/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 61 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 113 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(x1))))))))))))))))) 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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(x1))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) 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(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(x1))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(x1))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. The certificate found is represented by the following graph. "[89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539] {(89,90,[0_1|0, 1_1|0, 2_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0]), (89,91,[0_1|1, 1_1|1, 2_1|1, 3_1|1]), (89,92,[2_1|2]), (89,108,[3_1|2]), (89,124,[1_1|2]), (89,140,[1_1|2]), (89,156,[2_1|2]), (89,172,[2_1|2]), (89,188,[0_1|2]), (89,204,[3_1|2]), (89,220,[2_1|2]), (89,236,[1_1|2]), (89,252,[2_1|2]), (89,268,[1_1|2]), (89,284,[3_1|2]), (89,300,[0_1|2]), (89,316,[2_1|2]), (89,332,[2_1|2]), (89,348,[1_1|2]), (89,364,[2_1|2]), (89,380,[2_1|2]), (89,396,[2_1|2]), (89,412,[2_1|2]), (89,428,[3_1|2]), (89,444,[3_1|2]), (89,460,[3_1|2]), (89,476,[3_1|2]), (89,492,[2_1|2]), (89,508,[2_1|2]), (89,524,[3_1|2]), (90,90,[cons_0_1|0, cons_1_1|0, cons_2_1|0, cons_3_1|0]), (91,90,[encArg_1|1]), (91,91,[0_1|1, 1_1|1, 2_1|1, 3_1|1]), (91,92,[2_1|2]), (91,108,[3_1|2]), (91,124,[1_1|2]), (91,140,[1_1|2]), (91,156,[2_1|2]), (91,172,[2_1|2]), (91,188,[0_1|2]), (91,204,[3_1|2]), (91,220,[2_1|2]), (91,236,[1_1|2]), (91,252,[2_1|2]), (91,268,[1_1|2]), (91,284,[3_1|2]), (91,300,[0_1|2]), (91,316,[2_1|2]), (91,332,[2_1|2]), (91,348,[1_1|2]), (91,364,[2_1|2]), (91,380,[2_1|2]), (91,396,[2_1|2]), (91,412,[2_1|2]), (91,428,[3_1|2]), (91,444,[3_1|2]), (91,460,[3_1|2]), (91,476,[3_1|2]), (91,492,[2_1|2]), (91,508,[2_1|2]), (91,524,[3_1|2]), (92,93,[3_1|2]), (93,94,[3_1|2]), (94,95,[1_1|2]), (95,96,[1_1|2]), (96,97,[2_1|2]), (97,98,[3_1|2]), (98,99,[1_1|2]), (99,100,[2_1|2]), (100,101,[1_1|2]), (101,102,[1_1|2]), (102,103,[2_1|2]), (103,104,[2_1|2]), (104,105,[1_1|2]), (105,106,[0_1|2]), (106,107,[2_1|2]), (106,332,[2_1|2]), (106,348,[1_1|2]), (106,364,[2_1|2]), (106,380,[2_1|2]), (107,91,[3_1|2]), (107,108,[3_1|2]), (107,204,[3_1|2]), (107,284,[3_1|2]), (107,428,[3_1|2]), (107,444,[3_1|2]), (107,460,[3_1|2]), 