/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 62 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 498 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(1(2(0(2(0(x1)))))) 0(1(0(0(x1)))) -> 1(2(0(0(0(x1))))) 0(1(0(1(x1)))) -> 1(1(0(3(0(2(x1)))))) 0(4(1(0(x1)))) -> 0(3(1(4(0(x1))))) 0(4(1(0(x1)))) -> 1(4(0(0(5(x1))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(x1))))) 0(4(1(0(x1)))) -> 0(5(1(4(5(0(x1)))))) 0(4(1(0(x1)))) -> 0(5(1(5(4(0(x1)))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(5(x1)))))) 4(2(1(0(x1)))) -> 1(2(5(0(4(x1))))) 4(2(1(0(x1)))) -> 1(4(2(5(0(x1))))) 4(2(1(0(x1)))) -> 1(4(5(2(0(x1))))) 4(2(1(0(x1)))) -> 2(0(3(1(4(x1))))) 4(2(1(0(x1)))) -> 2(0(5(1(4(x1))))) 4(2(1(0(x1)))) -> 2(1(2(0(4(x1))))) 4(2(1(0(x1)))) -> 2(1(4(0(5(x1))))) 4(2(1(0(x1)))) -> 3(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 4(1(2(2(0(x1))))) 4(2(1(0(x1)))) -> 4(3(2(1(0(x1))))) 4(2(1(0(x1)))) -> 4(5(1(2(0(x1))))) 4(2(1(0(x1)))) -> 5(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 5(4(1(2(0(x1))))) 4(2(1(0(x1)))) -> 2(1(5(2(0(4(x1)))))) 4(2(1(0(x1)))) -> 4(1(2(5(2(0(x1)))))) 4(2(1(0(x1)))) -> 4(3(0(2(1(4(x1)))))) 4(3(0(0(x1)))) -> 3(0(2(0(4(x1))))) 4(3(0(0(x1)))) -> 3(2(0(4(0(x1))))) 4(4(1(0(x1)))) -> 4(0(5(1(4(x1))))) 4(4(1(0(x1)))) -> 4(1(2(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(4(0(5(x1))))) 4(4(1(0(x1)))) -> 4(1(5(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(5(4(0(x1))))) 0(0(2(1(0(x1))))) -> 0(1(2(0(2(0(x1)))))) 0(4(2(1(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(4(2(1(0(x1))))) -> 1(2(0(3(4(0(x1)))))) 0(4(2(1(0(x1))))) -> 4(0(3(1(0(2(x1)))))) 0(4(4(1(0(x1))))) -> 1(4(4(0(5(0(x1)))))) 0(4(4(1(0(x1))))) -> 4(0(1(2(0(4(x1)))))) 0(4(4(1(0(x1))))) -> 5(0(4(1(4(0(x1)))))) 1(0(1(1(4(x1))))) -> 1(1(1(4(0(5(x1)))))) 1(0(4(1(0(x1))))) -> 4(0(5(1(1(0(x1)))))) 1(1(3(0(0(x1))))) -> 1(3(1(0(2(0(x1)))))) 1(3(0(0(1(x1))))) -> 0(1(5(1(0(3(x1)))))) 1(4(1(0(0(x1))))) -> 5(1(1(4(0(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(2(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 1(4(3(0(0(x1))))) -> 3(2(1(0(4(0(x1)))))) 4(0(2(1(0(x1))))) -> 2(0(2(1(4(0(x1)))))) 4(1(0(1(0(x1))))) -> 1(0(4(3(1(0(x1)))))) 4(1(3(0(0(x1))))) -> 3(1(0(4(5(0(x1)))))) 4(1(3(0(0(x1))))) -> 4(3(1(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(4(3(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(5(0(2(0(4(x1)))))) 4(2(0(1(0(x1))))) -> 3(0(2(1(4(0(x1)))))) 4(2(1(0(0(x1))))) -> 2(1(4(2(0(0(x1)))))) 4(2(1(0(1(x1))))) -> 1(2(0(4(1(5(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(2(2(0(4(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(5(2(4(0(x1)))))) 4(2(1(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 4(2(2(1(0(x1))))) -> 1(2(5(0(2(4(x1)))))) 4(2(2(1(0(x1))))) -> 1(4(2(2(0(5(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(3(2(4(0(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(4(1(2(0(x1)))))) 4(2(3(0(0(x1))))) -> 3(2(5(4(0(0(x1)))))) 4(2(4(1(0(x1))))) -> 1(4(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(0(1(2(4(4(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(2(0(4(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(3(4(0(2(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(5(2(0(x1)))))) 4(3(1(2(1(x1))))) -> 3(2(2(1(4(1(x1)))))) 4(3(5(0(0(x1))))) -> 3(0(5(4(0(0(x1)))))) 4(3(5(0(0(x1))))) -> 5(0(3(1(4(0(x1)))))) 4(4(0(1(0(x1))))) -> 4(0(3(1(4(0(x1)))))) 4(4(1(0(1(x1))))) -> 4(0(3(1(4(1(x1)))))) 4(4(2(1(0(x1))))) -> 1(0(3(4(2(4(x1)))))) 4(4(2(1(0(x1))))) -> 4(0(3(1(4(2(x1)))))) 4(4(3(0(0(x1))))) -> 2(0(3(0(4(4(x1)))))) 4(4(3(0(0(x1))))) -> 4(3(0(4(5(0(x1)))))) 4(4(4(1(0(x1))))) -> 4(4(3(1(0(4(x1)))))) 4(5(2(1(0(x1))))) -> 3(2(1(4(5(0(x1)))))) 4(5(4(1(0(x1))))) -> 4(1(4(0(5(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(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(0(2(0(x1)))))) 0(1(0(0(x1)))) -> 1(2(0(0(0(x1))))) 0(1(0(1(x1)))) -> 1(1(0(3(0(2(x1)))))) 0(4(1(0(x1)))) -> 0(3(1(4(0(x1))))) 0(4(1(0(x1)))) -> 1(4(0(0(5(x1))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(x1))))) 0(4(1(0(x1)))) -> 0(5(1(4(5(0(x1)))))) 0(4(1(0(x1)))) -> 0(5(1(5(4(0(x1)))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(5(x1)))))) 4(2(1(0(x1)))) -> 1(2(5(0(4(x1))))) 4(2(1(0(x1)))) -> 1(4(2(5(0(x1))))) 4(2(1(0(x1)))) -> 1(4(5(2(0(x1))))) 4(2(1(0(x1)))) -> 2(0(3(1(4(x1))))) 4(2(1(0(x1)))) -> 2(0(5(1(4(x1))))) 4(2(1(0(x1)))) -> 2(1(2(0(4(x1))))) 4(2(1(0(x1)))) -> 2(1(4(0(5(x1))))) 4(2(1(0(x1)))) -> 3(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 4(1(2(2(0(x1))))) 4(2(1(0(x1)))) -> 4(3(2(1(0(x1))))) 4(2(1(0(x1)))) -> 4(5(1(2(0(x1))))) 4(2(1(0(x1)))) -> 5(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 5(4(1(2(0(x1))))) 4(2(1(0(x1)))) -> 2(1(5(2(0(4(x1)))))) 4(2(1(0(x1)))) -> 4(1(2(5(2(0(x1)))))) 4(2(1(0(x1)))) -> 4(3(0(2(1(4(x1)))))) 4(3(0(0(x1)))) -> 3(0(2(0(4(x1))))) 4(3(0(0(x1)))) -> 3(2(0(4(0(x1))))) 4(4(1(0(x1)))) -> 4(0(5(1(4(x1))))) 4(4(1(0(x1)))) -> 4(1(2(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(4(0(5(x1))))) 4(4(1(0(x1)))) -> 4(1(5(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(5(4(0(x1))))) 0(0(2(1(0(x1))))) -> 0(1(2(0(2(0(x1)))))) 0(4(2(1(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(4(2(1(0(x1))))) -> 1(2(0(3(4(0(x1)))))) 0(4(2(1(0(x1))))) -> 4(0(3(1(0(2(x1)))))) 0(4(4(1(0(x1))))) -> 