/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 91 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 143 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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(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(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_2(x_1)) -> 2(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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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 4. The certificate found is represented by the following graph. "[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] {(69,70,[0_1|0, 2_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]), (69,71,[0_1|1]), (69,75,[1_1|1]), (69,79,[0_1|1]), (69,84,[1_1|1]), (69,89,[1_1|1]), (69,94,[3_1|1]), (69,99,[5_1|1]), (69,104,[3_1|1]), (69,107,[1_1|1]), (69,111,[1_1|1]), (69,116,[1_1|1]), (69,121,[5_1|1]), (69,126,[1_1|1]), (69,131,[5_1|1]), (69,136,[1_1|1, 3_1|1, 4_1|1, 0_1|1, 2_1|1, 5_1|1]), (69,137,[2_1|2]), (69,142,[0_1|2]), 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cons_2_1|0, cons_5_1|0]), (71,72,[3_1|1]), (72,73,[1_1|1]), (73,74,[3_1|1]), (74,70,[1_1|1]), (75,76,[3_1|1]), (76,77,[0_1|1]), (77,78,[1_1|1]), (78,70,[4_1|1]), (79,80,[1_1|1]), (80,81,[3_1|1]), (81,82,[1_1|1]), (82,83,[3_1|1]), (83,70,[1_1|1]), (84,85,[3_1|1]), (85,86,[2_1|1]), (86,87,[1_1|1]), (87,88,[3_1|1]), (88,70,[0_1|1]), (88,71,[0_1|1]), (88,75,[1_1|1]), (88,79,[0_1|1]), (88,84,[1_1|1]), (88,89,[1_1|1]), (88,94,[3_1|1]), (88,99,[5_1|1]), (89,90,[3_1|1]), (90,91,[3_1|1]), (91,92,[1_1|1]), (92,93,[4_1|1]), (93,70,[0_1|1]), (93,71,[0_1|1]), (93,75,[1_1|1]), (93,79,[0_1|1]), (93,84,[1_1|1]), (93,89,[1_1|1]), (93,94,[3_1|1]), (93,99,[5_1|1]), (94,95,[0_1|1]), (95,96,[3_1|1]), (96,97,[1_1|1]), (97,98,[5_1|1]), (97,104,[3_1|1]), (97,107,[1_1|1]), (97,111,[1_1|1]), (97,116,[1_1|1]), (98,70,[1_1|1]), (99,100,[0_1|1]), (100,101,[3_1|1]), (101,102,[1_1|1]), (102,103,[5_1|1]), (102,104,[3_1|1]), (102,107,[1_1|1]), (102,111,[1_1|1]), (102,116,[1_1|1]), (103,70,[1_1|1]), (104,105,[1_1|1]), 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(422,161,[1_1|3]), (422,491,[1_1|3]), (422,495,[1_1|3]), (422,500,[1_1|3]), (423,424,[0_1|3]), (424,425,[3_1|3]), (425,426,[1_1|3]), (426,427,[5_1|3]), (427,156,[1_1|3]), (427,165,[1_1|3]), (427,170,[1_1|3]), (427,272,[1_1|3]), (427,300,[1_1|3]), (427,304,[1_1|3]), (427,309,[1_1|3]), (427,346,[1_1|3]), (427,143,[1_1|3]), (427,161,[1_1|3]), (427,491,[1_1|3]), (427,495,[1_1|3]), (427,500,[1_1|3]), (428,429,[0_1|3]), (429,430,[3_1|3]), (430,431,[3_1|3]), (431,432,[0_1|3]), (432,143,[1_1|3]), (432,161,[1_1|3]), (432,218,[1_1|3]), (432,491,[1_1|3]), (432,495,[1_1|3]), (432,500,[1_1|3]), (432,548,[1_1|3]), (433,434,[0_1|3]), (434,435,[3_1|3]), (435,436,[5_1|3]), (436,181,[0_1|3]), (436,194,[0_1|3]), (436,198,[0_1|3]), (436,357,[0_1|3]), (437,438