/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 34 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 507 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) 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(2(2(1(2(0(0(1(0(x1))))))))))))) -> 0(2(2(0(2(2(0(1(0(2(2(2(2(2(0(2(0(x1))))))))))))))))) 0(0(0(0(2(0(0(1(0(2(2(2(0(x1))))))))))))) -> 0(2(2(0(2(2(0(0(0(0(2(0(2(0(0(0(2(x1))))))))))))))))) 0(0(0(0(2(2(1(2(0(2(2(1(0(x1))))))))))))) -> 0(1(0(0(1(0(2(0(2(2(1(0(0(2(0(2(0(x1))))))))))))))))) 0(0(0(1(1(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(0(1(0(1(2(2(2(0(0(2(0(x1))))))))))))))))) 0(0(1(0(0(0(0(0(0(0(2(1(0(x1))))))))))))) -> 0(2(0(0(1(0(2(0(2(0(0(2(0(2(0(2(0(x1))))))))))))))))) 0(0(1(0(2(0(1(0(0(2(0(1(0(x1))))))))))))) -> 1(2(1(0(0(0(0(0(2(0(0(2(0(2(2(2(2(x1))))))))))))))))) 0(0(1(1(2(0(0(1(0(2(0(2(2(x1))))))))))))) -> 0(1(2(0(2(0(0(2(0(2(2(0(2(2(2(1(0(x1))))))))))))))))) 0(0(1(2(0(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(1(2(0(2(0(0(0(0(2(2(2(x1))))))))))))))))) 0(0(2(0(0(1(0(1(2(0(0(0(2(x1))))))))))))) -> 0(0(0(2(0(2(0(1(0(2(0(2(2(0(1(0(0(x1))))))))))))))))) 0(0(2(0(2(1(1(1(0(2(2(1(0(x1))))))))))))) -> 0(0(0(0(0(0(1(2(2(2(2(0(1(2(2(0(0(x1))))))))))))))))) 0(0(2(1(0(1(0(1(2(2(0(2(0(x1))))))))))))) -> 0(0(2(2(1(1(0(2(2(0(0(1(0(2(2(0(0(x1))))))))))))))))) 0(0(2(2(2(0(2(1(1(0(0(0(2(x1))))))))))))) -> 0(0(0(0(1(0(2(2(0(0(0(1(0(0(2(0(0(x1))))))))))))))))) 0(1(0(0(2(2(0(0(1(2(2(2(0(x1))))))))))))) -> 1(2(1(0(2(0(0(2(0(0(2(2(0(0(2(2(2(x1))))))))))))))))) 0(1(0(2(1(0(1(1(0(0(2(0(0(x1))))))))))))) -> 0(2(0(1(2(0(0(0(0(0(2(0(1(2(0(0(2(x1))))))))))))))))) 0(1(1(0(2(2(0(1(0(0(0(0(0(x1))))))))))))) -> 0(0(0(0(1(1(2(2(0(2(2(2(2(0(2(0(0(x1))))))))))))))))) 0(1(1(2(1(0(2(2(0(2(0(2(2(x1))))))))))))) -> 0(0(0(1(0(1(0(2(0(2(1(2(0(0(2(0(0(x1))))))))))))))))) 0(1(2(0(0(1(0(2(1(0(2(2(1(x1))))))))))))) -> 0(1(2(1(0(0(0(2(0(0(1(0(0(2(0(0(1(x1))))))))))))))))) 0(1(2(2(2(1(0(0(2(1(1(0(0(x1))))))))))))) -> 0(0(0(1(0(1(2(0(1(2(0(2(2(2(2(0(2(x1))))))))))))))))) 0(2(2(0(0(1(2(1(0(0(1(0(2(x1))))))))))))) -> 0(2(2(2(1(0(2(2(1(0(0(0(2(0(0(1(0(x1))))))))))))))))) 0(2(2(0(2(2(0(0(1(0(0(0(0(x1))))))))))))) -> 0(0(0(2(2(2(0(2(0(2(2(2(2(2(2(2(2(x1))))))))))))))))) 0(2(2(0(2(2(1(2(2(2(2(2(2(x1))))))))))))) -> 0(2(1(2(0(2(0(0(0(2(0(2(2(0(0(2(0(x1))))))))))))))))) 1(0(0(0(2(2(2(0(0(1(0(1(0(x1))))))))))))) -> 0(1(0(0(0(0(2(2(2(0(0(2(0(0(2(1(0(x1))))))))))))))))) 