WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 85 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 224 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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. "[49, 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, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784] {(49,50,[0_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]), (49,51,[1_1|1]), (49,55,[4_1|1]), (49,59,[0_1|1]), (49,64,[0_1|1]), (49,69,[4_1|1]), (49,74,[4_1|1]), (49,79,[4_1|1]), (49,83,[5_1|1]), (49,87,[0_1|1]), (49,92,[0_1|1]), (49,97,[1_1|1]), (49,102,[1_1|1]), (49,107,[3_1|1]), (49,112,[3_1|1]), (49,117,[3_1|1]), (49,122,[3_1|1]), (49,127,[4_1|1]), (49,132,[4_1|1]), (49,137,[4_1|1]), (49,142,[4_1|1]), (49,147,[4_1|1]), (49,152,[5_1|1]), (49,157,[4_1|1]), (49,162,[5_1|1]), (49,167,[4_1|1]), (49,172,[4_1|1]), (49,177,[0_1|1]), (49,182,[0_1|1]), (49,187,[2_1|1]), (49,192,[0_1|1]), (49,197,[0_1|1]), (49,201,[4_1|1]), (49,206,[5_1|1]), (49,211,[4_1|1]), (49,215,[4_1|1]), (49,219,[4_1|1]), (49,224,[0_1|1]), (49,229,[0_1|1]), (49,234,[0_1|1]), (49,239,[2_1|1]), (49,244,[2_1|1]), (49,249,[2_1|1]), (49,254,[2_1|1]), (49,259,[4_1|1]), (49,264,[4_1|1]), (49,269,[4_1|1]), (49,274,[0_1|1]), (49,279,[0_1|1]), (49,284,[0_1|1]), (49,289,[2_1|1]), (49,294,[4_1|1]), (49,299,[4_1|1]), (49,304,[5_1|1]), (49,309,[0_1|1]), (49,314,[0_1|1]), (49,319,[2_1|1]), (49,324,[2_1|1]), (49,329,[2_1|1]), (49,334,[3_1|1]), (49,339,[3_1|1]), (49,344,[4_1|1]), (49,349,[4_1|1]), (49,354,[5_1|1]), (49,359,[1_1|1, 2_1|1, 3_1|1, 5_1|1, 0_1|1]), (49,364,[1_1|2]), (49,368,[4_1|2]), (49,372,[0_1|2]), (49,377,[0_1|2]), (49,382,[4_1|2]), (49,387,[4_1|2]), (49,392,[0_1|2]), (49,397,[4_1|2]), (49,402,[4_1|2]), (49,406,[5_1|2]), (49,410,[0_1|2]), (49,415,[0_1|2]), (49,420,[1_1|2]), (49,425,[1_1|2]), (49,430,[3_1|2]), (49,435,[3_1|2]), (49,440,[3_1|2]), (49,445,[3_1|2]), (49,450,[4_1|2]), (49,455,[4_1|2]), (49,460,[4_1|2]), (49,465,[4_1|2]), (49,470,[4_1|2]), (49,475,[5_1|2]), (49,480,[1_1|2]), (49,485,[4_1|2]), (49,490,[5_1|2]), (49,495,[4_1|2]), (49,500,[4_1|2]), (49,505,[0_1|2]), (49,510,[0_1|2]), (49,515,[2_1|2]), (49,520,[0_1|2]), (49,525,[0_1|2]), (49,530,[0_1|2]), (49,535,[0_1|2]), (49,539,[4_1|2]), (49,544,[0_1|2]), (49,549,[0_1|2]), (49,554,[4_1|2]), (49,559,[5_1|2]), (49,564,[4_1|2]), (49,568,[4_1|2]), (49,572,[4_1|2]), (49,577,[1_1|2]), (49,582,[0_1|2]), (49,587,[0_1|2]), (49,592,[0_1|2]), (49,597,[0_1|2]), (49,602,[0_1|2]), (49,607,[2_1|2]), (49,612,[2_1|2]), (49,617,[2_1|2]), (49,622,[2_1|2]), (49,627,[4_1|2]), (49,632,[4_1|2]), (49,637,[4_1|2]), (49,642,[0_1|2]), (49,647,[0_1|2]), (49,652,[0_1|2]), (49,657,[2_1|2]), (49,662,[4_1|2]), (49,667,[4_1|2]), (49,672,[0_1|2]), (49,677,[0_1|2]), (49,682,[5_1|2]), (49,687,[0_1|2]), (49,692,[0_1|2]), (49,697,[0_1|2]), (49,702,[2_1|2]), (49,707,[2_1|2]), (49,712,[2_1|2]), (49,717,[3_1|2]), (49,722,[3_1|2]), (49,727,[4_1|2]), (49,732,[4_1|2]), (49,737,[5_1|2]), (49,742,[0_1|2]), (50,50,[1_1|0, 2_1|0, 3_1|0, 4_1|0, 5_1|0, cons_0_1|0]), (51,52,[0_1|1]), (52,53,[1_1|1]), (53,54,[3_1|1]), (54,50,[2_1|1]), (55,56,[1_1|1]), (55,50,[encArg_1|1]), (55,359,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (55,364,[1_1|2]), (55,368,[4_1|2]), (55,372,[0_1|2]), (55,377,[0_1|2]), (55,382,[4_1|2]), (55,387,[4_1|2]), (55,392,[0_1|2]), (55,397,[4_1|2]), (55,402,[4_1|2]), (55,406,[5_1|2]), (55,410,[0_1|2]), (55,415,[0_1|2]), (55,420,[1_1|2]), (55,425,[1_1|2]), (55,430,[3_1|2]), (55,435,[3_1|2]), (55,440,[3_1|2]), (55,445,[3_1|2]), (55,450,[4_1|2]), (55,455,[4_1|2]), (55,460,[4_1|2]), (55,465,[4_1|2]), (55,470,[4_1|2]), (55,475,[5_1|2]), 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(740,717,[3_1|2]), (740,722,[3_1|2]), (740,727,[4_1|2]), (740,732,[4_1|2]), (740,737,[5_1|2]), (741,359,[4_1|2]), (741,406,[4_1|2]), (741,475,[4_1|2]), (741,490,[4_1|2]), (741,559,[4_1|2]), (741,682,[4_1|2]), (741,737,[4_1|2]), (742,743,[4_1|2]), (743,744,[1_1|2]), (744,745,[4_1|2]), (745,746,[1_1|2]), (746,515,[2_1|2]), (746,607,[2_1|2]), (746,612,[2_1|2]), (746,617,[2_1|2]), (746,622,[2_1|2]), (746,657,[2_1|2]), (746,702,[2_1|2]), (746,707,[2_1|2]), (746,712,[2_1|2]), (746,501,[2_1|2]), (747,748,[2_1|3]), (748,749,[4_1|3]), (749,750,[1_1|3]), (750,751,[0_1|3]), (751,372,[3_1|3]), (751,377,[3_1|3]), (751,392,[3_1|3]), (751,410,[3_1|3]), (751,415,[3_1|3]), (751,505,[3_1|3]), (751,510,[3_1|3]), (751,520,[3_1|3]), (751,525,[3_1|3]), (751,530,[3_1|3]), (751,535,[3_1|3]), (751,544,[3_1|3]), (751,549,[3_1|3]), (751,582,[3_1|3]), (751,587,[3_1|3]), (751,592,[3_1|3]), (751,597,[3_1|3]), (751,602,[3_1|3]), (751,642,[3_1|3]), (751,647,[3_1|3]), (751,652,[3_1|3]), (751,672,[3_1|3]), (751,677,[3_1|3]), (751,687,[3_1|3]), (751,692,[3_1|3]), (751,697,[3_1|3]), (751,742,[3_1|3]), (751,608,[3_1|3]), (751,613,[3_1|3]), (752,753,[4_1|3]), (753,754,[0_1|3]), (754,755,[1_1|3]), (755,756,[2_1|3]), (756,521,[2_1|3]), (756,588,[2_1|3]), (756,598,[2_1|3]), (756,643,[2_1|3]), (756,693,[2_1|3]), (757,758,[3_1|3]), (758,759,[5_1|3]), (759,760,[4_1|3]), (760,446,[1_1|3]), (760,619,[1_1|3]), (761,762,[2_1|3]), (762,763,[4_1|3]), (763,764,[1_1|3]), (764,765,[0_1|3]), (765,608,[3_1|3]), (765,613,[3_1|3]), (766,767,[3_1|3]), (767,768,[5_1|3]), (768,769,[4_1|3]), (769,359,[1_1|3]), (769,406,[1_1|3]), (769,475,[1_1|3]), (769,490,[1_1|3]), (769,559,[1_1|3]), (769,682,[1_1|3]), (769,737,[1_1|3]), (769,446,[1_1|3]), (770,771,[4_1|2]), (771,772,[1_1|2]), (772,773,[2_1|2]), (773,774,[1_1|2]), (774,742,[0_1|2]), (775,776,[2_1|2]), (776,777,[5_1|2]), (777,778,[0_1|2]), (778,779,[3_1|2]), (779,742,[0_1|2]), (780,781,[4_1|3]), (781,782,[0_1|3]), (782,783,[0_1|3]), (783,784,[1_1|3]), (784,559,[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 (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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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 0(1(1(4(2(x1))))) ->^+ 4(1(3(1(2(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(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 (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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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(1(2(x1)))) -> 1(0(1(3(2(x1))))) 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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