/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 49 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 112 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(2(5(x1)))) -> 1(3(0(2(2(0(1(2(2(5(x1)))))))))) 2(1(4(2(x1)))) -> 3(2(1(0(4(1(2(3(3(2(x1)))))))))) 2(3(4(0(x1)))) -> 2(4(1(3(4(1(3(2(4(4(x1)))))))))) 5(2(4(0(x1)))) -> 5(1(1(3(3(0(4(1(0(0(x1)))))))))) 0(0(0(2(3(x1))))) -> 2(1(3(3(0(2(2(1(1(1(x1)))))))))) 0(1(5(0(2(x1))))) -> 0(5(4(5(1(0(1(0(2(2(x1)))))))))) 0(2(1(1(4(x1))))) -> 1(0(2(2(3(0(3(1(0(4(x1)))))))))) 1(1(3(4(4(x1))))) -> 2(4(5(4(3(4(1(3(0(4(x1)))))))))) 1(1(4(4(4(x1))))) -> 2(1(2(1(2(3(3(3(5(4(x1)))))))))) 1(4(2(4(2(x1))))) -> 1(2(5(5(2(3(1(3(0(2(x1)))))))))) 5(1(3(4(4(x1))))) -> 1(0(1(1(0(2(2(4(3(4(x1)))))))))) 5(2(3(4(0(x1))))) -> 5(1(2(1(0(3(0(2(2(0(x1)))))))))) 5(5(1(4(0(x1))))) -> 2(1(0(4(1(4(3(3(4(5(x1)))))))))) 0(0(1(1(4(1(x1)))))) -> 2(1(0(3(0(5(2(2(4(1(x1)))))))))) 0(0(1(5(0(3(x1)))))) -> 2(1(0(1(2(2(3(3(0(3(x1)))))))))) 0(0(2(3(4(2(x1)))))) -> 2(1(3(2(5(3(4(1(0(3(x1)))))))))) 0(0(3(4(0(0(x1)))))) -> 1(0(3(1(2(4(2(2(2(4(x1)))))))))) 1(4(2(2(0(1(x1)))))) -> 1(0(4(1(5(4(5(4(0(1(x1)))))))))) 2(3(3(4(0(2(x1)))))) -> 2(5(5(3(0(4(3(3(2(0(x1)))))))))) 3(1(1(4(0(0(x1)))))) -> 3(0(1(1(0(2(0(5(5(4(x1)))))))))) 3(4(5(5(1(4(x1)))))) -> 4(1(2(2(1(5(4(1(2(4(x1)))))))))) 5(0(0(5(1(1(x1)))))) -> 2(5(3(3(0(2(4(1(3(2(x1)))))))))) 5(0(5(5(5(0(x1)))))) -> 5(0(4(5(4(1(0(4(3(0(x1)))))))))) 0(0(0(2(0(0(2(x1))))))) -> 2(1(3(1(2(5(5(3(1(2(x1)))))))))) 0(0(0(3(1(4(3(x1))))))) -> 1(1(0(0(2(4(4(5(5(4(x1)))))))))) 0(0(2(4(0(5(3(x1))))))) -> 0(3(5(3(2(5(0(4(1(5(x1)))))))))) 0(0(3(5(4(5(5(x1))))))) -> 2(1(0(1(2(2(4(3(0(5(x1)))))))))) 0(1(5(1(4(4(3(x1))))))) -> 3(5(4(1(4(0(4(3(0(1(x1)))))))))) 0(2(0(5(0(0(2(x1))))))) -> 0(3(2(0(3(2(5(0(3(0(x1)))))))))) 0(5(2(4(2(4(0(x1))))))) -> 2(5(4(2(3(3(2(1(0(0(x1)))))))))) 1(3(1(5(5(5(5(x1))))))) -> 1(2(1(5(0(4(5(4(1(3(x1)))))))))) 2(0(2(3(4(0(2(x1))))))) -> 2(1(0(4(0(5(3(4(2(5(x1)))))))))) 2(0(2(4(0(0(5(x1))))))) -> 2(0(1(0(3(2(1(2(1(5(x1)))))))))) 2(1(4(2(2(0(5(x1))))))) -> 1(0(2(0(2(4(1(2(1(5(x1)))))))))) 2(1(4(5(2(5(1(x1))))))) -> 0(4(5(4(3(1(2(1(3(1(x1)))))))))) 2(3(5(3(5(3(3(x1))))))) -> 2(0(1(0(4(1(0(2(5(3(x1)))))))))) 2(3(5(4(1(4(4(x1))))))) -> 3(1(1(1(2(1(3(4(3(4(x1)))))))))) 3(1(4(1(1(4(0(x1))))))) -> 0(4(1(2(0(3(1(2(0(0(x1)))))))))) 3(1(4(3(5(4(2(x1))))))) -> 1(1(0(1(0(2(2(0(4(2(x1)))))))))) 3(2(3(3(4(0(5(x1))))))) -> 0(1(0(0(5(5(4(1(0(5(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 3(5(3(3(2(2(4(1(3(4(x1)))))))))) 3(4(3(4(3(4(0(x1))))))) -> 0(3(0(4(1(3(3(1(2(0(x1)))))))))) 3(4(3(5(2(4(0(x1))))))) -> 3(3(3(0(4(1(4(0(5(0(x1)))))))))) 3(4(4(0(0(5(1(x1))))))) -> 3(0(1(0(2(1(5(4(5(3(x1)))))))))) 5(0(5(3(3(5(1(x1))))))) -> 5(3(1(1(0(2(4(1(2(1(x1)))))))))) 5(1(4(4(4(2(2(x1))))))) -> 1(3(3(5(4(1(3(5(1(0(x1)))))))))) 5(1(4(4(5(5(1(x1))))))) -> 4(1(0(1(1(0(0(4(3(3(x1)))))))))) 5(5(1(5(4(3(1(x1))))))) -> 1(1(2(2(4(1(1(0(2(1(x1)))))))))) 5(5(5(1(4(0(0(x1))))))) -> 1(1(0(3(0(5(0(1(5(4(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(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(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(2(5(x1)))) -> 1(3(0(2(2(0(1(2(2(5(x1)))))))))) 2(1(4(2(x1)))) -> 3(2(1(0(4(1(2(3(3(2(x1)))))))))) 2(3(4(0(x1)))) -> 2(4(1(3(4(1(3(2(4(4(x1)))))))))) 5(2(4(0(x1)))) -> 5(1(1(3(3(0(4(1(0(0(x1)))))))))) 0(0(0(2(3(x1))))) -> 2(1(3(3(0(2(2(1(1(1(x1)))))))))) 0(1(5(0(2(x1))))) -> 0(5(4(5(1(0(1(0(2(2(x1)))))))))) 0(2(1(1(4(x1))))) -> 1(0(2(2(3(0(3(1(0(4(x1)))))))))) 1(1(3(4(4(x1))))) -> 2(4(5(4(3(4(1(3(0(4(x1)))))))))) 1(1(4(4(4(x1))))) -> 2(1(2(1(2(3(3(3(5(4(x1)))))))))) 1(4(2(4(2(x1))))) -> 1(2(5(5(2(3(1(3(0(2(x1)))))))))) 5(1(3(4(4(x1))))) -> 1(0(1(1(0(2(2(4(3(4(x1)))))))))) 5(2(3(4(0(x1))))) -> 5(1(2(1(0(3(0(2(2(0(x1)))))))))) 5(5(1(4(0(x1))))) -> 2(1(0(4(1(4(3(3(4(5(x1)))))))))) 0(0(1(1(4(1(x1)))))) -> 2(1(0(3(0(5(2(2(4(1(x1)))))))))) 0(0(1(5(0(3(x1)))))) -> 2(1(0(1(2(2(3(3(0(3(x1)))))))))) 0(0(2(3(4(2(x1)))))) -> 2(1(3(2(5(3(4(1(0(3(x1)))))))))) 0(0(3(4(0(0(x1)))))) -> 1(0(3(1(2(4(2(2(2(4(x1)))))))))) 1(4(2(2(0(1(x1)))))) -> 1(0(4(1(5(4(5(4(0(1(x1)))))))))) 2(3(3(4(0(2(x1)))))) -> 2(5(5(3(0(4(3(3(2(0(x1)))))))))) 3(1(1(4(0(0(x1)))))) -> 3(0(1(1(0(2(0(5(5(4(x1)))))))))) 3(4(5(5(1(4(x1)))))) -> 4(1(2(2(1(5(4(1(2(4(x1)))))))))) 5(0(0(5(1(1(x1)))))) -> 2(5(3(3(0(2(4(1(3(2(x1)))))))))) 5(0(5(5(5(0(x1)))))) -> 5(0(4(5(4(1(0(4(3(0(x1)))))))))) 0(0(0(2(0(0(2(x1))))))) -> 2(1(3(1(2(5(5(3(1(2(x1)))))))))) 0(0(0(3(1(4(3(x1))))))) -> 1(1(0(0(2(4(4(5(5(4(x1)))))))))) 0(0(2(4(0(5(3(x1))))))) -> 0(3(5(3(2(5(0(4(1(5(x1)))))))))) 0(0(3(5(4(5(5(x1))))))) -> 2(1(0(1(2(2(4(3(0(5(x1)))))))))) 0(1(5(1(4(4(3(x1))))))) -> 3(5(4(1(4(0(4(3(0(1(x1)))))))))) 0(2(0(5(0(0(2(x1))))))) -> 0(3(2(0(3(2(5(0(3(0(x1)))))))))) 0(5(2(4(2(4(0(x1))))))) -> 2(5(4(2(3(3(2(1(0(0(x1)))))))))) 1(3(1(5(5(5(5(x1))))))) -> 1(2(1(5(0(4(5(4(1(3(x1)))))))))) 2(0(2(3(4(0(2(x1))))))) -> 2(1(0(4(0(5(3(4(2(5(x1)))))))))) 2(0(2(4(0(0(5(x1))))))) -> 2(0(1(0(3(2(1(2(1(5(x1)))))))))) 2(1(4(2(2(0(5(x1))))))) -> 1(0(2(0(2(4(1(2(1(5(x1)))))))))) 2(1(4(5(2(5(1(x1))))))) -> 0(4(5(4(3(1(2(1(3(1(x1)))))))))) 2(3(5(3(5(3(3(x1))))))) -> 2(0(1(0(4(1(0(2(5(3(x1)))))))))) 2(3(5(4(1(4(4(x1))))))) -> 3(1(1(1(2(1(3(4(3(4(x1)))))))))) 3(1(4(1(1(4(0(x1))))))) -> 0(4(1(2(0(3(1(2(0(0(x1)))))))))) 3(1(4(3(5(4(2(x1))))))) -> 1(1(0(1(0(2(2(0(4(2(x1)))))))))) 3(2(3(3(4(0(5(x1))))))) -> 0(1(0(0(5(5(4(1(0(5(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 3(5(3(3(2(2(4(1(3(4(x1)))))))))) 3(4(3(4(3(4(0(x1))))))) -> 0(3(0(4(1(3(3(1(2(0(x1)))))))))) 3(4(3(5(2(4(0(x1))))))) -> 3(3(3(0(4(1(4(0(5(0(x1)))))))))) 3(4(4(0(0(5(1(x1))))))) -> 3(0(1(0(2(1(5(4(5(3(x1)))))))))) 5(0(5(3(3(5(1(x1))))))) -> 5(3(1(1(0(2(4(1(2(1(x1)))))))))) 5(1(4(4(4(2(2(x1))))))) -> 1(3(3(5(4(1(3(5(1(0(x1)))))))))) 5(1(4(4(5(5(1(x1))))))) -> 4(1(0(1(1(0(0(4(3(3(x1)))))))))) 5(5(1(5(4(3(1(x1))))))) -> 1(1(2(2(4(1(1(0(2(1(x1)))))))))) 5(5(5(1(4(0(0(x1))))))) -> 1(1(0(3(0(5(0(1(5(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(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(2(5(x1)))) -> 1(3(0(2(2(0(1(2(2(5(x1)))))))))) 2(1(4(2(x1)))) -> 3(2(1(0(4(1(2(3(3(2(x1)))))))))) 2(3(4(0(x1)))) -> 2(4(1(3(4(1(3(2(4(4(x1)))))))))) 5(2(4(0(x1)))) -> 5(1(1(3(3(0(4(1(0(0(x1)))))))))) 0(0(0(2(3(x1))))) -> 2(1(3(3(0(2(2(1(1(1(x1)))))))))) 0(1(5(0(2(x1))))) -> 0(5(4(5(1(0(1(0(2(2(x1)))))))))) 0(2(1(1(4(x1))))) -> 1(0(2(2(3(0(3(1(0(4(x1)))))))))) 1(1(3(4(4(x1))))) -> 2(4(5(4(3(4(1(3(0(4(x1)))))))))) 1(1(4(4(4(x1))))) -> 2(1(2(1(2(3(3(3(5(4(x1)))))))))) 1(4(2(4(2(x1))))) -> 1(2(5(5(2(3(1(3(0(2(x1)))))))))) 5(1(3(4(4(x1))))) -> 1(0(1(1(0(2(2(4(3(4(x1)))))))))) 5(2(3(4(0(x1))))) -> 5(1(2(1(0(3(0(2(2(0(x1)))))))))) 5(5(1(4(0(x1))))) -> 2(1(0(4(1(4(3(3(4(5(x1)))))))))) 0(0(1(1(4(1(x1)))))) -> 