Left Termination proof of /tmp/tmpoKxBcO/reverse-bf.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 189 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 11 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

reverse(X1s, X2s) :- reverse(X1s, [], X2s).
reverse([], Xs, Xs).
reverse(.(X, X1s), X2s, Ys) :- reverse(X1s, .(X, X2s), Ys).

Query: reverse(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: reverse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reverse(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: reverse(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T5, [], T7), SCOPE: 2, CLAUSE: 1 | reverse(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: reverse(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: reverse(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: reverse(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: reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1 | reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse(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: reverse(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: reverse(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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: reverse(T358, .(T357, T384), T367)\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: reverse(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1 | reverse(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: reverse(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: reverse(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: reverse(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\nreverse(X3, X4) :- reverse(X3, [], X4).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
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\nreverse([], X9, X9).\nand substitutionT5 -> [],\nX9 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 7;

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

84 [label="EVAL with clause\nreverse(.(X18, X19), X20, X21) :- reverse(X19, .(X18, X20), X21).\nand substitutionX18 -> T14,\nX19 -> T15,\nT5 -> .(T14, T15),\nX20 -> [],\nT7 -> T17,\nX21 -> T17,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X28, X28).\nand substitutionT15 -> [],\nT14 -> T24,\nX28 -> .(T24, []),\nT17 -> .(T24, [])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 15;

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

92 [label="EVAL with clause\nreverse(.(X37, X38), X39, X40) :- reverse(X38, .(X37, X39), X40).\nand substitutionX37 -> T33,\nX38 -> T34,\nT15 -> .(T33, T34),\nT14 -> T35,\nX39 -> .(T35, []),\nT17 -> T37,\nX40 -> T37,\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 18;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X47, X47).\nand substitutionT34 -> [],\nT33 -> T50,\nT35 -> T51,\nX47 -> .(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 = lightgreen];
21 -> 99 [arrowhead = none ];
99 -> 24;

100 [label="EVAL with clause\nreverse(.(X56, X57), X58, X59) :- reverse(X57, .(X56, X58), X59).\nand substitutionX56 -> T62,\nX57 -> T63,\nT34 -> .(T62, T63),\nT33 -> T64,\nT35 -> T65,\nX58 -> .(T64, .(T65, [])),\nT37 -> T67,\nX59 -> T67,\nT66 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X66, X66).\nand substitutionT63 -> [],\nT62 -> T86,\nT64 -> T87,\nT65 -> T88,\nX66 -> .(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 = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="EVAL with clause\nreverse(.(X75, X76), X77, X78) :- reverse(X76, .(X75, X77), X78).\nand substitutionX75 -> T101,\nX76 -> T102,\nT63 -> .(T101, T102),\nT62 -> T103,\nT64 -> T104,\nT65 -> T105,\nX77 -> .(T103, .(T104, .(T105, []))),\nT67 -> T107,\nX78 -> T107,\nT106 -> T107", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 34;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X85, X85).\nand substitutionT102 -> [],\nT101 -> T132,\nT103 -> T133,\nT104 -> T134,\nT105 -> T135,\nX85 -> .(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 = lightgreen];
37 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\nreverse(.(X94, X95), X96, X97) :- reverse(X95, .(X94, X96), X97).\nand substitutionX94 -> T150,\nX95 -> T151,\nT102 -> .(T150, T151),\nT101 -> T152,\nT103 -> T153,\nT104 -> T154,\nT105 -> T155,\nX96 -> .(T152, .(T153, .(T154, .(T155, [])))),\nT107 -> T157,\nX97 -> T157,\nT156 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X104, X104).\nand substitutionT151 -> [],\nT150 -> T188,\nT152 -> T189,\nT153 -> T190,\nT154 -> T191,\nT155 -> T192,\nX104 -> .(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 = lightgreen];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nreverse(.(X113, X114), X115, X116) :- reverse(X114, .(X113, X115), X116).\nand substitutionX113 -> T209,\nX114 -> T210,\nT151 -> .(T209, T210),\nT150 -> T211,\nT152 -> T212,\nT153 -> T213,\nT154 -> T214,\nT155 -> T215,\nX115 -> .(T211, .(T212, .(T213, .(T214, .(T215, []))))),\nT157 -> T217,\nX116 -> T217,\nT216 -> T217", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X123, X123).\nand substitutionT210 -> [],\nT209 -> T254,\nT211 -> T255,\nT212 -> T256,\nT213 -> T257,\nT214 -> T258,\nT215 -> T259,\nX123 -> .(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 = lightgreen];
53 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL with clause\nreverse(.(X132, X133), X134, X135) :- reverse(X133, .(X132, X134), X135).\nand substitutionX132 -> T278,\nX133 -> T279,\nT210 -> .(T278, T279),\nT209 -> T280,\nT211 -> T281,\nT212 -> T282,\nT213 -> T283,\nT214 -> T284,\nT215 -> T285,\nX134 -> .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, [])))))),\nT217 -> T287,\nX135 -> T287,\nT286 -> T287", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 132 [arrowhead = none ];
132 -> 58;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X142, X142).\nand substitutionT279 -> [],\nT278 -> T330,\nT280 -> T331,\nT281 -> T332,\nT282 -> T333,\nT283 -> T334,\nT284 -> T335,\nT285 -> T336,\nX142 -> .(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 = lightgreen];
61 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="EVAL with clause\nreverse(.(X151, X152), X153, X154) :- reverse(X152, .(X151, X153), X154).\nand substitutionX151 -> T357,\nX152 -> T358,\nT279 -> .(T357, T358),\nT278 -> T359,\nT280 -> T360,\nT281 -> T361,\nT282 -> T362,\nT283 -> T363,\nT284 -> T364,\nT285 -> T365,\nX153 -> .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))),\nT287 -> T367,\nX154 -> T367,\nT366 -> T367", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nreverse([], X163, X163).\nand substitutionT358 -> [],\nT357 -> T397,\nT384 -> T398,\nX163 -> .(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 = lightgreen];
70 -> 148 [arrowhead = none ];
148 -> 73;

149 [label="EVAL with clause\nreverse(.(X172, X173), X174, X175) :- reverse(X173, .(X172, X174), X175).\nand substitutionX172 -> T409,\nX173 -> T410,\nT358 -> .(T409, T410),\nT357 -> T411,\nT384 -> T412,\nX174 -> .(T411, T412),\nT367 -> T414,\nX175 -> T414,\nT413 -> T414", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 149 [arrowhead = none ];
149 -> 75;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:

Clauses:

reverseA([], T397, T398, .(T397, T398)).
reverseA(.(T409, T410), T411, T412, T414) :- reverseA(T410, T409, .(T411, T412), T414).
reverseB([], []).
reverseB(.(T24, []), .(T24, [])).
reverseB(.(T51, .(T50, [])), .(T50, .(T51, []))).
reverseB(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))).
reverseB(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))).
reverseB(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))).
reverseB(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))).
reverseB(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))).
reverseB(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) :- reverseA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367).

Query: reverseB(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
reverseB_in: (b,f)
reverseA_in: (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reverseB_in_ga([], []) → reverseB_out_ga([], [])
reverseB_in_ga(.(T24, []), .(T24, [])) → reverseB_out_ga(.(T24, []), .(T24, []))
reverseB_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseB_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseB_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseB_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseB_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseB_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseB_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseB_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseB_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseB_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseB_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseB_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseB_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseA_in_ggga([], T397, T398, .(T397, T398)) → reverseA_out_ggga([], T397, T398, .(T397, T398))
reverseA_in_ggga(.(T409, T410), T411, T412, T414) → U1_ggga(T409, T410, T411, T412, T414, reverseA_in_ggga(T410, T409, .(T411, T412), T414))
U1_ggga(T409, T410, T411, T412, T414, reverseA_out_ggga(T410, T409, .(T411, T412), T414)) → reverseA_out_ggga(.(T409, T410), T411, T412, T414)
U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseB_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

reverseB_in_ga([], []) → reverseB_out_ga([], [])
reverseB_in_ga(.(T24, []), .(T24, [])) → reverseB_out_ga(.(T24, []), .(T24, []))
reverseB_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseB_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseB_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseB_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseB_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseB_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseB_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseB_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseB_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseB_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseB_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseB_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseB_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseA_in_ggga([], T397, T398, .(T397, T398)) → reverseA_out_ggga([], T397, T398, .(T397, T398))
reverseA_in_ggga(.(T409, T410), T411, T412, T414) → U1_ggga(T409, T410, T411, T412, T414, reverseA_in_ggga(T410, T409, .(T411, T412), T414))
U1_ggga(T409, T410, T411, T412, T414, reverseA_out_ggga(T410, T409, .(T411, T412), T414)) → reverseA_out_ggga(.(T409, T410), T411, T412, T414)
U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseB_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

reverseB_in_ga([], []) → reverseB_out_ga([], [])
reverseB_in_ga(.(T24, []), .(T24, [])) → reverseB_out_ga(.(T24, []), .(T24, []))
reverseB_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseB_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseB_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseB_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseB_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseB_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseB_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseB_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseB_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseB_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseB_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseB_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseB_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseA_in_ggga([], T397, T398, .(T397, T398)) → reverseA_out_ggga([], T397, T398, .(T397, T398))
reverseA_in_ggga(.(T409, T410), T411, T412, T414) → U1_ggga(T409, T410, T411, T412, T414, reverseA_in_ggga(T410, T409, .(T411, T412), T414))
U1_ggga(T409, T410, T411, T412, T414, reverseA_out_ggga(T410, T409, .(T411, T412), T414)) → reverseA_out_ggga(.(T409, T410), T411, T412, T414)
U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseB_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
reverseB_in_ga(x1, x2) = reverseB_in_ga(x1)
[] = []
reverseB_out_ga(x1, x2) = reverseB_out_ga(x2)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ga(x11)
reverseA_in_ggga(x1, x2, x3, x4) = reverseA_in_ggga(x1, x2, x3)
reverseA_out_ggga(x1, x2, x3, x4) = reverseA_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6) = U1_ggga(x6)
REVERSEB_IN_GA(x1, x2) = REVERSEB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x11)
REVERSEA_IN_GGGA(x1, x2, x3, x4) = REVERSEA_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6) = U1_GGGA(x6)

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

reverseB_in_ga([], []) → reverseB_out_ga([], [])
reverseB_in_ga(.(T24, []), .(T24, [])) → reverseB_out_ga(.(T24, []), .(T24, []))
reverseB_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseB_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseB_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseB_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseB_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseB_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseB_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseB_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseB_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseB_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseB_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseB_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseB_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseA_in_ggga([], T397, T398, .(T397, T398)) → reverseA_out_ggga([], T397, T398, .(T397, T398))
reverseA_in_ggga(.(T409, T410), T411, T412, T414) → U1_ggga(T409, T410, T411, T412, T414, reverseA_in_ggga(T410, T409, .(T411, T412), T414))
U1_ggga(T409, T410, T411, T412, T414, reverseA_out_ggga(T410, T409, .(T411, T412), T414)) → reverseA_out_ggga(.(T409, T410), T411, T412, T414)
U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseB_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

reverseB_in_ga([], []) → reverseB_out_ga([], [])
reverseB_in_ga(.(T24, []), .(T24, [])) → reverseB_out_ga(.(T24, []), .(T24, []))
reverseB_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseB_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseB_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseB_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseB_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseB_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseB_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseB_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseB_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseB_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseB_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseB_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseB_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseA_in_ggga([], T397, T398, .(T397, T398)) → reverseA_out_ggga([], T397, T398, .(T397, T398))
reverseA_in_ggga(.(T409, T410), T411, T412, T414) → U1_ggga(T409, T410, T411, T412, T414, reverseA_in_ggga(T410, T409, .(T411, T412), T414))
U1_ggga(T409, T410, T411, T412, T414, reverseA_out_ggga(T410, T409, .(T411, T412), T414)) → reverseA_out_ggga(.(T409, T410), T411, T412, T414)
U2_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseA_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseB_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
reverseB_in_ga(x1, x2) = reverseB_in_ga(x1)
[] = []
reverseB_out_ga(x1, x2) = reverseB_out_ga(x2)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ga(x11)
reverseA_in_ggga(x1, x2, x3, x4) = reverseA_in_ggga(x1, x2, x3)
reverseA_out_ggga(x1, x2, x3, x4) = reverseA_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6) = U1_ggga(x6)
REVERSEA_IN_GGGA(x1, x2, x3, x4) = REVERSEA_IN_GGGA(x1, x2, x3)

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

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

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

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

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

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

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

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

(13) 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:

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES