WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 63 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 142 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 2 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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: 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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: 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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: 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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663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682] {(54,55,[0_1|0, 5_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0]), (54,56,[1_1|1]), (54,60,[1_1|1]), (54,64,[2_1|1]), (54,69,[0_1|1]), (54,73,[1_1|1]), (54,77,[1_1|1]), (54,81,[2_1|1]), (54,85,[1_1|1]), (54,90,[2_1|1]), (54,95,[4_1|1]), (54,100,[4_1|1]), (54,105,[1_1|1]), (54,110,[4_1|1]), (54,115,[4_1|1]), (54,120,[3_1|1]), (54,124,[5_1|1]), (54,128,[5_1|1]), (54,132,[0_1|1]), (54,137,[5_1|1]), (54,142,[3_1|1]), (54,146,[2_1|1]), (54,151,[3_1|1]), (54,156,[2_1|1]), (54,161,[1_1|1]), (54,166,[3_1|1]), (54,171,[4_1|1]), (54,176,[5_1|1]), (54,181,[5_1|1]), (54,186,[1_1|1]), (54,191,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 0_1|1, 5_1|1]), (54,216,[1_1|2]), (54,220,[1_1|2]), (54,224,[2_1|2]), (54,229,[1_1|2]), (54,233,[0_1|2]), (54,238,[2_1|2]), (54,243,[4_1|2]), (54,248,[0_1|2]), (54,252,[1_1|2]), (54,256,[1_1|2]), (54,260,[2_1|2]), 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(542,543,[3_1|3]), (543,544,[0_1|3]), (544,545,[0_1|3]), (545,546,[5_1|3]), (546,239,[2_1|3]), (547,548,[4_1|3]), (548,549,[2_1|3]), (549,550,[3_1|3]), (550,322,[0_1|3]), (551,552,[3_1|3]), (552,553,[0_1|3]), (553,554,[4_1|3]), (554,555,[0_1|3]), (555,322,[0_1|3]), (556,557,[4_1|3]), (557,558,[0_1|3]), (558,559,[2_1|3]), (559,560,[3_1|3]), (560,322,[0_1|3]), (561,562,[2_1|3]), (562,563,[3_1|3]), (563,564,[0_1|3]), (564,565,[4_1|3]), (565,322,[0_1|3]), (566,567,[3_1|3]), (567,568,[0_1|3]), (568,569,[0_1|3]), (569,570,[5_1|3]), (570,322,[2_1|3]), (571,572,[4_1|3]), (572,573,[5_1|3]), (573,574,[2_1|3]), (574,224,[1_1|3]), (574,238,[1_1|3]), (574,260,[1_1|3]), (574,269,[1_1|3]), (574,293,[1_1|3]), (574,323,[1_1|3]), (574,331,[1_1|3]), (574,372,[1_1|3]), (574,382,[1_1|3]), (574,304,[1_1|3]), (574,551,[1_1|3]), (575,576,[1_1|3]), (576,577,[3_1|3]), (577,578,[4_1|3]), (578,579,[5_1|3]), (578,636,[3_1|3]), (578,641,[4_1|3]), (578,646,[5_1|3]), (578,651,[5_1|3]), (578,656,[3_1|3]), (579,224,[4_1|3]), (579,238,[4_1|3]), (579,260,[4_1|3]), (579,269,[4_1|3]), (579,293,[4_1|3]), (579,323,[4_1|3]), (579,331,[4_1|3]), (579,372,[4_1|3]), (579,382,[4_1|3]), (579,304,[4_1|3]), (579,551,[4_1|3]), (580,581,[3_1|3]), (581,582,[4_1|3]), (582,583,[2_1|3]), (583,584,[1_1|3]), (584,224,[5_1|3]), (584,238,[5_1|3]), (584,260,[5_1|3]), (584,269,[5_1|3]), (584,293,[5_1|3]), (584,323,[5_1|3]), (584,331,[5_1|3]), (584,372,[5_1|3]), (584,382,[5_1|3]), (584,304,[5_1|3]), (584,551,[5_1|3]), (585,586,[1_1|3]), (586,587,[5_1|3]), (587,588,[3_1|3]), (588,589,[4_1|3]), (589,257,[1_1|3]), (589,265,[1_1|3]), (589,285,[1_1|3]), (589,276,[1_1|3]), (590,591,[4_1|3]), (591,592,[2_1|3]), (592,593,[3_1|3]), (593,239,[0_1|3]), (594,595,[1_1|3]), (595,596,[2_1|3]), (596,597,[3_1|3]), (597,598,[4_1|3]), (598,239,[5_1|3]), (599,600,[5_1|3]), (600,601,[3_1|3]), (601,602,[3_1|3]), (602,603,[0_1|3]), (603,239,[2_1|3]), (604,605,[1_1|3]), (605,606,[3_1|3]), (606,607,[5_1|3]), (607,239,[0_1|3]), (608,609,[1_1|3]), (609,610,[0_1|3]), (610,611,[2_1|3]), (611,239,[5_1|3]), (612,613,[3_1|3]), (613,614,[0_1|3]), (614,615,[0_1|3]), (615,616,[1_1|3]), (616,239,[5_1|3]), (617,618,[1_1|3]), (618,619,[5_1|3]), (619,620,[3_1|3]), (620,621,[4_1|3]), (621,276,[1_1|3]), (622,623,[4_1|3]), (623,624,[5_1|3]), (624,625,[2_1|3]), (625,304,[1_1|3]), (625,551,[1_1|3]), (625,291,[1_1|3]), (625,321,[1_1|3]), (625,549,[1_1|3]), (626,627,[1_1|3]), (627,628,[3_1|3]), (628,629,[4_1|3]), (629,630,[5_1|3]), (629,678,[5_1|3]), (630,304,[4_1|3]), (630,551,[4_1|3]), (630,291,[4_1|3]), (630,321,[4_1|3]), (630,549,[4_1|3]), (631,632,[3_1|3]), (632,633,[4_1|3]), (633,634,[2_1|3]), (634,635,[1_1|3]), (635,304,[5_1|3]), (635,551,[5_1|3]), (635,291,[5_1|3]), (635,321,[5_1|3]), (635,549,[5_1|3]), (636,637,[4_1|3]), (637,638,[4_1|3]), (638,639,[1_1|3]), (639,640,[5_1|3]), (640,262,[4_1|3]), (641,642,[1_1|3]), (642,643,[3_1|3]), (643,644,[5_1|3]), (644,645,[0_1|3]), (645,262,[4_1|3]), (646,647,[1_1|3]), (647,648,[3_1|3]), (648,649,[4_1|3]), (649,650,[5_1|3]), (650,262,[4_1|3]), (651,652,[1_1|3]), (652,653,[5_1|3]), (653,654,[3_1|3]), (654,655,[4_1|3]), (655,262,[4_1|3]), (656,657,[4_1|3]), (657,658,[0_1|3]), (658,659,[4_1|3]), (659,660,[5_1|3]), (659,387,[1_1|2]), (660,263,[1_1|3]), (661,662,[2_1|3]), (662,663,[5_1|3]), (663,664,[1_1|3]), (664,235,[4_1|3]), (665,666,[2_1|3]), (666,667,[1_1|3]), (667,668,[3_1|3]), (668,235,[4_1|3]), (669,670,[5_1|3]), (670,671,[2_1|3]), (671,672,[1_1|3]), (672,235,[4_1|3]), (673,674,[3_1|3]), (674,675,[4_1|3]), (675,676,[5_1|3]), (676,677,[2_1|3]), (677,235,[1_1|3]), (678,679,[4_1|3]), (679,680,[0_1|3]), (680,681,[3_1|3]), (681,682,[2_1|3]), (682,239,[3_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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 5(1(4(2(x1)))) ->^+ 3(3(4(2(1(5(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [x1 / 1(4(2(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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: 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(2(x1)))) -> 1(3(0(1(2(x1))))) 0(1(1(2(x1)))) -> 1(3(1(0(2(x1))))) 0(1(1(2(x1)))) -> 2(3(1(3(0(1(x1)))))) 0(1(2(0(x1)))) -> 1(0(0(2(2(x1))))) 0(1(2(4(x1)))) -> 0(2(3(4(1(x1))))) 0(1(2(4(x1)))) -> 1(0(4(2(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(0(4(x1))))) 0(1(2(4(x1)))) -> 2(1(4(0(3(x1))))) 0(1(2(4(x1)))) -> 1(2(3(4(0(5(x1)))))) 0(1(2(4(x1)))) -> 2(1(1(0(3(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(2(3(0(x1)))))) 0(1(2(4(x1)))) -> 4(5(0(2(3(1(x1)))))) 0(4(2(0(x1)))) -> 0(4(2(3(0(x1))))) 0(4(2(0(x1)))) -> 2(3(0(4(0(0(x1)))))) 0(4(2(0(x1)))) -> 3(4(0(2(3(0(x1)))))) 0(4(2(0(x1)))) -> 4(2(3(0(4(0(x1)))))) 0(4(2(0(x1)))) -> 4(3(0(0(5(2(x1)))))) 0(4(2(4(x1)))) -> 4(0(2(3(1(4(x1)))))) 0(4(2(4(x1)))) -> 4(0(4(2(0(3(x1)))))) 5(1(2(0(x1)))) -> 2(1(3(5(0(x1))))) 5(1(2(0(x1)))) -> 3(1(0(2(5(x1))))) 5(1(2(0(x1)))) -> 2(3(0(0(1(5(x1)))))) 5(1(2(4(x1)))) -> 3(2(5(1(4(x1))))) 5(1(2(4(x1)))) -> 5(2(1(3(4(x1))))) 5(1(2(4(x1)))) -> 5(5(2(1(4(x1))))) 5(1(2(4(x1)))) -> 0(3(4(5(2(1(x1)))))) 5(1(4(2(x1)))) -> 3(4(5(2(1(x1))))) 5(1(4(2(x1)))) -> 2(1(3(4(5(4(x1)))))) 5(1(4(2(x1)))) -> 3(3(4(2(1(5(x1)))))) 5(4(1(4(x1)))) -> 3(4(4(1(5(4(x1)))))) 5(4(1(4(x1)))) -> 4(1(3(5(0(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(3(4(5(4(x1)))))) 5(4(1(4(x1)))) -> 5(1(5(3(4(4(x1)))))) 5(4(2(0(x1)))) -> 5(4(2(3(0(x1))))) 5(4(2(0(x1)))) -> 0(1(2(3(4(5(x1)))))) 5(4(2(0(x1)))) -> 4(5(3(3(0(2(x1)))))) 0(1(2(0(4(x1))))) -> 0(2(4(1(3(0(x1)))))) 0(1(2(0(4(x1))))) -> 2(0(3(4(0(1(x1)))))) 0(1(2(0(4(x1))))) -> 4(0(2(3(1(0(x1)))))) 0(1(4(2(2(x1))))) -> 1(2(3(4(0(2(x1)))))) 5(0(1(2(4(x1))))) -> 3(0(2(1(4(5(x1)))))) 5(1(2(2(4(x1))))) -> 5(2(2(1(4(5(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(2(1(3(4(x1)))))) 5(1(2(4(0(x1))))) -> 5(0(3(2(1(4(x1)))))) 5(1(3(1(4(x1))))) -> 1(5(1(3(4(0(x1)))))) 5(1(4(1(2(x1))))) -> 2(1(5(3(4(1(x1)))))) 5(4(1(1(4(x1))))) -> 1(1(3(4(5(4(x1)))))) 5(4(1(4(0(x1))))) -> 3(4(0(4(5(1(x1)))))) 5(4(3(2(0(x1))))) -> 5(4(0(3(2(3(x1)))))) 5(4(5(2(0(x1))))) -> 5(0(5(4(4(2(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL