/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 57 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 310 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(1(0(x1)))) -> 0(2(0(0(3(1(x1)))))) 0(0(1(0(x1)))) -> 0(2(0(4(1(0(x1)))))) 0(0(1(0(x1)))) -> 2(0(0(0(2(1(x1)))))) 3(0(1(0(x1)))) -> 0(2(3(1(0(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(2(x1))))) 3(0(1(0(x1)))) -> 3(1(1(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(2(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(x1))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(x1))))) 3(0(1(0(x1)))) -> 5(0(3(1(0(x1))))) 3(0(1(0(x1)))) -> 2(0(2(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 2(2(0(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(2(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(2(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(2(2(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(2(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 5(1(1(3(0(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(4(0(x1))))) 3(4(1(0(x1)))) -> 3(1(4(0(2(x1))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(x1))))) 3(4(1(0(x1)))) -> 3(4(2(1(0(x1))))) 3(4(1(0(x1)))) -> 3(1(1(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(1(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(4(2(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(2(1(1(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(5(1(2(0(x1)))))) 0(1(4(1(0(x1))))) -> 0(1(1(4(0(2(x1)))))) 0(2(0(1(0(x1))))) -> 0(2(0(0(3(1(x1)))))) 0(2(0(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(1(3(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(3(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(5(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 2(0(0(3(1(0(x1)))))) 0(3(4(1(0(x1))))) -> 0(2(0(4(3(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(0(1(5(2(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(1(5(1(0(x1)))))) 0(5(0(1(0(x1))))) -> 0(2(0(0(1(5(x1)))))) 3(0(1(0(0(x1))))) -> 3(1(3(0(0(0(x1)))))) 3(0(1(1(0(x1))))) -> 3(1(0(1(2(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(3(1(5(0(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(5(0(x1)))))) 3(0(5(1(0(x1))))) -> 3(1(5(2(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(1(1(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(2(1(0(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(2(1(4(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(5(1(4(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(2(3(1(5(0(x1)))))) 3(2(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(0(3(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(3(0(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(2(4(3(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(3(4(0(2(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(4(3(1(0(x1)))))) 3(4(0(1(0(x1))))) -> 0(2(4(1(3(0(x1)))))) 3(4(0(1(0(x1))))) -> 3(1(4(0(0(2(x1)))))) 3(4(0(1(0(x1))))) -> 3(2(0(4(1(0(x1)))))) 3(4(4(1(0(x1))))) -> 3(1(1(4(4(0(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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(1(0(x1)))) -> 0(2(0(0(3(1(x1)))))) 0(0(1(0(x1)))) -> 0(2(0(4(1(0(x1)))))) 0(0(1(0(x1)))) -> 2(0(0(0(2(1(x1)))))) 3(0(1(0(x1)))) -> 0(2(3(1(0(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(2(x1))))) 3(0(1(0(x1)))) -> 3(1(1(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(2(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(x1))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(x1))))) 3(0(1(0(x1)))) -> 5(0(3(1(0(x1))))) 3(0(1(0(x1)))) -> 2(0(2(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 2(2(0(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(2(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(2(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(2(2(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(2(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 5(1(1(3(0(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(4(0(x1))))) 3(4(1(0(x1)))) -> 3(1(4(0(2(x1))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(x1))))) 3(4(1(0(x1)))) -> 3(4(2(1(0(x1))))) 3(4(1(0(x1)))) -> 3(1(1(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(1(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(4(2(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(2(1(1(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(5(1(2(0(x1)))))) 0(1(4(1(0(x1))))) -> 0(1(1(4(0(2(x1)))))) 0(2(0(1(0(x1))))) -> 0(2(0(0(3(1(x1)))))) 0(2(0(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(1(3(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(3(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(5(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 2(0(0(3(1(0(x1)))))) 0(3(4(1(0(x1))))) -> 0(2(0(4(3(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(0(1(5(2(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(1(5(1(0(x1)))))) 0(5(0(1(0(x1))))) -> 0(2(0(0(1(5(x1)))))) 3(0(1(0(0(x1))))) -> 3(1(3(0(0(0(x1)))))) 3(0(1(1(0(x1))))) -> 3(1(0(1(2(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(3(1(5(0(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(5(0(x1)))))) 3(0(5(1(0(x1))))) -> 3(1(5(2(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(1(1(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(2(1(0(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(2(1(4(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(5(1(4(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(2(3(1(5(0(x1)))))) 3(2(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(0(3(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(3(0(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(2(4(3(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(3(4(0(2(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(4(3(1(0(x1)))))) 3(4(0(1(0(x1))))) -> 0(2(4(1(3(0(x1)))))) 3(4(0(1(0(x1))))) -> 3(1(4(0(0(2(x1)))))) 3(4(0(1(0(x1))))) -> 3(2(0(4(1(0(x1)))))) 3(4(4(1(0(x1))))) -> 3(1(1(4(4(0(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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(1(0(x1)))) -> 0(2(0(0(3(1(x1)))))) 0(0(1(0(x1)))) -> 0(2(0(4(1(0(x1)))))) 0(0(1(0(x1)))) -> 2(0(0(0(2(1(x1)))))) 3(0(1(0(x1)))) -> 0(2(3(1(0(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(2(x1))))) 3(0(1(0(x1)))) -> 3(1(1(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(2(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(x1))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(x1))))) 3(0(1(0(x1)))) -> 5(0(3(1(0(x1))))) 3(0(1(0(x1)))) -> 2(0(2(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 2(2(0(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(2(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(2(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(2(2(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(2(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 5(1(1(3(0(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(4(0(x1))))) 3(4(1(0(x1)))) -> 