/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 79 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 2(5(3(0(x1)))) -> 1(0(0(1(3(0(4(5(1(2(x1)))))))))) 1(3(5(4(3(x1))))) -> 2(1(4(1(4(0(3(0(1(1(x1)))))))))) 5(1(3(5(0(x1))))) -> 5(1(4(3(0(4(4(5(2(1(x1)))))))))) 5(4(4(2(5(x1))))) -> 4(3(1(1(1(1(5(3(3(5(x1)))))))))) 2(2(5(0(5(4(x1)))))) -> 2(1(4(1(3(3(2(2(5(4(x1)))))))))) 3(0(5(5(4(3(x1)))))) -> 3(3(0(3(2(3(5(5(1(0(x1)))))))))) 3(5(4(3(5(2(x1)))))) -> 2(0(5(2(0(5(2(2(3(2(x1)))))))))) 4(4(2(5(5(0(x1)))))) -> 4(4(0(0(3(3(3(2(2(3(x1)))))))))) 4(5(3(5(5(0(x1)))))) -> 4(2(2(3(0(2(4(1(1(5(x1)))))))))) 5(4(5(1(1(2(x1)))))) -> 5(4(0(3(3(3(3(2(5(5(x1)))))))))) 5(5(5(5(5(3(x1)))))) -> 5(5(0(1(4(0(0(5(0(1(x1)))))))))) 3(5(0(0(5(4(3(x1))))))) -> 0(1(2(1(1(5(5(2(1(0(x1)))))))))) 3(5(4(2(5(2(3(x1))))))) -> 4(0(4(0(0(2(2(3(4(4(x1)))))))))) 3(5(4(5(1(4(0(x1))))))) -> 1(1(1(0(0(3(3(1(2(5(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(0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 2(5(3(0(x1)))) -> 1(0(0(1(3(0(4(5(1(2(x1)))))))))) 1(3(5(4(3(x1))))) -> 2(1(4(1(4(0(3(0(1(1(x1)))))))))) 5(1(3(5(0(x1))))) -> 5(1(4(3(0(4(4(5(2(1(x1)))))))))) 5(4(4(2(5(x1))))) -> 4(3(1(1(1(1(5(3(3(5(x1)))))))))) 2(2(5(0(5(4(x1)))))) -> 2(1(4(1(3(3(2(2(5(4(x1)))))))))) 3(0(5(5(4(3(x1)))))) -> 3(3(0(3(2(3(5(5(1(0(x1)))))))))) 3(5(4(3(5(2(x1)))))) -> 2(0(5(2(0(5(2(2(3(2(x1)))))))))) 4(4(2(5(5(0(x1)))))) -> 4(4(0(0(3(3(3(2(2(3(x1)))))))))) 4(5(3(5(5(0(x1)))))) -> 4(2(2(3(0(2(4(1(1(5(x1)))))))))) 5(4(5(1(1(2(x1)))))) -> 5(4(0(3(3(3(3(2(5(5(x1)))))))))) 5(5(5(5(5(3(x1)))))) -> 5(5(0(1(4(0(0(5(0(1(x1)))))))))) 3(5(0(0(5(4(3(x1))))))) -> 0(1(2(1(1(5(5(2(1(0(x1)))))))))) 3(5(4(2(5(2(3(x1))))))) -> 4(0(4(0(0(2(2(3(4(4(x1)))))))))) 3(5(4(5(1(4(0(x1))))))) -> 1(1(1(0(0(3(3(1(2(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(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(1, n^1). The TRS R consists of the following rules: 2(5(3(0(x1)))) -> 1(0(0(1(3(0(4(5(1(2(x1)))))))))) 1(3(5(4(3(x1))))) -> 2(1(4(1(4(0(3(0(1(1(x1)))))))))) 5(1(3(5(0(x1))))) -> 5(1(4(3(0(4(4(5(2(1(x1)))))))))) 5(4(4(2(5(x1))))) -> 4(3(1(1(1(1(5(3(3(5(x1)))))))))) 2(2(5(0(5(4(x1)))))) -> 2(1(4(1(3(3(2(2(5(4(x1)))))))))) 3(0(5(5(4(3(x1)))))) -> 3(3(0(3(2(3(5(5(1(0(x1)))))))))) 3(5(4(3(5(2(x1)))))) -> 2(0(5(2(0(5(2(2(3(2(x1)))))))))) 4(4(2(5(5(0(x1)))))) -> 4(4(0(0(3(3(3(2(2(3(x1)))))))))) 4(5(3(5(5(0(x1)))))) -> 4(2(2(3(0(2(4(1(1(5(x1)))))))))) 5(4(5(1(1(2(x1)))))) -> 5(4(0(3(3(3(3(2(5(5(x1)))))))))) 5(5(5(5(5(3(x1)))))) -> 5(5(0(1(4(0(0(5(0(1(x1)))))))))) 3(5(0(0(5(4(3(x1))))))) -> 0(1(2(1(1(5(5(2(1(0(x1)))))))))) 3(5(4(2(5(2(3(x1))))))) -> 4(0(4(0(0(2(2(3(4(4(x1)))))))))) 3(5(4(5(1(4(0(x1))))))) -> 1(1(1(0(0(3(3(1(2(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(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: 