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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 102 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: 5(4(0(x1))) -> 0(2(2(1(1(3(0(4(3(0(x1)))))))))) 4(1(4(0(x1)))) -> 4(0(2(2(1(1(0(0(5(2(x1)))))))))) 5(4(4(5(x1)))) -> 1(1(2(3(2(0(3(1(2(0(x1)))))))))) 0(3(4(0(4(x1))))) -> 2(0(2(1(3(0(4(2(3(4(x1)))))))))) 0(3(4(5(4(x1))))) -> 1(0(1(2(4(2(3(1(2(4(x1)))))))))) 1(1(5(5(4(x1))))) -> 1(3(0(0(0(2(3(3(4(1(x1)))))))))) 1(4(5(5(4(x1))))) -> 2(0(1(0(2(3(4(2(5(1(x1)))))))))) 2(5(4(5(0(x1))))) -> 1(1(2(4(2(0(2(0(5(0(x1)))))))))) 3(1(5(1(1(x1))))) -> 3(5(0(5(3(1(2(5(5(1(x1)))))))))) 4(0(0(4(5(x1))))) -> 3(3(2(0(0(1(4(3(3(5(x1)))))))))) 4(1(4(2(0(x1))))) -> 0(4(2(1(3(1(0(3(3(0(x1)))))))))) 5(4(1(4(1(x1))))) -> 0(1(0(2(0(1(2(4(4(2(x1)))))))))) 0(3(4(1(0(4(x1)))))) -> 2(4(1(4(3(3(3(0(0(4(x1)))))))))) 0(3(4(5(1(3(x1)))))) -> 4(4(4(3(1(3(3(3(0(3(x1)))))))))) 0(4(0(5(1(4(x1)))))) -> 0(0(3(3(1(2(0(3(2(4(x1)))))))))) 1(3(2(5(4(0(x1)))))) -> 3(0(4(0(0(4(3(1(4(0(x1)))))))))) 1(3(5(4(5(4(x1)))))) -> 2(3(2(0(1(2(5(2(1(5(x1)))))))))) 1(4(2(1(4(0(x1)))))) -> 0(3(3(0(2(0(3(2(0(5(x1)))))))))) 1(5(3(4(1(3(x1)))))) -> 1(1(3(1(1(2(0(1(2(3(x1)))))))))) 1(5(4(5(0(4(x1)))))) -> 4(3(2(3(2(0(1(3(4(0(x1)))))))))) 1(5(5(4(1(3(x1)))))) -> 3(5(1(3(2(1(1(4(3(3(x1)))))))))) 2(3(5(4(5(1(x1)))))) -> 5(4(3(5(5(5(1(2(5(1(x1)))))))))) 3(4(5(3(5(0(x1)))))) -> 3(4(3(3(2(3(5(3(3(0(x1)))))))))) 5(4(5(4(4(5(x1)))))) -> 1(0(3(2(0(3(3(4(4(5(x1)))))))))) 0(2(5(2(4(0(4(x1))))))) -> 4(3(0(1(2(1(0(2(0(1(x1)))))))))) 1(1(5(5(4(0(5(x1))))))) -> 4(2(2(2(1(1(4(2(4(2(x1)))))))))) 1(4(0(3(1(4(1(x1))))))) -> 0(4(1(3(3(3(4(2(2(0(x1)))))))))) 1(4(1(0(4(0(3(x1))))))) -> 2(4(4(3(3(2(4(0(2(3(x1)))))))))) 1(4(4(5(1(0(2(x1))))))) -> 2(3(0(1(4(0(1(4(2(3(x1)))))))))) 1(4(4(5(4(2(4(x1))))))) -> 1(0(0(1(5(1(2(1(5(4(x1)))))))))) 1(5(4(1(1(3(3(x1))))))) -> 0(5(3(2(0(1(3(3(2(3(x1)))))))))) 1(5(4(4(1(4(5(x1))))))) -> 2(3(2(0(2(5(3(4(3(5(x1)))))))))) 2(0(2(5(4(5(1(x1))))))) -> 0(0(5(3(0(2(1(2(4(0(x1)))))))))) 2(3(5(4(4(0(5(x1))))))) -> 2(5(1(3(0(2(3(0(2(5(x1)))))))))) 2(4(2(4(4(1(3(x1))))))) -> 0(3(2(2(1(0(1(1(0(3(x1)))))))))) 2(5(4(4(5(4(1(x1))))))) -> 2(2(1(2(4(5(3(2(4(2(x1)))))))))) 3(5(4(1(4(1(5(x1))))))) -> 3(1(5(2(5(3(2(5(2(5(x1)))))))))) 4(2(4(4(4(5(4(x1))))))) -> 4(3(3(4(1(0(4(5(5(4(x1)))))))))) 4(5(4(0(4(4(0(x1))))))) -> 0(3(1(3(4(0(0(5(5(5(x1)))))))))) 5(4(0(3(2(3(5(x1))))))) -> 1(0(3(2(1(0(1(2(3(5(x1)))))))))) 5(4(0(3(2(5(0(x1))))))) -> 2(4(3(2(4(1(1(0(5(5(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(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: 5(4(0(x1))) -> 0(2(2(1(1(3(0(4(3(0(x1)))))))))) 4(1(4(0(x1)))) -> 4(0(2(2(1(1(0(0(5(2(x1)))))))))) 5(4(4(5(x1)))) -> 1(1(2(3(2(0(3(1(2(0(x1)))))))))) 0(3(4(0(4(x1))))) -> 2(0(2(1(3(0(4(2(3(4(x1)))))))))) 0(3(4(5(4(x1))))) -> 1(0(1(2(4(2(3(1(2(4(x1)))))))))) 1(1(5(5(4(x1))))) -> 1(3(0(0(0(2(3(3(4(1(x1)))))))))) 1(4(5(5(4(x1))))) -> 2(0(1(0(2(3(4(2(5(1(x1)))))))))) 2(5(4(5(0(x1))))) -> 1(1(2(4(2(0(2(0(5(0(x1)))))))))) 3(1(5(1(1(x1))))) -> 3(5(0(5(3(1(2(5(5(1(x1)))))))))) 4(0(0(4(5(x1))))) -> 3(3(2(0(0(1(4(3(3(5(x1)))))))))) 4(1(4(2(0(x1))))) -> 0(4(2(1(3(1(0(3(3(0(x1)))))))))) 5(4(1(4(1(x1))))) -> 0(1(0(2(0(1(2(4(4(2(x1)))))))))) 0(3(4(1(0(4(x1)))))) -> 2(4(1(4(3(3(3(0(0(4(x1)))))))))) 0(3(4(5(1(3(x1)))))) -> 4(4(4(3(1(3(3(3(0(3(x1)))))))))) 0(4(0(5(1(4(x1)))))) -> 0(0(3(3(1(2(0(3(2(4(x1)))))))))) 1(3(2(5(4(0(x1)))))) -> 3(0(4(0(0(4(3(1(4(0(x1)))))))))) 1(3(5(4(5(4(x1)))))) -> 2(3(2(0(1(2(5(2(1(5(x1)))))))))) 1(4(2(1(4(0(x1)))))) -> 0(3(3(0(2(0(3(2(0(5(x1)))))))))) 1(5(3(4(1(3(x1)))))) -> 1(1(3(1(1(2(0(1(2(3(x1)))))))))) 1(5(4(5(0(4(x1)))))) -> 4(3(2(3(2(0(1(3(4(0(x1)))))))))) 1(5(5(4(1(3(x1)))))) -> 3(5(1(3(2(1(1(4(3(3(x1)))))))))) 2(3(5(4(5(1(x1)))))) -> 5(4(3(5(5(5(1(2(5(1(x1)))))))))) 3(4(5(3(5(0(x1)))))) -> 3(4(3(3(2(3(5(3(3(0(x1)))))))))) 5(4(5(4(4(5(x1)))))) -> 1(0(3(2(0(3(3(4(4(5(x1)))))))))) 0(2(5(2(4(0(4(x1))))))) -> 4(3(0(1(2(1(0(2(0(1(x1)))))))))) 1(1(5(5(4(0(5(x1))))))) -> 4(2(2(2(1(1(4(2(4(2(x1)))))))))) 1(4(0(3(1(4(1(x1))))))) -> 0(4(1(3(3(3(4(2(2(0(x1)))))))))) 1(4(1(0(4(0(3(x1))))))) -> 2(4(4(3(3(2(4(0(2(3(x1)))))))))) 1(4(4(5(1(0(2(x1))))))) -> 2(3(0(1(4(0(1(4(2(3(x1)))))))))) 1(4(4(5(4(2(4(x1))))))) -> 1(0(0(1(5(1(2(1(5(4(x1)))))))))) 1(5(4(1(1(3(3(x1))))))) -> 0(5(3(2(0(1(3(3(2(3(x1)))))))))) 1(5(4(4(1(4(5(x1))))))) -> 2(3(2(0(2(5(3(4(3(5(x1)))))))))) 2(0(2(5(4(5(1(x1))))))) -> 0(0(5(3(0(2(1(2(4(0(x1)))))))))) 2(3(5(4(4(0(5(x1))))))) -> 2(5(1(3(0(2(3(0(2(5(x1)))))))))) 2(4(2(4(4(1(3(x1))))))) -> 0(3(2(2(1(0(1(1(0(3(x1)))))))))) 2(5(4(4(5(4(1(x1))))))) -> 2(2(1(2(4(5(3(2(4(2(x1)))))))))) 3(5(4(1(4(1(5(x1))))))) -> 3(1(5(2(5(3(2(5(2(5(x1)))))))))) 4(2(4(4(4(5(4(x1))))))) -> 4(3(3(4(1(0(4(5(5(4(x1)))))))))) 4(5(4(0(4(4(0(x1))))))) -> 0(3(1(3(4(0(0(5(5(5(x1)))))))))) 5(4(0(3(2(3(5(x1))))))) -> 1(0(3(2(1(0(1(2(3(5(x1)))))))))) 5(4(0(3(2(5(0(x1))))))) -> 2(4(3(2(4(1(1(0(5(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(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: 5(4(0(x1))) -> 0(2(2(1(1(3(0(4(3(0(x1)))))))))) 4(1(4(0(x1)))) -> 4(0(2(2(1(1(0(0(5(2(x1)))))))))) 5(4(4(5(x1)))) -> 1(1(2(3(2(0(3(1(2(0(x1)))))))))) 0(3(4(0(4(x1))))) -> 2(0(2(1(3(0(4(2(3(4(x1)))))))))) 0(3(4(5(4(x1))))) -> 1(0(1(2(4(2(3(1(2(4(x1)))))))))) 1(1(5(5(4(x1))))) -> 1(3(0(0(0(2(3(3(4(1(x1)))))))))) 1(4(5(5(4(x1))))) -> 2(0(1(0(2(3(4(2(5(1(x1)))))))))) 2(5(4(5(0(x1))))) -> 1(1(2(4(2(0(2(0(5(0(x1)))))))))) 3(1(5(1(1(x1))))) -> 3(5(0(5(3(1(2(5(5(1(x1)))))))))) 4(0(0(4(5(x1))))) -> 3(3(2(0(0(1(4(3(3(5(x1)))))))))) 4(1(4(2(0(x1))))) -> 0(4(2(1(3(1(0(3(3(0(x1)))))))))) 5(4(1(4(1(x1))))) -> 0(1(0(2(0(1(2(4(4(2(x1)))))))))) 0(3(4(1(0(4(x1)))))) -> 2(4(1(4(3(3(3(0(0(4(x1)))))))))) 0(3(4(5(1(3(x1)))))) -> 4(4(4(3(1(3(3(3(0(3(x1)))))))))) 0(4(0(5(1(4(x1)))))) -> 0(0(3(3(1(2(0(3(2(4(x1)))))))))) 1(3(2(5(4(0(x1)))))) -> 3(0(4(0(0(4(3(1(4(0(x1)))))))))) 1(3(5(4(5(4(x1)))))) -> 2(3(2(0(1(2(5(2(1(5(x1)))))))))) 1(4(2(1(4(0(x1)))))) -> 0(3(3(0(2(0(3(2(0(5(x1)))))))))) 1(5(3(4(1(3(x1)))))) -> 1(1(3(1(1(2(0(1(2(3(x1)))))))))) 1(5(4(5(0(4(x1)))))) -> 4(3(2(3(2(0(1(3(4(0(x1)))))))))) 1(5(5(4(1(3(x1)))))) -> 3(5(1(3(2(1(1(4(3(3(x1)))))))))) 2(3(5(4(5(1(x1)))))) -> 5(4(3(5(5(5(1(2(5(1(x1)))))))))) 3(4(5(3(5(0(x1)))))) -> 3(4(3(3(2(3(5(3(3(0(x1)))))))))) 5(4(5(4(4(5(x1)))))) -> 1(0(3(2(0(3(3(4(4(5(x1)))))))))) 0(2(5(2(4(0(4(x1))))))) -> 4(3(0(1(2(1(0(2(0(1(x1)))))))))) 1(1(5(5(4(0(5(x1))))))) -> 4(2(2(2(1(1(4(2(4(2(x1)))))))))) 1(4(0(3(1(4(1(x1))))))) -> 0(4(1(3(3(3(4(2(2(0(x1)))))))))) 1(4(1(0(4(0(3(x1))))))) -> 2(4(4(3(3(2(4(0(2(3(x1)))))))))) 1(4(4(5(1(0(2(x1))))))) -> 2(3(0(1(4(0(1(4(2(3(x1)))))))))) 1(4(4(5(4(2(4(x1))))))) -> 1(0(0(1(5(1(2(1(5(4(x1)))))))))) 1(5(4(1(1(3(3(x1))))))) -> 0(5(3(2(0(1(3(3(2(3(x1)))))))))) 1(5(4(4(1(4(5(x1))))))) -> 2(3(2(0(2(5(3(4(3(5(x1)))))))))) 2(0(2(5(4(5(1(x1))))))) -> 0(0(5(3(0(2(1(2(4(0(x1)))))))))) 2(3(5(4(4(0(5(x1))))))) -> 2(5(1(3(0(2(3(0(2(5(x1)))))))))) 2(4(2(4(4(1(3(x1))))))) -> 0(3(2(2(1(0(1(1(0(3(x1)))))))))) 2(5(4(4(5(4(1(x1))))))) -> 2(2(1(2(4(5(3(2(4(2(x1)))))))))) 3(5(4(1(4(1(5(x1))))))) -> 3(1(5(2(5(3(2(5(2(5(x1)))))))))) 4(2(4(4(4(5(4(x1))))))) -> 4(3(3(4(1(0(4(5(5(4(x1)))))))))) 4(5(4(0(4(4(0(x1))))))) -> 0(3(1(3(4(0(0(5(5(5(x1)))))))))) 5(4(0(3(2(3(5(x1))))))) -> 1(0(3(2(1(0(1(2(3(5(x1)))))))))) 5(4(0(3(2(5(0(x1))))))) -> 2(4(3(2(4(1(1(0(5(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(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: 5(4(0(x1))) -> 0(2(2(1(1(3(0(4(3(0(x1)))))))))) 4(1(4(0(x1)))) -> 4(0(2(2(1(1(0(0(5(2(x1)))))))))) 5(4(4(5(x1)))) -> 1(1(2(3(2(0(3(1(2(0(x1)))))))))) 0(3(4(0(4(x1))))) -> 2(0(2(1(3(0(4(2(3(4(x1)))))))))) 0(3(4(5(4(x1))))) -> 1(0(1(2(4(2(3(1(2(4(x1)))))))))) 1(1(5(5(4(x1))))) -> 1(3(0(0(0(2(3(3(4(1(x1)))))))))) 1(4(5(5(4(x1))))) -> 2(0(1(0(2(3(4(2(5(1(x1)))))))))) 2(5(4(5(0(x1))))) -> 1(1(2(4(2(0(2(0(5(0(x1)))))))))) 3(1(5(1(1(x1))))) -> 3(5(0(5(3(1(2(5(5(1(x1)))))))))) 4(0(0(4(5(x1))))) -> 3(3(2(0(0(1(4(3(3(5(x1)))))))))) 4(1(4(2(0(x1))))) -> 0(4(2(1(3(1(0(3(3(0(x1)))))))))) 5(4(1(4(1(x1))))) -> 0(1(0(2(0(1(2(4(4(2(x1)))))))))) 0(3(4(1(0(4(x1)))))) -> 2(4(1(4(3(3(3(0(0(4(x1)))))))))) 0(3(4(5(1(3(x1)))))) -> 4(4(4(3(1(3(3(3(0(3(x1)))))))))) 0(4(0(5(1(4(x1)))))) -> 0(0(3(3(1(2(0(3(2(4(x1)))))))))) 1(3(2(5(4(0(x1)))))) -> 3(0(4(0(0(4(3(1(4(0(x1)))))))))) 1(3(5(4(5(4(x1)))))) -> 2(3(2(0(1(2(5(2(1(5(x1)))))))))) 1(4(2(1(4(0(x1)))))) -> 0(3(3(0(2(0(3(2(0(5(x1)))))))))) 1(5(3(4(1(3(x1)))))) -> 1(1(3(1(1(2(0(1(2(3(x1)))))))))) 1(5(4(5(0(4(x1)))))) -> 4(3(2(3(2(0(1(3(4(0(x1)))))))))) 1(5(5(4(1(3(x1)))))) -> 3(5(1(3(2(1(1(4(3(3(x1)))))))))) 2(3(5(4(5(1(x1)))))) -> 5(4(3(5(5(5(1(2(5(1(x1)))))))))) 3(4(5(3(5(0(x1)))))) -> 3(4(3(3(2(3(5(3(3(0(x1)))))))))) 5(4(5(4(4(5(x1)))))) -> 1(0(3(2(0(3(3(4(4(5(x1)))))))))) 0(2(5(2(4(0(4(x1))))))) -> 4(3(0(1(2(1(0(2(0(1(x1)))))))))) 1(1(5(5(4(0(5(x1))))))) -> 4(2(2(2(1(1(4(2(4(2(x1)))))))))) 1(4(0(3(1(4(1(x1))))))) -> 0(4(1(3(3(3(4(2(2(0(x1)))))))))) 1(4(1(0(4(0(3(x1))))))) -> 2(4(4(3(3(2(4(0(2(3(x1)))))))))) 1(4(4(5(1(0(2(x1))))))) -> 2(3(0(1(4(0(1(4(2(3(x1)))))))))) 1(4(4(5(4(2(4(x1))))))) -> 1(0(0(1(5(1(2(1(5(4(x1)))))))))) 1(5(4(1(1(3(3(x1))))))) -> 0(5(3(2(0(1(3(3(2(3(x1)))))))))) 1(5(4(4(1(4(5(x1))))))) -> 2(3(2(0(2(5(3(4(3(5(x1)))))))))) 2(0(2(5(4(5(1(x1))))))) -> 0(0(5(3(0(2(1(2(4(0(x1)))))))))) 2(3(5(4(4(0(5(x1))))))) -> 2(5(1(3(0(2(3(0(2(5(x1)))))))))) 2(4(2(4(4(1(3(x1))))))) -> 