Left Termination of the query pattern reverse_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
reverse(.(X, Xs), Ys) :- ','(reverse(Xs, Zs), app(Zs, .(X, []), Ys)).
reverse([], []).

Queries:

reverse(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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: 2 | reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(reverse(T7, X7), app(X7, .(T6, []), T9)) | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nreverse(T1, T2) !=? reverse(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2 | ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(reverse(T19, X32), ','(app(X32, .(T18, []), X33), app(X33, .(T6, []), T9)))\n\nT6 is ground\nT18 is ground\nT19 is ground\nX33 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: reverse(T19, X32)\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(T22, .(T18, []), X33), app(X33, .(T6, []), T9))\n\nT6 is ground\nT18 is ground\nT22 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse(T19, X32), SCOPE: 3, CLAUSE: 2 | reverse(T19, X32), SCOPE: 3, CLAUSE: 3\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse(T19, X32), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse(T19, X32), SCOPE: 3, CLAUSE: 3\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(reverse(T34, X68), app(X68, .(T33, []), X69))\n\nT33 is ground\nT34 is ground\nX69 is free; \nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: reverse(T34, X68)\n\nT34 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T37, .(T33, []), X69)\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 0 | app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 0\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T56, .(T57, []), X112)\n\nT56 is ground\nT57 is ground\nX112 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T22, .(T18, []), X33)\n\nT18 is ground\nT22 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T67, .(T6, []), T9)\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 0 | app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 1\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 0\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 1\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T92, .(T93, []), T95)\n\nT92 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app([], .(T6, []), T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 0 | app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 54 [arrowhead = none ];
54 -> 2;

55 [label="EVAL with clause\nreverse(.(X4, X5), X6) :- ','(reverse(X5, X7), app(X7, .(X4, []), X6)).\nand substitution\nX4 -> T6\nX5 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX6 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 3;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 56 [arrowhead = none ];
56 -> 4;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 57 [arrowhead = none ];
57 -> 5;

58 [label="EVAL with clause\nreverse([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 58 [arrowhead = none ];
58 -> 51;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 59 [arrowhead = none ];
59 -> 52;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 60 [arrowhead = none ];
60 -> 6;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="EVAL with clause\nreverse(.(X29, X30), X31) :- ','(reverse(X30, X32), app(X32, .(X29, []), X31)).\nand substitution\nX29 -> T18\nX30 -> T19\nT7 -> .(T18, T19)\nX7 -> X33\nX31 -> X33", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 62 [arrowhead = none ];
62 -> 8;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 66 [arrowhead = none ];
66 -> 10;

67 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX32 -> T22", fontsize=14, style = filled, fillcolor = violet];
8 -> 67 [arrowhead = none ];
67 -> 11;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 68 [arrowhead = none ];
68 -> 12;

69 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 69 [arrowhead = none ];
69 -> 30;

70 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nreplacements:\nX33 -> T67", fontsize=14, style = filled, fillcolor = violet];
11 -> 70 [arrowhead = none ];
70 -> 31;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 71 [arrowhead = none ];
71 -> 13;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 72 [arrowhead = none ];
72 -> 14;

73 [label="EVAL with clause\nreverse(.(X65, X66), X67) :- ','(reverse(X66, X68), app(X68, .(X65, []), X67)).\nand substitution\nX65 -> T33\nX66 -> T34\nT19 -> .(T33, T34)\nX32 -> X69\nX67 -> X69", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 73 [arrowhead = none ];
73 -> 15;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 74 [arrowhead = none ];
74 -> 16;

75 [label="EVAL with clause\nreverse([], []).\nand substitution\nT19 -> []\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 75 [arrowhead = none ];
75 -> 27;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 76 [arrowhead = none ];
76 -> 28;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 77 [arrowhead = none ];
77 -> 17;

78 [label="SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:\nX68 -> T37", fontsize=14, style = filled, fillcolor = violet];
15 -> 78 [arrowhead = none ];
78 -> 18;

79 [label="INSTANCE with matching:\nT19 -> T34\nX32 -> X68", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 79 [arrowhead = none , style=dashed];
79 -> 10[style=dashed];

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 80 [arrowhead = none ];
80 -> 19;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 81 [arrowhead = none ];
81 -> 20;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="EVAL with clause\napp(.(X108, X109), X110, .(X108, X111)) :- app(X109, X110, X111).\nand substitution\nX108 -> T55\nX109 -> T56\nT37 -> .(T55, T56)\nT33 -> T57\nX110 -> .(T57, [])\nX111 -> X112\nX69 -> .(T55, X112)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 84 [arrowhead = none ];
84 -> 23;

85 [label="EVAL with clause\napp([], X124, X124).\nand substitution\nT37 -> []\nT33 -> T63\nX124 -> .(T63, [])\nX69 -> .(T63, [])", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="INSTANCE with matching:\nT37 -> T56\nT33 -> T57\nX69 -> X112", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 87 [arrowhead = none , style=dashed];
87 -> 18[style=dashed];

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 89 [arrowhead = none ];
89 -> 29;

90 [label="INSTANCE with matching:\nT37 -> T22\nT33 -> T18\nX69 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 90 [arrowhead = none , style=dashed];
90 -> 18[style=dashed];

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 91 [arrowhead = none ];
91 -> 32;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 92 [arrowhead = none ];
92 -> 33;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 93 [arrowhead = none ];
93 -> 34;

94 [label="EVAL with clause\napp(.(X156, X157), X158, .(X156, X159)) :- app(X157, X158, X159).\nand substitution\nX156 -> T91\nX157 -> T92\nT67 -> .(T91, T92)\nT6 -> T93\nX158 -> .(T93, [])\nX159 -> T95\nT9 -> .(T91, T95)\nT94 -> T95", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 94 [arrowhead = none ];
94 -> 35;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 95 [arrowhead = none ];
95 -> 36;

96 [label="EVAL with clause\napp([], X170, X170).\nand substitution\nT67 -> []\nT6 -> T102\nX170 -> .(T102, [])\nT9 -> .(T102, [])", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 96 [arrowhead = none ];
96 -> 37;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 97 [arrowhead = none ];
97 -> 38;

98 [label="INSTANCE with matching:\nT67 -> T92\nT6 -> T93\nT9 -> T95", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 98 [arrowhead = none , style=dashed];
98 -> 31[style=dashed];

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="EVAL with clause\nreverse([], []).\nand substitution\nT7 -> []\nX7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 100 [arrowhead = none ];
100 -> 42;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 101 [arrowhead = none ];
101 -> 43;

102 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 102 [arrowhead = none ];
102 -> 49;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 103 [arrowhead = none ];
103 -> 44;

104 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 104 [arrowhead = none ];
104 -> 45;

105 [label="EVAL with clause\napp([], X184, X184).\nand substitution\nT6 -> T107\nX184 -> .(T107, [])\nT9 -> .(T107, [])", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 105 [arrowhead = none ];
105 -> 46;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 106 [arrowhead = none ];
106 -> 47;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 107 [arrowhead = none ];
107 -> 48;

108 [label="BACKTRACK\nfor clause: reverse([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 108 [arrowhead = none ];
108 -> 50;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 109 [arrowhead = none ];
109 -> 53;

}

(2) Obligation:

Triples:

reverse10(.(T33, T34), X69) :- reverse10(T34, X68).
reverse10(.(T33, T34), X69) :- ','(reversec10(T34, T37), app18(T37, T33, X69)).
app18(.(T55, T56), T57, .(T55, X112)) :- app18(T56, T57, X112).
app31(.(T91, T92), T93, .(T91, T95)) :- app31(T92, T93, T95).
reverse1(.(T6, .(T18, T19)), T9) :- reverse10(T19, X32).
reverse1(.(T6, .(T18, T19)), T9) :- ','(reversec10(T19, T22), app18(T22, T18, X33)).
reverse1(.(T6, .(T18, T19)), T9) :- ','(reversec10(T19, T22), ','(appc18(T22, T18, T67), app31(T67, T6, T9))).

Clauses:

reversec10(.(T33, T34), X69) :- ','(reversec10(T34, T37), appc18(T37, T33, X69)).
reversec10([], []).
appc18(.(T55, T56), T57, .(T55, X112)) :- appc18(T56, T57, X112).
appc18([], T63, .(T63, [])).
appc31(.(T91, T92), T93, .(T91, T95)) :- appc31(T92, T93, T95).
appc31([], T102, .(T102, [])).

Afs:

reverse1(x1, x2)  =  reverse1(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:
reverse1_in: (b,f)
reverse10_in: (b,f)
reversec10_in: (b,f)
appc18_in: (b,b,f)
app18_in: (b,b,f)
app31_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reversec10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reversec10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_gga(T37, T33, X69))
U2_GA(T33, T34, X69, reversec10_out_ga(T34, T37)) → APP18_IN_GGA(T37, T33, X69)
APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → U4_GGA(T55, T56, T57, X112, app18_in_gga(T56, T57, X112))
APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_GGA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reversec10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_gga(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → APP18_IN_GGA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, appc18_in_gga(T22, T18, T67))
U9_GA(T6, T18, T19, T9, appc18_out_gga(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gga(T67, T6, T9))
U9_GA(T6, T18, T19, T9, appc18_out_gga(T22, T18, T67)) → APP31_IN_GGA(T67, T6, T9)
APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → U5_GGA(T91, T92, T93, T95, app31_in_gga(T92, T93, T95))
APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GGA(T92, T93, T95)

The TRS R consists of the following rules:

reversec10_in_ga(.(T33, T34), X69) → U12_ga(T33, T34, X69, reversec10_in_ga(T34, T37))
reversec10_in_ga([], []) → reversec10_out_ga([], [])
U12_ga(T33, T34, X69, reversec10_out_ga(T34, T37)) → U13_ga(T33, T34, X69, appc18_in_gga(T37, T33, X69))
appc18_in_gga(.(T55, T56), T57, .(T55, X112)) → U14_gga(T55, T56, T57, X112, appc18_in_gga(T56, T57, X112))
appc18_in_gga([], T63, .(T63, [])) → appc18_out_gga([], T63, .(T63, []))
U14_gga(T55, T56, T57, X112, appc18_out_gga(T56, T57, X112)) → appc18_out_gga(.(T55, T56), T57, .(T55, X112))
U13_ga(T33, T34, X69, appc18_out_gga(T37, T33, X69)) → reversec10_out_ga(.(T33, T34), X69)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reverse10_in_ga(x1, x2)  =  reverse10_in_ga(x1)
reversec10_in_ga(x1, x2)  =  reversec10_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversec10_out_ga(x1, x2)  =  reversec10_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc18_in_gga(x1, x2, x3)  =  appc18_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appc18_out_gga(x1, x2, x3)  =  appc18_out_gga(x1, x2, x3)
app18_in_gga(x1, x2, x3)  =  app18_in_gga(x1, x2)
app31_in_gga(x1, x2, x3)  =  app31_in_gga(x1, x2)
REVERSE1_IN_GA(x1, x2)  =  REVERSE1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVERSE10_IN_GA(x1, x2)  =  REVERSE10_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APP18_IN_GGA(x1, x2, x3)  =  APP18_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APP31_IN_GGA(x1, x2, x3)  =  APP31_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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:

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reversec10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reversec10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_gga(T37, T33, X69))
U2_GA(T33, T34, X69, reversec10_out_ga(T34, T37)) → APP18_IN_GGA(T37, T33, X69)
APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → U4_GGA(T55, T56, T57, X112, app18_in_gga(T56, T57, X112))
APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_GGA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reversec10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_gga(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → APP18_IN_GGA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reversec10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, appc18_in_gga(T22, T18, T67))
U9_GA(T6, T18, T19, T9, appc18_out_gga(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gga(T67, T6, T9))
U9_GA(T6, T18, T19, T9, appc18_out_gga(T22, T18, T67)) → APP31_IN_GGA(T67, T6, T9)
APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → U5_GGA(T91, T92, T93, T95, app31_in_gga(T92, T93, T95))
APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GGA(T92, T93, T95)

The TRS R consists of the following rules:

reversec10_in_ga(.(T33, T34), X69) → U12_ga(T33, T34, X69, reversec10_in_ga(T34, T37))
reversec10_in_ga([], []) → reversec10_out_ga([], [])
U12_ga(T33, T34, X69, reversec10_out_ga(T34, T37)) → U13_ga(T33, T34, X69, appc18_in_gga(T37, T33, X69))
appc18_in_gga(.(T55, T56), T57, .(T55, X112)) → U14_gga(T55, T56, T57, X112, appc18_in_gga(T56, T57, X112))
appc18_in_gga([], T63, .(T63, [])) → appc18_out_gga([], T63, .(T63, []))
U14_gga(T55, T56, T57, X112, appc18_out_gga(T56, T57, X112)) → appc18_out_gga(.(T55, T56), T57, .(T55, X112))
U13_ga(T33, T34, X69, appc18_out_gga(T37, T33, X69)) → reversec10_out_ga(.(T33, T34), X69)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reverse10_in_ga(x1, x2)  =  reverse10_in_ga(x1)
reversec10_in_ga(x1, x2)  =  reversec10_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversec10_out_ga(x1, x2)  =  reversec10_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc18_in_gga(x1, x2, x3)  =  appc18_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appc18_out_gga(x1, x2, x3)  =  appc18_out_gga(x1, x2, x3)
app18_in_gga(x1, x2, x3)  =  app18_in_gga(x1, x2)
app31_in_gga(x1, x2, x3)  =  app31_in_gga(x1, x2)
REVERSE1_IN_GA(x1, x2)  =  REVERSE1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVERSE10_IN_GA(x1, x2)  =  REVERSE10_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APP18_IN_GGA(x1, x2, x3)  =  APP18_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APP31_IN_GGA(x1, x2, x3)  =  APP31_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GGA(T92, T93, T95)

The TRS R consists of the following rules:

reversec10_in_ga(.(T33, T34), X69) → U12_ga(T33, T34, X69, reversec10_in_ga(T34, T37))
reversec10_in_ga([], []) → reversec10_out_ga([], [])
U12_ga(T33, T34, X69, reversec10_out_ga(T34, T37)) → U13_ga(T33, T34, X69, appc18_in_gga(T37, T33, X69))
appc18_in_gga(.(T55, T56), T57, .(T55, X112)) → U14_gga(T55, T56, T57, X112, appc18_in_gga(T56, T57, X112))
appc18_in_gga([], T63, .(T63, [])) → appc18_out_gga([], T63, .(T63, []))
U14_gga(T55, T56, T57, X112, appc18_out_gga(T56, T57, X112)) → appc18_out_gga(.(T55, T56), T57, .(T55, X112))
U13_ga(T33, T34, X69, appc18_out_gga(T37, T33, X69)) → reversec10_out_ga(.(T33, T34), X69)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversec10_in_ga(x1, x2)  =  reversec10_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversec10_out_ga(x1, x2)  =  reversec10_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc18_in_gga(x1, x2, x3)  =  appc18_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appc18_out_gga(x1, x2, x3)  =  appc18_out_gga(x1, x2, x3)
APP31_IN_GGA(x1, x2, x3)  =  APP31_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APP31_IN_GGA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GGA(T92, T93, T95)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP31_IN_GGA(x1, x2, x3)  =  APP31_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP31_IN_GGA(.(T91, T92), T93) → APP31_IN_GGA(T92, T93)

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

(12) 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:

• APP31_IN_GGA(.(T91, T92), T93) → APP31_IN_GGA(T92, T93)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_GGA(T56, T57, X112)

The TRS R consists of the following rules:

reversec10_in_ga(.(T33, T34), X69) → U12_ga(T33, T34, X69, reversec10_in_ga(T34, T37))
reversec10_in_ga([], []) → reversec10_out_ga([], [])
U12_ga(T33, T34, X69, reversec10_out_ga(T34, T37)) → U13_ga(T33, T34, X69, appc18_in_gga(T37, T33, X69))
appc18_in_gga(.(T55, T56), T57, .(T55, X112)) → U14_gga(T55, T56, T57, X112, appc18_in_gga(T56, T57, X112))
appc18_in_gga([], T63, .(T63, [])) → appc18_out_gga([], T63, .(T63, []))
U14_gga(T55, T56, T57, X112, appc18_out_gga(T56, T57, X112)) → appc18_out_gga(.(T55, T56), T57, .(T55, X112))
U13_ga(T33, T34, X69, appc18_out_gga(T37, T33, X69)) → reversec10_out_ga(.(T33, T34), X69)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversec10_in_ga(x1, x2)  =  reversec10_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversec10_out_ga(x1, x2)  =  reversec10_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc18_in_gga(x1, x2, x3)  =  appc18_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appc18_out_gga(x1, x2, x3)  =  appc18_out_gga(x1, x2, x3)
APP18_IN_GGA(x1, x2, x3)  =  APP18_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APP18_IN_GGA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_GGA(T56, T57, X112)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP18_IN_GGA(x1, x2, x3)  =  APP18_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APP18_IN_GGA(.(T55, T56), T57) → APP18_IN_GGA(T56, T57)

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

(19) 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:

• APP18_IN_GGA(.(T55, T56), T57) → APP18_IN_GGA(T56, T57)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

The TRS R consists of the following rules:

reversec10_in_ga(.(T33, T34), X69) → U12_ga(T33, T34, X69, reversec10_in_ga(T34, T37))
reversec10_in_ga([], []) → reversec10_out_ga([], [])
U12_ga(T33, T34, X69, reversec10_out_ga(T34, T37)) → U13_ga(T33, T34, X69, appc18_in_gga(T37, T33, X69))
appc18_in_gga(.(T55, T56), T57, .(T55, X112)) → U14_gga(T55, T56, T57, X112, appc18_in_gga(T56, T57, X112))
appc18_in_gga([], T63, .(T63, [])) → appc18_out_gga([], T63, .(T63, []))
U14_gga(T55, T56, T57, X112, appc18_out_gga(T56, T57, X112)) → appc18_out_gga(.(T55, T56), T57, .(T55, X112))
U13_ga(T33, T34, X69, appc18_out_gga(T37, T33, X69)) → reversec10_out_ga(.(T33, T34), X69)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
reversec10_in_ga(x1, x2)  =  reversec10_in_ga(x1)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
[]  =  []
reversec10_out_ga(x1, x2)  =  reversec10_out_ga(x1, x2)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc18_in_gga(x1, x2, x3)  =  appc18_in_gga(x1, x2)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appc18_out_gga(x1, x2, x3)  =  appc18_out_gga(x1, x2, x3)
REVERSE10_IN_GA(x1, x2)  =  REVERSE10_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REVERSE10_IN_GA(x1, x2)  =  REVERSE10_IN_GA(x1)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

REVERSE10_IN_GA(.(T33, T34)) → REVERSE10_IN_GA(T34)

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

(26) 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:

• REVERSE10_IN_GA(.(T33, T34)) → REVERSE10_IN_GA(T34)
  The graph contains the following edges 1 > 1

(27) YES