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

(0) Obligation:

Clauses:

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

Query: rev(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="D: 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 | rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(rev(T7, X10), app(X10, .(T6, []), T9))\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 0 | ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 0\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app([], .(T6, []), T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 2 | app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(rev(T22, X34), app(X34, .(T21, []), X35)), app(X35, .(T6, []), T9))\n\nT6 is ground\nT21 is ground\nT22 is ground\nX35 is free\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="A: rev(T22, X34)\n\nT22 is ground\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(app(T23, .(T21, []), X35), app(X35, .(T6, []), T9))\n\nT6 is ground\nT21 is ground\nT23 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rev(T22, X34), SCOPE: 4, CLAUSE: 0 | rev(T22, X34), SCOPE: 4, CLAUSE: 1\n\nT22 is ground\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: rev(T22, X34), SCOPE: 4, CLAUSE: 0\n\nT22 is ground\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: rev(T22, X34), SCOPE: 4, CLAUSE: 1\n\nT22 is ground\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(rev(T29, X48), app(X48, .(T28, []), X49))\n\nT28 is ground\nT29 is ground\nX49 is free\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: rev(T29, X48)\n\nT29 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="B: app(T30, .(T28, []), X49)\n\nT28 is ground\nT30 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T30, .(T28, []), X49), SCOPE: 5, CLAUSE: 2 | app(T30, .(T28, []), X49), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T30, .(T28, []), X49), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nT30 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app(T30, .(T28, []), X49), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T45, .(T46, []), X71)\n\nT45 is ground\nT46 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T23, .(T21, []), X35)\n\nT21 is ground\nT23 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="C: app(T51, .(T6, []), T9)\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 2 | app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 2\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app(T70, .(T71, []), T73)\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 53 [arrowhead = none ];
53 -> 2;

54 [label="EVAL with clause\nrev([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 54 [arrowhead = none ];
54 -> 3;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 55 [arrowhead = none ];
55 -> 4;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 56 [arrowhead = none ];
56 -> 5;

57 [label="EVAL with clause\nrev(.(X7, X8), X9) :- ','(rev(X8, X10), app(X10, .(X7, []), X9)).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX9 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="EVAL with clause\nrev([], []).\nand substitutionT7 -> [],\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="EVAL with clause\nrev(.(X31, X32), X33) :- ','(rev(X32, X34), app(X34, .(X31, []), X33)).\nand substitutionX31 -> T21,\nX32 -> T22,\nT7 -> .(T21, T22),\nX10 -> X35,\nX33 -> X35", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\napp([], X17, X17).\nand substitutionT6 -> T16,\nX17 -> .(T16, []),\nT9 -> .(T16, [])", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 73 [arrowhead = none ];
73 -> 19;

74 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 74 [arrowhead = none ];
74 -> 23;

75 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground\nreplacements:X34 -> T23", fontsize=14, style = filled, fillcolor = violet];
21 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 77 [arrowhead = none ];
77 -> 43;

78 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT21 is ground\nT51 is ground\nreplacements:X35 -> T51", fontsize=14, style = filled, fillcolor = violet];
24 -> 78 [arrowhead = none ];
78 -> 44;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="EVAL with clause\nrev([], []).\nand substitutionT22 -> [],\nX34 -> []", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="EVAL with clause\nrev(.(X45, X46), X47) :- ','(rev(X46, X48), app(X48, .(X45, []), X47)).\nand substitutionX45 -> T28,\nX46 -> T29,\nT22 -> .(T28, T29),\nX34 -> X49,\nX47 -> X49", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 83 [arrowhead = none ];
83 -> 31;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 85 [arrowhead = none ];
85 -> 30;

86 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nreplacements:X48 -> T30", fontsize=14, style = filled, fillcolor = violet];
31 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="INSTANCE with matching:\nT22 -> T29\nX34 -> X48", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 88 [arrowhead = none , style=dashed];
88 -> 23[style=dashed];

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 90 [arrowhead = none ];
90 -> 36;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 91 [arrowhead = none ];
91 -> 37;

