1)) encode_nil -> nil encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) encode_lowers(x_1, x_2) -> lowers(encArg(x_1), encArg(x_2)) encode_greaters(x_1, x_2) -> greaters(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_<=(x_1, x_2) -> <=(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence greaters(x, .(y, z)) ->^+ if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z / .(y, z)]. The result substitution is [ ]. The rewrite sequence greaters(x, .(y, z)) ->^+ if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1]. The pumping substitution is [z / .(y, z)]. The result substitution is [ ]. ---------------------------------------- (8) BOUNDS(EXP, INF)