WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 22 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 563 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(1(x1)))) -> 0(1(0(0(2(3(x1)))))) 0(0(4(1(x1)))) -> 4(2(0(1(0(x1))))) 0(0(4(1(x1)))) -> 0(2(1(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 1(0(5(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 3(4(2(0(1(0(x1)))))) 0(3(4(0(x1)))) -> 0(3(0(2(4(2(x1)))))) 0(3(4(0(x1)))) -> 0(3(4(2(0(2(x1)))))) 0(3(4(0(x1)))) -> 0(4(5(3(0(5(x1)))))) 0(4(4(0(x1)))) -> 4(4(2(0(0(x1))))) 0(4(4(0(x1)))) -> 0(4(0(2(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(x1))))) 0(4(4(1(x1)))) -> 1(4(4(2(0(x1))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(3(x1)))))) 0(4(4(1(x1)))) -> 0(1(5(4(4(2(x1)))))) 0(4(4(1(x1)))) -> 1(5(4(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(0(1(5(4(x1)))))) 0(4(4(1(x1)))) -> 2(4(4(1(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(5(4(0(1(x1)))))) 0(4(4(1(x1)))) -> 4(2(4(2(1(0(x1)))))) 1(4(4(0(x1)))) -> 5(4(4(2(0(1(x1)))))) 1(4(4(1(x1)))) -> 2(1(4(1(2(4(x1)))))) 1(4(4(1(x1)))) -> 2(3(4(1(4(1(x1)))))) 1(4(4(1(x1)))) -> 2(4(1(2(1(4(x1)))))) 1(4(4(1(x1)))) -> 2(4(2(1(4(1(x1)))))) 4(0(0(0(x1)))) -> 0(4(2(0(0(5(x1)))))) 4(0(0(1(x1)))) -> 2(3(0(1(0(4(x1)))))) 4(0(4(0(x1)))) -> 2(4(2(0(0(4(x1)))))) 4(0(4(0(x1)))) -> 3(4(2(0(4(0(x1)))))) 4(0(4(1(x1)))) -> 2(4(2(0(4(1(x1)))))) 4(3(4(0(x1)))) -> 4(0(3(4(2(x1))))) 4(3(4(0(x1)))) -> 4(4(2(0(3(x1))))) 4(3(4(0(x1)))) -> 0(4(2(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 2(3(0(4(4(5(x1)))))) 4(3(4(0(x1)))) -> 2(3(4(4(0(2(x1)))))) 4(3(4(0(x1)))) -> 2(4(0(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(2(0(4(2(3(x1)))))) 4(3(4(0(x1)))) -> 4(3(0(5(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(2(3(0(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(5(3(0(3(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(1(x1))))) 4(3(4(1(x1)))) -> 1(4(2(4(2(3(x1)))))) 4(3(4(1(x1)))) -> 2(3(1(4(5(4(x1)))))) 4(3(4(1(x1)))) -> 2(3(4(2(1(4(x1)))))) 4(3(4(1(x1)))) -> 2(4(2(3(1(4(x1)))))) 4(3(4(1(x1)))) -> 3(1(4(2(4(2(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(5(1(x1)))))) 4(3(4(1(x1)))) -> 3(4(5(3(4(1(x1)))))) 4(4(4(0(x1)))) -> 2(4(2(4(4(0(x1)))))) 4(4(4(1(x1)))) -> 2(4(2(4(1(4(x1)))))) 4(4(4(1(x1)))) -> 4(2(4(1(2(4(x1)))))) 4(4(4(1(x1)))) -> 4(5(4(1(5(4(x1)))))) 0(0(4(0(0(x1))))) -> 0(4(2(0(0(0(x1)))))) 0(1(3(5(0(x1))))) -> 0(1(3(0(5(3(x1)))))) 0(3(4(0(0(x1))))) -> 3(4(2(0(0(0(x1)))))) 0(3(5(0(1(x1))))) -> 0(1(0(5(2(3(x1)))))) 0(3(5(4(1(x1))))) -> 5(1(0(4(5(3(x1)))))) 0(4(3(4(0(x1))))) -> 0(2(3(4(4(0(x1)))))) 1(4(4(0(0(x1))))) -> 4(1(4(2(0(0(x1)))))) 1(4(4(0(1(x1))))) -> 1(4(4(1(2(0(x1)))))) 1(4(4(4(1(x1))))) -> 2(4(1(4(4(1(x1)))))) 4(0(0(4(1(x1))))) -> 4(4(2(0(1(0(x1)))))) 4(0(4(4(0(x1))))) -> 4(0(4(2(4(0(x1)))))) 4(1(0(0(1(x1))))) -> 4(2(0(1(1(0(x1)))))) 4(1(3(4(0(x1))))) -> 3(1(4(2(4(0(x1)))))) 4(3(1(0(0(x1))))) -> 1(4(2(3(0(0(x1)))))) 4(3(1(0(0(x1))))) -> 3(1(4(2(0(0(x1)))))) 4(3(1(5(0(x1))))) -> 1(4(5(3(0(1(x1)))))) 4(3(2(4(0(x1))))) -> 4(2(0(3(4(2(x1)))))) 4(3(4(4(1(x1))))) -> 4(4(3(1(4(2(x1)))))) 4(3(5(0(0(x1))))) -> 4(0(5(2(3(0(x1)))))) 4(3(5(0(1(x1))))) -> 2(3(5(4(0(1(x1)))))) 4(3(5(4(0(x1))))) -> 0(3(4(5(4(5(x1)))))) 4(3(5(4(0(x1))))) -> 0(4(5(3(4(2(x1)))))) 4(3(5(4(0(x1))))) -> 4(4(5(3(0(2(x1)))))) 4(3(5(4(0(x1))))) -> 5(4(3(4(2(0(x1)))))) 4(3(5(4(1(x1))))) -> 3(4(2(4(5(1(x1)))))) 4(3(5(5(0(x1))))) -> 0(2(3(5(4(5(x1)))))) 4(3(5(5(1(x1))))) -> 1(5(4(2(5(3(x1)))))) 4(4(0(0(1(x1))))) -> 4(0(4(2(0(1(x1)))))) 4(4(4(4(1(x1))))) -> 4(4(4(1(2(4(x1)))))) 4(4(5(0(1(x1))))) -> 4(5(0(1(4(2(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(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) 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)) 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 (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(1(x1)))) -> 0(1(0(0(2(3(x1)))))) 0(0(4(1(x1)))) -> 4(2(0(1(0(x1))))) 0(0(4(1(x1)))) -> 0(2(1(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 1(0(5(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 3(4(2(0(1(0(x1)))))) 0(3(4(0(x1)))) -> 0(3(0(2(4(2(x1)))))) 0(3(4(0(x1)))) -> 0(3(4(2(0(2(x1)))))) 0(3(4(0(x1)))) -> 0(4(5(3(0(5(x1)))))) 0(4(4(0(x1)))) -> 4(4(2(0(0(x1))))) 0(4(4(0(x1)))) -> 0(4(0(2(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(x1))))) 0(4(4(1(x1)))) -> 1(4(4(2(0(x1))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(3(x1)))))) 0(4(4(1(x1)))) -> 0(1(5(4(4(2(x1)))))) 0(4(4(1(x1)))) -> 1(5(4(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(0(1(5(4(x1)))))) 0(4(4(1(x1)))) -> 2(4(4(1(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(5(4(0(1(x1)))))) 0(4(4(1(x1)))) -> 4(2(4(2(1(0(x1)))))) 1(4(4(0(x1)))) -> 5(4(4(2(0(1(x1)))))) 1(4(4(1(x1)))) -> 2(1(4(1(2(4(x1)))))) 1(4(4(1(x1)))) -> 2(3(4(1(4(1(x1)))))) 1(4(4(1(x1)))) -> 2(4(1(2(1(4(x1)))))) 1(4(4(1(x1)))) -> 2(4(2(1(4(1(x1)))))) 4(0(0(0(x1)))) -> 0(4(2(0(0(5(x1)))))) 4(0(0(1(x1)))) -> 2(3(0(1(0(4(x1)))))) 4(0(4(0(x1)))) -> 2(4(2(0(0(4(x1)))))) 4(0(4(0(x1)))) -> 3(4(2(0(4(0(x1)))))) 4(0(4(1(x1)))) -> 2(4(2(0(4(1(x1)))))) 4(3(4(0(x1)))) -> 4(0(3(4(2(x1))))) 4(3(4(0(x1)))) -> 4(4(2(0(3(x1))))) 4(3(4(0(x1)))) -> 0(4(2(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 2(3(0(4(4(5(x1)))))) 4(3(4(0(x1)))) -> 2(3(4(4(0(2(x1)))))) 4(3(4(0(x1)))) -> 2(4(0(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(2(0(4(2(3(x1)))))) 4(3(4(0(x1)))) -> 4(3(0(5(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(2(3(0(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(5(3(0(3(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(1(x1))))) 4(3(4(1(x1)))) -> 1(4(2(4(2(3(x1)))))) 4(3(4(1(x1)))) -> 2(3(1(4(5(4(x1)))))) 4(3(4(1(x1)))) -> 2(3(4(2(1(4(x1)))))) 4(3(4(1(x1)))) -> 2(4(2(3(1(4(x1)))))) 4(3(4(1(x1)))) -> 3(1(4(2(4(2(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(5(1(x1)))))) 4(3(4(1(x1)))) -> 3(4(5(3(4(1(x1)))))) 4(4(4(0(x1)))) -> 2(4(2(4(4(0(x1)))))) 4(4(4(1(x1)))) -> 2(4(2(4(1(4(x1)))))) 4(4(4(1(x1)))) -> 4(2(4(1(2(4(x1)))))) 4(4(4(1(x1)))) -> 4(5(4(1(5(4(x1)))))) 0(0(4(0(0(x1))))) -> 0(4(2(0(0(0(x1)))))) 0(1(3(5(0(x1))))) -> 0(1(3(0(5(3(x1)))))) 0(3(4(0(0(x1))))) -> 3(4(2(0(0(0(x1)))))) 0(3(5(0(1(x1))))) -> 0(1(0(5(2(3(x1)))))) 0(3(5(4(1(x1))))) -> 5(1(0(4(5(3(x1)))))) 0(4(3(4(0(x1))))) -> 0(2(3(4(4(0(x1)))))) 1(4(4(0(0(x1))))) -> 4(1(4(2(0(0(x1)))))) 1(4(4(0(1(x1))))) -> 1(4(4(1(2(0(x1)))))) 1(4(4(4(1(x1))))) -> 2(4(1(4(4(1(x1)))))) 4(0(0(4(1(x1))))) -> 4(4(2(0(1(0(x1)))))) 4(0(4(4(0(x1))))) -> 4(0(4(2(4(0(x1)))))) 4(1(0(0(1(x1))))) -> 4(2(0(1(1(0(x1)))))) 4(1(3(4(0(x1))))) -> 3(1(4(2(4(0(x1)))))) 4(3(1(0(0(x1))))) -> 1(4(2(3(0(0(x1)))))) 4(3(1(0(0(x1))))) -> 3(1(4(2(0(0(x1)))))) 4(3(1(5(0(x1))))) -> 1(4(5(3(0(1(x1)))))) 4(3(2(4(0(x1))))) -> 4(2(0(3(4(2(x1)))))) 4(3(4(4(1(x1))))) -> 4(4(3(1(4(2(x1)))))) 4(3(5(0(0(x1))))) -> 4(0(5(2(3(0(x1)))))) 4(3(5(0(1(x1))))) -> 2(3(5(4(0(1(x1)))))) 4(3(5(4(0(x1))))) -> 0(3(4(5(4(5(x1)))))) 4(3(5(4(0(x1))))) -> 0(4(5(3(4(2(x1)))))) 4(3(5(4(0(x1))))) -> 4(4(5(3(0(2(x1)))))) 4(3(5(4(0(x1))))) -> 5(4(3(4(2(0(x1)))))) 4(3(5(4(1(x1))))) -> 3(4(2(4(5(1(x1)))))) 4(3(5(5(0(x1))))) -> 0(2(3(5(4(5(x1)))))) 4(3(5(5(1(x1))))) -> 1(5(4(2(5(3(x1)))))) 4(4(0(0(1(x1))))) -> 4(0(4(2(0(1(x1)))))) 4(4(4(4(1(x1))))) -> 4(4(4(1(2(4(x1)))))) 4(4(5(0(1(x1))))) -> 4(5(0(1(4(2(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(5(x_1)) -> 5(encArg(x_1)) 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)) 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: 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(1(x1)))) -> 0(1(0(0(2(3(x1)))))) 0(0(4(1(x1)))) -> 4(2(0(1(0(x1))))) 0(0(4(1(x1)))) -> 0(2(1(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 1(0(5(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 3(4(2(0(1(0(x1)))))) 0(3(4(0(x1)))) -> 0(3(0(2(4(2(x1)))))) 0(3(4(0(x1)))) -> 0(3(4(2(0(2(x1)))))) 0(3(4(0(x1)))) -> 0(4(5(3(0(5(x1)))))) 0(4(4(0(x1)))) -> 4(4(2(0(0(x1))))) 0(4(4(0(x1)))) -> 0(4(0(2(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(x1))))) 0(4(4(1(x1)))) -> 1(4(4(2(0(x1))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(3(x1)))))) 0(4(4(1(x1)))) -> 0(1(5(4(4(2(x1)))))) 0(4(4(1(x1)))) -> 1(5(4(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(0(1(5(4(x1)))))) 0(4(4(1(x1)))) -> 2(4(4(1(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(5(4(0(1(x1)))))) 0(4(4(1(x1)))) -> 4(2(4(2(1(0(x1)))))) 1(4(4(0(x1)))) -> 5(4(4(2(0(1(x1)))))) 1(4(4(1(x1)))) -> 2(1(4(1(2(4(x1)))))) 1(4(4(1(x1)))) -> 2(3(4(1(4(1(x1)))))) 1(4(4(1(x1)))) -> 2(4(1(2(1(4(x1)))))) 1(4(4(1(x1)))) -> 2(4(2(1(4(1(x1)))))) 4(0(0(0(x1)))) -> 0(4(2(0(0(5(x1)))))) 4(0(0(1(x1)))) -> 2(3(0(1(0(4(x1)))))) 4(0(4(0(x1)))) -> 2(4(2(0(0(4(x1)))))) 4(0(4(0(x1)))) -> 3(4(2(0(4(0(x1)))))) 4(0(4(1(x1)))) -> 2(4(2(0(4(1(x1)))))) 4(3(4(0(x1)))) -> 4(0(3(4(2(x1))))) 4(3(4(0(x1)))) -> 4(4(2(0(3(x1))))) 4(3(4(0(x1)))) -> 0(4(2(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 2(3(0(4(4(5(x1)))))) 4(3(4(0(x1)))) -> 2(3(4(4(0(2(x1)))))) 4(3(4(0(x1)))) -> 2(4(0(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(2(0(4(2(3(x1)))))) 4(3(4(0(x1)))) -> 4(3(0(5(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(2(3(0(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(5(3(0(3(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(1(x1))))) 4(3(4(1(x1)))) -> 1(4(2(4(2(3(x1)))))) 4(3(4(1(x1)))) -> 2(3(1(4(5(4(x1)))))) 4(3(4(1(x1)))) -> 2(3(4(2(1(4(x1)))))) 4(3(4(1(x1)))) -> 2(4(2(3(1(4(x1)))))) 4(3(4(1(x1)))) -> 3(1(4(2(4(2(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(5(1(x1)))))) 4(3(4(1(x1)))) -> 3(4(5(3(4(1(x1)))))) 4(4(4(0(x1)))) -> 2(4(2(4(4(0(x1)))))) 4(4(4(1(x1)))) -> 2(4(2(4(1(4(x1)))))) 4(4(4(1(x1)))) -> 4(2(4(1(2(4(x1)))))) 4(4(4(1(x1)))) -> 4(5(4(1(5(4(x1)))))) 0(0(4(0(0(x1))))) -> 0(4(2(0(0(0(x1)))))) 