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(187,221,[1_1|2]), (187,317,[1_1|2]), (187,333,[1_1|2]), (187,397,[1_1|2]), (187,509,[1_1|2]), (187,142,[1_1|2]), (187,204,[3_1|2]), (187,220,[2_1|2]), (187,252,[2_1|2]), (188,189,[2_1|2]), (189,190,[2_1|2]), (190,191,[0_1|2]), (191,192,[1_1|2]), (192,193,[2_1|2]), (193,194,[1_1|2]), (194,195,[2_1|2]), (195,196,[1_1|2]), (196,197,[3_1|2]), (197,198,[1_1|2]), (198,199,[3_1|2]), (199,200,[0_1|2]), (200,201,[3_1|2]), (201,202,[3_1|2]), (201,396,[2_1|2]), (201,412,[2_1|2]), (201,428,[3_1|2]), (202,203,[0_1|2]), (202,156,[2_1|2]), (202,172,[2_1|2]), (202,188,[0_1|2]), (203,91,[3_1|2]), (203,108,[3_1|2]), (203,204,[3_1|2]), (203,284,[3_1|2]), (203,428,[3_1|2]), (203,444,[3_1|2]), (203,460,[3_1|2]), (203,476,[3_1|2]), (203,524,[3_1|2]), (203,396,[2_1|2]), (203,412,[2_1|2]), (203,492,[2_1|2]), (203,508,[2_1|2]), (204,205,[3_1|2]), (205,206,[3_1|2]), (206,207,[1_1|2]), (207,208,[3_1|2]), (208,209,[3_1|2]), (209,210,[0_1|2]), (210,211,[1_1|2]), (211,212,[0_1|2]), (212,213,[1_1|2]), 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(369,370,[1_1|2]), (370,371,[3_1|2]), (371,372,[1_1|2]), (372,373,[3_1|2]), (373,374,[3_1|2]), (374,375,[3_1|2]), (375,376,[3_1|2]), (376,377,[0_1|2]), (377,378,[3_1|2]), (378,379,[0_1|2]), (378,92,[2_1|2]), (378,108,[3_1|2]), (378,124,[1_1|2]), (378,140,[1_1|2]), (379,91,[0_1|2]), (379,124,[0_1|2, 1_1|2]), (379,140,[0_1|2, 1_1|2]), (379,236,[0_1|2]), (379,268,[0_1|2]), (379,348,[0_1|2]), (379,302,[0_1|2]), (379,92,[2_1|2]), (379,108,[3_1|2]), (379,156,[2_1|2]), (379,172,[2_1|2]), (379,188,[0_1|2]), (380,381,[2_1|2]), (381,382,[1_1|2]), (382,383,[0_1|2]), (383,384,[2_1|2]), (384,385,[1_1|2]), (385,386,[1_1|2]), (386,387,[1_1|2]), (387,388,[2_1|2]), (388,389,[2_1|2]), (389,390,[1_1|2]), (390,391,[1_1|2]), (391,392,[3_1|2]), (391,524,[3_1|2]), (392,393,[3_1|2]), (393,394,[2_1|2]), (394,395,[1_1|2]), (394,268,[1_1|2]), (395,91,[3_1|2]), (395,124,[3_1|2]), (395,140,[3_1|2]), (395,236,[3_1|2]), (395,268,[3_1|2]), (395,348,[3_1|2]), (395,396,[2_1|2]), (395,412,[2_1|2]), (395,428,[3_1|2]), (395,444,[3_1|2]), (395,460,[3_1|2]), (395,476,[3_1|2]), (395,492,[2_1|2]), (395,508,[2_1|2]), (395,524,[3_1|2]), (396,397,[1_1|2]), (397,398,[2_1|2]), (398,399,[1_1|2]), (399,400,[3_1|2]), (400,401,[2_1|2]), (401,402,[1_1|2]), (402,403,[3_1|2]), (403,404,[2_1|2]), (404,405,[1_1|2]), (405,406,[2_1|2]), (406,407,[1_1|2]), (407,408,[3_1|2]), (408,409,[1_1|2]), (409,410,[3_1|2]), (409,508,[2_1|2]), (410,411,[3_1|2]), (410,444,[3_1|2]), (411,91,[1_1|2]), (411,124,[1_1|2]), (411