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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 70 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 947 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 7 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 (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(x1)))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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(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)) 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 (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST ---------------------------------------- (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, 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(891,260,[1_1|3]), (891,265,[1_1|3]), (891,270,[1_1|3]), (891,290,[1_1|3]), (891,315,[1_1|3]), (891,320,[1_1|3]), (891,378,[1_1|3]), (891,383,[1_1|3]), (891,388,[1_1|3]), (891,453,[1_1|3]), (891,458,[1_1|3]), (891,463,[1_1|3]), (891,478,[1_1|3]), (891,483,[1_1|3]), (891,488,[1_1|3]), (891,523,[1_1|3]), (891,528,[1_1|3]), (891,533,[1_1|3]), (891,573,[1_1|3]), (891,682,[1_1|3]), (891,687,[1_1|3]), (891,316,[1_1|3]), (892,893,[1_1|3]), (893,894,[3_1|3]), (894,895,[1_1|3]), (895,896,[3_1|3]), (896,316,[1_1|3]), (897,898,[0_1|2]), (898,899,[2_1|2]), (899,900,[4_1|2]), (899,882,[4_1|2]), (900,901,[1_1|2]), (901,176,[3_1|2]), (901,242,[3_1|2]), (901,260,[3_1|2]), (901,265,[3_1|2]), (901,270,[3_1|2]), (901,290,[3_1|2]), (901,315,[3_1|2]), (901,320,[3_1|2]), (901,378,[3_1|2]), (901,383,[3_1|2]), (901,388,[3_1|2]), (901,453,[3_1|2]), (901,458,[3_1|2]), (901,463,[3_1|2]), (901,478,[3_1|2]), (901,483,[3_1|2]), (901,488,[3_1|2]), (901,523,[3_1|2]), (901,528,[3_1|2]), (901,533,[3_1|2]), (901,573,[3_1|2]), (901,682,[3_1|2]), (901,687,[3_1|2]), (902,903,[4_1|2]), (903,904,[2_1|2]), (904,905,[1_1|2]), (905,906,[3_1|2]), (906,176,[1_1|2]), (906,242,[1_1|2]), (906,260,[1_1|2]), (906,265,[1_1|2]), (906,270,[1_1|2]), (906,290,[1_1|2]), (906,315,[1_1|2]), (906,320,[1_1|2]), (906,378,[1_1|2]), (906,383,[1_1|2]), (906,388,[1_1|2]), (906,453,[1_1|2]), (906,458,[1_1|2]), (906,463,[1_1|2]), (906,478,[1_1|2]), (906,483,[1_1|2]), (906,488,[1_1|2]), (906,523,[1_1|2]), (906,528,[1_1|2]), (906,533,[1_1|2]), (906,573,[1_1|2]), (906,682,[1_1|2]), (906,687,[1_1|2]), (907,908,[0_1|2]), (908,909,[2_1|2]), (909,910,[1_1|2]), (910,911,[3_1|2]), (911,176,[1_1|2]), (911,242,[1_1|2]), (911,260,[1_1|2]), (911,265,[1_1|2]), (911,270,[1_1|2]), (911,290,[1_1|2]), (911,315,[1_1|2]), (911,320,[1_1|2]), (911,378,[1_1|2]), (911,383,[1_1|2]), (911,388,[1_1|2]), (911,453,[1_1|2]), (911,458,[1_1|2]), (911,463,[1_1|2]), (911,478,[1_1|2]), (911,483,[1_1|2]), (911,488,[1_1|2]), (911,523,[1_1|2]), (911,528,[1_1|2]), (911,533,[1_1|2]), (911,573,[1_1|2]), (911,682,[1_1|2]), (911,687,[1_1|2]), (912,913,[4_1|3]), (913,914,[0_1|3]), (914,915,[4_1|3]), (915,916,[2_1|3]), (916,234,[1_1|3]), (917,918,[4_1|3]), (918,919,[0_1|3]), (919,920,[2_1|3]), (920,921,[1_1|3]), (921,234,[3_1|3]), (922,923,[4_1|2]), (923,924,[2_1|2]), (924,925,[1_1|2]), (925,926,[3_1|2]), (926,176,[1_1|2]), (926,242,[1_1|2]), (926,260,[1_1|2]), (926,265,[1_1|2]), (926,270,[1_1|2]), (926,290,[1_1|2]), (926,315,[1_1|2]), (926,320,[1_1|2]), (926,378,[1_1|2]), (926,383,[1_1|2]), (926,388,[1_1|2]), (926,453,[1_1|2]), (926,458,[1_1|2]), (926,463,[1_1|2]), (926,478,[1_1|2]), (926,483,[1_1|2]), (926,488,[1_1|2]), (926,523,[1_1|2]), (926,528,[1_1|2]), (926,533,[1_1|2]), (926,573,[1_1|2]), (926,682,[1_1|2]), (926,687,[1_1|2]), (927,928,[0_1|2]), (928,929,[2_1|2]), (929,930,[4_1|2]), (929,882,[4_1|2]), (930,931,[1_1|2]), (931,176,[3_1|2]), (931,242,[3_1|2]), (931,260,[3_1|2]), (931,265,[3_1|2]), (931,270,[3_1|2]), (931,290,[3_1|2]), (931,315,[3_1|2]), (931,320,[3_1|2]), (931,378,[3_1|2]), (931,383,[3_1|2]), (931,388,[3_1|2]), (931,453,[3_1|2]), (931,458,[3_1|2]), (931,463,[3_1|2]), (931,478,[3_1|2]), (931,483,[3_1|2]), (931,488,[3_1|2]), (931,523,[3_1|2]), (931,528,[3_1|2]), (931,533,[3_1|2]), (931,573,[3_1|2]), (931,682,[3_1|2]), (931,687,[3_1|2]), (932,933,[0_1|2]), (933,934,[4_1|2]), (934,935,[5_1|2]), (935,936,[2_1|2]), (936,176,[1_1|2]), (936,242,[1_1|2]), (936,260,[1_1|2]), (936,265,[1_1|2]), (936,270,[1_1|2]), (936,290,[1_1|2]), (936,315,[1_1|2]), (936,320,[1_1|2]), (936,378,[1_1|2]), (936,383,[1_1|2]), (936,388,[1_1|2]), (936,453,[1_1|2]), (936,458,[1_1|2]), (936,463,[1_1|2]), (936,478,[1_1|2]), (936,483,[1_1|2]), (936,488,[1_1|2]), (936,523,[1_1|2]), (936,528,[1_1|2]), (936,533,[1_1|2]), (936,573,[1_1|2]), (936,682,[1_1|2]), (936,687,[1_1|2]), (937,938,[0_1|2]), (938,939,[4_1|2]), (939,940,[5_1|2]), (940,941,[2_1|2]), (941,176,[1_1|2]), (941,242,[1_1|2]), (941,260,[1_1|2]), (941,265,[1_1|2]), (941,270,[1_1|2]), (941,290,[1_1|2]), (941,315,[1_1|2]), (941,320,[1_1|2]), (941,378,[1_1|2]), (941,383,[1_1|2]), (941,388,[1_1|2]), (941,453,[1_1|2]), (941,458,[1_1|2]), (941,463,[1_1|2]), (941,478,[1_1|2]), (941,483,[1_1|2]), (941,488,[1_1|2]), (941,523,[1_1|2]), (941,528,[1_1|2]), (941,533,[1_1|2]), (941,573,[1_1|2]), (941,682,[1_1|2]), (941,687,[1_1|2]), (942,943,[5_1|3]), (943,944,[2_1|3]), (944,945,[1_1|3]), (945,316,[1_1|3]), (946,947,[2_1|3]), (947,948,[5_1|3]), (948,949,[2_1|3]), (949,950,[1_1|3]), (950,316,[1_1|3]), (951,952,[5_1|3]), (952,953,[2_1|3]), (953,954,[1_1|3]), (954,955,[3_1|3]), (955,316,[1_1|3]), (956,957,[5_1|3]), (957,958,[2_1|3]), (958,959,[2_1|3]), (959,960,[1_1|3]), (960,316,[1_1|3]), (961,962,[5_1|3]), (962,963,[5_1|3]), (963,964,[2_1|3]), (964,965,[1_1|3]), (965,316,[1_1|3]), (966,967,[4_1|3]), (967,968,[1_1|3]), (968,969,[3_1|3]), (969,970,[1_1|3]), (970,580,[5_1|3]), (971,972,[5_1|3]), (972,973,[2_1|3]), (973,974,[1_1|3]), (974,975,[3_1|3]), (975,580,[1_1|3]), (976,977,[1_1|3]), (977,978,[3_1|3]), (978,979,[4_1|3]), (979,980,[1_1|3]), (980,580,[2_1|3]), (981,982,[2_1|3]), (982,983,[1_1|3]), (983,984,[4_1|3]), (984,985,[1_1|3]), (985,580,[3_1|3]), (986,987,[4_1|3]), (987,988,[2_1|3]), (988,989,[1_1|3]), (989,990,[3_1|3]), (990,580,[1_1|3]), (991,992,[1_1|3]), (992,993,[3_1|3]), (993,994,[0_1|3]), (994,995,[2_1|3]), (995,176,[1_1|3]), (995,242,[1_1|3]), (995,260,[1_1|3]), (995,265,[1_1|3]), (995,270,[1_1|3]), (995,290,[1_1|3]), (995,315,[1_1|3]), (995,320,[1_1|3]), (995,378,[1_1|3]), (995,383,[1_1|3]), (995,388,[1_1|3]), (995,453,[1_1|3]), (995,458,[1_1|3]), (995,463,[1_1|3]), (995,478,[1_1|3]), (995,483,[1_1|3]), (995,488,[1_1|3]), (995,523,[1_1|3]), (995,528,[1_1|3]), (995,533,[1_1|3]), (995,573,[1_1|3]), (995,682,[1_1|3]), (995,687,[1_1|3]), (996,997,[1_1|3]), (997,998,[3_1|3]), (998,999,[1_1|3]), (999,1000,[3_1|3]), (1000,242,[1_1|3]), (1000,260,[1_1|3]), (1000,265,[1_1|3]), (1000,270,[1_1|3]), (1000,290,[1_1|3]), (1000,315,[1_1|3]), (1000,320,[1_1|3]), (1000,378,[1_1|3]), (1000,383,[1_1|3]), (1000,388,[1_1|3]), (1000,453,[1_1|3]), (1000,458,[1_1|3]), (1000,463,[1_1|3]), (1000,478,[1_1|3]), (1000,483,[1_1|3]), (1000,488,[1_1|3]), (1000,523,[1_1|3]), (1000,528,[1_1|3]), (1000,533,[1_1|3]), (1000,573,[1_1|3]), (1000,682,[1_1|3]), (1000,687,[1_1|3]), (1000,316,[1_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 (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 4(5(5(1(1(x1))))) ->^+ 5(5(2(1(1(4(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 / 5(5(1(1(x1))))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) 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(1(x1)))) -> 0(0(2(1(3(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 2(1(3(1(0(0(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(1(x1))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(x1))))) 4(1(0(1(x1)))) -> 0(2(1(3(4(1(x1)))))) 4(1(0(1(x1)))) -> 0(4(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(2(2(1(x1)))))) 4(1(0(1(x1)))) -> 1(0(4(5(2(1(x1)))))) 4(1(0(1(x1)))) -> 4(0(2(1(3(1(x1)))))) 4(1(0(1(x1)))) -> 4(1(0(2(1(3(x1)))))) 4(1(0(1(x1)))) -> 4(1(3(0(2(1(x1)))))) 4(1(1(1(x1)))) -> 4(1(3(1(3(1(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(x1))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(x1))))) 4(3(0(1(x1)))) -> 0(4(2(1(3(x1))))) 4(3(0(1(x1)))) -> 3(0(4(2(1(x1))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(x1))))) 4(3(0(1(x1)))) -> 4(0(2(1(3(x1))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(x1))))) 4(3(0(1(x1)))) -> 0(2(1(3(3(4(x1)))))) 4(3(0(1(x1)))) -> 0(2(1(3(4(3(x1)))))) 4(3(0(1(x1)))) -> 0(2(4(1(3(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(2(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 0(4(5(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(1(3(4(x1)))))) 4(3(0(1(x1)))) -> 1(0(2(4(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(0(4(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 1(3(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(0(4(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(1(0(4(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(3(4(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(2(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 3(4(3(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(2(1(3(x1)))))) 4(3(0(1(x1)))) -> 4(3(0(5(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(1(0(2(1(x1)))))) 4(3(0(1(x1)))) -> 4(3(3(0(2(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(1(x1))))) 4(5(1(1(x1)))) -> 4(2(5(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(1(3(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(2(2(1(1(x1)))))) 4(5(1(1(x1)))) -> 4(5(5(2(1(1(x1)))))) 0(0(1(0(1(x1))))) -> 0(0(1(0(2(1(x1)))))) 0(0(3(1(1(x1))))) -> 2(1(3(1(0(0(x1)))))) 0(0(5(1(1(x1))))) -> 5(0(0(2(1(1(x1)))))) 0(4(1(0(1(x1))))) -> 2(1(0(4(0(1(x1)))))) 0(4(3(0(1(x1))))) -> 0(0(2(1(3(4(x1)))))) 0(4(3(0(1(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(4(3(0(1(x1))))) -> 2(3(1(0(4(0(x1)))))) 0(4(3(1(1(x1))))) -> 2(1(4(1(3(0(x1)))))) 0(4(3(1(1(x1))))) -> 4(2(0(3(1(1(x1)))))) 0(4(5(1(1(x1))))) -> 4(0(1(5(2(1(x1)))))) 0(4(5(1(1(x1))))) -> 5(2(1(4(0(1(x1)))))) 4(0(3(0(1(x1))))) -> 0(0(4(2(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(1(3(x1)))))) 4(0(3(0(1(x1))))) -> 2(0(4(0(3(1(x1)))))) 4(1(0(1(1(x1))))) -> 1(0(4(2(1(1(x1)))))) 4(1(1(0(1(x1))))) -> 4(1(0(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(1(2(0(1(x1))))) -> 1(0(2(4(1(3(x1)))))) 4(1(2(0(1(x1))))) -> 4(0(2(2(1(1(x1)))))) 4(1(2(0(1(x1))))) -> 4(1(0(2(2(1(x1)))))) 4(1(3(0(1(x1))))) -> 4(1(3(0(2(1(x1)))))) 4(1(5(0(1(x1))))) -> 1(1(5(0(4(2(x1)))))) 4(1(5(1(1(x1))))) -> 1(4(5(2(1(1(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(0(2(1(4(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(0(2(x1)))))) 4(3(0(1(1(x1))))) -> 1(3(1(4(2(0(x1)))))) 4(3(0(1(1(x1))))) -> 3(1(0(2(4(1(x1)))))) 4(3(0(1(1(x1))))) -> 4(3(1(1(0(2(x1)))))) 4(3(1(0(1(x1))))) -> 0(4(2(1(3(1(x1)))))) 4(3(1(0(1(x1))))) -> 1(3(4(0(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(2(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 3(4(0(5(2(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(0(2(1(3(1(x1)))))) 4(3(2(0(1(x1))))) -> 4(3(0(2(1(3(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(2(4(2(x1)))))) 4(3(2(1(1(x1))))) -> 1(3(1(4(5(2(x1)))))) 4(3(2(1(1(x1))))) -> 3(1(3(4(1(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(1(2(2(x1)))))) 4(3(2(1(1(x1))))) -> 4(1(3(3(1(2(x1)))))) 4(3(3(0(1(x1))))) -> 4(3(3(0(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 3(4(0(4(2(1(x1)))))) 4(3(4(0(1(x1))))) -> 4(4(0(2(1(3(x1)))))) 4(3(5(0(1(x1))))) -> 0(5(2(3(1(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(2(5(4(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 1(3(4(0(5(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(0(4(2(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(1(5(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 3(4(1(5(0(2(x1)))))) 4(3(5(0(1(x1))))) -> 3(5(1(0(4(2(x1)))))) 4(3(5(0(1(x1))))) -> 4(3(0(2(1(5(x1)))))) 4(3(5(0(1(x1))))) -> 5(1(3(4(0(2(x1)))))) 4(3(5(1(1(x1))))) -> 1(4(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(1(5(4(2(x1)))))) 4(3(5(1(1(x1))))) -> 3(1(4(5(2(1(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(3(1(3(5(x1)))))) 4(3(5(1(1(x1))))) -> 4(1(5(2(1(3(x1)))))) 4(3(5(1(1(x1))))) -> 4(3(1(5(2(1(x1)))))) 4(4(1(0(1(x1))))) -> 4(4(1(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 0(4(4(2(1(3(x1)))))) 4(4(3(0(1(x1))))) -> 4(0(2(1(3(4(x1)))))) 4(4(3(0(1(x1))))) -> 4(3(4(0(2(1(x1)))))) 4(4(3(0(1(x1))))) -> 4(4(3(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(1(3(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 0(5(4(1(3(1(x1)))))) 4(5(1(0(1(x1))))) -> 1(0(4(5(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(0(5(2(1(1(x1)))))) 4(5(1(0(1(x1))))) -> 4(5(1(0(2(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(1(0(2(4(1(x1)))))) 4(5(1(0(1(x1))))) -> 5(4(1(0(2(1(x1)))))) 4(5(1(1(1(x1))))) -> 4(5(2(1(1(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(2(1(3(4(5(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 0(5(4(2(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 1(0(5(1(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 1(5(0(3(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 2(1(4(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 2(4(1(3(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 3(0(4(5(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(1(5(2(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(3(3(5(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(0(5(4(1(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(0(3(x1)))))) 4(5(3(0(1(x1))))) -> 4(1(3(5(5(0(x1)))))) 4(5(3(0(1(x1))))) -> 4(2(5(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 4(5(2(0(3(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(1(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(0(3(4(4(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(1(3(0(3(4(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(1(3(4(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(2(4(1(3(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(3(0(4(2(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(0(1(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(1(3(2(0(x1)))))) 4(5(3(0(1(x1))))) -> 5(4(3(0(2(1(x1)))))) 4(5(3(1(1(x1))))) -> 2(4(1(3(1(5(x1)))))) 4(5(3(1(1(x1))))) -> 4(5(2(1(3(1(x1)))))) 4(5(3(1(1(x1))))) -> 5(1(3(4(1(2(x1)))))) 4(5(3(1(1(x1))))) -> 5(2(1(4(1(3(x1)))))) 4(5(3(1(1(x1))))) -> 5(4(2(1(3(1(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(1(1(4(x1)))))) 4(5(5(1(1(x1))))) -> 5(5(2(4(1(1(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(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)) 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: INNERMOST