0(3(2(2(1(0(1(1(0(3(x1)))))))))) 2(5(4(4(5(4(1(x1))))))) -> 2(2(1(2(4(5(3(2(4(2(x1)))))))))) 3(5(4(1(4(1(5(x1))))))) -> 3(1(5(2(5(3(2(5(2(5(x1)))))))))) 4(2(4(4(4(5(4(x1))))))) -> 4(3(3(4(1(0(4(5(5(4(x1)))))))))) 4(5(4(0(4(4(0(x1))))))) -> 0(3(1(3(4(0(0(5(5(5(x1)))))))))) 5(4(0(3(2(3(5(x1))))))) -> 1(0(3(2(1(0(1(2(3(5(x1)))))))))) 5(4(0(3(2(5(0(x1))))))) -> 2(4(3(2(4(1(1(0(5(5(x1)))))))))) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(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 3. The certificate found is represented by the following graph. 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{(96,97,[5_1|0, 4_1|0, 0_1|0, 1_1|0, 2_1|0, 3_1|0, encArg_1|0, encode_5_1|0, encode_4_1|0, encode_0_1|0, encode_2_1|0, encode_1_1|0, encode_3_1|0]), (96,98,[5_1|1, 4_1|1, 0_1|1, 1_1|1, 2_1|1, 3_1|1]), (96,99,[0_1|2]), (96,108,[1_1|2]), (96,117,[2_1|2]), (96,126,[1_1|2]), (96,135,[0_1|2]), (96,144,[1_1|2]), (96,153,[4_1|2]), (96,162,[0_1|2]), (96,171,[3_1|2]), (96,180,[4_1|2]), (96,189,[0_1|2]), (96,198,[2_1|2]), (96,207,[1_1|2]), (96,216,[4_1|2]), (96,225,[2_1|2]), (96,234,[0_1|2]), (96,243,[4_1|2]), (96,252,[1_1|2]), (96,261,[4_1|2]), (96,270,[2_1|2]), (96,279,[0_1|2]), (96,288,[0_1|2]), (96,297,[2_1|2]), (96,306,[2_1|2]), (96,315,[1_1|2]), (96,324,[3_1|2]), (96,333,[2_1|2]), (96,342,[1_1|2]), (96,351,[4_1|2]), (96,360,[0_1|2]), (96,369,[2_1|2]), (96,378,[3_1|2]), (96,387,[1_1|2]), (96,396,[2_1|2]), (96,405,[5_1|2]), (96,414,[2_1|2]), (96,423,[0_1|2]), (96,432,[0_1|2]), (96,441,[3_1|2]), (96,450,[3_1|2]), (96,459,[3_1|2]), (97,97,[cons_5_1|0, cons_4_1|0, cons_0_1|0, cons_1_1|0, 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(323,298,[4_1|2]), (323,162,[0_1|2]), (323,171,[3_1|2]), (323,189,[0_1|2]), (324,325,[0_1|2]), (325,326,[4_1|2]), (326,327,[0_1|2]), (327,328,[0_1|2]), (328,329,[4_1|2]), (329,330,[3_1|2]), (330,331,[1_1|2]), (330,288,[0_1|2]), (331,332,[4_1|2]), (331,171,[3_1|2]), (332,98,[0_1|2]), (332,99,[0_1|2]), (332,135,[0_1|2]), (332,162,[0_1|2]), (332,189,[0_1|2]), (332,234,[0_1|2]), (332,279,[0_1|2]), (332,288,[0_1|2]), (332,360,[0_1|2]), (332,423,[0_1|2]), (332,432,[0_1|2]), (332,154,[0_1|2]), (332,198,[2_1|2]), (332,207,[1_1|2]), (332,216,[4_1|2]), (332,225,[2_1|2]), (332,243,[4_1|2]), (333,334,[3_1|2]), (334,335,[2_1|2]), (335,336,[0_1|2]), (336,337,[1_1|2]), (337,338,[2_1|2]), (338