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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 75 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 30 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(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: 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(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 (full) 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: 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(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: 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(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: 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 3. The certificate found is represented by the following graph. "[72, 73, 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, 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, 278, 279, 280, 281, 282] {(72,73,[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]), (72,92,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (72,93,[3_1|2]), (72,98,[0_1|2]), (72,103,[0_1|2]), (72,108,[0_1|2]), (72,113,[4_1|2]), (72,118,[2_1|2]), (72,123,[5_1|2]), (72,128,[1_1|2]), (72,133,[3_1|2]), (72,138,[4_1|2]), (72,143,[3_1|2]), (72,148,[2_1|2]), (72,153,[1_1|2]), (72,158,[0_1|2]), (72,163,[0_1|2]), (72,168,[4_1|2]), (72,173,[3_1|2]), (72,178,[3_1|2]), (72,183,[0_1|2]), (73,73,[cons_1_1|0, cons_5_1|0, cons_0_1|0, cons_3_1|0, cons_4_1|0, cons_2_1|0]), (92,73,[encArg_1|1]), (92,92,[1_1|1, 5_1|1, 0_1|1, 3_1|1, 4_1|1, 2_1|1]), (92,93,[3_1|2]), (92,98,[0_1|2]), (92,103,[0_1|2]), (92,108,[0_1|2]), (92,113,[4_1|2]), (92,118,[2_1|2]), (92,123,[5_1|2]), (92,128,[1_1|2]), (92,133,[3_1|2]), (92,138,[4_1|2]), (92,143,[3_1|2]), (92,148,[2_1|2]), (92,153,[1_1|2]), (92,158,[0_1|2]), (92,163,[0_1|2]), (92,168,[4_1|2]), (92,173,[3_1|2]), (92,178,[3_1|2]), (92,183,[0_1|2]), (93,94,[1_1|2]), (94,95,[1_1|2]), (95,96,[2_1|2]), (96,97,[2_1|2]), (97,92,[4_1|2]), (97,113,[4_1|2]), (97,138,[4_1|2]), (97,168,[4_1|2]), (97,173,[3_1|2]), (97,178,[3_1|2]), (98,99,[4_1|2]), (99,100,[5_1|2]), (100,101,[0_1|2]), (101,102,[2_1|2]), (102,92,[1_1|2]), (102,113,[1_1|2]), (102,138,[1_1|2]), (102,168,[1_1|2]), (102,93,[3_1|2]), (102,98,[0_1|2]), (102,103,[0_1|2]), (102,108,[0_1|2]), (102,222,[3_1|3]), (103,104,[0_1|2]), (104,105,[1_1|2]), (105,106,[3_1|2]), (106,107,[4_1|2]), (106,173,[3_1|2]), (106,227,[3_1|3]), (107,92,[1_1|2]), (107,123,[1_1|2]), (107,93,[3_1|2]), (107,98,[0_1|2]), (107,103,[0_1|2]), (107,108,[0_1|2]), (107,222,[3_1|3]), (108,109,[2_1|2]), (109,110,[5_1|2]), (110,111,[2_1|2]), (111,112,[0_1|2]), (112,92,[4_1|2]), (112,113,[4_1|2]), (112,138,[4_1|2]), (112,168,[4_1|2]), (112,173,[3_1|2]), (112,178,[3_1|2]), (113,114,[2_1|2]), (114,115,[3_1|2]), (115,116,[1_1|2]), (116,117,[1_1|2]), (117,92,[1_1|2]), (117,113,[1_1|2]), (117,138,[1_1|2]), (117,168,[1_1|2]), (117,93,[3_1|2]), (117,98,[0_1|2]), (117,103,[0_1|2]), (117,108,[0_1|2]), (117,222,[3_1|3]), (118,119,[4_1|2]), (119,120,[0_1|2]), (120,121,[4_1|2]), (121,122,[4_1|2]), (122,92,[0_1|2]), (122,98,[0_1|2]), (122,103,[0_1|2]), (122,108,[0_1|2]), (122,158,[0_1|2]), (122,163,[0_1|2]), (122,183,[0_1|2]), (122,148,[2_1|2]), (122,153,[1_1|2]), (122,232,[2_1|3]), (123,124,[1_1|2]), (124,125,[5_1|2]), (125,126,[2_1|2]), (126,127,[1_1|2]), (127,92,[0_1|2]), (127,98,[0_1|2]), (127,103,[0_1|2]), (127,108,[0_1|2]), (127,158,[0_1|2]), (127,163,[0_1|2]), (127,183,[0_1|2]), (127,148,[2_1|2]), (127,153,[1_1|2]), (127,232,[2_1|3]), (128,129,[0_1|2]), (129,130,[4_1|2]), (130,131,[0_1|2]), (131,132,[2_1|2]), (132,92,[2_1|2]), (132,98,[2_1|2]), (132,103,[2_1|2]), (132,108,[2_1|2]), (132,158,[2_1|2]), (132,163,[2_1|2]), (132,183,[2_1|2, 0_1|2]), (132,104,[2_1|2]), (133,134,[0_1|2]), (134,135,[4_1|2]), (135,136,[5_1|2]), (136,137,[0_1|2]), (137,92,[2_1|2]), (137,118,[2_1|2]), (137,148,[2_1|2]), (137,109,[2_1|2]), (137,159,[2_1|2]), (137,183,[0_1|2]), (138,139,[1_1|2]), (139,140,[1_1|2]), (140,141,[3_1|2]), (141,142,[2_1|2]), (142,92,[4_1|2]), (142,113,[4_1|2]), (142,138,[4_1|2]), (142,168,[4_1|2]), (142,173,[3_1|2]), (142,178,[3_1|2]), (143,144,[4_1|2]), (144,145,[4_1|2]), (145,146,[1_1|2]), (146,147,[2_1|2]), (147,92,[2_1|2]), (147,113,[2_1|2]), (147,138,[2_1|2]), (147,168,[2_1|2]), (147,183,[0_1|2]), (148,149,[1_1|2]), (149,150,[1_1|2]), (150,151,[0_1|2]), (151,152,[2_1|2]), (152,92,[0_1|2]), (152,98,[0_1|2]), (152,103,[0_1|2]), (152,108,[0_1|2]), (152,158,[0_1|2]), (152,163,[0_1|2]), (152,183,[0_1|2]), (152,134,[0_1|2]), (152,148,[2_1|2]), (152,153,[1_1|2]), (152,232,[2_1|3]), (153,154,[0_1|2]), (154,155,[1_1|2]), (155,156,[3_1|2]), (156,157,[4_1|2]), (157,92,[2_1|2]), (157,123,[2_1|2]), (157,183,[0_1|2]), (158,159,[2_1|2]), (159,160,[0_1|2]), (160,161,[0_1|2]), (160,237,[2_1|3]), (161,162,[3_1|2]), (162,92,[0_1|2]), (162,98,[0_1|2]), (162,103,[0_1|2]), (162,108,[0_1|2]), (162,158,[0_1|2]), (162,163,[0_1|2]), (162,183,[0_1|2]), (162,148,[2_1|2]), (162,153,[1_1|2]), (162,232,[2_1|3]), (163,164,[1_1|2]), (164,165,[3_1|2]), (165,166,[4_1|2]), (166,167,[3_1|2]), (167,92,[4_1|2]), (167,113,[4_1|2]), (167,138,[4_1|2]), (167,168,[4_1|2]), (167,173,[3_1|2]), (167,178,[3_1|2]), (168,169,[1_1|2]), (169,170,[3_1|2]), (170,171,[4_1|2]), (171,172,[2_1|2]), (172,92,[3_1|2]), (172,113,[3_1|2]), (172,138,[3_1|2]), (172,168,[3_1|2, 4_1|2]), (173,174,[3_1|2]), (174,175,[2_1|2]), (175,176,[2_1|2]), (176,177,[3_1|2]), (177,92,[1_1|2]), (177,113,[1_1|2]), (177,138,[1_1|2]), (177,168,[1_1|2]), (177,93,[3_1|2]), (177,98,[0_1|2]), (177,103,[0_1|2]), (177,108,[0_1|2]), (177,222,[3_1|3]), (178,179,[3_1|2]), (179,180,[2_1|2]), (180,181,[3_1|2]), (180,278,[4_1|3]), (181,182,[5_1|2]), (181,143,[3_1|2]), (181,242,[3_1|3]), (181,247,[4_1|3]), (182,92,[5_1|2]), (182,123,[5_1|2]), (182,113,[4_1|2]), (182,118,[2_1|2]), (182,128,[1_1|2]), (182,133,[3_1|2]), (182,138,[4_1|2]), (182,143,[3_1|2]), (182,247,[4_1|3]), (183,184,[4_1|2]), (184,185,[1_1|2]), (185,186,[2_1|2]), (186,187,[4_1|2]), (187,92,[0_1|2]), (187,98,[0_1|2]), (187,103,[0_1|2]), (187,108,[0_1|2]), (187,158,[0_1|2]), (187,163,[0_1|2]), (187,183,[0_1|2]), (187,148,[2_1|2]), (187,153,[1_1|2]), (187,232,[2_1|3]), (222,223,[1_1|3]), (223,224,[1_1|3]), (224,225,[2_1|3]), (225,226,[2_1|3]), (226,113,[4_1|3]), (226,138,[4_1|3]), (226,168,[4_1|3]), (227,228,[3_1|3]), (228,229,[2_1|3]), (229,230,[2_1|3]), (230,231,[3_1|3]), (231,113,[1_1|3]), (231,138,[1_1|3]), (231,168,[1_1|3]), (232,233,[1_1|3]), (233,234,[1_1|3]), (234,235,[0_1|3]), (235,236,[2_1|3]), (236,134,[0_1|3]), (237,238,[1_1|3]), (238,239,[1_1|3]), (239,240,[0_1|3]), (240,241,[2_1|3]), (241,92,[0_1|3]), (241,98,[0_1|3]), (241,103,[0_1|3]), (241,108,[0_1|3]), (241,158,[0_1|3, 0_1|2]), (241,163,[0_1|3, 0_1|2]), (241,183,[0_1|3]), (241,148,[2_1|2]), (241,153,[1_1|2]), (241,232,[2_1|3]), (242,243,[4_1|3]), (243,244,[4_1|3]), (244,245,[1_1|3]), (245,246,[2_1|3]), (246,113,[2_1|3]), (246,138,[2_1|3]), (246,168,[2_1|3]), (247,248,[2_1|3]), (248,249,[3_1|3]), (249,250,[1_1|3]), (250,251,[1_1|3]), (251,113,[1_1|3]), (251,138,[1_1|3]), (251,168,[1_1|3]), (251,247,[1_1|3]), (278,279,[1_1|3]), (279,280,[3_1|3]), (280,281,[4_1|3]), (281,282,[2_1|3]), (282,113,[3_1|3]), (282,138,[3_1|3]), (282,247,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)