2(5(3(0(x1)))) -> 1(0(0(1(3(0(4(5(1(2(x1)))))))))) 1(3(5(4(3(x1))))) -> 2(1(4(1(4(0(3(0(1(1(x1)))))))))) 5(1(3(5(0(x1))))) -> 5(1(4(3(0(4(4(5(2(1(x1)))))))))) 5(4(4(2(5(x1))))) -> 4(3(1(1(1(1(5(3(3(5(x1)))))))))) 2(2(5(0(5(4(x1)))))) -> 2(1(4(1(3(3(2(2(5(4(x1)))))))))) 3(0(5(5(4(3(x1)))))) -> 3(3(0(3(2(3(5(5(1(0(x1)))))))))) 3(5(4(3(5(2(x1)))))) -> 2(0(5(2(0(5(2(2(3(2(x1)))))))))) 4(4(2(5(5(0(x1)))))) -> 4(4(0(0(3(3(3(2(2(3(x1)))))))))) 4(5(3(5(5(0(x1)))))) -> 4(2(2(3(0(2(4(1(1(5(x1)))))))))) 5(4(5(1(1(2(x1)))))) -> 5(4(0(3(3(3(3(2(5(5(x1)))))))))) 5(5(5(5(5(3(x1)))))) -> 5(5(0(1(4(0(0(5(0(1(x1)))))))))) 3(5(0(0(5(4(3(x1))))))) -> 0(1(2(1(1(5(5(2(1(0(x1)))))))))) 3(5(4(2(5(2(3(x1))))))) -> 4(0(4(0(0(2(2(3(4(4(x1)))))))))) 3(5(4(5(1(4(0(x1))))))) -> 1(1(1(0(0(3(3(1(2(5(x1)))))))))) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(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 2. The certificate found is represented by the following graph. "[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] {(129,130,[2_1|0, 1_1|0, 5_1|0, 3_1|0, 4_1|0, encArg_1|0, encode_2_1|0, encode_5_1|0, encode_3_1|0, encode_0_1|0, encode_1_1|0, encode_4_1|0]), (129,131,[0_1|1, 2_1|1, 1_1|1, 5_1|1, 3_1|1, 4_1|1]), (129,132,[1_1|2]), (129,141,[2_1|2]), (129,150,[2_1|2]), (129,159,[5_1|2]), (129,168,[4_1|2]), (129,177,[5_1|2]), (129,186,[5_1|2]), (129,195,[3_1|2]), (129,204,[2_1|2]), (129,213,[4_1|2]), (129,222,[1_1|2]), (129,231,[0_1|2]), (129,240,[4_1|2]), (129,249,[4_1|2]), (130,130,[0_1|0, cons_2_1|0, cons_1_1|0, cons_5_1|0, cons_3_1|0, cons_4_1|0]), (131,130,[encArg_1|1]), (131,131,[0_1|1, 2_1|1, 1_1|1, 5_1|1, 3_1|1, 4_1|1]), (131,132,[1_1|2]), (131,141,[2_1|2]), (131,150,[2_1|2]), (131,159,[5_1|2]), (131,168,[4_1|2]), (131,177,[5_1|2]), (131,186,[5_1|2]), (131,195,[3_1|2]), (131,204,[2_1|2]), (131,213,[4_1|2]), (131,222,[1_1|2]), (131,231,[0_1|2]), (131,240,[4_1|2]), (131,249,[4_1|2]), (132,133,[0_1|2]), (133,134,[0_1|2]), (134,135,[1_1|2]), (135,136,[3_1|2]), (136,137,[0_1|2]), (137,138,[4_1|2]), (138,139,[5_1|2]), (139,140,[1_1|2]), (140,131,[2_1|2]), (140,231,[2_1|2]), (140,132,[1_1|2]), (140,141,[2_1|2]), (141,142,[1_1|2]), (142,143,[4_1|2]), (143,144,[1_1|2]), (144,145,[3_1|2]), (145,146,[3_1|2]), (146,147,[2_1|2]), (147,148,[2_1|2]), (148,149,[5_1|2]), (148,168,[4_1|2]), (148,177,[5_1|2]), (149,131,[4_1|2]), (149,168,[4_1|2]), (149,213,[4_1|2]), (149,240,[4_1|2]), (149,249,[4_1|2]), (149,178,[4_1|2]), (150,151,[1_1|2]), (151,152,[4_1|2]), (152,153,[1_1|2]), (153,154,[4_1|2]), (154,155,[0_1|2]), (155,156,[3_1|2]), (156,157,[0_1|2]), (157,158,[1_1|2]), (158,131,[1_1|2]), (158,195,[1_1|2]), (158,169,[1_1|2]), (158,150,[2_1|2]), (159,160,[1_1|2]), (160,161,[4_1|2]), (161,162,[3_1|2]), (162,163,[0_1|2]), (163,164,[4_1|2]), (164,165,[4