/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 65 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 111 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(1(1(1(2(3(3(1(3(3(2(x1))))))))))))) -> 0(2(2(1(0(1(2(3(0(0(3(2(2(3(1(2(2(x1))))))))))))))))) 0(0(0(3(3(1(1(1(3(3(1(3(2(x1))))))))))))) -> 3(0(2(1(0(3(3(3(1(2(2(2(1(2(1(2(2(x1))))))))))))))))) 0(0(2(0(3(1(3(2(1(0(0(2(3(x1))))))))))))) -> 0(2(2(2(1(3(3(3(2(2(0(3(1(3(2(1(2(x1))))))))))))))))) 0(1(0(0(3(3(2(1(2(0(1(2(3(x1))))))))))))) -> 3(2(2(1(2(2(0(3(0(2(2(2(1(2(2(3(1(x1))))))))))))))))) 0(1(2(0(1(1(0(2(2(1(2(0(2(x1))))))))))))) -> 2(0(3(3(2(2(1(0(3(3(3(3(2(2(2(3(3(x1))))))))))))))))) 0(1(2(1(1(1(2(2(0(1(3(2(0(x1))))))))))))) -> 2(3(2(0(1(2(3(0(3(3(2(1(3(3(1(3(3(x1))))))))))))))))) 0(2(2(0(0(3(1(0(3(2(1(3(0(x1))))))))))))) -> 3(1(2(2(1(3(3(2(2(3(0(2(2(1(1(1(0(x1))))))))))))))))) 0(2(3(3(1(0(3(3(0(2(3(1(1(x1))))))))))))) -> 2(1(3(2(2(2(0(2(2(2(3(3(2(2(2(0(3(x1))))))))))))))))) 0(3(0(3(0(2(3(0(0(3(1(2(1(x1))))))))))))) -> 3(1(1(2(2(3(0(1(2(2(2(2(3(2(2(2(0(x1))))))))))))))))) 0(3(0(3(1(0(1(2(2(0(3(1(3(x1))))))))))))) -> 2(3(2(2(3(0(3(0(3(3(2(2(1(2(2(0(3(x1))))))))))))))))) 1(0(0(1(2(2(2(3(2(3(2(0(1(x1))))))))))))) -> 1(2(1(0(2(2(1(2(1(0(3(3(2(2(2(3(3(x1))))))))))))))))) 1(0(2(3(0(3(2(3(2(2(3(2(3(x1))))))))))))) -> 1(3(2(2(1(2(2(2(3(3(2(2(3(1(2(1(2(x1))))))))))))))))) 1(3(1(0(1(1(3(2(2(1(1(2(1(x1))))))))))))) -> 1(2(3(2(3(2(1(2(2(2(2(2(2(0(1(2(2(x1))))))))))))))))) 2(0(0(1(3(0(3(1(3(0(1(2(1(x1))))))))))))) -> 2(2(3(3(0(1(0(0(3(3(3(1(0(2(2(1(2(x1))))))))))))))))) 2(0(0(2(3(0(3(1(0(0(2(1(3(x1))))))))))))) -> 2(0(2(1(2(2(2(2(3(2(3(1(3(3(1(3(1(x1))))))))))))))))) 2(1(0(2(2(0(0(1(3(2(0(3(3(x1))))))))))))) -> 2(0(2(2(1(3(2(1(1(1(2(2(1(3(3(3(3(x1))))))))))))))))) 2(1(0(3(0(3(0(3(3(0(2(1(1(x1))))))))))))) -> 2(2(0(1(2(1(1(0(2(2(2(3(2(3(0(2(2(x1))))))))))))))))) 2(1(1(2(0(1(1(3(0(2(3(0(1(x1))))))))))))) -> 2(2(3(3(3(3(3(2(1(0(1(2(2(3(3(2(2(x1))))))))))))))))) 2(1(1(3(3(0(3(2(3(2(1(1(3(x1))))))))))))) -> 2(3(2(3(2(2(3(3(2(1(2(2(3(3(0(1(3(x1))))))))))))))))) 2(1(1(3(3(3(0(3(0(3(0(0(2(x1))))))))))))) -> 2(0(2(2(0(2(1(3(3(3(2(3(3(2(3(3(2(x1))))))))))))))))) 2(2(0(0(1(0(2(3(0(3(0(1(0(x1))))))))))))) -> 2(2(2(1(0(2(0(1(3(1(3(0(3(3(3(3(2(x1))))))))))))))))) 2(2(0(0(3(0(2(2(3(0(1(3(3(x1))))))))))))) -> 2(2(1(1(0(1(2(1(2(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(2(0(3(0(1(0(2(3(2(3(1(2(x1))))))))))))) -> 