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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 43 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 124 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 (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(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(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(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(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. 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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] {(59,60,[0_1|0, 3_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]), (59,61,[2_1|1]), (59,66,[1_1|1]), (59,71,[4_1|1]), (59,76,[5_1|1]), (59,81,[5_1|1]), (59,86,[4_1|1]), (59,91,[5_1|1]), (59,96,[3_1|1]), (59,101,[3_1|1]), (59,106,[5_1|1]), (59,111,[5_1|1]), (59,116,[5_1|1]), (59,121,[5_1|1]), (59,126,[1_1|1, 2_1|1, 4_1|1, 5_1|1, 0_1|1, 3_1|1]), (59,132,[0_1|2]), (59,136,[0_1|2]), (59,140,[0_1|2]), (59,144,[0_1|2]), (59,148,[0_1|2]), (59,153,[0_1|2]), (59,157,[2_1|2]), 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(423,362,[0_1|3]), (423,366,[0_1|3]), (424,425,[0_1|3]), (425,426,[1_1|3]), (426,427,[2_1|3]), (427,428,[2_1|3]), (428,132,[4_1|3]), (428,136,[4_1|3]), (428,140,[4_1|3]), (428,144,[4_1|3]), (428,148,[4_1|3]), (428,153,[4_1|3]), (428,161,[4_1|3]), (428,166,[4_1|3]), (428,171,[4_1|3]), (428,176,[4_1|3]), (428,180,[4_1|3]), (428,200,[4_1|3]), (428,275,[4_1|3]), (428,280,[4_1|3]), (428,285,[4_1|3]), (428,290,[4_1|3]), (428,295,[4_1|3]), (428,314,[4_1|3]), (428,343,[4_1|3]), (428,348,[4_1|3]), (428,357,[4_1|3]), (428,361,[4_1|3]), (428,365,[4_1|3]), (428,133,[4_1|3]), (428,137,[4_1|3]), (428,141,[4_1|3]), (428,370,[4_1|3]), (428,374,[4_1|3]), (428,145,[4_1|3]), (428,149,[4_1|3]), (428,154,[4_1|3]), (428,162,[4_1|3]), (428,167,[4_1|3]), (428,172,[4_1|3]), (428,181,[4_1|3]), (428,286,[4_1|3]), (428,291,[4_1|3]), (428,296,[4_1|3]), (428,217,[4_1|3]), (428,349,[4_1|3]), (428,358,[4_1|3]), (428,362,[4_1|3]), (428,366,[4_1|3]), (429,430,[0_1|3]), (430,431,[2_1|3]), (431,432,[4_1|3]), (432,433,[1_1|3]), (433,132,[4_1|3]), (433,136,[4_1|3]), (433,140,[4_1|3]), (433,144,[4_1|3]), (433,148,[4_1|3]), (433,153,[4_1|3]), (433,161,[4_1|3]), (433,166,[4_1|3]), (433,171,[4_1|3]), (433,176,[4_1|3]), (433,180,[4_1|3]), (433,200,[4_1|3]), (433,275,[4_1|3]), (433,280,[4_1|3]), (433,285,[4_1|3]), (433,290,[4_1|3]), (433,295,[4_1|3]), (433,314,[4_1|3]), (433,343,[4_1|3]), (433,348,[4_1|3]), (433,357,[4_1|3]), (433,361,[4_1|3]), (433,365,[4_1|3]), (433,133,[4_1|3]), (433,137,[4_1|3]), (433,141,[4_1|3]), (433,370,[4_1|3]), (433,374,[4_1|3]), (433,145,[4_1|3]), (433,149,[4_1|3]), (433,154,[4_1|3]), (433,162,[4_1|3]), (433,167,[4_1|3]), (433,172,[4_1|3]), (433,181,[4_1|3]), (433,286,[4_1|3]), (433,291,[4_1|3]), (433,296,[4_1|3]), (433,217,[4_1|3]), (433,349,[4_1|3]), (433,358,[4_1|3]), (433,362,[4_1|3]), (433,366