_1|2]), (165,166,[5_1|2]), (166,167,[2_1|2]), (167,131,[1_1|2]), (167,231,[1_1|2]), (167,150,[2_1|2]), (168,169,[3_1|2]), (169,170,[1_1|2]), (170,171,[1_1|2]), (171,172,[1_1|2]), (172,173,[1_1|2]), (173,174,[5_1|2]), (174,175,[3_1|2]), (175,176,[3_1|2]), (175,204,[2_1|2]), (175,213,[4_1|2]), (175,222,[1_1|2]), (175,231,[0_1|2]), (176,131,[5_1|2]), (176,159,[5_1|2]), (176,177,[5_1|2]), (176,186,[5_1|2]), (176,168,[4_1|2]), (177,178,[4_1|2]), (178,179,[0_1|2]), (179,180,[3_1|2]), (180,181,[3_1|2]), (181,182,[3_1|2]), (182,183,[3_1|2]), (183,184,[2_1|2]), (184,185,[5_1|2]), (184,186,[5_1|2]), (185,131,[5_1|2]), (185,141,[5_1|2]), (185,150,[5_1|2]), (185,204,[5_1|2]), (185,159,[5_1|2]), (185,168,[4_1|2]), (185,177,[5_1|2]), (185,186,[5_1|2]), (186,187,[5_1|2]), (187,188,[0_1|2]), (188,189,[1_1|2]), (189,190,[4_1|2]), (190,191,[0_1|2]), (191,192,[0_1|2]), (192,193,[5_1|2]), (193,194,[0_1|2]), (194,131,[1_1|2]), (194,195,[1_1|2]), (194,150,[2_1|2]), (195,196,[3_1|2]), (196,197,[0_1|2]), (197,198,[3_1|2]), (198,199,[2_1|2]), (199,200,[3_1|2]), (200,201,[5_1|2]), (201,202,[5_1|2]), (202,203,[1_1|2]), (203,131,[0_1|2]), (203,195,[0_1|2]), (203,169,[0_1|2]), (204,205,[0_1|2]), (205,206,[5_1|2]), (206,207,[2_1|2]), (207,208,[0_1|2]), (208,209,[5_1|2]), (209,210,[2_1|2]), (210,211,[2_1|2]), (211,212,[3_1|2]), (212,131,[2_1|2]), (212,141,[2_1|2]), (212,150,[2_1|2]), (212,204,[2_1|2]), (212,132,[1_1|2]), (213,214,[0_1|2]), (214,215,[4_1|2]), (215,216,[0_1|2]), (216,217,[0_1|2]), (217,218,[2_1|2]), (218,219,[2_1|2]), (219,220,[3_1|2]), (220,221,[4_1|2]), (220,240,[4_1|2]), (221,131,[4_1|2]), (221,195,[4_1|2]), (221,240,[4_1|2]), (221,249,[4_1|2]), (222,223,[1_1|2]), (223,224,[1_1|2]), (224,225,[0_1|2]), (225,226,[0_1|2]), (226,227,[3_1|2]), (227,228,[3_1|2]), (228,229,[1_1|2]), (229,230,[2_1|2]), (229,132,[1_1|2]), (230,131,[5_1|2]), (230,231,[5_1|2]), (230,214,[5_1|2]), (230,159,[5_1|2]), (230,168,[4_1|2]), (230,177,[5_1|2]), (230,186,[5_1|2]), (231,232,[1_1|2]), (232,233,[2_1|2]), (233,234,[1_1|2]), (234,235,[1_1|2]), (235,236,[5_1|2]), (236,237,[5_1|2]), (237,238,[2_1|2]), (238,239,[1_1|2]), (239,131,[0_1|2]), (239,195,[0_1|2]), (239,169,[0_1|2]), (240,241,[4_1|2]), (241,242,[0_1|2]), (242,243,[0_1|2]), (243,244,[3_1|2]), (244,245,[3_1|2]), (245,246,[3_1|2]), (246,247,[2_1|2]), (247,248,[2_1|2]), (248,131,[3_1|2]), (248,231,[3_1|2, 0_1|2]), (248,188,[3_1|2]), (248,195,[3_1|2]), (248,204,[2_1|2]), (248,213,[4_1|2]), (248,222,[1_1|2]), (249,250,[2_1|2]), (250,251,[2_1|2]), (251,252,[3_1|2]), (252,253,[0_1|2]), (253,254,[2_1|2]), (254,255,[4_1|2]), (255,256,[1_1|2]), (256,257,[1_1|2]), (257,131,[5_1|2]), (257,231,[5_1|2]), (257,188,[5_1|2]), (257,159,[5_1|2]), (257,168,[4_1|2]), (257,177,[5_1|2]), (257,186,[5_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)