,[5_1|3]), (438,439,[0_1|3]), (439,440,[0_1|3]), (440,181,[3_1|3]), (440,194,[3_1|3]), (440,198,[3_1|3]), (440,357,[3_1|3]), (441,442,[0_1|3]), (442,443,[3_1|3]), (443,444,[0_1|3]), (444,181,[2_1|3]), (444,194,[2_1|3]), (444,198,[2_1|3]), (444,357,[2_1|3]), (445,446,[0_1|3]), (446,447,[3_1|3]), (447,448,[3_1|3]), (448,181,[0_1|3]), (448,194,[0_1|3]), (448,198,[0_1|3]), (448,357,[0_1|3]), (449,450,[5_1|3]), (450,451,[0_1|3]), (451,452,[3_1|3]), (452,453,[3_1|3]), (453,181,[0_1|3]), (453,194,[0_1|3]), (453,198,[0_1|3]), (453,357,[0_1|3]), (454,455,[5_1|3]), (455,456,[0_1|3]), (456,457,[3_1|3]), (457,458,[5_1|3]), (458,181,[0_1|3]), (458,194,[0_1|3]), (458,198,[0_1|3]), (458,357,[0_1|3]), (459,460,[3_1|3]), (460,461,[0_1|3]), (461,462,[1_1|3]), (462,463,[3_1|3]), (463,181,[0_1|3]), (463,194,[0_1|3]), (463,198,[0_1|3]), (463,357,[0_1|3]), (464,465,[0_1|3]), (465,466,[1_1|3]), (466,467,[3_1|3]), (467,329,[5_1|3]), (467,292,[5_1|3]), (467,314,[5_1|3]), (467,483,[5_1|3]), (467,546,[5_1|3]), (467,533,[5_1|3]), (467,560,[5_1|3]), (468,469,[1_1|3]), (469,470,[3_1|3]), (470,471,[5_1|3]), (471,472,[0_1|3]), (472,138,[2_1|3]), (472,148,[2_1|3]), (472,366,[2_1|3]), (472,576,[2_1|3]), (473,474,[5_1|3]), (474,475,[0_1|3]), (475,476,[1_1|3]), (476,477,[4_1|3]), (477,142,[3_1|3]), (477,152,[3_1|3]), (477,160,[3_1|3]), (477,216,[3_1|3]), (477,220,[3_1|3]), (477,229,[3_1|3]), (477,239,[3_1|3]), (477,243,[3_1|3]), (477,248,[3_1|3]), (477,263,[3_1|3]), (477,292,[3_1|3]), (477,314,[3_1|3]), (477,331,[3_1|3]), (477,483,[3_1|3]), (477,217,[3_1|3]), (477,264,[3_1|3]), (477,546,[3_1|3]), (477,547,[3_1|3]), (478,479,[5_1|3]), (479,480,[1_1|3]), (480,481,[4_1|3]), (481,482,[0_1|3]), (482,142,[3_1|3]), (482,152,[3_1|3]), (482,160,[3_1|3]), (482,216,[3_1|3]), (482,220,[3_1|3]), (482,229,[3_1|3]), (482,239,[3_1|3]), (482,243,[3_1|3]), (482,248,[3_1|3]), (482,263,[3_1|3]), (482,292,[3_1|3]), (482,314,[3_1|3]), (482,331,[3_1|3]), (482,483,[3_1|3]), (482,217,[3_1|3]), (482,264,[3_1|3]), (482,546,[3_1|3]), (482,547,[3_1|3]), (483,484,[5_1|3]), (484,485,[1_1|3]), (485,486,[4_1|3]), (486,487,[3_1|3]), (487,143,[1_1|3]), (487,161,[1_1|3]), (487,218,[1_1|3]), (487,491,[1_1|3]), (487,495,[1_1|3]), (487,500,[1_1|3]), (487,548,[1_1|3]), (488,489,[1_1|3]), (489,490,[5_1|3]), (490,156,[1_1|3]), (490,165,[1_1|3]), (490,170,[1_1|3]), (490,272,[1_1|3]), (490,300,[1_1|3]), (490,304,[1_1|3]), (490,309,[1_1|3]), (490,346,[1_1|3]), (490,491,[1_1|3]), (490,495,[1_1|3]), (490,500,[1_1|3]), (490,143,[1_1|3]), (490,161,[1_1|3]), (491,492,[3_1|3]), (492,493,[1_1|3]), (493,494,[3_1|3]), (494,156,[5_1|3]), (494,165,[5_1|3]), (494,170,[5_1|3]), (494,272,[5_1|3]), (494,300,[5_1|3]), (494,304,[5_1|3]), (494,309,[5_1|3]), (494,346,[5_1|3]), (494,491,[5_1|3]), (494,495,[5_1|3]), (494,500,[5_1|3]), (494,143,[5_1|3]), (494,161,[5_1|3]), (495,496,[3_1|3]), (496,497,[3_1|3]), (497,498,[3_1|3]), (498,499,