WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 51 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 0 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 (innermost) 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(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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 (innermost) 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(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: 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(n^1, n^1). The TRS R consists of the following rules: 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: 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: 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: 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 4. The certificate found is represented by the following graph. "[9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 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, 145, 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] {(9,10,[0_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]), (9,11,[1_1|1]), (9,29,[1_1|1]), (9,53,[1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1]), (9,54,[1_1|2]), (9,72,[1_1|2]), (10,10,[1_1|0, 2_1|0, 3_1|0, 4_1|0, 5_1|0, cons_0_1|0]), (11,12,[2_1|1]), (12,13,[3_1|1]), (13,14,[4_1|1]), (14,15,[5_1|1]), (15,16,[1_1|1]), (16,17,[1_1|1]), 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(96,97,[2_1|2]), (97,98,[3_1|2]), (98,99,[4_1|2]), (99,100,[5_1|2]), (100,101,[1_1|2]), (101,102,[1_1|2]), (102,103,[0_1|2]), (103,104,[1_1|2]), (104,105,[2_1|2]), (105,106,[3_1|2]), (106,107,[4_1|2]), (107,108,[5_1|2]), (108,109,[0_1|2]), (109,110,[1_1|2]), (110,111,[2_1|2]), (111,112,[3_1|2]), (112,113,[4_1|2]), (113,11,[5_1|2]), (113,29,[5_1|2]), (114,115,[2_1|2]), (115,116,[3_1|2]), (116,117,[4_1|2]), (117,118,[5_1|2]), (118,119,[1_1|2]), (119,120,[1_1|2]), (120,121,[0_1|2]), (121,122,[1_1|2]), (122,123,[2_1|2]), (123,124,[3_1|2]), (124,125,[4_1|2]), (125,126,[5_1|2]), (126,127,[0_1|2]), (127,128,[1_1|2]), (128,129,[2_1|2]), (129,130,[3_1|2]), (130,131,[4_1|2]), (131,132,[5_1|2]), (132,133,[0_1|2]), (133,134,[1_1|2]), (134,135,[2_1|2]), (135,136,[3_1|2]), (136,137,[4_1|2]), (137,11,[5_1|2]), (137,29,[5_1|2]), (145,147,[2_1|2]), (147,148,[3_1|2]), (148,149,[4_1|2]), (149,150,[5_1|2]), (150,151,[1_1|2]), (151,152,[1_1|2]), (152,153,[0_1|2]), (153,154,[1_1|2]), (154,155,[2_1|2]), (155,156,[3_1|2]), (156,157,[4_1|2]), (157,158,[5_1|2]), (158,159,[0_1|2]), (159,160,[1_1|2]), (160,161,[2_1|2]), (161,162,[3_1|2]), (162,163,[4_1|2]), (163,96,[5_1|2]), (163,114,[5_1|2]), (164,165,[2_1|2]), (165,166,[3_1|2]), (166,167,[4_1|2]), (167,168,[5_1|2]), (168,169,[1_1|2]), (169,170,[1_1|2]), (170,171,[0_1|2]), (171,172,[1_1|2]), (172,173,[2_1|2]), (173,174,[3_1|2]), (174,175,[4_1|2]), (175,176,[5_1|2]), (176,177,[0_1|2]), (177,178,[1_1|2]), (178,179,[2_1|2]), (179,180,[3_1|2]), (180,181,[4_1|2]), (181,182,[5_1|2]), (182,183,[0_1|2]), (183,184,[1_1|2]), (184,185,[2_1|2]), (185,186,[3_1|2]), (186,187,[4_1|2]), (187,96,[5_1|2]), (187,114,[5_1|2]), (188,189,[2_1|3]), (189,190,[3_1|3]), (190,191,[4_1|3]), (191,192,[5_1|3]), (192,193,[1_1|3]), (193,194,[1_1|3]), (194,195,[0_1|3]), (194,272,[1_1|4]), (194,290,[1_1|4]), (195,196,[1_1|3]), (196,197,[2_1|3]), (197,198,[3_1|3]), (198,199,[4_1|3]), (199,200,[5_1|3]), (200,201,[0_1|3]), (200,230,[1_1|4]), (200,248,[1_1|4]), (201,202,[1_1|3]), (202,203,[2_1|3]), (203,204,[3_1|3]), (204,205,[4_1|3]), (205,54,[5_1|3]), (205,72,[5_1|3]), (205,188,[5_1|3]), (205,206,[5_1|3]), (205,60,[5_1|3]), (205,78,[5_1|3]), (206,207,[2_1|3]), (207,208,[3_1|3]), (208,209,[4_1|3]), (209,210,[5_1|3]), (210,211,[1_1|3]), (211,212,[1_1|3]), (212,213,[0_1|3]), (212,314,[1_1|4]), (212,332,[1_1|4]), (213,214,[1_1|3]), (214,215,[2_1|3]), (215,216,[3_1|3]), (216,217,[4_1|3]), (217,218,[5_1|3]), (218,219,[0_1|3]), (218,272,[1_1|4]), (218,290,[1_1|4]), (219,220,[1_1|3]), (220,221,[2_1|3]), (221,222,[3_1|3]), (222,223,[4_1|3]), (223,224,[5_1|3]), (224,225,[0_1|3]), (224,230,[1_1|4]), (224,248,[1_1|4]), (225,226,[1_1|3]), (226,227,[2_1|3]), (227,228,[3_1|3]), (228,229,[4_1|3]), (229,54,[5_1|3]), (229,72,[5_1|3]), (229,188,[5_1|3]), (229,206,[5_1|3]), (229,60,[5_1|3]), (229,78,[5_1|3]), (230,231,[2_1|4]), (231,232,[3_1|4]), (232,233,[4_1|4]), (233,234,[5_1|4]), (234,235,[1_1|4]), (235,236,[1_1|4]), (236,237,[0_1|4]), (237,238,[1_1|4]), (238,239,[2_1|4]), (239,240,[3_1|4]), (240,241,[4_1|4]), (241,242,[5_1|4]), (242,243,[0_1|4]), (243,244,[1_1|4]), (244,245,[2_1|4]), (245,246,[3_1|4]), (246,247,[4_1|4]), (247,188,[5_1|4]), (247,206,[5_1|4]), (248,249,[2_1|4]), (249,250,[3_1|4]), (250,251,[4_1|4]), (251,252,[5_1|4]), (252,253,[1_1|4]), (253,254,[1_1|4]), (254,255,[0_1|4]), (255,256,[1_1|4]), (256,257,[2_1|4]), (257,258,[3_1|4]), (258,259,[4_1|4]), (259,260,[5_1|4]), (260,261,[0_1|4]), (261,262,[1_1|4]), (262,263,[2_1|4]), (263,264,[3_1|4]), (264,265,[4_1|4]), (265,266,[5_1|4]), (266,267,[0_1|4]), (267,268,[1_1|4]), (268,269,[2_1|4]), (269,270,[3_1|4]), (270,271,[4_1|4]), (271,188,[5_1|4]), (271,206,[5_1|4]), (272,273,[2_1|4]), (273,274,[3_1|4]), (274,275,[4_1|4]), (275,276,[5_1|4]), (276,277,[1_1|4]), (277,278,[1_1|4]), (278,279,[0_1|4]), (279,280,[1_1|4]), (280,281,[2_1|4]), (281,282,[3_1|4]), (282,283,[4_1|4]), (283,284,[5_1|4]), (284,285,[0_1|4]), (285,286,[1_1|4]), (286,287,[2_1|4]), (287,288,[3_1|4]), (288,289,[4_1|4]), (289,230,[5_1|4]), (289,248,[5_1|4]), (290,291,[2_1|4]), (291,292,[3_1|4]), (292,293,[4_1|4]), (293,294,[5_1|4]), (294,295,[1_1|4]), (295,296,[1_1|4]), (296,297,[0_1|4]), (297,298,[1_1|4]), (298,299,[2_1|4]), (299,300,[3_1|4]), (300,301,[4_1|4]), (301,302,[5_1|4]), (302,303,[0_1|4]), (303,304,[1_1|4]), (304,305,[2_1|4]), (305,306,[3_1|4]), (306,307,[4_1|4]), (307,308,[5_1|4]), (308,309,[0_1|4]), (309,310,[1_1|4]), (310,311,[2_1|4]), (311,312,[3_1|4]), (312,313,[4_1|4]), (313,230,[5_1|4]), (313,248,[5_1|4]), (314,315,[2_1|4]), (315,316,[3_1|4]), (316,317,[4_1|4]), (317,318,[5_1|4]), (318,319,[1_1|4]), (319,320,[1_1|4]), (320,321,[0_1|4]), (321,322,[1_1|4]), (322,323,[2_1|4]), (323,324,[3_1|4]), (324,325,[4_1|4]), (325,326,[5_1|4]), (326,327,[0_1|4]), (327,328,[1_1|4]), (328,329,[2_1|4]), (329,330,[3_1|4]), (330,331,[4_1|4]), (331,272,[5_1|4]), (331,290,[5_1|4]), (332,333,[2_1|4]), (333,334,[3_1|4]), (334,335,[4_1|4]), (335,336,[5_1|4]), (336,337,[1_1|4]), (337,338,[1_1|4]), (338,339,[0_1|4]), (339,340,[1_1|4]), (340,341,[2_1|4]), (341,342,[3_1|4]), (342,343,[4_1|4]), (343,344,[5_1|4]), (344,345,[0_1|4]), (345,346,[1_1|4]), (346,347,[2_1|4]), (347,348,[3_1|4]), (348,349,[4_1|4]), (349,350,[5_1|4]), (350,351,[0_1|4]), (351,352,[1_1|4]), (352,353,[2_1|4]), (353,354,[3_1|4]), (354,355,[4_1|4]), (355,272,[5_1|4]), (355,290,[5_1|4])}" ---------------------------------------- (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 (innermost) 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(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: INNERMOST ---------------------------------------- (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(3(4(5(1(x1))))))) ->^+ 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / 1(x1)]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) 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(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) 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(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x1))))))))))))))))))) 0(1(2(3(4(5(1(x1))))))) -> 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(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(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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: INNERMOST