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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 41 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 262 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(1(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(1(0(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(2(x1)))))) 0(1(2(4(x1)))) -> 0(3(2(3(1(4(x1)))))) 0(5(0(5(x1)))) -> 0(3(0(5(5(x1))))) 0(5(1(2(x1)))) -> 1(0(1(5(2(x1))))) 0(5(1(2(x1)))) -> 0(1(0(1(5(2(x1)))))) 0(5(1(2(x1)))) -> 0(3(2(3(1(5(x1)))))) 0(5(4(2(x1)))) -> 0(4(5(3(2(x1))))) 0(5(5(2(x1)))) -> 5(0(1(5(2(x1))))) 1(0(0(5(x1)))) -> 1(1(0(0(1(5(4(x1))))))) 1(0(1(2(x1)))) -> 1(1(3(0(2(x1))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(2(x1)))))) 1(0(1(2(x1)))) -> 1(1(0(3(2(3(x1)))))) 1(0(5(4(x1)))) -> 0(1(1(5(4(x1))))) 1(2(0(5(x1)))) -> 0(3(2(3(1(5(x1)))))) 1(2(0(5(x1)))) -> 5(0(3(3(2(1(x1)))))) 1(5(0(2(x1)))) -> 1(1(0(1(1(5(2(x1))))))) 1(5(1(2(x1)))) -> 0(1(1(5(2(x1))))) 1(5(1(2(x1)))) -> 1(0(1(5(3(2(x1)))))) 5(0(0(2(x1)))) -> 5(0(3(0(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(x1))))) 5(0(1(2(x1)))) -> 5(1(0(3(2(3(x1)))))) 0(0(0(1(2(x1))))) -> 0(2(0(1(0(3(3(4(x1)))))))) 0(0(2(5(2(x1))))) -> 0(3(2(0(5(2(x1)))))) 0(1(2(5(0(x1))))) -> 3(3(2(2(0(0(1(5(x1)))))))) 0(1(2(5(2(x1))))) -> 0(3(2(1(5(3(2(x1))))))) 0(3(5(2(2(x1))))) -> 0(4(5(3(2(2(x1)))))) 0(4(2(0(5(x1))))) -> 0(4(0(3(2(1(5(x1))))))) 0(4(2(5(2(x1))))) -> 0(5(4(3(3(2(2(x1))))))) 0(5(0(2(2(x1))))) -> 0(2(5(0(3(2(x1)))))) 0(5(0(5(1(x1))))) -> 0(1(0(3(5(5(x1)))))) 0(5(1(3(0(x1))))) -> 0(0(1(1(5(3(x1)))))) 0(5(2(2(4(x1))))) -> 0(5(3(2(2(4(x1)))))) 0(5(2(3(1(x1))))) -> 0(1(5(3(2(2(2(x1))))))) 0(5(2(4(1(x1))))) -> 0(4(3(2(5(1(x1)))))) 0(5(3(5(2(x1))))) -> 0(0(3(5(5(2(x1)))))) 0(5(5(3(1(x1))))) -> 5(0(1(5(3(3(2(x1))))))) 1(0(5(5(1(x1))))) -> 0(4(5(1(5(1(x1)))))) 1(1(2(2(0(x1))))) -> 1(1(3(2(2(0(x1)))))) 1(1(2(3(4(x1))))) -> 1(1(3(2(2(4(x1)))))) 1(1(3(5(2(x1))))) -> 1(1(5(3(3(2(x1)))))) 1(5(0(5(0(x1))))) -> 0(1(5(3(5(1(0(x1))))))) 1(5(5(1(2(x1))))) -> 1(5(1(1(5(3(2(2(x1)))))))) 5(0(2(0(5(x1))))) -> 5(0(3(3(2(0(5(x1))))))) 5(0(2(3(4(x1))))) -> 5(0(3(2(3(4(x1)))))) 5(5(0(1(2(x1))))) -> 5(5(3(0(2(1(x1)))))) 0(0(0(5(1(2(x1)))))) -> 0(0(1(5(0(2(4(x1))))))) 0(0(1(2(4(1(x1)))))) -> 1(3(0(2(3(0(4(1(x1)))))))) 0(0(2(1(2(0(x1)))))) -> 0(1(0(3(2(2(2(0(x1)))))))) 0(0(2(3(0(5(x1)))))) -> 0(0(3(0(3(2(5(x1))))))) 0(0(5(2(3(4(x1)))))) -> 0(4(0(3(1(2(5(x1))))))) 0(0(5(5(3(4(x1)))))) -> 1(4(1(0(0(3(5(5(x1)))))))) 0(1(2(0(1(2(x1)))))) -> 1(0(3(2(2(1(1(0(x1)))))))) 0(1(2(2(0(5(x1)))))) -> 0(4(1(5(0(3(2(2(x1)))))))) 0(1(2(5(5(5(x1)))))) -> 0(2(5(1(5(3(5(x1))))))) 0(1(3(1(5(2(x1)))))) -> 1(0(1(5(3(3(2(x1))))))) 0(1(4(4(0(5(x1)))))) -> 4(3(0(0(1(5(4(x1))))))) 0(2(5(3(5(1(x1)))))) -> 0(3(3(2(2(1(5(5(x1)))))))) 0(5(0(0(5(4(x1)))))) -> 0(1(0(1(0(4(5(5(x1)))))))) 0(5(1(2(1(4(x1)))))) -> 1(1(5(3(2(2(0(4(x1)))))))) 0(5(5(1(2(5(x1)))))) -> 0(2(1(5(5(4(5(x1))))))) 1(0(0(2(3(4(x1)))))) -> 1(4(0(0(3(3(2(x1))))))) 1(0(1(3(5(1(x1)))))) -> 0(1(1(1(5(3(2(2(x1)))))))) 1(0(5(4(2(1(x1)))))) -> 0(1(1(5(3(4(2(x1))))))) 1(1(0(1(2(2(x1)))))) -> 1(0(1(1(3(1(2(2(x1)))))))) 1(2(1(2(0(0(x1)))))) -> 0(3(2(2(1(0(1(x1))))))) 1(4(1(0(0(5(x1)))))) -> 0(0(1(5(4(2(1(x1))))))) 1(4(3(5(0(2(x1)))))) -> 0(3(3(2(1(5(4(x1))))))) 0(0(1(2(3(5(5(x1))))))) -> 5(1(5(0(3(0(2(1(x1)))))))) 0(0(2(3(4(2(1(x1))))))) -> 0(0(4(1(3(2(3(2(x1)))))))) 0(1(0(0(5(3(4(x1))))))) -> 0(3(0(5(0(4(3(1(x1)))))))) 0(1(2(4(4(0(5(x1))))))) -> 5(4(0(1(0(3(2(4(x1)))))))) 0(5(2(5(1(3(4(x1))))))) -> 1(4(5(2(0(3(1(5(x1)))))))) 0(5(5(0(2(5(1(x1))))))) -> 5(3(0(0(1(5(2(5(x1)))))))) 0(5(5(2(5(3(4(x1))))))) -> 0(3(2(1(4(5(5(5(x1)))))))) 1(0(1(2(3(4(5(x1))))))) -> 3(0(2(1(5(1(3(4(x1)))))))) 1(1(0(2(0(2(2(x1))))))) -> 1(1(0(2(0(3(2(2(x1)))))))) 1(2(4(3(5(3(5(x1))))))) -> 5(1(3(2(2(4(3(5(x1)))))))) 1(4(3(5(2(5(2(x1))))))) -> 5(1(3(2(2(1(5(4(x1)))))))) 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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[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] {(55,56,[0_1|0, 1_1|0, 5_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]), (55,57,[2_1|1, 3_1|1, 4_1|1, 0_1|1, 1_1|1, 5_1|1]), (55,58,[0_1|2]), (55,62,[1_1|2]), (55,69,[5_1|2]), (55,76,[0_1|2]), (55,83,[0_1|2]), (55,89,[0_1|2]), (55,94,[0_1|2]), (55,101,[0_1|2]), (55,107,[0_1|2]), (55,114,[0_1|2]), (55,120,[1_1|2]), (55,127,[1_1|2]), (55,132,[0_1|2]), (55,139,[0_1|2]), (55,144,[5_1|2]), (55,151,[3_1|2]), (55,158,[0_1|2]), (55,164,[0_1|2]), (55,170,[1_1|2]), (55,177,[1_1|2]), (55,183,[4_1|2]), (55,189,[0_1|2]), (55,196,[0_1|2]), (55,200,[0_1|2]), (55,205,[0_1|2]), (55,210,[0_1|2]), (55,217,[1_1|2]), (55,221,[0_1|2]), (55,226,[0_1|2]), (55,231,[1_1|2]), (55,238,[0_1|2]), (55,243,[0_1|2]), (55,247,[5_1|2]), (55,251,[0_1|2]), (55,258,[5_1|2]), (55,264,[0_1|2]), (55,270,[5_1|2]), (55,277,[0_1|2]), (55,282,[0_1|2]), (55,288,[0_1|2]), (55,293,[1_1|2]), (55,300,[0_1|2]), (55,305,[0_1|2]), (55,310,[0_1|2]), (55,316,[0_1|2]), (55,322,[0_1|2]), (55,329,[1_1|2]), (55,335,[1_1|2]), (55,341,[1_1|2]), (55,345,[1_1|2]), (55,350,[1_1|2]), (55,355,[3_1|2]), (55,362,[0_1|2]), (55,369,[0_1|2]), (55,373,[0_1|2]), (55,379,[0_1|2]), 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(506,430,[1_1|2]), (506,435,[1_1|2]), (506,440,[1_1|2]), (506,445,[1_1|2]), (506,452,[1_1|2]), (506,459,[0_1|2]), (506,465,[0_1|2]), (506,471,[5_1|2]), (506,517,[0_1|3]), (507,508,[3_1|2]), (508,509,[2_1|2]), (509,510,[3_1|2]), (510,511,[1_1|2]), (511,69,[5_1|2]), (511,144,[5_1|2]), (511,247,[5_1|2]), (511,258,[5_1|2]), (511,270,[5_1|2]), (511,384,[5_1|2]), (511,395,[5_1|2]), (511,471,[5_1|2]), (511,478,[5_1|2]), (511,482,[5_1|2]), (511,486,[5_1|2]), (511,491,[5_1|2]), (511,497,[5_1|2]), (511,502,[5_1|2]), (511,278,[5_1|2]), (511,317,[5_1|2]), (512,513,[3_1|2]), (513,514,[2_1|2]), (514,515,[3_1|2]), (515,516,[1_1|2]), (515,402,[1_1|2]), (515,408,[0_1|2]), (515,414,[0_1|2]), (515,418,[1_1|2]), (515,423,[1_1|2]), (515,527,[1_1|3]), (515,533,[0_1|3]), (515,537,[1_1|3]), (516,57,[5_1|2]), (516,69,[5_1|2]), (516,144,[5_1|2]), (516,247,[5_1|2]), (516,258,[5_1|2]), (516,270,[5_1|2]), (516,384,[5_1|2]), (516,395,[5_1|2]), (516,471,[5_1|2]), (516,478,[5_1|2]), (516,482,[5_1|2]), (516,486,[5_1|2]), (516,491,[5_1|2]), (516,497,[5_1|2]), (516,502,[5_1|2]), (516,278,[5_1|2]), (516,317,[5_1|2]), (517,518,[1_1|3]), (518,519,[1_1|3]), (519,520,[5_1|3]), (520,318,[4_1|3]), (521,522,[1_1|3]), (522,523,[0_1|3]), (523,524,[1_1|3]), (524,525,[1_1|3]), (525,526,[5_1|3]), (526,88,[2_1|3]), (527,528,[1_1|3]), (528,529,[0_1|3]), (529,530,[1_1|3]), (530,531,[1_1|3]), (531,532,[5_1|3]), (532,77,[2_1|3]), (532,165,[2_1|3]), (532,206,[2_1|3]), (532,265,[2_1|3]), (533,534,[1_1|3]), (534,535,[1_1|3]), (535,536,[5_1|3]), (536,128,[2_1|3]), (537,538,[0_1|3]), (538,539,[1_1|3]), (539,540,[5_1|3]), (540,541,[3_1|3]), (541,128,[2_1|3]), (542,543,[1_1|3]), (543,544,[3_1|3]), (544,545,[0_1|3]), (545,128,[2_1|3]), (546,547,[1_1|3]), (547,548,[0_1|3]), (548,549,[3_1|3]), (549,550,[2_1|3]), (550,128,[2_1|3]), (551,552,[1_1|3]), (552,553,[0_1|3]), (553,554,[3_1|3]), (554,555,[2_1|3]), (555,128,[3_1|3]), (556,557,[1_1|3]), (557,558,[1_1|3]), (558,559,[5_1|3]), (559,145,[4_1|3]), (559,318,[4_1|3]), (560,561,[1_1|3]), (561,562,[0_1|3]), (562,563,[0_1|3]), (563,564,[1_1|3]), (564,565,[5_1|3]), (565,278,[4_1|3]), (565,317,[4_1|3]), (566,567,[3_1|3]), (567,568,[0_1|3]), (568,569,[0_1|3]), (569,570,[1_1|3]), (570,571,[5_1|3]), (571,572,[2_1|3]), (572,167,[5_1|3]), (573,574,[5_1|3]), (574,575,[1_1|3]), (575,576,[1_1|3]), (576,577,[5_1|3]), (577,578,[3_1|3]), (578,579,[2_1|3]), (579,128,[2_1|3]), (580,581,[0_1|3]), (581,582,[1_1|3]), (582,583,[1_1|3]), (583,584,[5_1|3]), (584,64,[3_1|3]), (585,586,[0_1|3]), (586,587,[1_1|3]), (587,588,[5_1|3]), (588,128,[2_1|3]), (589,590,[1_1|3]), (590,591,[0_1|3]), (591,592,[1_1|3]), (592,593,[5_1|3]), (593,128,[2_1|3]), (594,595,[3_1|3]), (595,596,[2_1|3]), (596,597,[3_1|3]), (597,598,[1_1|3]), (598,128,[5_1|3]), (599,600,[3_1|3]), (600,601,[0_1|3]), (601,602,[5_1|3]), (602,278,[5_1|3]), (602,317,[5_1|3]), (603,604,[1_1|3]), (604,605,[3_1|3]), (605,606,[0_1|3]), (606,77,[2_1|3]), (606,165,[2_1|3]), (606,206,[2_1|3]), (606,265,[2_1|3]), (607,608,[1_1|3]), (608,609,[0_1|3]), (609,610,[3_1|3]), (610,611,[2_1|3]), (611,77,[2_1|3]), (611,165,[2_1|3]), (611,206,[2_1|3]), (611,265,[2_1|3]), (612,613,[1_1|3]), (613,614,[0_1|3]), (614,615,[3_1|3]), (615,616,[2_1|3]), (616,77,[3_1|3]), (616,165,[3_1|3]), (616,206,[3_1|3]), (616,265,[3_1|3]), (617,618,[3_1|3]), (618,619,[2_1|3]), (619,620,[2_1|3]), (620,621,[0_1|3]), (621,622,[0_1|3]), (622,623,[1_1|3]), (623,208,[5_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)