WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 77 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 426 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[45, 46, 47, 48, 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] {(45,46,[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]), (45,47,[1_1|1]), (45,51,[4_1|1]), (45,55,[0_1|1]), (45,60,[0_1|1]), (45,65,[4_1|1]), (45,70,[4_1|1]), (45,75,[4_1|1]), (45,79,[5_1|1]), (45,83,[0_1|1]), (45,88,[0_1|1]), (45,93,[1_1|1]), (45,98,[1_1|1]), (45,103,[3_1|1]), (45,108,[3_1|1]), (45,113,[3_1|1]), (45,118,[3_1|1]), (45,123,[4_1|1]), (45,128,[4_1|1]), (45,133,[4_1|1]), (45,138,[4_1|1]), (45,143,[4_1|1]), (45,148,[5_1|1]), (45,153,[4_1|1]), (45,158,[5_1|1]), (45,163,[4_1|1]), (45,168,[4_1|1]), (45,173,[0_1|1]), (45,178,[0_1|1]), (45,183,[2_1|1]), (45,188,[0_1|1]), (45,193,[0_1|1]), (45,197,[4_1|1]), (45,202,[5_1|1]), (45,207,[4_1|1]), (45,211,[4_1|1]), (45,215,[4_1|1]), (45,220,[0_1|1]), (45,225,[0_1|1]), (45,230,[0_1|1]), (45,235,[2_1|1]), (45,240,[2_1|1]), (45,245,[2_1|1]), (45,250,[2_1|1]), (45,255,[4_1|1]), (45,260,[4_1|1]), (45,265,[4_1|1]), (45,270,[0_1|1]), (45,275,[0_1|1]), (45,280,[0_1|1]), (45,285,[2_1|1]), (45,290,[4_1|1]), (45,295,[4_1|1]), (45,300,[5_1|1]), (45,305,[0_1|1]), (45,310,[0_1|1]), (45,315,[2_1|1]), (45,320,[2_1|1]), (45,325,[2_1|1]), (45,330,[3_1|1]), (45,335,[3_1|1]), (45,340,[4_1|1]), (45,345,[4_1|1]), (45,350,[5_1|1]), (45,355,[1_1|1, 2_1|1, 3_1|1, 5_1|1, 0_1|1]), (45,360,[1_1|2]), (45,364,[4_1|2]), (45,368,[0_1|2]), (45,373,[0_1|2]), (45,378,[4_1|2]), (45,383,[4_1|2]), (45,388,[0_1|2]), (45,393,[4_1|2]), (45,398,[4_1|2]), (45,402,[5_1|2]), (45,406,[0_1|2]), (45,411,[0_1|2]), (45,416,[1_1|2]), (45,421,[1_1|2]), (45,426,[3_1|2]), (45,431,[3_1|2]), (45,436,[3_1|2]), (45,441,[3_1|2]), (45,446,[4_1|2]), (45,451,[4_1|2]), (45,456,[4_1|2]), (45,461,[4_1|2]), (45,466,[4_1|2]), (45,471,[5_1|2]), (45,476,[1_1|2]), (45,481,[4_1|2]), (45,486,[5_1|2]), (45,491,[4_1|2]), (45,496,[4_1|2]), (45,501,[0_1|2]), (45,506,[0_1|2]), (45,511,[2_1|2]), (45,516,[0_1|2]), (45,521,[0_1|2]), (45,526,[0_1|2]), (45,531,[0_1|2]), (45,535,[4_1|2]), (45,540,[0_1|2]), (45,545,[0_1|2]), (45,550,[4_1|2]), (45,555,[5_1|2]), (45,560,[4_1|2]), (45,564,[4_1|2]), (45,568,[4_1|2]), (45,573,[1_1|2]), (45,578,[0_1|2]), (45,583,[0_1|2]), (45,588,[0_1|2]), (45,593,[0_1|2]), (45,598,[0_1|2]), (45,603,[2_1|2]), (45,608,[2_1|2]), (45,613,[2_1|2]), (45,618,[2_1|2]), (45,623,[4_1|2]), (45,628,[4_1|2]), (45,633,[4_1|2]), (45,638,[0_1|2]), (45,643,[0_1|2]), (45,648,[0_1|2]), (45,653,[2_1|2]), (45,658,[4_1|2]), (45,663,[4_1|2]), (45,668,[0_1|2]), (45,673,[0_1|2]), (45,678,[5_1|2]), (45,683,[0_1|2]), (45,688,[0_1|2]), (45,693,[0_1|2]), (45,698,[2_1|2]), (45,703,[2_1|2]), (45,708,[2_1|2]), (45,713,[3_1|2]), (45,718,[3_1|2]), (45,723,[4_1|2]), (45,728,[4_1|2]), (45,733,[5_1|2]), (45,738,[0_1|2]), (46,46,[1_1|0, 2_1|0, 3_1|0, 4_1|0, 5_1|0, cons_0_1|0]), (47,48,[0_1|1]), (48,49,[1_1|1]), (49,50,[3_1|1]), (50,46,[2_1|1]), (51,52,[1_1|1]), (51,46,[encArg_1|1]), (51,355,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (51,360,[1_1|2]), (51,364,[4_1|2]), (51,368,[0_1|2]), (51,373,[0_1|2]), (51,378,[4_1|2]), (51,383,[4_1|2]), (51,388,[0_1|2]), (51,393,[4_1|2]), (51,398,[4_1|2]), (51,402,[5_1|2]), (51,406,[0_1|2]), (51,411,[0_1|2]), (51,416,[1_1|2]), (51,421,[1_1|2]), (51,426,[3_1|2]), (51,431,[3_1|2]), (51,436,[3_1|2]), (51,441,[3_1|2]), (51,446,[4_1|2]), (51,451,[4_1|2]), (51,456,[4_1|2]), (51,461,[4_1|2]), (51,466,[4_1|2]), (51,471,[5_1|2]), 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(736,713,[3_1|2]), (736,718,[3_1|2]), (736,723,[4_1|2]), (736,728,[4_1|2]), (736,733,[5_1|2]), (737,355,[4_1|2]), (737,402,[4_1|2]), (737,471,[4_1|2]), (737,486,[4_1|2]), (737,555,[4_1|2]), (737,678,[4_1|2]), (737,733,[4_1|2]), (738,739,[4_1|2]), (739,740,[1_1|2]), (740,741,[4_1|2]), (741,742,[1_1|2]), (742,511,[2_1|2]), (742,603,[2_1|2]), (742,608,[2_1|2]), (742,613,[2_1|2]), (742,618,[2_1|2]), (742,653,[2_1|2]), (742,698,[2_1|2]), (742,703,[2_1|2]), (742,708,[2_1|2]), (742,497,[2_1|2]), (743,744,[2_1|3]), (744,745,[4_1|3]), (745,746,[1_1|3]), (746,747,[0_1|3]), (747,368,[3_1|3]), (747,373,[3_1|3]), (747,388,[3_1|3]), (747,406,[3_1|3]), (747,411,[3_1|3]), (747,501,[3_1|3]), (747,506,[3_1|3]), (747,516,[3_1|3]), (747,521,[3_1|3]), (747,526,[3_1|3]), (747,531,[3_1|3]), (747,540,[3_1|3]), (747,545,[3_1|3]), (747,578,[3_1|3]), (747,583,[3_1|3]), (747,588,[3_1|3]), (747,593,[3_1|3]), (747,598,[3_1|3]), (747,638,[3_1|3]), (747,643,[3_1|3]), (747,648,[3_1|3]), (747,668,[3_1|3]), (747,673,[3_1|3]), (747,683,[3_1|3]), (747,688,[3_1|3]), (747,693,[3_1|3]), (747,738,[3_1|3]), (747,604,[3_1|3]), (747,609,[3_1|3]), (748,749,[4_1|3]), (749,750,[0_1|3]), (750,751,[1_1|3]), (751,752,[2_1|3]), (752,517,[2_1|3]), (752,584,[2_1|3]), (752,594,[2_1|3]), (752,639,[2_1|3]), (752,689,[2_1|3]), (753,754,[3_1|3]), (754,755,[5_1|3]), (755,756,[4_1|3]), (756,442,[1_1|3]), (756,615,[1_1|3]), (757,758,[2_1|3]), (758,759,[4_1|3]), (759,760,[1_1|3]), (760,761,[0_1|3]), (761,604,[3_1|3]), (761,609,[3_1|3]), (762,763,[3_1|3]), (763,764,[5_1|3]), (764,765,[4_1|3]), (765,355,[1_1|3]), (765,402,[1_1|3]), (765,471,[1_1|3]), (765,486,[1_1|3]), (765,555,[1_1|3]), (765,678,[1_1|3]), (765,733,[1_1|3]), (765,442,[1_1|3]), (766,767,[4_1|2]), (767,768,[1_1|2]), (768,769,[2_1|2]), (769,770,[1_1|2]), (770,738,[0_1|2]), (771,772,[2_1|2]), (772,773,[5_1|2]), (773,774,[0_1|2]), (774,775,[3_1|2]), (775,738,[0_1|2]), (776,777,[4_1|3]), (777,778,[0_1|3]), (778,779,[0_1|3]), (779,780,[1_1|3]), (780,555,[5_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(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: FULL