Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

tappend(nil, T, T).
tappend(node(nil, X, T2), T1, node(T1, X, T2)).
tappend(node(T1, X, nil), T2, node(T1, X, T2)).
tappend(node(T1, X, T2), T3, node(U, X, T2)) :- tappend(T1, T3, U).
tappend(node(T1, X, T2), T3, node(T1, X, U)) :- tappend(T2, T3, U).
s2t(s(X), node(T, Y, T)) :- s2t(X, T).
s2t(s(X), node(nil, Y, T)) :- s2t(X, T).
s2t(s(X), node(T, Y, nil)) :- s2t(X, T).
s2t(s(X), node(nil, Y, nil)).
s2t(0, nil).
goal(X) :- ','(s2t(X, T1), tappend(T1, T2, T3)).

Queries:

goal(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T4), SCOPE: 1, CLAUSE: 10\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2t(T6, X3), tappend(X3, X4, X5))\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 5 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 6 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 7 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 8 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 6 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 7 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 8 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(s2t(T11, X29), tappend(node(X29, X30, X29), X4, X5))\n\nT11 is ground\nX4 is free; \nX5 is free; \nX29 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: s2t(T11, X29)\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: tappend(node(T13, X30, T13), X4, X5)\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: s2t(T11, X29), SCOPE: 3, CLAUSE: 5 | s2t(T11, X29), SCOPE: 3, CLAUSE: 6 | s2t(T11, X29), SCOPE: 3, CLAUSE: 7 | s2t(T11, X29), SCOPE: 3, CLAUSE: 8 | s2t(T11, X29), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: s2t(T11, X29), SCOPE: 3, CLAUSE: 5\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2t(T11, X29), SCOPE: 3, CLAUSE: 6 | s2t(T11, X29), SCOPE: 3, CLAUSE: 7 | s2t(T11, X29), SCOPE: 3, CLAUSE: 8 | s2t(T11, X29), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: s2t(T19, X67)\n\nT19 is ground\nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2t(T11, X29), SCOPE: 3, CLAUSE: 6\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: s2t(T11, X29), SCOPE: 3, CLAUSE: 7 | s2t(T11, X29), SCOPE: 3, CLAUSE: 8 | s2t(T11, X29), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: s2t(T25, X101)\n\nT25 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: s2t(T11, X29), SCOPE: 3, CLAUSE: 7\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2t(T11, X29), SCOPE: 3, CLAUSE: 8 | s2t(T11, X29), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: s2t(T31, X133)\n\nT31 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: s2t(T11, X29), SCOPE: 3, CLAUSE: 8\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2t(T11, X29), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 0 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 1 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 2 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 3 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 4\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 1 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 2 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 3 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 4\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 1\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 2 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 3 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 4\n\nX4 is free\nX5 is free; \nX30 is free; ", 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: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 2\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 3 | tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 4\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 3\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: tappend(node(T13, X30, T13), X4, X5), SCOPE: 4, CLAUSE: 4\n\nX4 is free\nX5 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: tappend(T50, X275, X276)\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 0 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 1 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 2 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 3 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 4\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 0\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 1 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 2 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 3 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 4\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 1\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 2 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 3 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 4\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 2\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 3 | tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 4\n\nX275 is free\nX276 is free; ", 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: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 3\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: tappend(T50, X275, X276), SCOPE: 5, CLAUSE: 4\n\nX275 is free\nX276 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: tappend(T86, X371, X372)\n\nX371 is free\nX372 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: tappend(T96, X401, X402)\n\nX401 is free\nX402 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: tappend(T101, X438, X439)\n\nX438 is free\nX439 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 6\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 7 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 8 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(s2t(T106, X468), tappend(node(nil, X467, X468), X4, X5))\n\nT106 is ground\nX4 is free; \nX5 is free; \nX467 is free; \nX468 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: s2t(T106, X468)\n\nT106 is ground\nX468 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: tappend(node(nil, X467, T108), X4, X5)\n\nX4 is free\nX5 is free; \nX467 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 7\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 8 | ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(s2t(T115, X512), tappend(node(X512, X513, nil), X4, X5))\n\nT115 is ground\nX4 is free; \nX5 is free; \nX512 is free; \nX513 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: s2t(T115, X512)\n\nT115 is ground\nX512 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: tappend(node(T117, X513, nil), X4, X5)\n\nX4 is free\nX5 is free; \nX513 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 8\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(s2t(T6, X3), tappend(X3, X4, X5)), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nX3 is free; \nX4 is free; \nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: tappend(node(nil, X553, nil), X4, X5)\n\nX4 is free\nX5 is free; \nX553 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: tappend(nil, X4, X5)\n\nX4 is free\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="ONLY EVAL with clause\ngoal(X2) :- ','(s2t(X2, X3), tappend(X3, X4, X5)).\nand substitution\nT4 -> T6\nX2 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 89 [arrowhead = none ];
89 -> 3;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 90 [arrowhead = none ];
90 -> 4;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 91 [arrowhead = none ];
91 -> 5;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 92 [arrowhead = none ];
92 -> 6;

93 [label="EVAL with clause\ns2t(s(X26), node(X27, X28, X27)) :- s2t(X26, X27).\nand substitution\nX26 -> T11\nT6 -> s(T11)\nX27 -> X29\nX28 -> X30\nX3 -> node(X29, X30, X29)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 93 [arrowhead = none ];
93 -> 7;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 94 [arrowhead = none ];
94 -> 8;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 95 [arrowhead = none ];
95 -> 70;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 96 [arrowhead = none ];
96 -> 71;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 97 [arrowhead = none ];
97 -> 9;

98 [label="SPLIT 2\nreplacements:\nX29 -> T13", fontsize=14, style = filled, fillcolor = violet];
7 -> 98 [arrowhead = none ];
98 -> 10;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 100 [arrowhead = none ];
100 -> 32;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 101 [arrowhead = none ];
101 -> 12;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 102 [arrowhead = none ];
102 -> 13;

103 [label="EVAL with clause\ns2t(s(X64), node(X65, X66, X65)) :- s2t(X64, X65).\nand substitution\nX64 -> T19\nT11 -> s(T19)\nX65 -> X67\nX66 -> X68\nX29 -> node(X67, X68, X67)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 103 [arrowhead = none ];
103 -> 14;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 104 [arrowhead = none ];
104 -> 15;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 105 [arrowhead = none ];
105 -> 16;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 106 [arrowhead = none ];
106 -> 17;

107 [label="INSTANCE with matching:\nT11 -> T19\nX29 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 107 [arrowhead = none , style=dashed];
107 -> 9[style=dashed];

108 [label="EVAL with clause\ns2t(s(X97), node(nil, X98, X99)) :- s2t(X97, X99).\nand substitution\nX97 -> T25\nT11 -> s(T25)\nX98 -> X100\nX99 -> X101\nX29 -> node(nil, X100, X101)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 108 [arrowhead = none ];
108 -> 18;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 109 [arrowhead = none ];
109 -> 19;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 110 [arrowhead = none ];
110 -> 20;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 111 [arrowhead = none ];
111 -> 21;

112 [label="INSTANCE with matching:\nT11 -> T25\nX29 -> X101", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 112 [arrowhead = none , style=dashed];
112 -> 9[style=dashed];

113 [label="EVAL with clause\ns2t(s(X130), node(X131, X132, nil)) :- s2t(X130, X131).\nand substitution\nX130 -> T31\nT11 -> s(T31)\nX131 -> X133\nX132 -> X134\nX29 -> node(X133, X134, nil)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 113 [arrowhead = none ];
113 -> 22;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 114 [arrowhead = none ];
114 -> 23;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 115 [arrowhead = none ];
115 -> 24;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 116 [arrowhead = none ];
116 -> 25;

117 [label="INSTANCE with matching:\nT11 -> T31\nX29 -> X133", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 117 [arrowhead = none , style=dashed];
117 -> 9[style=dashed];

118 [label="EVAL with clause\ns2t(s(X155), node(nil, X156, nil)).\nand substitution\nX155 -> T37\nT11 -> s(T37)\nX156 -> X157\nX29 -> node(nil, X157, nil)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 118 [arrowhead = none ];
118 -> 26;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 119 [arrowhead = none ];
119 -> 27;

120 [label="EVAL with clause\ns2t(0, nil).\nand substitution\nT11 -> 0\nX29 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 120 [arrowhead = none ];
120 -> 29;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 121 [arrowhead = none ];
121 -> 30;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 122 [arrowhead = none ];
122 -> 28;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 123 [arrowhead = none ];
123 -> 31;

124 [label="BACKTRACK\nfor clause: tappend(nil, T, T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 124 [arrowhead = none ];
124 -> 33;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 125 [arrowhead = none ];
125 -> 34;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 126 [arrowhead = none ];
126 -> 35;

127 [label="EVAL with clause\ntappend(node(nil, X193, X194), X195, node(X195, X193, X194)).\nand substitution\nT13 -> nil\nX30 -> X196\nX193 -> X196\nX194 -> nil\nX4 -> X197\nX195 -> X197\nX5 -> node(X197, X196, nil)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 127 [arrowhead = none ];
127 -> 36;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 128 [arrowhead = none ];
128 -> 37;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 129 [arrowhead = none ];
129 -> 39;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 131 [arrowhead = none ];
131 -> 38;

132 [label="EVAL with clause\ntappend(node(X221, X222, nil), X223, node(X221, X222, X223)).\nand substitution\nT13 -> nil\nX221 -> nil\nX30 -> X224\nX222 -> X224\nT42 -> nil\nX4 -> X225\nX223 -> X225\nX5 -> node(nil, X224, X225)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 132 [arrowhead = none ];
132 -> 41;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 133 [arrowhead = none ];
133 -> 42;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 134 [arrowhead = none ];
134 -> 44;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 135 [arrowhead = none ];
135 -> 45;

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 136 [arrowhead = none ];
136 -> 43;

137 [label="ONLY EVAL with clause\ntappend(node(X269, X270, X271), X272, node(X273, X270, X271)) :- tappend(X269, X272, X273).\nand substitution\nT13 -> T50\nX269 -> T50\nX30 -> X274\nX270 -> X274\nX271 -> T50\nX4 -> X275\nX272 -> X275\nX273 -> X276\nX5 -> node(X276, X274, T50)\nT49 -> T50", fontsize=14, style = filled, fillcolor = white];
44 -> 137 [arrowhead = none ];
137 -> 46;

138 [label="ONLY EVAL with clause\ntappend(node(X432, X433, X434), X435, node(X432, X433, X436)) :- tappend(X434, X435, X436).\nand substitution\nT13 -> T101\nX432 -> T101\nX30 -> X437\nX433 -> X437\nX434 -> T101\nX4 -> X438\nX435 -> X438\nX436 -> X439\nX5 -> node(T101, X437, X439)\nT100 -> T101", fontsize=14, style = filled, fillcolor = white];
45 -> 138 [arrowhead = none ];
138 -> 69;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 139 [arrowhead = none ];
139 -> 47;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 140 [arrowhead = none ];
140 -> 48;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 141 [arrowhead = none ];
141 -> 49;

142 [label="EVAL with clause\ntappend(nil, X289, X289).\nand substitution\nT50 -> nil\nX275 -> X290\nX289 -> X290\nX276 -> X290", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 142 [arrowhead = none ];
142 -> 50;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 143 [arrowhead = none ];
143 -> 51;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 144 [arrowhead = none ];
144 -> 53;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 145 [arrowhead = none ];
145 -> 54;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 146 [arrowhead = none ];
146 -> 52;

147 [label="EVAL with clause\ntappend(node(nil, X308, X309), X310, node(X310, X308, X309)).\nand substitution\nX308 -> T59\nX309 -> T60\nT50 -> node(nil, T59, T60)\nX275 -> X311\nX310 -> X311\nX276 -> node(X311, T59, T60)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 147 [arrowhead = none ];
147 -> 55;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 148 [arrowhead = none ];
148 -> 56;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 149 [arrowhead = none ];
149 -> 58;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 150 [arrowhead = none ];
150 -> 59;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 151 [arrowhead = none ];
151 -> 57;

152 [label="EVAL with clause\ntappend(node(X331, X332, nil), X333, node(X331, X332, X333)).\nand substitution\nX331 -> T69\nX332 -> T70\nT50 -> node(T69, T70, nil)\nX275 -> X334\nX333 -> X334\nX276 -> node(T69, T70, X334)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 152 [arrowhead = none ];
152 -> 60;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 153 [arrowhead = none ];
153 -> 61;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 154 [arrowhead = none ];
154 -> 63;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 155 [arrowhead = none ];
155 -> 64;

156 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 156 [arrowhead = none ];
156 -> 62;

157 [label="EVAL with clause\ntappend(node(X366, X367, X368), X369, node(X370, X367, X368)) :- tappend(X366, X369, X370).\nand substitution\nX366 -> T86\nX367 -> T84\nX368 -> T85\nT50 -> node(T86, T84, T85)\nX275 -> X371\nX369 -> X371\nX370 -> X372\nX276 -> node(X372, T84, T85)\nT83 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 157 [arrowhead = none ];
157 -> 65;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 158 [arrowhead = none ];
158 -> 66;

159 [label="EVAL with clause\ntappend(node(X396, X397, X398), X399, node(X396, X397, X400)) :- tappend(X398, X399, X400).\nand substitution\nX396 -> T93\nX397 -> T94\nX398 -> T96\nT50 -> node(T93, T94, T96)\nX275 -> X401\nX399 -> X401\nX400 -> X402\nX276 -> node(T93, T94, X402)\nT95 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 159 [arrowhead = none ];
159 -> 67;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 160 [arrowhead = none ];
160 -> 68;

161 [label="INSTANCE with matching:\nT50 -> T86\nX275 -> X371\nX276 -> X372", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 161 [arrowhead = none , style=dashed];
161 -> 46[style=dashed];

162 [label="INSTANCE with matching:\nT50 -> T96\nX275 -> X401\nX276 -> X402", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 162 [arrowhead = none , style=dashed];
162 -> 46[style=dashed];

163 [label="INSTANCE with matching:\nT50 -> T101\nX275 -> X438\nX276 -> X439", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 163 [arrowhead = none , style=dashed];
163 -> 46[style=dashed];

164 [label="EVAL with clause\ns2t(s(X464), node(nil, X465, X466)) :- s2t(X464, X466).\nand substitution\nX464 -> T106\nT6 -> s(T106)\nX465 -> X467\nX466 -> X468\nX3 -> node(nil, X467, X468)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 164 [arrowhead = none ];
164 -> 72;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 165 [arrowhead = none ];
165 -> 73;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 166 [arrowhead = none ];
166 -> 76;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 167 [arrowhead = none ];
167 -> 77;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 168 [arrowhead = none ];
168 -> 74;

169 [label="SPLIT 2\nreplacements:\nX468 -> T108", fontsize=14, style = filled, fillcolor = violet];
72 -> 169 [arrowhead = none ];
169 -> 75;

170 [label="INSTANCE with matching:\nT11 -> T106\nX29 -> X468", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 170 [arrowhead = none , style=dashed];
170 -> 9[style=dashed];

171 [label="INSTANCE with matching:\nT50 -> node(nil, X467, T108)\nX275 -> X4\nX276 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 171 [arrowhead = none , style=dashed];
171 -> 46[style=dashed];

172 [label="EVAL with clause\ns2t(s(X509), node(X510, X511, nil)) :- s2t(X509, X510).\nand substitution\nX509 -> T115\nT6 -> s(T115)\nX510 -> X512\nX511 -> X513\nX3 -> node(X512, X513, nil)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 172 [arrowhead = none ];
172 -> 78;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 173 [arrowhead = none ];
173 -> 79;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 174 [arrowhead = none ];
174 -> 82;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 175 [arrowhead = none ];
175 -> 83;

176 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
78 -> 176 [arrowhead = none ];
176 -> 80;

177 [label="SPLIT 2\nreplacements:\nX512 -> T117", fontsize=14, style = filled, fillcolor = violet];
78 -> 177 [arrowhead = none ];
177 -> 81;

178 [label="INSTANCE with matching:\nT11 -> T115\nX29 -> X512", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 178 [arrowhead = none , style=dashed];
178 -> 9[style=dashed];

179 [label="INSTANCE with matching:\nT50 -> node(T117, X513, nil)\nX275 -> X4\nX276 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 179 [arrowhead = none , style=dashed];
179 -> 46[style=dashed];

180 [label="EVAL with clause\ns2t(s(X551), node(nil, X552, nil)).\nand substitution\nX551 -> T123\nT6 -> s(T123)\nX552 -> X553\nX3 -> node(nil, X553, nil)", 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\ns2t(0, nil).\nand substitution\nT6 -> 0\nX3 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 182 [arrowhead = none ];
182 -> 86;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 183 [arrowhead = none ];
183 -> 87;

184 [label="INSTANCE with matching:\nT13 -> nil\nX30 -> X553", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 184 [arrowhead = none , style=dashed];
184 -> 10[style=dashed];

185 [label="INSTANCE with matching:\nT50 -> nil\nX275 -> X4\nX276 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 185 [arrowhead = none , style=dashed];
185 -> 46[style=dashed];

}

(2) Obligation:

Triples:

s2t9(s(T19), node(X67, X68, X67)) :- s2t9(T19, X67).
s2t9(s(T25), node(nil, X100, X101)) :- s2t9(T25, X101).
s2t9(s(T31), node(X133, X134, nil)) :- s2t9(T31, X133).
tappend46(node(T86, T84, T85), X371, node(X372, T84, T85)) :- tappend46(T86, X371, X372).
tappend46(node(T93, T94, T96), X401, node(T93, T94, X402)) :- tappend46(T96, X401, X402).
tappend10(T50, X274, X275, node(X276, X274, T50)) :- tappend46(T50, X275, X276).
tappend10(T101, X437, X438, node(T101, X437, X439)) :- tappend46(T101, X438, X439).
goal1(s(T11)) :- s2t9(T11, X29).
goal1(s(T11)) :- ','(s2tc9(T11, T13), tappend10(T13, X30, X4, X5)).
goal1(s(T106)) :- s2t9(T106, X468).
goal1(s(T106)) :- ','(s2tc9(T106, T108), tappend46(node(nil, X467, T108), X4, X5)).
goal1(s(T115)) :- s2t9(T115, X512).
goal1(s(T115)) :- ','(s2tc9(T115, T117), tappend46(node(T117, X513, nil), X4, X5)).
goal1(s(T123)) :- tappend10(nil, X553, X4, X5).
goal1(0) :- tappend46(nil, X4, X5).

Clauses:

s2tc9(s(T19), node(X67, X68, X67)) :- s2tc9(T19, X67).
s2tc9(s(T25), node(nil, X100, X101)) :- s2tc9(T25, X101).
s2tc9(s(T31), node(X133, X134, nil)) :- s2tc9(T31, X133).
s2tc9(s(T37), node(nil, X157, nil)).
s2tc9(0, nil).
tappendc46(nil, X290, X290).
tappendc46(node(nil, T59, T60), X311, node(X311, T59, T60)).
tappendc46(node(T69, T70, nil), X334, node(T69, T70, X334)).
tappendc46(node(T86, T84, T85), X371, node(X372, T84, T85)) :- tappendc46(T86, X371, X372).
tappendc46(node(T93, T94, T96), X401, node(T93, T94, X402)) :- tappendc46(T96, X401, X402).
tappendc10(nil, X196, X197, node(X197, X196, nil)).
tappendc10(nil, X224, X225, node(nil, X224, X225)).
tappendc10(T50, X274, X275, node(X276, X274, T50)) :- tappendc46(T50, X275, X276).
tappendc10(T101, X437, X438, node(T101, X437, X439)) :- tappendc46(T101, X438, X439).

Afs:

