WORST_CASE(?, 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 34 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 749 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(2(1(0(1(0(1(2(2(2(0(1(2(2(0(1(1(1(x1))))))))))))))))))))))) -> 0(0(0(1(2(2(1(1(0(2(1(1(2(2(2(2(1(2(0(0(1(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(0(1(1(1(1(2(1(1(2(1(2(1(2(1(2(1(1(1(0(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(2(1(2(1(2(1(1(1(0(0(2(2(1(1(0(1(0(0(2(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(1(2(1(2(1(2(1(1(1(0(2(0(1(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 0(1(1(2(0(1(1(1(1(1(1(0(2(0(1(0(1(1(0(1(0(1(2(2(0(2(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(0(2(1(2(2(1(1(1(2(0(2(0(1(2(0(1(1(2(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(1(1(2(0(1(0(1(0(2(2(2(0(1(2(2(2(1(0(0(1(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(1(2(2(0(0(1(1(0(2(2(1(0(1(1(2(0(1(0(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(2(0(1(1(1(2(0(1(1(1(2(1(2(1(0(1(2(2(1(1(0(x1))))))))))))))))))))))))))) 0(1(0(1(1(2(0(1(1(2(2(0(0(2(1(0(2(0(2(2(2(0(2(x1))))))))))))))))))))))) -> 2(1(2(1(1(2(1(1(0(1(0(1(0(1(2(1(0(0(2(1(1(1(2(1(1(1(2(x1))))))))))))))))))))))))))) 0(1(1(0(2(1(0(2(1(0(2(0(2(2(0(1(1(0(1(2(2(1(1(x1))))))))))))))))))))))) -> 2(1(0(1(0(0(1(1(1(2(0(0(2(2(2(1(1(2(1(1(0(1(1(0(1(1(1(x1))))))))))))))))))))))))))) 0(1(1(2(1(1(2(2(2(1(1(2(0(0(1(2(0(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 0(0(2(2(1(0(2(1(2(1(1(2(1(2(2(1(1(1(1(1(0(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 0(1(2(1(2(0(0(1(0(1(1(2(1(1(2(2(1(1(1(2(2(0(2(x1))))))))))))))))))))))) -> 0(1(2(0(1(1(0(1(2(1(1(1(2(0(1(0(1(1(0(1(2(2(1(2(1(2(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(0(0(2(0(0(1(0(0(2(2(0(1(1(0(1(0(1(x1))))))))))))))))))))))) -> 0(1(0(2(0(0(2(1(1(1(0(0(2(1(0(2(2(2(1(1(0(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(1(0(0(1(1(2(1(1(0(1(0(1(0(2(2(2(1(x1))))))))))))))))))))))) -> 1(2(1(2(2(1(1(1(1(1(1(0(2(2(2(1(1(2(1(2(1(0(2(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(1(0(2(0(0(2(0(1(1(1(2(0(1(1(2(1(1(0(1(0(0(x1))))))))))))))))))))))) -> 1(2(2(2(0(1(1(2(0(0(1(1(0(1(0(1(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))))))))))))))))) 0(2(1(2(1(1(0(0(1(1(0(2(2(0(1(2(0(2(0(1(1(0(0(x1))))))))))))))))))))))) -> 2(1(1(1(2(2(0(1(1(1(1(2(2(2(2(1(0(0(2(0(2(1(2(1(1(2(2(x1))))))))))))))))))))))))))) 0(2(1(2(2(0(1(0(1(0(1(0(2(1(0(0(1(1(1(2(1(0(0(x1))))))))))))))))))))))) -> 2(2(2(2(2(1(1(2(2(1(0(1(1(1(1(0(1(1(2(2(1(1(1(2(0(1(2(x1))))))))))))))))))))))))))) 0(2(2(1(1(1(1(2(2(1(2(1(2(0(0(1(2(1(0(0(2(0(0(x1))))))))))))))))))))))) -> 2(2(2(1(1(2(2(2(0(2(0(1(0(1(1(2(2(2(0(1(2(1(1(0(0(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(0(1(2(2(1(2(1(0(1(0(2(2(1(2(1(0(2(1(2(x1))))))))))))))))))))))) -> 1(0(2(2(1(2(2(2(1(2(0(2(1(0(1(0(1(2(2(1(1(0(1(1(1(1(2(x1))))))))))))))))))))))))))) 0(2(2(2(1(2(2(0(0(2(1(2(2(1(0(1(1(1(1(1(2(0(0(x1))))))))))))))))))))))) -> 0(1(1(1(2(0(0(0(1(1(2(2(0(1(1(2(1(1(1(1(1(1(1(2(2(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(2(0(1(1(2(1(0(1(2(2(1(1(2(2(0(2(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(2(2(1(1(1(0(1(1(0(1(2(2(1(2(1(2(0(0(1(1(1(1(x1))))))))))))))))))))))))))) 1(0(0(0(0(2(1(1(1(2(2(2(1(2(1(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))) -> 1(1(2(1(1(2(2(0(2(0(1(1(0(1(1(0(0(1(0(1(1(1(2(1(2(1(0(x1))))))))))))))))))))))))))) 1(0(0(1(1(1(1(0(1(0(0(1(0(2(0(2(1(1(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(2(2(1(1(2(2(1(0(2(2(2(1(2(2(1(2(0(2(1(2(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 1(0(2(2(0(2(2(2(1(0(2(2(2(1(0(0(2(0(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(0(0(1(1(1(0(2(0(0(0(1(2(1(2(1(1(1(1(1(1(0(0(0(2(1(1(x1))))))))))))))))))))))))))) 1(0(2(2(1(1(0(0(2(1(2(1(0(2(1(0(0(2(2(0(1(2(1(x1))))))))))))))))))))))) -> 1(1(2(1(0(2(1(1(1(2(1(0(2(1(2(2(1(0(1(1(0(2(2(1(1(0(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(0(1(0(0(0(1(2(0(1(1(1(1(2(1(1(1(2(0(x1))))))))))))))))))))))) -> 1(2(1(0(0(1(2(1(1(1(2(0(1(2(1(1(1(2(2(0(1(2(2(1(0(2(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(2(0(2(0(1(1(0(2(2(1(1(2(1(1(0(0(2(0(x1))))))))))))))))))))))) -> 1(2(1(2(1(2(2(1(2(1(1(1(0(2(2(1(1(2(0(1(1(0(1(1(0(0(2(x1))))))))))))))))))))))))))) 1(1(0(2(1(1(1(1(2(0(0(2(2(1(0(2(2(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 1(1(0(0(1(0(0(2(1(1(2(1(0(2(2(2(2(1(1(1(0(1(1(2(1(1(0(x1))))))))))))))))))))))))))) 1(1(1(0(1(0(2(0(0(0(1(2(1(1(1(2(2(0(0(1(1(1(1(x1))))))))))))))))))))))) -> 1(0(1(2(1(0(1(1(0(1(1(1(1(1(0(1(2(1(0(2(0(1(1(1(2(1(1(x1))))))))))))))))))))))))))) 1(1(1(2(1(2(2(1(2(2(2(1(0(0(0(0(2(1(0(1(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(1(1(2(0(1(2(1(1(2(2(2(1(1(1(1(0(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 1(1(2(0(0(0(2(0(1(2(2(2(1(1(1(2(0(0(2(1(2(0(1(x1))))))))))))))))))))))) -> 1(1(2(1(2(2(1(1(0(2(2(1(1(2(0(2(0(2(1(1(1(2(1(2(0(2(1(x1))))))))))))))))))))))))))) 1(1(2(0(0(2(1(0(0(0(2(2(2(2(2(2(1(0(1(1(0(0(2(x1))))))))))))))))))))))) -> 1(1(1(2(1(0(1(2(1(0(1(1(2(2(2(0(2(1(1(0(2(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 1(1(2(0(2(2(0(1(1(2(0(1(2(2(2(1(2(2(2(0(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(2(2(2(2(0(1(1(2(1(1(1(2(0(0(2(1(0(0(2(2(0(1(1(x1))))))))))))))))))))))))))) 1(1(2(2(2(2(2(1(2(2(0(0(1(2(0(0(0(2(2(1(0(2(2(x1))))))))))))))))))))))) -> 1(0(2(1(2(2(1(1(1(2(2(2(2(2(2(1(1(0(0(1(0(0(0(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(0(1(1(1(0(0(2(1(1(2(1(2(2(0(2(0(1(1(2(x1))))))))))))))))))))))) -> 1(2(2(0(2(1(2(1(1(0(1(1(0(2(2(2(1(1(0(2(1(2(1(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(1(0(1(1(0(0(2(2(2(2(0(0(2(2(0(0(2(2(0(x1))))))))))))))))))))))) -> 1(2(2(0(1(2(1(1(1(1(2(2(0(2(0(0(2(0(1(1(1(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 1(2(0(2(2(2(0(0(1(1(0(1(1(2(1(0(2(2(0(2(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(0(1(2(0(0(0(0(1(1(1(2(1(1(2(1(1(2(0(2(2(2(0(1(x1))))))))))))))))))))))))))) 1(2(1(2(1(1(1(0(2(0(1(2(2(2(0(1(1(1(2(2(0(0(0(x1))))))))))))))))))))))) -> 1(2(0(1(2(2(1(2(1(1(0(1(1(1(0(1(0(1(2(1(1(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 2(0(2(0(1(1(0(0(0(1(1(0(0(1(2(2(1(1(2(0(2(0(1(x1))))))))))))))))))))))) -> 2(2(1(0(2(1(2(2(1(0(0(2(0(2(2(1(1(1(0(2(2(2(2(2(1(0(1(x1))))))))))))))))))))))))))) 2(1(1(0(2(0(2(1(1(2(1(1(0(0(0(0(0(2(2(1(1(1(2(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(0(2(1(0(2(1(1(1(1(0(2(2(2(1(1(1(1(0(2(2(2(x1))))))))))))))))))))))))))) 2(1(1(1(2(2(1(0(1(1(2(2(2(0(0(0(0(2(0(2(0(1(2(x1))))))))))))))))))))))) -> 2(1(0(1(2(0(0(2(2(1(1(0(1(1(2(0(1(1(0(0(0(2(1(1(1(1(1(x1))))))))))))))))))))))))))) 2(2(0(0(1(2(0(0(0(1(1(0(2(2(2(2(1(1(2(1(2(0(2(x1))))))))))))))))))))))) -> 2(0(1(1(1(0(0(0(1(2(2(2(0(1(1(0(2(2(0(1(1(1(0(2(2(2(0(x1))))))))))))))))))))))))))) 2(2(0(1(2(2(1(2(2(0(2(0(1(1(2(2(1(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 2(2(1(0(1(1(0(0(2(2(0(1(0(1(1(1(2(1(1(1(0(2(1(2(2(1(2(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(2(1(0(1(0(1(2(2(2(0(1(2(2(0(1(1(1(x1))))))))))))))))))))))) -> 0(0(0(1(2(2(1(1(0(2(1(1(2(2(2(2(1(2(0(0(1(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(0(1(1(1(1(2(1(1(2(1(2(1(2(1(2(1(1(1(0(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(2(1(2(1(2(1(1(1(0(0(2(2(1(1(0(1(0(0(2(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(1(2(1(2(1(2(1(1(1(0(2(0(1(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 0(1(1(2(0(1(1(1(1(1(1(0(2(0(1(0(1(1(0(1(0(1(2(2(0(2(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(0(2(1(2(2(1(1(1(2(0(2(0(1(2(0(1(1(2(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(1(1(2(0(1(0(1(0(2(2(2(0(1(2(2(2(1(0(0(1(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(1(2(2(0(0(1(1(0(2(2(1(0(1(1(2(0(1(0(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(2(0(1(1(1(2(0(1(1(1(2(1(2(1(0(1(2(2(1(1(0(x1))))))))))))))))))))))))))) 0(1(0(1(1(2(0(1(1(2(2(0(0(2(1(0(2(0(2(2(2(0(2(x1))))))))))))))))))))))) -> 2(1(2(1(1(2(1(1(0(1(0(1(0(1(2(1(0(0(2(1(1(1(2(1(1(1(2(x1))))))))))))))))))))))))))) 0(1(1(0(2(1(0(2(1(0(2(0(2(2(0(1(1(0(1(2(2(1(1(x1))))))))))))))))))))))) -> 2(1(0(1(0(0(1(1(1(2(0(0(2(2(2(1(1(2(1(1(0(1(1(0(1(1(1(x1))))))))))))))))))))))))))) 0(1(1(2(1(1(2(2(2(1(1(2(0(0(1(2(0(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 0(0(2(2(1(0(2(1(2(1(1(2(1(2(2(1(1(1(1(1(0(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 0(1(2(1(2(0(0(1(0(1(1(2(1(1(2(2(1(1(1(2(2(0(2(x1))))))))))))))))))))))) -> 0(1(2(0(1(1(0(1(2(1(1(1(2(0(1(0(1(1(0(1(2(2(1(2(1(2(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(0(0(2(0(0(1(0(0(2(2(0(1(1(0(1(0(1(x1))))))))))))))))))))))) -> 0(1(0(2(0(0(2(1(1(1(0(0(2(1(0(2(2(2(1(1(0(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(1(0(0(1(1(2(1(1(0(1(0(1(0(2(2(2(1(x1))))))))))))))))))))))) -> 1(2(1(2(2(1(1(1(1(1(1(0(2(2(2(1(1(2(1(2(1(0(2(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(1(0(2(0(0(2(0(1(1(1(2(0(1(1(2(1(1(0(1(0(0(x1))))))))))))))))))))))) -> 1(2(2(2(0(1(1(2(0(0(1(1(0(1(0(1(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))))))))))))))))) 0(2(1(2(1(1(0(0(1(1(0(2(2(0(1(2(0(2(0(1(1(0(0(x1))))))))))))))))))))))) -> 2(1(1(1(2(2(0(1(1(1(1(2(2(2(2(1(0(0(2(0(2(1(2(1(1(2(2(x1))))))))))))))))))))))))))) 0(2(1(2(2(0(1(0(1(0(1(0(2(1(0(0(1(1(1(2(1(0(0(x1))))))))))))))))))))))) -> 2(2(2(2(2(1(1(2(2(1(0(1(1(1(1(0(1(1(2(2(1(1(1(2(0(1(2(x1))))))))))))))))))))))))))) 0(2(2(1(1(1(1(2(2(1(2(1(2(0(0(1(2(1(0(0(2(0(0(x1))))))))))))))))))))))) -> 2(2(2(1(1(2(2(2(0(2(0(1(0(1(1(2(2(2(0(1(2(1(1(0(0(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(0(1(2(2(1(2(1(0(1(0(2(2(1(2(1(0(2(1(2(x1))))))))))))))))))))))) -> 1(0(2(2(1(2(2(2(1(2(0(2(1(0(1(0(1(2(2(1(1(0(1(1(1(1(2(x1))))))))))))))))))))))))))) 0(2(2(2(1(2(2(0(0(2(1(2(2(1(0(1(1(1(1(1(2(0(0(x1))))))))))))))))))))))) -> 0(1(1(1(2(0(0(0(1(1(2(2(0(1(1(2(1(1(1(1(1(1(1(2(2(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(2(0(1(1(2(1(0(1(2(2(1(1(2(2(0(2(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(2(2(1(1(1(0(1(1(0(1(2(2(1(2(1(2(0(0(1(1(1(1(x1))))))))))))))))))))))))))) 1(0(0(0(0(2(1(1(1(2(2(2(1(2(1(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))) -> 1(1(2(1(1(2(2(0(2(0(1(1(0(1(1(0(0(1(0(1(1(1(2(1(2(1(0(x1))))))))))))))))))))))))))) 1(0(0(1(1(1(1(0(1(0(0(1(0(2(0(2(1(1(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(2(2(1(1(2(2(1(0(2(2(2(1(2(2(1(2(0(2(1(2(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 1(0(2(2(0(2(2(2(1(0(2(2(2(1(0(0(2(0(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(0(0(1(1(1(0(2(0(0(0(1(2(1(2(1(1(1(1(1(1(0(0(0(2(1(1(x1))))))))))))))))))))))))))) 1(0(2(2(1(1(0(0(2(1(2(1(0(2(1(0(0(2(2(0(1(2(1(x1))))))))))))))))))))))) -> 1(1(2(1(0(2(1(1(1(2(1(0(2(1(2(2(1(0(1(1(0(2(2(1(1(0(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(0(1(0(0(0(1(2(0(1(1(1(1(2(1(1(1(2(0(x1))))))))))))))))))))))) -> 1(2(1(0(0(1(2(1(1(1(2(0(1(2(1(1(1(2(2(0(1(2(2(1(0(2(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(2(0(2(0(1(1(0(2(2(1(1(2(1(1(0(0(2(0(x1))))))))))))))))))))))) -> 1(2(1(2(1(2(2(1(2(1(1(1(0(2(2(1(1(2(0(1(1(0(1(1(0(0(2(x1))))))))))))))))))))))))))) 1(1(0(2(1(1(1(1(2(0(0(2(2(1(0(2(2(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 1(1(0(0(1(0(0(2(1(1(2(1(0(2(2(2(2(1(1(1(0(1(1(2(1(1(0(x1))))))))))))))))))))))))))) 1(1(1(0(1(0(2(0(0(0(1(2(1(1(1(2(2(0(0(1(1(1(1(x1))))))))))))))))))))))) -> 1(0(1(2(1(0(1(1(0(1(1(1(1(1(0(1(2(1(0(2(0(1(1(1(2(1(1(x1))))))))))))))))))))))))))) 1(1(1(2(1(2(2(1(2(2(2(1(0(0(0(0(2(1(0(1(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(1(1(2(0(1(2(1(1(2(2(2(1(1(1(1(0(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 1(1(2(0(0(0(2(0(1(2(2(2(1(1(1(2(0(0(2(1(2(0(1(x1))))))))))))))))))))))) -> 1(1(2(1(2(2(1(1(0(2(2(1(1(2(0(2(0(2(1(1(1(2(1(2(0(2(1(x1))))))))))))))))))))))))))) 1(1(2(0(0(2(1(0(0(0(2(2(2(2(2(2(1(0(1(1(0(0(2(x1))))))))))))))))))))))) -> 1(1(1(2(1(0(1(2(1(0(1(1(2(2(2(0(2(1(1(0(2(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 1(1(2(0(2(2(0(1(1(2(0(1(2(2(2(1(2(2(2(0(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(2(2(2(2(0(1(1(2(1(1(1(2(0(0(2(1(0(0(2(2(0(1(1(x1))))))))))))))))))))))))))) 1(1(2(2(2(2(2(1(2(2(0(0(1(2(0(0(0(2(2(1(0(2(2(x1))))))))))))))))))))))) -> 1(0(2(1(2(2(1(1(1(2(2(2(2(2(2(1(1(0(0(1(0(0(0(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(0(1(1(1(0(0(2(1(1(2(1(2(2(0(2(0(1(1(2(x1))))))))))))))))))))))) -> 1(2(2(0(2(1(2(1(1(0(1(1(0(2(2(2(1(1(0(2(1(2(1(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(1(0(1(1(0(0(2(2(2(2(0(0(2(2(0(0(2(2(0(x1))))))))))))))))))))))) -> 1(2(2(0(1(2(1(1(1(1(2(2(0(2(0(0(2(0(1(1(1(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 1(2(0(2(2(2(0(0(1(1(0(1(1(2(1(0(2(2(0(2(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(0(1(2(0(0(0(0(1(1(1(2(1(1(2(1(1(2(0(2(2(2(0(1(x1))))))))))))))))))))))))))) 1(2(1(2(1(1(1(0(2(0(1(2(2(2(0(1(1(1(2(2(0(0(0(x1))))))))))))))))))))))) -> 1(2(0(1(2(2(1(2(1(1(0(1(1(1(0(1(0(1(2(1(1(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 2(0(2(0(1(1(0(0(0(1(1(0(0(1(2(2(1(1(2(0(2(0(1(x1))))))))))))))))))))))) -> 2(2(1(0(2(1(2(2(1(0(0(2(0(2(2(1(1(1(0(2(2(2(2(2(1(0(1(x1))))))))))))))))))))))))))) 2(1(1(0(2(0(2(1(1(2(1(1(0(0(0(0(0(2(2(1(1(1(2(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(0(2(1(0(2(1(1(1(1(0(2(2(2(1(1(1(1(0(2(2(2(x1))))))))))))))))))))))))))) 2(1(1(1(2(2(1(0(1(1(2(2(2(0(0(0(0(2(0(2(0(1(2(x1))))))))))))))))))))))) -> 2(1(0(1(2(0(0(2(2(1(1(0(1(1(2(0(1(1(0(0(0(2(1(1(1(1(1(x1))))))))))))))))))))))))))) 2(2(0(0(1(2(0(0(0(1(1(0(2(2(2(2(1(1(2(1(2(0(2(x1))))))))))))))))))))))) -> 2(0(1(1(1(0(0(0(1(2(2(2(0(1(1(0(2(2(0(1(1(1(0(2(2(2(0(x1))))))))))))))))))))))))))) 2(2(0(1(2(2(1(2(2(0(2(0(1(1(2(2(1(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 2(2(1(0(1(1(0(0(2(2(0(1(0(1(1(1(2(1(1(1(0(2(1(2(2(1(2(x1))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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)) 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(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(2(1(0(1(0(1(2(2(2(0(1(2(2(0(1(1(1(x1))))))))))))))))))))))) -> 0(0(0(1(2(2(1(1(0(2(1(1(2(2(2(2(1(2(0(0(1(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(0(1(1(1(1(2(1(1(2(1(2(1(2(1(2(1(1(1(0(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(2(1(2(1(2(1(1(1(0(0(2(2(1(1(0(1(0(0(2(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(1(2(1(2(1(2(1(1(1(0(2(0(1(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 0(1(1(2(0(1(1(1(1(1(1(0(2(0(1(0(1(1(0(1(0(1(2(2(0(2(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(0(2(1(2(2(1(1(1(2(0(2(0(1(2(0(1(1(2(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(1(1(2(0(1(0(1(0(2(2(2(0(1(2(2(2(1(0(0(1(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(1(2(2(0(0(1(1(0(2(2(1(0(1(1(2(0(1(0(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(2(0(1(1(1(2(0(1(1(1(2(1(2(1(0(1(2(2(1(1(0(x1))))))))))))))))))))))))))) 0(1(0(1(1(2(0(1(1(2(2(0(0(2(1(0(2(0(2(2(2(0(2(x1))))))))))))))))))))))) -> 2(1(2(1(1(2(1(1(0(1(0(1(0(1(2(1(0(0(2(1(1(1(2(1(1(1(2(x1))))))))))))))))))))))))))) 0(1(1(0(2(1(0(2(1(0(2(0(2(2(0(1(1(0(1(2(2(1(1(x1))))))))))))))))))))))) -> 2(1(0(1(0(0(1(1(1(2(0(0(2(2(2(1(1(2(1(1(0(1(1(0(1(1(1(x1))))))))))))))))))))))))))) 0(1(1(2(1(1(2(2(2(1(1(2(0(0(1(2(0(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 0(0(2(2(1(0(2(1(2(1(1(2(1(2(2(1(1(1(1(1(0(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 0(1(2(1(2(0(0(1(0(1(1(2(1(1(2(2(1(1(1(2(2(0(2(x1))))))))))))))))))))))) -> 0(1(2(0(1(1(0(1(2(1(1(1(2(0(1(0(1(1(0(1(2(2(1(2(1(2(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(0(0(2(0(0(1(0(0(2(2(0(1(1(0(1(0(1(x1))))))))))))))))))))))) -> 0(1(0(2(0(0(2(1(1(1(0(0(2(1(0(2(2(2(1(1(0(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(1(0(0(1(1(2(1(1(0(1(0(1(0(2(2(2(1(x1))))))))))))))))))))))) -> 1(2(1(2(2(1(1(1(1(1(1(0(2(2(2(1(1(2(1(2(1(0(2(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(1(0(2(0(0(2(0(1(1(1(2(0(1(1(2(1(1(0(1(0(0(x1))))))))))))))))))))))) -> 1(2(2(2(0(1(1(2(0(0(1(1(0(1(0(1(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))))))))))))))))) 0(2(1(2(1(1(0(0(1(1(0(2(2(0(1(2(0(2(0(1(1(0(0(x1))))))))))))))))))))))) -> 