WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 48 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 120 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(1(x1)) -> 2(0(3(4(3(1(1(2(1(3(x1)))))))))) 0(1(x1)) -> 4(4(2(4(2(3(3(1(3(3(x1)))))))))) 0(1(x1)) -> 4(4(4(2(1(0(3(4(1(4(x1)))))))))) 0(0(1(x1))) -> 5(4(2(1(4(1(3(4(2(2(x1)))))))))) 0(2(0(x1))) -> 3(0(0(3(2(2(2(4(1(3(x1)))))))))) 1(5(1(x1))) -> 1(5(3(2(4(3(0(3(2(4(x1)))))))))) 0(1(2(5(x1)))) -> 1(4(3(0(3(4(4(2(3(5(x1)))))))))) 0(2(2(1(x1)))) -> 3(0(5(2(1(1(2(2(3(1(x1)))))))))) 0(4(5(4(x1)))) -> 1(1(5(4(1(2(4(1(4(3(x1)))))))))) 4(5(1(0(x1)))) -> 2(0(3(2(4(2(5(2(2(2(x1)))))))))) 0(2(0(0(1(x1))))) -> 0(2(2(0(5(4(4(3(4(1(x1)))))))))) 0(2(0(4(1(x1))))) -> 2(3(0(3(2(4(2(0(4(1(x1)))))))))) 3(0(0(1(5(x1))))) -> 3(2(4(3(2(2(1(2(0(5(x1)))))))))) 3(0(2(2(1(x1))))) -> 3(2(0(5(4(1(3(1(4(3(x1)))))))))) 3(3(5(0(1(x1))))) -> 3(2(4(1(1(0(5(1(2(4(x1)))))))))) 5(3(3(5(1(x1))))) -> 5(3(2(5(4(4(1(1(4(4(x1)))))))))) 5(4(5(0(2(x1))))) -> 5(4(0(3(4(5(2(1(3(1(x1)))))))))) 0(0(0(2(2(5(x1)))))) -> 1(3(2(1(4(2(0(1(4(0(x1)))))))))) 0(0(2(0(2(4(x1)))))) -> 5(2(3(4(4(5(2(1(4(1(x1)))))))))) 0(0(2(2(5(4(x1)))))) -> 1(4(5(5(5(2(4(4(4(3(x1)))))))))) 0(1(0(4(0(1(x1)))))) -> 0(3(0(3(3(3(0(1(1(4(x1)))))))))) 0(2(4(0(2(4(x1)))))) -> 5(0(5(4(2(2(0(1(4(2(x1)))))))))) 0(2(5(2(5(2(x1)))))) -> 3(2(1(1(1(1(5(1(4(3(x1)))))))))) 0(3(0(5(0(2(x1)))))) -> 4(2(1(2(4(4(0(0(1(1(x1)))))))))) 0(4(0(0(4(2(x1)))))) -> 0(2(2(3(5(4(4(3(1(4(x1)))))))))) 1(1(0(4(0(4(x1)))))) -> 1(2(1(3(4(4(5(4(4(2(x1)))))))))) 2(0(0(1(3(5(x1)))))) -> 4(2(4(0(2(4(4(1(3(5(x1)))))))))) 2(5(0(0(3(5(x1)))))) -> 4(1(3(3(3(5(4(0(5(5(x1)))))))))) 2(5(0(4(3(0(x1)))))) -> 2(4(1(2(2(0(5(5(2(0(x1)))))))))) 2(5(4(0(0(1(x1)))))) -> 4(4(2(4(1(0(1(1(0(1(x1)))))))))) 3(1(5(1(1(5(x1)))))) -> 3(3(0(3(3(0(3(2(2(0(x1)))))))))) 3(5(1(0(5(1(x1)))))) -> 5(3(0(1(3(1(0(5(3(4(x1)))))))))) 0(0(2(0(0(4(2(x1))))))) -> 1(3(4(1(0(0(5(5(2(4(x1)))))))))) 0(0(4(3(0(1(5(x1))))))) -> 2(5(4(5(1(2(4(1(0(5(x1)))))))))) 0(1(0(0(0(4(2(x1))))))) -> 3(2(5(2(0(4(2(3(4(1(x1)))))))))) 0(1(0(4(0(0(2(x1))))))) -> 5(5(3(4(2(0(5(2(2(3(x1)))))))))) 0(2(3(1(0(2(4(x1))))))) -> 3(3(1(0(4(2(1(2(4(3(x1)))))))))) 0(2(3(1(5(0(1(x1))))))) -> 4(5(5(3(4(1(0(4(2(2(x1)))))))))) 2(4(5(1(0(0(2(x1))))))) -> 4(3(1(2(1(3(4(1(5(1(x1)))))))))) 3(0(3(0(5(2(5(x1))))))) -> 3(3(1(2(3(3(1(2(1(5(x1)))))))))) 3(3(0(4(0(0(4(x1))))))) -> 3(3(3(1(1(5(4(4(5(4(x1)))))))))) 3(3(5(0(1(5(2(x1))))))) -> 3(1(5(1(1(2(4(3(1(3(x1)))))))))) 5(0(2(0(5(1(5(x1))))))) -> 5(0(0(0(0(3(2(4(3(5(x1)))))))))) 5(1(1(3(1(0(4(x1))))))) -> 5(1(2(3(1(4(4(5(3(3(x1)))))))))) 5(2(3(5(2(0(2(x1))))))) -> 5(2(2(4(4(1(2(5(5(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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)) 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(1(x1)) -> 2(0(3(4(3(1(1(2(1(3(x1)))))))))) 0(1(x1)) -> 4(4(2(4(2(3(3(1(3(3(x1)))))))))) 0(1(x1)) -> 4(4(4(2(1(0(3(4(1(4(x1)))))))))) 0(0(1(x1))) -> 5(4(2(1(4(1(3(4(2(2(x1)))))))))) 0(2(0(x1))) -> 3(0(0(3(2(2(2(4(1(3(x1)))))))))) 1(5(1(x1))) -> 1(5(3(2(4(3(0(3(2(4(x1)))))))))) 0(1(2(5(x1)))) -> 1(4(3(0(3(4(4(2(3(5(x1)))))))))) 0(2(2(1(x1)))) -> 3(0(5(2(1(1(2(2(3(1(x1)))))))))) 0(4(5(4(x1)))) -> 1(1(5(4(1(2(4(1(4(3(x1)))))))))) 4(5(1(0(x1)))) -> 2(0(3(2(4(2(5(2(2(2(x1)))))))))) 0(2(0(0(1(x1))))) -> 0(2(2(0(5(4(4(3(4(1(x1)))))))))) 0(2(0(4(1(x1))))) -> 2(3(0(3(2(4(2(0(4(1(x1)))))))))) 3(0(0(1(5(x1))))) -> 3(2(4(3(2(2(1(2(0(5(x1)))))))))) 3(0(2(2(1(x1))))) -> 3(2(0(5(4(1(3(1(4(3(x1)))))))))) 3(3(5(0(1(x1))))) -> 3(2(4(1(1(0(5(1(2(4(x1)))))))))) 5(3(3(5(1(x1))))) -> 5(3(2(5(4(4(1(1(4(4(x1)))))))))) 5(4(5(0(2(x1))))) -> 5(4(0(3(4(5(2(1(3(1(x1)))))))))) 0(0(0(2(2(5(x1)))))) -> 1(3(2(1(4(2(0(1(4(0(x1)))))))))) 0(0(2(0(2(4(x1)))))) -> 5(2(3(4(4(5(2(1(4(1(x1)))))))))) 0(0(2(2(5(4(x1)))))) -> 1(4(5(5(5(2(4(4(4(3(x1)))))))))) 0(1(0(4(0(1(x1)))))) -> 0(3(0(3(3(3(0(1(1(4(x1)))))))))) 0(2(4(0(2(4(x1)))))) -> 5(0(5(4(2(2(0(1(4(2(x1)))))))))) 0(2(5(2(5(2(x1)))))) -> 3(2(1(1(1(1(5(1(4(3(x1)))))))))) 0(3(0(5(0(2(x1)))))) -> 4(2(1(2(4(4(0(0(1(1(x1)))))))))) 0(4(0(0(4(2(x1)))))) -> 0(2(2(3(5(4(4(3(1(4(x1)))))))))) 1(1(0(4(0(4(x1)))))) -> 1(2(1(3(4(4(5(4(4(2(x1)))))))))) 2(0(0(1(3(5(x1)))))) -> 