Left Termination of the query pattern g_in_1(a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

g(W) :- ','(eq(X, .(.(a, []), .(.(R, []), []))), ','(eq(Y, .(.(S, .(c, [])), .([], []))), ','(app_1(X, Y, Z), ','(eq(Z, .(U, V)), app_2(U, U, W))))).
app_1([], X, X).
app_1(.(X, Xs), Ys, .(X, Zs)) :- app_1(Xs, Ys, Zs).
app_2([], X, X).
app_2(.(X, Xs), Ys, .(X, Zs)) :- app_2(Xs, Ys, Zs).
eq(X, X).

Queries:

g(a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: g(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: g(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(eq(X3, .(.(a, []), .(.(X4, []), []))), ','(eq(X5, .(.(X6, .(c, [])), .([], []))), ','(app_1(X3, X5, X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, T4)))))\n\nX3 is free\nX4 is free; \nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(eq(X3, .(.(a, []), .(.(X4, []), []))), ','(eq(X5, .(.(X6, .(c, [])), .([], []))), ','(app_1(X3, X5, X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, T4))))), SCOPE: 2, CLAUSE: 5\n\nX3 is free\nX4 is free; \nX5 is free; \nX6 is free; \nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(eq(X5, .(.(X6, .(c, [])), .([], []))), ','(app_1(.(.(a, []), .(.(X27, []), [])), X5, X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, T4))))\n\nX5 is free\nX6 is free; \nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(eq(X5, .(.(X6, .(c, [])), .([], []))), ','(app_1(.(.(a, []), .(.(X27, []), [])), X5, X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, T4)))), SCOPE: 3, CLAUSE: 5\n\nX5 is free\nX6 is free; \nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app_1(.(.(a, []), .(.(X27, []), [])), .(.(X54, .(c, [])), .([], [])), X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, T4)))\n\nX7 is free\nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: app_1(.(.(a, []), .(.(X27, []), [])), .(.(X54, .(c, [])), .([], [])), X7)\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(eq(T5, .(X8, X9)), app_2(X8, X8, T6))\n\nX8 is free\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app_1(.(.(a, []), .(.(X27, []), [])), .(.(X54, .(c, [])), .([], [])), X7), SCOPE: 4, CLAUSE: 1 | app_1(.(.(a, []), .(.(X27, []), [])), .(.(X54, .(c, [])), .([], [])), X7), SCOPE: 4, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app_1(.(.(a, []), .(.(X27, []), [])), .(.(X54, .(c, [])), .([], [])), X7), SCOPE: 4, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app_1(.(.(X101, []), []), .(.(X102, .(c, [])), .([], [])), X103)\n\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app_1(.(.(X101, []), []), .(.(X102, .(c, [])), .([], [])), X103), SCOPE: 5, CLAUSE: 1 | app_1(.(.(X101, []), []), .(.(X102, .(c, [])), .([], [])), X103), SCOPE: 5, CLAUSE: 2\n\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app_1(.(.(X101, []), []), .(.(X102, .(c, [])), .([], [])), X103), SCOPE: 5, CLAUSE: 2\n\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app_1([], .(.(X135, .(c, [])), .([], [])), X136)\n\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app_1([], .(.(X135, .(c, [])), .([], [])), X136), SCOPE: 6, CLAUSE: 1 | app_1([], .(.(X135, .(c, [])), .([], [])), X136), SCOPE: 6, CLAUSE: 2\n\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app_1([], .(.(X135, .(c, [])), .([], [])), X136), SCOPE: 6, CLAUSE: 1\n\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app_1([], .(.(X135, .(c, [])), .([], [])), X136), SCOPE: 6, CLAUSE: 2\n\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(eq(T5, .(X8, X9)), app_2(X8, X8, T6)), SCOPE: 7, CLAUSE: 5\n\nX8 is free\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app_2(T16, T16, T17)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app_2(T16, T16, T17), SCOPE: 8, CLAUSE: 3 | app_2(T16, T16, T17), SCOPE: 8, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app_2(T16, T16, T17), SCOPE: 8, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app_2(T16, T16, T17), SCOPE: 8, CLAUSE: 4\n\n", 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: app_2(T27, .(T29, T27), T28)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app_2(T27, .(T29, T27), T28), SCOPE: 9, CLAUSE: 3 | app_2(T27, .(T29, T27), T28), SCOPE: 9, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app_2(T27, .(T29, T27), T28), SCOPE: 9, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app_2(T27, .(T29, T27), T28), SCOPE: 9, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app_2(T49, .(T51, .(T52, T49)), T50)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app_2(T49, .(T51, .