2(1(0(3(0(5(2(2(4(1(x1)))))))))) 0(0(1(5(0(3(x1)))))) -> 2(1(0(1(2(2(3(3(0(3(x1)))))))))) 0(0(2(3(4(2(x1)))))) -> 2(1(3(2(5(3(4(1(0(3(x1)))))))))) 0(0(3(4(0(0(x1)))))) -> 1(0(3(1(2(4(2(2(2(4(x1)))))))))) 1(4(2(2(0(1(x1)))))) -> 1(0(4(1(5(4(5(4(0(1(x1)))))))))) 2(3(3(4(0(2(x1)))))) -> 2(5(5(3(0(4(3(3(2(0(x1)))))))))) 3(1(1(4(0(0(x1)))))) -> 3(0(1(1(0(2(0(5(5(4(x1)))))))))) 3(4(5(5(1(4(x1)))))) -> 4(1(2(2(1(5(4(1(2(4(x1)))))))))) 5(0(0(5(1(1(x1)))))) -> 2(5(3(3(0(2(4(1(3(2(x1)))))))))) 5(0(5(5(5(0(x1)))))) -> 5(0(4(5(4(1(0(4(3(0(x1)))))))))) 0(0(0(2(0(0(2(x1))))))) -> 2(1(3(1(2(5(5(3(1(2(x1)))))))))) 0(0(0(3(1(4(3(x1))))))) -> 1(1(0(0(2(4(4(5(5(4(x1)))))))))) 0(0(2(4(0(5(3(x1))))))) -> 0(3(5(3(2(5(0(4(1(5(x1)))))))))) 0(0(3(5(4(5(5(x1))))))) -> 2(1(0(1(2(2(4(3(0(5(x1)))))))))) 0(1(5(1(4(4(3(x1))))))) -> 3(5(4(1(4(0(4(3(0(1(x1)))))))))) 0(2(0(5(0(0(2(x1))))))) -> 0(3(2(0(3(2(5(0(3(0(x1)))))))))) 0(5(2(4(2(4(0(x1))))))) -> 2(5(4(2(3(3(2(1(0(0(x1)))))))))) 1(3(1(5(5(5(5(x1))))))) -> 1(2(1(5(0(4(5(4(1(3(x1)))))))))) 2(0(2(3(4(0(2(x1))))))) -> 2(1(0(4(0(5(3(4(2(5(x1)))))))))) 2(0(2(4(0(0(5(x1))))))) -> 2(0(1(0(3(2(1(2(1(5(x1)))))))))) 2(1(4(2(2(0(5(x1))))))) -> 1(0(2(0(2(4(1(2(1(5(x1)))))))))) 2(1(4(5(2(5(1(x1))))))) -> 0(4(5(4(3(1(2(1(3(1(x1)))))))))) 2(3(5(3(5(3(3(x1))))))) -> 2(0(1(0(4(1(0(2(5(3(x1)))))))))) 2(3(5(4(1(4(4(x1))))))) -> 3(1(1(1(2(1(3(4(3(4(x1)))))))))) 3(1(4(1(1(4(0(x1))))))) -> 0(4(1(2(0(3(1(2(0(0(x1)))))))))) 3(1(4(3(5(4(2(x1))))))) -> 1(1(0(1(0(2(2(0(4(2(x1)))))))))) 3(2(3(3(4(0(5(x1))))))) -> 0(1(0(0(5(5(4(1(0(5(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 3(5(3(3(2(2(4(1(3(4(x1)))))))))) 3(4(3(4(3(4(0(x1))))))) -> 0(3(0(4(1(3(3(1(2(0(x1)))))))))) 3(4(3(5(2(4(0(x1))))))) -> 3(3(3(0(4(1(4(0(5(0(x1)))))))))) 3(4(4(0(0(5(1(x1))))))) -> 3(0(1(0(2(1(5(4(5(3(x1)))))))))) 5(0(5(3(3(5(1(x1))))))) -> 5(3(1(1(0(2(4(1(2(1(x1)))))))))) 5(1(4(4(4(2(2(x1))))))) -> 1(3(3(5(4(1(3(5(1(0(x1)))))))))) 5(1(4(4(5(5(1(x1))))))) -> 4(1(0(1(1(0(0(4(3(3(x1)))))))))) 5(5(1(5(4(3(1(x1))))))) -> 1(1(2(2(4(1(1(0(2(1(x1)))))))))) 5(5(5(1(4(0(0(x1))))))) -> 1(1(0(3(0(5(0(1(5(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(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(2(5(x1)))) -> 1(3(0(2(2(0(1(2(2(5(x1)))))))))) 