92 [label="EVAL with clause\napp([], X56, X56).\nand substitutionT30 -> [],\nT28 -> T37,\nX56 -> .(T37, []),\nX49 -> .(T37, [])", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 92 [arrowhead = none ];
92 -> 38;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="EVAL with clause\napp(.(X67, X68), X69, .(X67, X70)) :- app(X68, X69, X70).\nand substitutionX67 -> T44,\nX68 -> T45,\nT30 -> .(T44, T45),\nT28 -> T46,\nX69 -> .(T46, []),\nX70 -> X71,\nX49 -> .(T44, X71)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 94 [arrowhead = none ];
94 -> 41;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 95 [arrowhead = none ];
95 -> 42;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 96 [arrowhead = none ];
96 -> 40;

97 [label="INSTANCE with matching:\nT30 -> T45\nT28 -> T46\nX49 -> X71", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 97 [arrowhead = none , style=dashed];
97 -> 34[style=dashed];

98 [label="INSTANCE with matching:\nT30 -> T23\nT28 -> T21\nX49 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 98 [arrowhead = none , style=dashed];
98 -> 34[style=dashed];

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 101 [arrowhead = none ];
101 -> 47;

102 [label="EVAL with clause\napp([], X84, X84).\nand substitutionT51 -> [],\nT6 -> T60,\nX84 -> .(T60, []),\nT9 -> .(T60, [])", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 103 [arrowhead = none ];
103 -> 49;

104 [label="EVAL with clause\napp(.(X93, X94), X95, .(X93, X96)) :- app(X94, X95, X96).\nand substitutionX93 -> T69,\nX94 -> T70,\nT51 -> .(T69, T70),\nT6 -> T71,\nX95 -> .(T71, []),\nX96 -> T73,\nT9 -> .(T69, T73),\nT72 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 104 [arrowhead = none ];
104 -> 51;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 105 [arrowhead = none ];
105 -> 52;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="INSTANCE with matching:\nT51 -> T70\nT6 -> T71\nT9 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 107 [arrowhead = none , style=dashed];
107 -> 44[style=dashed];

}

(2) Obligation:

Triples:

revA(.(X1, X2), X3) :- revA(X2, X4).
revA(.(X1, X2), X3) :- ','(revcA(X2, X4), appB(X4, X1, X3)).
appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
appC(.(X1, X2), X3, .(X1, X4)) :- appC(X2, X3, X4).
revD(.(X1, .(X2, X3)), X4) :- revA(X3, X5).
revD(.(X1, .(X2, X3)), X4) :- ','(revcA(X3, X5), appB(X5, X2, X6)).
revD(.(X1, .(X2, X3)), X4) :- ','(revcA(X3, X5), ','(appcB(X5, X2, X6), appC(X6, X1, X4))).

Clauses:

revcA([], []).
revcA(.(X1, X2), X3) :- ','(revcA(X2, X4), appcB(X4, X1, X3)).
appcB([], X1, .(X1, [])).
appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
appcC([], X1, .(X1, [])).
appcC(.(X1, X2), X3, .(X1, X4)) :- appcC(X2, X3, X4).

Afs:

revD(x1, x2)  =  revD(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
revD_in: (b,f)
revA_in: (b,f)
revcA_in: (b,f)
appcB_in: (b,b,f)
appB_in: (b,b,f)
appC_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVD_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, revA_in_ga(X3, X5))
REVD_IN_GA(.(X1, .(X2, X3)), X4) → REVA_IN_GA(X3, X5)
REVA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, revA_in_ga(X2, X4))
REVA_IN_GA(.(X1, X2), X3) → REVA_IN_GA(X2, X4)
REVA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, revcA_in_ga(X2, X4))
U2_GA(X1, X2, X3, revcA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, appB_in_gga(X4, X1, X3))
U2_GA(X1, X2, X3, revcA_out_ga(X2, X4)) → APPB_IN_GGA(X4, X1, X3)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U4_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
REVD_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, revcA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, appB_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → APPB_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, appcB_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, appC_in_gga(X6, X1, X4))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → APPC_IN_GGA(X6, X1, X4)
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U5_GGA(X1, X2, X3, X4, appC_in_gga(X2, X3, X4))
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

revcA_in_ga([], []) → revcA_out_ga([], [])
revcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, revcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, revcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → revcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revA_in_ga(x1, x2)  =  revA_in_ga(x1)
revcA_in_ga(x1, x2)  =  revcA_in_ga(x1)
[]  =  []
revcA_out_ga(x1, x2)  =  revcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appC_in_gga(x1, x2, x3)  =  appC_in_gga(x1, x2)
REVD_IN_GA(x1, x2)  =  REVD_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVA_IN_GA(x1, x2)  =  REVA_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)
APPB_IN_GGA(x1, x2, x3)  =  APPB_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)
APPC_IN_GGA(x1, x2, x3)  =  APPC_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:

REVD_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, revA_in_ga(X3, X5))
REVD_IN_GA(.(X1, .(X2, X3)), X4) → REVA_IN_GA(X3, X5)
REVA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, revA_in_ga(X2, X4))
REVA_IN_GA(.(X1, X2), X3) → REVA_IN_GA(X2, X4)
REVA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, revcA_in_ga(X2, X4))
U2_GA(X1, X2, X3, revcA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, appB_in_gga(X4, X1, X3))
U2_GA(X1, X2, X3, revcA_out_ga(X2, X4)) → APPB_IN_GGA(X4, X1, X3)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U4_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
REVD_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, revcA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, appB_in_gga(X5, X2, X6))
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → APPB_IN_GGA(X5, X2, X6)
U7_GA(X1, X2, X3, X4, revcA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, appcB_in_gga(X5, X2, X6))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → U10_GA(X1, X2, X3, X4, appC_in_gga(X6, X1, X4))
U9_GA(X1, X2, X3, X4, appcB_out_gga(X5, X2, X6)) → APPC_IN_GGA(X6, X1, X4)
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U5_GGA(X1, X2, X3, X4, appC_in_gga(X2, X3, X4))
APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

revcA_in_ga([], []) → revcA_out_ga([], [])
revcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, revcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, revcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → revcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revA_in_ga(x1, x2)  =  revA_in_ga(x1)
revcA_in_ga(x1, x2)  =  revcA_in_ga(x1)
[]  =  []
revcA_out_ga(x1, x2)  =  revcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appC_in_gga(x1, x2, x3)  =  appC_in_gga(x1, x2)
REVD_IN_GA(x1, x2)  =  REVD_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REVA_IN_GA(x1, x2)  =  REVA_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)
APPB_IN_GGA(x1, x2, x3)  =  APPB_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)
APPC_IN_GGA(x1, x2, x3)  =  APPC_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:

APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

revcA_in_ga([], []) → revcA_out_ga([], [])
revcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, revcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, revcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → revcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revcA_in_ga(x1, x2)  =  revcA_in_ga(x1)
[]  =  []
revcA_out_ga(x1, x2)  =  revcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
APPC_IN_GGA(x1, x2, x3)  =  APPC_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:

APPC_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPC_IN_GGA(X2, X3, X4)

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

APPC_IN_GGA(.(X1, X2), X3) → APPC_IN_GGA(X2, X3)

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:

• APPC_IN_GGA(.(X1, X2), X3) → APPC_IN_GGA(X2, X3)
  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:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

revcA_in_ga([], []) → revcA_out_ga([], [])
revcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, revcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, revcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → revcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revcA_in_ga(x1, x2)  =  revcA_in_ga(x1)
[]  =  []
revcA_out_ga(x1, x2)  =  revcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
APPB_IN_GGA(x1, x2, x3)  =  APPB_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:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

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

APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)

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:

• APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)
  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:

REVA_IN_GA(.(X1, X2), X3) → REVA_IN_GA(X2, X4)

The TRS R consists of the following rules:

revcA_in_ga([], []) → revcA_out_ga([], [])
revcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, revcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, revcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, appcB_in_gga(X4, X1, X3))
appcB_in_gga([], X1, .(X1, [])) → appcB_out_gga([], X1, .(X1, []))
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U14_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U14_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
U13_ga(X1, X2, X3, appcB_out_gga(X4, X1, X3)) → revcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revcA_in_ga(x1, x2)  =  revcA_in_ga(x1)
[]  =  []
revcA_out_ga(x1, x2)  =  revcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
REVA_IN_GA(x1, x2)  =  REVA_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:

REVA_IN_GA(.(X1, X2), X3) → REVA_IN_GA(X2, X4)

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

REVA_IN_GA(.(X1, X2)) → REVA_IN_GA(X2)

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:

• REVA_IN_GA(.(X1, X2)) → REVA_IN_GA(X2)
  The graph contains the following edges 1 > 1

(27) YES