/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 69 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 236 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(4(2(x1))) -> 5(3(3(3(5(0(3(1(1(3(x1)))))))))) 2(3(0(4(x1)))) -> 2(3(3(5(3(1(1(5(3(2(x1)))))))))) 3(0(0(4(x1)))) -> 3(0(3(4(3(0(3(0(3(2(x1)))))))))) 1(1(0(0(2(x1))))) -> 1(1(5(0(4(4(1(4(1(2(x1)))))))))) 1(2(0(3(2(x1))))) -> 4(2(0(3(4(3(4(3(5(3(x1)))))))))) 4(0(5(3(2(x1))))) -> 2(4(1(3(1(4(1(2(3(2(x1)))))))))) 5(2(0(0(4(x1))))) -> 5(3(0(3(5(3(0(3(0(2(x1)))))))))) 5(4(2(0(0(x1))))) -> 5(4(5(0(1(3(5(3(1(1(x1)))))))))) 0(1(0(0(0(0(x1)))))) -> 0(2(2(5(0(1(5(5(2(1(x1)))))))))) 1(2(5(3(0(0(x1)))))) -> 4(3(0(5(0(2(4(3(3(5(x1)))))))))) 1(3(2(4(1(0(x1)))))) -> 4(3(2(1(2(4(4(1(5(0(x1)))))))))) 1(3(4(0(0(0(x1)))))) -> 1(5(3(2(2(3(3(1(4(1(x1)))))))))) 2(0(2(0(0(0(x1)))))) -> 2(5(1(0(1(3(1(3(0(0(x1)))))))))) 2(4(0(4(1(0(x1)))))) -> 2(1(4(1(5(2(5(0(4(1(x1)))))))))) 3(0(0(0(0(1(x1)))))) -> 2(3(3(0(5(1(5(1(0(1(x1)))))))))) 3(0(0(1(5(2(x1)))))) -> 2(4(4(4(1(4(1(0(1(3(x1)))))))))) 3(2(4(1(0(5(x1)))))) -> 3(3(3(1(1(4(4(1(0(5(x1)))))))))) 3(3(4(0(0(0(x1)))))) -> 3(0(3(3(3(4(3(3(0(0(x1)))))))))) 4(3(2(0(0(5(x1)))))) -> 1(0(5(0(2(4(1(3(0(5(x1)))))))))) 4(5(3(0(1(0(x1)))))) -> 4(0(3(1(3(1(0(4(2(1(x1)))))))))) 5(1(2(0(0(4(x1)))))) -> 5(3(3(5(1(0(3(5(2(2(x1)))))))))) 0(3(3(0(0(2(0(x1))))))) -> 0(3(3(1(1(3(5(4(3(0(x1)))))))))) 0(5(4(0(0(1(0(x1))))))) -> 4(4(4(4(1(4(5(1(1(0(x1)))))))))) 1(0(0(4(4(0(2(x1))))))) -> 5(4(5(5(0(4(5(2(2(4(x1)))))))))) 1(2(0(0(5(1(2(x1))))))) -> 4(2(4(0(3(5(3(0(4(2(x1)))))))))) 1(3(2(4(4(2(0(x1))))))) -> 1(1(3(1(5(5(2(1(3(0(x1)))))))))) 1(3(4(0(0(0(2(x1))))))) -> 0(1(2(1(4(1(3(1(2(4(x1)))))))))) 2(0(0(0(1(4(5(x1))))))) -> 5(0(4(0(2(5(4(5(5(4(x1)))))))))) 2(0(1(0(0(0(2(x1))))))) -> 3(2(3(1(3(0(3(0(2(4(x1)))))))))) 2(0(1(0(0(5(4(x1))))))) -> 2(4(1(2(2(0(5(0(1(4(x1)))))))))) 2(0(4(5(3(2(0(x1))))))) -> 2(3(2(3(1(0(3(0(3(4(x1)))))))))) 2(0(5(1(3(1(2(x1))))))) -> 2(3(2(5(5(1(2(3(3(0(x1)))))))))) 2(0(5(1(3(3(2(x1))))))) -> 3(4(3(3(3(4(4(1(4(2(x1)))))))))) 2(2(3(0(0(2(0(x1))))))) -> 2(1(5(0(4(5(2(3(0(0(x1)))))))))) 2(2(4(4(0(0(4(x1))))))) -> 2(3(3(4(5(5(0(4(0(2(x1)))))))))) 2(5(2(0(1(0(4(x1))))))) -> 3(4(5(0(0(3(1(3(0(4(x1)))))))))) 3(0(0(3(3(1(2(x1))))))) -> 3(5(2(5(1(2(1(4(1(2(x1)))))))))) 