(T52, T49)), T50), SCOPE: 10, CLAUSE: 3 | app_2(T49, .(T51, .(T52, T49)), T50), SCOPE: 10, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app_2(T49, .(T51, .(T52, T49)), T50), SCOPE: 10, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app_2(T49, .(T51, .(T52, T49)), T50), SCOPE: 10, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app_2(T82, .(T84, .(T85, .(T86, T82))), T83)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: app_2(T82, .(T84, .(T85, .(T86, T82))), T83), SCOPE: 11, CLAUSE: 3 | app_2(T82, .(T84, .(T85, .(T86, T82))), T83), SCOPE: 11, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app_2(T82, .(T84, .(T85, .(T86, T82))), T83), SCOPE: 11, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app_2(T82, .(T84, .(T85, .(T86, T82))), T83), SCOPE: 11, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: app_2(T126, .(T128, .(T129, .(T130, .(T131, T126)))), T127)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: app_2(T126, .(T128, .(T129, .(T130, .(T131, T126)))), T127), SCOPE: 12, CLAUSE: 3 | app_2(T126, .(T128, .(T129, .(T130, .(T131, T126)))), T127), SCOPE: 12, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: app_2(T126, .(T128, .(T129, .(T130, .(T131, T126)))), T127), SCOPE: 12, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: app_2(T126, .(T128, .(T129, .(T130, .(T131, T126)))), T127), SCOPE: 12, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: app_2(T181, .(T183, .(T184, .(T185, .(T186, .(T187, T181))))), T182)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: app_2(T181, .(T183, .(T184, .(T185, .(T186, .(T187, T181))))), T182), SCOPE: 13, CLAUSE: 3 | app_2(T181, .(T183, .(T184, .(T185, .(T186, .(T187, T181))))), T182), SCOPE: 13, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: app_2(T181, .(T183, .(T184, .(T185, .(T186, .(T187, T181))))), T182), SCOPE: 13, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app_2(T181, .(T183, .(T184, .(T185, .(T186, .(T187, T181))))), T182), SCOPE: 13, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: app_2(T247, .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247)))))), T248)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app_2(T247, .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247)))))), T248), SCOPE: 14, CLAUSE: 3 | app_2(T247, .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247)))))), T248), SCOPE: 14, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app_2(T247, .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247)))))), T248), SCOPE: 14, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app_2(T247, .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247)))))), T248), SCOPE: 14, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app_2(T324, .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324))))))), T325)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: app_2(T324, .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324))))))), T325), SCOPE: 15, CLAUSE: 3 | app_2(T324, .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324))))))), T325), SCOPE: 15, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: app_2(T324, .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324))))))), T325), SCOPE: 15, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: app_2(T324, .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324))))))), T325), SCOPE: 15, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: app_2(T412, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T413)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: app_2(T412, .(T414, T438), T413)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: app_2(T412, .(T414, T438), T413), SCOPE: 16, CLAUSE: 3 | app_2(T412, .(T414, T438), T413), SCOPE: 16, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: app_2(T412, .(T414, T438), T413), SCOPE: 16, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: app_2(T412, .(T414, T438), T413), SCOPE: 16, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: app_2(T468, .(T470, T471), T469)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 98 [arrowhead = none ];
98 -> 2;

99 [label="ONLY EVAL with clause\ng(X2) :- ','(eq(X3, .(.(a, []), .(.(X4, []), []))), ','(eq(X5, .(.(X6, .(c, [])), .([], []))), ','(app_1(X3, X5, X7), ','(eq(X7, .(X8, X9)), app_2(X8, X8, X2))))).\nand substitution\nT1 -> T4\nX2 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = white];
2 -> 99 [arrowhead = none ];
99 -> 3;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 100 [arrowhead = none ];
100 -> 4;

101 [label="ONLY EVAL with clause\neq(X25, X25).\nand substitution\nX3 -> .(.(a, []), .(.(X27, []), []))\nX25 -> .(.(a, []), .(.(X27, []), []))\nX4 -> X27\nX26 -> .(.(a, []), .(.(X27, []), []))", fontsize=14, style = filled, fillcolor = white];
4 -> 101 [arrowhead = none ];
101 -> 5;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 102 [arrowhead = none ];
102 -> 6;

103 [label="ONLY EVAL with clause\neq(X52, X52).\nand substitution\nX5 -> .(.(X54, .(c, [])), .([], []))\nX52 -> .(.(X54, .(c, [])), .([], []))\nX6 -> X54\nX53 -> .(.(X54, .(c, [])), .([], []))", fontsize=14, style = filled, fillcolor = white];
6 -> 103 [arrowhead = none ];
103 -> 7;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 104 [arrowhead = none ];
104 -> 8;

105 [label="SPLIT 2\nreplacements:\nX7 -> T5\nT4 -> T6", fontsize=14, style = filled, fillcolor = violet];
7 -> 105 [arrowhead = none ];
105 -> 9;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 106 [arrowhead = none ];
106 -> 10;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 107 [arrowhead = none ];
107 -> 22;

108 [label="BACKTRACK\nfor clause: app_1([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 108 [arrowhead = none ];
108 -> 11;

109 [label="ONLY EVAL with clause\napp_1(.(X97, X98), X99, .(X97, X100)) :- app_1(X98, X99, X100).\nand substitution\nX97 -> .(a, [])\nX27 -> X101\nX98 -> .(.(X101, []), [])\nX54 -> X102\nX99 -> .(.(X102, .(c, [])), .([], []))\nX100 -> X103\nX7 -> .(.(a, []), X103)", fontsize=14, style = filled, fillcolor = white];
11 -> 109 [arrowhead = none ];
109 -> 12;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 110 [arrowhead = none ];
110 -> 13;

111 [label="BACKTRACK\nfor clause: app_1([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 111 [arrowhead = none ];
111 -> 14;

112 [label="ONLY EVAL with clause\napp_1(.(X130, X131), X132, .(X130, X133)) :- app_1(X131, X132, X133).\nand substitution\nX101 -> X134\nX130 -> .(X134, [])\nX131 -> []\nX102 -> X135\nX132 -> .(.(X135, .(c, [])), .([], []))\nX133 -> X136\nX103 -> .(.(X134, []), X136)", fontsize=14, style = filled, fillcolor = white];
14 -> 112 [arrowhead = none ];
112 -> 15;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 113 [arrowhead = none ];
113 -> 16;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 114 [arrowhead = none ];
114 -> 17;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 115 [arrowhead = none ];
115 -> 18;

116 [label="ONLY EVAL with clause\napp_1([], X147, X147).\nand substitution\nX135 -> X148\nX147 -> .(.(X148, .(c, [])), .([], []))\nX136 -> .(.(X148, .(c, [])), .([], []))", fontsize=14, style = filled, fillcolor = white];
17 -> 116 [arrowhead = none ];
116 -> 19;

117 [label="BACKTRACK\nfor clause: app_1(.(X, Xs), Ys, .(X, Zs)) :- app_1(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
18 -> 117 [arrowhead = none ];
117 -> 21;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 118 [arrowhead = none ];
118 -> 20;

119 [label="EVAL with clause\neq(X157, X157).\nand substitution\nT5 -> .(T16, T15)\nX157 -> .(T16, T15)\nX8 -> T16\nX9 -> T15\nT13 -> .(T16, T15)\nT14 -> T16\nT6 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 119 [arrowhead = none ];
119 -> 23;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 120 [arrowhead = none ];
120 -> 24;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 121 [arrowhead = none ];
121 -> 25;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 122 [arrowhead = none ];
122 -> 26;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 123 [arrowhead = none ];
123 -> 27;

124 [label="EVAL with clause\napp_2([], X165, X165).\nand substitution\nT16 -> []\nX165 -> []\nT17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 124 [arrowhead = none ];
124 -> 28;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 125 [arrowhead = none ];
125 -> 29;

126 [label="EVAL with clause\napp_2(.(X175, X176), X177, .(X175, X178)) :- app_2(X176, X177, X178).\nand substitution\nX175 -> T29\nX176 -> T27\nT16 -> .(T29, T27)\nX177 -> .(T29, T27)\nX178 -> T28\nT17 -> .(T29, T28)\nT25 -> T27\nT26 -> T28\nT24 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 126 [arrowhead = none ];
126 -> 31;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 127 [arrowhead = none ];
127 -> 32;

128 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 128 [arrowhead = none ];
128 -> 30;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 129 [arrowhead = none ];
129 -> 33;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 130 [arrowhead = none ];
130 -> 34;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 131 [arrowhead = none ];
131 -> 35;

132 [label="EVAL with clause\napp_2([], X189, X189).\nand substitution\nT27 -> []\nT29 -> T36\nX189 -> .(T36, [])\nT28 -> .(T36, [])", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 132 [arrowhead = none ];
132 -> 36;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 133 [arrowhead = none ];
133 -> 37;

134 [label="EVAL with clause\napp_2(.(X199, X200), X201, .(X199, X202)) :- app_2(X200, X201, X202).\nand substitution\nX199 -> T52\nX200 -> T49\nT27 -> .(T52, T49)\nT29 -> T51\nX201 -> .(T51, .(T52, T49))\nX202 -> T50\nT28 -> .(T52, T50)\nT46 -> T49\nT48 -> T50\nT47 -> T51\nT45 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 134 [arrowhead = none ];
134 -> 39;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 135 [arrowhead = none ];
135 -> 40;

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 136 [arrowhead = none ];
136 -> 38;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 137 [arrowhead = none ];
137 -> 41;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 138 [arrowhead = none ];
138 -> 42;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 139 [arrowhead = none ];
139 -> 43;

140 [label="EVAL with clause\napp_2([], X213, X213).\nand substitution\nT49 -> []\nT51 -> T65\nT52 -> T66\nX213 -> .(T65, .(T66, []))\nT50 -> .(T65, .(T66, []))", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 140 [arrowhead = none ];
140 -> 44;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 141 [arrowhead = none ];
141 -> 45;

142 [label="EVAL with clause\napp_2(.(X223, X224), X225, .(X223, X226)) :- app_2(X224, X225, X226).\nand substitution\nX223 -> T86\nX224 -> T82\nT49 -> .(T86, T82)\nT51 -> T84\nT52 -> T85\nX225 -> .(T84, .(T85, .(T86, T82)))\nX226 -> T83\nT50 -> .(T86, T83)\nT78 -> T82\nT81 -> T83\nT79 -> T84\nT80 -> T85\nT77 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 142 [arrowhead = none ];
142 -> 47;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 143 [arrowhead = none ];
143 -> 48;

144 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 144 [arrowhead = none ];
144 -> 46;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 145 [arrowhead = none ];
145 -> 49;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 146 [arrowhead = none ];
146 -> 50;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 147 [arrowhead = none ];
147 -> 51;

148 [label="EVAL with clause\napp_2([], X237, X237).\nand substitution\nT82 -> []\nT84 -> T105\nT85 -> T106\nT86 -> T107\nX237 -> .(T105, .(T106, .(T107, [])))\nT83 -> .(T105, .(T106, .(T107, [])))", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 148 [arrowhead = none ];
148 -> 52;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 149 [arrowhead = none ];
149 -> 53;

150 [label="EVAL with clause\napp_2(.(X247, X248), X249, .(X247, X250)) :- app_2(X248, X249, X250).\nand substitution\nX247 -> T131\nX248 -> T126\nT82 -> .(T131, T126)\nT84 -> T128\nT85 -> T129\nT86 -> T130\nX249 -> .(T128, .(T129, .(T130, .(T131, T126))))\nX250 -> T127\nT83 -> .(T131, T127)\nT121 -> T126\nT125 -> T127\nT122 -> T128\nT123 -> T129\nT124 -> T130\nT120 -> T131", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 150 [arrowhead = none ];
150 -> 55;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 151 [arrowhead = none ];
151 -> 56;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 152 [arrowhead = none ];
152 -> 54;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 153 [arrowhead = none ];
153 -> 57;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 154 [arrowhead = none ];
154 -> 58;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 155 [arrowhead = none ];
155 -> 59;

156 [label="EVAL with clause\napp_2([], X261, X261).\nand substitution\nT126 -> []\nT128 -> T156\nT129 -> T157\nT130 -> T158\nT131 -> T159\nX261 -> .(T156, .(T157, .(T158, .(T159, []))))\nT127 -> .(T156, .(T157, .(T158, .(T159, []))))", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 156 [arrowhead = none ];
156 -> 60;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 157 [arrowhead = none ];
157 -> 61;

158 [label="EVAL with clause\napp_2(.(X271, X272), X273, .(X271, X274)) :- app_2(X272, X273, X274).\nand substitution\nX271 -> T187\nX272 -> T181\nT126 -> .(T187, T181)\nT128 -> T183\nT129 -> T184\nT130 -> T185\nT131 -> T186\nX273 -> .(T183, .(T184, .(T185, .(T186, .(T187, T181)))))\nX274 -> T182\nT127 -> .(T187, T182)\nT175 -> T181\nT180 -> T182\nT176 -> T183\nT177 -> T184\nT178 -> T185\nT179 -> T186\nT174 -> T187", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 158 [arrowhead = none ];
158 -> 63;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 159 [arrowhead = none ];
159 -> 64;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 160 [arrowhead = none ];
160 -> 62;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 161 [arrowhead = none ];
161 -> 65;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 162 [arrowhead = none ];
162 -> 66;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 163 [arrowhead = none ];
163 -> 67;

164 [label="EVAL with clause\napp_2([], X285, X285).\nand substitution\nT181 -> []\nT183 -> T218\nT184 -> T219\nT185 -> T220\nT186 -> T221\nT187 -> T222\nX285 -> .(T218, .(T219, .(T220, .(T221, .(T222, [])))))\nT182 -> .(T218, .(T219, .(T220, .(T221, .(T222, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 164 [arrowhead = none ];
164 -> 68;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 165 [arrowhead = none ];
165 -> 69;

166 [label="EVAL with clause\napp_2(.(X295, X296), X297, .(X295, X298)) :- app_2(X296, X297, X298).\nand substitution\nX295 -> T254\nX296 -> T247\nT181 -> .(T254, T247)\nT183 -> T249\nT184 -> T250\nT185 -> T251\nT186 -> T252\nT187 -> T253\nX297 -> .(T249, .(T250, .(T251, .(T252, .(T253, .(T254, T247))))))\nX298 -> T248\nT182 -> .(T254, T248)\nT240 -> T247\nT246 -> T248\nT241 -> T249\nT242 -> T250\nT243 -> T251\nT244 -> T252\nT245 -> T253\nT239 -> T254", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 166 [arrowhead = none ];
166 -> 71;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 167 [arrowhead = none ];
167 -> 72;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 168 [arrowhead = none ];
168 -> 70;

169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 169 [arrowhead = none ];
169 -> 73;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 170 [arrowhead = none ];
170 -> 74;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 171 [arrowhead = none ];
171 -> 75;

172 [label="EVAL with clause\napp_2([], X309, X309).\nand substitution\nT247 -> []\nT249 -> T291\nT250 -> T292\nT251 -> T293\nT252 -> T294\nT253 -> T295\nT254 -> T296\nX309 -> .(T291, .(T292, .(T293, .(T294, .(T295, .(T296, []))))))\nT248 -> .(T291, .(T292, .(T293, .(T294, .(T295, .(T296, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 172 [arrowhead = none ];
172 -> 76;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 173 [arrowhead = none ];
173 -> 77;

174 [label="EVAL with clause\napp_2(.(X319, X320), X321, .(X319, X322)) :- app_2(X320, X321, X322).\nand substitution\nX319 -> T332\nX320 -> T324\nT247 -> .(T332, T324)\nT249 -> T326\nT250 -> T327\nT251 -> T328\nT252 -> T329\nT253 -> T330\nT254 -> T331\nX321 -> .(T326, .(T327, .(T328, .(T329, .(T330, .(T331, .(T332, T324)))))))\nX322 -> T325\nT248 -> .(T332, T325)\nT316 -> T324\nT323 -> T325\nT317 -> T326\nT318 -> T327\nT319 -> T328\nT320 -> T329\nT321 -> T330\nT322 -> T331\nT315 -> T332", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 174 [arrowhead = none ];
174 -> 79;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 175 [arrowhead = none ];
175 -> 80;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 176 [arrowhead = none ];
176 -> 78;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 177 [arrowhead = none ];
177 -> 81;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 178 [arrowhead = none ];
178 -> 82;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 179 [arrowhead = none ];
179 -> 83;

180 [label="EVAL with clause\napp_2([], X333, X333).\nand substitution\nT324 -> []\nT326 -> T375\nT327 -> T376\nT328 -> T377\nT329 -> T378\nT330 -> T379\nT331 -> T380\nT332 -> T381\nX333 -> .(T375, .(T376, .(T377, .(T378, .(T379, .(T380, .(T381, [])))))))\nT325 -> .(T375, .(T376, .(T377, .(T378, .(T379, .(T380, .(T381, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 180 [arrowhead = none ];
180 -> 84;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 181 [arrowhead = none ];
181 -> 85;

182 [label="EVAL with clause\napp_2(.(X343, X344), X345, .(X343, X346)) :- app_2(X344, X345, X346).\nand substitution\nX343 -> T421\nX344 -> T412\nT324 -> .(T421, T412)\nT326 -> T414\nT327 -> T415\nT328 -> T416\nT329 -> T417\nT330 -> T418\nT331 -> T419\nT332 -> T420\nX345 -> .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))))\nX346 -> T413\nT325 -> .(T421, T413)\nT403 -> T412\nT411 -> T413\nT404 -> T414\nT405 -> T415\nT406 -> T416\nT407 -> T417\nT408 -> T418\nT409 -> T419\nT410 -> T420\nT402 -> T421", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 182 [arrowhead = none ];
182 -> 87;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 183 [arrowhead = none ];
183 -> 88;

184 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 184 [arrowhead = none ];
184 -> 86;

185 [label="GENERALIZATION\nT438 <-- .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))", fontsize=14, style = filled, fillcolor = violet];
87 -> 185 [arrowhead = none ];
185 -> 89;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
89 -> 186 [arrowhead = none ];
186 -> 90;

187 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 187 [arrowhead = none ];
187 -> 91;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 188 [arrowhead = none ];
188 -> 92;

189 [label="EVAL with clause\napp_2([], X359, X359).\nand substitution\nT412 -> []\nT414 -> T451\nT438 -> T452\nX359 -> .(T451, T452)\nT413 -> .(T451, T452)", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 189 [arrowhead = none ];
189 -> 93;

190 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 190 [arrowhead = none ];
190 -> 94;

191 [label="EVAL with clause\napp_2(.(X369, X370), X371, .(X369, X372)) :- app_2(X370, X371, X372).\nand substitution\nX369 -> T463\nX370 -> T468\nT412 -> .(T463, T468)\nT414 -> T470\nT438 -> T471\nX371 -> .(T470, T471)\nX372 -> T469\nT413 -> .(T463, T469)\nT464 -> T468\nT467 -> T469\nT465 -> T470\nT466 -> T471", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 191 [arrowhead = none ];
191 -> 96;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 192 [arrowhead = none ];
192 -> 97;

193 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 193 [arrowhead = none ];
193 -> 95;

194 [label="INSTANCE with matching:\nT412 -> T468\nT414 -> T470\nT438 -> T471\nT413 -> T469", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 194 [arrowhead = none , style=dashed];
194 -> 89[style=dashed];

}

(2) Obligation:

Triples:

app_289(.(T463, T468), T470, T471, .(T463, T469)) :- app_289(T468, T470, T471, T469).
g1(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T413))))))))) :- ','(app_1c8(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15)), app_289(T412, T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))), T413)).

