WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 52 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 18 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 1(3(3(x1))) -> 3(5(3(2(5(0(2(4(5(4(x1)))))))))) 0(3(3(3(x1)))) -> 5(4(3(5(3(0(5(4(4(0(x1)))))))))) 0(3(3(3(1(x1))))) -> 5(4(4(0(3(1(0(5(1(0(x1)))))))))) 1(2(3(3(3(x1))))) -> 4(1(1(2(3(5(0(4(0(5(x1)))))))))) 1(4(4(2(2(x1))))) -> 1(1(2(0(1(1(1(0(2(2(x1)))))))))) 0(3(3(1(4(3(x1)))))) -> 4(4(3(0(2(3(0(3(0(0(x1)))))))))) 3(3(3(3(4(0(x1)))))) -> 3(0(0(2(1(0(5(3(5(4(x1)))))))))) 4(0(1(3(4(0(x1)))))) -> 2(2(3(0(0(0(5(0(0(0(x1)))))))))) 4(1(4(4(4(1(x1)))))) -> 4(1(0(3(3(5(5(5(4(1(x1)))))))))) 0(1(3(5(2(2(3(x1))))))) -> 0(3(0(0(5(0(0(4(4(3(x1)))))))))) 0(2(3(1(3(2(5(x1))))))) -> 0(4(3(1(2(3(2(3(2(0(x1)))))))))) 1(1(3(3(5(3(1(x1))))))) -> 3(5(0(5(3(2(5(0(0(1(x1)))))))))) 3(5(2(0(1(3(3(x1))))))) -> 3(4(3(2(3(2(4(4(5(5(x1)))))))))) 4(1(4(2(4(0(1(x1))))))) -> 5(2(2(1(0(5(5(4(5(1(x1)))))))))) 4(5(1(2(4(4(4(x1))))))) -> 4(1(1(4(5(3(0(1(0(4(x1)))))))))) 5(1(4(5(3(3(3(x1))))))) -> 5(1(4(5(3(4(4(2(3(2(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(2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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_5(x_1)) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 1(3(3(x1))) -> 3(5(3(2(5(0(2(4(5(4(x1)))))))))) 0(3(3(3(x1)))) -> 5(4(3(5(3(0(5(4(4(0(x1)))))))))) 0(3(3(3(1(x1))))) -> 5(4(4(0(3(1(0(5(1(0(x1)))))))))) 1(2(3(3(3(x1))))) -> 4(1(1(2(3(5(0(4(0(5(x1)))))))))) 1(4(4(2(2(x1))))) -> 1(1(2(0(1(1(1(0(2(2(x1)))))))))) 0(3(3(1(4(3(x1)))))) -> 4(4(3(0(2(3(0(3(0(0(x1)))))))))) 3(3(3(3(4(0(x1)))))) -> 3(0(0(2(1(0(5(3(5(4(x1)))))))))) 4(0(1(3(4(0(x1)))))) -> 2(2(3(0(0(0(5(0(0(0(x1)))))))))) 4(1(4(4(4(1(x1)))))) -> 4(1(0(3(3(5(5(5(4(1(x1)))))))))) 0(1(3(5(2(2(3(x1))))))) -> 0(3(0(0(5(0(0(4(4(3(x1)))))))))) 0(2(3(1(3(2(5(x1))))))) -> 0(4(3(1(2(3(2(3(2(0(x1)))))))))) 1(1(3(3(5(3(1(x1))))))) -> 3(5(0(5(3(2(5(0(0(1(x1)))))))))) 3(5(2(0(1(3(3(x1))))))) -> 3(4(3(2(3(2(4(4(5(5(x1)))))))))) 4(1(4(2(4(0(1(x1))))))) -> 5(2(2(1(0(5(5(4(5(1(x1)))))))))) 4(5(1(2(4(4(4(x1))))))) -> 4(1(1(4(5(3(0(1(0(4(x1)))))))))) 5(1(4(5(3(3(3(x1))))))) -> 5(1(4(5(3(4(4(2(3(2(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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_5(x_1)) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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(1, n^1). The TRS R consists of the following rules: 1(3(3(x1))) -> 3(5(3(2(5(0(2(4(5(4(x1)))))))))) 0(3(3(3(x1)))) -> 5(4(3(5(3(0(5(4(4(0(x1)))))))))) 0(3(3(3(1(x1))))) -> 5(4(4(0(3(1(0(5(1(0(x1)))))))))) 1(2(3(3(3(x1))))) -> 4(1(1(2(3(5(0(4(0(5(x1)))))))))) 1(4(4(2(2(x1))))) -> 1(1(2(0(1(1(1(0(2(2(x1)))))))))) 0(3(3(1(4(3(x1)))))) -> 4(4(3(0(2(3(0(3(0(0(x1)))))))))) 3(3(3(3(4(0(x1)))))) -> 3(0(0(2(1(0(5(3(5(4(x1)))))))))) 4(0(1(3(4(0(x1)))))) -> 2(2(3(0(0(0(5(0(0(0(x1)))))))))) 4(1(4(4(4(1(x1)))))) -> 4(1(0(3(3(5(5(5(4(1(x1)))))))))) 0(1(3(5(2(2(3(x1))))))) -> 0(3(0(0(5(0(0(4(4(3(x1)))))))))) 0(2(3(1(3(2(5(x1))))))) -> 0(4(3(1(2(3(2(3(2(0(x1)))))))))) 1(1(3(3(5(3(1(x1))))))) -> 3(5(0(5(3(2(5(0(0(1(x1)))))))))) 3(5(2(0(1(3(3(x1))))))) -> 3(4(3(2(3(2(4(4(5(5(x1)))))))))) 4(1(4(2(4(0(1(x1))))))) -> 5(2(2(1(0(5(5(4(5(1(x1)))))))))) 4(5(1(2(4(4(4(x1))))))) -> 4(1(1(4(5(3(0(1(0(4(x1)))))))))) 5(1(4(5(3(3(3(x1))))))) -> 5(1(4(5(3(4(4(2(3(2(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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_5(x_1)) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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: 1(3(3(x1))) -> 3(5(3(2(5(0(2(4(5(4(x1)))))))))) 0(3(3(3(x1)))) -> 5(4(3(5(3(0(5(4(4(0(x1)))))))))) 0(3(3(3(1(x1))))) -> 5(4(4(0(3(1(0(5(1(0(x1)))))))))) 1(2(3(3(3(x1))))) -> 4(1(1(2(3(5(0(4(0(5(x1)))))))))) 1(4(4(2(2(x1))))) -> 1(1(2(0(1(1(1(0(2(2(x1)))))))))) 0(3(3(1(4(3(x1)))))) -> 4(4(3(0(2(3(0(3(0(0(x1)))))))))) 3(3(3(3(4(0(x1)))))) -> 3(0(0(2(1(0(5(3(5(4(x1)))))))))) 4(0(1(3(4(0(x1)))))) -> 2(2(3(0(0(0(5(0(0(0(x1)))))))))) 4(1(4(4(4(1(x1)))))) -> 4(1(0(3(3(5(5(5(4(1(x1)))))))))) 0(1(3(5(2(2(3(x1))))))) -> 0(3(0(0(5(0(0(4(4(3(x1)))))))))) 0(2(3(1(3(2(5(x1))))))) -> 0(4(3(1(2(3(2(3(2(0(x1)))))))))) 1(1(3(3(5(3(1(x1))))))) -> 3(5(0(5(3(2(5(0(0(1(x1)))))))))) 3(5(2(0(1(3(3(x1))))))) -> 3(4(3(2(3(2(4(4(5(5(x1)))))))))) 4(1(4(2(4(0(1(x1))))))) -> 5(2(2(1(0(5(5(4(5(1(x1)))))))))) 4(5(1(2(4(4(4(x1))))))) -> 4(1(1(4(5(3(0(1(0(4(x1)))))))))) 5(1(4(5(3(3(3(x1))))))) -> 5(1(4(5(3(4(4(2(3(2(x1)))))))))) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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_5(x_1)) -> 5(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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. "[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] {(127,128,[1_1|0, 0_1|0, 3_1|0, 4_1|0, 5_1|0, encArg_1|0, encode_1_1|0, encode_3_1|0, encode_5_1|0, encode_2_1|0, encode_0_1|0, encode_4_1|0]), (127,129,[2_1|1, 1_1|1, 0_1|1, 3_1|1, 4_1|1, 5_1|1]), (127,130,[3_1|2]), (127,139,[4_1|2]), (127,148,[1_1|2]), (127,157,[3_1|2]), (127,166,[5_1|2]), (127,175,[5_1|2]), (127,184,[4_1|2]), (127,193,[0_1|2]), (127,202,[0_1|2]), (127,211,[3_1|2]), (127,220,[3_1|2]), (127,229,[2_1|2]), (127,238,[4_1|2]), (127,247,[5_1|2]), (127,256,[4_1|2]), (127,265,[5_1|2]), (128,128,[2_1|0, cons_1_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0, cons_5_1|0]), (129,128,[encArg_1|1]), (129,129,[2_1|1, 1_1|1, 0_1|1, 3_1|1, 4_1|1, 5_1|1]), (129,130,[3_1|2]), (129,139,[4_1|2]), (129,148,[1_1|2]), (129,157,[3_1|2]), (129,166,[5_1|2]), (129,175,[5_1|2]), (129,184,[4_1|2]), (129,193,[0_1|2]), (129,202,[0_1|2]), (129,211,[3_1|2]), (129,220,[3_1|2]), (129,229,[2_1|2]), (129,238,[4_1|2]), (129,247,[5_1|2]), (129,256,[4_1|2]), (129,265,[5_1|2]), (130,131,[5_1|2]), (131,132,[3_1|2]), (132,133,[2_1|2]), (133,134,[5_1|2]), (134,135,[0_1|2]), (135,136,[2_1|2]), (136,137,[4_1|2]), (137,138,[5_1|2]), (138,129,[4_1|2]), (138,130,[4_1|2]), (138,157,[4_1|2]), (138,211,[4_1|2]), (138,220,[4_1|2]), (138,229,[2_1|2]), (138,238,[4_1|2]), (138,247,[5_1|2]), (138,256,[4_1|2]), (139,140,[1_1|2]), (140,141,[1_1|2]), (141,142,[2_1|2]), (142,143,[3_1|2]), (143,144,[5_1|2]), (144,145,[0_1|2]), (145,146,[4_1|2]), (146,147,[0_1|2]), (147,129,[5_1|2]), (147,130,[5_1|2]), (147,157,[5_1|2]), (147,211,[5_1|2]), (147,220,[5_1|2]), (147,265,[5_1|2]), (148,149,[1_1|2]), (149,150,[2_1|2]), (150,151,[0_1|2]), (151,152,[1_1|2]), (152,153,[1_1|2]), (153,154,[1_1|2]), (154,155,[0_1|2]), (155,156,[2_1|2]), (156,129,[2_1|2]), (156,229,[2_1|2]), (156,230,[2_1|2]), (157,158,[5_1|2]), (158,159,[0_1|2]), (159,160,[5_1|2]), (160,161,[3_1|2]), (161,162,[2_1|2]), (162,163,[5_1|2]), (163,164,[0_1|2]), (164,165,[0_1|2]), (164,193,[0_1|2]), (165,129,[1_1|2]), (165,148,[1_1|2]), (165,130,[3_1|2]), (165,139,[4_1|2]), (165,157,[3_1|2]), (166,167,[4_1|2]), (167,168,[3_1|2]), (168,169,[5_1|2]), (169,170,[3_1|2]), (170,171,[0_1|2]), (171,172,[5_1|2]), (172,173,[4_1|2]), (173,174,[4_1|2]), (173,229,[2_1|2]), (174,129,[0_1|2]), (174,130,[0_1|2]), (174,157,[0_1|2]), (174,211,[0_1|2]), (174,220,[0_1|2]), (174,166,[5_1|2]), (174,175,[5_1|2]), (174,184,[4_1|2]), (174,193,[0_1|2]), (174,202,[0_1|2]), (175,176,[4_1|2]), (176,177,[4_1|2]), (177,178,[0_1|2]), (178,179,[3_1|2]), (179,180,[1_1|2]), (180,181,[0_1|2]), (181,182,[5_1|2]), (182,183,[1_1|2]), (183,129,[0_1|2]), (183,148,[0_1|2]), (183,166,[5_1|2]), (183,175,[5_1|2]), (183,184,[4_1|2]), (183,193,[0_1|2]), (183,202,[0_1|2]), (184,185,[4_1|2]), (185,186,[3_1|2]), (186,187,[0_1|2]), (187,188,[2_1|2]), (188,189,[3_1|2]), (189,190,[0_1|2]), (190,191,[3_1|2]), (191,192,[0_1|2]), (192,129,[0_1|2]), (192,130,[0_1|2]), (192,157,[0_1|2]), (192,211,[0_1|2]), (192,220,[0_1|2]), (192,166,[5_1|2]), (192,175,[5_1|2]), (192,184,[4_1|2]), (192,193,[0_1|2]), (192,202,[0_1|2]), (193,194,[3_1|2]), (194,195,[0_1|2]), (195,196,[0_1|2]), (196,197,[5_1|2]), (197,198,[0_1|2]), (198,199,[0_1|2]), (199,200,[4_1|2]), (200,201,[4_1|2]), (201,129,[3_1|2]), (201,130,[3_1|2]), (201,157,[3_1|2]), (201,211,[3_1|2]), (201,220,[3_1|2]), (201,231,[3_1|2]), (202,203,[4_1|2]), (203,204,[3_1|2]), (204,205,[1_1|2]), (205,206,[2_1|2]), (206,207,[3_1|2]), (207,208,[2_1|2]), (208,209,[3_1|2]), (209,210,[2_1|2]), (210,129,[0_1|2]), (210,166,[0_1|2, 5_1|2]), (210,175,[0_1|2, 5_1|2]), (210,247,[0_1|2]), (210,265,[0_1|2]), (210,184,[4_1|2]), (210,193,[0_1|2]), (210,202,[0_1|2]), (211,212,[0_1|2]), (212,213,[0_1|2]), (213,214,[2_1|2]), (214,215,[1_1|2]), (215,216,[0_1|2]), (216,217,[5_1|2]), (217,218,[3_1|2]), (218,219,[5_1|2]), (219,129,[4_1|2]), (219,193,[4_1|2]), (219,202,[4_1|2]), (219,229,[2_1|2]), (219,238,[4_1|2]), (219,247,[5_1|2]), (219,256,[4_1|2]), (220,221,[4_1|2]), (221,222,[3_1|2]), (222,223,[2_1|2]), (223,224,[3_1|2]), (224,225,[2_1|2]), (225,226,[4_1|2]), (226,227,[4_1|2]), (227,228,[5_1|2]), (228,129,[5_1|2]), (228,130,[5_1|2]), (228,157,[5_1|2]), (228,211,[5_1|2]), (228,220,[5_1|2]), (228,265,[5_1|2]), (229,230,[2_1|2]), (230,231,[3_1|2]), (231,232,[0_1|2]), (232,233,[0_1|2]), (233,234,[0_1|2]), (234,235,[5_1|2]), (235,236,[0_1|2]), (236,237,[0_1|2]), (237,129,[0_1|2]), (237,193,[0_1|2]), (237,202,[0_1|2]), (237,166,[5_1|2]), (237,175,[5_1|2]), (237,184,[4_1|2]), (238,239,[1_1|2]), (239,240,[0_1|2]), (240,241,[3_1|2]), (241,242,[3_1|2]), (242,243,[5_1|2]), (243,244,[5_1|2]), (244,245,[5_1|2]), (245,246,[4_1|2]), (245,238,[4_1|2]), (245,247,[5_1|2]), (246,129,[1_1|2]), (246,148,[1_1|2]), (246,140,[1_1|2]), (246,239,[1_1|2]), (246,257,[1_1|2]), (246,130,[3_1|2]), (246,139,[4_1|2]), (246,157,[3_1|2]), (247,248,[2_1|2]), (248,249,[2_1|2]), (249,250,[1_1|2]), (250,251,[0_1|2]), (251,252,[5_1|2]), (252,253,[5_1|2]), (253,254,[4_1|2]), (253,256,[4_1|2]), (254,255,[5_1|2]), (254,265,[5_1|2]), (255,129,[1_1|2]), (255,148,[1_1|2]), (255,130,[3_1|2]), (255,139,[4_1|2]), (255,157,[3_1|2]), (256,257,[1_1|2]), (257,258,[1_1|2]), (258,259,[4_1|2]), (259,260,[5_1|2]), (260,261,[3_1|2]), (261,262,[0_1|2]), (262,263,[1_1|2]), (263,264,[0_1|2]), (264,129,[4_1|2]), (264,139,[4_1|2]), (264,184,[4_1|2]), (264,238,[4_1|2]), (264,256,[4_1|2]), (264,185,[4_1|2]), (264,229,[2_1|2]), (264,247,[5_1|2]), (265,266,[1_1|2]), (266,267,[4_1|2]), (267,268,[5_1|2]), (268,269,[3_1|2]), (269,270,[4_1|2]), (270,271,[4_1|2]), (271,272,[2_1|2]), (272,273,[3_1|2]), (273,129,[2_1|2]), (273,130,[2_1|2]), (273,157,[2_1|2]), (273,211,[2_1|2]), (273,220,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)