WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 49 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 109 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) 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(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(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(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)) encArg(cons_3(x_1)) -> 3(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)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) 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(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(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)) encArg(cons_3(x_1)) -> 3(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)) 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(1, n^1). The TRS R consists of the following rules: 0(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(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)) encArg(cons_3(x_1)) -> 3(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)) 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(0(0(0(0(1(1(2(0(0(3(1(3(x1))))))))))))) -> 2(3(3(1(1(2(3(1(2(1(1(2(2(1(0(2(3(x1))))))))))))))))) 0(0(0(3(1(2(2(3(2(3(1(0(1(x1))))))))))))) -> 3(2(1(3(0(1(3(1(1(2(1(3(3(3(3(1(3(x1))))))))))))))))) 0(0(3(0(1(1(2(1(0(1(3(3(3(x1))))))))))))) -> 1(3(3(1(0(2(1(1(2(2(3(2(1(2(1(1(3(x1))))))))))))))))) 0(0(3(2(1(3(2(2(3(2(0(0(0(x1))))))))))))) -> 1(2(1(3(2(3(2(2(3(3(1(0(0(2(1(3(3(x1))))))))))))))))) 0(3(0(1(3(1(3(2(2(1(2(2(0(x1))))))))))))) -> 2(3(3(0(3(3(3(1(3(2(0(2(2(1(3(1(2(x1))))))))))))))))) 0(3(2(2(0(0(3(2(3(1(1(2(1(x1))))))))))))) -> 2(0(3(3(1(2(0(3(1(2(1(2(1(1(0(3(1(x1))))))))))))))))) 0(3(3(3(0(1(2(3(1(1(0(0(3(x1))))))))))))) -> 0(2(2(0(1(2(1(2(1(3(1(3(0(3(3(0(3(x1))))))))))))))))) 1(0(1(3(1(1(1(1(3(2(0(3(2(x1))))))))))))) -> 3(3(3(1(3(3(0(1(0(1(2(1(2(1(1(0(3(x1))))))))))))))))) 1(0(3(2(3(3(0(0(1(2(1(0(1(x1))))))))))))) -> 2(1(1(1(1(3(2(2(2(1(3(3(1(3(2(2(1(x1))))))))))))))))) 1(1(0(0(0(2(3(0(1(2(0(1(3(x1))))))))))))) -> 1(2(2(2(1(3(3(2(1(1(2(1(0(2(0(2(3(x1))))))))))))))))) 1(1(2(1(0(2(2(0(3(0(0(0(0(x1))))))))))))) -> 2(2(1(3(3(3(1(2(0(1(0(2(3(1(1(0(3(x1))))))))))))))))) 1(3(2(3(1(1(2(0(1(2(3(1(0(x1))))))))))))) -> 1(3(2(2(0(2(3(3(0(1(2(1(2(2(1(3(1(x1))))))))))))))))) 2(0(3(3(3(2(2(3(2(0(0(3(3(x1))))))))))))) -> 3(3(2(2(0(1(1(2(1(1(2(1(2(2(1(3(3(x1))))))))))))))))) 2(1(2(0(2(1(0(1(1(0(1(2(0(x1))))))))))))) -> 0(0(1(1(3(3(1(1(0(0(3(3(3(3(3(3(2(x1))))))))))))))))) 2(2(3(1(3(2(2(0(2(2(2(2(0(x1))))))))))))) -> 2(1(3(0(1(3(2(2(1(0(0(2(1(1(3(0(0(x1))))))))))))))))) 2(3(0(3(0(2(1(2(0(3(2(0(1(x1))))))))))))) -> 2(1(3(3(0(2(2(1(3(2(0(2(2(2(1(2(1(x1))))))))))))))))) 2(3(1(0(1(0(0(1(1(0(1(2(1(x1))))))))))))) -> 1(3(1(3(3(3(3(2(2(1(3(1(2(1(1(0(3(x1))))))))))))))))) 2(3(3(1(2(2(3(0(2(2(0(0(1(x1))))))))))))) -> 2(3(1(2(2(1(1(3(1(3(3(3(3(0(3(0(0(x1))))))))))))))))) 2(3(3(1(2(2(3(2(3(1(1(0(1(x1))))))))))))) -> 2(2(1(0(2(1(1(1(2(2(1(1(3(3(2(1(3(x1))))))))))))))))) 3(0(3(0(2(3(2(3(3(3(1(2(1(x1))))))))))))) -> 2(1(2(1(3(2(1(3(2(1(2(1(3(1(3(3(1(x1))))))))))))))))) 3(0(3(0(2(3(3(2(3(2(2(2(1(x1))))))))))))) -> 2(2(2(1(3(3(3(2(1(1(2(1(1(1(1(3(1(x1))))))))))))))))) 3(0(3(1(3(2(0(0(1(2(1(0(0(x1))))))))))))) -> 3(1(3(0(3(3(2(2(1(1(2(2(2(1(1(0(2(x1))))))))))))))))) 3(1(2(3(1(1(3(1(1(3(0(1(1(x1))))))))))))) -> 3(3(2(1(3(1(2(3(3(3(2(1(1(2(1(0(1(x1))))))))))))))))) 3(2(2(1(0(2(0(3(3(3(3(2(1(x1))))))))))))) -> 3(3(3(2(3(1(1(3(3(2(1(2(1(2(1(3(1(x1))))))))))))))))) 3(2(3(0(1(1(3(0(0(0(0(2(1(x1))))))))))))) -> 3(3(2(1(1(2(3(2(1(2(0(3(1(3(2(2(1(x1))))))))))))))))) 3(2(3(2(3(2(3(0(3(2(0(1(1(x1))))))))))))) -> 2(2(1(1(2(1(3(3(2(0(0(0(0(0(1(3(3(x1))))))))))))))))) 3(3(1(3(1(2(3(2(0(2(2(0(0(x1))))))))))))) -> 2(1(2(1(3(0(2(2(0(2(2(1(1(3(3(3(3(x1))))))))))))))))) 3(3(2(1(3(1(2(0(0(3(2(3(3(x1))))))))))))) -> 3(2(1(3(2(2(1(3(3(3(1(0(2(1(3(3(3(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)) encArg(cons_3(x_1)) -> 3(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)) 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 2. The certificate found is represented by the following graph. "[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] {(86,87,[0_1|0, 1_1|0, 2_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0]), (86,88,[0_1|1, 1_1|1, 2_1|1, 3_1|1]), (86,89,[2_1|2]), (86,105,[3_1|2]), (86,121,[1_1|2]), (86,137,[1_1|2]), (86,153,[2_1|2]), (86,169,[2_1|2]), (86,185,[0_1|2]), (86,201,[3_1|2]), (86,217,[2_1|2]), (86,233,[1_1|2]), (86,249,[2_1|2]), (86,265,[1_1|2]), (86,281,[3_1|2]), (86,297,[0_1|2]), (86,313,[2_1|2]), (86,329,[2_1|2]), (86,345,[1_1|2]), (86,361,[2_1|2]), (86,377,[2_1|2]), (86,393,[2_1|2]), (86,409,[2_1|2]), (86,425,[3_1|2]), (86,441,[3_1|2]), (86,457,[3_1|2]), (86,473,[3_1|2]), (86,489,[2_1|2]), (86,505,[2_1|2]), (86,521,[3_1|2]), (87,87,[cons_0_1|0, cons_1_1|0, cons_2_1|0, cons_3_1|0]), (88,87,[encArg_1|1]), (88,88,[0_1|1, 