2(2(0(2(2(2(1(0(0(3(1(3(1(3(3(2(2(x1))))))))))))))))) 2(2(1(0(2(1(2(1(1(0(1(2(0(x1))))))))))))) -> 2(2(0(2(0(3(1(2(2(0(1(2(2(2(2(2(2(x1))))))))))))))))) 2(3(1(1(0(2(3(1(2(3(3(1(1(x1))))))))))))) -> 2(2(2(1(2(1(1(2(0(2(0(0(3(0(1(3(3(x1))))))))))))))))) 2(3(2(0(3(0(1(3(2(2(2(0(2(x1))))))))))))) -> 2(1(2(2(3(0(0(1(3(2(2(3(2(2(3(3(2(x1))))))))))))))))) 2(3(2(1(1(1(3(2(3(2(3(2(1(x1))))))))))))) -> 2(2(2(0(3(2(2(0(2(3(2(3(0(2(2(0(2(x1))))))))))))))))) 3(0(2(3(0(1(0(3(3(0(0(1(0(x1))))))))))))) -> 3(0(0(1(2(2(3(3(3(2(0(1(2(0(3(3(2(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)) 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 (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(1(1(1(2(3(3(1(3(3(2(x1))))))))))))) -> 0(2(2(1(0(1(2(3(0(0(3(2(2(3(1(2(2(x1))))))))))))))))) 0(0(0(3(3(1(1(1(3(3(1(3(2(x1))))))))))))) -> 3(0(2(1(0(3(3(3(1(2(2(2(1(2(1(2(2(x1))))))))))))))))) 0(0(2(0(3(1(3(2(1(0(0(2(3(x1))))))))))))) -> 0(2(2(2(1(3(3(3(2(2(0(3(1(3(2(1(2(x1))))))))))))))))) 0(1(0(0(3(3(2(1(2(0(1(2(3(x1))))))))))))) -> 3(2(2(1(2(2(0(3(0(2(2(2(1(2(2(3(1(x1))))))))))))))))) 0(1(2(0(1(1(0(2(2(1(2(0(2(x1))))))))))))) -> 2(0(3(3(2(2(1(0(3(3(3(3(2(2(2(3(3(x1))))))))))))))))) 0(1(2(1(1(1(2(2(0(1(3(2(0(x1))))))))))))) -> 2(3(2(0(1(2(3(0(3(3(2(1(3(3(1(3(3(x1))))))))))))))))) 0(2(2(0(0(3(1(0(3(2(1(3(0(x1))))))))))))) -> 3(1(2(2(1(3(3(2(2(3(0(2(2(1(1(1(0(x1))))))))))))))))) 0(2(3(3(1(0(3(3(0(2(3(1(1(x1))))))))))))) -> 2(1(3(2(2(2(0(2(2(2(3(3(2(2(2(0(3(x1))))))))))))))))) 0(3(0(3(0(2(3(0(0(3(1(2(1(x1))))))))))))) -> 3(1(1(2(2(3(0(1(2(2(2(2(3(2(2(2(0(x1))))))))))))))))) 0(3(0(3(1(0(1(2(2(0(3(1(3(x1))))))))))))) -> 2(3(2(2(3(0(3(0(3(3(2(2(1(2(2(0(3(x1))))))))))))))))) 1(0(0(1(2(2(2(3(2(3(2(0(1(x1))))))))))))) -> 1(2(1(0(2(2(1(2(1(0(3(3(2(2(2(3(3(x1))))))))))))))))) 1(0(2(3(0(3(2(3(2(2(3(2(3(x1))))))))))))) -> 1(3(2(2(1(2(2(2(3(3(2(2(3(1(2(1(2(x1))))))))))))))))) 1(3(1(0(1(1(3(2(2(1(1(2(1(x1))))))))))))) -> 1(2(3(2(3(2(1(2(2(2(2(2(2(0(1(2(2(x1))))))))))))))))) 2(0(0(1(3(0(3(1(3(0(1(2(1(x1))))))))))))) -> 2(2(3(3(0(1(0(0(3(3(3(1(0(2(2(1(2(x1))))))))))))))))) 2(0(0(2(3(0(3(1(0(0(2(1(3(x1))))))))))))) -> 2(0(2(1(2(2(2(2(3(2(3(1(3(3(1(3(1(x1))))))))))))))))) 2(1(0(2(2(0(0(1(3(2(0(3(3(x1))))))))))))) -> 2(0(2(2(1(3(2(1(1(1(2(2(1(3(3(3(3(x1))))))))))))))))) 2(1(0(3(0(3(0(3(3(0(2(1(1(x1))))))))))))) -> 2(2(0(1(2(1(1(0(2(2(2(3(2(3(0(2(2(x1))))))))))))))))) 2(1(1(2(0(1(1(3(0(2(3(0(1(x1))))))))))))) -> 2(2(3(3(3(3(3(2(1(0(1(2(2(3(3(2(2(x1))))))))))))))))) 2(1(1(3(3(0(3(2(3(2(1(1(3(x1))))))))))))) -> 2(3(2(3(2(2(3(3(2(1(2(2(3(3(0(1(3(x1))))))))))))))))) 2(1(1(3(3(3(0(3(0(3(0(0(2(x1))))))))))))) -> 2(0(2(2(0(2(1(3(3(3(2(3(3(2(3(3(2(x1))))))))))))))))) 2(2(0(0(1(0(2(3(0(3(0(1(0(x1))))))))))))) -> 2(2(2(1(0(2(0(1(3(1(3(0(3(3(3(3(2(x1))))))))))))))))) 2(2(0(0(3(0(2(2(3(0(1(3(3(x1))))))))))))) -> 2(2(1(1(0(1(2(1(2(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(2(0(3(0(1(0(2(3(2(3(1(2(x1))))))))))))) -> 2(2(0(2(2(2(1(0(0(3(1(3(1(3(3(2(2(x1))))))))))))))))) 2(2(1(0(2(1(2(1(1(0(1(2(0(x1))))))))))))) -> 2(2(0(2(0(3(1(2(2(0(1(2(2(2(2(2(2(x1))))))))))))))))) 2(3(1(1(0(2(3(1(2(3(3(1(1(x1))))))))))))) -> 2(2(2(1(2(1(1(2(0(2(0(0(3(0(1(3(3(x1))))))))))))))))) 2(3(2(0(3(0(1(3(2(2(2(0(2(x1))))))))))))) -> 2(1(2(2(3(0(0(1(3(2(2(3(2(2(3(3(2(x1))))))))))))))))) 2(3(2(1(1(1(3(2(3(2(3(2(1(x1))))))))))))) -> 2(2(2(0(3(2(2(0(2(3(2(3(0(2(2(0(2(x1))))))))))))))))) 3(0(2(3(0(1(0(3(3(0(0(1(0(x1))))))))))))) -> 3(0(0(1(2(2(3(3(3(2(0(1(2(0(3(3(2(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: 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(1(1(1(2(3(3(1(3(3(2(x1))))))))))))) -> 0(2(2(1(0(1(2(3(0(0(3(2(2(3(1(2(2(x1))))))))))))))))) 0(0(0(3(3(1(1(1(3(3(1(3(2(x1))))))))))))) -> 3(0(2(1(0(3(3(3(1(2(2(2(1(2(1(2(2(x1))))))))))))))))) 0(0(2(0(3(1(3(2(1(0(0(2(3(x1))))))))))))) -> 0(2(2(2(1(3(3(3(2(2(0(3(1(3(2(1(2(x1))))))))))))))))) 0(1(0(0(3(3(2(1(2(0(1(2(3(x1))))))))))))) -> 3(2(2(1(2(2(0(3(0(2(2(2(1(2(2(3(1(x1))))))))))))))))) 0(1(2(0(1(1(0(2(2(1(2(0(2(x1))))))))))))) -> 2(0(3(3(2(2(1(0(3(3(3(3(2(2(2(3(3(x1))))))))))))))))) 0(1(2(1(1(1(2(2(0(1(3(2(0(x1))))))))))))) -> 2(3(2(0(1(2(3(0(3(3(2(1(3(3(1(3(3(x1))))))))))))))))) 0(2(2(0(0(3(1(0(3(2(1(3(0(x1))))))))))))) -> 3(1(2(2(1(3(3(2(2(3(0(2(2(1(1(1(0(x1))))))))))))))))) 0(2(3(3(1(0(3(3(0(2(3(1(1(x1))))))))))))) -> 2(1(3(2(2(2(0(2(2(2(3(3(2(2(2(0(3(x1))))))))))))))))) 0(3(0(3(0(2(3(0(0(3(1(2(1(x1))))))))))))) -> 3(1(1(2(2(3(0(1(2(2(2(2(3(2(2(2(0(x1))))))))))))))))) 0(3(0(3(1(0(1(2(2(0(3(1(3(x1))))))))))))) -> 2(3(2(2(3(0(3(0(3(3(2(2(1(2(2(0(3(x1))))))))))))))))) 1(0(0(1(2(2(2(3(2(3(2(0(1(x1))))))))))))) -> 1(2(1(0(2(2(1(2(1(0(3(3(2(2(2(3(3(x1))))))))))))))))) 1(0(2(3(0(3(2(3(2(2(3(2(3(x1))))))))))))) -> 1(3(2(2(1(2(2(2(3(3(2(2(3(1(2(1(2(x1))))))))))))))))) 1(3(1(0(1(1(3(2(2(1(1(2(1(x1))))))))))))) -> 1(2(3(2(3(2(1(2(2(2(2(2(2(0(1(2(2(x1))))))))))))))))) 2(0(0(1(3(0(3(1(3(0(1(2(1(x1))))))))))))) -> 2(2(3(3(0(1(0(0(3(3(3(1(0(2(2(1(2(x1))))))))))))))))) 2(0(0(2(3(0(3(1(0(0(2(1(3(x1))))))))))))) -> 2(0(2(1(2(2(2(2(3(2(3(1(3(3(1(3(1(x1))))))))))))))))) 2(1(0(2(2(0(0(1(3(2(0(3(3(x1))))))))))))) -> 2(0(2(2(1(3(2(1(1(1(2(2(1(3(3(3(3(x1))))))))))))))))) 2(1(0(3(0(3(0(3(3(0(2(1(1(x1))))))))))))) -> 2(2(0(1(2(1(1(0(2(2(2(3(2(3(0(2(2(x1))))))))))))))))) 2(1(1(2(0(1(1(3(0(2(3(0(1(x1))))))))))))) -> 2(2(3(3(3(3(3(2(1(0(1(2(2(3(3(2(2(x1))))))))))))))))) 2(1(1(3(3(0(3(2(3(2(1(1(3(x1))))))))))))) -> 2(3(2(3(2(2(3(3(2(1(2(2(3(3(0(1(3(x1))))))))))))))))) 2(1(1(3(3(3(0(3(0(3(0(0(2(x1))))))))))))) -> 2(0(2(2(0(2(1(3(3(3(2(3(3(2(3(3(2(x1))))))))))))))))) 2(2(0(0(1(0(2(3(0(3(0(1(0(x1))))))))))))) -> 2(2(2(1(0(2(0(1(3(1(3(0(3(3(3(3(2(x1))))))))))))))))) 2(2(0(0(3(0(2(2(3(0(1(3(3(x1))))))))))))) -> 2(2(1(1(0(1(2(1(2(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(2(0(3(0(1(0(2(3(2(3(1(2(x1))))))))))))) -> 2(2(0(2(2(2(1(0(0(3(1(3(1(3(3(2(2(x1))))))))))))))))) 2(2(1(0(2(1(2(1(1(0(1(2(0(x1))))))))))))) -> 2(2(0(2(0(3(1(2(2(0(1(2(2(2(2(2(2(x1))))))))))))))))) 2(3(1(1(0(2(3(1(2(3(3(1(1(x1))))))))))))) -> 2(2(2(1(2(1(1(2(0(2(0(0(3(0(1(3(3(x1))))))))))))))))) 2(3(2(0(3(0(1(3(2(2(2(0(2(x1))))))))))))) -> 2(1(2(2(3(0(0(1(3(2(2(3(2(2(3(3(2(x1))))))))))))))))) 2(3(2(1(1(1(3(2(3(2(3(2(1(x1))))))))))))) -> 2(2(2(0(3(2(2(0(2(3(2(3(0(2(2(0(2(x1))))))))))))))))) 3(0(2(3(0(1(0(3(3(0(0(1(0(x1))))))))))))) -> 3(0(0(1(2(2(3(3(3(2(0(1(2(0(3(3(2(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: 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(1(1(1(2(3(3(1(3(3(2(x1))))))))))))) -> 0(2(2(1(0(1(2(3(0(0(3(2(2(3(1(2(2(x1))))))))))))))))) 0(0(0(3(3(1(1(1(3(3(1(3(2(x1))))))))))))) -> 3(0(2(1(0(3(3(3(1(2(2(2(1(2(1(2(2(x1))))))))))))))))) 0(0(2(0(3(1(3(2(1(0(0(2(3(x1))))))))))))) -> 0(2(2(2(1(3(3(3(2(2(0(3(1(3(2(1(2(x1))))))))))))))))) 0(1(0(0(3(3(2(1(2(0(1(2(3(x1))))))))))))) -> 3(2(2(1(2(2(0(3(0(2(2(2(1(2(2(3(1(x1))))))))))))))))) 0(1(2(0(1(1(0(2(2(1(2(0(2(x1))))))))))))) -> 2(0(3(3(2(2(1(0(3(3(3(3(2(2(2(3(3(x1))))))))))))))))) 0(1(2(1(1(1(2(2(0(1(3(2(0(x1))))))))))))) -> 2(3(2(0(1(2(3(0(3(3(2(1(3(3(1(3(3(x1))))))))))))))))) 0(2(2(0(0(3(1(0(3(2(1(3(0(x1))))))))))))) -> 3(1(2(2(1(3(3(2(2(3(0(2(2(1(1(1(0(x1))))))))))))))))) 0(2(3(3(1(0(3(3(0(2(3(1(1(x1))))))))))))) -> 2(1(3(2(2(2(0(2(2(2(3(3(2(2(2(0(3(x1))))))))))))))))) 0(3(0(3(0(2(3(0(0(3(1(2(1(x1))))))))))))) -> 3(1(1(2(2(3(0(1(2(2(2(2(3(2(2(2(0(x1))))))))))))))))) 0(3(0(3(1(0(1(2(2(0(3(1(3(x1))))))))))))) -> 2(3(2(2(3(0(3(0(3(3(2(2(1(2(2(0(3(x1))))))))))))))))) 1(0(0(1(2(2(2(3(2(3(2(0(1(x1))))))))))))) -> 1(2(1(0(2(2(1(2(1(0(3(3(2(2(2(3(3(x1))))))))))))))))) 1(0(2(3(0(3(2(3(2(2(3(2(3(x1))))))))))))) -> 1(3(2(2(1(2(2(2(3(3(2(2(3(1(2(1(2(x1))))))))))))))))) 1(3(1(0(1(1(3(2(2(1(1(2(1(x1))))))))))))) -> 1(2(3(2(3(2(1(2(2(2(2(2(2(0(1(2(2(x1))))))))))))))))) 2(0(0(1(3(0(3(1(3(0(1(2(1(x1))))))))))))) -> 2(2(3(3(0(1(0(0(3(3(3(1(0(2(2(1(2(x1))))))))))))))))) 2(0(0(2(3(0(3(1(0(0(2(1(3(x1))))))))))))) -> 2(0(2(1(2(2(2(2(3(2(3(1(3(3(1(3(1(x1))))))))))))))))) 2(1(0(2(2(0(0(1(3(2(0(3(3(x1))))))))))))) -> 2(0(2(2(1(3(2(1(1(1(2(2(1(3(3(3(3(x1))))))))))))))))) 2(1(0(3(0(3(0(3(3(0(2(1(1(x1))))))))))))) -> 2(2(0(1(2(1(1(0(2(2(2(3(2(3(0(2(2(x1))))))))))))))))) 2(1(1(2(0(1(1(3(0(2(3(0(1(x1))))))))))))) -> 2(2(3(3(3(3(3(2(1(0(1(2(2(3(3(2(2(x1))))))))))))))))) 2(1(1(3(3(0(3(2(3(2(1(1(3(x1))))))))))))) -> 2(3(2(3(2(2(3(3(2(1(2(2(3(3(0(1(3(x1))))))))))))))))) 2(1(1(3(3(3(0(3(0(3(0(0(2(x1))))))))))))) -> 2(0(2(2(0(2(1(3(3(3(2(3(3(2(3(3(2(x1))))))))))))))))) 2(2(0(0(1(0(2(3(0(3(0(1(0(x1))))))))))))) -> 2(2(2(1(0(2(0(1(3(1(3(0(3(3(3(3(2(x1))))))))))))))))) 2(2(0(0(3(0(2(2(3(0(1(3(3(x1))))))))))))) -> 2(2(1(1(0(1(2(1(2(0(2(2(2(0(2(2(2(x1))))))))))))))))) 2(2(0(3(0(1(0(2(3(2(3(1(2(x1))))))))))))) -> 2(2(0(2(2(2(1(0(0(3(1(3(1(3(3(2(2(x1))))))))))))))))) 2(2(1(0(2(1(2(1(1(0(1(2(0(x1))))))))))))) -> 2(2(0(2(0(3(1(2(2(0(1(2(2(2(2(2(2(x1))))))))))))))))) 2(3(1(1(0(2(3(1(2(3(3(1(1(x1))))))))))))) -> 2(2(2(1(2(1(1(2(0(2(0(0(3(0(1(3(3(x1))))))))))))))))) 2(3(2(0(3(0(1(3(2(2(2(0(2(x1))))))))))))) -> 2(1(2(2(3(0(0(1(3(2(2(3(2(2(3(3(2(x1))))))))))))))))) 2(3(2(1(1(1(3(2(3(2(3(2(1(x1))))))))))))) -> 2(2(2(0(3(2(2(0(2(3(2(3(0(2(2(0(2(x1))))))))))))))))) 3(0(2(3(0(1(0(3(3(0(0(1(0(x1))))))))))))) -> 3(0(0(1(2(2(3(3(3(2(0(1(2(0(3(3(2(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: 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 2. The certificate found is represented by the following graph. "[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, 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(334,335,[1_1|2]), (335,336,[0_1|2]), (336,337,[1_1|2]), (337,338,[2_1|2]), (338,339,[2_1|2]), (339,340,[3_1|2]), (340,341,[3_1|2]), (341,342,[2_1|2]), (341,375,[2_1|2]), (341,391,[2_1|2]), (341,407,[2_1|2]), (341,423,[2_1|2]), (342,54,[2_1|2]), (342,215,[2_1|2]), (342,231,[2_1|2]), (342,247,[2_1|2]), (342,263,[2_1|2]), (342,279,[2_1|2]), (342,295,[2_1|2]), (342,311,[2_1|2]), (342,327,[2_1|2]), (342,343,[2_1|2]), (342,359,[2_1|2]), (342,375,[2_1|2]), (342,391,[2_1|2]), (342,407,[2_1|2]), (342,423,[2_1|2]), (342,439,[2_1|2]), (342,455,[2_1|2]), (342,471,[2_1|2]), (343,344,[3_1|2]), (344,345,[2_1|2]), (345,346,[3_1|2]), (346,347,[2_1|2]), (347,348,[2_1|2]), (348,349,[3_1|2]), (349,350,[3_1|2]), (350,351,[2_1|2]), (351,352,[1_1|2]), (352,353,[2_1|2]), (353,354,[2_1|2]), (354,355,[3_1|2]), (355,356,[3_1|2]), (356,357,[0_1|2]), (357,358,[1_1|2]), (357,247,[1_1|2]), (358,54,[3_1|2]), (358,71,[3_1|2]), (358,103,[3_1|2]), (358,151,[3_1|2]), (358,183,[3_1|2]), (358,487,[3_1|2]), (358,232,[3_1|2]), (359,360,[0_1|2]), (360,361,[2_1|2]), (361,362,[2_1|2]), (362,363,[0_1|2]), (363,364,[2_1|2]), (364,365,[1_1|2]), (365,366,[3_1|2]), (366,367,[3_1|2]), (367,368,[3_1|2]), (368,369,[2_1|2]), (369,370,[3_1|2]), (370,371,[3_1|2]), (371,372,[2_1|2]), (372,373,[3_1|2]), (373,374,[3_1|2]), (374,54,[2_1|2]), (374,119,[2_1|2]), (374,135,[2_1|2]), (374,167,[2_1|2]), (374,199,[2_1|2]), (374,263,[2_1|2]), (374,279,[2_1|2]), (374,295,[2_1|2]), (374,311,[2_1|2]), (374,327,[2_1|2]), (374,343,[2_1|2]), (374,359,[2_1|2]), (374,375,[2_1|2]), (374,391,[2_1|2]), (374,407,[2_1|2]), (374,423,[2_1|2]), (374,439,[2_1|2]), (374,455,[2_1|2]), (374,471,[2_1|2]), (374,56,[2_1|2]), (374,88,[2_1|2]), (375,376,[2_1|2]), (376,377,[2_1|2]), (377,378,[1_1|2]), (378,379,[0_1|2]), (379,380,[2_1|2]), (380,381,[0_1|2]), (381,382,[1_1|2]), (382,383,[3_1|2]), (383,384,[1_1|2]), (384,385,[3_1|2]), (385,386,[0_1|2]), (386,387,[3_1|2]), (387,388,[3_1|2]), (388,389,[3_1|2]), (389,390,[3_1|2]), (390,54,[2_1|2]), (390,55,[2_1|2]), (390,87,[2_1|2]), (390,263,[2_1|2]), (390,279,[2_1|2]), (390,295,[2_1|2]), (390,311,[2_1|2]), (390,327,[2_1|2]), (390,343,[2_1|2]), (390,359,[2_1|2]), (390,375,[2_1|2]), (390,391,[2_1|2]), (390,407,[2_1|2]), (390,423,[2_1|2]), (390,439,[2_1|2]), (390,455,[2_1|2]), (390,471,[2_1|2]), (391,392,[2_1|2]), (392,393,[1_1|2]), (393,394,[1_1|2]), (394