0(1(3(5(0(x1))))) -> 0(1(3(0(5(3(x1)))))) 0(3(4(0(0(x1))))) -> 3(4(2(0(0(0(x1)))))) 0(3(5(0(1(x1))))) -> 0(1(0(5(2(3(x1)))))) 0(3(5(4(1(x1))))) -> 5(1(0(4(5(3(x1)))))) 0(4(3(4(0(x1))))) -> 0(2(3(4(4(0(x1)))))) 1(4(4(0(0(x1))))) -> 4(1(4(2(0(0(x1)))))) 1(4(4(0(1(x1))))) -> 1(4(4(1(2(0(x1)))))) 1(4(4(4(1(x1))))) -> 2(4(1(4(4(1(x1)))))) 4(0(0(4(1(x1))))) -> 4(4(2(0(1(0(x1)))))) 4(0(4(4(0(x1))))) -> 4(0(4(2(4(0(x1)))))) 4(1(0(0(1(x1))))) -> 4(2(0(1(1(0(x1)))))) 4(1(3(4(0(x1))))) -> 3(1(4(2(4(0(x1)))))) 4(3(1(0(0(x1))))) -> 1(4(2(3(0(0(x1)))))) 4(3(1(0(0(x1))))) -> 3(1(4(2(0(0(x1)))))) 4(3(1(5(0(x1))))) -> 1(4(5(3(0(1(x1)))))) 4(3(2(4(0(x1))))) -> 4(2(0(3(4(2(x1)))))) 4(3(4(4(1(x1))))) -> 4(4(3(1(4(2(x1)))))) 4(3(5(0(0(x1))))) -> 4(0(5(2(3(0(x1)))))) 4(3(5(0(1(x1))))) -> 2(3(5(4(0(1(x1)))))) 4(3(5(4(0(x1))))) -> 0(3(4(5(4(5(x1)))))) 4(3(5(4(0(x1))))) -> 0(4(5(3(4(2(x1)))))) 4(3(5(4(0(x1))))) -> 4(4(5(3(0(2(x1)))))) 4(3(5(4(0(x1))))) -> 5(4(3(4(2(0(x1)))))) 4(3(5(4(1(x1))))) -> 3(4(2(4(5(1(x1)))))) 4(3(5(5(0(x1))))) -> 0(2(3(5(4(5(x1)))))) 4(3(5(5(1(x1))))) -> 1(5(4(2(5(3(x1)))))) 4(4(0(0(1(x1))))) -> 4(0(4(2(0(1(x1)))))) 4(4(4(4(1(x1))))) -> 4(4(4(1(2(4(x1)))))) 4(4(5(0(1(x1))))) -> 4(5(0(1(4(2(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(5(x_1)) -> 5(encArg(x_1)) 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)) 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: 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(1(x1)))) -> 0(1(0(0(2(3(x1)))))) 0(0(4(1(x1)))) -> 4(2(0(1(0(x1))))) 0(0(4(1(x1)))) -> 0(2(1(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 1(0(5(4(2(0(x1)))))) 0(0(4(1(x1)))) -> 3(4(2(0(1(0(x1)))))) 0(3(4(0(x1)))) -> 0(3(0(2(4(2(x1)))))) 0(3(4(0(x1)))) -> 0(3(4(2(0(2(x1)))))) 0(3(4(0(x1)))) -> 0(4(5(3(0(5(x1)))))) 0(4(4(0(x1)))) -> 4(4(2(0(0(x1))))) 0(4(4(0(x1)))) -> 0(4(0(2(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(0(4(2(x1)))))) 0(4(4(0(x1)))) -> 4(2(0(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(x1))))) 0(4(4(1(x1)))) -> 1(4(4(2(0(x1))))) 0(4(4(1(x1)))) -> 0(1(4(4(2(3(x1)))))) 0(4(4(1(x1)))) -> 0(1(5(4(4(2(x1)))))) 0(4(4(1(x1)))) -> 1(5(4(4(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(0(1(5(4(x1)))))) 0(4(4(1(x1)))) -> 2(4(4(1(2(0(x1)))))) 0(4(4(1(x1)))) -> 2(4(5(4(0(1(x1)))))) 0(4(4(1(x1)))) -> 4(2(4(2(1(0(x1)))))) 1(4(4(0(x1)))) -> 5(4(4(2(0(1(x1)))))) 1(4(4(1(x1)))) -> 2(1(4(1(2(4(x1)))))) 1(4(4(1(x1)))) -> 2(3(4(1(4(1(x1)))))) 1(4(4(1(x1)))) -> 2(4(1(2(1(4(x1)))))) 1(4(4(1(x1)))) -> 2(4(2(1(4(1(x1)))))) 4(0(0(0(x1)))) -> 0(4(2(0(0(5(x1)))))) 4(0(0(1(x1)))) -> 2(3(0(1(0(4(x1)))))) 4(0(4(0(x1)))) -> 2(4(2(0(0(4(x1)))))) 4(0(4(0(x1)))) -> 3(4(2(0(4(0(x1)))))) 4(0(4(1(x1)))) -> 2(4(2(0(4(1(x1)))))) 4(3(4(0(x1)))) -> 4(0(3(4(2(x1))))) 4(3(4(0(x1)))) -> 4(4(2(0(3(x1))))) 4(3(4(0(x1)))) -> 0(4(2(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 2(3(0(4(4(5(x1)))))) 4(3(4(0(x1)))) -> 2(3(4(4(0(2(x1)))))) 4(3(4(0(x1)))) -> 2(4(0(3(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(2(0(4(2(3(x1)))))) 4(3(4(0(x1)))) -> 4(3(0(5(4(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(2(3(0(2(x1)))))) 4(3(4(0(x1)))) -> 4(4(5(3(0(3(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(1(x1))))) 4(3(4(1(x1)))) -> 1(4(2(4(2(3(x1)))))) 4(3(4(1(x1)))) -> 2(3(1(4(5(4(x1)))))) 4(3(4(1(x1)))) -> 2(3(4(2(1(4(x1)))))) 4(3(4(1(x1)))) -> 2(4(2(3(1(4(x1)))))) 4(3(4(1(x1)))) -> 3(1(4(2(4(2(x1)))))) 4(3(4(1(x1)))) -> 3(4(2(4(5(1(x1)))))) 4(3(4(1(x1)))) -> 3(4(5(3(4(1(x1)))))) 4(4(4(0(x1)))) -> 2(4(2(4(4(0(x1)))))) 4(4(4(1(x1)))) -> 2(4(2(4(1(4(x1)))))) 4(4(4(1(x1)))) -> 4(2(4(1(2(4(x1)))))) 4(4(4(1(x1)))) -> 4(5(4(1(5(4(x1)))))) 0(0(4(0(0(x1))))) -> 0(4(2(0(0(0(x1)))))) 0(1(3(5(0(x1))))) -> 0(1(3(0(5(3(x1)))))) 0(3(4(0(0(x1))))) -> 3(4(2(0(0(0(x1)))))) 0(3(5(0(1(x1))))) -> 0(1(0(5(2(3(x1)))))) 0(3(5(4(1(x1))))) -> 5(1(0(4(5(3(x1)))))) 0(4(3(4(0(x1))))) -> 0(2(3(4(4(0(x1)))))) 1(4(4(0(0(x1))))) -> 4(1(4(2(0(0(x1)))))) 1(4(4(0(1(x1))))) -> 1(4(4(1(2(0(x1)))))) 1(4(4(4(1(x1))))) -> 2(4(1(4(4(1(x1)))))) 4(0(0(4(1(x1))))) -> 4(4(2(0(1(0(x1)))))) 4(0(4(4(0(x1))))) -> 4(0(4(2(4(0(x1)))))) 4(1(0(0(1(x1))))) -> 4(2(0(1(1(0(x1)))))) 4(1(3(4(0(x1))))) -> 3(1(4(2(4(0(x1)))))) 4(3(1(0(0(x1))))) -> 1(4(2(3(0(0(x1)))))) 4(3(1(0(0(x1))))) -> 3(1(4(2(0(0(x1)))))) 4(3(1(5(0(x1))))) -> 1(4(5(3(0(1(x1)))))) 4(3(2(4(0(x1))))) -> 4(2(0(3(4(2(x1)))))) 4(3(4(4(1(x1))))) -> 4(4(3(1(4(2(x1)))))) 4(3(5(0(0(x1))))) -> 4(0(5(2(3(0(x1)))))) 4(3(5(0(1(x1))))) -> 2(3(5(4(0(1(x1)))))) 4(3(5(4(0(x1))))) -> 0(3(4(5(4(5(x1)))))) 4(3(5(4(0(x1))))) -> 0(4(5(3(4(2(x1)))))) 4(3(5(4(0(x1))))) -> 4(4(5(3(0(2(x1)))))) 4(3(5(4(0(x1))))) -> 5(4(3(4(2(0(x1)))))) 4(3(5(4(1(x1))))) -> 3(4(2(4(5(1(x1)))))) 4(3(5(5(0(x1))))) -> 0(2(3(5(4(5(x1)))))) 4(3(5(5(1(x1))))) -> 1(5(4(2(5(3(x1)))))) 4(4(0(0(1(x1))))) -> 4(0(4(2(0(1(x1)))))) 4(4(4(4(1(x1))))) -> 4(4(4(1(2(4(x1)))))) 4(4(5(0(1(x1))))) -> 4(5(0(1(4(2(x1)))))) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) 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)) 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: 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 5. The certificate found is represented by the following graph. "[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, 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, 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871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928] {(65,66,[0_1|0, 1_1|0, 4_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]), (65,67,[2_1|1, 3_1|1, 5_1|1, 0_1|1, 1_1|1, 4_1|1]), (65,68,[0_1|2]), (65,73,[4_1|2]), (65,77,[0_1|2]), (65,82,[1_1|2]), (65,87,[3_1|2]), (65,92,[0_1|2]), (65,97,[0_1|2]), (65,102,[0_1|2]), (65,107,[0_1|2]), (65,112,[3_1|2]), (65,117,[0_1|2]), (65,122,[5_1|2]), (65,127,[4_1|2]), (65,131,[0_1|2]), (65,136,[4_1|2]), (65,141,[4_1|2]), (65,146,[4_1|2]), (65,151,[0_1|2]), (65,155,[1_1|2]), (65,159,[0_1|2]), (65,164,[0_1|2]), (65,169,[1_1|2]), (65,174,[2_1|2]), (65,179,[2_1|2]), (65,184,[2_1|2]), (65,189,[4_1|2]), (65,194,[0_1|2]), (65,199,[0_1|2]), (65,204,[5_1|2]), 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(762,763,[4_1|3]), (763,217,[2_1|3]), (764,765,[5_1|3]), (765,766,[4_1|3]), (766,767,[4_1|3]), (767,768,[2_1|3]), (768,217,[0_1|3]), (769,770,[4_1|3]), (770,771,[0_1|3]), (771,772,[1_1|3]), (772,773,[5_1|3]), (773,217,[4_1|3]), (774,775,[4_1|3]), (775,776,[4_1|3]), (776,777,[1_1|3]), (777,778,[2_1|3]), (778,217,[0_1|3]), (779,780,[4_1|3]), (780,781,[5_1|3]), (781,782,[4_1|3]), (782,783,[0_1|3]), (783,217,[1_1|3]), (784,785,[2_1|3]), (785,786,[4_1|3]), (786,787,[2_1|3]), (787,788,[1_1|3]), (788,217,[0_1|3]), (789,790,[4_1|4]), (790,791,[2_1|4]), (791,792,[4_1|4]), (792,793,[1_1|4]), (793,558,[4_1|4]), (794,795,[2_1|4]), (795,796,[4_1|4]), (796,797,[1_1|4]), (797,798,[2_1|4]), (798,558,[4_1|4]), (799,800,[5_1|4]), (800,801,[4_1|4]), (801,802,[1_1|4]), (802,803,[5_1|4]), (803,558,[4_1|4]), (804,805,[4_1|3]), (805,806,[2_1|3]), (806,807,[4_1|3]), (807,808,[1_1|3]), (808,217,[4_1|3]), (808,454,[4_1|3]), (809,810,[2_1|3]), (810,811,[4_1|3]), (811,812,[1_1|3]), (812,813,[2_1|3]), (813,217,[4_1|3]), (813,454,[4_1|3]), (814,815,[5_1|3]), (815,816,[4_1|3]), (816,817,[1_1|3]), (817,818,[5_1|3]), (818,217,[4_1|3]), (818,454,[4_1|3]), (819,820,[4_1|3]), (820,821,[1_1|3]), (820,869,[2_1|4]), (820,874,[2_1|4]), (820,879,[2_1|4]), (820,884,[2_1|4]), (821,822,[4_1|3]), (822,823,[4_1|3]), (823,217,[1_1|3]), (823,454,[1_1|3]), (823,558,[1_1|3]), (824,825,[4_1|3]), (825,826,[1_1|3]), (825,889,[2_1|4]), (825,894,[2_1|4]), (825,899,[2_1|4]), (825,904,[2_1|4]), (826,827,[4_1|3]), (827,828,[4_1|3]), (828,210,[1_1|3]), (829,830,[4_1|3]), (830,831,[2_1|3]), (831,832,[4_1|3]), (832,833,[1_1|3]), (833,210,[4_1|3]), (834,835,[2_1|3]), (835,836,[4_1|3]), (836,837,[1_1|3]), (837,838,[2_1|3]), (838,210,[4_1|3]), (839,840,[5_1|3]), (840,841,[4_1|3]), (841,842,[1_1|3]), (842,843,[5_1|3]), (843,210,[4_1|3]), (844,845,[1_1|4]), (845,846,[4_1|4]), (846,847,[1_1|4]), (847,848,[2_1|4]), (848,454,[4_1|4]), (849,850,[3_1|4]), (850,851,[4_1|4]), (851,852,[1_1|4]), (852,853,[4_1|4]), (853,454,[1_1|4]), (854,855,[4_1|4]), (855,856,[1_1|4]), (856,857,[2_1|4]), (857,858,[1_1|4]), (858,454,[4_1|4]), (859,860,[4_1|4]), (860,861,[2_1|4]), (861,862,[1_1|4]), (862,863,[4_1|4]), (863,454,[1_1|4]), (864,865,[4_1|4]), (865,866,[1_1|4]), (865,909,[2_1|5]), (865,914,[2_1|5]), (865,919,[2_1|5]), (865,924,[2_1|5]), (866,867,[4_1|4]), (867,868,[4_1|4]), (868,558,[1_1|4]), (869,870,[1_1|4]), (870,871,[4_1|4]), (871,872,[1_1|4]), (872,873,[2_1|4]), (873,217,[4_1|4]), (873,454,[4_1|4]), (873,558,[4_1|4]), (874,875,[3_1|4]), (875,876,[4_1|4]), (876,877,[1_1|4]), (877,878,[4_1|4]), (878,217,[1_1|4]), (878,454,[1_1|4]), (878,558,[1_1|4]), (879,880,[4_1|4]), (880,881,[1_1|4]), (881,882,[2_1|4]), (882,883,[1_1|4]), (883,217,[4_1|4]), (883,454,[4_1|4]), (883,558,[4_1|4]), (884,885,[4_1|4]), (885,886,[2_1|4]), (886,887,[1_1|4]), (887,888,[4_1|4]), (888,217,[1_1|4]), (888,454,[1_1|4]), (888,558,[1_1|4]), (889,890,[1_1|4]), (890,891,[4_1|4]), (891,892,[1_1|4]), (892,893,[2_1|4]), (893,210,[4_1|4]), (894,895,[3_1|4]), (895,896,[4_1|4]), (896,897,[1_1|4]), (897,898,[4_1|4]), (898,210,[1_1|4]), (899,900,[4_1|4]), (900,901,[1_1|4]), (901,902,[2_1|4]), (902,903,[1_1|4]), (903,210,[4_1|4]), (904,905,[4_1|4]), (905,906,[2_1|4]), (906,907,[1_1|4]), (907,908,[4_1|4]), (908,210,[1_1|4]), (909,910,[1_1|5]), (910,911,[4_1|5]), (911,912,[1_1|5]), (912,913,[2_1|5]), (913,558,[4_1|5]), (914,915,[3_1|5]), (915,916,[4_1|5]), (916,917,[1_1|5]), (917,918,[4_1|5]), (918,558,[1_1|5]), (919,920,[4_1|5]), (920,921,[1_1|5]), (921,922,[2_1|5]), (922,923,[1_1|5]), (923,558,[4_1|5]), (924,925,[4_1|5]), (925,926,[2_1|5]), (926,927,[1_1|5]), (927,928,[4_1|5]), (928,558,[1_1|5])}" ---------------------------------------- (8) BOUNDS(1, n^1)