1_1|1, 2_1|1, 3_1|1]), (88,89,[2_1|2]), (88,105,[3_1|2]), (88,121,[1_1|2]), (88,137,[1_1|2]), (88,153,[2_1|2]), (88,169,[2_1|2]), (88,185,[0_1|2]), (88,201,[3_1|2]), (88,217,[2_1|2]), (88,233,[1_1|2]), (88,249,[2_1|2]), (88,265,[1_1|2]), (88,281,[3_1|2]), (88,297,[0_1|2]), (88,313,[2_1|2]), (88,329,[2_1|2]), (88,345,[1_1|2]), (88,361,[2_1|2]), (88,377,[2_1|2]), (88,393,[2_1|2]), (88,409,[2_1|2]), (88,425,[3_1|2]), (88,441,[3_1|2]), (88,457,[3_1|2]), (88,473,[3_1|2]), (88,489,[2_1|2]), (88,505,[2_1|2]), (88,521,[3_1|2]), (89,90,[3_1|2]), (90,91,[3_1|2]), (91,92,[1_1|2]), (92,93,[1_1|2]), (93,94,[2_1|2]), (94,95,[3_1|2]), (95,96,[1_1|2]), (96,97,[2_1|2]), (97,98,[1_1|2]), (98,99,[1_1|2]), (99,100,[2_1|2]), (100,101,[2_1|2]), (101,102,[1_1|2]), (102,103,[0_1|2]), (103,104,[2_1|2]), (103,329,[2_1|2]), (103,345,[1_1|2]), (103,361,[2_1|2]), (103,377,[2_1|2]), (104,88,[3_1|2]), (104,105,[3_1|2]), (104,201,[3_1|2]), (104,281,[3_1|2]), (104,425,[3_1|2]), (104,441,[3_1|2]), (104,457,[3_1|2]), 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(184,218,[1_1|2]), (184,314,[1_1|2]), (184,330,[1_1|2]), (184,394,[1_1|2]), (184,506,[1_1|2]), (184,139,[1_1|2]), (184,201,[3_1|2]), (184,217,[2_1|2]), (184,249,[2_1|2]), (185,186,[2_1|2]), (186,187,[2_1|2]), (187,188,[0_1|2]), (188,189,[1_1|2]), (189,190,[2_1|2]), (190,191,[1_1|2]), (191,192,[2_1|2]), (192,193,[1_1|2]), (193,194,[3_1|2]), (194,195,[1_1|2]), (195,196,[3_1|2]), (196,197,[0_1|2]), (197,198,[3_1|2]), (198,199,[3_1|2]), (198,393,[2_1|2]), (198,409,[2_1|2]), (198,425,[3_1|2]), (199,200,[0_1|2]), (199,153,[2_1|2]), (199,169,[2_1|2]), (199,185,[0_1|2]), (200,88,[3_1|2]), (200,105,[3_1|2]), (200,201,[3_1|2]), (200,281,[3_1|2]), (200,425,[3_1|2]), (200,441,[3_1|2]), (200,457,[3_1|2]), (200,473,[3_1|2]), (200,521,[3_1|2]), (200,393,[2_1|2]), (200,409,[2_1|2]), (200,489,[2_1|2]), (200,505,[2_1|2]), (201,202,[3_1|2]), (202,203,[3_1|2]), (203,204,[1_1|2]), (204,205,[3_1|2]), (205,206,[3_1|2]), (206,207,[0_1|2]), (207,208,[1_1|2]), (208,209,[0_1|2]), (209,210,[1_1|2]), 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(344,137,[1_1|2]), (344,233,[1_1|2]), (344,265,[1_1|2]), (344,345,[1_1|2]), (344,201,[3_1|2]), (344,217,[2_1|2]), (344,249,[2_1|2]), (345,346,[3_1|2]), (346,347,[1_1|2]), (347,348,[3_1|2]), (348,349,[3_1|2]), (349,350,[3_1|2]), (350,351,[3_1|2]), (351,352,[2_1|2]), (352,353,[2_1|2]), (353,354,[1_1|2]), (354,355,[3_1|2]), (355,356,[1_1|2]), (356,357,[2_1|2]), (357,358,[1_1|2]), (358,359,[1_1|2]), (358,217,[2_1|2]), (359,360,[0_1|2]), (359,153,[2_1|2]), (359,169,[2_1|2]), (359,185,[0_1|2]), (360,88,[3_1|2]), (360,121,[3_1|2]), (360,137,[3_1|2]), (360,233,[3_1|2]), (360,265,[3_1|2]), (360,345,[3_1|2]), (360,218,[3_1|2]), (360,314,[3_1|2]), (360,330,[3_1|2]), (360,394,[3_1|2]), (360,506,[3_1|2]), (360,139,[3_1|2]), (360,393,[2_1|2]), (360,409,[2_1|2]), (360,425,[3_1|2]), (360,441,[3_1|2]), (360,457,[3_1|2]), (360,473,[3_1|2]), (360,489,[2_1|2]), (360,505,[2_1|2]), (360,521,[3_1|2]), (361,362,[3_1|2]), (362,363,[1_1|2]), (363,364,[2_1|2]), (364,365,[2_1|2]), (365,366,[1_1|2]), (366,367,[1_1|2]), (367,368,[3_1|2]), (368,369,[1_1|2]), (369,370,[3_1|2]), (370,371,[3_1|2]), (371,372,[3_1|2]), (372,373,[3_1|2]), (373,374,[0_1|2]), (374,375,[3_1|2]), (375,376,[0_1|2]), (375,89,[2_1|2]), (375,105,[3_1|2]), (375,121,[1_1|2]), (375,137,[1_1|2]), (376,88,[0_1|2]), (376,121,[0_1|2, 1_1|2]), (376,137,[0_1|2, 1_1|2]), (376,233,[0_1|2]), (376,265,[0_1|2]), (376,345,[0_1|2]), (376,299,[0_1|2]), (376,89,[2_1|2]), (376,105,[3_1|2]), (376,153,[2_1|2]), (376,169,[2_1|2]), (376,185,[0_1|2]), (377,378,[2_1|2]), (378,379,[1_1|2]), (379,380,[0_1|2]), (380,381,[2_1|2]), (381,382,[1_1|2]), (382,383,[1_1|2]), (383,384,[1_1|2]), (384,385,[2_1|2]), (385,386,[2_1|2]), (386,387,[1_1|2]), (387,388,[1_1|2]), (388,389,[3_1|2]), (388,521,[3_1|2]), (389,390,[3_1|2]), (390,391,[2_1|2]), (391,392,[1_1|2]), (391,265,[1_1|2]), (392,88,[3_1|2]), (392,121,[3_1|2]), (392,137,[3_1|2]), (392,233,[3_1|2]), (392,265,[3_1|2]), (392,345,[3_1|2]), (392,393,[2_1|2]), (392,409,[2_1|2]), (392,425,[3_1|2]), (392,441,[3_1|2]), (392,457,[3_1|2]), (392,473,[3_1|2]), (392,489,[2_1|2]), (392,505,[2_1|2]), (392,521,[3_1|2]), (393,394,[1_1|2]), (394,395,[2_1|2]), (395,396,[1_1|2]), (396,397,[3_1|2]), (397,398,[2_1|2]), (398,399,[1_1|2]), (399,400,[3_1|2]), (400,401,[2_1|2]), (401,402,[1_1|2]), (402,403,[2_1|2]), (403,404,[1_1|2]), (404,405,[3_1|2]), (405,406,[1_1|2]), (406,407,[3_1|2]), (406,505,[2_1|2]), (407,408,[3_1|2]), (407,441,[3_1|2]), (408,88,[1_1|2]), (408,121,[1_1|2]), (408,137,[1_1|2]), (408,233,[1_1|2]), (408,265,[1_1|2]), (408,345,[1_1|2]), (408,218,[1_1|2]), (408,314,[1_1|2]), (408,330,[1_1|2]), (408,394,[1_1|2]), (408,506,[1_1|2]), (408,139,[1_1|2]), (408,201,[3_1|2]), (408,217,[2_1|2]), (408,249,[2_1|2]), (409,410,[2_1|2]), (410,411,[2_1|2]), (411,412,[1_1|2]), (412,413,[3_1|2]), (413,414,[3_1|2]), (414,415,[3_1|2]), (415,416,[2_1|2]), (416,417,[1_1|2]), (417,418,[1_1|2]), (418,419,[2_1|2]), (419,420,[1_1|2]), (420,421,[1_1|2]), (421,422,[1_1|2]), (422,423,[1_1|2]), (423,424,[3_1|2]), (423,441,[3_1|2]), (424,88,[1_1|2]), (424,121,[1_1|2]), (424,137,[1_1|2]), (424,233,[1_1|2]), (424,265,[1_1|2]), (424,345,[1_1|2]), (424,218,[1_1|2]), (424,314,[1_1|2]), (424,330,[1_1|2]), (424,394,[1_1|2]), (424,506,[1_1|2]), (424,251,[1_1|2]), (424,379,[1_1|2]), (424,491,[1_1|2]), (424,412,[1_1|2]), (424,201,[3_1|2]), (424,217,[2_1|2]), (424,249,[2_1|2]), (425,426,[1_1|2]), (426,427,[3_1|2]), (427,428,[0_1|2]), (428,429,[3_1|2]), (429,430,[3_1|2]), (430,431,[2_1|2]), (431,432,[2_1|2]), (432,433,[1_1|2]), (433,434,[1_1|2]), (434,435,[2_1|2]), (435,436,[2_1|2]), (436,437,[2_1|2]), (437,438,[1_1|2]), (438,439,[1_1|2]), (439,440,[0_1|2]), (440,88,[2_1|2]), (440,185,[2_1|2]), (440,297,[2_1|2, 0_1|2]), (440,298,[2_1|2]), (440,281,[3_1|2]), (440,313,[2_1|2]), (440,329,[2_1|2]), (440,345,[1_1|2]), (440,361,[2_1|2]), (440,377,[2_1|2]), (441,442,[3_1|2]), (442,443,[2_1|2]), (443,444,[1_1|2]), (444,445,[3_1|2]), (445,446,[1_1|2]), (446,447,[2_1|2]), (447,448,[3_1|2]), (448,449,[3_1|2]), (449,450,[3_1|2]), (450,451,[2_1|2]), (451,452,[1_1|2]), (452,453,[1_1|2]), (453,454,[2_1|2]), (454,455,[1_1|2]), (454,201,[3_1|2]), (455,456,[0_1|2]), (456,88,[1_1|2]), (456,121,[1_1|2]), (456,137,[1_1|2]), (456,233,[1_1|2]), (456,265,[1_1|2]), (456,345,[1_1|2]), (456,201,[3_1|2]), (456,217,[2_1|2]), (456,249,[2_1|2]), (457,458,[3_1|2]), (458,459,[3_1|2]), (459,460,[2_1|2]), (460,461,[3_1|2]), (461,462,[1_1|2]), (462,463,[1_1|2]), (463,464,[3_1|2]), (464,465,[3_1|2]), (465,466,[2_1|2]), (466,467,[1_1|2]), (467,468,[2_1|2]), (468,469,[1_1|2]), (469,470,[2_1|2]), (470,471,[1_1|2]), (471,472,[3_1|2]), (471,441,[3_1|2]), (472,88,[1_1|2]), (472,121,[1_1|2]), (472,137,[1_1|2]), (472,233,[1_1|2]), (472,265,[1_1|2]), (472,345,[1_1|2]), (472,218,[1_1|2]), (472,314,[1_1|2]), (472,330,[1_1|2]), (472,394,[1_1|2]), (472,506,[1_1|2]), (472,107,[1_1|2]), (472,523,[1_1|2]), (472,444,[1_1|2]), (472,476,[1_1|2]), (472,201,[3_1|2]), (472,217,[2_1|2]), (472,249,[2_1|2]), (473,474,[3_1|2]), (474,475,[2_1|2]), (475,476,[1_1|2]), (476,477,[1_1|2]), (477,478,[2_1|2]), (478,479,[3_1|2]), (479,480,[2_1|2]), (480,481,[1_1|2]), (481,482,[2_1|2]), (482,483,[0_1|2]), (483,484,[3_1|2]), (484,485,[1_1|2]), (485,486,[3_1|2]), (485,457,[3_1|2]), (486,487,[2_1|2]), (487,488,[2_1|2]), (487,297,[0_1|2]), (488,88,[1_1|2]), (488,121,[1_1|2]), (488,137,[1_1|2]), (488,233,[1_1|2]), (488,265,[1_1|2]), (488,345,[1_1|2]), (488,218,[1_1|2]), (488,314,[1_1|2]), (488,330,[1_1|2]), (488,394,[1_1|2]), (488,506,[1_1|2]), (488,201,[3_1|2]), (488,217,[2_1|2]), (488,249,[2_1|2]), (489,490,[2_1|2]), (490,491,[1_1|2]), (491,492,[1_1|2]), (492,493,[2_1|2]), (493,494,[1_1|2]), (494,495,[3_1|2]), (495,496,[3_1|2]), (496,497,[2_1|2]), (497,498,[0_1|2]), (498,499,[0_1|2]), (499,500,[0_1|2]), (500,501,[0_1|2]), (501,502,[0_1|2]), (502,503,[1_1|2]), (503,504,[3_1|2]), (503,505,[2_1|2]), (503,521,[3_1|2]), (504,88,[3_1|2]), (504,121,[3_1|2]), (504,137,[3_1|2]), (504,233,[3_1|2]), (504,265,[3_1|2]), (504,345,[3_1|2]), (504,393,[2_1|2]), (504,409,[2_1|2]), (504,425,[3_1|2]), (504,441,[3_1|2]), (504,457,[3_1|2]), (504,473,[3_1|2]), (504,489,[2_1|2]), (504,505,[2_1|2]), (504,521,[3_1|2]), (505,506,[1_1|2]), (506,507,[2_1|2]), (507,508,[1_1|2]), (508,509,[3_1|2]), (509,510,[0_1|2]), (510,511,[2_1|2]), (511,512,[2_1|2]), (512,513,[0_1|2]), (513,514,[2_1|2]), (514,515,[2_1|2]), (515,516,[1_1|2]), (516,517,[1_1|2]), (517,518,[3_1|2]), (518,519,[3_1|2]), (519,520,[3_1|2]), (519,505,[2_1|2]), (519,521,[3_1|2]), (520,88,[3_1|2]), (520,185,[3_1|2]), (520,297,[3_1|2]), (520,298,[3_1|2]), (520,393,[2_1|2]), (520,409,[2_1|2]), (520,425,[3_1|2]), (520,441,[3_1|2]), (520,457,[3_1|2]), (520,473,[3_1|2]), (520,489,[2_1|2]), (520,505,[2_1|2]), (520,521,[3_1|2]), (521,522,[2_1|2]), (522,523,[1_1|2]), (523,524,[3_1|2]), (524,525,[2_1|2]), (525,526,[2_1|2]), (526,527,[1_1|2]), (527,528,[3_1|2]), (528,529,[3_1|2]), (529,530,[3_1|2]), (530,531,[1_1|2]), (531,532,[0_1|2]), (532,533,[2_1|2]), (533,534,[1_1|2]), (534,535,[3_1|2]), (535,536,[3_1|2]), (535,505,[2_1|2]), (535,521,[3_1|2]), (536,88,[3_1|2]), (536,105,[3_1|2]), (536,201,[3_1|2]), (536,281,[3_1|2]), (536,425,[3_1|2]), (536,441,[3_1|2]), (536,457,[3_1|2]), (536,473,[3_1|2]), (536,521,[3_1|2]), (536,202,[3_1|2]), (536,282,[3_1|2]), (536,442,[3_1|2]), (536,458,[3_1|2]), (536,474,[3_1|2]), (536,91,[3_1|2]), (536,155,[3_1|2]), (536,393,[2_1|2]), (536,409,[2_1|2]), (536,489,[2_1|2]), (536,505,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)