,140,[1_1|2]), (411,236,[1_1|2]), (411,268,[1_1|2]), (411,348,[1_1|2]), (411,221,[1_1|2]), (411,317,[1_1|2]), (411,333,[1_1|2]), (411,397,[1_1|2]), (411,509,[1_1|2]), (411,142,[1_1|2]), (411,204,[3_1|2]), (411,220,[2_1|2]), (411,252,[2_1|2]), (412,413,[2_1|2]), (413,414,[2_1|2]), (414,415,[1_1|2]), (415,416,[3_1|2]), (416,417,[3_1|2]), (417,418,[3_1|2]), (418,419,[2_1|2]), (419,420,[1_1|2]), (420,421,[1_1|2]), (421,422,[2_1|2]), (422,423,[1_1|2]), (423,424,[1_1|2]), (424,425,[1_1|2]), (425,426,[1_1|2]), (426,427,[3_1|2]), (426,444,[3_1|2]), (427,91,[1_1|2]), (427,124,[1_1|2]), (427,140,[1_1|2]), (427,236,[1_1|2]), (427,268,[1_1|2]), (427,348,[1_1|2]), (427,221,[1_1|2]), (427,317,[1_1|2]), (427,333,[1_1|2]), (427,397,[1_1|2]), (427,509,[1_1|2]), (427,254,[1_1|2]), (427,382,[1_1|2]), (427,494,[1_1|2]), (427,415,[1_1|2]), (427,204,[3_1|2]), (427,220,[2_1|2]), (427,252,[2_1|2]), (428,429,[1_1|2]), (429,430,[3_1|2]), (430,431,[0_1|2]), (431,432,[3_1|2]), (432,433,[3_1|2]), (433,434,[2_1|2]), (434,435,[2_1|2]), (435,436,[1_1|2]), (436,437,[1_1|2]), (437,438,[2_1|2]), (438,439,[2_1|2]), (439,440,[2_1|2]), (440,441,[1_1|2]), (441,442,[1_1|2]), (442,443,[0_1|2]), (443,91,[2_1|2]), (443,188,[2_1|2]), (443,300,[2_1|2, 0_1|2]), (443,301,[2_1|2]), (443,284,[3_1|2]), (443,316,[2_1|2]), (443,332,[2_1|2]), (443,348,[1_1|2]), (443,364,[2_1|2]), (443,380,[2_1|2]), (444,445,[3_1|2]), (445,446,[2_1|2]), (446,447,[1_1|2]), (447,448,[3_1|2]), (448,449,[1_1|2]), (449,450,[2_1|2]), (450,451,[3_1|2]), (451,452,[3_1|2]), (452,453,[3_1|2]), (453,454,[2_1|2]), (454,455,[1_1|2]), (455,456,[1_1|2]), (456,457,[2_1|2]), (457,458,[1_1|2]), (457,204,[3_1|2]), (458,459,[0_1|2]), (459,91,[1_1|2]), (459,124,[1_1|2]), (459,140,[1_1|2]), (459,236,[1_1|2]), (459,268,[1_1|2]), (459,348,[1_1|2]), (459,204,[3_1|2]), (459,220,[2_1|2]), (459,252,[2_1|2]), (460,461,[3_1|2]), (461,462,[3_1|2]), (462,463,[2_1|2]), (463,464,[3_1|2]), (464,465,[1_1|2]), (465,466,[1_1|2]), (466,467,[3_1|2]), (467,468,[3_1|2]), (468,469,[2_1|2]), (469,470,[1_1|2]), (470,471,[2_1|2]), (471,472,[1_1|2]), (472,473,[2_1|2]), (473,474,[1_1|2]), (474,475,[3_1|2]), (474,444,[3_1|2]), (475,91,[1_1|2]), (475,124,[1_1|2]), (475,140,[1_1|2]), (475,236,[1_1|2]), (475,268,[1_1|2]), (475,348,[1_1|2]), (475,221,[1_1|2]), (475,317,[1_1|2]), (475,333,[1_1|2]), (475,397,[1_1|2]), (475,509,[1_1|2]), (475,110,[1_1|2]), (475,526,[1_1|2]), (475,447,[1_1|2]), (475,479,[1_1|2]), (475,204,[3_1|2]), (475,220,[2_1|2]), (475,252,[2_1|2]), (476,477,[3