1(0(1(0(2(2(0(2(0(0(0(0(1(x1))))))))))))) -> 2(1(0(0(0(2(0(2(0(0(0(2(2(2(2(0(1(x1))))))))))))))))) 1(0(1(2(0(1(0(2(1(2(1(0(0(x1))))))))))))) -> 1(1(2(2(1(2(2(0(0(2(0(1(0(1(0(2(2(x1))))))))))))))))) 1(0(2(2(0(2(0(2(2(2(0(1(0(x1))))))))))))) -> 2(2(2(0(2(0(2(2(0(2(0(2(0(2(2(2(2(x1))))))))))))))))) 1(1(0(0(0(0(2(1(0(0(2(2(0(x1))))))))))))) -> 2(2(2(0(0(0(0(0(0(2(2(0(0(0(1(0(2(x1))))))))))))))))) 1(2(2(2(2(1(0(0(0(2(2(0(2(x1))))))))))))) -> 0(0(2(2(1(1(0(0(0(0(1(0(0(0(2(0(2(x1))))))))))))))))) 2(0(0(0(2(0(1(2(2(2(0(2(2(x1))))))))))))) -> 2(0(0(2(2(2(0(2(0(2(0(2(0(2(1(1(0(x1))))))))))))))))) 2(0(0(2(0(0(1(1(0(0(2(0(0(x1))))))))))))) -> 1(0(2(2(0(2(2(2(1(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(0(0(2(2(1(0(1(0(0(2(2(0(x1))))))))))))) -> 2(2(0(0(2(1(0(2(0(2(0(0(0(0(2(0(2(x1))))))))))))))))) 2(2(0(0(0(2(1(0(0(0(2(0(2(x1))))))))))))) -> 0(0(2(0(2(2(2(0(2(0(2(0(0(2(0(2(0(x1))))))))))))))))) 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(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) 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(2(2(1(2(0(0(1(0(x1))))))))))))) -> 0(2(2(0(2(2(0(1(0(2(2(2(2(2(0(2(0(x1))))))))))))))))) 0(0(0(0(2(0(0(1(0(2(2(2(0(x1))))))))))))) -> 0(2(2(0(2(2(0(0(0(0(2(0(2(0(0(0(2(x1))))))))))))))))) 0(0(0(0(2(2(1(2(0(2(2(1(0(x1))))))))))))) -> 0(1(0(0(1(0(2(0(2(2(1(0(0(2(0(2(0(x1))))))))))))))))) 0(0(0(1(1(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(0(1(0(1(2(2(2(0(0(2(0(x1))))))))))))))))) 0(0(1(0(0(0(0(0(0(0(2(1(0(x1))))))))))))) -> 0(2(0(0(1(0(2(0(2(0(0(2(0(2(0(2(0(x1))))))))))))))))) 0(0(1(0(2(0(1(0(0(2(0(1(0(x1))))))))))))) -> 1(2(1(0(0(0(0(0(2(0(0(2(0(2(2(2(2(x1))))))))))))))))) 0(0(1(1(2(0(0(1(0(2(0(2(2(x1))))))))))))) -> 0(1(2(0(2(0(0(2(0(2(2(0(2(2(2(1(0(x1))))))))))))))))) 0(0(1(2(0(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(1(2(0(2(0(0(0(0(2(2(2(x1))))))))))))))))) 0(0(2(0(0(1(0(1(2(0(0(0(2(x1))))))))))))) -> 0(0(0(2(0(2(0(1(0(2(0(2(2(0(1(0(0(x1))))))))))))))))) 0(0(2(0(2(1(1(1(0(2(2(1(0(x1))))))))))))) -> 0(0(0(0(0(0(1(2(2(2(2(0(1(2(2(0(0(x1))))))))))))))))) 0(0(2(1(0(1(0(1(2(2(0(2(0(x1))))))))))))) -> 0(0(2(2(1(1(0(2(2(0(0(1(0(2(2(0(0(x1))))))))))))))))) 0(0(2(2(2(0(2(1(1(0(0(0(2(x1))))))))))))) -> 0(0(0(0(1(0(2(2(0(0(0(1(0(0(2(0(0(x1))))))))))))))))) 0(1(0(0(2(2(0(0(1(2(2(2(0(x1))))))))))))) -> 1(2(1(0(2(0(0(2(0(0(2(2(0(0(2(2(2(x1))))))))))))))))) 0(1(0(2(1(0(1(1(0(0(2(0(0(x1))))))))))))) -> 0(2(0(1(2(0(0(0(0(0(2(0(1(2(0(0(2(x1))))))))))))))))) 