/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 184 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt 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(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence isNat(s(x)) ->^+ isNat(x) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST