/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 57 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 325 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(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(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(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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 3. The certificate found is represented by the following graph. "[59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 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, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 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] {(59,60,[0_1|0, 1_1|0, 5_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]), (59,61,[2_1|1, 3_1|1, 4_1|1, 0_1|1, 1_1|1, 5_1|1]), (59,62,[0_1|2]), (59,66,[1_1|2]), (59,73,[5_1|2]), (59,80,[0_1|2]), (59,87,[0_1|2]), (59,93,[0_1|2]), (59,98,[0_1|2]), (59,105,[0_1|2]), (59,111,[0_1|2]), (59,118,[0_1|2]), (59,124,[1_1|2]), (59,131,[1_1|2]), (59,136,[0_1|2]), (59,143,[0_1|2]), (59,148,[5_1|2]), (59,155,[3_1|2]), (59,162,[0_1|2]), (59,168,[0_1|2]), (59,174,[1_1|2]), (59,181,[1_1|2]), (59,187,[4_1|2]), (59,193,[0_1|2]), (59,200,[0_1|2]), (59,204,[0_1|2]), (59,209,[0_1|2]), (59,214,[0_1|2]), (59,221,[1_1|2]), (59,225,[0_1|2]), (59,230,[0_1|2]), (59,235,[1_1|2]), (59,242,[0_1|2]), (59,247,[0_1|2]), (59,251,[5_1|2]), (59,255,[0_1|2]), (59,262,[5_1|2]), (59,268,[0_1|2]), (59,274,[5_1|2]), (59,281,[0_1|2]), (59,286,[0_1|2]), (59,292,[0_1|2]), (59,297,[1_1|2]), (59,304,[0_1|2]), (59,309,[0_1|2]), (59,314,[0_1|2]), (59,320,[0_1|2]), (59,326,[0_1|2]), (59,333,[1_1|2]), (59,339,[1_1|2]), (59,345,[1_1|2]), (59,349,[1_1|2]), (59,354,[1_1|2]), (59,359,[3_1|2]), (59,366,[0_1|2]), (59,373,[0_1|2]), (59,377,[0_1|2]), (59,383,[0_1|2]), 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(510,427,[1_1|2]), (510,434,[1_1|2]), (510,439,[1_1|2]), (510,444,[1_1|2]), (510,449,[1_1|2]), (510,456,[1_1|2]), (510,463,[0_1|2]), (510,469,[0_1|2]), (510,475,[5_1|2]), (510,521,[0_1|3]), (511,512,[3_1|2]), (512,513,[2_1|2]), (513,514,[3_1|2]), (514,515,[1_1|2]), (515,73,[5_1|2]), (515,148,[5_1|2]), (515,251,[5_1|2]), (515,262,[5_1|2]), (515,274,[5_1|2]), (515,388,[5_1|2]), (515,399,[5_1|2]), (515,475,[5_1|2]), (515,482,[5_1|2]), (515,486,[5_1|2]), (515,490,[5_1|2]), (515,495,[5_1|2]), (515,501,[5_1|2]), (515,506,[5_1|2]), (515,282,[5_1|2]), (515,321,[5_1|2]), (516,517,[3_1|2]), (517,518,[2_1|2]), (518,519,[3_1|2]), (519,520,[1_1|2]), (519,406,[1_1|2]), (519,412,[0_1|2]), (519,418,[0_1|2]), (519,422,[1_1|2]), (519,427,[1_1|2]), (519,531,[1_1|3]), (519,537,[0_1|3]), (519,541,[1_1|3]), (520,61,[5_1|2]), (520,73,[5_1|2]), (520,148,[5_1|2]), (520,251,[5_1|2]), (520,262,[5_1|2]), (520,274,[5_1|2]), (520,388,[5_1|2]), (520,399,[5_1|2]), (520,475,[5_1|2]), (520,482,[5_1|2]), (520,486,[5_1|2]), (520,490,[5_1|2]), (520,495,[5_1|2]), (520,501,[5_1|2]), (520,506,[5_1|2]), (520,282,[5_1|2]), (520,321,[5_1|2]), (521,522,[1_1|3]), (522,523,[1_1|3]), (523,524,[5_1|3]), (524,322,[4_1|3]), (525,526,[1_1|3]), (526,527,[0_1|3]), (527,528,[1_1|3]), (528,529,[1_1|3]), (529,530,[5_1|3]), (530,92,[2_1|3]), (531,532,[1_1|3]), (532,533,[0_1|3]), (533,534,[1_1|3]), (534,535,[1_1|3]), (535,536,[5_1|3]), (536,81,[2_1|3]), (536,169,[2_1|3]), (536,210,[2_1|3]), (536,269,[2_1|3]), (537,538,[1_1|3]), (538,539,[1_1|3]), (539,540,[5_1|3]), (540,132,[2_1|3]), (541,542,[0_1|3]), (542,543,[1_1|3]), (543,544,[5_1|3]), (544,545,[3_1|3]), (545,132,[2_1|3]), (546,547,[1_1|3]), (547,548,[3_1|3]), (548,549,[0_1|3]), (549,132,[2_1|3]), (550,551,[1_1|3]), (551,552,[0_1|3]), (552,553,[3_1|3]), (553,554,[2_1|3]), (554,132,[2_1|3]), (555,556,[1_1|3]), (556,557,[0_1|3]), (557,558,[3_1|3]), (558,559,[2_1|3]), (559,132,[3_1|3]), (560,561,[1_1|3]), (561,562,[1_1|3]), (562,563,[5_1|3]), (563,149,[4_1|3]), (563,322,[4_1|3]), (564,565,[1_1|3]), (565,566,[0_1|3]), (566,567,[0_1|3]), (567,568,[1_1|3]), (568,569,[5_1|3]), (569,282,[4_1|3]), (569,321,[4_1|3]), (570,571,[3_1|3]), (571,572,[0_1|3]), (572,573,[0_1|3]), (573,574,[1_1|3]), (574,575,[5_1|3]), (575,576,[2_1|3]), (576,171,[5_1|3]), (577,578,[5_1|3]), (578,579,[1_1|3]), (579,580,[1_1|3]), (580,581,[5_1|3]), (581,582,[3_1|3]), (582,583,[2_1|3]), (583,132,[2_1|3]), (584,585,[0_1|3]), (585,586,[1_1|3]), (586,587,[1_1|3]), (587,588,[5_1|3]), (588,68,[3_1|3]), (589,590,[0_1|3]), (590,591,[1_1|3]), (591,592,[5_1|3]), (592,132,[2_1|3]), (593,594,[1_1|3]), (594,595,[0_1|3]), (595,596,[1_1|3]), (596,597,[5_1|3]), (597,132,[2_1|3]), (598,599,[3_1|3]), (599,600,[2_1|3]), (600,601,[3_1|3]), (601,602,[1_1|3]), (602,132,[5_1|3]), (603,604,[3_1|3]), (604,605,[0_1|3]), (605,606,[5_1|3]), (606,282,[5_1|3]), (606,321,[5_1|3]), (607,608,[1_1|3]), (608,609,[3_1|3]), (609,610,[0_1|3]), (610,81,[2_1|3]), (610,169,[2_1|3]), (610,210,[2_1|3]), (610,269,[2_1|3]), (611,612,[1_1|3]), (612,613,[0_1|3]), (613,614,[3_1|3]), (614,615,[2_1|3]), (615,81,[2_1|3]), (615,169,[2_1|3]), (615,210,[2_1|3]), (615,269,[2_1|3]), (616,617,[1_1|3]), (617,618,[0_1|3]), (618,619,[3_1|3]), (619,620,[2_1|3]), (620,81,[3_1|3]), (620,169,[3_1|3]), (620,210,[3_1|3]), (620,269,[3_1|3]), (621,622,[3_1|3]), (622,623,[2_1|3]), (623,624,[2_1|3]), (624,625,[0_1|3]), (625,626,[0_1|3]), (626,627,[1_1|3]), (627,212,[5_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)