/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 51 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 138 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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 3. 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, 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] {(65,66,[0_1|0, 3_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,[1_1|1]), (65,71,[1_1|1]), (65,76,[0_1|1]), (65,81,[0_1|1]), (65,86,[1_1|1]), (65,91,[4_1|1]), (65,96,[1_1|1]), (65,101,[1_1|1, 2_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1]), (65,115,[1_1|2]), (65,119,[0_1|2]), (65,124,[4_1|2]), (65,129,[0_1|2]), (65,134,[1_1|2]), (65,139,[0_1|2]), (65,143,[1_1|2]), (65,147,[3_1|2]), (65,151,[1_1|2]), (65,156,[0_1|2]), (65,160,[1_1|2]), (65,165,[1_1|2]), (65,169,[0_1|2]), (65,174,[0_1|2]), (65,178,[2_1|2]), (65,182,[1_1|2]), (65,186,[2_1|2]), (65,191,[0_1|2]), (65,196,[2_1|2]), (65,201,[2_1|2]), (65,206,[1_1|2]), (65,211,[0_1|2]), (65,215,[1_1|2]), (65,220,[4_1|2]), (65,225,[0_1|2]), (65,230,[0_1|2]), (65,235,[0_1|2]), (65,240,[0_1|2]), (65,245,[1_1|2]), (65,250,[4_1|2]), (65,255,[3_1|2]), (65,260,[0_1|2]), (65,265,[4_1|2]), (65,270,[1_1|2]), (65,274,[3_1|2]), (65,279,[3_1|2]), (65,284,[1_1|2]), (65,289,[0_1|2]), (65,294,[3_1|2]), (65,299,[1_1|2]), (65,304,[4_1|2]), (65,309,[1_1|2]), (65,314,[4_1|2]), (65,318,[4_1|2]), (65,323,[0_1|2]), (65,328,[4_1|2]), (65,333,[4_1|2]), (65,338,[4_1|2]), (65,343,[4_1|2]), (65,348,[1_1|2]), (66,66,[1_1|0, 2_1|0, 5_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0]), (67,68,[2_1|1]), (68,69,[1_1|1]), (69,70,[2_1|1]), (70,66,[0_1|1]), (70,67,[1_1|1]), (70,71,[1_1|1]), (70,76,[0_1|1]), (70,81,[0_1|1]), (70,86,[1_1|1]), (71,72,[2_1|1]), (72,73,[2_1|1]), (73,74,[5_1|1]), (74,75,[0_1|1]), (74,67,[1_1|1]), (74,71,[1_1|1]), (75,66,[1_1|1]), (76,77,[1_1|1]), (77,78,[2_1|1]), (78,79,[2_1|1]), (79,80,[5_1|1]), (80,66,[5_1|1]), (81,82,[5_1|1]), (82,83,[1_1|1]), (83,84,[1_1|1]), (84,85,[2_1|1]), (85,66,[5_1|1]), (86,87,[2_1|1]), (87,88,[5_1|1]), (88,89,[5_1|1]), (89,90,[0_1|1]), (89,102,[2_1|1]), (89,106,[1_1|1]), (89,110,[2_1|1]), (89,353,[1_1|2]), (90,66,[4_1|1]), (90,91,[4_1|1]), (90,96,[1_1|1]), (91,92,[4_1|1]), (92,93,[5_1|1]), (93,94,[1_1|1]), (94,95,[2_1|1]), (95,66,[1_1|1]), (96,97,[1_1|1]), (97,98,[2_1|1]), (98,99,[5_1|1]), (99,100,[4_1|1]), (99,91,[4_1|1]), (99,96,[1_1|1]), (100,66,[1_1|1]), (101,66,[encArg_1|1]), (101,101,[1_1|1, 