3(1(4(0(2(x1))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(x1))))) 3(4(1(0(x1)))) -> 3(4(2(1(0(x1))))) 3(4(1(0(x1)))) -> 3(1(1(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(1(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(4(2(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(2(1(1(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(5(1(2(0(x1)))))) 0(1(4(1(0(x1))))) -> 0(1(1(4(0(2(x1)))))) 0(2(0(1(0(x1))))) -> 0(2(0(0(3(1(x1)))))) 0(2(0(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(1(3(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(3(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(5(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 2(0(0(3(1(0(x1)))))) 0(3(4(1(0(x1))))) -> 0(2(0(4(3(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(0(1(5(2(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(1(5(1(0(x1)))))) 0(5(0(1(0(x1))))) -> 0(2(0(0(1(5(x1)))))) 3(0(1(0(0(x1))))) -> 3(1(3(0(0(0(x1)))))) 3(0(1(1(0(x1))))) -> 3(1(0(1(2(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(3(1(5(0(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(5(0(x1)))))) 3(0(5(1(0(x1))))) -> 3(1(5(2(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(1(1(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(2(1(0(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(2(1(4(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(5(1(4(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(2(3(1(5(0(x1)))))) 3(2(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(0(3(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(3(0(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(2(4(3(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(3(4(0(2(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(4(3(1(0(x1)))))) 3(4(0(1(0(x1))))) -> 0(2(4(1(3(0(x1)))))) 3(4(0(1(0(x1))))) -> 3(1(4(0(0(2(x1)))))) 3(4(0(1(0(x1))))) -> 3(2(0(4(1(0(x1)))))) 3(4(4(1(0(x1))))) -> 3(1(1(4(4(0(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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(1(0(x1)))) -> 0(2(0(0(3(1(x1)))))) 0(0(1(0(x1)))) -> 0(2(0(4(1(0(x1)))))) 0(0(1(0(x1)))) -> 2(0(0(0(2(1(x1)))))) 3(0(1(0(x1)))) -> 0(2(3(1(0(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(2(x1))))) 3(0(1(0(x1)))) -> 3(1(1(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(2(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(x1))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(x1))))) 3(0(1(0(x1)))) -> 5(0(3(1(0(x1))))) 3(0(1(0(x1)))) -> 2(0(2(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 2(2(0(3(1(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(0(2(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(2(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(1(5(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(2(2(1(0(0(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(0(0(2(x1)))))) 3(0(1(0(x1)))) -> 3(5(1(5(0(0(x1)))))) 3(0(1(0(x1)))) -> 5(1(1(3(0(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(4(0(x1))))) 3(4(1(0(x1)))) -> 3(1(4(0(2(x1))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(x1))))) 3(4(1(0(x1)))) -> 3(4(2(1(0(x1))))) 3(4(1(0(x1)))) -> 3(1(1(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(1(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(2(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(1(4(2(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(4(0(2(x1)))))) 3(4(1(0(x1)))) -> 3(1(5(5(4(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(2(1(1(0(x1)))))) 3(4(1(0(x1)))) -> 3(4(5(1(2(0(x1)))))) 0(1(4(1(0(x1))))) -> 0(1(1(4(0(2(x1)))))) 0(2(0(1(0(x1))))) -> 0(2(0(0(3(1(x1)))))) 0(2(0(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(1(3(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(3(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 0(0(3(5(1(0(x1)))))) 0(3(0(1(0(x1))))) -> 2(0(0(3(1(0(x1)))))) 0(3(4(1(0(x1))))) -> 0(2(0(4(3(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(0(1(5(2(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(1(5(1(0(x1)))))) 0(5(0(1(0(x1))))) -> 0(2(0(0(1(5(x1)))))) 3(0(1(0(0(x1))))) -> 3(1(3(0(0(0(x1)))))) 3(0(1(1(0(x1))))) -> 3(1(0(1(2(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 2(3(1(5(0(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(1(0(x1)))))) 3(0(2(1(0(x1))))) -> 3(1(2(0(5(0(x1)))))) 3(0(5(1(0(x1))))) -> 3(1(5(2(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(1(1(0(0(x1)))))) 3(1(0(1(0(x1))))) -> 3(1(2(1(0(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(2(1(4(0(x1)))))) 3(1(4(1(0(x1))))) -> 3(1(5(1(4(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(2(3(1(5(0(x1)))))) 3(2(0(1(0(x1))))) -> 2(0(3(1(1(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(0(3(0(x1)))))) 3(3(0(1(0(x1))))) -> 3(1(2(3(0(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(2(4(3(0(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(3(4(0(2(x1)))))) 3(3(4(1(0(x1))))) -> 3(1(4(3(1(0(x1)))))) 3(4(0(1(0(x1))))) -> 0(2(4(1(3(0(x1)))))) 3(4(0(1(0(x1))))) -> 3(1(4(0(0(2(x1)))))) 3(4(0(1(0(x1))))) -> 3(2(0(4(1(0(x1)))))) 3(4(4(1(0(x1))))) -> 3(1(1(4(4(0(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(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. "[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] {(54,55,[0_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0]), (54,56,[1_1|1, 2_1|1, 4_1|1, 5_1|1, 0_1|1, 3_1|1]), (54,57,[0_1|2]), (54,62,[0_1|2]), (54,67,[2_1|2]), (54,72,[0_1|2]), (54,77,[2_1|2]), (54,82,[0_1|2]), (54,87,[0_1|2]), (54,92,[0_1|2]), (54,97,[2_1|2]), (54,102,[0_1|2]), (54,107,[0_1|2]), (54,112,[0_1|2]), (54,117,[0_1|2]), (54,122,[0_1|2]), (54,126,[3_1|2]), (54,130,[3_1|2]), 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(289,290,[2_1|2]), (290,56,[0_1|2]), (290,57,[0_1|2]), (290,62,[0_1|2]), (290,72,[0_1|2]), (290,82,[0_1|2]), (290,87,[0_1|2]), (290,92,[0_1|2]), (290,102,[0_1|2]), (290,107,[0_1|2]), (290,112,[0_1|2]), (290,117,[0_1|2]), (290,122,[0_1|2]), (290,291,[0_1|2]), (290,326,[0_1|2]), (290,67,[2_1|2]), (290,77,[2_1|2]), (290,97,[2_1|2]), (291,292,[2_1|2]), (292,293,[4_1|2]), (293,294,[1_1|2]), (294,295,[3_1|2]), (294,122,[0_1|2]), (294,126,[3_1|2]), (294,130,[3_1|2]), (294,134,[3_1|2]), (294,138,[3_1|2]), (294,142,[3_1|2]), (294,146,[5_1|2]), (294,150,[2_1|2]), (294,155,[2_1|2]), (294,160,[3_1|2]), (294,165,[3_1|2]), (294,170,[3_1|2]), (294,175,[3_1|2]), (294,180,[3_1|2]), (294,185,[3_1|2]), (294,190,[3_1|2]), (294,195,[3_1|2]), (294,200,[5_1|2]), (294,205,[3_1|2]), (294,210,[3_1|2]), (294,215,[2_1|2]), (294,220,[2_1|2]), (294,225,[3_1|2]), (294,230,[3_1|2]), (295,56,[0_1|2]), (295,57,[0_1|2]), (295,62,[0_1|2]), (295,72,[0_1|2]), (295,82,[0_1|2]), (295,87,[0_1|2]), (295,92,[0_1|2]), (295,102,[0_1|2]), (295,107,[0_1|2]), (295,112,[0_1|2]), (295,117,[0_1|2]), (295,122,[0_1|2]), (295,291,[0_1|2]), (295,326,[0_1|2]), (295,67,[2_1|2]), (295,77,[2_1|2]), (295,97,[2_1|2]), (296,297,[1_1|2]), (297,298,[4_1|2]), (298,299,[0_1|2]), (299,300,[0_1|2]), (299,366,[0_1|2]), (299,77,[2_1|2]), (300,56,[2_1|2]), (300,57,[2_1|2]), (300,62,[2_1|2]), (300,72,[2_1|2]), (300,82,[2_1|2]), (300,87,[2_1|2]), (300,92,[2_1|2]), (300,102,[2_1|2]), (300,107,[2_1|2]), (300,112,[2_1|2]), (300,117,[2_1|2]), (300,122,[2_1|2]), (300,291,[2_1|2]), (300,326,[2_1|2]), (301,302,[2_1|2]), (302,303,[0_1|2]), (303,304,[4_1|2]), (304,305,[1_1|2]), (305,56,[0_1|2]), (305,57,[0_1|2]), (305,62,[0_1|2]), (305,72,[0_1|2]), (305,82,[0_1|2]), (305,87,[0_1|2]), (305,92,[0_1|2]), (305,102,[0_1|2]), (305,107,[0_1|2]), (305,112,[0_1|2]), (305,117,[0_1|2]), (305,122,[0_1|2]), (305,291,[0_1|2]), (305,326,[0_1|2]), (305,67,[2_1|2]), (305,77,[2_1|2]), (305,97,[2_1|2]), (306,307,[1_1|2]), (307,308,[1_1|2]), (308,309,[4_1|2]), (309,310,[4_1|2]), (310,56,[0_1|2]), (310,57,[0_1|2]), (310,62,[0_1|2]), (310,72,[0_1|2]), (310,82,[0_1|2]), (310,87,[0_1|2]), (310,92,[0_1|2]), (310,102,[0_1|2]), (310,107,[0_1|2]), (310,112,[0_1|2]), (310,117,[0_1|2]), (310,122,[0_1|2]), (310,291,[0_1|2]), (310,326,[0_1|2]), (310,67,[2_1|2]), (310,77,[2_1|2]), (310,97,[2_1|2]), (311,312,[1_1|2]), (312,313,[1_1|2]), (313,314,[1_1|2]), (314,315,[0_1|2]), (314,57,[0_1|2]), (314,62,[0_1|2]), (314,67,[2_1|2]), (315,56,[0_1|2]), (315,57,[0_1|2]), (315,62,[0_1|2]), (315,72,[0_1|2]), (315,82,[0_1|2]), (315,87,[0_1|2]), (315,92,[0_1|2]), (315,102,[0_1|2]), (315,107,[0_1|2]), (315,112,[0_1|2]), (315,117,[0_1|2]), (315,122,[0_1|2]), (315,291,[0_1|2]), (315,326,[0_1|2]), (315,67,[2_1|2]), (315,77,[2_1|2]), (315,97,[2_1|2]), (316,317,[1_1|2]), (317,318,[2_1|2]), (318,319,[1_1|2]), (319,320,[0_1|2]), (319,57,[0_1|2]), (319,62,[0_1|2]), (319,67,[2_1|2]), (320,56,[0_1|2]), (320,57,[0_1|2]), (320,62,[0_1|2]), (320,72,[0_1|2]), (320,82,[0_1|2]), (320,87,[0_1|2]), (320,92,[0_1|2]), (320,102,[0_1|2]), (320,107,[0_1|2]), (320,112,[0_1|2]), (320,117,[0_1|2]), (320,122,[0_1|2]), (320,291,[0_1|2]), (320,326,[0_1|2]), (320,67,[2_1|2]), (320,77,[2_1|2]), (320,97,[2_1|2]), (321,322,[1_1|2]), (322,323,[5_1|2]), (323,324,[1_1|2]), (324,325,[4_1|2]), (325,56,[0_1|2]), (325,57,[0_1|2]), (325,62,[0_1|2]), (325,72,[0_1|2]), (325,82,[0_1|2]), (325,87,[0_1|2]), (325,92,[0_1|2]), (325,102,[0_1|2]), (325,107,[0_1|2]), (325,112,[0_1|2]), (325,117,[0_1|2]), (325,122,[0_1|2]), (325,291,[0_1|2]), (325,326,[0_1|2]), (325,67,[2_1|2]), (325,77,[2_1|2]), (325,97,[2_1|2]), (326,327,[2_1|2]), (327,328,[3_1|2]), (328,329,[1_1|2]), (329,330,[5_1|2]), (330,56,[0_1|2]), (330,57,[0_1|2]), (330,62,[0_1|2]), (330,72,[0_1|2]), (330,82,[0_1|2]), (330,87,[0_1|2]), (330,92,[0_1|2]), (330,102,[0_1|2]), (330,107,[0_1|2]), (330,112,[0_1|2]), (330,117,[0_1|2]), (330,122,[0_1|2]), (330,291,[0_1|2]), (330,326,[0_1|2]), (330,67,[2_1|2]), (330,77,[2_1|2]), (330,97,[2_1|2]), (331,332,[1_1|2]), (332,333,[2_1|2]), (333,334,[0_1|2]), (333,82,[0_1|2]), (333,87,[0_1|2]), (333,92,[0_1|2]), (333,97,[2_1|2]), (334,335,[3_1|2]), (334,122,[0_1|2]), (334,126,[3_1|2]), (334,130,[3_1|2]), (334,134,[3_1|2]), (334,138,[3_1|2]), (334,142,[3_1|2]), (334,146,[5_1|2]), (334,150,[2_1|2]), (334,155,[2_1|2]), (334,160,[3_1|2]), (334,165,[3_1|2]), (334,170,[3_1|2]), (334,175,[3_1|2]), (334,180,[3_1|2]), (334,185,[3_1|2]), (334,190,[3_1|2]), (334,195,[3_1|2]), (334,200,[5_1|2]), (334,205,[3_1|2]), (334,210,[3_1|2]), (334,215,[2_1|2]), (334,220,[2_1|2]), (334,225,[3_1|2]), (334,230,[3_1|2]), (335,56,[0_1|2]), (335,57,[0_1|2]), (335,62,[0_1|2]), (335,72,[0_1|2]), (335,82,[0_1|2]), (335,87,[0_1|2]), (335,92,[0_1|2]), (335,102,[0_1|2]), (335,107,[0_1|2]), (335,112,[0_1|2]), (335,117,[0_1|2]), (335,122,[0_1|2]), (335,291,[0_1|2]), (335,326,[0_1|2]), (335,67,[2_1|2]), (335,77,[2_1|2]), (335,97,[2_1|2]), (336,337,[1_1|2]), (337,338,[2_1|2]), (338,339,[3_1|2]), (339,340,[0_1|2]), (339,57,[0_1|2]), (339,62,[0_1|2]), (339,67,[2_1|2]), (340,56,[0_1|2]), (340,57,[0_1|2]), (340,62,[0_1|2]), (340,72,[0_1|2]), (340,82,[0_1|2]), (340,87,[0_1|2]), (340,92,[0_1|2]), (340,102,[0_1|2]), (340,107,[0_1|2]), (340,112,[0_1|2]), (340,117,[0_1|2]), (340,122,[0_1|2]), (340,291,[0_1|2]), (340,326,[0_1|2]), (340,67,[2_1|2]), (340,77,[2_1|2]), (340,97,[2_1|2]), (341,342,[1_1|2]), (342,343,[2_1|2]), (343,344,[4_1|2]), (344,345,[3_1|2]), (344,122,[0_1|2]), (344,126,[3_1|2]), (344,130,[3_1|2]), (344,134,[3_1|2]), (344,138,[3_1|2]), (344,142,[3_1|2]), (344,146,[5_1|2]), (344,150,[2_1|2]), (344,155,[2_1|2]), (344,160,[3_1|2]), (344,165,[3_1|2]), (344,170,[3_1|2]), (344,175,[3_1|2]), (344,180,[3_1|2]), (344,185,[3_1|2]), (344,190,[3_1|2]), (344,195,[3_1|2]), (344,200,[5_1|2]), (344,205,[3_1|2]), (344,210,[3_1|2]), (344,215,[2_1|2]), (344,220,[2_1|2]), (344,225,[3_1|2]), (344,230,[3_1|2]), (345,56,[0_1|2]), (345,57,[0_1|2]), (345,62,[0_1|2]), (345,72,[0_1|2]), (345,82,[0_1|2]), (345,87,[0_1|2]), (345,92,[0_1|2]), (345,102,[0_1|2]), (345,107,[0_1|2]), (345,112,[0_1|2]), (345,117,[0_1|2]), (345,122,[0_1|2]), (345,291,[0_1|2]), (345,326,[0_1|2]), (345,67,[2_1|2]), (345,77,[2_1|2]), (345,97,[2_1|2]), (346,347,[1_1|2]), (347,348,[3_1|2]), (348,349,[4_1|2]), (349,350,[0_1|2]), (349,366,[0_1|2]), (349,77,[2_1|2]), (350,56,[2_1|2]), (350,57,[2_1|2]), (350,62,[2_1|2]), (350,72,[2_1|2]), (350,82,[2_1|2]), (350,87,[2_1|2]), (350,92,[2_1|2]), (350,102,[2_1|2]), (350,107,[2_1|2]), (350,112,[2_1|2]), (350,117,[2_1|2]), (350,122,[2_1|2]), (350,291,[2_1|2]), (350,326,[2_1|2]), (351,352,[1_1|2]), (352,353,[4_1|2]), (353,354,[3_1|2]), (353,356,[2_1|2]), (353,311,[3_1|2]), (353,316,[3_1|2]), (354,355,[1_1|2]), (355,56,[0_1|2]), (355,57,[0_1|2]), (355,62,[0_1|2]), (355,72,[0_1|2]), (355,82,[0_1|2]), (355,87,[0_1|2]), (355,92,[0_1|2]), (355,102,[0_1|2]), (355,107,[0_1|2]), (355,112,[0_1|2]), (355,117,[0_1|2]), (355,122,[0_1|2]), (355,291,[0_1|2]), (355,326,[0_1|2]), (355,67,[2_1|2]), (355,77,[2_1|2]), (355,97,[2_1|2]), (356,357,[0_1|2]), (357,358,[3_1|2]), (358,359,[1_1|2]), (359,360,[1_1|2]), (360,56,[0_1|2]), (360,57,[0_1|2]), (360,62,[0_1|2]), (360,72,[0_1|2]), (360,82,[0_1|2]), (360,87,[0_1|2]), (360,92,[0_1|2]), (360,102,[0_1|2]), (360,107,[0_1|2]), (360,112,[0_1|2]), (360,117,[0_1|2]), (360,122,[0_1|2]), (360,291,[0_1|2]), (360,326,[0_1|2]), (360,67,[2_1|2]), (360,77,[2_1|2]), (360,97,[2_1|2]), (361,362,[1_1|2]), (362,363,[2_1|2]), (363,364,[1_1|2]), (364,365,[4_1|2]), (365,56,[0_1|2]), (365,57,[0_1|2]), (365,62,[0_1|2]), (365,72,[0_1|2]), (365,82,[0_1|2]), (365,87,[0_1|2]), (365,92,[0_1|2]), (365,102,[0_1|2]), (365,107,[0_1|2]), (365,112,[0_1|2]), (365,117,[0_1|2]), (365,122,[0_1|2]), (365,291,[0_1|2]), (365,326,[0_1|2]), (365,67,[2_1|2]), (365,77,[2_1|2]), (365,97,[2_1|2]), (366,367,[2_1|2]), (367,368,[0_1|2]), (368,369,[0_1|2]), (369,370,[3_1|2]), (369,356,[2_1|2]), (369,311,[3_1|2]), (369,316,[3_1|2]), (369,361,[3_1|2]), (369,321,[3_1|2]), (370,56,[1_1|2]), (370,57,[1_1|2]), (370,62,[1_1|2]), (370,72,[1_1|2]), (370,82,[1_1|2]), (370,87,[1_1|2]), (370,92,[1_1|2]), (370,102,[1_1|2]), (370,107,[1_1|2]), (370,112,[1_1|2]), (370,117,[1_1|2]), (370,122,[1_1|2]), (370,291,[1_1|2]), (370,326,[1_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)