,395,[0_1|2]), (395,396,[1_1|2]), (396,397,[2_1|2]), (397,398,[1_1|2]), (398,399,[2_1|2]), (399,400,[0_1|2]), (400,401,[2_1|2]), (401,402,[2_1|2]), (402,403,[2_1|2]), (403,404,[0_1|2]), (404,405,[2_1|2]), (405,406,[2_1|2]), (405,375,[2_1|2]), (405,391,[2_1|2]), (405,407,[2_1|2]), (405,423,[2_1|2]), (406,54,[2_1|2]), (406,71,[2_1|2]), (406,103,[2_1|2]), (406,151,[2_1|2]), (406,183,[2_1|2]), (406,487,[2_1|2]), (406,263,[2_1|2]), (406,279,[2_1|2]), (406,295,[2_1|2]), (406,311,[2_1|2]), (406,327,[2_1|2]), (406,343,[2_1|2]), (406,359,[2_1|2]), (406,375,[2_1|2]), (406,391,[2_1|2]), (406,407,[2_1|2]), (406,423,[2_1|2]), (406,439,[2_1|2]), (406,455,[2_1|2]), (406,471,[2_1|2]), (407,408,[2_1|2]), (408,409,[0_1|2]), (409,410,[2_1|2]), (410,411,[2_1|2]), (411,412,[2_1|2]), (412,413,[1_1|2]), (413,414,[0_1|2]), (414,415,[0_1|2]), (415,416,[3_1|2]), (416,417,[1_1|2]), (417,418,[3_1|2]), (418,419,[1_1|2]), (419,420,[3_1|2]), (420,421,[3_1|2]), (421,422,[2_1|2]), (421,375,[2_1|2]), (421,391,[2_1|2]), (421,407,[2_1|2]), (421,423,[2_1|2]), (422,54,[2_1|2]), (422,119,[2_1|2]), (422,135,[2_1|2]), (422,167,[2_1|2]), (422,199,[2_1|2]), (422,263,[2_1|2]), (422,279,[2_1|2]), (422,295,[2_1|2]), (422,311,[2_1|2]), (422,327,[2_1|2]), (422,343,[2_1|2]), (422,359,[2_1|2]), (422,375,[2_1|2]), (422,391,[2_1|2]), (422,407,[2_1|2]), (422,423,[2_1|2]), (422,439,[2_1|2]), (422,455,[2_1|2]), (422,471,[2_1|2]), (422,216,[2_1|2]), (422,248,[2_1|2]), (422,153,[2_1|2]), (423,424,[2_1|2]), (424,425,[0_1|2]), (425,426,[2_1|2]), (426,427,[0_1|2]), (427,428,[3_1|2]), (428,429,[1_1|2]), (429,430,[2_1|2]), (430,431,[2_1|2]), (431,432,[0_1|2]), (432,433,[1_1|2]), (433,434,[2_1|2]), (434,435,[2_1|2]), (435,436,[2_1|2]), (436,437,[2_1|2]), (437,438,[2_1|2]), (437,375,[2_1|2]), (437,391,[2_1|2]), (437,407,[2_1|2]), (437,423,[2_1|2]), (438,54,[2_1|2]), (438,55,[2_1|2]), (438,87,[2_1|2]), (438,120,[2_1|2]), (438,280,[2_1|2]), (438,296,[2_1|2]), (438,360,[2_1|2]), (438,263,[2_1|2]), (438,279,[2_1|2]), (438,295,[2_1|2]), (438,311,[2_1|2]), (438,327,[2_1|2]), (438,343,[2_1|2]), (438,359,[2_1|2]), (438,375,[2_1|2]), (438,391,[2_1|2]), (438,407,[2_1|2]), (438,423,[2_1|2]), (438,439,[2_1|2]), (438,455,[2_1|2]), (438,471,[2_1|2]), (439,440,[2_1|2]), (440,441,[2_1|2]), (441,442,[1_1|2]), (442,443,[2_1|2]), (443,444,[1_1|2]), (444,445,[1_1|2]), (445,446,[2_1|2]), (446,447,[0_1|2]), (447,448,[2_1|2]), (448,449,[0_1|2]), (449,450,[0_1|2]), (450,451,[3_1|2]), (451,452,[0_1|2]), (452,453,[1_1|2]), (453,454,[3_1|2]), (454,54,[3_1|2]), (454,215,[3_1|2]), (454,231,[3_1|2]), (454,247,[3_1|2]), (454,185,[3_1|2]), (454,487,[3_1|2]), (455,456,[1_1|2]), (456,457,[2_1|2]), (457,458,[2_1|2]), (458,459,[3_1|2]), (459,460,[0_1|2]), (460,461,[0_1|2]), (461,462,[1_1|2]), (462,463,[3_1|2]), (463,464,[2_1|2]), (464,465,[2_1|2]), (465,466,[3_1|2]), (466,467,[2_1|2]), (467,468,[2_1|2]), (468,469,[3_1|2]), (469,470,[3_1|2]), (470,54,[2_1|2]), (470,119,[2_1|2]), (470,135,[2_1|2]), (470,167,[2_1|2]), (470,199,[2_1|2]), (470,263,[2_1|2]), (470,279,[2_1|2]), (470,295,[2_1|2]), (470,311,[2_1|2]), (470,327,[2_1|2]), (470,343,[2_1|2]), (470,359,[2_1|2]), (470,375,[2_1|2]), (470,391,[2_1|2]), (470,407,[2_1|2]), (470,423,[2_1|2]), (470,439,[2_1|2]), (470,455,[2_1|2]), (470,471,[2_1|2]), (470,56,[2_1|2]), (470,88,[2_1|2]), (470,281,[2_1|2]), (470,297,[2_1|2]), (470,361,[2_1|2]), (470,410,[2_1|2]), (470,426,[2_1|2]), (471,472,[2_1|2]), (472,473,[2_1|2]), (473,474,[0_1|2]), (474,475,[3_1|2]), (475,476,[2_1|2]), (476,477,[2_1|2]), (477,478,[0_1|2]), (478,479,[2_1|2]), (479,480,[3_1|2]), (480,481,[2_1|2]), (481,482,[3_1|2]), (482,483,[0_1|2]), (483,484,[2_1|2]), (484,485,[2_1|2]), (485,486,[0_1|2]), (485,151,[3_1|2]), (485,167,[2_1|2]), (486,54,[2_1|2]), (486,215,[2_1|2]), (486,231,[2_1|2]), (486,247,[2_1|2]), (486,168,[2_1|2]), (486,456,[2_1|2]), (486,263,[2_1|2]), (486,279,[2_1|2]), (486,295,[2_1|2]), (486,311,[2_1|2]), (486,327,[2_1|2]), (486,343,[2_1|2]), (486,359,[2_1|2]), (486,375,[2_1|2]), (486,391,[2_1|2]), (486,407,[2_1|2]), (486,423,[2_1|2]), (486,439,[2_1|2]), (486,455,[2_1|2]), (486,471,[2_1|2]), (487,488,[0_1|2]), (488,489,[0_1|2]), (489,490,[1_1|2]), (490,491,[2_1|2]), (491,492,[2_1|2]), (492,493,[3_1|2]), (493,494,[3_1|2]), (494,495,[3_1|2]), (495,496,[2_1|2]), (496,497,[0_1|2]), (497,498,[1_1|2]), (498,499,[2_1|2]), (499,500,[0_1|2]), (500,501,[3_1|2]), (501,502,[3_1|2]), (502,54,[2_1|2]), (502,55,[2_1|2]), (502,87,[2_1|2]), (502,263,[2_1|2]), (502,279,[2_1|2]), (502,295,[2_1|2]), (502,311,[2_1|2]), (502,327,[2_1|2]), (502,343,[2_1|2]), (502,359,[2_1|2]), (502,375,[2_1|2]), (502,391,[2_1|2]), (502,407,[2_1|2]), (502,423,[2_1|2]), (502,439,[2_1|2]), (502,455,[2_1|2]), (502,471,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)