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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 15 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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(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(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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: 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_0(x_1) -> 0(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 3. The certificate found is represented by the following graph. "[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] {(148,149,[1_1|0, 5_1|0, 0_1|0, 3_1|0, 4_1|0, 2_1|0, encArg_1|0, encode_1_1|0, encode_4_1|0, encode_3_1|0, encode_2_1|0, encode_5_1|0, encode_0_1|0]), (148,150,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (148,151,[3_1|2]), (148,156,[0_1|2]), (148,161,[0_1|2]), (148,166,[0_1|2]), (148,171,[4_1|2]), (148,176,[2_1|2]), (148,181,[5_1|2]), (148,186,[1_1|2]), (148,191,[3_1|2]), (148,196,[4_1|2]), (148,201,[3_1|2]), (148,206,[2_1|2]), (148,211,[1_1|2]), (148,216,[0_1|2]), (148,221,[0_1|2]), (148,226,[4_1|2]), (148,231,[3_1|2]), (148,236,[3_1|2]), (148,241,[0_1|2]), (149,149,[cons_1_1|0, cons_5_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0, cons_2_1|0]), (150,149,[encArg_1|1]), (150,150,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (150,151,[3_1|2]), (150,156,[0_1|2]), (150,161,[0_1|2]), (150,166,[0_1|2]), (150,171,[4_1|2]), (150,176,[2_1|2]), (150,181,[5_1|2]), (150,186,[1_1|2]), (150,191,[3_1|2]), (150,196,[4_1|2]), (150,201,[3_1|2]), (150,206,[2_1|2]), (150,211,[1_1|2]), (150,216,[0_1|2]), (150,221,[0_1|2]), (150,226,[4_1|2]), (150,231,[3_1|2]), (150,236,[3_1|2]), (150,241,[0_1|2]), (151,152,[1_1|2]), (152,153,[1_1|2]), (153,154,[2_1|2]), (154,155,[2_1|2]), (155,150,[4_1|2]), (155,171,[4_1|2]), (155,196,[4_1|2]), (155,226,[4_1|2]), (155,231,[3_1|2]), (155,236,[3_1|2]), (156,157,[4_1|2]), (157,158,[5_1|2]), (158,159,[0_1|2]), (159,160,[2_1|2]), (160,150,[1_1|2]), (160,171,[1_1|2]), (160,196,[1_1|2]), (160,226,[1_1|2]), (160,151,[3_1|2]), (160,156,[0_1|2]), (160,161,[0_1|2]), (160,166,[0_1|2]), (160,246,[3_1|3]), (161,162,[0_1|2]), (162,163,[1_1|2]), (163,164,[3_1|2]), (164,165,[4_1|2]), (164,231,[3_1|2]), (164,251,[3_1|3]), (165,150,[1_1|2]), (165,181,[1_1|2]), (165,151,[3_1|2]), (165,156,[0_1|2]), (165,161,[0_1|2]), (165,166,[0_1|2]), (165,246,[3_1|3]), (166,167,[2_1|2]), (167,168,[5_1|2]), (168,169,[2_1|2]), (169,170,[0_1|2]), (170,150,[4_1|2]), (170,171,[4_1|2]), (170,196,[4_1|2]), (170,226,[4_1|2]), (170,231,[3_1|2]), (170,236,[3_1|2]), (171,172,[2_1|2]), (172,173,[3_1|2]), (173,174,[1_1|2]), (174,175,[1_1|2]), (175,150,[1_1|2]), (175,171,[1_1|2]), (175,196,[1_1|2]), (175,226,[1_1|2]), (175,151,[3_1|2]), (175,156,[0_1|2]), (175,161,[0_1|2]), (175,166,[0_1|2]), (175,246,[3_1|3]), (176,177,[4_1|2]), (177,178,[0_1|2]), (178,179,[4_1|2]), (179,180,[4_1|2]), (180,150,[0_1|2]), (180,156,[0_1|2]), (180,161,[0_1|2]), (180,166,[0_1|2]), (180,216,[0_1|2]), (180,221,[0_1|2]), (180,241,[0_1|2]), (180,206,[2_1|2]), (180,211,[1_1|2]), (180,256,[2_1|3]), (181,182,[1_1|2]), (182,183,[5_1|2]), (183,184,[2_1|2]), (184,185,[1_1|2]), (185,150,[0_1|2]), (185,156,[0_1|2]), (185,161,[0_1|2]), (185,166,[0_1|2]), (185,216,[0_1|2]), (185,221,[0_1|2]), (185,241,[0_1|2]), (185,206,[2_1|2]), (185,211,[1_1|2]), (185,256,[2_1|3]), (186,187,[0_1|2]), (187,188,[4_1|2]), (188,189,[0_1|2]), (189,190,[2_1|2]), (190,150,[2_1|2]), (190,156,[2_1|2]), (190,161,[2_1|2]), (190,166,[2_1|2]), (190,216,[2_1|2]), (190,221,[2_1|2]), (190,241,[2_1|2, 0_1|2]), (190,162,[2_1|2]), (191,192,[0_1|2]), (192,193,[4_1|2]), (193,194,[5_1|2]), (194,195,[0_1|2]), (195,150,[2_1|2]), (195,176,[2_1|2]), (195,206,[2_1|2]), (195,167,[2_1|2]), (195,217,[2_1|2]), (195,241,[0_1|2]), (196,197,[1_1|2]), (197,198,[1_1|2]), (198,199,[3_1|2]), (199,200,[2_1|2]), (200,150,[4_1|2]), (200,171,[4_1|2]), (200,196,[4_1|2]), (200,226,[4_1|2]), (200,231,[3_1|2]), (200,236,[3_1|2]), (201,202,[4_1|2]), (202,203,[4_1|2]), (203,204,[1_1|2]), (204,205,[2_1|2]), (205,150,[2_1|2]), (205,171,[2_1|2]), (205,196,[2_1|2]), (205,226,[2_1|2]), (205,241,[0_1|2]), (206,207,[1_1|2]), (207,208,[1_1|2]), (208,209,[0_1|2]), (209,210,[2_1|2]), (210,150,[0_1|2]), (210,156,[0_1|2]), (210,161,[0_1|2]), (210,166,[0_1|2]), (210,216,[0_1|2]), (210,221,[0_1|2]), (210,241,[0_1|2]), (210,192,[0_1|2]), (210,206,[2_1|2]), (210,211,[1_1|2]), (210,256,[2_1|3]), (211,212,[0_1|2]), (212,213,[1_1|2]), (213,214,[3_1|2]), (214,215,[4_1|2]), (215,150,[2_1|2]), (215,181,[2_1|2]), (215,241,[0_1|2]), (216,217,[2_1|2]), (217,218,[0_1|2]), (218,219,[0_1|2]), (218,261,[2_1|3]), (219,220,[3_1|2]), (220,150,[0_1|2]), (220,156,[0_1|2]), (220,161,[0_1|2]), (220,166,[0_1|2]), (220,216,[0_1|2]), (220,221,[0_1|2]), (220,241,[0_1|2]), (220,206,[2_1|2]), (220,211,[1_1|2]), (220,256,[2_1|3]), (221,222,[1_1|2]), (222,223,[3_1|2]), (223,224,[4_1|2]), (224,225,[3_1|2]), (225,150,[4_1|2]), (225,171,[4_1|2]), (225,196,[4_1|2]), (225,226,[4_1|2]), (225,231,[3_1|2]), (225,236,[3_1|2]), (226,227,[1_1|2]), (227,228,[3_1|2]), (228,229,[4_1|2]), (229,230,[2_1|2]), (230,150,[3_1|2]), (230,171,[3_1|2]), (230,196,[3_1|2]), (230,226,[3_1|2, 4_1|2]), (231,232,[3_1|2]), (232,233,[2_1|2]), (233,234,[2_1|2]), (234,235,[3_1|2]), (235,150,[1_1|2]), (235,171,[1_1|2]), (235,196,[1_1|2]), (235,226,[1_1|2]), (235,151,[3_1|2]), (235,156,[0_1|2]), (235,161,[0_1|2]), (235,166,[0_1|2]), (235,246,[3_1|3]), (236,237,[3_1|2]), (237,238,[2_1|2]), (238,239,[3_1|2]), (238,276,[4_1|3]), (239,240,[5_1|2]), (239,201,[3_1|2]), (239,266,[3_1|3]), (239,271,[4_1|3]), (240,150,[5_1|2]), (240,181,[5_1|2]), (240,171,[4_1|2]), (240,176,[2_1|2]), (240,186,[1_1|2]), (240,191,[3_1|2]), (240,196,[4_1|2]), (240,201,[3_1|2]), (240,271,[4_1|3]), (241,242,[4_1|2]), (242,243,[1_1|2]), (243,244,[2_1|2]), (244,245,[4_1|2]), (245,150,[0_1|2]), (245,156,[0_1|2]), (245,161,[0_1|2]), (245,166,[0_1|2]), (245,216,[0_1|2]), (245,221,[0_1|2]), (245,241,[0_1|2]), (245,206,[2_1|2]), (245,211,[1_1|2]), (245,256,[2_1|3]), (246,247,[1_1|3]), (247,248,[1_1|3]), (248,249,[2_1|3]), (249,250,[2_1|3]), (250,171,[4_1|3]), (250,196,[4_1|3]), (250,226,[4_1|3]), (251,252,[3_1|3]), (252,253,[2_1|3]), (253,254,[2_1|3]), (254,255,[3_1|3]), (255,171,[1_1|3]), (255,196,[1_1|3]), (255,226,[1_1|3]), (256,257,[1_1|3]), (257,258,[1_1|3]), (258,259,[0_1|3]), (259,260,[2_1|3]), (260,192,[0_1|3]), (261,262,[1_1|3]), (262,263,[1_1|3]), (263,264,[0_1|3]), (264,265,[2_1|3]), (265,150,[0_1|3]), (265,156,[0_1|3]), (265,161,[0_1|3]), (265,166,[0_1|3]), (265,216,[0_1|3, 0_1|2]), (265,221,[0_1|3, 0_1|2]), (265,241,[0_1|3]), (265,206,[2_1|2]), (265,211,[1_1|2]), (265,256,[2_1|3]), (266,267,[4_1|3]), (267,268,[4_1|3]), (268,269,[1_1|3]), (269,270,[2_1|3]), (270,171,[2_1|3]), (270,196,[2_1|3]), (270,226,[2_1|3]), (271,272,[2_1|3]), (272,273,[3_1|3]), (273,274,[1_1|3]), (274,275,[1_1|3]), (275,171,[1_1|3]), (275,196,[1_1|3]), (275,226,[1_1|3]), (275,271,[1_1|3]), (276,277,[1_1|3]), (277,278,[3_1|3]), (278,279,[4_1|3]), (279,280,[2_1|3]), (280,171,[3_1|3]), (280,196,[3_1|3]), (280,271,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)