,339,[5_1|2]), (339,340,[2_1|2]), (340,341,[1_1|2]), (340,342,[1_1|2]), (340,351,[4_1|2]), (340,360,[0_1|2]), (340,369,[2_1|2]), (340,378,[3_1|2]), (341,98,[5_1|2]), (341,153,[5_1|2]), (341,180,[5_1|2]), (341,216,[5_1|2]), (341,243,[5_1|2]), (341,261,[5_1|2]), (341,351,[5_1|2]), (341,406,[5_1|2]), (341,99,[0_1|2]), (341,108,[1_1|2]), (341,117,[2_1|2]), (341,126,[1_1|2]), (341,135,[0_1|2]), (341,144,[1_1|2]), (341,468,[0_1|3]), (342,343,[1_1|2]), (343,344,[3_1|2]), (344,345,[1_1|2]), (345,346,[1_1|2]), (346,347,[2_1|2]), (347,348,[0_1|2]), (348,349,[1_1|2]), (349,350,[2_1|2]), (349,405,[5_1|2]), (349,414,[2_1|2]), (350,98,[3_1|2]), (350,171,[3_1|2]), (350,324,[3_1|2]), (350,378,[3_1|2]), (350,441,[3_1|2]), (350,450,[3_1|2]), (350,459,[3_1|2]), (350,253,[3_1|2]), (351,352,[3_1|2]), (352,353,[2_1|2]), (353,354,[3_1|2]), (354,355,[2_1|2]), (355,356,[0_1|2]), (356,357,[1_1|2]), (357,358,[3_1|2]), (358,359,[4_1|2]), (358,171,[3_1|2]), (359,98,[0_1|2]), (359,153,[0_1|2]), (359,180,[0_1|2]), (359,216,[0_1|2, 4_1|2]), (359,243,[0_1|2, 4_1|2]), (359,261,[0_1|2]), (359,351,[0_1|2]), (359,163,[0_1|2]), (359,289,[0_1|2]), (359,198,[2_1|2]), (359,207,[1_1|2]), (359,225,[2_1|2]), (359,234,[0_1|2]), (360,361,[5_1|2]), (361,362,[3_1|2]), (362,363,[2_1|2]), (363,364,[0_1|2]), (364,365,[1_1|2]), (365,366,[3_1|2]), (366,367,[3_1|2]), (367,368,[2_1|2]), (367,405,[5_1|2]), (367,414,[2_1|2]), (368,98,[3_1|2]), (368,171,[3_1|2]), (368,324,[3_1|2]), (368,378,[3_1|2]), (368,441,[3_1|2]), (368,450,[3_1|2]), (368,459,[3_1|2]), (368,172,[3_1|2]), (369,370,[3_1|2]), (370,371,[2_1|2]), (371,372,[0_1|2]), (372,373,[2_1|2]), (373,374,[5_1|2]), (374,375,[3_1|2]), (375,376,[4_1|2]), (376,377,[3_1|2]), (376,459,[3_1|2]), (377,98,[5_1|2]), (377,405,[5_1|2]), (377,99,[0_1|2]), (377,108,[1_1|2]), (377,117,[2_1|2]), (377,126,[1_1|2]), (377,135,[0_1|2]), (377,144,[1_1|2]), (377,468,[0_1|3]), (378,379,[5_1|2]), (379,380,[1_1|2]), (380,381,[3_1|2]), (381,382,[2_1|2]), (382,383,[1_1|2]), (383,384,[1_1|2]), (384,385,[4_1|2]), (385,386,[3_1|2]), (386,98,[3_1|2]), (386,171,[3_1|2]), (386,324,[3_1|2]), (386,378,[3_1|2]), (386,441,[3_1|2]), (386,450,[3_1|2]), (386,459,[3_1|2]), (386,253,[3_1|2]), (387,388,[1_1|2]), (388,389,[2_1|2]), (389,390,[4_1|2]), (390,391,[2_1|2]), (391,392,[0_1|2]), (392,393,[2_1|2]), (393,394,[0_1|2]), (394,395,[5_1|2]), (395,98,[0_1