/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 190 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 788 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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. "[54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 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, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767] {(54,55,[0_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]), (54,56,[3_1|1]), (54,63,[3_1|1]), (54,70,[3_1|1]), (54,76,[3_1|1]), (54,80,[5_1|1]), (54,85,[3_1|1]), (54,90,[5_1|1]), (54,95,[3_1|1]), (54,102,[3_1|1]), (54,108,[4_1|1]), (54,115,[2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1, 1_1|1]), (54,116,[0_1|2]), (54,121,[3_1|2]), (54,128,[1_1|2]), (54,135,[0_1|2]), (54,141,[4_1|2]), (54,147,[5_1|2]), (54,154,[0_1|2]), 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(616,450,[4_1|2]), (616,457,[5_1|2]), (616,464,[5_1|2]), (616,629,[0_1|3]), (616,624,[2_1|3]), (617,618,[4_1|2]), (618,619,[2_1|2]), (619,620,[3_1|2]), (620,621,[1_1|2]), (621,622,[4_1|2]), (622,623,[1_1|2]), (622,484,[2_1|2]), (622,489,[2_1|2]), (622,494,[1_1|2]), (622,501,[2_1|2]), (622,507,[2_1|2]), (622,514,[0_1|2]), (622,521,[5_1|2]), (622,710,[2_1|3]), (623,115,[0_1|2]), (623,147,[0_1|2, 5_1|2]), (623,167,[0_1|2, 5_1|2]), (623,172,[0_1|2, 5_1|2]), (623,341,[0_1|2, 5_1|2]), (623,375,[0_1|2, 5_1|2]), (623,380,[0_1|2, 5_1|2]), (623,437,[0_1|2, 5_1|2]), (623,457,[0_1|2, 5_1|2]), (623,464,[0_1|2, 5_1|2]), (623,521,[0_1|2]), (623,547,[0_1|2]), (623,552,[0_1|2]), (623,584,[0_1|2]), (623,610,[0_1|2]), (623,617,[0_1|2]), (623,161,[0_1|2]), (623,116,[0_1|2]), (623,121,[3_1|2]), (623,128,[1_1|2]), (623,135,[0_1|2]), (623,141,[4_1|2]), (623,154,[0_1|2]), (623,160,[3_1|2]), (623,178,[0_1|2]), (623,185,[0_1|2]), (623,191,[3_1|2]), (623,195,[0_1|2]), (623,200,[0_1|2]), (623,206,[0_1|2]), (623,213,[0_1|2]), (623,218,[3_1|2]), (623,223,[3_1|2]), (623,230,[0_1|2]), (623,237,[3_1|2]), (623,242,[3_1|2]), (623,249,[3_1|2]), (623,256,[0_1|2]), (623,263,[3_1|2]), (623,270,[3_1|2]), (623,276,[3_1|2]), (623,280,[3_1|2]), (623,287,[3_1|2]), (623,294,[0_1|2]), (623,301,[0_1|2]), (623,308,[0_1|2]), (623,314,[0_1|2]), (623,320,[0_1|2]), (623,327,[2_1|2]), (623,334,[0_1|2]), (623,348,[0_1|2]), (623,354,[0_1|2]), (623,361,[0_1|2]), (623,368,[0_1|2]), (623,386,[0_1|2]), (623,392,[2_1|2]), (623,399,[3_1|2]), (623,405,[0_1|2]), (623,412,[3_1|2]), (623,419,[3_1|2]), (623,424,[3_1|2]), (623,430,[0_1|2]), (623,443,[3_1|2]), (623,450,[4_1|2]), (623,470,[0_1|2]), (623,477,[0_1|2]), (623,629,[0_1|3]), (623,624,[2_1|3]), (624,625,[3_1|3]), (625,626,[1_1|3]), (626,627,[3_1|3]), (627,628,[0_1|3]), (628,130,[0_1|3]), (629,630,[2_1|3]), (630,631,[3_1|3]), (631,632,[1_1|3]), (632,633,[3_1|3]), (633,129,[0_1|3]), (633,452,[0_1|3]), (633,149,[0_1|3]), (634,635,[2_1|3]), (635,636,[3_1|3]), (636,637,[1_1|3]), (637,638,[3_1|3]), (638,116,[0_1|3]), (638,135,[0_1|3]), (638,154,[0_1|3]), (638,178,[0_1|3]), (638,185,[0_1|3]), (638,195,[0_1|3]), (638,200,[0_1|3]), (638,206,[0_1|3]), (638,213,[0_1|3]), (638,230,[0_1|3]), (638,256,[0_1|3]), (638,294,[0_1|3]), (638,301,[0_1|3]), (638,308,[0_1|3]), (638,314,[0_1|3]), (638,320,[0_1|3]), (638,334,[0_1|3]), (638,348,[0_1|3]), (638,354,[0_1|3]), (638,361,[0_1|3]), (638,368,[0_1|3]), (638,386,[0_1|3]), (638,405,[0_1|3]), (638,430,[0_1|3]), (638,470,[0_1|3]), (638,477,[0_1|3]), (638,514,[0_1|3]), (638,533,[0_1|3]), (638,591,[0_1|3]), (638,129,[0_1|3]), (638,149,[0_1|3]), (639,640,[3_1|3]), (640,641,[1_1|3]), (641,642,[3_1|3]), (642,643,[0_1|3]), (642,629,[0_1|3]), (643,148,[1_1|3]), (643,585,[1_1|3]), (644,645,[5_1|3]), (645,646,[4_1|3]), (646,647,[4_1|3]), (647,648,[3_1|3]), (648,649,[1_1|3]), (649,161,[4_1|3]), (650,651,[5_1|3]), (651,652,[4_1|3]), (652,653,[3_1|3]), (653,654,[1_1|3]), (654,655,[3_1|3]), (655,656,[2_1|3]), (656,161,[0_1|3]), (657,658,[3_1|3]), (658,659,[1_1|3]), (659,660,[3_1|3]), (660,661,[0_1|3]), (661,231,[0_1|3]), (661,309,[0_1|3]), (661,335,[0_1|3]), (661,362,[0_1|3]), (661,369,[0_1|3]), (661,387,[0_1|3]), (661,130,[0_1|3]), (662,663,[1_1|3]), (663,664,[3_1|3]), (664,665,[0_1|3]), (664,662,[3_1|3]), (664,666,[0_1|3]), (664,671,[0_1|3]), (664,677,[0_1|3]), (665,128,[2_1|3]), (665,494,[2_1|3]), (665,528,[2_1|3]), (665,604,[2_1|3]), (665,605,[2_1|3]), (666,667,[2_1|3]), (667,668,[3_1|3]), (668,669,[3_1|3]), (669,670,[1_1|3]), (670,128,[3_1|3]), (670,494,[3_1|3]), (670,528,[3_1|3]), (670,604,[3_1|3]), (670,605,[3_1|3]), (671,672,[4_1|3]), (672,673,[3_1|3]), (673,674,[1_1|3]), (674,675,[3_1|3]), (675,676,[2_1|3]), (676,128,[2_1|3]), (676,494,[2_1|3]), (676,528,[2_1|3]), (676,604,[2_1|3]), (676,605,[2_1|3]), (677,678,[2_1|3]), (678,679,[3_1|3]), (679,680,[2_1|3]), (680,681,[2_1|3]), (681,682,[1_1|3]), (682,683,[3_1|3]), (683,128,[2_1|3]), (683,494,[2_1|3]), (683,528,[2_1|3]), 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(713,714,[0_1|3]), (714,116,[0_1|3]), (714,135,[0_1|3]), (714,154,[0_1|3]), (714,178,[0_1|3]), (714,185,[0_1|3]), (714,195,[0_1|3]), (714,200,[0_1|3]), (714,206,[0_1|3]), (714,213,[0_1|3]), (714,230,[0_1|3]), (714,256,[0_1|3]), (714,294,[0_1|3]), (714,301,[0_1|3]), (714,308,[0_1|3]), (714,314,[0_1|3]), (714,320,[0_1|3]), (714,334,[0_1|3]), (714,348,[0_1|3]), (714,354,[0_1|3]), (714,361,[0_1|3]), (714,368,[0_1|3]), (714,386,[0_1|3]), (714,405,[0_1|3]), (714,430,[0_1|3]), (714,470,[0_1|3]), (714,477,[0_1|3]), (714,514,[0_1|3]), (714,533,[0_1|3]), (714,591,[0_1|3]), (714,342,[0_1|3]), (714,231,[0_1|3]), (714,309,[0_1|3]), (714,335,[0_1|3]), (714,362,[0_1|3]), (714,369,[0_1|3]), (714,387,[0_1|3]), (714,130,[0_1|3]), (715,716,[0_1|3]), (716,717,[4_1|3]), (717,718,[4_1|3]), (718,719,[3_1|3]), (719,720,[2_1|3]), (720,328,[0_1|3]), (721,722,[2_1|3]), (722,723,[0_1|3]), (723,724,[4_1|3]), (724,725,[4_1|3]), (725,726,[5_1|3]), (726,141,[4_1|3]), (726,450,[4_1|3]), (726,540,[4_1|3]), (726,557,[4_1|3]), (726,597,[4_1|3]), (726,376,[4_1|3]), (726,381,[4_1|3]), (726,438,[4_1|3]), (726,458,[4_1|3]), (726,465,[4_1|3]), (726,522,[4_1|3]), (726,553,[4_1|3]), (726,618,[4_1|3]), (726,162,[4_1|3]), (727,728,[2_1|3]), (728,729,[0_1|3]), (729,730,[4_1|3]), (730,147,[5_1|3]), (730,167,[5_1|3]), (730,172,[5_1|3]), (730,341,[5_1|3]), (730,375,[5_1|3]), (730,380,[5_1|3]), (730,437,[5_1|3]), (730,457,[5_1|3]), (730,464,[5_1|3]), (730,521,[5_1|3]), (730,547,[5_1|3]), (730,552,[5_1|3]), (730,584,[5_1|3]), (730,610,[5_1|3]), (730,617,[5_1|3]), (731,732,[1_1|3]), (732,733,[3_1|3]), (733,734,[0_1|3]), (733,191,[3_1|2]), (733,195,[0_1|2]), (733,200,[0_1|2]), (733,206,[0_1|2]), (733,213,[0_1|2]), (733,218,[3_1|2]), (733,223,[3_1|2]), (733,230,[0_1|2]), (733,237,[3_1|2]), (733,242,[3_1|2]), (733,249,[3_1|2]), (733,256,[0_1|2]), (733,263,[3_1|2]), (733,270,[3_1|2]), (733,276,[3_1|2]), (733,280,[3_1|2]), (733,287,[3_1|2]), (733,294,[0_1|2]), (733,301,[0_1|2]), (733,662,[3_1|3]), (733,666,[0_1|3]), (733,671,[0_1|3]), (733,677,[0_1|3]), (733,684,[0_1|3]), (733,689,[3_1|3]), (734,115,[2_1|3]), (734,128,[2_1|3]), (734,494,[2_1|3]), (734,528,[2_1|3]), (734,604,[2_1|3]), (735,736,[2_1|3]), (736,737,[3_1|3]), (737,738,[3_1|3]), (738,739,[1_1|3]), (739,115,[3_1|3]), (739,128,[3_1|3]), (739,494,[3_1|3]), (739,528,[3_1|3]), (739,604,[3_1|3]), (740,741,[4_1|3]), (741,742,[3_1|3]), (742,743,[1_1|3]), (743,744,[3_1|3]), (744,745,[2_1|3]), (745,115,[2_1|3]), (745,128,[2_1|3]), (745,494,[2_1|3]), (745,528,[2_1|3]), (745,604,[2_1|3]), (746,747,[2_1|3]), (747,748,[3_1|3]), (748,749,[2_1|3]), (749,750,[2_1|3]), (750,751,[1_1|3]), (751,752,[3_1|3]), (752,115,[2_1|3]), (752,128,[2_1|3]), (752,494,[2_1|3]), (752,528,[2_1|3]), (752,604,[2_1|3]), (753,754,[3_1|3]), (754,755,[2_1|3]), (755,756,[2_1|3]), (756,757,[1_1|3]), (757,141,[4_1|3]), (757,450,[4_1|3]), (757,540,[4_1|3]), (757,557,[4_1|3]), (757,597,[4_1|3]), (757,495,[4_1|3]), (758,759,[2_1|3]), (759,760,[1_1|3]), (760,761,[0_1|3]), (761,762,[4_1|3]), (762,141,[3_1|3]), (762,450,[3_1|3]), (762,540,[3_1|3]), (762,557,[3_1|3]), (762,597,[3_1|3]), (762,495,[3_1|3]), (763,764,[2_1|3]), (764,765,[3_1|3]), (765,766,[1_1|3]), (766,767,[3_1|3]), (767,342,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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 ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(3(3(4(x1)))) ->^+ 5(4(3(2(3(0(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [x1 / 3(3(4(x1)))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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 ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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