_1|2]), (477,478,[2_1|2]), (478,479,[1_1|2]), (479,480,[1_1|2]), (480,481,[2_1|2]), (481,482,[3_1|2]), (482,483,[2_1|2]), (483,484,[1_1|2]), (484,485,[2_1|2]), (485,486,[0_1|2]), (486,487,[3_1|2]), (487,488,[1_1|2]), (488,489,[3_1|2]), (488,460,[3_1|2]), (489,490,[2_1|2]), (490,491,[2_1|2]), (490,300,[0_1|2]), (491,91,[1_1|2]), (491,124,[1_1|2]), (491,140,[1_1|2]), (491,236,[1_1|2]), (491,268,[1_1|2]), (491,348,[1_1|2]), (491,221,[1_1|2]), (491,317,[1_1|2]), (491,333,[1_1|2]), (491,397,[1_1|2]), (491,509,[1_1|2]), (491,204,[3_1|2]), (491,220,[2_1|2]), (491,252,[2_1|2]), (492,493,[2_1|2]), (493,494,[1_1|2]), (494,495,[1_1|2]), (495,496,[2_1|2]), (496,497,[1_1|2]), (497,498,[3_1|2]), (498,499,[3_1|2]), (499,500,[2_1|2]), (500,501,[0_1|2]), (501,502,[0_1|2]), (502,503,[0_1|2]), (503,504,[0_1|2]), (504,505,[0_1|2]), (505,506,[1_1|2]), (506,507,[3_1|2]), (506,508,[2_1|2]), (506,524,[3_1|2]), (507,91,[3_1|2]), (507,124,[3_1|2]), (507,140,[3_1|2]), (507,236,[3_1|2]), (507,268,[3_1|2]), (507,348,[3_1|2]), (507,396,[2_1|2]), (507,412,[2_1|2]), (507,428,[3_1|2]), (507,444,[3_1|2]), (507,460,[3_1|2]), (507,476,[3_1|2]), (507,492,[2_1|2]), (507,508,[2_1|2]), (507,524,[3_1|2]), (508,509,[1_1|2]), (509,510,[2_1|2]), (510,511,[1_1|2]), (511,512,[3_1|2]), (512,513,[0_1|2]), (513,514,[2_1|2]), (514,515,[2_1|2]), (515,516,[0_1|2]), (516,517,[2_1|2]), (517,518,[2_1|2]), (518,519,[1_1|2]), (519,520,[1_1|2]), (520,521,[3_1|2]), (521,522,[3_1|2]), (522,523,[3_1|2]), (522,508,[2_1|2]), (522,524,[3_1|2]), (523,91,[3_1|2]), (523,188,[3_1|2]), (523,300,[3_1|2]), (523,301,[3_1|2]), (523,396,[2_1|2]), (523,412,[2_1|2]), (523,428,[3_1|2]), (523,444,[3_1|2]), (523,460,[3_1|2]), (523,476,[3_1|2]), (523,492,[2_1|2]), (523,508,[2_1|2]), (523,524,[3_1|2]), (524,525,[2_1|2]), (525,526,[1_1|2]), (526,527,[3_1|2]), (527,528,[2_1|2]), (528,529,[2_1|2]), (529,530,[1_1|2]), (530,531,[3_1|2]), (531,532,[3_1|2]), (532,533,[3_1|2]), (533,534,[1_1|2]), (534,535,[0_1|2]), (535,536,[2_1|2]), (536,537,[1_1|2]), (537,538,[3_1|2]), (538,539,[3_1|2]), (538,508,[2_1|2]), (538,524,[3_1|2]), (539,91,[3_1|2]), (539,108,[3_1|2]), (539,204,[3_1|2]), (539,284,[3_1|2]), (539,428,[3_1|2]), (539,444,[3_1|2]), (539,460,[3_1|2]), (539,476,[3_1|2]), (539,524,[3_1|2]), (539,205,[3_1|2]), (539,285,[3_1|2]), (539,445,[3_1|2]), (539,461,[3_1|2]), (539,477,[3_1|2]), (539,94,[3_1|2]), (539,158,[3_1|2]), (539,396,[2_1|2]), (539,412,[2_1|2]), (539,492,[2_1|2]), (539,508,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)