,[4_1|3]), (434,435,[0_1|3]), (435,436,[3_1|3]), (436,437,[1_1|3]), (437,305,[4_1|3]), (437,178,[4_1|3]), (438,439,[3_1|3]), (439,440,[0_1|3]), (440,441,[1_1|3]), (441,305,[2_1|3]), (441,178,[2_1|3]), (442,443,[0_1|3]), (443,444,[1_1|3]), (444,445,[3_1|3]), (445,446,[2_1|3]), (446,305,[5_1|3]), (446,178,[5_1|3]), (447,448,[1_1|3]), (448,449,[1_1|3]), (449,450,[4_1|3]), (450,451,[5_1|3]), (451,310,[0_1|3]), (451,329,[0_1|3]), (451,316,[0_1|3]), (452,453,[0_1|3]), (453,454,[3_1|3]), (454,455,[1_1|3]), (455,244,[4_1|3]), (456,457,[0_1|2]), (457,458,[2_1|2]), (458,459,[1_1|2]), (459,126,[4_1|2]), (459,190,[4_1|2]), (459,195,[4_1|2]), (459,210,[4_1|2]), (459,352,[4_1|2]), (460,461,[2_1|2]), (461,462,[4_1|2]), (462,463,[2_1|2]), (463,464,[5_1|2]), (464,126,[0_1|2]), (464,190,[0_1|2, 1_1|2]), (464,195,[0_1|2, 1_1|2]), (464,210,[0_1|2, 1_1|2]), (464,352,[0_1|2, 1_1|2]), (464,132,[0_1|2]), (464,136,[0_1|2]), (464,140,[0_1|2]), (464,144,[0_1|2]), (464,148,[0_1|2]), (464,153,[0_1|2]), (464,157,[2_1|2]), (464,161,[0_1|2]), (464,166,[0_1|2]), (464,171,[0_1|2]), (464,176,[0_1|2]), (464,180,[0_1|2]), (464,185,[3_1|2]), (464,348,[0_1|2]), (464,200,[0_1|2]), (464,205,[2_1|2]), (464,215,[4_1|2]), (464,220,[5_1|2]), (464,225,[5_1|2]), (464,230,[4_1|2]), (464,235,[5_1|2]), (464,240,[5_1|2]), (464,245,[3_1|2]), (464,250,[3_1|2]), (464,255,[3_1|2]), (464,260,[5_1|2]), (464,265,[5_1|2]), (464,270,[5_1|2]), (464,275,[0_1|2]), (464,280,[0_1|2]), (464,285,[0_1|2]), (464,290,[0_1|2]), (464,295,[0_1|2]), (464,299,[5_1|2]), (464,357,[0_1|3]), (464,361,[0_1|3]), (464,365,[0_1|3]), (465,466,[4_1|3]), (466,467,[3_1|3]), (467,468,[2_1|3]), (468,469,[0_1|3]), (468,357,[0_1|3]), (468,361,[0_1|3]), (469,132,[0_1|3]), (469,136,[0_1|3]), (469,140,[0_1|3]), (469,144,[0_1|3]), (469,148,[0_1|3]), (469,153,[0_1|3]), (469,161,[0_1|3]), (469,166,[0_1|3]), (469,171,[0_1|3]), (469,176,[0_1|3]), (469,180,[0_1|3]), (469,200,[0_1|3]), (469,275,[0_1|3]), (469,280,[0_1|3]), (469,285,[0_1|3]), (469,290,[0_1|3]), (469,295,[0_1|3]), (469,314,[0_1|3]), (469,343,[0_1|3]), (469,348,[0_1|3]), (469,357,[0_1|3]), (469,361,[0_1|3]), (469,365,[0_1|3]), (469,133,[0_1|3]), (469,137,[0_1|3]), (469,141,[0_1|3]), (469,370,[0_1|3]), (469,374,[0_1|3]), (469,145,[0_1|3]), (469,149,[0_1|3]), (469,154,[0_1|3]), (469,162,[0_1|3]), (469,167,[0_1|3]), (469,172,[0_1|3]), (469,181,[0_1|3]), (469,286,[0_1|3]), (469,291,[0_1|3]), (469,296,[0_1|3]), (469,349,[0_1|3]), (469,358,[0_1|3]), (469,362,[0_1|3]), (469,366,[0_1|3]), (470,471,[2_1|3]), (471,472,[4_1|3]), (472,473,[2_1|3]), (473,474,[5_1|3]), (474,310,[0_1|3]), (474,329,[0_1|3]), (474,316,[0_1|3]), (475,476,[1_1|3]), (476,477,[4_1|3]), (477,478,[5_1|3]), (478,479,[0_1|3]), (479,310,[2_1|3]), (479,329,[2_1|3]), (479,316,[2_1|3]), (480,481,[2_1|3]), (481,482,[5_1|3]), (482,483,[1_1|3]), (483,484,[4_1|3]), (484,310,[0_1|3]), (484,329,[0_1|3]), (484,316,[0_1|3]), (485,486,[3_1|3]), (486,487,[0_1|3]), (487,488,[5_1|3]), (488,489,[1_1|3]), (489,310,[4_1|3]), (489,329,[4_1|3]), (489,316,[4_1|3]), (490,491,[5_1|3]), (491,492,[0_1|3]), (492,493,[5_1|3]), (493,494,[1_1|3]), (494,310,[4_1|3]), (494,329,[4_1|3]), (494,316,[4_1|3]), (495,496,[0_1|2]), (496,497,[3_1|2]), (497,498,[1_1|2]), (498,126,[4_1|2]), (498,132,[4_1|2]), (498,136,[4_1|2]), (498,140,[4_1|2]), (498,144,[4_1|2]), (498,148,[4_1|2]), (498,153,[4_1|2]), (498,161,[4_1|2]), (498,166,[4_1|2]), (498,171,[4_1|2]), (498,176,[4_1|2]), (498,180,[4_1|2]), (498,200,[4_1|2]), (498,275,[4_1|2]), (498,280,[4_1|2]), (498,285,[4_1|2]), (498,290,[4_1|2]), (498,295,[4_1|2]), (498,314,[4_1|2]), (498,343,[4_1|2]), (498,348,[4_1|2]), (498,357,[4_1|2]), (498,361,[4_1|2]), (498,365,[4_1|2]), (499,500,[0_1|4]), (500,501,[2_1|4]), (501,502,[1_1|4]), (502,418,[2_1|4]), (502,426,[2_1|4]), (502,444,[2_1|4]), (503,504,[0_1|4]), (504,505,[3_1|4]), (505,506,[1_1|4]), (506,418,[4_1|4]), (506,426,[4_1|4]), (506,444,[4_1|4]), (507,508,[0_1|4]), (508,509,[3_1|4]), (509,510,[1_1|4]), (510,511,[2_1|4]), (511,441,[2_1|4]), (512,513,[0_1|3]), (513,514,[1_1|3]), (514,515,[2_1|3]), (515,217,[4_1|3]), (516,517,[1_1|3]), (517,518,[2_1|3]), (518,519,[0_1|3]), (519,217,[0_1|3]), (520,521,[0_1|3]), (521,522,[1_1|3]), (522,523,[2_1|3]), (523,524,[2_1|3]), (524,217,[4_1|3]), (525,526,[0_1|3]), (526,527,[2_1|3]), (527,528,[4_1|3]), (528,529,[1_1|3]), (529,217,[4_1|3]), (530,531,[0_1|3]), (531,532,[2_1|3]), (532,533,[1_1|3]), (533,155,[4_1|3]), (533,163,[4_1|3]), (533,182,[4_1|3]), (534,535,[0_1|3]), (535,536,[3_1|3]), (536,537,[1_1|3]), (537,538,[2_1|3]), (538,155,[2_1|3]), (538,163,[2_1|3]), (538,182,[2_1|3]), (539,540,[0_1|4]), (540,541,[2_1|4]), (541,542,[1_1|4]), (542,411,[2_1|4]), (543,544,[0_1|4]), (544,545,[3_1|4]), (545,546,[1_1|4]), (546,411,[4_1|4]), (547,548,[2_1|4]), (548,549,[0_1|4]), (549,550,[3_1|4]), (550,551,[1_1|4]), (551,370,[4_1|4]), (551,374,[4_1|4]), (551,358,[4_1|4]), (551,362,[4_1|4]), (551,366,[4_1|4]), (551,378,[4_1|4]), (551,371,[4_1|4]), (551,375,[4_1|4]), (552,553,[0_1|4]), (553,554,[2_1|4]), (554,555,[1_1|4]), (555,514,[2_1|4]), (555,522,[2_1|4]), (556,557,[0_1|4]), (557,558,[3_1|4]), (558,559,[1_1|4]), (559,514,[4_1|4]), (559,522,[4_1|4])}" ---------------------------------------- (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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(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(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (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(4(x1)))) ->^+ 1(2(4(2(5(0(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(5(4(x1)))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(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(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(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(4(x_1)) -> 4(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)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST