/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).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), 63 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 83 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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(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)) 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 (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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: 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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: 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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: 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 2. The certificate found is represented by the following graph. "[67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431] {(67,68,[0_1|0, 5_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]), (67,69,[0_1|1]), (67,73,[0_1|1]), (67,77,[0_1|1]), (67,81,[1_1|1]), 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(406,268,[2_1|2]), (406,273,[2_1|2]), (406,278,[2_1|2]), (406,283,[2_1|2]), (406,288,[2_1|2]), (406,298,[2_1|2]), (406,367,[2_1|2]), (406,402,[2_1|2]), (406,284,[2_1|2]), (407,408,[4_1|2]), (408,409,[2_1|2]), (409,410,[5_1|2]), (410,411,[5_1|2]), (411,204,[5_1|2]), (411,307,[5_1|2]), (411,311,[5_1|2]), (411,320,[5_1|2]), (411,330,[5_1|2]), (411,382,[5_1|2]), (411,387,[5_1|2]), (411,350,[1_1|2]), (411,354,[1_1|2]), (411,358,[1_1|2]), (411,362,[1_1|2]), (411,367,[2_1|2]), (411,372,[3_1|2]), (411,377,[4_1|2]), (411,392,[1_1|2]), (411,397,[1_1|2]), (411,402,[2_1|2]), (411,407,[0_1|2]), (411,412,[0_1|2]), (412,413,[4_1|2]), (413,414,[5_1|2]), (414,415,[4_1|2]), (415,416,[2_1|2]), (416,204,[5_1|2]), (416,307,[5_1|2]), (416,311,[5_1|2]), (416,320,[5_1|2]), (416,330,[5_1|2]), (416,382,[5_1|2]), (416,387,[5_1|2]), (416,350,[1_1|2]), (416,354,[1_1|2]), (416,358,[1_1|2]), (416,362,[1_1|2]), (416,367,[2_1|2]), (416,372,[3_1|2]), (416,377,[4_1|2]), (416,392,[1_1|2]), (416,397,[1_1|2]), (416,402,[2_1|2]), (416,407,[0_1|2]), (416,412,[0_1|2]), (417,418,[3_1|2]), (418,419,[2_1|2]), (419,420,[0_1|2]), (420,421,[3_1|2]), (421,204,[2_1|2]), (421,258,[2_1|2]), (421,263,[2_1|2]), (421,268,[2_1|2]), (421,273,[2_1|2]), (421,278,[2_1|2]), (421,283,[2_1|2]), (421,288,[2_1|2]), (421,298,[2_1|2]), (421,367,[2_1|2]), (421,402,[2_1|2]), (421,284,[2_1|2]), (422,423,[5_1|2]), (423,424,[4_1|2]), (424,425,[5_1|2]), (425,426,[0_1|2]), (426,204,[4_1|2]), (426,307,[4_1|2]), (426,311,[4_1|2]), (426,320,[4_1|2]), (426,330,[4_1|2]), (426,382,[4_1|2]), (426,387,[4_1|2]), (427,428,[5_1|2]), (428,429,[0_1|2]), (429,430,[4_1|2]), (430,431,[2_1|2]), (431,307,[2_1|2]), (431,311,[2_1|2]), (431,320,[2_1|2]), (431,330,[2_1|2]), (431,382,[2_1|2]), (431,387,[2_1|2]), (431,299,[2_1|2])}" ---------------------------------------- (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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: 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(2(2(x1)))) ->^+ 2(1(3(0(2(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(2(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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: 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(2(2(x1)))) -> 0(1(0(2(2(x1))))) 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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: FULL