/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 48 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 243 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(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: 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(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 (full) 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: 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(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: 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(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: 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] {(52,53,[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]), (52,54,[1_1|1, 2_1|1, 4_1|1, 5_1|1, 0_1|1, 3_1|1]), (52,55,[0_1|2]), (52,60,[0_1|2]), (52,65,[2_1|2]), (52,70,[0_1|2]), (52,75,[2_1|2]), (52,80,[0_1|2]), (52,85,[0_1|2]), (52,90,[0_1|2]), (52,95,[2_1|2]), (52,100,[0_1|2]), (52,105,[0_1|2]), (52,110,[0_1|2]), (52,115,[0_1|2]), (52,120,[0_1|2]), (52,124,[3_1|2]), (52,128,[3_1|2]), 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(288,54,[0_1|2]), (288,55,[0_1|2]), (288,60,[0_1|2]), (288,70,[0_1|2]), (288,80,[0_1|2]), (288,85,[0_1|2]), (288,90,[0_1|2]), (288,100,[0_1|2]), (288,105,[0_1|2]), (288,110,[0_1|2]), (288,115,[0_1|2]), (288,120,[0_1|2]), (288,289,[0_1|2]), (288,324,[0_1|2]), (288,65,[2_1|2]), (288,75,[2_1|2]), (288,95,[2_1|2]), (289,290,[2_1|2]), (290,291,[4_1|2]), (291,292,[1_1|2]), (292,293,[3_1|2]), (292,120,[0_1|2]), (292,124,[3_1|2]), (292,128,[3_1|2]), (292,132,[3_1|2]), (292,136,[3_1|2]), (292,140,[3_1|2]), (292,144,[5_1|2]), (292,148,[2_1|2]), (292,153,[2_1|2]), (292,158,[3_1|2]), (292,163,[3_1|2]), (292,168,[3_1|2]), (292,173,[3_1|2]), (292,178,[3_1|2]), (292,183,[3_1|2]), (292,188,[3_1|2]), (292,193,[3_1|2]), (292,198,[5_1|2]), (292,203,[3_1|2]), (292,208,[3_1|2]), (292,213,[2_1|2]), (292,218,[2_1|2]), (292,223,[3_1|2]), (292,228,[3_1|2]), (293,54,[0_1|2]), (293,55,[0_1|2]), (293,60,[0_1|2]), (293,70,[0_1|2]), (293,80,[0_1|2]), (293,85,[0_1|2]), (293,90,[0_1|2]), (293,100,[0_1|2]), (293,105,[0_1|2]), (293,110,[0_1|2]), (293,115,[0_1|2]), (293,120,[0_1|2]), (293,289,[0_1|2]), (293,324,[0_1|2]), (293,65,[2_1|2]), (293,75,[2_1|2]), (293,95,[2_1|2]), (294,295,[1_1|2]), (295,296,[4_1|2]), (296,297,[0_1|2]), (297,298,[0_1|2]), (297,364,[0_1|2]), (297,75,[2_1|2]), (298,54,[2_1|2]), (298,55,[2_1|2]), (298,60,[2_1|2]), (298,70,[2_1|2]), (298,80,[2_1|2]), (298,85,[2_1|2]), (298,90,[2_1|2]), (298,100,[2_1|2]), (298,105,[2_1|2]), (298,110,[2_1|2]), (298,115,[2_1|2]), (298,120,[2_1|2]), (298,289,[2_1|2]), (298,324,[2_1|2]), (299,300,[2_1|2]), (300,301,[0_1|2]), (301,302,[4_1|2]), (302,303,[1_1|2]), (303,54,[0_1|2]), (303,55,[0_1|2]), (303,60,[0_1|2]), (303,70,[0_1|2]), (303,80,[0_1|2]), (303,85,[0_1|2]), (303,90,[0_1|2]), (303,100,[0_1|2]), (303,105,[0_1|2]), (303,110,[0_1|2]), (303,115,[0_1|2]), (303,120,[0_1|2]), (303,289,[0_1|2]), (303,324,[0_1|2]), (303,65,[2_1|2]), (303,75,[2_1|2]), (303,95,[2_1|2]), (304,305,[1_1|2]), (305,306,[1_1|2]), (306,307,[4_1|2]), (307,308,[4_1|2]), (308,54,[0_1|2]), (308,55,[0_1|2]), (308,60,[0_1|2]), (308,70,[0_1|2]), (308,80,[0_1|2]), (308,85,[0_1|2]), (308,90,[0_1|2]), (308,100,[0_1|2]), (308,105,[0_1|2]), (308,110,[0_1|2]), (308,115,[0_1|2]), (308,120,[0_1|2]), (308,289,[0_1|2]), (308,324,[0_1|2]), (308,65,[2_1|2]), (308,75,[2_1|2]), (308,95,[2_1|2]), (309,310,[1_1|2]), (310,311,[1_1|2]), (311,312,[1_1|2]), (312,313,[0_1|2]), (312,55,[0_1|2]), (312,60,[0_1|2]), (312,65,[2_1|2]), (313,54,[0_1|2]), (313,55,[0_1|2]), (313,60,[0_1|2]), (313,70,[0_1|2]), (313,80,[0_1|2]), (313,85,[0_1|2]), (313,90,[0_1|2]), (313,100,[0_1|2]), (313,105,[0_1|2]), (313,110,[0_1|2]), (313,115,[0_1|2]), (313,120,[0_1|2]), (313,289,[0_1|2]), (313,324,[0_1|2]), (313,65,[2_1|2]), (313,75,[2_1|2]), (313,95,[2_1|2]), (314,315,[1_1|2]), (315,316,[2_1|2]), (316,317,[1_1|2]), (317,318,[0_1|2]), (317,55,[0_1|2]), (317,60,[0_1|2]), (317,65,[2_1|2]), (318,54,[0_1|2]), (318,55,[0_1|2]), (318,60,[0_1|2]), (318,70,[0_1|2]), (318,80,[0_1|2]), (318,85,[0_1|2]), (318,90,[0_1|2]), (318,100,[0_1|2]), (318,105,[0_1|2]), (318,110,[0_1|2]), (318,115,[0_1|2]), (318,120,[0_1|2]), (318,289,[0_1|2]), (318,324,[0_1|2]), (318,65,[2_1|2]), (318,75,[2_1|2]), (318,95,[2_1|2]), (319,320,[1_1|2]), (320,321,[5_1|2]), (321,322,[1_1|2]), (322,323,[4_1|2]), (323,54,[0_1|2]), (323,55,[0_1|2]), (323,60,[0_1|2]), (323,70,[0_1|2]), (323,80,[0_1|2]), (323,85,[0_1|2]), (323,90,[0_1|2]), (323,100,[0_1|2]), (323,105,[0_1|2]), (323,110,[0_1|2]), (323,115,[0_1|2]), (323,120,[0_1|2]), (323,289,[0_1|2]), (323,324,[0_1|2]), (323,65,[2_1|2]), (323,75,[2_1|2]), (323,95,[2_1|2]), (324,325,[2_1|2]), (325,326,[3_1|2]), (326,327,[1_1|2]), (327,328,[5_1|2]), (328,54,[0_1|2]), (328,55,[0_1|2]), (328,60,[0_1|2]), (328,70,[0_1|2]), (328,80,[0_1|2]), (328,85,[0_1|2]), (328,90,[0_1|2]), (328,100,[0_1|2]), (328,105,[0_1|2]), (328,110,[0_1|2]), (328,115,[0_1|2]), (328,120,[0_1|2]), (328,289,[0_1|2]), (328,324,[0_1|2]), (328,65,[2_1|2]), (328,75,[2_1|2]), (328,95,[2_1|2]), (329,330,[1_1|2]), (330,331,[2_1|2]), (331,332,[0_1|2]), (331,80,[0_1|2]), (331,85,[0_1|2]), (331,90,[0_1|2]), (331,95,[2_1|2]), (332,333,[3_1|2]), (332,120,[0_1|2]), (332,124,[3_1|2]), (332,128,[3_1|2]), (332,132,[3_1|2]), (332,136,[3_1|2]), (332,140,[3_1|2]), (332,144,[5_1|2]), (332,148,[2_1|2]), (332,153,[2_1|2]), (332,158,[3_1|2]), (332,163,[3_1|2]), (332,168,[3_1|2]), (332,173,[3_1|2]), (332,178,[3_1|2]), (332,183,[3_1|2]), (332,188,[3_1|2]), (332,193,[3_1|2]), (332,198,[5_1|2]), (332,203,[3_1|2]), (332,208,[3_1|2]), (332,213,[2_1|2]), (332,218,[2_1|2]), (332,223,[3_1|2]), (332,228,[3_1|2]), (333,54,[0_1|2]), (333,55,[0_1|2]), (333,60,[0_1|2]), (333,70,[0_1|2]), (333,80,[0_1|2]), (333,85,[0_1|2]), (333,90,[0_1|2]), (333,100,[0_1|2]), (333,105,[0_1|2]), (333,110,[0_1|2]), (333,115,[0_1|2]), (333,120,[0_1|2]), (333,289,[0_1|2]), (333,324,[0_1|2]), (333,65,[2_1|2]), (333,75,[2_1|2]), (333,95,[2_1|2]), (334,335,[1_1|2]), (335,336,[2_1|2]), (336,337,[3_1|2]), (337,338,[0_1|2]), (337,55,[0_1|2]), (337,60,[0_1|2]), (337,65,[2_1|2]), (338,54,[0_1|2]), (338,55,[0_1|2]), (338,60,[0_1|2]), (338,70,[0_1|2]), (338,80,[0_1|2]), (338,85,[0_1|2]), (338,90,[0_1|2]), (338,100,[0_1|2]), (338,105,[0_1|2]), (338,110,[0_1|2]), (338,115,[0_1|2]), (338,120,[0_1|2]), (338,289,[0_1|2]), (338,324,[0_1|2]), (338,65,[2_1|2]), (338,75,[2_1|2]), (338,95,[2_1|2]), (339,340,[1_1|2]), (340,341,[2_1|2]), (341,342,[4_1|2]), (342,343,[3_1|2]), (342,120,[0_1|2]), (342,124,[3_1|2]), (342,128,[3_1|2]), (342,132,[3_1|2]), (342,136,[3_1|2]), (342,140,[3_1|2]), (342,144,[5_1|2]), (342,148,[2_1|2]), (342,153,[2_1|2]), (342,158,[3_1|2]), (342,163,[3_1|2]), (342,168,[3_1|2]), (342,173,[3_1|2]), (342,178,[3_1|2]), (342,183,[3_1|2]), (342,188,[3_1|2]), (342,193,[3_1|2]), (342,198,[5_1|2]), (342,203,[3_1|2]), (342,208,[3_1|2]), (342,213,[2_1|2]), (342,218,[2_1|2]), (342,223,[3_1|2]), (342,228,[3_1|2]), (343,54,[0_1|2]), (343,55,[0_1|2]), (343,60,[0_1|2]), (343,70,[0_1|2]), (343,80,[0_1|2]), (343,85,[0_1|2]), (343,90,[0_1|2]), (343,100,[0_1|2]), (343,105,[0_1|2]), (343,110,[0_1|2]), (343,115,[0_1|2]), (343,120,[0_1|2]), (343,289,[0_1|2]), (343,324,[0_1|2]), (343,65,[2_1|2]), (343,75,[2_1|2]), (343,95,[2_1|2]), (344,345,[1_1|2]), (345,346,[3_1|2]), (346,347,[4_1|2]), (347,348,[0_1|2]), (347,364,[0_1|2]), (347,75,[2_1|2]), (348,54,[2_1|2]), (348,55,[2_1|2]), (348,60,[2_1|2]), (348,70,[2_1|2]), (348,80,[2_1|2]), (348,85,[2_1|2]), (348,90,[2_1|2]), (348,100,[2_1|2]), (348,105,[2_1|2]), (348,110,[2_1|2]), (348,115,[2_1|2]), (348,120,[2_1|2]), (348,289,[2_1|2]), (348,324,[2_1|2]), (349,350,[1_1|2]), (350,351,[4_1|2]), (351,352,[3_1|2]), (351,354,[2_1|2]), (351,309,[3_1|2]), (351,314,[3_1|2]), (352,353,[1_1|2]), (353,54,[0_1|2]), (353,55,[0_1|2]), (353,60,[0_1|2]), (353,70,[0_1|2]), (353,80,[0_1|2]), (353,85,[0_1|2]), (353,90,[0_1|2]), (353,100,[0_1|2]), (353,105,[0_1|2]), (353,110,[0_1|2]), (353,115,[0_1|2]), (353,120,[0_1|2]), (353,289,[0_1|2]), (353,324,[0_1|2]), (353,65,[2_1|2]), (353,75,[2_1|2]), (353,95,[2_1|2]), (354,355,[0_1|2]), (355,356,[3_1|2]), (356,357,[1_1|2]), (357,358,[1_1|2]), (358,54,[0_1|2]), (358,55,[0_1|2]), (358,60,[0_1|2]), (358,70,[0_1|2]), (358,80,[0_1|2]), (358,85,[0_1|2]), (358,90,[0_1|2]), (358,100,[0_1|2]), (358,105,[0_1|2]), (358,110,[0_1|2]), (358,115,[0_1|2]), (358,120,[0_1|2]), (358,289,[0_1|2]), (358,324,[0_1|2]), (358,65,[2_1|2]), (358,75,[2_1|2]), (358,95,[2_1|2]), (359,360,[1_1|2]), (360,361,[2_1|2]), (361,362,[1_1|2]), (362,363,[4_1|2]), (363,54,[0_1|2]), (363,55,[0_1|2]), (363,60,[0_1|2]), (363,70,[0_1|2]), (363,80,[0_1|2]), (363,85,[0_1|2]), (363,90,[0_1|2]), (363,100,[0_1|2]), (363,105,[0_1|2]), (363,110,[0_1|2]), (363,115,[0_1|2]), (363,120,[0_1|2]), (363,289,[0_1|2]), (363,324,[0_1|2]), (363,65,[2_1|2]), (363,75,[2_1|2]), (363,95,[2_1|2]), (364,365,[2_1|2]), (365,366,[0_1|2]), (366,367,[0_1|2]), (367,368,[3_1|2]), (367,354,[2_1|2]), (367,309,[3_1|2]), (367,314,[3_1|2]), (367,359,[3_1|2]), (367,319,[3_1|2]), (368,54,[1_1|2]), (368,55,[1_1|2]), (368,60,[1_1|2]), (368,70,[1_1|2]), (368,80,[1_1|2]), (368,85,[1_1|2]), (368,90,[1_1|2]), (368,100,[1_1|2]), (368,105,[1_1|2]), (368,110,[1_1|2]), (368,115,[1_1|2]), (368,120,[1_1|2]), (368,289,[1_1|2]), (368,324,[1_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)