WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 42 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 523 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 6 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(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(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(1(x_1)) -> 1(encArg(x_1)) 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_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(n^1, n^1). The TRS R consists of the following rules: 0(1(0(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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(n^1, n^1). The TRS R consists of the following rules: 0(1(0(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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(1(0(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) 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_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. "[50, 51, 52, 53, 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] {(50,51,[0_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]), (50,52,[2_1|1]), (50,56,[2_1|1]), (50,60,[3_1|1]), (50,64,[3_1|1]), (50,68,[4_1|1]), (50,72,[4_1|1]), (50,76,[4_1|1]), (50,80,[4_1|1]), (50,84,[2_1|1]), (50,89,[2_1|1]), (50,94,[2_1|1]), (50,99,[2_1|1]), (50,104,[2_1|1]), (50,109,[3_1|1]), (50,114,[3_1|1]), (50,119,[3_1|1]), (50,124,[3_1|1]), (50,129,[4_1|1]), (50,134,[4_1|1]), (50,139,[4_1|1]), (50,144,[4_1|1]), (50,149,[4_1|1]), (50,154,[4_1|1]), (50,159,[4_1|1]), (50,164,[4_1|1]), (50,169,[4_1|1]), (50,174,[2_1|1]), (50,179,[4_1|1]), (50,184,[2_1|1]), (50,189,[3_1|1]), (50,194,[4_1|1]), (50,199,[4_1|1]), (50,204,[0_1|1]), (50,209,[4_1|1]), (50,214,[3_1|1]), (50,219,[4_1|1]), (50,224,[0_1|1]), (50,229,[5_1|1]), (50,234,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 0_1|1, 5_1|1]), (50,235,[2_1|2]), 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(645,406,[2_1|2]), (645,451,[2_1|2]), (645,516,[2_1|2]), (645,546,[2_1|2]), (645,551,[2_1|2]), (645,566,[2_1|2]), (645,571,[2_1|2]), (645,472,[2_1|2]), (645,656,[2_1|2]), (645,661,[2_1|2]), (645,666,[2_1|2]), (646,647,[4_1|2]), (647,648,[4_1|2]), (648,649,[3_1|2]), (649,650,[2_1|2]), (650,234,[2_1|2]), (650,235,[2_1|2]), (650,239,[2_1|2]), (650,244,[2_1|2]), (650,249,[2_1|2]), (650,254,[2_1|2]), (650,259,[2_1|2]), (650,274,[2_1|2]), (650,278,[2_1|2]), (650,306,[2_1|2]), (650,311,[2_1|2]), (650,316,[2_1|2]), (650,321,[2_1|2]), (650,391,[2_1|2]), (650,396,[2_1|2]), (650,406,[2_1|2]), (650,451,[2_1|2]), (650,516,[2_1|2]), (650,546,[2_1|2]), (650,551,[2_1|2]), (650,566,[2_1|2]), (650,571,[2_1|2]), (650,472,[2_1|2]), (650,656,[2_1|2]), (650,661,[2_1|2]), (650,666,[2_1|2]), (651,652,[5_1|2]), (652,653,[0_1|2]), (653,654,[4_1|2]), (654,655,[5_1|2]), (655,234,[2_1|2]), (655,235,[2_1|2]), (655,239,[2_1|2]), (655,244,[2_1|2]), (655,249,[2_1|2]), (655,254,[2_1|2]), (655,259,[2_1|2]), (655,274,[2_1|2]), (655,278,[2_1|2]), (655,306,[2_1|2]), (655,311,[2_1|2]), (655,316,[2_1|2]), (655,321,[2_1|2]), (655,391,[2_1|2]), (655,396,[2_1|2]), (655,406,[2_1|2]), (655,451,[2_1|2]), (655,516,[2_1|2]), (655,546,[2_1|2]), (655,551,[2_1|2]), (655,566,[2_1|2]), (655,571,[2_1|2]), (655,472,[2_1|2]), (655,638,[2_1|2]), (655,656,[2_1|2]), (655,661,[2_1|2]), (655,666,[2_1|2]), (656,657,[0_1|2]), (657,658,[3_1|2]), (658,659,[1_1|2]), (659,660,[0_1|2]), (660,427,[4_1|2]), (660,527,[4_1|2]), (660,532,[4_1|2]), (660,458,[4_1|2]), (661,662,[4_1|2]), (662,663,[3_1|2]), (663,664,[0_1|2]), (664,665,[4_1|2]), (665,472,[1_1|2]), (666,667,[0_1|2]), (667,668,[3_1|2]), (668,669,[1_1|2]), (669,670,[0_1|2]), (670,472,[4_1|2]), (671,672,[2_1|2]), (672,673,[2_1|2]), (673,674,[1_1|2]), (674,675,[4_1|2]), (675,234,[0_1|2]), (675,235,[0_1|2, 2_1|2]), (675,239,[0_1|2, 2_1|2]), (675,244,[0_1|2, 2_1|2]), (675,249,[0_1|2, 2_1|2]), (675,254,[0_1|2, 2_1|2]), (675,259,[0_1|2, 2_1|2]), (675,274,[0_1|2, 2_1|2]), (675,278,[0_1|2, 2_1|2]), (675,306,[0_1|2, 2_1|2]), (675,311,[0_1|2, 2_1|2]), (675,316,[0_1|2, 2_1|2]), (675,321,[0_1|2, 2_1|2]), (675,391,[0_1|2, 2_1|2]), (675,396,[0_1|2, 2_1|2]), (675,406,[0_1|2, 2_1|2]), (675,451,[0_1|2, 2_1|2]), (675,516,[0_1|2, 2_1|2]), (675,546,[0_1|2, 2_1|2]), (675,551,[0_1|2, 2_1|2]), (675,566,[0_1|2]), (675,571,[0_1|2]), (675,472,[0_1|2]), (675,656,[0_1|2, 2_1|2]), (675,661,[0_1|2, 2_1|2]), (675,666,[0_1|2, 2_1|2]), (675,264,[4_1|2]), (675,269,[4_1|2]), (675,282,[3_1|2]), (675,286,[3_1|2]), (675,290,[4_1|2]), (675,294,[4_1|2]), (675,298,[4_1|2]), (675,302,[4_1|2]), (675,326,[3_1|2]), (675,331,[3_1|2]), (675,336,[3_1|2]), (675,341,[3_1|2]), (675,346,[4_1|2]), (675,351,[4_1|2]), (675,356,[4_1|2]), (675,361,[4_1|2]), (675,366,[4_1|2]), (675,371,[4_1|2]), (675,376,[4_1|2]), (675,381,[4_1|2]), (675,386,[4_1|2]), (675,401,[4_1|2]), (675,411,[3_1|2]), (675,416,[4_1|2]), (675,421,[4_1|2]), (675,426,[0_1|2]), (675,431,[3_1|2]), (675,436,[5_1|2]), (675,441,[0_1|2]), (675,446,[1_1|2]), (675,456,[4_1|2]), (675,461,[4_1|2]), (675,466,[4_1|2]), (675,471,[4_1|2]), (675,476,[4_1|2]), (675,481,[4_1|2]), (675,486,[4_1|2]), (675,491,[5_1|2]), (675,496,[5_1|2]), (675,501,[5_1|2]), (675,506,[1_1|2]), (675,511,[1_1|2]), (675,521,[0_1|2]), (675,526,[0_1|2]), (675,531,[0_1|2]), (675,536,[0_1|2]), (675,541,[1_1|2]), (675,556,[0_1|2]), (675,561,[4_1|2]), (676,677,[1_1|2]), (677,678,[0_1|2]), (678,679,[5_1|2]), (679,680,[3_1|2]), (680,234,[2_1|2]), (680,235,[2_1|2]), (680,239,[2_1|2]), (680,244,[2_1|2]), (680,249,[2_1|2]), (680,254,[2_1|2]), (680,259,[2_1|2]), (680,274,[2_1|2]), (680,278,[2_1|2]), (680,306,[2_1|2]), (680,311,[2_1|2]), (680,316,[2_1|2]), (680,321,[2_1|2]), (680,391,[2_1|2]), (680,396,[2_1|2]), (680,406,[2_1|2]), (680,451,[2_1|2]), (680,516,[2_1|2]), (680,546,[2_1|2]), (680,551,[2_1|2]), (680,566,[2_1|2]), (680,571,[2_1|2]), (680,472,[2_1|2]), (680,656,[2_1|2]), (680,661,[2_1|2]), (680,666,[2_1|2]), (681,682,[2_1|3]), (682,683,[1_1|3]), (683,684,[0_1|3]), (684,685,[4_1|3]), (685,638,[5_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(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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 ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(4(2(x1)))) ->^+ 3(2(2(1(4(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 / 1(4(2(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(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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 ---------------------------------------- (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(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) 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_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