2(1(4(2(x1)))) -> 3(2(1(0(4(1(2(3(3(2(x1)))))))))) 2(3(4(0(x1)))) -> 2(4(1(3(4(1(3(2(4(4(x1)))))))))) 5(2(4(0(x1)))) -> 5(1(1(3(3(0(4(1(0(0(x1)))))))))) 0(0(0(2(3(x1))))) -> 2(1(3(3(0(2(2(1(1(1(x1)))))))))) 0(1(5(0(2(x1))))) -> 0(5(4(5(1(0(1(0(2(2(x1)))))))))) 0(2(1(1(4(x1))))) -> 1(0(2(2(3(0(3(1(0(4(x1)))))))))) 1(1(3(4(4(x1))))) -> 2(4(5(4(3(4(1(3(0(4(x1)))))))))) 1(1(4(4(4(x1))))) -> 2(1(2(1(2(3(3(3(5(4(x1)))))))))) 1(4(2(4(2(x1))))) -> 1(2(5(5(2(3(1(3(0(2(x1)))))))))) 5(1(3(4(4(x1))))) -> 1(0(1(1(0(2(2(4(3(4(x1)))))))))) 5(2(3(4(0(x1))))) -> 5(1(2(1(0(3(0(2(2(0(x1)))))))))) 5(5(1(4(0(x1))))) -> 2(1(0(4(1(4(3(3(4(5(x1)))))))))) 0(0(1(1(4(1(x1)))))) -> 2(1(0(3(0(5(2(2(4(1(x1)))))))))) 0(0(1(5(0(3(x1)))))) -> 2(1(0(1(2(2(3(3(0(3(x1)))))))))) 0(0(2(3(4(2(x1)))))) -> 2(1(3(2(5(3(4(1(0(3(x1)))))))))) 0(0(3(4(0(0(x1)))))) -> 1(0(3(1(2(4(2(2(2(4(x1)))))))))) 1(4(2(2(0(1(x1)))))) -> 1(0(4(1(5(4(5(4(0(1(x1)))))))))) 2(3(3(4(0(2(x1)))))) -> 2(5(5(3(0(4(3(3(2(0(x1)))))))))) 3(1(1(4(0(0(x1)))))) -> 3(0(1(1(0(2(0(5(5(4(x1)))))))))) 3(4(5(5(1(4(x1)))))) -> 4(1(2(2(1(5(4(1(2(4(x1)))))))))) 5(0(0(5(1(1(x1)))))) -> 2(5(3(3(0(2(4(1(3(2(x1)))))))))) 5(0(5(5(5(0(x1)))))) -> 5(0(4(5(4(1(0(4(3(0(x1)))))))))) 0(0(0(2(0(0(2(x1))))))) -> 2(1(3(1(2(5(5(3(1(2(x1)))))))))) 0(0(0(3(1(4(3(x1))))))) -> 1(1(0(0(2(4(4(5(5(4(x1)))))))))) 0(0(2(4(0(5(3(x1))))))) -> 0(3(5(3(2(5(0(4(1(5(x1)))))))))) 0(0(3(5(4(5(5(x1))))))) -> 2(1(0(1(2(2(4(3(0(5(x1)))))))))) 0(1(5(1(4(4(3(x1))))))) -> 3(5(4(1(4(0(4(3(0(1(x1)))))))))) 0(2(0(5(0(0(2(x1))))))) -> 0(3(2(0(3(2(5(0(3(0(x1)))))))))) 0(5(2(4(2(4(0(x1))))))) -> 2(5(4(2(3(3(2(1(0(0(x1)))))))))) 1(3(1(5(5(5(5(x1))))))) -> 1(2(1(5(0(4(5(4(1(3(x1)))))))))) 2(0(2(3(4(0(2(x1))))))) -> 2(1(0(4(0(5(3(4(2(5(x1)))))))))) 2(0(2(4(0(0(5(x1))))))) -> 2(0(1(0(3(2(1(2(1(5(x1)))))))))) 2(1(4(2(2(0(5(x1))))))) -> 1(0(2(0(2(4(1(2(1(5(x1)))))))))) 2(1(4(5(2(5(1(x1))))))) -> 0(4(5(4(3(1(2(1(3(1(x1)))))))))) 2(3(5(3(5(3(3(x1))))))) -> 2(0(1(0(4(1(0(2(5(3(x1)))))))))) 2(3(5(4(1(4(4(x1))))))) -> 3(1(1(1(2(1(3(4(3(4(x1)))))))))) 3(1(4(1(1(4(0(x1))))))) -> 0(4(1(2(0(3(1(2(0(0(x1)))))))))) 3(1(4(3(5(4(2(x1))))))) -> 1(1(0(1(0(2(2(0(4(2(x1)))))))))) 3(2(3(3(4(0(5(x1))))))) -> 0(1(0(0(5(5(4(1(0(5(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 3(5(3(3(2(2(4(1(3(4(x1)))))))))) 3(4(3(4(3(4(0(x1))))))) -> 0(3(0(4(1(3(3(1(2(0(x1)))))))))) 3(4(3(5(2(4(0(x1))))))) -> 3(3(3(0(4(1(4(0(5(0(x1)))))))))) 3(4(4(0(0(5(1(x1))))))) -> 3(0(1(0(2(1(5(4(5(3(x1)))))))))) 5(0(5(3(3(5(1(x1))))))) -> 5(3(1(1(0(2(4(1(2(1(x1)))))))))) 5(1(4(4(4(2(2(x1))))))) -> 1(3(3(5(4(1(3(5(1(0(x1)))))))))) 5(1(4(4(5(5(1(x1))))))) -> 4(1(0(1(1(0(0(4(3(3(x1)))))))))) 5(5(1(5(4(3(1(x1))))))) -> 1(1(2(2(4(1(1(0(2(1(x1)))))))))) 5(5(5(1(4(0(0(x1))))))) -> 1(1(0(3(0(5(0(1(5(4(x1)))))))))) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(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. "[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] {(105,106,[0_1|0, 2_1|0, 5_1|0, 1_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_2_1|0, encode_5_1|0, encode_1_1|0, encode_3_1|0, encode_4_1|0]), (105,107,[4_1|1, 0_1|1, 2_1|1, 5_1|1, 1_1|1, 3_1|1]), (105,108,[1_1|2]), (105,117,[2_1|2]), (105,126,[0_1|2]), (105,135,[2_1|2]), (105,144,[2_1|2]), (105,153,[1_1|2]), (105,162,[2_1|2]), (105,171,[2_1|2]), (105,180,[1_1|2]), (105,189,[2_1|2]), (105,198,[0_1|2]), (105,207,[3_1|2]), (105,216,[1_1|2]), (105,225,[0_1|2]), (105,234,[2_1|2]), (105,243,[3_1|2]), (105,252,[1_1|2]), (105,261,[0_1|2]), (105,270,[2_1|2]), (105,279,[2_1|2]), (105,288,[2_1|2]), (105,297,[3_1|2]), (105,306,[2_1|2]), (105,315,[2_1|2]), (105,324,[5_1|2]), (105,333,[5_1|2]), (105,342,[1_1|2]), (105,351,[1_1|2]), 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(496,497,[2_1|2]), (497,498,[2_1|2]), (498,499,[1_1|2]), (499,500,[5_1|2]), (500,501,[4_1|2]), (501,502,[1_1|2]), (502,503,[2_1|2]), (503,107,[4_1|2]), (503,360,[4_1|2]), (503,495,[4_1|2]), (504,505,[5_1|2]), (505,506,[3_1|2]), (506,507,[3_1|2]), (507,508,[2_1|2]), (508,509,[2_1|2]), (509,510,[4_1|2]), (510,511,[1_1|2]), (511,512,[3_1|2]), (511,495,[4_1|2]), (511,504,[3_1|2]), (511,513,[0_1|2]), (511,522,[3_1|2]), (511,531,[3_1|2]), (512,107,[4_1|2]), (512,207,[4_1|2]), (512,243,[4_1|2]), (512,297,[4_1|2]), (512,468,[4_1|2]), (512,504,[4_1|2]), (512,522,[4_1|2]), (512,531,[4_1|2]), (513,514,[3_1|2]), (514,515,[0_1|2]), (515,516,[4_1|2]), (516,517,[1_1|2]), (517,518,[3_1|2]), (518,519,[3_1|2]), (519,520,[1_1|2]), (520,521,[2_1|2]), (520,306,[2_1|2]), (520,315,[2_1|2]), (521,107,[0_1|2]), (521,126,[0_1|2]), (521,198,[0_1|2]), (521,225,[0_1|2]), (521,261,[0_1|2]), (521,477,[0_1|2]), (521,513,[0_1|2]), (521,540,[0_1|2]), (521,108,[1_1|2]), (521,117,[2_1|2]), (521,135,[2_1|2]), (521,144,[2_1|2]), (521,153,[1_1|2]), (521,162,[2_1|2]), (521,171,[2_1|2]), (521,180,[1_1|2]), (521,189,[2_1|2]), (521,207,[3_1|2]), (521,216,[1_1|2]), (521,234,[2_1|2]), (522,523,[3_1|2]), (523,524,[3_1|2]), (524,525,[0_1|2]), (525,526,[4_1|2]), (526,527,[1_1|2]), (527,528,[4_1|2]), (528,529,[0_1|2]), (529,530,[5_1|2]), (529,396,[2_1|2]), (529,405,[5_1|2]), (529,414,[5_1|2]), (530,107,[0_1|2]), (530,126,[0_1|2]), (530,198,[0_1|2]), (530,225,[0_1|2]), (530,261,[0_1|2]), (530,477,[0_1|2]), (530,513,[0_1|2]), (530,540,[0_1|2]), (530,108,[1_1|2]), (530,117,[2_1|2]), (530,135,[2_1|2]), (530,144,[2_1|2]), (530,153,[1_1|2]), (530,162,[2_1|2]), (530,171,[2_1|2]), (530,180,[1_1|2]), (530,189,[2_1|2]), (530,207,[3_1|2]), (530,216,[1_1|2]), (530,234,[2_1|2]), (531,532,[0_1|2]), (532,533,[1_1|2]), (533,534,[0_1|2]), (534,535,[2_1|2]), (535,536,[1_1|2]), (536,537,[5_1|2]), (537,538,[4_1|2]), (538,539,[5_1|2]), (539,107,[3_1|2]), (539,108,[3_1|2]), (539,153,[3_1|2]), (539,180,[3_1|2]), (539,216,[3_1|2]), (539,252,[3_1|2]), (539,342,[3_1|2]), (539,351,[3_1|2]), (539,378,[3_1|2]), (539,387,[3_1|2]), (539,441,[3_1|2]), (539,450,[3_1|2]), (539,459,[3_1|2]), (539,486,[3_1|2, 1_1|2]), (539,325,[3_1|2]), (539,334,[3_1|2]), (539,468,[3_1|2]), (539,477,[0_1|2]), (539,495,[4_1|2]), (539,504,[3_1|2]), (539,513,[0_1|2]), (539,522,[3_1|2]), (539,531,[3_1|2]), (539,540,[0_1|2]), (540,541,[1_1|2]), (541,542,[0_1|2]), (542,543,[0_1|2]), (543,544,[5_1|2]), (544,545,[5_1|2]), (545,546,[4_1|2]), (546,547,[1_1|2]), (547,548,[0_1|2]), (547,234,[2_1|2]), (548,107,[5_1|2]), (548,324,[5_1|2]), (548,333,[5_1|2]), (548,405,[5_1|2]), (548,414,[5_1|2]), (548,199,[5_1|2]), (548,342,[1_1|2]), (548,351,[1_1|2]), (548,360,[4_1|2]), (548,369,[2_1|2]), (548,378,[1_1|2]), (548,387,[1_1|2]), (548,396,[2_1|2]), (549,550,[3_1|3]), (550,551,[0_1|3]), (551,552,[2_1|3]), (552,553,[2_1|3]), (553,554,[0_1|3]), (554,555,[1_1|3]), (555,556,[2_1|3]), (556,557,[2_1|3]), (557,235,[5_1|3]), (557,280,[5_1|3]), (557,397,[5_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)