|2]), (395,99,[0_1|2]), (395,135,[0_1|2]), (395,162,[0_1|2]), (395,189,[0_1|2]), (395,234,[0_1|2]), (395,279,[0_1|2]), (395,288,[0_1|2]), (395,360,[0_1|2]), (395,423,[0_1|2]), (395,432,[0_1|2]), (395,198,[2_1|2]), (395,207,[1_1|2]), (395,216,[4_1|2]), (395,225,[2_1|2]), (395,243,[4_1|2]), (396,397,[2_1|2]), (397,398,[1_1|2]), (398,399,[2_1|2]), (399,400,[4_1|2]), (400,401,[5_1|2]), (401,402,[3_1|2]), (402,403,[2_1|2]), (402,432,[0_1|2]), (403,404,[4_1|2]), (403,180,[4_1|2]), (404,98,[2_1|2]), (404,108,[2_1|2]), (404,126,[2_1|2]), (404,144,[2_1|2]), (404,207,[2_1|2]), (404,252,[2_1|2]), (404,315,[2_1|2]), (404,342,[2_1|2]), (404,387,[2_1|2, 1_1|2]), (404,396,[2_1|2]), (404,405,[5_1|2]), (404,414,[2_1|2]), (404,423,[0_1|2]), (404,432,[0_1|2]), (405,406,[4_1|2]), (406,407,[3_1|2]), (407,408,[5_1|2]), (408,409,[5_1|2]), (409,410,[5_1|2]), (410,411,[1_1|2]), (411,412,[2_1|2]), (412,413,[5_1|2]), (413,98,[1_1|2]), (413,108,[1_1|2]), (413,126,[1_1|2]), (413,144,[1_1|2]), (413,207,[1_1|2]), (413,252,[1_1|2]), (413,315,[1_1|2]), (413,342,[1_1|2]), (413,387,[1_1|2]), (413,261,[4_1|2]), (413,270,[2_1|2]), (413,279,[0_1|2]), (413,288,[0_1|2]), (413,297,[2_1|2]), (413,306,[2_1|2]), (413,324,[3_1|2]), (413,333,[2_1|2]), (413,351,[4_1|2]), (413,360,[0_1|2]), (413,369,[2_1|2]), (413,378,[3_1|2]), (414,415,[5_1|2]), (415,416,[1_1|2]), (416,417,[3_1|2]), (417,418,[0_1|2]), (418,419,[2_1|2]), (419,420,[3_1|2]), (420,421,[0_1|2]), (420,243,[4_1|2]), (421,422,[2_1|2]), (421,387,[1_1|2]), (421,396,[2_1|2]), (422,98,[5_1|2]), (422,405,[5_1|2]), (422,361,[5_1|2]), (422,99,[0_1|2]), (422,108,[1_1|2]), (422,117,[2_1|2]), (422,126,[1_1|2]), (422,135,[0_1|2]), (422,144,[1_1|2]), (422,468,[0_1|3]), (423,424,[0_1|2]), (424,425,[5_1|2]), (425,426,[3_1|2]), (426,427,[0_1|2]), (427,428,[2_1|2]), (428,429,[1_1|2]), (429,430,[2_1|2]), (430,431,[4_1|2]), (430,171,[3_1|2]), (431,98,[0_1|2]), (431,108,[0_1|2]), (431,126,[0_1|2]), (431,144,[0_1|2]), (431,207,[0_1|2, 1_1|2]), (431,252,[0_1|2]), (431,315,[0_1|2]), (431,342,[0_1|2]), (431,387,[0_1|2]), (431,198,[2_1|2]), (431,216,[4_1|2]), (431,225,[2_1|2]), (431,234,[0_1|2]), (431,243,[4_1|2]), (432,433,[3_1|2]), (433,434,[2_1|2]), (434,435,[2_1|2]), (435,436,[1_1|2]), (436,437,[0_1|2]), (437,438,[1_1|2]), (438,439,[1_1|2]), (439,440,[0_1|2]), (439,198,[2_1|2]), (439,207,[1_1|2]), (439,216,[4_1|2]), (439,225,[2_1|2]), (440,98,[3_1|2]), (440,171,[3_1|2]), (440,324,[3_1|2]), (440,378,[3_1|2]), (440,441,[3_1|2]), (440,450,[3_1|2]), (440,459,[3_1|2]), (440,253,[3_1|2]), (441,442,[5_1|2]), (442,443,[0_1|2]), (443,444,[5_1|2]), (444,445,[3_1|2]), (445,446,[1_1|2]), (446,447,[2_1|2]), (447,448,[5_1|2]), (448,449,[5_1|2]), (449,98,[1_1|2]), (449,108,[1_1|2]), (449,126,[1_1|2]), (449,144,[1_1|2]), (449,207,[1_1|2]), (449,252,[1_1|2]), (449,315,[1_1|2]), (449,342,[1_1|2]), (449,387,[1_1|2]), (449,127,[1_1|2]), (449,343,[1_1|2]), (449,388,[1_1|2]), (449,261,[4_1|2]), (449,270,[2_1|2]), (449,279,[0_1|2]), (449,288,[0_1|2]), (449,297,[2_1|2]), (449,306,[2_1|2]), (449,324,[3_1|2]), (449,333,[2_1|2]), (449,351,[4_1|2]), (449,360,[0_1|2]), (449,369,[2_1|2]), (449,378,[3_1|2]), (450,451,[4_1|2]), (451,452,[3_1|2]), (452,453,[3_1|2]), (453,454,[2_1|2]), (454,455,[3_1|2]), (455,456,[5_1|2]), (456,457,[3_1|2]), (457,458,[3_1|2]), (458,98,[0_1|2]), (458,99,[0_1|2]), (458,135,[0_1|2]), (458,162,[0_1|2]), (458,189,[0_1|2]), (458,234,[0_1|2]), (458,279,[0_1|2]), (458,288,[0_1|2]), (458,360,[0_1|2]), (458,423,[0_1|2]), (458,432,[0_1|2]), (458,443,[0_1|2]), (458,198,[2_1|2]), (458,207,[1_1|2]), (458,216,[4_1|2]), (458,225,[2_1|2]), (458,243,[4_1|2]), (459,460,[1_1|2]), (460,461,[5_1|2]), (461,462,[2_1|2]), (462,463,[5_1|2]), (463,464,[3_1|2]), (464,465,[2_1|2]), (465,466,[5_1|2]), (466,467,[2_1|2]), (466,387,[1_1|2]), (466,396,[2_1|2]), (467,98,[5_1|2]), (467,405,[5_1|2]), (467,99,[0_1|2]), (467,108,[1_1|2]), (467,117,[2_1|2]), (467,126,[1_1|2]), (467,135,[0_1|2]), (467,144,[1_1|2]), (467,468,[0_1|3]), (468,469,[2_1|3]), (469,470,[2_1|3]), (470,471,[1_1|3]), (471,472,[1_1|3]), (472,473,[3_1|3]), (473,474,[0_1|3]), (474,475,[4_1|3]), (475,476,[3_1|3]), (476,154,[0_1|3]), (477,478,[2_1|3]), (478,479,[2_1|3]), (479,480,[1_1|3]), (480,481,[1_1|3]), (481,482,[3_1|3]), (482,483,[0_1|3]), (483,484,[4_1|3]), (484,485,[3_1|3]), (485,99,[0_1|3]), (485,135,[0_1|3]), (485,162,[0_1|3]), (485,189,[0_1|3]), (485,234,[0_1|3]), (485,279,[0_1|3]), (485,288,[0_1|3]), (485,360,[0_1|3]), (485,423,[0_1|3]), (485,432,[0_1|3]), (485,154,[0_1|3]), (486,487,[0_1|3]), (487,488,[2_1|3]), (488,489,[2_1|3]), (489,490,[1_1|3]), (490,491,[1_1|3]), (491,492,[0_1|3]), (492,493,[0_1|3]), (493,494,[5_1|3]), (494,154,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)