1(4(4(0(5(0(x1)))))) 0(4(4(1(0(x1))))) -> 4(0(1(2(0(4(x1)))))) 0(4(4(1(0(x1))))) -> 5(0(4(1(4(0(x1)))))) 1(0(1(1(4(x1))))) -> 1(1(1(4(0(5(x1)))))) 1(0(4(1(0(x1))))) -> 4(0(5(1(1(0(x1)))))) 1(1(3(0(0(x1))))) -> 1(3(1(0(2(0(x1)))))) 1(3(0(0(1(x1))))) -> 0(1(5(1(0(3(x1)))))) 1(4(1(0(0(x1))))) -> 5(1(1(4(0(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(2(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 1(4(3(0(0(x1))))) -> 3(2(1(0(4(0(x1)))))) 4(0(2(1(0(x1))))) -> 2(0(2(1(4(0(x1)))))) 4(1(0(1(0(x1))))) -> 1(0(4(3(1(0(x1)))))) 4(1(3(0(0(x1))))) -> 3(1(0(4(5(0(x1)))))) 4(1(3(0(0(x1))))) -> 4(3(1(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(4(3(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(5(0(2(0(4(x1)))))) 4(2(0(1(0(x1))))) -> 3(0(2(1(4(0(x1)))))) 4(2(1(0(0(x1))))) -> 2(1(4(2(0(0(x1)))))) 4(2(1(0(1(x1))))) -> 1(2(0(4(1(5(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(2(2(0(4(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(5(2(4(0(x1)))))) 4(2(1(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 4(2(2(1(0(x1))))) -> 1(2(5(0(2(4(x1)))))) 4(2(2(1(0(x1))))) -> 1(4(2(2(0(5(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(3(2(4(0(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(4(1(2(0(x1)))))) 4(2(3(0(0(x1))))) -> 3(2(5(4(0(0(x1)))))) 4(2(4(1(0(x1))))) -> 1(4(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(0(1(2(4(4(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(2(0(4(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(3(4(0(2(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(5(2(0(x1)))))) 4(3(1(2(1(x1))))) -> 3(2(2(1(4(1(x1)))))) 4(3(5(0(0(x1))))) -> 3(0(5(4(0(0(x1)))))) 4(3(5(0(0(x1))))) -> 5(0(3(1(4(0(x1)))))) 4(4(0(1(0(x1))))) -> 4(0(3(1(4(0(x1)))))) 4(4(1(0(1(x1))))) -> 4(0(3(1(4(1(x1)))))) 4(4(2(1(0(x1))))) -> 1(0(3(4(2(4(x1)))))) 4(4(2(1(0(x1))))) -> 4(0(3(1(4(2(x1)))))) 4(4(3(0(0(x1))))) -> 2(0(3(0(4(4(x1)))))) 4(4(3(0(0(x1))))) -> 4(3(0(4(5(0(x1)))))) 4(4(4(1(0(x1))))) -> 4(4(3(1(0(4(x1)))))) 4(5(2(1(0(x1))))) -> 3(2(1(4(5(0(x1)))))) 4(5(4(1(0(x1))))) -> 4(1(4(0(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(0(2(0(x1)))))) 0(1(0(0(x1)))) -> 1(2(0(0(0(x1))))) 0(1(0(1(x1)))) -> 1(1(0(3(0(2(x1)))))) 0(4(1(0(x1)))) -> 0(3(1(4(0(x1))))) 0(4(1(0(x1)))) -> 1(4(0(0(5(x1))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(x1))))) 0(4(1(0(x1)))) -> 0(5(1(4(5(0(x1)))))) 0(4(1(0(x1)))) -> 0(5(1(5(4(0(x1)))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(5(x1)))))) 4(2(1(0(x1)))) -> 1(2(5(0(4(x1))))) 4(2(1(0(x1)))) -> 1(4(2(5(0(x1))))) 4(2(1(0(x1)))) -> 1(4(5(2(0(x1))))) 4(2(1(0(x1)))) -> 2(0(3(1(4(x1))))) 4(2(1(0(x1)))) -> 2(0(5(1(4(x1))))) 4(2(1(0(x1)))) -> 2(1(2(0(4(x1))))) 4(2(1(0(x1)))) -> 2(1(4(0(5(x1))))) 4(2(1(0(x1)))) -> 3(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 4(1(2(2(0(x1))))) 4(2(1(0(x1)))) -> 4(3(2(1(0(x1))))) 4(2(1(0(x1)))) -> 4(5(1(2(0(x1))))) 4(2(1(0(x1)))) -> 5(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 5(4(1(2(0(x1))))) 4(2(1(0(x1)))) -> 2(1(5(2(0(4(x1)))))) 4(2(1(0(x1)))) -> 4(1(2(5(2(0(x1)))))) 4(2(1(0(x1)))) -> 4(3(0(2(1(4(x1)))))) 4(3(0(0(x1)))) -> 3(0(2(0(4(x1))))) 4(3(0(0(x1)))) -> 3(2(0(4(0(x1))))) 4(4(1(0(x1)))) -> 4(0(5(1(4(x1))))) 4(4(1(0(x1)))) -> 4(1(2(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(4(0(5(x1))))) 4(4(1(0(x1)))) -> 4(1(5(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(5(4(0(x1))))) 0(0(2(1(0(x1))))) -> 0(1(2(0(2(0(x1)))))) 0(4(2(1(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(4(2(1(0(x1))))) -> 1(2(0(3(4(0(x1)))))) 0(4(2(1(0(x1))))) -> 4(0(3(1(0(2(x1)))))) 0(4(4(1(0(x1))))) -> 1(4(4(0(5(0(x1)))))) 0(4(4(1(0(x1))))) -> 4(0(1(2(0(4(x1)))))) 0(4(4(1(0(x1))))) -> 5(0(4(1(4(0(x1)))))) 1(0(1(1(4(x1))))) -> 1(1(1(4(0(5(x1)))))) 1(0(4(1(0(x1))))) -> 4(0(5(1(1(0(x1)))))) 1(1(3(0(0(x1))))) -> 1(3(1(0(2(0(x1)))))) 1(3(0(0(1(x1))))) -> 0(1(5(1(0(3(x1)))))) 1(4(1(0(0(x1))))) -> 5(1(1(4(0(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(2(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 1(4(3(0(0(x1))))) -> 3(2(1(0(4(0(x1)))))) 4(0(2(1(0(x1))))) -> 2(0(2(1(4(0(x1)))))) 4(1(0(1(0(x1))))) -> 1(0(4(3(1(0(x1)))))) 4(1(3(0(0(x1))))) -> 3(1(0(4(5(0(x1)))))) 4(1(3(0(0(x1))))) -> 4(3(1(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(4(3(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(5(0(2(0(4(x1)))))) 4(2(0(1(0(x1))))) -> 3(0(2(1(4(0(x1)))))) 4(2(1(0(0(x1))))) -> 2(1(4(2(0(0(x1)))))) 4(2(1(0(1(x1))))) -> 1(2(0(4(1(5(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(2(2(0(4(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(5(2(4(0(x1)))))) 4(2(1(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 4(2(2(1(0(x1))))) -> 1(2(5(0(2(4(x1)))))) 4(2(2(1(0(x1))))) -> 1(4(2(2(0(5(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(3(2(4(0(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(4(1(2(0(x1)))))) 4(2(3(0(0(x1))))) -> 3(2(5(4(0(0(x1)))))) 4(2(4(1(0(x1))))) -> 1(4(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(0(1(2(4(4(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(2(0(4(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(3(4(0(2(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(5(2(0(x1)))))) 4(3(1(2(1(x1))))) -> 3(2(2(1(4(1(x1)))))) 4(3(5(0(0(x1))))) -> 3(0(5(4(0(0(x1)))))) 4(3(5(0(0(x1))))) -> 5(0(3(1(4(0(x1)))))) 4(4(0(1(0(x1))))) -> 4(0(3(1(4(0(x1)))))) 4(4(1(0(1(x1))))) -> 4(0(3(1(4(1(x1)))))) 4(4(2(1(0(x1))))) -> 1(0(3(4(2(4(x1)))))) 4(4(2(1(0(x1))))) -> 4(0(3(1(4(2(x1)))))) 4(4(3(0(0(x1))))) -> 2(0(3(0(4(4(x1)))))) 4(4(3(0(0(x1))))) -> 4(3(0(4(5(0(x1)))))) 4(4(4(1(0(x1))))) -> 4(4(3(1(0(4(x1)))))) 4(5(2(1(0(x1))))) -> 3(2(1(4(5(0(x1)))))) 4(5(4(1(0(x1))))) -> 4(1(4(0(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(0(2(0(x1)))))) 0(1(0(0(x1)))) -> 1(2(0(0(0(x1))))) 0(1(0(1(x1)))) -> 1(1(0(3(0(2(x1)))))) 0(4(1(0(x1)))) -> 0(3(1(4(0(x1))))) 0(4(1(0(x1)))) -> 1(4(0(0(5(x1))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(x1))))) 0(4(1(0(x1)))) -> 0(5(1(4(5(0(x1)))))) 0(4(1(0(x1)))) -> 0(5(1(5(4(0(x1)))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(5(x1)))))) 4(2(1(0(x1)))) -> 1(2(5(0(4(x1))))) 4(2(1(0(x1)))) -> 1(4(2(5(0(x1))))) 4(2(1(0(x1)))) -> 1(4(5(2(0(x1))))) 4(2(1(0(x1)))) -> 2(0(3(1(4(x1))))) 4(2(1(0(x1)))) -> 2(0(5(1(4(x1))))) 4(2(1(0(x1)))) -> 2(1(2(0(4(x1))))) 4(2(1(0(x1)))) -> 2(1(4(0(5(x1))))) 4(2(1(0(x1)))) -> 3(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 4(1(2(2(0(x1))))) 4(2(1(0(x1)))) -> 4(3(2(1(0(x1))))) 4(2(1(0(x1)))) -> 4(5(1(2(0(x1))))) 4(2(1(0(x1)))) -> 5(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 5(4(1(2(0(x1))))) 4(2(1(0(x1)))) -> 2(1(5(2(0(4(x1)))))) 4(2(1(0(x1)))) -> 4(1(2(5(2(0(x1)))))) 4(2(1(0(x1)))) -> 4(3(0(2(1(4(x1)))))) 4(3(0(0(x1)))) -> 3(0(2(0(4(x1))))) 4(3(0(0(x1)))) -> 3(2(0(4(0(x1))))) 4(4(1(0(x1)))) -> 4(0(5(1(4(x1))))) 4(4(1(0(x1)))) -> 4(1(2(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(4(0(5(x1))))) 4(4(1(0(x1)))) -> 4(1(5(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(5(4(0(x1))))) 0(0(2(1(0(x1))))) -> 0(1(2(0(2(0(x1)))))) 0(4(2(1(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(4(2(1(0(x1))))) -> 1(2(0(3(4(0(x1)))))) 0(4(2(1(0(x1))))) -> 4(0(3(1(0(2(x1)))))) 0(4(4(1(0(x1))))) -> 1(4(4(0(5(0(x1)))))) 0(4(4(1(0(x1))))) -> 4(0(1(2(0(4(x1)))))) 0(4(4(1(0(x1))))) -> 5(0(4(1(4(0(x1)))))) 1(0(1(1(4(x1))))) -> 1(1(1(4(0(5(x1)))))) 1(0(4(1(0(x1))))) -> 4(0(5(1(1(0(x1)))))) 1(1(3(0(0(x1))))) -> 1(3(1(0(2(0(x1)))))) 1(3(0(0(1(x1))))) -> 0(1(5(1(0(3(x1)))))) 1(4(1(0(0(x1))))) -> 5(1(1(4(0(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(2(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 1(4(3(0(0(x1))))) -> 3(2(1(0(4(0(x1)))))) 4(0(2(1(0(x1))))) -> 2(0(2(1(4(0(x1)))))) 4(1(0(1(0(x1))))) -> 1(0(4(3(1(0(x1)))))) 4(1(3(0(0(x1))))) -> 3(1(0(4(5(0(x1)))))) 4(1(3(0(0(x1))))) -> 4(3(1(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(4(3(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(5(0(2(0(4(x1)))))) 4(2(0(1(0(x1))))) -> 