4(2(4(0(2(4(4(1(3(5(x1)))))))))) 2(5(0(0(3(5(x1)))))) -> 4(1(3(3(3(5(4(0(5(5(x1)))))))))) 2(5(0(4(3(0(x1)))))) -> 2(4(1(2(2(0(5(5(2(0(x1)))))))))) 2(5(4(0(0(1(x1)))))) -> 4(4(2(4(1(0(1(1(0(1(x1)))))))))) 3(1(5(1(1(5(x1)))))) -> 3(3(0(3(3(0(3(2(2(0(x1)))))))))) 3(5(1(0(5(1(x1)))))) -> 5(3(0(1(3(1(0(5(3(4(x1)))))))))) 0(0(2(0(0(4(2(x1))))))) -> 1(3(4(1(0(0(5(5(2(4(x1)))))))))) 0(0(4(3(0(1(5(x1))))))) -> 2(5(4(5(1(2(4(1(0(5(x1)))))))))) 0(1(0(0(0(4(2(x1))))))) -> 3(2(5(2(0(4(2(3(4(1(x1)))))))))) 0(1(0(4(0(0(2(x1))))))) -> 5(5(3(4(2(0(5(2(2(3(x1)))))))))) 0(2(3(1(0(2(4(x1))))))) -> 3(3(1(0(4(2(1(2(4(3(x1)))))))))) 0(2(3(1(5(0(1(x1))))))) -> 4(5(5(3(4(1(0(4(2(2(x1)))))))))) 2(4(5(1(0(0(2(x1))))))) -> 4(3(1(2(1(3(4(1(5(1(x1)))))))))) 3(0(3(0(5(2(5(x1))))))) -> 3(3(1(2(3(3(1(2(1(5(x1)))))))))) 3(3(0(4(0(0(4(x1))))))) -> 3(3(3(1(1(5(4(4(5(4(x1)))))))))) 3(3(5(0(1(5(2(x1))))))) -> 3(1(5(1(1(2(4(3(1(3(x1)))))))))) 5(0(2(0(5(1(5(x1))))))) -> 5(0(0(0(0(3(2(4(3(5(x1)))))))))) 5(1(1(3(1(0(4(x1))))))) -> 5(1(2(3(1(4(4(5(3(3(x1)))))))))) 5(2(3(5(2(0(2(x1))))))) -> 5(2(2(4(4(1(2(5(5(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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)) 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(1(x1)) -> 2(0(3(4(3(1(1(2(1(3(x1)))))))))) 0(1(x1)) -> 4(4(2(4(2(3(3(1(3(3(x1)))))))))) 0(1(x1)) -> 4(4(4(2(1(0(3(4(1(4(x1)))))))))) 0(0(1(x1))) -> 5(4(2(1(4(1(3(4(2(2(x1)))))))))) 0(2(0(x1))) -> 3(0(0(3(2(2(2(4(1(3(x1)))))))))) 1(5(1(x1))) -> 1(5(3(2(4(3(0(3(2(4(x1)))))))))) 0(1(2(5(x1)))) -> 1(4(3(0(3(4(4(2(3(5(x1)))))))))) 0(2(2(1(x1)))) -> 3(0(5(2(1(1(2(2(3(1(x1)))))))))) 0(4(5(4(x1)))) -> 1(1(5(4(1(2(4(1(4(3(x1)))))))))) 4(5(1(0(x1)))) -> 2(0(3(2(4(2(5(2(2(2(x1)))))))))) 0(2(0(0(1(x1))))) -> 0(2(2(0(5(4(4(3(4(1(x1)))))))))) 0(2(0(4(1(x1))))) -> 2(3(0(3(2(4(2(0(4(1(x1)))))))))) 3(0(0(1(5(x1))))) -> 3(2(4(3(2(2(1(2(0(5(x1)))))))))) 3(0(2(2(1(x1))))) -> 3(2(0(5(4(1(3(1(4(3(x1)))))))))) 3(3(5(0(1(x1))))) -> 3(2(4(1(1(0(5(1(2(4(x1)))))))))) 5(3(3(5(1(x1))))) -> 5(3(2(5(4(4(1(1(4(4(x1)))))))))) 5(4(5(0(2(x1))))) -> 