2_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1]), (101,115,[1_1|2]), (101,119,[0_1|2]), (101,124,[4_1|2]), (101,129,[0_1|2]), (101,134,[1_1|2]), (101,139,[0_1|2]), (101,143,[1_1|2]), (101,147,[3_1|2]), (101,151,[1_1|2]), (101,156,[0_1|2]), (101,160,[1_1|2]), (101,165,[1_1|2]), (101,169,[0_1|2]), (101,174,[0_1|2]), (101,178,[2_1|2]), (101,182,[1_1|2]), (101,186,[2_1|2]), (101,191,[0_1|2]), (101,196,[2_1|2]), (101,201,[2_1|2]), (101,206,[1_1|2]), (101,211,[0_1|2]), (101,215,[1_1|2]), (101,220,[4_1|2]), (101,225,[0_1|2]), (101,230,[0_1|2]), (101,235,[0_1|2]), (101,240,[0_1|2]), (101,245,[1_1|2]), (101,250,[4_1|2]), (101,255,[3_1|2]), (101,260,[0_1|2]), (101,265,[4_1|2]), (101,270,[1_1|2]), (101,274,[3_1|2]), (101,279,[3_1|2]), (101,284,[1_1|2]), (101,289,[0_1|2]), (101,294,[3_1|2]), (101,299,[1_1|2]), (101,304,[4_1|2]), (101,309,[1_1|2]), (101,314,[4_1|2]), (101,318,[4_1|2]), (101,323,[0_1|2]), (101,328,[4_1|2]), (101,333,[4_1|2]), (101,338,[4_1|2]), (101,343,[4_1|2]), (101,348,[1_1|2]), (102,103,[1_1|1]), (103,104,[2_1|1]), (104,105,[0_1|1]), (104,102,[2_1|1]), (104,106,[1_1|1]), (104,110,[2_1|1]), (104,353,[1_1|2]), (105,66,[4_1|1]), (105,91,[4_1|1]), (105,96,[1_1|1]), (106,107,[2_1|1]), (107,108,[5_1|1]), (108,109,[0_1|1]), (108,102,[2_1|1]), (108,106,[1_1|1]), (108,110,[2_1|1]), (108,353,[1_1|2]), (109,66,[4_1|1]), (109,91,[4_1|1]), (109,96,[1_1|1]), (110,111,[5_1|1]), (111,112,[4_1|1]), (112,113,[4_1|1]), (112,357,[0_1|1]), (113,114,[0_1|1]), (113,67,[1_1|1]), (113,71,[1_1|1]), (114,66,[1_1|1]), (115,116,[2_1|2]), (116,117,[1_1|2]), (117,118,[2_1|2]), (118,101,[0_1|2]), (118,115,[0_1|2, 1_1|2]), (118,134,[0_1|2, 1_1|2]), (118,143,[0_1|2, 1_1|2]), (118,151,[0_1|2, 1_1|2]), (118,160,[0_1|2, 1_1|2]), (118,165,[0_1|2, 1_1|2]), (118,182,[0_1|2, 1_1|2]), (118,206,[0_1|2, 1_1|2]), (118,215,[0_1|2, 1_1|2]), (118,245,[0_1|2, 1_1|2]), (118,270,[0_1|2]), (118,284,[0_1|2]), (118,299,[0_1|2]), (118,309,[0_1|2]), (118,348,[0_1|2]), (118,310,[0_1|2]), (118,119,[0_1|2]), (118,124,[4_1|2]), (118,129,[0_1|2]), (118,139,[0_1|2]), (118,147,[3_1|2]), (118,156,[0_1|2]), (118,169,[0_1|2]), (118,174,[0_1|2]), (118,178,[2_1|2]), (118,186,[2_1|2]), (118,191,[0_1|2]), (118,196,[2_1|2]), (118,201,[2_1|2]), (118,211,[0_1|2]), (118,220,[4_1|2]), (118,225,[0_1|2]), (118,230,[0_1|2]), (118,235,[0_1|2]), (118,240,[0_1|2]), (118,250,[4_1|2]), (118,372,[2_1|3]), (118,377,[1_1|3]), (118,381,[3_1|3]), (118,385,[1_1|3]), (118,390,[1_1|3]), (118,394,[2_1|3]), (119,120,[1_1|2]), (120,121,[2_1|2]), (121,122,[2_1|2]), (122,123,[4_1|2]), (122,299,[1_1|2]), (122,304,[4_1|2]), (122,309,[1_1|2]), (123,101,[1_1|2]), (123,115,[1_1|2]), (123,134,[1_1|2]), (123,143,[1_1|2]), (123,151,[1_1|2]), (123,160,[1_1|2]), (123,165,[1_1|2]), (123,182,[1_1|2]), (123,206,[1_1|2]), (123,215,[1_1|2]), (123,245,[1_1|2]), (123,270,[1_1|2]), (123,284,[1_1|2]), (123,299,[1_1|2]), (123,309,[1_1|2]), (123,348,[1_1|2]), (123,315,[1_1|2]), (123,344,[1_1|2]), (124,125,[0_1|2]), (125,126,[1_1|2]), (126,127,[2_1|2]), (127,128,[5_1|2]), (128,101,[4_1|2]), (128,251,[4_1|2]), 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(303,101,[4_1|2]), (303,299,[1_1|2]), (303,304,[4_1|2]), (303,309,[1_1|2]), (303,314,[4_1|2]), (303,318,[4_1|2]), (303,323,[0_1|2]), (303,328,[4_1|2]), (303,333,[4_1|2]), (303,338,[4_1|2]), (303,343,[4_1|2]), (303,348,[1_1|2]), (304,305,[4_1|2]), (305,306,[5_1|2]), (306,307,[1_1|2]), (307,308,[2_1|2]), (308,101,[1_1|2]), (308,115,[1_1|2]), (308,134,[1_1|2]), (308,143,[1_1|2]), (308,151,[1_1|2]), (308,160,[1_1|2]), (308,165,[1_1|2]), (308,182,[1_1|2]), (308,206,[1_1|2]), (308,215,[1_1|2]), (308,245,[1_1|2]), (308,270,[1_1|2]), (308,284,[1_1|2]), (308,299,[1_1|2]), (308,309,[1_1|2]), (308,348,[1_1|2]), (309,310,[1_1|2]), (310,311,[2_1|2]), (311,312,[5_1|2]), (312,313,[4_1|2]), (312,299,[1_1|2]), (312,304,[4_1|2]), (312,309,[1_1|2]), (313,101,[1_1|2]), (313,115,[1_1|2]), (313,134,[1_1|2]), (313,143,[1_1|2]), (313,151,[1_1|2]), (313,160,[1_1|2]), (313,165,[1_1|2]), (313,182,[1_1|2]), (313,206,[1_1|2]), (313,215,[1_1|2]), (313,245,[1_1|2]), (313,270,[1_1|2]), (313,284,[1_1|2]), (313,299,[1_1|2]), (313,309,[1_1|2]), (313,348,[1_1|2]), (314,315,[1_1|2]), (315,316,[2_1|2]), (316,317,[5_1|2]), (317,101,[4_1|2]), (317,299,[1_1|2]), (317,304,[4_1|2]), (317,309,[1_1|2]), (317,314,[4_1|2]), (317,318,[4_1|2]), (317,323,[0_1|2]), (317,328,[4_1|2]), (317,333,[4_1|2]), (317,338,[4_1|2]), (317,343,[4_1|2]), (317,348,[1_1|2]), (318,319,[4_1|2]), (319,320,[0_1|2]), (320,321,[1_1|2]), (321,322,[3_1|2]), (322,101,[1_1|2]), (322,115,[1_1|2]), (322,134,[1_1|2]), (322,143,[1_1|2]), (322,151,[1_1|2]), (322,160,[1_1|2]), (322,165,[1_1|2]), (322,182,[1_1|2]), (322,206,[1_1|2]), (322,215,[1_1|2]), (322,245,[1_1|2]), (322,270,[1_1|2]), (322,284,[1_1|2]), (322,299,[1_1|2]), (322,309,[1_1|2]), (322,348,[1_1|2]), (322,315,[1_1|2]), (322,344,[1_1|2]), (323,324,[1_1|2]), (324,325,[2_1|2]), (325,326,[5_1|2]), (326,327,[4_1|2]), (326,299,[1_1|2]), (326,304,[4_1|2]), (326,309,[1_1|2]), (327,101,[1_1|2]), (327,115,[1_1|2]), (327,134,[1_1|2]), (327,143,[1_1|2]), (327,151,[1_1|2]), (327,160,[1_1|2]), (327,165,[1_1|2]), (327,182,[1_1|2]), (327,206,[1_1|2]), (327,215,[1_1|2]), (327,245,[1_1|2]), (327,270,[1_1|2]), (327,284,[1_1|2]), (327,299,[1_1|2]), (327,309,[1_1|2]), (327,348,[1_1|2]), (328,329,[0_1|2]), (329,330,[2_1|2]), (330,331,[5_1|2]), (331,332,[0_1|2]), (331,178,[2_1|2]), (331,182,[1_1|2]), (331,186,[2_1|2]), (331,191,[0_1|2]), (331,196,[2_1|2]), (331,201,[2_1|2]), (331,206,[1_1|2]), (331,412,[2_1|3]), (331,416,[2_1|3]), (331,390,[1_1|3]), (332,101,[4_1|2]), (332,251,[4_1|2]), (332,299,[1_1|2]), (332,304,[4_1|2]), (332,309,[1_1|2]), (332,314,[4_1|2]), (332,318,[4_1|2]), (332,323,[0_1|2]), (332,328,[4_1|2]), (332,333,[4_1|2]), (332,338,[4_1|2]), (332,343,[4_1|2]), (332,348,[1_1|2]), (333,334,[4_1|2]), (334,335,[1_1|2]), (335,336,[2_1|2]), (336,337,[1_1|2]), (337,101,[5_1|2]), (337,115,[5_1|2]), (337,134,[5_1|2]), (337,143,[5_1|2]), (337,151,[5_1|2]), (337,160,[5_1|2]), (337,165,[5_1|2]), (337,182,[5_1|2]), (337,206,[5_1|2]), (337,215,[5_1|2]), (337,245,[5_1|2]), (337,270,[5_1|2]), (337,284,[5_1|2]), (337,299,[5_1|2]), (337,309,[5_1|2]), (337,348,[5_1|2]), (337,315,[5_1|2]), (337,344,[5_1|2]), (338,339,[3_1|2]), (339,340,[1_1|2]), (340,341,[2_1|2]), (341,342,[2_1|2]), (342,101,[5_1|2]), (342,115,[5_1|2]), (342,134,[5_1|2]), (342,143,[5_1|2]), (342,151,[5_1|2]), (342,160,[5_1|2]), (342,165,[5_1|2]), (342,182,[5_1|2]), (342,206,[5_1|2]), (342,215,[5_1|2]), (342,245,[5_1|2]), (342,270,[5_1|2]), (342,284,[5_1|2]), (342,299,[5_1|2]), (342,309,[5_1|2]), (342,348,[5_1|2]), (342,256,[5_1|2]), (343,344,[1_1|2]), (344,345,[2_1|2]), (345,346,[5_1|2]), (346,347,[3_1|2]), (346,294,[3_1|2]), (347,101,[4_1|2]), (347,115,[4_1|2]), (347,134,[4_1|2]), (347,143,[4_1|2]), (347,151,[4_1|2]), (347,160,[4_1|2]), (347,165,[4_1|2]), (347,182,[4_1|2]), (347,206,[4_1|2]), (347,215,[4_1|2]), (347,245,[4_1|2]), (347,270,[4_1|2]), (347,284,[4_1|2]), (347,299,[4_1|2, 1_1|2]), (347,309,[4_1|2, 1_1|2]), (347,348,[4_1|2, 1_1|2]), (347,256,[4_1|2]), (347,340,[4_1|2]), (347,304,[4_1|2]), (347,314,[4_1|2]), (347,318,[4_1|2]), (347,323,[0_1|2]), (347,328,[4_1|2]), (347,333,[4_1|2]), (347,338,[4_1|2]), (347,343,[4_1|2]), (348,349,[3_1|2]), (349,350,[2_1|2]), (350,351,[5_1|2]), (351,352,[5_1|2]), (352,101,[4_1|2]), (352,115,[4_1|2]), (352,134,[4_1|2]), (352,143,[4_1|2]), (352,151,[4_1|2]), (352,160,[4_1|2]), (352,165,[4_1|2]), (352,182,[4_1|2]), (352,206,[4_1|2]), (352,215,[4_1|2]), (352,245,[4_1|2]), (352,270,[4_1|2]), (352,284,[4_1|2]), (352,299,[4_1|2, 1_1|2]), (352,309,[4_1|2, 1_1|2]), (352,348,[4_1|2, 1_1|2]), (352,256,[4_1|2]), (352,304,[4_1|2]), (352,314,[4_1|2]), (352,318,[4_1|2]), (352,323,[0_1|2]), (352,328,[4_1|2]), (352,333,[4_1|2]), (352,338,[4_1|2]), (352,343,[4_1|2]), (353,354,[2_1|2]), (354,355,[1_1|2]), (355,356,[2_1|2]), (356,97,[0_1|2]), (357,358,[1_1|1]), (358,359,[2_1|1]), (359,360,[5_1|1]), (360,361,[4_1|1]), (360,91,[4_1|1]), (360,96,[1_1|1]), (361,66,[1_1|1]), (372,373,[0_1|3]), (373,374,[4_1|3]), (374,375,[4_1|3]), (375,376,[0_1|3]), (376,126,[1_1|3]), (377,378,[3_1|3]), (378,379,[2_1|3]), (379,380,[2_1|3]), (380,256,[0_1|3]), (381,382,[2_1|3]), (382,383,[1_1|3]), (383,384,[2_1|3]), (384,256,[0_1|3]), (385,386,[3_1|3]), (386,387,[3_1|3]), (387,388,[3_1|3]), (388,389,[2_1|3]), (389,256,[0_1|3]), (390,391,[2_1|3]), (391,392,[1_1|3]), (392,393,[2_1|3]), (393,310,[0_1|3]), (394,395,[1_1|3]), (395,396,[2_1|3]), (396,397,[0_1|3]), (397,315,[4_1|3]), (397,344,[4_1|3]), (398,399,[2_1|3]), (399,400,[1_1|3]), (400,401,[2_1|3]), (401,115,[0_1|3]), (401,134,[0_1|3]), (401,143,[0_1|3]), (401,151,[0_1|3]), (401,160,[0_1|3]), (401,165,[0_1|3]), (401,182,[0_1|3]), (401,206,[0_1|3]), (401,215,[0_1|3]), (401,245,[0_1|3]), (401,270,[0_1|3]), (401,284,[0_1|3]), (401,299,[0_1|3]), (401,309,[0_1|3]), (401,348,[0_1|3]), (401,310,[0_1|3]), (402,403,[0_1|3]), (403,404,[1_1|3]), (404,405,[2_1|3]), (405,406,[5_1|3]), (406,251,[4_1|3]), (407,408,[1_1|3]), (408,409,[2_1|3]), (409,410,[2_1|3]), (410,411,[4_1|3]), (411,315,[1_1|3]), (411,344,[1_1|3]), (412,413,[1_1|3]), (413,414,[2_1|3]), (414,415,[0_1|3]), (414,412,[2_1|3]), (415,115,[4_1|3]), (415,134,[4_1|3]), (415,143,[4_1|3]), (415,151,[4_1|3]), (415,160,[4_1|3]), (415,165,[4_1|3]), (415,182,[4_1|3]), (415,206,[4_1|3]), (415,215,[4_1|3]), (415,245,[4_1|3]), (415,270,[4_1|3]), (415,284,[4_1|3]), (415,299,[4_1|3]), (415,309,[4_1|3]), (415,348,[4_1|3]), (415,315,[4_1|3]), (415,344,[4_1|3]), (415,310,[4_1|3]), (416,417,[0_1|3]), (417,418,[4_1|3]), (418,419,[4_1|3]), (419,420,[0_1|3]), (420,120,[1_1|3]), (420,157,[1_1|3]), (420,236,[1_1|3]), (420,290,[1_1|3]), (420,324,[1_1|3]), (421,422,[3_1|3]), (422,423,[3_1|3]), (423,424,[2_1|3]), (424,256,[0_1|3]), (425,426,[4_1|3]), (426,427,[0_1|3]), (427,428,[1_1|3]), (428,429,[3_1|3]), (429,315,[1_1|3]), (429,344,[1_1|3]), (430,431,[2_1|3]), (431,432,[1_1|3]), (432,433,[2_1|3]), (433,101,[0_1|3]), (433,115,[0_1|3, 1_1|2]), (433,134,[0_1|3, 1_1|2]), (433,143,[0_1|3, 1_1|2]), (433,151,[0_1|3, 1_1|2]), (433,160,[0_1|3, 1_1|2]), (433,165,[0_1|3, 1_1|2]), (433,182,[0_1|3, 1_1|2]), (433,206,[0_1|3, 1_1|2]), (433,215,[0_1|3, 1_1|2]), (433,245,[0_1|3, 1_1|2]), (433,270,[0_1|3]), (433,284,[0_1|3]), (433,299,[0_1|3]), (433,309,[0_1|3]), (433,348,[0_1|3]), (433,315,[0_1|3]), (433,344,[0_1|3]), (433,119,[0_1|2]), (433,124,[4_1|2]), (433,129,[0_1|2]), (433,139,[0_1|2]), (433,147,[3_1|2]), (433,156,[0_1|2]), (433,169,[0_1|2]), (433,174,[0_1|2]), (433,178,[2_1|2]), (433,186,[2_1|2]), (433,191,[0_1|2]), (433,196,[2_1|2]), (433,201,[2_1|2]), (433,211,[0_1|2]), (433,220,[4_1|2]), (433,225,[0_1|2]), (433,230,[0_1|2]), (433,235,[0_1|2]), (433,240,[0_1|2]), (433,250,[4_1|2]), (433,372,[2_1|3]), (433,377,[1_1|3]), (433,381,[3_1|3]), (433,385,[1_1|3]), (433,390,[1_1|3]), (433,394,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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 ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(5(1(x1)))) ->^+ 1(2(2(5(0(1(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. The pumping substitution is [x1 / 5(1(x1))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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 ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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