Left Termination proof of /tmp/tmpNJrkgc/NJ1.pl

Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 PiDP

↳7 PiDPToQDPProof (⇐)

↳8 QDP

↳9 QDPSizeChangeProof (⇔)

↳10 YES

(0) Obligation:

Clauses:

rev(LS, RES) :- r1(LS, [], RES).
r1([], RES, RES).
r1(.(X, Xs), Accm, RES) :- r1(Xs, .(X, Accm), RES).

Queries:

rev(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rev(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: r1(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: r1(T5, [], T7), SCOPE: 2, CLAUSE: 1 | r1(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: r1(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: r1(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: r1(T15, .(T14, []), T17)\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1 | r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: r1(T34, .(T33, .(T35, [])), T37)\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1 | r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: r1(T63, .(T62, .(T64, .(T65, []))), T67)\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1 | r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107)\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1 | r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157)\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1 | r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217)\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1 | r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287)\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1 | r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: r1(T358, .(T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))), T367)\n\nT357 is ground\nT358 is ground\nT359 is ground\nT360 is ground\nT361 is ground\nT362 is ground\nT363 is ground\nT364 is ground\nT365 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: r1(T358, .(T357, T384), T367)\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1 | r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: r1(T410, .(T409, .(T411, T412)), T414)\n\nT409 is ground\nT410 is ground\nT411 is ground\nT412 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="ONLY EVAL with clause\nrev(X3, X4) :- r1(X3, [], X4).\nand substitution\nT1 -> T5\nX3 -> T5\nT2 -> T7\nX4 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 81 [arrowhead = none ];
81 -> 6;

82 [label="EVAL with clause\nr1([], X9, X9).\nand substitution\nT5 -> []\nX9 -> []\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 7;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 83 [arrowhead = none ];
83 -> 8;

84 [label="EVAL with clause\nr1(.(X19, X20), X21, X22) :- r1(X20, .(X19, X21), X22).\nand substitution\nX19 -> T14\nX20 -> T15\nT5 -> .(T14, T15)\nX21 -> []\nT7 -> T17\nX22 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 85 [arrowhead = none ];
85 -> 11;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 88 [arrowhead = none ];
88 -> 13;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="EVAL with clause\nr1([], X33, X33).\nand substitution\nT15 -> []\nT14 -> T24\nX33 -> .(T24, [])\nT17 -> .(T24, [])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="EVAL with clause\nr1(.(X43, X44), X45, X46) :- r1(X44, .(X43, X45), X46).\nand substitution\nX43 -> T33\nX44 -> T34\nT15 -> .(T33, T34)\nT14 -> T35\nX45 -> .(T35, [])\nT17 -> T37\nX46 -> T37\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 18;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 93 [arrowhead = none ];
93 -> 19;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 95 [arrowhead = none ];
95 -> 20;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 97 [arrowhead = none ];
97 -> 22;

98 [label="EVAL with clause\nr1([], X57, X57).\nand substitution\nT34 -> []\nT33 -> T50\nT35 -> T51\nX57 -> .(T50, .(T51, []))\nT37 -> .(T50, .(T51, []))", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 98 [arrowhead = none ];
98 -> 23;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 99 [arrowhead = none ];
99 -> 24;

100 [label="EVAL with clause\nr1(.(X67, X68), X69, X70) :- r1(X68, .(X67, X69), X70).\nand substitution\nX67 -> T62\nX68 -> T63\nT34 -> .(T62, T63)\nT33 -> T64\nT35 -> T65\nX69 -> .(T64, .(T65, []))\nT37 -> T67\nX70 -> T67\nT66 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 104 [arrowhead = none ];
104 -> 29;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 105 [arrowhead = none ];
105 -> 30;

106 [label="EVAL with clause\nr1([], X81, X81).\nand substitution\nT63 -> []\nT62 -> T86\nT64 -> T87\nT65 -> T88\nX81 -> .(T86, .(T87, .(T88, [])))\nT67 -> .(T86, .(T87, .(T88, [])))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 31;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="EVAL with clause\nr1(.(X91, X92), X93, X94) :- r1(X92, .(X91, X93), X94).\nand substitution\nX91 -> T101\nX92 -> T102\nT63 -> .(T101, T102)\nT62 -> T103\nT64 -> T104\nT65 -> T105\nX93 -> .(T103, .(T104, .(T105, [])))\nT67 -> T107\nX94 -> T107\nT106 -> T107", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 34;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 109 [arrowhead = none ];
109 -> 35;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 111 [arrowhead = none ];
111 -> 36;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 112 [arrowhead = none ];
112 -> 37;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="EVAL with clause\nr1([], X105, X105).\nand substitution\nT102 -> []\nT101 -> T132\nT103 -> T133\nT104 -> T134\nT105 -> T135\nX105 -> .(T132, .(T133, .(T134, .(T135, []))))\nT107 -> .(T132, .(T133, .(T134, .(T135, []))))", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 114 [arrowhead = none ];
114 -> 39;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\nr1(.(X115, X116), X117, X118) :- r1(X116, .(X115, X117), X118).\nand substitution\nX115 -> T150\nX116 -> T151\nT102 -> .(T150, T151)\nT101 -> T152\nT103 -> T153\nT104 -> T154\nT105 -> T155\nX117 -> .(T152, .(T153, .(T154, .(T155, []))))\nT107 -> T157\nX118 -> T157\nT156 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 118 [arrowhead = none ];
118 -> 41;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="EVAL with clause\nr1([], X129, X129).\nand substitution\nT151 -> []\nT150 -> T188\nT152 -> T189\nT153 -> T190\nT154 -> T191\nT155 -> T192\nX129 -> .(T188, .(T189, .(T190, .(T191, .(T192, [])))))\nT157 -> .(T188, .(T189, .(T190, .(T191, .(T192, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 122 [arrowhead = none ];
122 -> 47;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nr1(.(X139, X140), X141, X142) :- r1(X140, .(X139, X141), X142).\nand substitution\nX139 -> T209\nX140 -> T210\nT151 -> .(T209, T210)\nT150 -> T211\nT152 -> T212\nT153 -> T213\nT154 -> T214\nT155 -> T215\nX141 -> .(T211, .(T212, .(T213, .(T214, .(T215, [])))))\nT157 -> T217\nX142 -> T217\nT216 -> T217", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 125 [arrowhead = none ];
125 -> 51;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 126 [arrowhead = none ];
126 -> 49;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 127 [arrowhead = none ];
127 -> 52;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 128 [arrowhead = none ];
128 -> 53;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 129 [arrowhead = none ];
129 -> 54;

130 [label="EVAL with clause\nr1([], X153, X153).\nand substitution\nT210 -> []\nT209 -> T254\nT211 -> T255\nT212 -> T256\nT213 -> T257\nT214 -> T258\nT215 -> T259\nX153 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))\nT217 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL with clause\nr1(.(X163, X164), X165, X166) :- r1(X164, .(X163, X165), X166).\nand substitution\nX163 -> T278\nX164 -> T279\nT210 -> .(T278, T279)\nT209 -> T280\nT211 -> T281\nT212 -> T282\nT213 -> T283\nT214 -> T284\nT215 -> T285\nX165 -> .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))\nT217 -> T287\nX166 -> T287\nT286 -> T287", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 132 [arrowhead = none ];
132 -> 58;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 133 [arrowhead = none ];
133 -> 59;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 134 [arrowhead = none ];
134 -> 57;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 135 [arrowhead = none ];
135 -> 60;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 136 [arrowhead = none ];
136 -> 61;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 137 [arrowhead = none ];
137 -> 62;

138 [label="EVAL with clause\nr1([], X177, X177).\nand substitution\nT279 -> []\nT278 -> T330\nT280 -> T331\nT281 -> T332\nT282 -> T333\nT283 -> T334\nT284 -> T335\nT285 -> T336\nX177 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))\nT287 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="EVAL with clause\nr1(.(X187, X188), X189, X190) :- r1(X188, .(X187, X189), X190).\nand substitution\nX187 -> T357\nX188 -> T358\nT279 -> .(T357, T358)\nT278 -> T359\nT280 -> T360\nT281 -> T361\nT282 -> T362\nT283 -> T363\nT284 -> T364\nT285 -> T365\nX189 -> .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))\nT287 -> T367\nX190 -> T367\nT366 -> T367", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 141 [arrowhead = none ];
141 -> 67;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 142 [arrowhead = none ];
142 -> 65;

143 [label="GENERALIZATION\nT384 <-- .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))\n\nNew Knowledge:\nT384 is ground", fontsize=14, style = filled, fillcolor = violet];
66 -> 143 [arrowhead = none ];
143 -> 68;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 144 [arrowhead = none ];
144 -> 69;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 145 [arrowhead = none ];
145 -> 70;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 146 [arrowhead = none ];
146 -> 71;

147 [label="EVAL with clause\nr1([], X203, X203).\nand substitution\nT358 -> []\nT357 -> T397\nT384 -> T398\nX203 -> .(T397, T398)\nT367 -> .(T397, T398)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 147 [arrowhead = none ];
147 -> 72;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 148 [arrowhead = none ];
148 -> 73;

149 [label="EVAL with clause\nr1(.(X213, X214), X215, X216) :- r1(X214, .(X213, X215), X216).\nand substitution\nX213 -> T409\nX214 -> T410\nT358 -> .(T409, T410)\nT357 -> T411\nT384 -> T412\nX215 -> .(T411, T412)\nT367 -> T414\nX216 -> T414\nT413 -> T414", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 149 [arrowhead = none ];
149 -> 75;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 150 [arrowhead = none ];
150 -> 76;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 151 [arrowhead = none ];
151 -> 74;

152 [label="INSTANCE with matching:\nT358 -> T410\nT357 -> T409\nT384 -> .(T411, T412)\nT367 -> T414", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 152 [arrowhead = none , style=dashed];
152 -> 68[style=dashed];

}

(2) Obligation:

Triples:

r168(.(T409, T410), T411, T412, T414) :- r168(T410, T409, .(T411, T412), T414).
rev1(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) :- r168(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367).

Clauses:

r1c68([], T397, T398, .(T397, T398)).
r1c68(.(T409, T410), T411, T412, T414) :- r1c68(T410, T409, .(T411, T412), T414).

Afs:

rev1(x1, x2) = rev1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev1_in: (b,f)
r168_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REV1_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r168_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REV1_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → R168_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
R168_IN_GGGA(.(T409, T410), T411, T412, T414) → U1_GGGA(T409, T410, T411, T412, T414, r168_in_ggga(T410, T409, .(T411, T412), T414))
R168_IN_GGGA(.(T409, T410), T411, T412, T414) → R168_IN_GGGA(T410, T409, .(T411, T412), T414)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
r168_in_ggga(x1, x2, x3, x4) = r168_in_ggga(x1, x2, x3)
[] = []
REV1_IN_GA(x1, x2) = REV1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
R168_IN_GGGA(x1, x2, x3, x4) = R168_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6) = U1_GGGA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

REV1_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r168_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REV1_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → R168_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
R168_IN_GGGA(.(T409, T410), T411, T412, T414) → U1_GGGA(T409, T410, T411, T412, T414, r168_in_ggga(T410, T409, .(T411, T412), T414))
R168_IN_GGGA(.(T409, T410), T411, T412, T414) → R168_IN_GGGA(T410, T409, .(T411, T412), T414)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

R168_IN_GGGA(.(T409, T410), T411, T412, T414) → R168_IN_GGGA(T410, T409, .(T411, T412), T414)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
R168_IN_GGGA(x1, x2, x3, x4) = R168_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains

(7) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

R168_IN_GGGA(.(T409, T410), T411, T412) → R168_IN_GGGA(T410, T409, .(T411, T412))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(9) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

R168_IN_GGGA(.(T409, T410), T411, T412) → R168_IN_GGGA(T410, T409, .(T411, T412))
The graph contains the following edges 1 > 1, 1 > 2