2(1(1(1(2(2(0(1(1(1(1(2(2(2(2(1(0(0(2(0(2(1(2(1(1(2(2(x1))))))))))))))))))))))))))) 0(2(1(2(2(0(1(0(1(0(1(0(2(1(0(0(1(1(1(2(1(0(0(x1))))))))))))))))))))))) -> 2(2(2(2(2(1(1(2(2(1(0(1(1(1(1(0(1(1(2(2(1(1(1(2(0(1(2(x1))))))))))))))))))))))))))) 0(2(2(1(1(1(1(2(2(1(2(1(2(0(0(1(2(1(0(0(2(0(0(x1))))))))))))))))))))))) -> 2(2(2(1(1(2(2(2(0(2(0(1(0(1(1(2(2(2(0(1(2(1(1(0(0(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(0(1(2(2(1(2(1(0(1(0(2(2(1(2(1(0(2(1(2(x1))))))))))))))))))))))) -> 1(0(2(2(1(2(2(2(1(2(0(2(1(0(1(0(1(2(2(1(1(0(1(1(1(1(2(x1))))))))))))))))))))))))))) 0(2(2(2(1(2(2(0(0(2(1(2(2(1(0(1(1(1(1(1(2(0(0(x1))))))))))))))))))))))) -> 0(1(1(1(2(0(0(0(1(1(2(2(0(1(1(2(1(1(1(1(1(1(1(2(2(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(2(0(1(1(2(1(0(1(2(2(1(1(2(2(0(2(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(2(2(1(1(1(0(1(1(0(1(2(2(1(2(1(2(0(0(1(1(1(1(x1))))))))))))))))))))))))))) 1(0(0(0(0(2(1(1(1(2(2(2(1(2(1(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))) -> 1(1(2(1(1(2(2(0(2(0(1(1(0(1(1(0(0(1(0(1(1(1(2(1(2(1(0(x1))))))))))))))))))))))))))) 1(0(0(1(1(1(1(0(1(0(0(1(0(2(0(2(1(1(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(2(2(1(1(2(2(1(0(2(2(2(1(2(2(1(2(0(2(1(2(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 1(0(2(2(0(2(2(2(1(0(2(2(2(1(0(0(2(0(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(0(0(1(1(1(0(2(0(0(0(1(2(1(2(1(1(1(1(1(1(0(0(0(2(1(1(x1))))))))))))))))))))))))))) 1(0(2(2(1(1(0(0(2(1(2(1(0(2(1(0(0(2(2(0(1(2(1(x1))))))))))))))))))))))) -> 1(1(2(1(0(2(1(1(1(2(1(0(2(1(2(2(1(0(1(1(0(2(2(1(1(0(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(0(1(0(0(0(1(2(0(1(1(1(1(2(1(1(1(2(0(x1))))))))))))))))))))))) -> 1(2(1(0(0(1(2(1(1(1(2(0(1(2(1(1(1(2(2(0(1(2(2(1(0(2(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(2(0(2(0(1(1(0(2(2(1(1(2(1(1(0(0(2(0(x1))))))))))))))))))))))) -> 1(2(1(2(1(2(2(1(2(1(1(1(0(2(2(1(1(2(0(1(1(0(1(1(0(0(2(x1))))))))))))))))))))))))))) 1(1(0(2(1(1(1(1(2(0(0(2(2(1(0(2(2(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 1(1(0(0(1(0(0(2(1(1(2(1(0(2(2(2(2(1(1(1(0(1(1(2(1(1(0(x1))))))))))))))))))))))))))) 1(1(1(0(1(0(2(0(0(0(1(2(1(1(1(2(2(0(0(1(1(1(1(x1))))))))))))))))))))))) -> 1(0(1(2(1(0(1(1(0(1(1(1(1(1(0(1(2(1(0(2(0(1(1(1(2(1(1(x1))))))))))))))))))))))))))) 1(1(1(2(1(2(2(1(2(2(2(1(0(0(0(0(2(1(0(1(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(1(1(2(0(1(2(1(1(2(2(2(1(1(1(1(0(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 1(1(2(0(0(0(2(0(1(2(2(2(1(1(1(2(0(0(2(1(2(0(1(x1))))))))))))))))))))))) -> 1(1(2(1(2(2(1(1(0(2(2(1(1(2(0(2(0(2(1(1(1(2(1(2(0(2(1(x1))))))))))))))))))))))))))) 1(1(2(0(0(2(1(0(0(0(2(2(2(2(2(2(1(0(1(1(0(0(2(x1))))))))))))))))))))))) -> 1(1(1(2(1(0(1(2(1(0(1(1(2(2(2(0(2(1(1(0(2(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 1(1(2(0(2(2(0(1(1(2(0(1(2(2(2(1(2(2(2(0(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(2(2(2(2(0(1(1(2(1(1(1(2(0(0(2(1(0(0(2(2(0(1(1(x1))))))))))))))))))))))))))) 1(1(2(2(2(2(2(1(2(2(0(0(1(2(0(0(0(2(2(1(0(2(2(x1))))))))))))))))))))))) -> 1(0(2(1(2(2(1(1(1(2(2(2(2(2(2(1(1(0(0(1(0(0(0(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(0(1(1(1(0(0(2(1(1(2(1(2(2(0(2(0(1(1(2(x1))))))))))))))))))))))) -> 1(2(2(0(2(1(2(1(1(0(1(1(0(2(2(2(1(1(0(2(1(2(1(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(1(0(1(1(0(0(2(2(2(2(0(0(2(2(0(0(2(2(0(x1))))))))))))))))))))))) -> 1(2(2(0(1(2(1(1(1(1(2(2(0(2(0(0(2(0(1(1(1(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 1(2(0(2(2(2(0(0(1(1(0(1(1(2(1(0(2(2(0(2(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(0(1(2(0(0(0(0(1(1(1(2(1(1(2(1(1(2(0(2(2(2(0(1(x1))))))))))))))))))))))))))) 1(2(1(2(1(1(1(0(2(0(1(2(2(2(0(1(1(1(2(2(0(0(0(x1))))))))))))))))))))))) -> 1(2(0(1(2(2(1(2(1(1(0(1(1(1(0(1(0(1(2(1(1(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 2(0(2(0(1(1(0(0(0(1(1(0(0(1(2(2(1(1(2(0(2(0(1(x1))))))))))))))))))))))) -> 2(2(1(0(2(1(2(2(1(0(0(2(0(2(2(1(1(1(0(2(2(2(2(2(1(0(1(x1))))))))))))))))))))))))))) 2(1(1(0(2(0(2(1(1(2(1(1(0(0(0(0(0(2(2(1(1(1(2(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(0(2(1(0(2(1(1(1(1(0(2(2(2(1(1(1(1(0(2(2(2(x1))))))))))))))))))))))))))) 2(1(1(1(2(2(1(0(1(1(2(2(2(0(0(0(0(2(0(2(0(1(2(x1))))))))))))))))))))))) -> 2(1(0(1(2(0(0(2(2(1(1(0(1(1(2(0(1(1(0(0(0(2(1(1(1(1(1(x1))))))))))))))))))))))))))) 2(2(0(0(1(2(0(0(0(1(1(0(2(2(2(2(1(1(2(1(2(0(2(x1))))))))))))))))))))))) -> 2(0(1(1(1(0(0(0(1(2(2(2(0(1(1(0(2(2(0(1(1(1(0(2(2(2(0(x1))))))))))))))))))))))))))) 2(2(0(1(2(2(1(2(2(0(2(0(1(1(2(2(1(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 2(2(1(0(1(1(0(0(2(2(0(1(0(1(1(1(2(1(1(1(0(2(1(2(2(1(2(x1))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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)) 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(0(0(0(0(1(2(1(0(1(0(1(2(2(2(0(1(2(2(0(1(1(1(x1))))))))))))))))))))))) -> 0(0(0(1(2(2(1(1(0(2(1(1(2(2(2(2(1(2(0(0(1(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(0(1(1(1(1(2(1(1(2(1(2(1(2(1(2(1(1(1(0(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(2(1(2(1(2(1(1(1(0(0(2(2(1(1(0(1(0(0(2(1(1(x1))))))))))))))))))))))))))) 0(0(0(2(1(2(1(2(1(2(1(1(1(0(2(0(1(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 0(1(1(2(0(1(1(1(1(1(1(0(2(0(1(0(1(1(0(1(0(1(2(2(0(2(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(0(2(1(2(2(1(1(1(2(0(2(0(1(2(0(1(1(2(x1))))))))))))))))))))))) -> 0(1(1(1(1(1(1(1(2(0(1(0(1(0(2(2(2(0(1(2(2(2(1(0(0(1(1(x1))))))))))))))))))))))))))) 0(0(1(1(1(1(2(2(0(0(1(1(0(2(2(1(0(1(1(2(0(1(0(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(2(0(1(1(1(2(0(1(1(1(2(1(2(1(0(1(2(2(1(1(0(x1))))))))))))))))))))))))))) 0(1(0(1(1(2(0(1(1(2(2(0(0(2(1(0(2(0(2(2(2(0(2(x1))))))))))))))))))))))) -> 2(1(2(1(1(2(1(1(0(1(0(1(0(1(2(1(0(0(2(1(1(1(2(1(1(1(2(x1))))))))))))))))))))))))))) 0(1(1(0(2(1(0(2(1(0(2(0(2(2(0(1(1(0(1(2(2(1(1(x1))))))))))))))))))))))) -> 2(1(0(1(0(0(1(1(1(2(0(0(2(2(2(1(1(2(1(1(0(1(1(0(1(1(1(x1))))))))))))))))))))))))))) 0(1(1(2(1(1(2(2(2(1(1(2(0(0(1(2(0(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 0(0(2(2(1(0(2(1(2(1(1(2(1(2(2(1(1(1(1(1(0(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 0(1(2(1(2(0(0(1(0(1(1(2(1(1(2(2(1(1(1(2(2(0(2(x1))))))))))))))))))))))) -> 0(1(2(0(1(1(0(1(2(1(1(1(2(0(1(0(1(1(0(1(2(2(1(2(1(2(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(0(0(2(0(0(1(0(0(2(2(0(1(1(0(1(0(1(x1))))))))))))))))))))))) -> 0(1(0(2(0(0(2(1(1(1(0(0(2(1(0(2(2(2(1(1(0(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(0(2(1(2(1(0(0(1(1(2(1(1(0(1(0(1(0(2(2(2(1(x1))))))))))))))))))))))) -> 1(2(1(2(2(1(1(1(1(1(1(0(2(2(2(1(1(2(1(2(1(0(2(1(1(0(1(x1))))))))))))))))))))))))))) 0(2(1(0(2(0(0(2(0(1(1(1(2(0(1(1(2(1(1(0(1(0(0(x1))))))))))))))))))))))) -> 1(2(2(2(0(1(1(2(0(0(1(1(0(1(0(1(1(1(0(1(1(1(1(2(2(1(1(x1))))))))))))))))))))))))))) 0(2(1(2(1(1(0(0(1(1(0(2(2(0(1(2(0(2(0(1(1(0(0(x1))))))))))))))))))))))) -> 2(1(1(1(2(2(0(1(1(1(1(2(2(2(2(1(0(0(2(0(2(1(2(1(1(2(2(x1))))))))))))))))))))))))))) 0(2(1(2(2(0(1(0(1(0(1(0(2(1(0(0(1(1(1(2(1(0(0(x1))))))))))))))))))))))) -> 2(2(2(2(2(1(1(2(2(1(0(1(1(1(1(0(1(1(2(2(1(1(1(2(0(1(2(x1))))))))))))))))))))))))))) 0(2(2(1(1(1(1(2(2(1(2(1(2(0(0(1(2(1(0(0(2(0(0(x1))))))))))))))))))))))) -> 2(2(2(1(1(2(2(2(0(2(0(1(0(1(1(2(2(2(0(1(2(1(1(0(0(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(0(1(2(2(1(2(1(0(1(0(2(2(1(2(1(0(2(1(2(x1))))))))))))))))))))))) -> 1(0(2(2(1(2(2(2(1(2(0(2(1(0(1(0(1(2(2(1(1(0(1(1(1(1(2(x1))))))))))))))))))))))))))) 0(2(2(2(1(2(2(0(0(2(1(2(2(1(0(1(1(1(1(1(2(0(0(x1))))))))))))))))))))))) -> 0(1(1(1(2(0(0(0(1(1(2(2(0(1(1(2(1(1(1(1(1(1(1(2(2(2(2(x1))))))))))))))))))))))))))) 0(2(2(2(2(0(1(1(2(1(0(1(2(2(1(1(2(2(0(2(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(2(2(1(1(1(0(1(1(0(1(2(2(1(2(1(2(0(0(1(1(1(1(x1))))))))))))))))))))))))))) 1(0(0(0(0(2(1(1(1(2(2(2(1(2(1(0(0(1(0(0(2(1(0(x1))))))))))))))))))))))) -> 1(1(2(1(1(2(2(0(2(0(1(1(0(1(1(0(0(1(0(1(1(1(2(1(2(1(0(x1))))))))))))))))))))))))))) 1(0(0(1(1(1(1(0(1(0(0(1(0(2(0(2(1(1(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(2(2(1(1(2(2(1(0(2(2(2(1(2(2(1(2(0(2(1(2(1(1(1(1(0(1(x1))))))))))))))))))))))))))) 1(0(2(2(0(2(2(2(1(0(2(2(2(1(0(0(2(0(2(0(2(1(1(x1))))))))))))))))))))))) -> 1(0(0(1(1(1(0(2(0(0(0(1(2(1(2(1(1(1(1(1(1(0(0(0(2(1(1(x1))))))))))))))))))))))))))) 1(0(2(2(1(1(0(0(2(1(2(1(0(2(1(0(0(2(2(0(1(2(1(x1))))))))))))))))))))))) -> 1(1(2(1(0(2(1(1(1(2(1(0(2(1(2(2(1(0(1(1(0(2(2(1(1(0(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(0(1(0(0(0(1(2(0(1(1(1(1(2(1(1(1(2(0(x1))))))))))))))))))))))) -> 1(2(1(0(0(1(2(1(1(1(2(0(1(2(1(1(1(2(2(0(1(2(2(1(0(2(1(x1))))))))))))))))))))))))))) 1(1(0(0(0(2(0(2(0(1(1(0(2(2(1(1(2(1(1(0(0(2(0(x1))))))))))))))))))))))) -> 1(2(1(2(1(2(2(1(2(1(1(1(0(2(2(1(1(2(0(1(1(0(1(1(0(0(2(x1))))))))))))))))))))))))))) 1(1(0(2(1(1(1(1(2(0(0(2(2(1(0(2(2(1(2(2(0(1(2(x1))))))))))))))))))))))) -> 1(1(0(0(1(0(0(2(1(1(2(1(0(2(2(2(2(1(1(1(0(1(1(2(1(1(0(x1))))))))))))))))))))))))))) 1(1(1(0(1(0(2(0(0(0(1(2(1(1(1(2(2(0(0(1(1(1(1(x1))))))))))))))))))))))) -> 1(0(1(2(1(0(1(1(0(1(1(1(1(1(0(1(2(1(0(2(0(1(1(1(2(1(1(x1))))))))))))))))))))))))))) 1(1(1(2(1(2(2(1(2(2(2(1(0(0(0(0(2(1(0(1(1(0(0(x1))))))))))))))))))))))) -> 1(2(1(1(1(1(1(2(0(1(2(1(1(2(2(2(1(1(1(1(0(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 1(1(2(0(0(0(2(0(1(2(2(2(1(1(1(2(0(0(2(1(2(0(1(x1))))))))))))))))))))))) -> 1(1(2(1(2(2(1(1(0(2(2(1(1(2(0(2(0(2(1(1(1(2(1(2(0(2(1(x1))))))))))))))))))))))))))) 1(1(2(0(0(2(1(0(0(0(2(2(2(2(2(2(1(0(1(1(0(0(2(x1))))))))))))))))))))))) -> 1(1(1(2(1(0(1(2(1(0(1(1(2(2(2(0(2(1(1(0(2(1(2(2(1(1(1(x1))))))))))))))))))))))))))) 1(1(2(0(2(2(0(1(1(2(0(1(2(2(2(1(2(2(2(0(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(2(2(2(2(0(1(1(2(1(1(1(2(0(0(2(1(0(0(2(2(0(1(1(x1))))))))))))))))))))))))))) 1(1(2(2(2(2(2(1(2(2(0(0(1(2(0(0(0(2(2(1(0(2(2(x1))))))))))))))))))))))) -> 1(0(2(1(2(2(1(1(1(2(2(2(2(2(2(1(1(0(0(1(0(0(0(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(0(1(1(1(0(0(2(1(1(2(1(2(2(0(2(0(1(1(2(x1))))))))))))))))))))))) -> 1(2(2(0(2(1(2(1(1(0(1(1(0(2(2(2(1(1(0(2(1(2(1(1(1(1(2(x1))))))))))))))))))))))))))) 1(2(0(1(1(0(1(1(0(0(2(2(2(2(0(0(2(2(0(0(2(2(0(x1))))))))))))))))))))))) -> 1(2(2(0(1(2(1(1(1(1(2(2(0(2(0(0(2(0(1(1(1(0(1(2(1(2(0(x1))))))))))))))))))))))))))) 1(2(0(2(2(2(0(0(1(1(0(1(1(2(1(0(2(2(0(2(2(0(1(x1))))))))))))))))))))))) -> 1(1(1(1(0(1(2(0(0(0(0(1(1(1(2(1(1(2(1(1(2(0(2(2(2(0(1(x1))))))))))))))))))))))))))) 1(2(1(2(1(1(1(0(2(0(1(2(2(2(0(1(1(1(2(2(0(0(0(x1))))))))))))))))))))))) -> 1(2(0(1(2(2(1(2(1(1(0(1(1(1(0(1(0(1(2(1(1(0(2(2(0(2(2(x1))))))))))))))))))))))))))) 2(0(2(0(1(1(0(0(0(1(1(0(0(1(2(2(1(1(2(0(2(0(1(x1))))))))))))))))))))))) -> 2(2(1(0(2(1(2(2(1(0(0(2(0(2(2(1(1(1(0(2(2(2(2(2(1(0(1(x1))))))))))))))))))))))))))) 2(1(1(0(2(0(2(1(1(2(1(1(0(0(0(0(0(2(2(1(1(1(2(x1))))))))))))))))))))))) -> 2(2(1(1(2(2(0(2(1(0(2(1(1(1(1(0(2(2(2(1(1(1(1(0(2(2(2(x1))))))))))))))))))))))))))) 2(1(1(1(2(2(1(0(1(1(2(2(2(0(0(0(0(2(0(2(0(1(2(x1))))))))))))))))))))))) -> 2(1(0(1(2(0(0(2(2(1(1(0(1(1(2(0(1(1(0(0(0(2(1(1(1(1(1(x1))))))))))))))))))))))))))) 2(2(0(0(1(2(0(0(0(1(1(0(2(2(2(2(1(1(2(1(2(0(2(x1))))))))))))))))))))))) -> 2(0(1(1(1(0(0(0(1(2(2(2(0(1(1(0(2(2(0(1(1(1(0(2(2(2(0(x1))))))))))))))))))))))))))) 2(2(0(1(2(2(1(2(2(0(2(0(1(1(2(2(1(0(1(0(0(2(0(x1))))))))))))))))))))))) -> 2(2(1(0(1(1(0(0(2(2(0(1(0(1(1(1(2(1(1(1(0(2(1(2(2(1(2(x1))))))))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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)) 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 2. The certificate found is represented by the following graph. 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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, 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(996,997,[2_1|2]), (997,998,[1_1|2]), (998,999,[1_1|2]), (999,1000,[1_1|2]), (1000,1001,[1_1|2]), (1001,1002,[0_1|2]), (1001,433,[1_1|2]), (1001,459,[0_1|2]), (1001,485,[1_1|2]), (1002,1003,[2_1|2]), (1003,1004,[2_1|2]), (1003,1031,[2_1|2]), (1003,1057,[2_1|2]), (1004,42,[2_1|2]), (1004,147,[2_1|2]), (1004,173,[2_1|2]), (1004,199,[2_1|2]), (1004,355,[2_1|2]), (1004,381,[2_1|2]), (1004,407,[2_1|2]), (1004,953,[2_1|2]), (1004,979,[2_1|2]), (1004,1005,[2_1|2]), (1004,1031,[2_1|2]), (1004,1057,[2_1|2]), (1004,304,[2_1|2]), (1004,330,[2_1|2]), (1004,486,[2_1|2]), (1004,538,[2_1|2]), (1004,616,[2_1|2]), (1004,642,[2_1|2]), (1004,720,[2_1|2]), (1004,850,[2_1|2]), (1004,876,[2_1|2]), (1004,928,[2_1|2]), (1004,513,[2_1|2]), (1004,591,[2_1|2]), (1004,747,[2_1|2]), (1004,774,[2_1|2]), (1004,359,[2_1|2]), (1005,1006,[1_1|2]), (1006,1007,[0_1|2]), (1007,1008,[1_1|2]), (1008,1009,[2_1|2]), (1009,1010,[0_1|2]), (1010,1011,[0_1|2]), (1011,1012,[2_1|2]), (1012,1013,[2_1|2]), (1013,1014,[1_1|2]), (1014,1015,[1_1|2]), (1015,1016,[0_1|2]), (1016,1017,[1_1|2]), (1017,1018,[1_1|2]), (1018,1019,[2_1|2]), (1019,1020,[0_1|2]), (1020,1021,[1_1|2]), (1021,1022,[1_1|2]), (1022,1023,[0_1|2]), (1023,1024,[0_1|2]), (1024,1025,[0_1|2]), (1025,1026,[2_1|2]), (1026,1027,[1_1|2]), (1027,1028,[1_1|2]), (1028,1029,[1_1|2]), (1028,693,[1_1|2]), (1028,719,[1_1|2]), (1029,1030,[1_1|2]), (1029,615,[1_1|2]), (1029,641,[1_1|2]), (1029,667,[1_1|2]), (1029,693,[1_1|2]), (1029,719,[1_1|2]), (1029,745,[1_1|2]), (1029,771,[1_1|2]), (1029,797,[1_1|2]), (1029,823,[1_1|2]), (1030,42,[1_1|2]), (1030,147,[1_1|2]), (1030,173,[1_1|2]), (1030,199,[1_1|2]), (1030,355,[1_1|2]), (1030,381,[1_1|2]), (1030,407,[1_1|2]), (1030,953,[1_1|2]), (1030,979,[1_1|2]), (1030,1005,[1_1|2]), (1030,1031,[1_1|2]), (1030,1057,[1_1|2]), (1030,304,[1_1|2]), (1030,330,[1_1|2]), (1030,486,[1_1|2]), (1030,538,[1_1|2]), (1030,616,[1_1|2]), (1030,642,[1_1|2]), (1030,720,[1_1|2]), (1030,850,[1_1|2]), (1030,876,[1_1|2]), (1030,928,[1_1|2]), (1030,253,[1_1|2]), (1030,511,[1_1|2]), (1030,537,[1_1|2]), (1030,563,[1_1|2]), (1030,589,[1_1|2]), (1030,615,[1_1|2]), (1030,641,[1_1|2]), (1030,667,[1_1|2]), (1030,693,[1_1|2]), (1030,719,[1_1|2]), (1030,745,[1_1|2]), (1030,771,[1_1|2]), (1030,797,[1_1|2]), (1030,823,[1_1|2]), (1030,849,[1_1|2]), (1030,875,[1_1|2]), (1030,901,[1_1|2]), (1030,927,[1_1|2]), (1031,1032,[0_1|2]), (1032,1033,[1_1|2]), (1033,1034,[1_1|2]), (1034,1035,[1_1|2]), (1035,1036,[0_1|2]), (1036,1037,[0_1|2]), (1037,1038,[0_1|2]), (1038,1039,[1_1|2]), (1039,1040,[2_1|2]), (1040,1041,[2_1|2]), (1041,1042,[2_1|2]), (1042,1043,[0_1|2]), (1043,1044,[1_1|2]), (1044,1045,[1_1|2]), (1045,1046,[0_1|2]), (1046,1047,[2_1|2]), (1047,1048,[2_1|2]), (1048,1049,[0_1|2]), (1049,1050,[1_1|2]), (1050,1051,[1_1|2]), (1051,1052,[1_1|2]), (1052,1053,[0_1|2]), (1052,433,[1_1|2]), (1053,1054,[2_1|2]), (1054,1055,[2_1|2]), (1054,1031,[2_1|2]), (1054,1057,[2_1|2]), (1055,1056,[2_1|2]), (1055,953,[2_1|2]), (1056,42,[0_1|2]), (1056,147,[0_1|2, 2_1|2]), (1056,173,[0_1|2, 2_1|2]), (1056,199,[0_1|2, 2_1|2]), (1056,355,[0_1|2, 2_1|2]), (1056,381,[0_1|2, 2_1|2]), (1056,407,[0_1|2, 2_1|2]), (1056,953,[0_1|2]), (1056,979,[0_1|2]), (1056,1005,[0_1|2]), (1056,1031,[0_1|2]), (1056,1057,[0_1|2]), (1056,43,[0_1|2]), (1056,69,[0_1|2]), (1056,95,[0_1|2]), (1056,121,[0_1|2]), (1056,225,[0_1|2]), (1056,251,[0_1|2]), (1056,277,[0_1|2]), (1056,303,[1_1|2]), (1056,329,[1_1|2]), (1056,433,[1_1|2]), (1056,459,[0_1|2]), (1056,485,[1_1|2]), (1057,1058,[2_1|2]), (1058,1059,[1_1|2]), (1059,1060,[0_1|2]), (1060,1061,[1_1|2]), (1061,1062,[1_1|2]), (1062,1063,[0_1|2]), (1063,1064,[0_1|2]), (1064,1065,[2_1|2]), (1065,1066,[2_1|2]), (1066,1067,[0_1|2]), (1067,1068,[1_1|2]), (1068,1069,[0_1|2]), (1069,1070,[1_1|2]), (1070,1071,[1_1|2]), (1071,1072,[1_1|2]), (1072,1073,[2_1|2]), (1073,1074,[1_1|2]), (1074,1075,[1_1|2]), (1075,1076,[1_1|2]), (1076,1077,[0_1|2]), (1077,1078,[2_1|2]), (1078,1079,[1_1|2]), (1079,1080,[2_1|2]), (1080,1081,[2_1|2]), (1081,1082,[1_1|2]), (1081,849,[1_1|2]), (1081,875,[1_1|2]), (1081,901,[1_1|2]), (1081,927,[1_1|2]), (1082,42,[2_1|2]), (1082,43,[2_1|2]), (1082,69,[2_1|2]), (1082,95,[2_1|2]), (1082,121,[2_1|2]), (1082,225,[2_1|2]), (1082,251,[2_1|2]), (1082,277,[2_1|2]), (1082,459,[2_1|2]), (1082,1032,[2_1|2]), (1082,953,[2_1|2]), (1082,979,[2_1|2]), (1082,1005,[2_1|2]), (1082,1031,[2_1|2]), (1082,1057,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)