3(0(0(5(3(1(2(x1))))))) -> 3(1(5(4(3(0(4(1(4(4(x1)))))))))) 3(2(0(2(2(0(5(x1))))))) -> 3(2(0(1(3(4(3(4(1(5(x1)))))))))) 3(2(2(4(1(0(0(x1))))))) -> 3(5(0(4(5(0(4(0(3(1(x1)))))))))) 3(2(5(0(0(0(0(x1))))))) -> 2(4(5(5(5(1(3(5(4(1(x1)))))))))) 3(4(1(0(0(0(1(x1))))))) -> 3(2(4(5(0(2(4(3(3(1(x1)))))))))) 3(4(1(1(4(1(3(x1))))))) -> 3(5(1(0(5(0(1(3(1(5(x1)))))))))) 4(0(0(3(4(4(5(x1))))))) -> 2(4(1(3(1(3(4(2(2(4(x1)))))))))) 4(4(0(2(0(0(0(x1))))))) -> 0(2(0(3(0(4(1(3(0(1(x1)))))))))) 5(1(2(0(0(1(2(x1))))))) -> 0(2(5(4(1(5(0(2(2(4(x1)))))))))) 5(2(0(0(1(1(1(x1))))))) -> 0(2(1(1(0(0(3(1(1(1(x1)))))))))) 5(2(0(4(3(4(4(x1))))))) -> 5(4(1(0(5(0(2(4(4(4(x1)))))))))) 5(2(4(0(0(2(0(x1))))))) -> 5(2(1(4(5(3(0(2(1(1(x1)))))))))) 5(4(0(1(4(2(5(x1))))))) -> 5(2(4(4(3(3(3(0(4(5(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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(4(2(x1))) -> 5(3(3(3(5(0(3(1(1(3(x1)))))))))) 2(3(0(4(x1)))) -> 2(3(3(5(3(1(1(5(3(2(x1)))))))))) 3(0(0(4(x1)))) -> 3(0(3(4(3(0(3(0(3(2(x1)))))))))) 1(1(0(0(2(x1))))) -> 1(1(5(0(4(4(1(4(1(2(x1)))))))))) 1(2(0(3(2(x1))))) -> 4(2(0(3(4(3(4(3(5(3(x1)))))))))) 4(0(5(3(2(x1))))) -> 2(4(1(3(1(4(1(2(3(2(x1)))))))))) 5(2(0(0(4(x1))))) -> 5(3(0(3(5(3(0(3(0(2(x1)))))))))) 5(4(2(0(0(x1))))) -> 5(4(5(0(1(3(5(3(1(1(x1)))))))))) 0(1(0(0(0(0(x1)))))) -> 0(2(2(5(0(1(5(5(2(1(x1)))))))))) 1(2(5(3(0(0(x1)))))) -> 4(3(0(5(0(2(4(3(3(5(x1)))))))))) 1(3(2(4(1(0(x1)))))) -> 4(3(2(1(2(4(4(1(5(0(x1)))))))))) 1(3(4(0(0(0(x1)))))) -> 1(5(3(2(2(3(3(1(4(1(x1)))))))))) 2(0(2(0(0(0(x1)))))) -> 2(5(1(0(1(3(1(3(0(0(x1)))))))))) 2(4(0(4(1(0(x1)))))) -> 2(1(4(1(5(2(5(0(4(1(x1)))))))))) 3(0(0(0(0(1(x1)))))) -> 2(3(3(0(5(1(5(1(0(1(x1)))))))))) 3(0(0(1(5(2(x1)))))) -> 2(4(4(4(1(4(1(0(1(3(x1)))))))))) 3(2(4(1(0(5(x1)))))) -> 3(3(3(1(1(4(4(1(0(5(x1)))))))))) 3(3(4(0(0(0(x1)))))) -> 3(0(3(3(3(4(3(3(0(0(x1)))))))))) 4(3(2(0(0(5(x1)))))) -> 1(0(5(0(2(4(1(3(0(5(x1)))))))))) 4(5(3(0(1(0(x1)))))) -> 4(0(3(1(3(1(0(4(2(1(x1)))))))))) 5(1(2(0(0(4(x1)))))) -> 5(3(3(5(1(0(3(5(2(2(x1)))))))))) 0(3(3(0(0(2(0(x1))))))) -> 0(3(3(1(1(3(5(4(3(0(x1)))))))))) 0(5(4(0(0(1(0(x1))))))) -> 4(4(4(4(1(4(5(1(1(0(x1)))))))))) 1(0(0(4(4(0(2(x1))))))) -> 5(4(5(5(0(4(5(2(2(4(x1)))))))))) 1(2(0(0(5(1(2(x1))))))) -> 4(2(4(0(3(5(3(0(4(2(x1)))))))))) 1(3(2(4(4(2(0(x1))))))) -> 