/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 142 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 (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 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(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) The (relative) TRS S consists of the following rules: encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) The (relative) TRS S consists of the following rules: encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) The (relative) TRS S consists of the following rules: encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: FULL ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence h(c(s(x), c(s(0), y)), z) ->^+ h(y, c(s(0), c(x, z))) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [y / c(s(x), c(s(0), y))]. The result substitution is [z / c(s(0), c(x, z))]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) The (relative) TRS S consists of the following rules: encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: h(x, c(y, z)) -> h(c(s(y), x), z) h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) The (relative) TRS S consists of the following rules: encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2)) encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: FULL