3(0(2(1(4(0(x1)))))) 4(2(1(0(0(x1))))) -> 2(1(4(2(0(0(x1)))))) 4(2(1(0(1(x1))))) -> 1(2(0(4(1(5(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(2(2(0(4(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(5(2(4(0(x1)))))) 4(2(1(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 4(2(2(1(0(x1))))) -> 1(2(5(0(2(4(x1)))))) 4(2(2(1(0(x1))))) -> 1(4(2(2(0(5(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(3(2(4(0(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(4(1(2(0(x1)))))) 4(2(3(0(0(x1))))) -> 3(2(5(4(0(0(x1)))))) 4(2(4(1(0(x1))))) -> 1(4(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(0(1(2(4(4(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(2(0(4(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(3(4(0(2(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(5(2(0(x1)))))) 4(3(1(2(1(x1))))) -> 3(2(2(1(4(1(x1)))))) 4(3(5(0(0(x1))))) -> 3(0(5(4(0(0(x1)))))) 4(3(5(0(0(x1))))) -> 5(0(3(1(4(0(x1)))))) 4(4(0(1(0(x1))))) -> 4(0(3(1(4(0(x1)))))) 4(4(1(0(1(x1))))) -> 4(0(3(1(4(1(x1)))))) 4(4(2(1(0(x1))))) -> 1(0(3(4(2(4(x1)))))) 4(4(2(1(0(x1))))) -> 4(0(3(1(4(2(x1)))))) 4(4(3(0(0(x1))))) -> 2(0(3(0(4(4(x1)))))) 4(4(3(0(0(x1))))) -> 4(3(0(4(5(0(x1)))))) 4(4(4(1(0(x1))))) -> 4(4(3(1(0(4(x1)))))) 4(5(2(1(0(x1))))) -> 3(2(1(4(5(0(x1)))))) 4(5(4(1(0(x1))))) -> 4(1(4(0(5(0(x1)))))) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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. "[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, 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, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734] {(67,68,[0_1|0, 4_1|0, 1_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]), (67,69,[2_1|1, 3_1|1, 5_1|1, 0_1|1, 4_1|1, 1_1|1]), (67,70,[0_1|2]), (67,75,[1_1|2]), (67,79,[1_1|2]), (67,84,[0_1|2]), (67,88,[1_1|2]), (67,92,[1_1|2]), (67,96,[0_1|2]), (67,101,[0_1|2]), (67,106,[1_1|2]), (67,111,[0_1|2]), (67,116,[1_1|2]), (67,121,[4_1|2]), (67,126,[1_1|2]), (67,131,[4_1|2]), (67,136,[5_1|2]), (67,141,[1_1|2]), (67,145,[1_1|2]), (67,149,[1_1|2]), (67,153,[2_1|2]), (67,157,[2_1|2]), (67,161,[2_1|2]), (67,165,[2_1|2]), (67,169,[3_1|2]), (67,173,[4_1|2]), (67,177,[4_1|2]), (67,181,[4_1|2]), (67,185,[5_1|2]), (67,189,[5_1|2]), (67,193,[2_1|2]), (67,198,[4_1|2]), (67,203,[4_1|2]), (67,208,[2_1|2]), (67,213,[1_1|2]), (67,218,[4_1|2]), (67,223,[4_1|2]), (67,228,[4_1|2]), (67,233,[1_1|2]), (67,238,[1_1|2]), (67,243,[3_1|2]), (67,248,[1_1|2]), (67,253,[1_1|2]), (67,258,[2_1|2]), (67,263,[2_1|2]), (67,268,[3_1|2]), (67,273,[1_1|2]), (67,278,[4_1|2]), (67,283,[4_1|2]), (67,288,[4_1|2]), (67,293,[4_1|2]), (67,298,[4_1|2]), (67,303,[3_1|2]), (67,307,[3_1|2]), (67,311,[3_1|2]), (67,316,[3_1|2]), (67,321,[5_1|2]), (67,326,[4_1|2]), (67,330,[4_1|2]), (67,334,[4_1|2]), (67,338,[4_1|2]), (67,342,[4_1|2]), (67,346,[4_1|2]), (67,351,[4_1|2]), (67,356,[1_1|2]), (67,361,[4_1|2]), (67,366,[2_1|2]), (67,371,[4_1|2]), (67,376,[4_1|2]), (67,381,[2_1|2]), (67,386,[1_1|2]), (67,391,[3_1|2]), (67,396,[4_1|2]), (67,401,[3_1|2]), (67,406,[4_1|2]), (67,411,[1_1|2]), (67,416,[4_1|2]), (67,421,[1_1|2]), (67,426,[0_1|2]), (67,431,[5_1|2]), (67,436,[4_1|2]), (67,441,[3_1|2]), (67,446,[0_1|2]), (68,68,[2_1|0, 3_1|0, 5_1|0, cons_0_1|0, cons_4_1|0, cons_1_1|0]), (69,68,[encArg_1|1]), (69,69,[2_1|1, 3_1|1, 5_1|1, 0_1|1, 4_1|1, 1_1|1]), (69,70,[0_1|2]), (69,75,[1_1|2]), (69,79,[1_1|2]), (69,84,[0_1|2]), (69,88,[1_1|2]), (69,92,[1_1|2]), (69,96,[0_1|2]), (69,101,[0_1|2]), (69,106,[1_1|2]), (69,111,[0_1|2]), (69,116,[1_1|2]), (69,121,[4_1|2]), (69,126,[1_1|2]), (69,131,[4_1|2]), (69,136,[5_1|2]), (69,141,[1_1|2]), (69,145,[1_1|2]), (69,149,[1_1|2]), (69,153,[2_1|2]), (69,157,[2_1|2]), (69,161,[2_1|2]), (69,165,[2_1|2]), (69,169,[3_1|2]), (69,173,[4_1|2]), (69,177,[4_1|2]), (69,181,[4_1|2]), (69,185,[5_1|2]), (69,189,[5_1|2]), (69,193,[2_1|2]), (69,198,[4_1|2]), (69,203,[4_1|2]), (69,208,[2_1|2]), (69,213,[1_1|2]), (69,218,[4_1|2]), (69,223,[4_1|2]), (69,228,[4_1|2]), (69,233,[1_1|2]), (69,238,[1_1|2]), (69,243,[3_1|2]), (69,248,[1_1|2]), (69,253,[1_1|2]), (69,258,[2_1|2]), (69,263,[2_1|2]), (69,268,[3_1|2]), (69,273,[1_1|2]), (69,278,[4_1|2]), (69,283,[4_1|2]), (69,288,[4_1|2]), (69,293,[4_1|2]), (69,298,[4_1|2]), (69,303,[3_1|2]), (69,307,[3_1|2]), (69,311,[3_1|2]), (69,316,[3_1|2]), (69,321,[5_1|2]), (69,326,[4_1|2]), (69,330,[4_1|2]), (69,334,[4_1|2]), (69,338,[4_1|2]), (69,342,[4_1|2]), (69,346,[4_1|2]), (69,351,[4_1|2]), (69,356,[1_1|2]), (69,361,[4_1|2]), (69,366,[2_1|2]), (69,371,[4_1|2]), (69,376,[4_1|2]), (69,381,[2_1|2]), (69,386,[1_1|2]), (69,391,[3_1|2]), (69,396,[4_1|2]), (69,401,[3_1|2]), (69,406,[4_1|2]), (69,411,[1_1|2]), (69,416,[4_1|2]), (69,421,[1_1|2]), (69,426,[0_1|2]), (69,431,[5_1|2]), (69,436,[4_1|2]), (69,441,[3_1|2]), (69,446,[0_1|2]), (70,71,[1_1|2]), (71,72,[2_1|2]), (72,73,[0_1|2]), (73,74,[2_1|2]), (74,69,[0_1|2]), (74,70,[0_1|2]), (74,84,[0_1|2]), (74,96,[0_1|2]), (74,101,[0_1|2]), (74,111,[0_1|2]), (74,426,[0_1|2]), (74,446,[0_1|2]), (74,75,[1_1|2]), (74,79,[1_1|2]), (74,88,[1_1|2]), (74,92,[1_1|2]), (74,106,[1_1|2]), (74,116,[1_1|2]), (74,121,[4_1|2]), (74,126,[1_1|2]), (74,131,[4_1|2]), (74,136,[5_1|2]), (74,655,[0_1|2]), (74,660,[0_1|2]), (74,664,[0_1|2]), 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---------------------------------------- (8) BOUNDS(1, n^1)