1(1(3(1(5(5(2(1(3(0(x1)))))))))) 1(3(4(0(0(0(2(x1))))))) -> 0(1(2(1(4(1(3(1(2(4(x1)))))))))) 2(0(0(0(1(4(5(x1))))))) -> 5(0(4(0(2(5(4(5(5(4(x1)))))))))) 2(0(1(0(0(0(2(x1))))))) -> 3(2(3(1(3(0(3(0(2(4(x1)))))))))) 2(0(1(0(0(5(4(x1))))))) -> 2(4(1(2(2(0(5(0(1(4(x1)))))))))) 2(0(4(5(3(2(0(x1))))))) -> 2(3(2(3(1(0(3(0(3(4(x1)))))))))) 2(0(5(1(3(1(2(x1))))))) -> 2(3(2(5(5(1(2(3(3(0(x1)))))))))) 2(0(5(1(3(3(2(x1))))))) -> 3(4(3(3(3(4(4(1(4(2(x1)))))))))) 2(2(3(0(0(2(0(x1))))))) -> 2(1(5(0(4(5(2(3(0(0(x1)))))))))) 2(2(4(4(0(0(4(x1))))))) -> 2(3(3(4(5(5(0(4(0(2(x1)))))))))) 2(5(2(0(1(0(4(x1))))))) -> 3(4(5(0(0(3(1(3(0(4(x1)))))))))) 3(0(0(3(3(1(2(x1))))))) -> 3(5(2(5(1(2(1(4(1(2(x1)))))))))) 3(0(0(5(3(1(2(x1))))))) -> 3(1(5(4(3(0(4(1(4(4(x1)))))))))) 3(2(0(2(2(0(5(x1))))))) -> 3(2(0(1(3(4(3(4(1(5(x1)))))))))) 3(2(2(4(1(0(0(x1))))))) -> 3(5(0(4(5(0(4(0(3(1(x1)))))))))) 3(2(5(0(0(0(0(x1))))))) -> 2(4(5(5(5(1(3(5(4(1(x1)))))))))) 3(4(1(0(0(0(1(x1))))))) -> 3(2(4(5(0(2(4(3(3(1(x1)))))))))) 3(4(1(1(4(1(3(x1))))))) -> 3(5(1(0(5(0(1(3(1(5(x1)))))))))) 4(0(0(3(4(4(5(x1))))))) -> 2(4(1(3(1(3(4(2(2(4(x1)))))))))) 4(4(0(2(0(0(0(x1))))))) -> 0(2(0(3(0(4(1(3(0(1(x1)))))))))) 5(1(2(0(0(1(2(x1))))))) -> 0(2(5(4(1(5(0(2(2(4(x1)))))))))) 5(2(0(0(1(1(1(x1))))))) -> 0(2(1(1(0(0(3(1(1(1(x1)))))))))) 5(2(0(4(3(4(4(x1))))))) -> 5(4(1(0(5(0(2(4(4(4(x1)))))))))) 5(2(4(0(0(2(0(x1))))))) -> 5(2(1(4(5(3(0(2(1(1(x1)))))))))) 5(4(0(1(4(2(5(x1))))))) -> 5(2(4(4(3(3(3(0(4(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(4(2(x1))) -> 5(3(3(3(5(0(3(1(1(3(x1)))))))))) 2(3(0(4(x1)))) -> 2(3(3(5(3(1(1(5(3(2(x1)))))))))) 3(0(0(4(x1)))) -> 3(0(3(4(3(0(3(0(3(2(x1)))))))))) 1(1(0(0(2(x1))))) -> 1(1(5(0(4(4(1(4(1(2(x1)))))))))) 1(2(0(3(2(x1))))) -> 4(2(0(3(4(3(4(3(5(3(x1)))))))))) 4(0(5(3(2(x1))))) -> 2(4(1(3(1(4(1(2(3(2(x1)))))))))) 5(2(0(0(4(x1))))) -> 5(3(0(3(5(3(0(3(0(2(x1)))))))))) 5(4(2(0(0(x1))))) -> 5(4(5(0(1(3(5(3(1(1(x1)))))))))) 0(1(0(0(0(0(x1)))))) -> 0(2(2(5(0(1(5(5(2(1(x1)))))))))) 1(2(5(3(0(0(x1)))))) -> 4(3(0(5(0(2(4(3(3(5(x1)))))))))) 1(3(2(4(1(0(x1)))))) -> 4(3(2(1(2(4(4(1(5(0(x1)))))))))) 1(3(4(0(0(0(x1)))))) -> 1(5(3(2(2(3(3(1(4(1(x1)))))))))) 2(0(2(0(0(0(x1)))))) -> 2(5(1(0(1(3(1(3(0(0(x1)))))))))) 2(4(0(4(1(0(x1)))))) -> 2(1(4(1(5(2(5(0(4(1(x1)))))))))) 3(0(0(0(0(1(x1)))))) -> 2(3(3(0(5(1(5(1(0(1(x1)))))))))) 3(0(0(1(5(2(x1)))))) -> 2(4(4(4(1(4(1(0(1(3(x1)))))))))) 3(2(4(1(0(5(x1)))))) -> 3(3(3(1(1(4(4(1(0(5(x1)))))))))) 3(3(4(0(0(0(x1)))))) -> 3(0(3(3(3(4(3(3(0(0(x1)))))))))) 4(3(2(0(0(5(x1)))))) -> 1(0(5(0(2(4(1(3(0(5(x1)))))))))) 4(5(3(0(1(0(x1)))))) -> 4(0(3(1(3(1(0(4(2(1(x1)))))))))) 5(1(2(0(0(4(x1)))))) -> 5(3(3(5(1(0(3(5(2(2(x1)))))))))) 0(3(3(0(0(2(0(x1))))))) -> 0(3(3(1(1(3(5(4(3(0(x1)))))))))) 0(5(4(0(0(1(0(x1))))))) -> 4(4(4(4(1(4(5(1(1(0(x1)))))))))) 1(0(0(4(4(0(2(x1))))))) -> 5(4(5(5(0(4(5(2(2(4(x1)))))))))) 1(2(0(0(5(1(2(x1))))))) -> 4(2(4(0(3(5(3(0(4(2(x1)))))))))) 1(3(2(4(4(2(0(x1))))))) -> 1(1(3(1(5(5(2(1(3(0(x1)))))))))) 1(3(4(0(0(0(2(x1))))))) -> 0(1(2(1(4(1(3(1(2(4(x1)))))))))) 2(0(0(0(1(4(5(x1))))))) -> 5(0(4(0(2(5(4(5(5(4(x1)))))))))) 2(0(1(0(0(0(2(x1))))))) -> 3(2(3(1(3(0(3(0(2(4(x1)))))))))) 2(0(1(0(0(5(4(x1))))))) -> 2(4(1(2(2(0(5(0(1(4(x1)))))))))) 2(0(4(5(3(2(0(x1))))))) -> 2(3(2(3(1(0(3(0(3(4(x1)))))))))) 2(0(5(1(3(1(2(x1))))))) -> 2(3(2(5(5(1(2(3(3(0(x1)))))))))) 2(0(5(1(3(3(2(x1))))))) -> 3(4(3(3(3(4(4(1(4(2(x1)))))))))) 2(2(3(0(0(2(0(x1))))))) -> 2(1(5(0(4(5(2(3(0(0(x1)))))))))) 2(2(4(4(0(0(4(x1))))))) -> 2(3(3(4(5(5(0(4(0(2(x1)))))))))) 2(5(2(0(1(0(4(x1))))))) -> 3(4(5(0(0(3(1(3(0(4(x1)))))))))) 3(0(0(3(3(1(2(x1))))))) -> 3(5(2(5(1(2(1(4(1(2(x1)))))))))) 3(0(0(5(3(1(2(x1))))))) -> 3(1(5(4(3(0(4(1(4(4(x1)))))))))) 3(2(0(2(2(0(5(x1))))))) -> 3(2(0(1(3(4(3(4(1(5(x1)))))))))) 3(2(2(4(1(0(0(x1))))))) -> 3(5(0(4(5(0(4(0(3(1(x1)))))))))) 3(2(5(0(0(0(0(x1))))))) -> 2(4(5(5(5(1(3(5(4(1(x1)))))))))) 3(4(1(0(0(0(1(x1))))))) -> 3(2(4(5(0(2(4(3(3(1(x1)))))))))) 3(4(1(1(4(1(3(x1))))))) -> 3(5(1(0(5(0(1(3(1(5(x1)))))))))) 4(0(0(3(4(4(5(x1))))))) -> 2(4(1(3(1(3(4(2(2(4(x1)))))))))) 4(4(0(2(0(0(0(x1))))))) -> 0(2(0(3(0(4(1(3(0(1(x1)))))))))) 5(1(2(0(0(1(2(x1))))))) -> 0(2(5(4(1(5(0(2(2(4(x1)))))))))) 5(2(0(0(1(1(1(x1))))))) -> 0(2(1(1(0(0(3(1(1(1(x1)))))))))) 5(2(0(4(3(4(4(x1))))))) -> 5(4(1(0(5(0(2(4(4(4(x1)))))))))) 5(2(4(0(0(2(0(x1))))))) -> 5(2(1(4(5(3(0(2(1(1(x1)))))))))) 5(4(0(1(4(2(5(x1))))))) -> 5(2(4(4(3(3(3(0(4(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(4(2(x1))) -> 5(3(3(3(5(0(3(1(1(3(x1)))))))))) 2(3(0(4(x1)))) -> 2(3(3(5(3(1(1(5(3(2(x1)))))))))) 3(0(0(4(x1)))) -> 3(0(3(4(3(0(3(0(3(2(x1)))))))))) 1(1(0(0(2(x1))))) -> 1(1(5(0(4(4(1(4(1(2(x1)))))))))) 1(2(0(3(2(x1))))) -> 4(2(0(3(4(3(4(3(5(3(x1)))))))))) 4(0(5(3(2(x1))))) -> 2(4(1(3(1(4(1(2(3(2(x1)))))))))) 5(2(0(0(4(x1))))) -> 5(3(0(3(5(3(0(3(0(2(x1)))))))))) 5(4(2(0(0(x1))))) -> 5(4(5(0(1(3(5(3(1(1(x1)))))))))) 0(1(0(0(0(0(x1)))))) -> 0(2(2(5(0(1(5(5(2(1(x1)))))))))) 1(2(5(3(0(0(x1)))))) -> 4(3(0(5(0(2(4(3(3(5(x1)))))))))) 1(3(2(4(1(0(x1)))))) -> 4(3(2(1(2(4(4(1(5(0(x1)))))))))) 1(3(4(0(0(0(x1)))))) -> 1(5(3(2(2(3(3(1(4(1(x1)))))))))) 2(0(2(0(0(0(x1)))))) -> 2(5(1(0(1(3(1(3(0(0(x1)))))))))) 2(4(0(4(1(0(x1)))))) -> 2(1(4(1(5(2(5(0(4(1(x1)))))))))) 3(0(0(0(0(1(x1)))))) -> 2(3(3(0(5(1(5(1(0(1(x1)))))))))) 3(0(0(1(5(2(x1)))))) -> 2(4(4(4(1(4(1(0(1(3(x1)))))))))) 3(2(4(1(0(5(x1)))))) -> 3(3(3(1(1(4(4(1(0(5(x1)))))))))) 3(3(4(0(0(0(x1)))))) -> 3(0(3(3(3(4(3(3(0(0(x1)))))))))) 4(3(2(0(0(5(x1)))))) -> 1(0(5(0(2(4(1(3(0(5(x1)))))))))) 4(5(3(0(1(0(x1)))))) -> 4(0(3(1(3(1(0(4(2(1(x1)))))))))) 5(1(2(0(0(4(x1)))))) -> 5(3(3(5(1(0(3(5(2(2(x1)))))))))) 0(3(3(0(0(2(0(x1))))))) -> 0(3(3(1(1(3(5(4(3(0(x1)))))))))) 0(5(4(0(0(1(0(x1))))))) -> 4(4(4(4(1(4(5(1(1(0(x1)))))))))) 1(0(0(4(4(0(2(x1))))))) -> 5(4(5(5(0(4(5(2(2(4(x1)))))))))) 1(2(0(0(5(1(2(x1))))))) -> 4(2(4(0(3(5(3(0(4(2(x1)))))))))) 1(3(2(4(4(2(0(x1))))))) -> 1(1(3(1(5(5(2(1(3(0(x1)))))))))) 1(3(4(0(0(0(2(x1))))))) -> 0(1(2(1(4(1(3(1(2(4(x1)))))))))) 2(0(0(0(1(4(5(x1))))))) -> 5(0(4(0(2(5(4(5(5(4(x1)))))))))) 2(0(1(0(0(0(2(x1))))))) -> 3(2(3(1(3(0(3(0(2(4(x1)))))))))) 2(0(1(0(0(5(4(x1))))))) -> 2(4(1(2(2(0(5(0(1(4(x1)))))))))) 2(0(4(5(3(2(0(x1))))))) -> 2(3(2(3(1(0(3(0(3(4(x1)))))))))) 2(0(5(1(3(1(2(x1))))))) -> 2(3(2(5(5(1(2(3(3(0(x1)))))))))) 2(0(5(1(3(3(2(x1))))))) -> 3(4(3(3(3(4(4(1(4(2(x1)))))))))) 2(2(3(0(0(2(0(x1))))))) -> 2(1(5(0(4(5(2(3(0(0(x1)))))))))) 2(2(4(4(0(0(4(x1))))))) -> 2(3(3(4(5(5(0(4(0(2(x1)))))))))) 2(5(2(0(1(0(4(x1))))))) -> 3(4(5(0(0(3(1(3(0(4(x1)))))))))) 3(0(0(3(3(1(2(x1))))))) -> 3(5(2(5(1(2(1(4(1(2(x1)))))))))) 3(0(0(5(3(1(2(x1))))))) -> 3(1(5(4(3(0(4(1(4(4(x1)))))))))) 3(2(0(2(2(0(5(x1))))))) -> 3(2(0(1(3(4(3(4(1(5(x1)))))))))) 3(2(2(4(1(0(0(x1))))))) -> 3(5(0(4(5(0(4(0(3(1(x1)))))))))) 3(2(5(0(0(0(0(x1))))))) -> 2(4(5(5(5(1(3(5(4(1(x1)))))))))) 3(4(1(0(0(0(1(x1))))))) -> 3(2(4(5(0(2(4(3(3(1(x1)))))))))) 3(4(1(1(4(1(3(x1))))))) -> 3(5(1(0(5(0(1(3(1(5(x1)))))))))) 4(0(0(3(4(4(5(x1))))))) -> 2(4(1(3(1(3(4(2(2(4(x1)))))))))) 4(4(0(2(0(0(0(x1))))))) -> 0(2(0(3(0(4(1(3(0(1(x1)))))))))) 5(1(2(0(0(1(2(x1))))))) -> 0(2(5(4(1(5(0(2(2(4(x1)))))))))) 5(2(0(0(1(1(1(x1))))))) -> 0(2(1(1(0(0(3(1(1(1(x1)))))))))) 5(2(0(4(3(4(4(x1))))))) -> 5(4(1(0(5(0(2(4(4(4(x1)))))))))) 5(2(4(0(0(2(0(x1))))))) -> 5(2(1(4(5(3(0(2(1(1(x1)))))))))) 5(4(0(1(4(2(5(x1))))))) -> 5(2(4(4(3(3(3(0(4(5(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. 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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, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653] {(86,87,[0_1|0, 2_1|0, 3_1|0, 1_1|0, 4_1|0, 5_1|0, encArg_1|0, encode_0_1|0, encode_4_1|0, encode_2_1|0, encode_5_1|0, encode_3_1|0, encode_1_1|0]), (86,88,[0_1|1, 2_1|1, 3_1|1, 1_1|1, 4_1|1, 5_1|1]), (86,89,[5_1|2]), (86,98,[0_1|2]), (86,107,[0_1|2]), (86,116,[4_1|2]), (86,125,[2_1|2]), (86,134,[2_1|2]), (86,143,[5_1|2]), (86,152,[3_1|2]), (86,161,[2_1|2]), (86,170,[2_1|2]), (86,179,[2_1|2]), (86,188,[3_1|2]), 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(475,350,[2_1|2]), (475,359,[2_1|2]), (475,368,[2_1|2]), (475,377,[2_1|2]), (475,449,[2_1|2]), (475,125,[2_1|2]), (475,134,[2_1|2]), (475,143,[5_1|2]), (475,152,[3_1|2]), (475,161,[2_1|2]), (475,170,[2_1|2]), (475,179,[2_1|2]), (475,188,[3_1|2]), (475,197,[2_1|2]), (475,206,[2_1|2]), (475,215,[2_1|2]), (475,224,[3_1|2]), (476,477,[2_1|2]), (477,478,[1_1|2]), (478,479,[1_1|2]), (479,480,[0_1|2]), (480,481,[0_1|2]), (481,482,[3_1|2]), (482,483,[1_1|2]), (483,484,[1_1|2]), (483,341,[1_1|2]), (484,88,[1_1|2]), (484,341,[1_1|2]), (484,386,[1_1|2]), (484,395,[1_1|2]), (484,440,[1_1|2]), (484,342,[1_1|2]), (484,387,[1_1|2]), (484,350,[4_1|2]), (484,359,[4_1|2]), (484,368,[4_1|2]), (484,377,[4_1|2]), (484,404,[0_1|2]), (484,413,[5_1|2]), (485,486,[4_1|2]), (486,487,[1_1|2]), (487,488,[0_1|2]), (488,489,[5_1|2]), (489,490,[0_1|2]), (490,491,[2_1|2]), (491,492,[4_1|2]), (492,493,[4_1|2]), (492,458,[0_1|2]), (493,88,[4_1|2]), (493,116,[4_1|2]), (493,350,[4_1|2]), (493,359,[4_1|2]), (493,368,[4_1|2]), (493,377,[4_1|2]), (493,449,[4_1|2]), (493,117,[4_1|2]), (493,422,[2_1|2]), (493,431,[2_1|2]), (493,440,[1_1|2]), (493,458,[0_1|2]), (494,495,[2_1|2]), (495,496,[1_1|2]), (496,497,[4_1|2]), (497,498,[5_1|2]), (498,499,[3_1|2]), (499,500,[0_1|2]), (500,501,[2_1|2]), (501,502,[1_1|2]), (501,341,[1_1|2]), (502,88,[1_1|2]), (502,98,[1_1|2]), (502,107,[1_1|2]), (502,404,[1_1|2, 0_1|2]), (502,458,[1_1|2]), (502,476,[1_1|2]), (502,530,[1_1|2]), (502,460,[1_1|2]), (502,341,[1_1|2]), (502,350,[4_1|2]), (502,359,[4_1|2]), (502,368,[4_1|2]), (502,377,[4_1|2]), (502,386,[1_1|2]), (502,395,[1_1|2]), (502,413,[5_1|2]), (503,504,[4_1|2]), (504,505,[5_1|2]), (505,506,[0_1|2]), (506,507,[1_1|2]), (507,508,[3_1|2]), (508,509,[5_1|2]), (509,510,[3_1|2]), (510,511,[1_1|2]), (510,341,[1_1|2]), (511,88,[1_1|2]), (511,98,[1_1|2]), (511,107,[1_1|2]), (511,404,[1_1|2, 0_1|2]), (511,458,[1_1|2]), (511,476,[1_1|2]), (511,530,[1_1|2]), (511,341,[1_1|2]), (511,350,[4_1|2]), (511,359,[4_1|2]), (511,368,[4_1|2]), (511,377,[4_1|2]), (511,386,[1_1|2]), (511,395,[1_1|2]), (511,413,[5_1|2]), (512,513,[2_1|2]), (513,514,[4_1|2]), (514,515,[4_1|2]), (515,516,[3_1|2]), (516,517,[3_1|2]), (517,518,[3_1|2]), (518,519,[0_1|2]), (519,520,[4_1|2]), (519,449,[4_1|2]), (520,88,[5_1|2]), (520,89,[5_1|2]), (520,143,[5_1|2]), (520,413,[5_1|2]), (520,467,[5_1|2]), (520,485,[5_1|2]), (520,494,[5_1|2]), (520,503,[5_1|2]), (520,512,[5_1|2]), (520,521,[5_1|2]), (520,135,[5_1|2]), (520,476,[0_1|2]), (520,530,[0_1|2]), (521,522,[3_1|2]), (522,523,[3_1|2]), (523,524,[5_1|2]), (524,525,[1_1|2]), (525,526,[0_1|2]), (526,527,[3_1|2]), (527,528,[5_1|2]), (528,529,[2_1|2]), (528,206,[2_1|2]), (528,215,[2_1|2]), (529,88,[2_1|2]), (529,116,[2_1|2]), (529,350,[2_1|2]), (529,359,[2_1|2]), (529,368,[2_1|2]), (529,377,[2_1|2]), (529,449,[2_1|2]), (529,125,[2_1|2]), (529,134,[2_1|2]), (529,143,[5_1|2]), (529,152,[3_1|2]), (529,161,[2_1|2]), (529,170,[2_1|2]), (529,179,[2_1|2]), (529,188,[3_1|2]), (529,197,[2_1|2]), (529,206,[2_1|2]), (529,215,[2_1|2]), (529,224,[3_1|2]), (530,531,[2_1|2]), (531,532,[5_1|2]), (532,533,[4_1|2]), (533,534,[1_1|2]), (534,535,[5_1|2]), (535,536,[0_1|2]), (536,537,[2_1|2]), (536,215,[2_1|2]), (537,538,[2_1|2]), (537,197,[2_1|2]), (538,88,[4_1|2]), (538,125,[4_1|2]), (538,134,[4_1|2]), (538,161,[4_1|2]), (538,170,[4_1|2]), (538,179,[4_1|2]), (538,197,[4_1|2]), (538,206,[4_1|2]), (538,215,[4_1|2]), (538,242,[4_1|2]), (538,251,[4_1|2]), (538,305,[4_1|2]), (538,422,[4_1|2, 2_1|2]), (538,431,[4_1|2, 2_1|2]), (538,406,[4_1|2]), (538,440,[1_1|2]), (538,449,[4_1|2]), (538,458,[0_1|2]), (591,592,[3_1|3]), (592,593,[3_1|3]), (593,594,[3_1|3]), (594,595,[5_1|3]), (595,596,[0_1|3]), (596,597,[3_1|3]), (597,598,[1_1|3]), (598,599,[1_1|3]), (599,351,[3_1|3]), (599,360,[3_1|3]), (600,601,[1_1|3]), (601,602,[5_1|3]), (602,603,[0_1|3]), (603,604,[4_1|3]), (604,605,[4_1|3]), (605,606,[1_1|3]), (606,607,[4_1|3]), (607,608,[1_1|3]), (608,99,[2_1|3]), (608,459,[2_1|3]), (608,477,[2_1|3]), (608,531,[2_1|3]), (609,610,[0_1|3]), (610,611,[3_1|3]), (611,612,[4_1|3]), (612,613,[3_1|3]), (613,614,[0_1|3]), (614,615,[3_1|3]), (615,616,[0_1|3]), (616,617,[3_1|3]), (617,116,[2_1|3]), (617,350,[2_1|3]), (617,359,[2_1|3]), (617,368,[2_1|3]), (617,377,[2_1|3]), (617,449,[2_1|3]), (618,619,[3_1|3]), (619,620,[3_1|3]), (620,621,[3_1|3]), (621,622,[5_1|3]), (622,623,[0_1|3]), (623,624,[3_1|3]), (624,625,[1_1|3]), (625,626,[1_1|3]), (626,125,[3_1|3]), (626,134,[3_1|3]), (626,161,[3_1|3]), (626,170,[3_1|3]), (626,179,[3_1|3]), (626,197,[3_1|3]), (626,206,[3_1|3]), (626,215,[3_1|3]), (626,242,[3_1|3]), (626,251,[3_1|3]), (626,305,[3_1|3]), (626,422,[3_1|3]), (626,431,[3_1|3]), (626,351,[3_1|3]), (626,360,[3_1|3]), (627,628,[3_1|3]), (628,629,[3_1|3]), (629,630,[3_1|3]), (630,631,[5_1|3]), (631,632,[0_1|3]), (632,633,[3_1|3]), (633,634,[1_1|3]), (634,635,[1_1|3]), (634,377,[4_1|2]), (634,386,[1_1|2]), (634,395,[1_1|2]), (634,404,[0_1|2]), (635,88,[3_1|3]), (635,125,[3_1|3]), (635,134,[3_1|3]), (635,161,[3_1|3]), (635,170,[3_1|3]), (635,179,[3_1|3]), (635,197,[3_1|3]), (635,206,[3_1|3]), (635,215,[3_1|3]), (635,242,[3_1|3, 2_1|2]), (635,251,[3_1|3, 2_1|2]), (635,305,[3_1|3, 2_1|2]), (635,422,[3_1|3]), (635,431,[3_1|3]), (635,233,[3_1|2]), (635,260,[3_1|2]), (635,269,[3_1|2]), (635,278,[3_1|2]), (635,287,[3_1|2]), (635,296,[3_1|2]), (635,314,[3_1|2]), (635,323,[3_1|2]), (635,332,[3_1|2]), (636,637,[3_1|3]), (637,638,[3_1|3]), (638,639,[3_1|3]), (639,640,[5_1|3]), (640,641,[0_1|3]), (641,642,[3_1|3]), (642,643,[1_1|3]), (643,644,[1_1|3]), (644,457,[3_1|3]), (645,646,[3_1|3]), (646,647,[3_1|3]), (647,648,[5_1|3]), (648,649,[3_1|3]), (649,650,[1_1|3]), (650,651,[1_1|3]), (651,652,[5_1|3]), (652,653,[3_1|3]), (653,116,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)