5(4(0(3(4(5(2(1(3(1(x1)))))))))) 0(0(0(2(2(5(x1)))))) -> 1(3(2(1(4(2(0(1(4(0(x1)))))))))) 0(0(2(0(2(4(x1)))))) -> 5(2(3(4(4(5(2(1(4(1(x1)))))))))) 0(0(2(2(5(4(x1)))))) -> 1(4(5(5(5(2(4(4(4(3(x1)))))))))) 0(1(0(4(0(1(x1)))))) -> 0(3(0(3(3(3(0(1(1(4(x1)))))))))) 0(2(4(0(2(4(x1)))))) -> 5(0(5(4(2(2(0(1(4(2(x1)))))))))) 0(2(5(2(5(2(x1)))))) -> 3(2(1(1(1(1(5(1(4(3(x1)))))))))) 0(3(0(5(0(2(x1)))))) -> 4(2(1(2(4(4(0(0(1(1(x1)))))))))) 0(4(0(0(4(2(x1)))))) -> 0(2(2(3(5(4(4(3(1(4(x1)))))))))) 1(1(0(4(0(4(x1)))))) -> 1(2(1(3(4(4(5(4(4(2(x1)))))))))) 2(0(0(1(3(5(x1)))))) -> 4(2(4(0(2(4(4(1(3(5(x1)))))))))) 2(5(0(0(3(5(x1)))))) -> 4(1(3(3(3(5(4(0(5(5(x1)))))))))) 2(5(0(4(3(0(x1)))))) -> 2(4(1(2(2(0(5(5(2(0(x1)))))))))) 2(5(4(0(0(1(x1)))))) -> 4(4(2(4(1(0(1(1(0(1(x1)))))))))) 3(1(5(1(1(5(x1)))))) -> 3(3(0(3(3(0(3(2(2(0(x1)))))))))) 3(5(1(0(5(1(x1)))))) -> 5(3(0(1(3(1(0(5(3(4(x1)))))))))) 0(0(2(0(0(4(2(x1))))))) -> 1(3(4(1(0(0(5(5(2(4(x1)))))))))) 0(0(4(3(0(1(5(x1))))))) -> 2(5(4(5(1(2(4(1(0(5(x1)))))))))) 0(1(0(0(0(4(2(x1))))))) -> 3(2(5(2(0(4(2(3(4(1(x1)))))))))) 0(1(0(4(0(0(2(x1))))))) -> 5(5(3(4(2(0(5(2(2(3(x1)))))))))) 0(2(3(1(0(2(4(x1))))))) -> 3(3(1(0(4(2(1(2(4(3(x1)))))))))) 0(2(3(1(5(0(1(x1))))))) -> 4(5(5(3(4(1(0(4(2(2(x1)))))))))) 2(4(5(1(0(0(2(x1))))))) -> 4(3(1(2(1(3(4(1(5(1(x1)))))))))) 3(0(3(0(5(2(5(x1))))))) -> 3(3(1(2(3(3(1(2(1(5(x1)))))))))) 3(3(0(4(0(0(4(x1))))))) -> 3(3(3(1(1(5(4(4(5(4(x1)))))))))) 3(3(5(0(1(5(2(x1))))))) -> 3(1(5(1(1(2(4(3(1(3(x1)))))))))) 5(0(2(0(5(1(5(x1))))))) -> 5(0(0(0(0(3(2(4(3(5(x1)))))))))) 5(1(1(3(1(0(4(x1))))))) -> 5(1(2(3(1(4(4(5(3(3(x1)))))))))) 5(2(3(5(2(0(2(x1))))))) -> 5(2(2(4(4(1(2(5(5(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(x1)) -> 2(0(3(4(3(1(1(2(1(3(x1)))))))))) 0(1(x1)) -> 4(4(2(4(2(3(3(1(3(3(x1)))))))))) 0(1(x1)) -> 4(4(4(2(1(0(3(4(1(4(x1)))))))))) 0(0(1(x1))) -> 5(4(2(1(4(1(3(4(2(2(x1)))))))))) 0(2(0(x1))) -> 3(0(0(3(2(2(2(4(1(3(x1)))))))))) 1(5(1(x1))) -> 1(5(3(2(4(3(0(3(2(4(x1)))))))))) 0(1(2(5(x1)))) -> 1(4(3(0(3(4(4(2(3(5(x1)))))))))) 0(2(2(1(x1)))) -> 3(0(5(2(1(1(2(2(3(1(x1)))))))))) 0(4(5(4(x1)))) -> 1(1(5(4(1(2(4(1(4(3(x1)))))))))) 4(5(1(0(x1)))) -> 2(0(3(2(4(2(5(2(2(2(x1)))))))))) 0(2(0(0(1(x1))))) -> 0(2(2(0(5(4(4(3(4(1(x1)))))))))) 0(2(0(4(1(x1))))) -> 2(3(0(3(2(4(2(0(4(1(x1)))))))))) 3(0(0(1(5(x1))))) -> 3(2(4(3(2(2(1(2(0(5(x1)))))))))) 3(0(2(2(1(x1))))) -> 3(2(0(5(4(1(3(1(4(3(x1)))))))))) 3(3(5(0(1(x1))))) -> 3(2(4(1(1(0(5(1(2(4(x1)))))))))) 5(3(3(5(1(x1))))) -> 5(3(2(5(4(4(1(1(4(4(x1)))))))))) 5(4(5(0(2(x1))))) -> 5(4(0(3(4(5(2(1(3(1(x1)))))))))) 0(0(0(2(2(5(x1)))))) -> 1(3(2(1(4(2(0(1(4(0(x1)))))))))) 0(0(2(0(2(4(x1)))))) -> 5(2(3(4(4(5(2(1(4(1(x1)))))))))) 0(0(2(2(5(4(x1)))))) -> 1(4(5(5(5(2(4(4(4(3(x1)))))))))) 0(1(0(4(0(1(x1)))))) -> 0(3(0(3(3(3(0(1(1(4(x1)))))))))) 0(2(4(0(2(4(x1)))))) -> 5(0(5(4(2(2(0(1(4(2(x1)))))))))) 0(2(5(2(5(2(x1)))))) -> 3(2(1(1(1(1(5(1(4(3(x1)))))))))) 0(3(0(5(0(2(x1)))))) -> 4(2(1(2(4(4(0(0(1(1(x1)))))))))) 0(4(0(0(4(2(x1)))))) -> 0(2(2(3(5(4(4(3(1(4(x1)))))))))) 1(1(0(4(0(4(x1)))))) -> 1(2(1(3(4(4(5(4(4(2(x1)))))))))) 2(0(0(1(3(5(x1)))))) -> 4(2(4(0(2(4(4(1(3(5(x1)))))))))) 2(5(0(0(3(5(x1)))))) -> 4(1(3(3(3(5(4(0(5(5(x1)))))))))) 2(5(0(4(3(0(x1)))))) -> 2(4(1(2(2(0(5(5(2(0(x1)))))))))) 2(5(4(0(0(1(x1)))))) -> 4(4(2(4(1(0(1(1(0(1(x1)))))))))) 3(1(5(1(1(5(x1)))))) -> 3(3(0(3(3(0(3(2(2(0(x1)))))))))) 3(5(1(0(5(1(x1)))))) -> 5(3(0(1(3(1(0(5(3(4(x1)))))))))) 0(0(2(0(0(4(2(x1))))))) -> 1(3(4(1(0(0(5(5(2(4(x1)))))))))) 0(0(4(3(0(1(5(x1))))))) -> 2(5(4(5(1(2(4(1(0(5(x1)))))))))) 0(1(0(0(0(4(2(x1))))))) -> 3(2(5(2(0(4(2(3(4(1(x1)))))))))) 0(1(0(4(0(0(2(x1))))))) -> 5(5(3(4(2(0(5(2(2(3(x1)))))))))) 0(2(3(1(0(2(4(x1))))))) -> 3(3(1(0(4(2(1(2(4(3(x1)))))))))) 0(2(3(1(5(0(1(x1))))))) -> 4(5(5(3(4(1(0(4(2(2(x1)))))))))) 2(4(5(1(0(0(2(x1))))))) -> 4(3(1(2(1(3(4(1(5(1(x1)))))))))) 3(0(3(0(5(2(5(x1))))))) -> 3(3(1(2(3(3(1(2(1(5(x1)))))))))) 3(3(0(4(0(0(4(x1))))))) -> 3(3(3(1(1(5(4(4(5(4(x1)))))))))) 3(3(5(0(1(5(2(x1))))))) -> 3(1(5(1(1(2(4(3(1(3(x1)))))))))) 5(0(2(0(5(1(5(x1))))))) -> 5(0(0(0(0(3(2(4(3(5(x1)))))))))) 5(1(1(3(1(0(4(x1))))))) -> 5(1(2(3(1(4(4(5(3(3(x1)))))))))) 5(2(3(5(2(0(2(x1))))))) -> 5(2(2(4(4(1(2(5(5(2(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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)) 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. "[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, 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(584,126,[3_1|3]), (584,171,[3_1|3]), (584,189,[3_1|3]), (584,198,[3_1|3]), (584,288,[3_1|3]), (584,315,[3_1|3]), (584,324,[3_1|3]), (584,442,[3_1|3]), (585,586,[4_1|3]), (586,587,[2_1|3]), (587,588,[4_1|3]), (588,589,[2_1|3]), (589,590,[3_1|3]), (590,591,[3_1|3]), (591,592,[1_1|3]), (592,593,[3_1|3]), (593,126,[3_1|3]), (593,171,[3_1|3]), (593,189,[3_1|3]), (593,198,[3_1|3]), (593,288,[3_1|3]), (593,315,[3_1|3]), (593,324,[3_1|3]), (593,442,[3_1|3]), (594,595,[4_1|3]), (595,596,[4_1|3]), (596,597,[2_1|3]), (597,598,[1_1|3]), (598,599,[0_1|3]), (599,600,[3_1|3]), (600,601,[4_1|3]), (601,602,[1_1|3]), (602,126,[4_1|3]), (602,171,[4_1|3]), (602,189,[4_1|3]), (602,198,[4_1|3]), (602,288,[4_1|3]), (602,315,[4_1|3]), (602,324,[4_1|3]), (602,442,[4_1|3]), (603,604,[0_1|3]), (604,605,[3_1|3]), (605,606,[4_1|3]), (606,607,[3_1|3]), (607,608,[1_1|3]), (608,609,[1_1|3]), (609,610,[2_1|3]), (610,611,[1_1|3]), (611,259,[3_1|3]), (612,613,[4_1|3]), (613,614,[2_1|3]), (614,615,[4_1|3]), (615,616,[2_1|3]), (616,617,[3_1|3]), (617,618,[3_1|3]), (618,619,[1_1|3]), (619,620,[3_1|3]), (620,259,[3_1|3]), (621,622,[4_1|3]), (622,623,[4_1|3]), (623,624,[2_1|3]), (624,625,[1_1|3]), (625,626,[0_1|3]), (626,627,[3_1|3]), (627,628,[4_1|3]), (628,629,[1_1|3]), (629,259,[4_1|3]), (630,631,[5_1|3]), (631,632,[3_1|3]), (632,633,[2_1|3]), (633,634,[4_1|3]), (634,635,[3_1|3]), (635,636,[0_1|3]), (636,637,[3_1|3]), (637,638,[2_1|3]), (638,268,[4_1|3]), (639,640,[4_1|3]), (640,641,[2_1|3]), (641,642,[1_1|3]), (642,643,[4_1|3]), (643,644,[1_1|3]), (644,645,[3_1|3]), (645,646,[4_1|3]), (646,647,[2_1|3]), (647,314,[2_1|3]), (647,324,[2_1|3]), (648,649,[0_1|3]), (649,650,[3_1|3]), (650,651,[4_1|3]), (651,652,[3_1|3]), (652,653,[1_1|3]), (653,654,[1_1|3]), (654,655,[2_1|3]), (655,656,[1_1|3]), (656,314,[3_1|3]), (656,324,[3_1|3]), (656,396,[3_1|2]), (657,658,[4_1|3]), (658,659,[2_1|3]), (659,660,[4_1|3]), (660,661,[2_1|3]), (661,662,[3_1|3]), (662,663,[3_1|3]), (663,664,[1_1|3]), (664,665,[3_1|3]), (665,314,[3_1|3]), (665,324,[3_1|3]), (665,396,[3_1|2]), (666,667,[4_1|3]), (667,668,[4_1|3]), (668,669,[2_1|3]), (669,670,[1_1|3]), (670,671,[0_1|3]), (671,672,[3_1|3]), (672,673,[4_1|3]), (673,674,[1_1|3]), (674,314,[4_1|3]), (674,324,[4_1|3]), (675,676,[5_1|3]), (676,677,[3_1|3]), (677,678,[2_1|3]), (678,679,[4_1|3]), (679,680,[3_1|3]), (680,681,[0_1|3]), (681,682,[3_1|3]), (682,683,[2_1|3]), (683,126,[4_1|3]), (683,171,[4_1|3]), (683,189,[4_1|3]), (683,198,[4_1|3]), (683,288,[4_1|3]), (683,315,[4_1|3]), (683,324,[4_1|3]), (683,442,[4_1|3]), (684,685,[5_1|3]), (685,686,[3_1|3]), (686,687,[2_1|3]), (687,688,[4_1|3]), (688,689,[3_1|3]), (689,690,[0_1|3]), (690,691,[3_1|3]), (691,692,[2_1|3]), (692,381,[4_1|3]), (693,694,[0_1|3]), (694,695,[3_1|3]), (695,696,[4_1|3]), (696,697,[3_1|3]), (697,698,[1_1|3]), (698,699,[1_1|3]), (699,700,[2_1|3]), (700,701,[1_1|3]), (701,408,[3_1|3]), (702,703,[4_1|3]), (703,704,[2_1|3]), (704,705,[4_1|3]), (705,706,[2_1|3]), (706,707,[3_1|3]), (707,708,[3_1|3]), (708,709,[1_1|3]), (709,710,[3_1|3]), (710,408,[3_1|3]), (711,712,[4_1|3]), (712,713,[4_1|3]), (713,714,[2_1|3]), (714,715,[1_1|3]), (715,716,[0_1|3]), (716,717,[3_1|3]), (717,718,[4_1|3]), (718,719,[1_1|3]), (719,408,[4_1|3]), (720,721,[3_1|3]), (721,722,[2_1|3]), (722,723,[5_1|3]), (723,724,[4_1|3]), (724,725,[4_1|3]), (725,726,[1_1|3]), (726,727,[1_1|3]), (727,728,[4_1|3]), (728,442,[4_1|3]), (729,730,[0_1|3]), (730,731,[3_1|3]), (731,732,[4_1|3]), (732,733,[3_1|3]), (733,734,[1_1|3]), (734,735,[1_1|3]), (735,736,[2_1|3]), (736,737,[1_1|3]), (737,492,[3_1|3]), (738,739,[4_1|3]), (739,740,[2_1|3]), (740,741,[4_1|3]), (741,742,[2_1|3]), (742,743,[3_1|3]), (743,744,[3_1|3]), (744,745,[1_1|3]), (745,746,[3_1|3]), (746,492,[3_1|3]), (747,748,[4_1|3]), (748,749,[4_1|3]), (749,750,[2_1|3]), (750,751,[1_1|3]), (751,752,[0_1|3]), (752,753,[3_1|3]), (753,754,[4_1|3]), (754,755,[1_1|3]), (755,492,[4_1|3]), (756,757,[0_1|3]), (757,758,[3_1|3]), (758,759,[4_1|3]), (759,760,[3_1|3]), (760,761,[1_1|3]), (761,762,[1_1|3]), (762,763,[2_1|3]), (763,764,[1_1|3]), (764,98,[3_1|3]), (764,126,[3_1|3]), (764,171,[3_1|3]), (764,189,[3_1|3]), (764,198,[3_1|3]), (764,288,[3_1|3]), (764,315,[3_1|3]), (764,324,[3_1|3]), (764,531,[3_1|3]), (764,342,[3_1|2]), (764,351,[3_1|2]), (764,360,[3_1|2]), (764,369,[3_1|2]), (764,378,[3_1|2]), (764,387,[3_1|2]), (764,396,[3_1|2]), (764,405,[5_1|2]), (765,766,[4_1|3]), (766,767,[2_1|3]), (767,768,[4_1|3]), (768,769,[2_1|3]), (769,770,[3_1|3]), (770,771,[3_1|3]), (771,772,[1_1|3]), (772,773,[3_1|3]), (772,369,[3_1|2]), (772,378,[3_1|2]), (772,387,[3_1|2]), (773,98,[3_1|3]), (773,126,[3_1|3]), (773,171,[3_1|3]), (773,189,[3_1|3]), (773,198,[3_1|3]), (773,288,[3_1|3]), (773,315,[3_1|3]), (773,324,[3_1|3]), (773,531,[3_1|3]), (773,342,[3_1|2]), (773,351,[3_1|2]), (773,360,[3_1|2]), (773,369,[3_1|2]), (773,378,[3_1|2]), (773,387,[3_1|2]), (773,396,[3_1|2]), (773,405,[5_1|2]), (774,775,[4_1|3]), (775,776,[4_1|3]), (776,777,[2_1|3]), (777,778,[1_1|3]), (778,779,[0_1|3]), (779,780,[3_1|3]), (780,781,[4_1|3]), (781,782,[1_1|3]), (782,98,[4_1|3]), (782,126,[4_1|3]), (782,171,[4_1|3]), (782,189,[4_1|3]), (782,198,[4_1|3]), (782,288,[4_1|3]), (782,315,[4_1|3]), (782,324,[4_1|3]), (782,531,[4_1|3]), (782,333,[2_1|2]), (783,784,[4_1|3]), (784,785,[3_1|3]), (785,786,[0_1|3]), (786,787,[3_1|3]), (787,788,[4_1|3]), (788,789,[4_1|3]), (789,790,[2_1|3]), (790,791,[3_1|3]), (791,208,[5_1|3]), (792,793,[5_1|3]), (793,794,[3_1|3]), (794,795,[2_1|3]), (795,796,[4_1|3]), (796,797,[3_1|3]), (797,798,[0_1|3]), (798,799,[3_1|3]), (799,800,[2_1|3]), (799,495,[4_1|2]), (800,98,[4_1|3]), (800,99,[4_1|3]), (800,207,[4_1|3]), (800,234,[4_1|3]), (800,333,[4_1|3, 2_1|2]), (800,477,[4_1|3]), (800,226,[4_1|3]), (800,298,[4_1|3]), (800,315,[4_1|3]), (800,324,[4_1|3]), (800,531,[4_1|3]), (800,442,[4_1|3]), (801,802,[0_1|3]), (802,803,[0_1|3]), (803,804,[3_1|3]), (804,805,[2_1|3]), (805,806,[2_1|3]), (806,807,[2_1|3]), (807,808,[4_1|3]), (808,809,[1_1|3]), (809,649,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)