Clauses:

app_2c89([], T451, T452, .(T451, T452)).
app_2c89(.(T463, T468), T470, T471, .(T463, T469)) :- app_2c89(T468, T470, T471, T469).
app_1c8(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], []))))).

Afs:

g1(x1)  =  g1

(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:
g1_in: (f)
app_289_in: (f,f,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

G1_IN_A(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T413))))))))) → U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_in_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15)))
U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_out_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15))) → U3_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_289_in_aaaa(T412, T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))), T413))
U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_out_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15))) → APP_289_IN_AAAA(T412, T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))), T413)
APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → U1_AAAA(T463, T468, T470, T471, T469, app_289_in_aaaa(T468, T470, T471, T469))
APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → APP_289_IN_AAAA(T468, T470, T471, T469)

The TRS R consists of the following rules:

app_1c8_in_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], []))))) → app_1c8_out_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], [])))))

The argument filtering Pi contains the following mapping:
app_1c8_in_aaa(x1, x2, x3)  =  app_1c8_in_aaa
app_1c8_out_aaa(x1, x2, x3)  =  app_1c8_out_aaa
app_289_in_aaaa(x1, x2, x3, x4)  =  app_289_in_aaaa
.(x1, x2)  =  .(x1, x2)
G1_IN_A(x1)  =  G1_IN_A
U2_A(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U2_A(x10)
U3_A(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_A(x10)
APP_289_IN_AAAA(x1, x2, x3, x4)  =  APP_289_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

G1_IN_A(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T413))))))))) → U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_in_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15)))
U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_out_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15))) → U3_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_289_in_aaaa(T412, T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))), T413))
U2_A(T414, T415, T416, T417, T418, T419, T420, T421, T413, app_1c8_out_aaa(X27, X54, .(.(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412)))))))), T15))) → APP_289_IN_AAAA(T412, T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, .(T421, T412))))))), T413)
APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → U1_AAAA(T463, T468, T470, T471, T469, app_289_in_aaaa(T468, T470, T471, T469))
APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → APP_289_IN_AAAA(T468, T470, T471, T469)

The TRS R consists of the following rules:

app_1c8_in_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], []))))) → app_1c8_out_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], [])))))

The argument filtering Pi contains the following mapping:
app_1c8_in_aaa(x1, x2, x3)  =  app_1c8_in_aaa
app_1c8_out_aaa(x1, x2, x3)  =  app_1c8_out_aaa
app_289_in_aaaa(x1, x2, x3, x4)  =  app_289_in_aaaa
.(x1, x2)  =  .(x1, x2)
G1_IN_A(x1)  =  G1_IN_A
U2_A(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U2_A(x10)
U3_A(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_A(x10)
APP_289_IN_AAAA(x1, x2, x3, x4)  =  APP_289_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → APP_289_IN_AAAA(T468, T470, T471, T469)

The TRS R consists of the following rules:

app_1c8_in_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], []))))) → app_1c8_out_aaa(X134, X148, .(.(a, []), .(.(X134, []), .(.(X148, .(c, [])), .([], [])))))

The argument filtering Pi contains the following mapping:
app_1c8_in_aaa(x1, x2, x3)  =  app_1c8_in_aaa
app_1c8_out_aaa(x1, x2, x3)  =  app_1c8_out_aaa
.(x1, x2)  =  .(x1, x2)
APP_289_IN_AAAA(x1, x2, x3, x4)  =  APP_289_IN_AAAA

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

APP_289_IN_AAAA(.(T463, T468), T470, T471, .(T463, T469)) → APP_289_IN_AAAA(T468, T470, T471, T469)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP_289_IN_AAAA(x1, x2, x3, x4)  =  APP_289_IN_AAAA

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

APP_289_IN_AAAA → APP_289_IN_AAAA

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

(11) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP_289_IN_AAAA evaluates to t =APP_289_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP_289_IN_AAAA to APP_289_IN_AAAA.

(12) NO