[5_1|3]), (499,156,[1_1|3]), (499,165,[1_1|3]), (499,170,[1_1|3]), (499,272,[1_1|3]), (499,300,[1_1|3]), (499,304,[1_1|3]), (499,309,[1_1|3]), (499,346,[1_1|3]), (499,491,[1_1|3]), (499,495,[1_1|3]), (499,500,[1_1|3]), (499,143,[1_1|3]), (499,161,[1_1|3]), (500,501,[3_1|3]), (501,502,[5_1|3]), (502,503,[5_1|3]), (503,504,[1_1|3]), (504,156,[4_1|3]), (504,165,[4_1|3]), (504,170,[4_1|3]), (504,272,[4_1|3]), (504,300,[4_1|3]), (504,304,[4_1|3]), (504,309,[4_1|3]), (504,346,[4_1|3]), (504,491,[4_1|3]), (504,495,[4_1|3]), (504,500,[4_1|3]), (504,143,[4_1|3]), (504,161,[4_1|3]), (505,506,[3_1|3]), (506,507,[5_1|3]), (507,508,[0_1|3]), (508,181,[1_1|3]), (508,194,[1_1|3]), (508,198,[1_1|3]), (508,357,[1_1|3]), (509,510,[2_1|3]), (510,511,[3_1|3]), (511,512,[3_1|3]), (512,513,[0_1|3]), (513,143,[1_1|3]), (513,161,[1_1|3]), (513,218,[1_1|3]), (513,491,[1_1|3]), (513,495,[1_1|3]), (513,500,[1_1|3]), (513,548,[1_1|3]), (514,515,[3_1|3]), (515,516,[0_1|3]), (516,517,[3_1|3]), (517,217,[2_1|3]), (517,264,[2_1|3]), (517,547,[2_1|3]), (517,483,[2_1|3]), (518,519,[3_1|3]), (519,520,[3_1|3]), (520,521,[0_1|3]), (521,522,[2_1|3]), (522,217,[3_1|3]), (522,264,[3_1|3]), (522,547,[3_1|3]), (522,483,[3_1|3]), (523,524,[3_1|3]), (524,525,[5_1|3]), (525,526,[2_1|3]), (526,527,[0_1|3]), (527,217,[3_1|3]), (527,264,[3_1|3]), (527,547,[3_1|3]), (527,483,[3_1|3]), (528,529,[1_1|2]), (529,530,[3_1|2]), (530,531,[5_1|2]), (531,532,[0_1|2]), (531,142,[0_1|2]), (531,147,[2_1|2]), (531,152,[0_1|2]), (531,156,[1_1|2]), (531,160,[0_1|2]), (531,165,[1_1|2]), (531,170,[1_1|2]), (531,175,[3_1|2]), (531,180,[5_1|2]), (531,385,[0_1|3]), (531,390,[2_1|3]), (531,395,[0_1|3]), (531,399,[1_1|3]), (531,403,[0_1|3]), (531,408,[1_1|3]), (531,413,[1_1|3]), (531,418,[3_1|3]), (531,423,[5_1|3]), (532,136,[1_1|2]), (532,156,[1_1|2]), (532,165,[1_1|2]), (532,170,[1_1|2]), (532,272,[1_1|2]), (532,300,[1_1|2]), (532,304,[1_1|2]), (532,309,[1_1|2]), (532,346,[1_1|2]), (533,534,[0_1|3]), (534,535,[5_1|3]), (535,536,[1_1|3]), (536,329,[3_1|3]), (536,292,[3_1|3]), (536,314,[3_1|3]), (536,483,[3_1|3]), (536,546,[3_1|3]), (537,538,[0_1|3]), (538,539,[3_1|3]), (539,540,[3_1|3]), (540,541,[0_1|3]), (541,238,[1_1|3]), (542,543,[1_1|3]), (543,544,[0_1|3]), (544,545,[3_1|3]), (545,143,[2_1|3]), (545,161,[2_1|3]), (545,218,[2_1|3]), (545,491,[2_1|3]), (545,495,[2_1|3]), (545,500,[2_1|3]), (545,548,[2_1|3]), (546,547,[0_1|3]), (547,548,[1_1|3]), (548,549,[3_1|3]), (549,292,[5_1|3]), (549,314,[5_1|3]), (549,483,[5_1|3]), (549,546,[5_1|3]), (549,533,[5_1|3]), (549,560,[5_1|3]), (550,551,[0_1|4]), (551,552,[3_1|4]), (552,553,[3_1|4]), (553,554,[0_1|4]), (554,466,[1_1|4]), (555,556,[3_1|3]), (556,557,[1_1|3]), (557,558,[3_1|3]), (558,559,[1_1|3]), (559,300,[5_1|3]), (559,304,[5_1|3]), (559,309,[5_1|3]), (559,491,[5_1|3]), (559,495,[5_1|3]), (559,500,[5_1|3]), (560,561,[1_1|3]), (561,562,[3_1|3]), (562,563,[5_1|3]), (563,564,[0_1|3]), (564,300,[1_1|3]), (564,304,[1_1|3]), (564,309,[1_1|3]), (564,491,[1_1|3]), (564,495,[1_1|3]), (564,500,[1_1|3]), (565,566,[5_1|4]), (566,567,[0_1|4]), (567,568,[1_1|4]), (568,569,[4_1|4]), (569,544,[3_1|4]), (570,571,[5_1|4]), (571,572,[1_1|4]), (572,573,[4_1|4]), (573,574,[0_1|4]), (574,544,[3_1|4]), (575,576,[0_1|4]), (576,577,[3_1|4]), (577,578,[3_1|4]), (578,579,[0_1|4]), (579,548,[1_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 (full) 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))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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(1(x1))) ->^+ 1(3(3(3(5(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 / 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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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(0(1(x1))) -> 2(0(3(3(0(1(x1)))))) 0(1(0(x1))) -> 0(1(3(4(0(3(x1)))))) 0(1(0(x1))) -> 2(0(3(0(1(4(x1)))))) 0(1(1(x1))) -> 0(3(1(3(1(x1))))) 0(1(1(x1))) -> 1(3(0(1(4(x1))))) 0(1(1(x1))) -> 0(1(3(1(3(1(x1)))))) 0(1(1(x1))) -> 1(3(2(1(3(0(x1)))))) 0(1(1(x1))) -> 1(3(3(1(4(0(x1)))))) 0(1(1(x1))) -> 3(0(3(1(5(1(x1)))))) 0(1(1(x1))) -> 5(0(3(1(5(1(x1)))))) 0(5(0(x1))) -> 3(0(3(5(0(x1))))) 0(5(0(x1))) -> 3(5(0(0(3(x1))))) 0(5(0(x1))) -> 5(0(3(0(2(x1))))) 0(5(0(x1))) -> 5(0(3(3(0(x1))))) 0(5(0(x1))) -> 4(5(0(3(3(0(x1)))))) 0(5(0(x1))) -> 4(5(0(3(5(0(x1)))))) 0(5(0(x1))) -> 5(3(0(1(3(0(x1)))))) 2(0(0(x1))) -> 0(3(0(3(2(x1))))) 2(0(0(x1))) -> 0(3(3(0(2(3(x1)))))) 2(0(0(x1))) -> 0(3(5(2(0(3(x1)))))) 5(1(0(x1))) -> 3(5(0(1(4(3(x1)))))) 5(1(0(x1))) -> 3(5(1(4(0(3(x1)))))) 5(1(1(x1))) -> 3(1(5(1(x1)))) 5(1(1(x1))) -> 1(3(1(3(5(x1))))) 5(1(1(x1))) -> 1(3(3(3(5(1(x1)))))) 5(1(1(x1))) -> 1(3(5(5(1(4(x1)))))) 0(2(0(1(x1)))) -> 0(2(3(3(0(1(x1)))))) 0(5(1(0(x1)))) -> 0(0(1(3(5(x1))))) 0(5(4(0(x1)))) -> 0(4(5(0(3(x1))))) 2(0(2(0(x1)))) -> 3(0(3(0(2(2(x1)))))) 2(0(4(1(x1)))) -> 2(3(0(1(4(4(x1)))))) 2(0(5(0(x1)))) -> 0(0(3(5(2(x1))))) 2(2(4(1(x1)))) -> 3(2(4(3(2(1(x1)))))) 5(1(0(1(x1)))) -> 0(5(1(4(3(1(x1)))))) 5(1(1(0(x1)))) -> 0(5(1(5(1(x1))))) 5(1(2(0(x1)))) -> 3(1(3(5(0(2(x1)))))) 5(1(5(0(x1)))) -> 5(3(5(0(1(x1))))) 5(2(0(1(x1)))) -> 5(1(0(3(2(x1))))) 5(3(1(1(x1)))) -> 5(3(1(3(1(5(x1)))))) 5(4(1(1(x1)))) -> 5(1(4(1(4(5(x1)))))) 5(5(1(0(x1)))) -> 5(0(5(1(3(x1))))) 5(5(1(1(x1)))) -> 5(1(3(5(0(1(x1)))))) 0(2(4(1(0(x1))))) -> 2(4(0(0(1(3(x1)))))) 0(5(5(1(1(x1))))) -> 5(1(3(5(0(1(x1)))))) 2(2(2(4(1(x1))))) -> 1(2(2(1(4(2(x1)))))) 2(5(0(1(1(x1))))) -> 5(1(2(0(1(3(x1)))))) 5(0(2(4(1(x1))))) -> 5(1(4(0(3(2(x1)))))) 5(2(4(1(0(x1))))) -> 0(2(3(4(5(1(x1)))))) 5(3(0(4(1(x1))))) -> 5(3(0(1(4(1(x1)))))) 5(3(4(1(1(x1))))) -> 1(4(3(5(2(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(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_2(x_1)) -> 2(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