0(1(1(0(2(2(0(1(0(0(0(0(0(x1))))))))))))) -> 0(0(0(0(1(1(2(2(0(2(2(2(2(0(2(0(0(x1))))))))))))))))) 0(1(1(2(1(0(2(2(0(2(0(2(2(x1))))))))))))) -> 0(0(0(1(0(1(0(2(0(2(1(2(0(0(2(0(0(x1))))))))))))))))) 0(1(2(0(0(1(0(2(1(0(2(2(1(x1))))))))))))) -> 0(1(2(1(0(0(0(2(0(0(1(0(0(2(0(0(1(x1))))))))))))))))) 0(1(2(2(2(1(0(0(2(1(1(0(0(x1))))))))))))) -> 0(0(0(1(0(1(2(0(1(2(0(2(2(2(2(0(2(x1))))))))))))))))) 0(2(2(0(0(1(2(1(0(0(1(0(2(x1))))))))))))) -> 0(2(2(2(1(0(2(2(1(0(0(0(2(0(0(1(0(x1))))))))))))))))) 0(2(2(0(2(2(0(0(1(0(0(0(0(x1))))))))))))) -> 0(0(0(2(2(2(0(2(0(2(2(2(2(2(2(2(2(x1))))))))))))))))) 0(2(2(0(2(2(1(2(2(2(2(2(2(x1))))))))))))) -> 0(2(1(2(0(2(0(0(0(2(0(2(2(0(0(2(0(x1))))))))))))))))) 1(0(0(0(2(2(2(0(0(1(0(1(0(x1))))))))))))) -> 0(1(0(0(0(0(2(2(2(0(0(2(0(0(2(1(0(x1))))))))))))))))) 1(0(1(0(2(2(0(2(0(0(0(0(1(x1))))))))))))) -> 2(1(0(0(0(2(0(2(0(0(0(2(2(2(2(0(1(x1))))))))))))))))) 1(0(1(2(0(1(0(2(1(2(1(0(0(x1))))))))))))) -> 1(1(2(2(1(2(2(0(0(2(0(1(0(1(0(2(2(x1))))))))))))))))) 1(0(2(2(0(2(0(2(2(2(0(1(0(x1))))))))))))) -> 2(2(2(0(2(0(2(2(0(2(0(2(0(2(2(2(2(x1))))))))))))))))) 1(1(0(0(0(0(2(1(0(0(2(2(0(x1))))))))))))) -> 2(2(2(0(0(0(0(0(0(2(2(0(0(0(1(0(2(x1))))))))))))))))) 1(2(2(2(2(1(0(0(0(2(2(0(2(x1))))))))))))) -> 0(0(2(2(1(1(0(0(0(0(1(0(0(0(2(0(2(x1))))))))))))))))) 2(0(0(0(2(0(1(2(2(2(0(2(2(x1))))))))))))) -> 2(0(0(2(2(2(0(2(0(2(0(2(0(2(1(1(0(x1))))))))))))))))) 2(0(0(2(0(0(1(1(0(0(2(0(0(x1))))))))))))) -> 1(0(2(2(0(2(2(2(1(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(0(0(2(2(1(0(1(0(0(2(2(0(x1))))))))))))) -> 2(2(0(0(2(1(0(2(0(2(0(0(0(0(2(0(2(x1))))))))))))))))) 2(2(0(0(0(2(1(0(0(0(2(0(2(x1))))))))))))) -> 0(0(2(0(2(2(2(0(2(0(2(0(0(2(0(2(0(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) 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(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(2(2(1(2(0(0(1(0(x1))))))))))))) -> 0(2(2(0(2(2(0(1(0(2(2(2(2(2(0(2(0(x1))))))))))))))))) 0(0(0(0(2(0(0(1(0(2(2(2(0(x1))))))))))))) -> 0(2(2(0(2(2(0(0(0(0(2(0(2(0(0(0(2(x1))))))))))))))))) 0(0(0(0(2(2(1(2(0(2(2(1(0(x1))))))))))))) -> 0(1(0(0(1(0(2(0(2(2(1(0(0(2(0(2(0(x1))))))))))))))))) 0(0(0(1(1(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(0(1(0(1(2(2(2(0(0(2(0(x1))))))))))))))))) 0(0(1(0(0(0(0(0(0(0(2(1(0(x1))))))))))))) -> 0(2(0(0(1(0(2(0(2(0(0(2(0(2(0(2(0(x1))))))))))))))))) 0(0(1(0(2(0(1(0(0(2(0(1(0(x1))))))))))))) -> 1(2(1(0(0(0(0(0(2(0(0(2(0(2(2(2(2(x1))))))))))))))))) 0(0(1(1(2(0(0(1(0(2(0(2(2(x1))))))))))))) -> 0(1(2(0(2(0(0(2(0(2(2(0(2(2(2(1(0(x1))))))))))))))))) 0(0(1(2(0(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(1(2(0(2(0(0(0(0(2(2(2(x1))))))))))))))))) 0(0(2(0(0(1(0(1(2(0(0(0(2(x1))))))))))))) -> 0(0(0(2(0(2(0(1(0(2(0(2(2(0(1(0(0(x1))))))))))))))))) 0(0(2(0(2(1(1(1(0(2(2(1(0(x1))))))))))))) -> 0(0(0(0(0(0(1(2(2(2(2(0(1(2(2(0(0(x1))))))))))))))))) 0(0(2(1(0(1(0(1(2(2(0(2(0(x1))))))))))))) -> 0(0(2(2(1(1(0(2(2(0(0(1(0(2(2(0(0(x1))))))))))))))))) 0(0(2(2(2(0(2(1(1(0(0(0(2(x1))))))))))))) -> 0(0(0(0(1(0(2(2(0(0(0(1(0(0(2(0(0(x1))))))))))))))))) 0(1(0(0(2(2(0(0(1(2(2(2(0(x1))))))))))))) -> 1(2(1(0(2(0(0(2(0(0(2(2(0(0(2(2(2(x1))))))))))))))))) 0(1(0(2(1(0(1(1(0(0(2(0(0(x1))))))))))))) -> 0(2(0(1(2(0(0(0(0(0(2(0(1(2(0(0(2(x1))))))))))))))))) 0(1(1(0(2(2(0(1(0(0(0(0(0(x1))))))))))))) -> 0(0(0(0(1(1(2(2(0(2(2(2(2(0(2(0(0(x1))))))))))))))))) 0(1(1(2(1(0(2(2(0(2(0(2(2(x1))))))))))))) -> 0(0(0(1(0(1(0(2(0(2(1(2(0(0(2(0(0(x1))))))))))))))))) 0(1(2(0(0(1(0(2(1(0(2(2(1(x1))))))))))))) -> 0(1(2(1(0(0(0(2(0(0(1(0(0(2(0(0(1(x1))))))))))))))))) 0(1(2(2(2(1(0(0(2(1(1(0(0(x1))))))))))))) -> 0(0(0(1(0(1(2(0(1(2(0(2(2(2(2(0(2(x1))))))))))))))))) 0(2(2(0(0(1(2(1(0(0(1(0(2(x1))))))))))))) -> 0(2(2(2(1(0(2(2(1(0(0(0(2(0(0(1(0(x1))))))))))))))))) 0(2(2(0(2(2(0(0(1(0(0(0(0(x1))))))))))))) -> 0(0(0(2(2(2(0(2(0(2(2(2(2(2(2(2(2(x1))))))))))))))))) 0(2(2(0(2(2(1(2(2(2(2(2(2(x1))))))))))))) -> 0(2(1(2(0(2(0(0(0(2(0(2(2(0(0(2(0(x1))))))))))))))))) 1(0(0(0(2(2(2(0(0(1(0(1(0(x1))))))))))))) -> 0(1(0(0(0(0(2(2(2(0(0(2(0(0(2(1(0(x1))))))))))))))))) 1(0(1(0(2(2(0(2(0(0(0(0(1(x1))))))))))))) -> 2(1(0(0(0(2(0(2(0(0(0(2(2(2(2(0(1(x1))))))))))))))))) 1(0(1(2(0(1(0(2(1(2(1(0(0(x1))))))))))))) -> 1(1(2(2(1(2(2(0(0(2(0(1(0(1(0(2(2(x1))))))))))))))))) 1(0(2(2(0(2(0(2(2(2(0(1(0(x1))))))))))))) -> 2(2(2(0(2(0(2(2(0(2(0(2(0(2(2(2(2(x1))))))))))))))))) 1(1(0(0(0(0(2(1(0(0(2(2(0(x1))))))))))))) -> 2(2(2(0(0(0(0(0(0(2(2(0(0(0(1(0(2(x1))))))))))))))))) 1(2(2(2(2(1(0(0(0(2(2(0(2(x1))))))))))))) -> 0(0(2(2(1(1(0(0(0(0(1(0(0(0(2(0(2(x1))))))))))))))))) 2(0(0(0(2(0(1(2(2(2(0(2(2(x1))))))))))))) -> 2(0(0(2(2(2(0(2(0(2(0(2(0(2(1(1(0(x1))))))))))))))))) 2(0(0(2(0(0(1(1(0(0(2(0(0(x1))))))))))))) -> 1(0(2(2(0(2(2(2(1(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(0(0(2(2(1(0(1(0(0(2(2(0(x1))))))))))))) -> 2(2(0(0(2(1(0(2(0(2(0(0(0(0(2(0(2(x1))))))))))))))))) 2(2(0(0(0(2(1(0(0(0(2(0(2(x1))))))))))))) -> 0(0(2(0(2(2(2(0(2(0(2(0(0(2(0(2(0(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) 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(2(2(1(2(0(0(1(0(x1))))))))))))) -> 0(2(2(0(2(2(0(1(0(2(2(2(2(2(0(2(0(x1))))))))))))))))) 0(0(0(0(2(0(0(1(0(2(2(2(0(x1))))))))))))) -> 0(2(2(0(2(2(0(0(0(0(2(0(2(0(0(0(2(x1))))))))))))))))) 0(0(0(0(2(2(1(2(0(2(2(1(0(x1))))))))))))) -> 0(1(0(0(1(0(2(0(2(2(1(0(0(2(0(2(0(x1))))))))))))))))) 0(0(0(1(1(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(0(1(0(1(2(2(2(0(0(2(0(x1))))))))))))))))) 0(0(1(0(0(0(0(0(0(0(2(1(0(x1))))))))))))) -> 0(2(0(0(1(0(2(0(2(0(0(2(0(2(0(2(0(x1))))))))))))))))) 0(0(1(0(2(0(1(0(0(2(0(1(0(x1))))))))))))) -> 1(2(1(0(0(0(0(0(2(0(0(2(0(2(2(2(2(x1))))))))))))))))) 0(0(1(1(2(0(0(1(0(2(0(2(2(x1))))))))))))) -> 0(1(2(0(2(0(0(2(0(2(2(0(2(2(2(1(0(x1))))))))))))))))) 0(0(1(2(0(0(2(0(1(0(0(2(2(x1))))))))))))) -> 0(0(2(0(0(0(1(2(0(2(0(0(0(0(2(2(2(x1))))))))))))))))) 0(0(2(0(0(1(0(1(2(0(0(0(2(x1))))))))))))) -> 0(0(0(2(0(2(0(1(0(2(0(2(2(0(1(0(0(x1))))))))))))))))) 0(0(2(0(2(1(1(1(0(2(2(1(0(x1))))))))))))) -> 0(0(0(0(0(0(1(2(2(2(2(0(1(2(2(0(0(x1))))))))))))))))) 0(0(2(1(0(1(0(1(2(2(0(2(0(x1))))))))))))) -> 0(0(2(2(1(1(0(2(2(0(0(1(0(2(2(0(0(x1))))))))))))))))) 0(0(2(2(2(0(2(1(1(0(0(0(2(x1))))))))))))) -> 0(0(0(0(1(0(2(2(0(0(0(1(0(0(2(0(0(x1))))))))))))))))) 0(1(0(0(2(2(0(0(1(2(2(2(0(x1))))))))))))) -> 1(2(1(0(2(0(0(2(0(0(2(2(0(0(2(2(2(x1))))))))))))))))) 0(1(0(2(1(0(1(1(0(0(2(0(0(x1))))))))))))) -> 0(2(0(1(2(0(0(0(0(0(2(0(1(2(0(0(2(x1))))))))))))))))) 0(1(1(0(2(2(0(1(0(0(0(0(0(x1))))))))))))) -> 0(0(0(0(1(1(2(2(0(2(2(2(2(0(2(0(0(x1))))))))))))))))) 0(1(1(2(1(0(2(2(0(2(0(2(2(x1))))))))))))) -> 0(0(0(1(0(1(0(2(0(2(1(2(0(0(2(0(0(x1))))))))))))))))) 0(1(2(0(0(1(0(2(1(0(2(2(1(x1))))))))))))) -> 0(1(2(1(0(0(0(2(0(0(1(0(0(2(0(0(1(x1))))))))))))))))) 0(1(2(2(2(1(0(0(2(1(1(0(0(x1))))))))))))) -> 0(0(0(1(0(1(2(0(1(2(0(2(2(2(2(0(2(x1))))))))))))))))) 0(2(2(0(0(1(2(1(0(0(1(0(2(x1))))))))))))) -> 0(2(2(2(1(0(2(2(1(0(0(0(2(0(0(1(0(x1))))))))))))))))) 0(2(2(0(2(2(0(0(1(0(0(0(0(x1))))))))))))) -> 0(0(0(2(2(2(0(2(0(2(2(2(2(2(2(2(2(x1))))))))))))))))) 0(2(2(0(2(2(1(2(2(2(2(2(2(x1))))))))))))) -> 0(2(1(2(0(2(0(0(0(2(0(2(2(0(0(2(0(x1))))))))))))))))) 1(0(0(0(2(2(2(0(0(1(0(1(0(x1))))))))))))) -> 0(1(0(0(0(0(2(2(2(0(0(2(0(0(2(1(0(x1))))))))))))))))) 1(0(1(0(2(2(0(2(0(0(0(0(1(x1))))))))))))) -> 2(1(0(0(0(2(0(2(0(0(0(2(2(2(2(0(1(x1))))))))))))))))) 1(0(1(2(0(1(0(2(1(2(1(0(0(x1))))))))))))) -> 1(1(2(2(1(2(2(0(0(2(0(1(0(1(0(2(2(x1))))))))))))))))) 1(0(2(2(0(2(0(2(2(2(0(1(0(x1))))))))))))) -> 2(2(2(0(2(0(2(2(0(2(0(2(0(2(2(2(2(x1))))))))))))))))) 1(1(0(0(0(0(2(1(0(0(2(2(0(x1))))))))))))) -> 2(2(2(0(0(0(0(0(0(2(2(0(0(0(1(0(2(x1))))))))))))))))) 1(2(2(2(2(1(0(0(0(2(2(0(2(x1))))))))))))) -> 0(0(2(2(1(1(0(0(0(0(1(0(0(0(2(0(2(x1))))))))))))))))) 2(0(0(0(2(0(1(2(2(2(0(2(2(x1))))))))))))) -> 2(0(0(2(2(2(0(2(0(2(0(2(0(2(1(1(0(x1))))))))))))))))) 2(0(0(2(0(0(1(1(0(0(2(0(0(x1))))))))))))) -> 1(0(2(2(0(2(2(2(1(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(0(0(2(2(1(0(1(0(0(2(2(0(x1))))))))))))) -> 2(2(0(0(2(1(0(2(0(2(0(0(0(0(2(0(2(x1))))))))))))))))) 2(2(0(0(0(2(1(0(0(0(2(0(2(x1))))))))))))) -> 0(0(2(0(2(2(2(0(2(0(2(0(0(2(0(2(0(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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 3. The certificate found is represented by the following graph. "[49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 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, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563] {(49,50,[0_1|0, 1_1|0, 2_1|0, encArg_1|0, encode_0_1|0, encode_2_1|0, encode_1_1|0]), (49,51,[0_1|1, 1_1|1, 2_1|1]), (49,52,[0_1|2]), (49,68,[0_1|2]), (49,84,[0_1|2]), (49,100,[0_1|2]), (49,116,[0_1|2]), (49,132,[1_1|2]), (49,148,[0_1|2]), (49,164,[0_1|2]), (49,180,[0_1|2]), (49,196,[0_1|2]), (49,212,[0_1|2]), (49,228,[0_1|2]), (49,244,[1_1|2]), (49,260,[0_1|2]), (49,276,[0_1|2]), (49,292,[0_1|2]), (49,308,[0_1|2]), (49,324,[0_1|2]), (49,340,[0_1|2]), (49,356,[0_1|2]), 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(115,70,[0_1|2]), (115,342,[0_1|2]), (115,215,[0_1|2]), (115,471,[0_1|2]), (115,52,[0_1|2]), (115,68,[0_1|2]), (115,84,[0_1|2]), (115,100,[0_1|2]), (115,116,[0_1|2]), (115,132,[1_1|2]), (115,148,[0_1|2]), (115,164,[0_1|2]), (115,180,[0_1|2]), (115,196,[0_1|2]), (115,212,[0_1|2]), (115,228,[0_1|2]), (115,244,[1_1|2]), (115,260,[0_1|2]), (115,276,[0_1|2]), (115,292,[0_1|2]), (115,308,[0_1|2]), (115,324,[0_1|2]), (115,340,[0_1|2]), (115,356,[0_1|2]), (115,372,[0_1|2]), (116,117,[2_1|2]), (117,118,[0_1|2]), (118,119,[0_1|2]), (119,120,[1_1|2]), (120,121,[0_1|2]), (121,122,[2_1|2]), (122,123,[0_1|2]), (123,124,[2_1|2]), (124,125,[0_1|2]), (125,126,[0_1|2]), (126,127,[2_1|2]), (127,128,[0_1|2]), (128,129,[2_1|2]), (129,130,[0_1|2]), (130,131,[2_1|2]), (130,484,[2_1|2]), (130,500,[1_1|2]), (130,516,[2_1|2]), (131,51,[0_1|2]), (131,52,[0_1|2]), (131,68,[0_1|2]), (131,84,[0_1|2]), (131,100,[0_1|2]), (131,116,[0_1|2]), (131,148,[0_1|2]), (131,164,[0_1|2]), (131,180,[0_1|2]), (131,196,[0_1|2]), 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(483,261,[2_1|2]), (483,341,[2_1|2]), (483,373,[2_1|2]), (483,56,[2_1|2]), (483,72,[2_1|2]), (483,500,[1_1|2]), (483,532,[0_1|2]), (484,485,[0_1|2]), (485,486,[0_1|2]), (486,487,[2_1|2]), (487,488,[2_1|2]), (488,489,[2_1|2]), (489,490,[0_1|2]), (490,491,[2_1|2]), (491,492,[0_1|2]), (492,493,[2_1|2]), (493,494,[0_1|2]), (494,495,[2_1|2]), (495,496,[0_1|2]), (496,497,[2_1|2]), (497,498,[1_1|2]), (497,452,[2_1|2]), (498,499,[1_1|2]), (498,388,[0_1|2]), (498,404,[2_1|2]), (498,420,[1_1|2]), (498,436,[2_1|2]), (499,51,[0_1|2]), (499,404,[0_1|2]), (499,436,[0_1|2]), (499,452,[0_1|2]), (499,484,[0_1|2]), (499,516,[0_1|2]), (499,437,[0_1|2]), (499,453,[0_1|2]), (499,517,[0_1|2]), (499,54,[0_1|2]), (499,70,[0_1|2]), (499,342,[0_1|2]), (499,52,[0_1|2]), (499,68,[0_1|2]), (499,84,[0_1|2]), (499,100,[0_1|2]), (499,116,[0_1|2]), (499,132,[1_1|2]), (499,148,[0_1|2]), (499,164,[0_1|2]), (499,180,[0_1|2]), (499,196,[0_1|2]), (499,212,[0_1|2]), (499,228,[0_1|2]), (499,244,[1_1|2]), (499,260,[0_1|2]), (499,276,[0_1|2]), (499,292,[0_1|2]), (499,308,[0_1|2]), (499,324,[0_1|2]), (499,340,[0_1|2]), (499,356,[0_1|2]), (499,372,[0_1|2]), (500,501,[0_1|2]), (501,502,[2_1|2]), (502,503,[2_1|2]), (503,504,[0_1|2]), (504,505,[2_1|2]), (505,506,[2_1|2]), (506,507,[2_1|2]), (507,508,[1_1|2]), (508,509,[0_1|2]), (509,510,[2_1|2]), (510,511,[2_1|2]), (511,512,[2_1|2]), (512,513,[0_1|2]), (513,514,[2_1|2]), (514,515,[2_1|2]), (514,532,[0_1|2]), (515,51,[2_1|2]), (515,52,[2_1|2]), (515,68,[2_1|2]), (515,84,[2_1|2]), (515,100,[2_1|2]), (515,116,[2_1|2]), (515,148,[2_1|2]), (515,164,[2_1|2]), (515,180,[2_1|2]), (515,196,[2_1|2]), (515,212,[2_1|2]), (515,228,[2_1|2]), (515,260,[2_1|2]), (515,276,[2_1|2]), (515,292,[2_1|2]), (515,308,[2_1|2]), (515,324,[2_1|2]), (515,340,[2_1|2]), (515,356,[2_1|2]), (515,372,[2_1|2]), (515,388,[2_1|2]), (515,468,[2_1|2]), (515,532,[2_1|2, 0_1|2]), (515,101,[2_1|2]), (515,165,[2_1|2]), (515,181,[2_1|2]), (515,197,[2_1|2]), (515,213,[2_1|2]), (515,229,[2_1|2]), (515,277,[2_1|2]), (515,293,[2_1|2]), (515,325,[2_1|2]), (515,357,[2_1|2]), (515,469,[2_1|2]), (515,533,[2_1|2]), (515,486,[2_1|2]), (515,119,[2_1|2]), (515,104,[2_1|2]), (515,168,[2_1|2]), (515,484,[2_1|2]), (515,500,[1_1|2]), (515,516,[2_1|2]), (516,517,[2_1|2]), (517,518,[0_1|2]), (518,519,[0_1|2]), (519,520,[2_1|2]), (520,521,[1_1|2]), (521,522,[0_1|2]), (522,523,[2_1|2]), (523,524,[0_1|2]), (524,525,[2_1|2]), (525,526,[0_1|2]), (526,527,[0_1|2]), (527,528,[0_1|2]), (527,196,[0_1|2]), (528,529,[0_1|2]), (529,530,[2_1|2]), (530,531,[0_1|2]), (530,340,[0_1|2]), (530,356,[0_1|2]), (530,372,[0_1|2]), (531,51,[2_1|2]), (531,52,[2_1|2]), (531,68,[2_1|2]), (531,84,[2_1|2]), (531,100,[2_1|2]), (531,116,[2_1|2]), (531,148,[2_1|2]), (531,164,[2_1|2]), (531,180,[2_1|2]), (531,196,[2_1|2]), (531,212,[2_1|2]), (531,228,[2_1|2]), (531,260,[2_1|2]), (531,276,[2_1|2]), (531,292,[2_1|2]), (531,308,[2_1|2]), (531,324,[2_1|2]), (531,340,[2_1|2]), (531,356,[2_1|2]), (531,372,[2_1|2]), (531,388,[2_1|2]), (531,468,[2_1|2]), (531,532,[2_1|2, 0_1|2]), (531,485,[2_1|2]), (531,518,[2_1|2]), (531,55,[2_1|2]), (531,71,[2_1|2]), (531,484,[2_1|2]), (531,500,[1_1|2]), (531,516,[2_1|2]), (532,533,[0_1|2]), (533,534,[2_1|2]), (534,535,[0_1|2]), (535,536,[2_1|2]), (536,537,[2_1|2]), (537,538,[2_1|2]), (538,539,[0_1|2]), (539,540,[2_1|2]), (540,541,[0_1|2]), (541,542,[2_1|2]), (542,543,[0_1|2]), (543,544,[0_1|2]), (544,545,[2_1|2]), (545,546,[0_1|2]), (546,547,[2_1|2]), (546,484,[2_1|2]), (546,500,[1_1|2]), (546,516,[2_1|2]), (547,51,[0_1|2]), (547,404,[0_1|2]), (547,436,[0_1|2]), (547,452,[0_1|2]), (547,484,[0_1|2]), (547,516,[0_1|2]), (547,53,[0_1|2]), (547,69,[0_1|2]), (547,117,[0_1|2]), (547,261,[0_1|2]), (547,341,[0_1|2]), (547,373,[0_1|2]), (547,536,[0_1|2]), (547,185,[0_1|2]), (547,411,[0_1|2]), (547,52,[0_1|2]), (547,68,[0_1|2]), (547,84,[0_1|2]), (547,100,[0_1|2]), (547,116,[0_1|2]), (547,132,[1_1|2]), (547,148,[0_1|2]), (547,164,[0_1|2]), (547,180,[0_1|2]), (547,196,[0_1|2]), (547,212,[0_1|2]), (547,228,[0_1|2]), (547,244,[1_1|2]), (547,260,[0_1|2]), (547,276,[0_1|2]), (547,292,[0_1|2]), (547,308,[0_1|2]), (547,324,[0_1|2]), (547,340,[0_1|2]), (547,356,[0_1|2]), (547,372,[0_1|2]), (548,549,[1_1|3]), (549,550,[0_1|3]), (550,551,[0_1|3]), (551,552,[0_1|3]), (552,553,[2_1|3]), (553,554,[0_1|3]), (554,555,[2_1|3]), (555,556,[0_1|3]), (556,557,[0_1|3]), (557,558,[0_1|3]), (558,559,[2_1|3]), (559,560,[2_1|3]), (560,561,[2_1|3]), (561,562,[2_1|3]), (562,563,[0_1|3]), (563,107,[1_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)