goal1(x1)  =  goal1(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:
goal1_in: (b)
s2t9_in: (b,f)
s2tc9_in: (b,f)
tappend10_in: (b,f,f,f)
tappend46_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T11)) → U8_G(T11, s2t9_in_ga(T11, X29))
GOAL1_IN_G(s(T11)) → S2T9_IN_GA(T11, X29)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → U1_GA(T19, X67, X68, s2t9_in_ga(T19, X67))
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → U2_GA(T25, X100, X101, s2t9_in_ga(T25, X101))
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → U3_GA(T31, X133, X134, s2t9_in_ga(T31, X133))
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)
GOAL1_IN_G(s(T11)) → U9_G(T11, s2tc9_in_ga(T11, T13))
U9_G(T11, s2tc9_out_ga(T11, T13)) → U10_G(T11, tappend10_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tc9_out_ga(T11, T13)) → TAPPEND10_IN_GAAA(T13, X30, X4, X5)
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → U6_GAAA(T50, X274, X275, X276, tappend46_in_gaa(T50, X275, X276))
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → TAPPEND46_IN_GAA(T50, X275, X276)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → U4_GAA(T86, T84, T85, X371, X372, tappend46_in_gaa(T86, X371, X372))
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → U5_GAA(T93, T94, T96, X401, X402, tappend46_in_gaa(T96, X401, X402))
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → U7_GAAA(T101, X437, X438, X439, tappend46_in_gaa(T101, X438, X439))
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → TAPPEND46_IN_GAA(T101, X438, X439)
GOAL1_IN_G(s(T106)) → U11_G(T106, s2t9_in_ga(T106, X468))
GOAL1_IN_G(s(T106)) → U12_G(T106, s2tc9_in_ga(T106, T108))
U12_G(T106, s2tc9_out_ga(T106, T108)) → U13_G(T106, tappend46_in_gaa(node(nil, X467, T108), X4, X5))
U12_G(T106, s2tc9_out_ga(T106, T108)) → TAPPEND46_IN_GAA(node(nil, X467, T108), X4, X5)
GOAL1_IN_G(s(T115)) → U14_G(T115, s2t9_in_ga(T115, X512))
GOAL1_IN_G(s(T115)) → U15_G(T115, s2tc9_in_ga(T115, T117))
U15_G(T115, s2tc9_out_ga(T115, T117)) → U16_G(T115, tappend46_in_gaa(node(T117, X513, nil), X4, X5))
U15_G(T115, s2tc9_out_ga(T115, T117)) → TAPPEND46_IN_GAA(node(T117, X513, nil), X4, X5)
GOAL1_IN_G(s(T123)) → U17_G(T123, tappend10_in_gaaa(nil, X553, X4, X5))
GOAL1_IN_G(s(T123)) → TAPPEND10_IN_GAAA(nil, X553, X4, X5)
GOAL1_IN_G(0) → U18_G(tappend46_in_gaa(nil, X4, X5))
GOAL1_IN_G(0) → TAPPEND46_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2t9_in_ga(x1, x2)  =  s2t9_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
tappend10_in_gaaa(x1, x2, x3, x4)  =  tappend10_in_gaaa(x1)
tappend46_in_gaa(x1, x2, x3)  =  tappend46_in_gaa(x1)
nil  =  nil
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x1, x2)
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
TAPPEND10_IN_GAAA(x1, x2, x3, x4)  =  TAPPEND10_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x1, x5)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x1, x3, x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x3, x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x1, x5)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1)  =  U18_G(x1)

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:

GOAL1_IN_G(s(T11)) → U8_G(T11, s2t9_in_ga(T11, X29))
GOAL1_IN_G(s(T11)) → S2T9_IN_GA(T11, X29)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → U1_GA(T19, X67, X68, s2t9_in_ga(T19, X67))
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → U2_GA(T25, X100, X101, s2t9_in_ga(T25, X101))
S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → U3_GA(T31, X133, X134, s2t9_in_ga(T31, X133))
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)
GOAL1_IN_G(s(T11)) → U9_G(T11, s2tc9_in_ga(T11, T13))
U9_G(T11, s2tc9_out_ga(T11, T13)) → U10_G(T11, tappend10_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tc9_out_ga(T11, T13)) → TAPPEND10_IN_GAAA(T13, X30, X4, X5)
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → U6_GAAA(T50, X274, X275, X276, tappend46_in_gaa(T50, X275, X276))
TAPPEND10_IN_GAAA(T50, X274, X275, node(X276, X274, T50)) → TAPPEND46_IN_GAA(T50, X275, X276)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → U4_GAA(T86, T84, T85, X371, X372, tappend46_in_gaa(T86, X371, X372))
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → U5_GAA(T93, T94, T96, X401, X402, tappend46_in_gaa(T96, X401, X402))
TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → U7_GAAA(T101, X437, X438, X439, tappend46_in_gaa(T101, X438, X439))
TAPPEND10_IN_GAAA(T101, X437, X438, node(T101, X437, X439)) → TAPPEND46_IN_GAA(T101, X438, X439)
GOAL1_IN_G(s(T106)) → U11_G(T106, s2t9_in_ga(T106, X468))
GOAL1_IN_G(s(T106)) → U12_G(T106, s2tc9_in_ga(T106, T108))
U12_G(T106, s2tc9_out_ga(T106, T108)) → U13_G(T106, tappend46_in_gaa(node(nil, X467, T108), X4, X5))
U12_G(T106, s2tc9_out_ga(T106, T108)) → TAPPEND46_IN_GAA(node(nil, X467, T108), X4, X5)
GOAL1_IN_G(s(T115)) → U14_G(T115, s2t9_in_ga(T115, X512))
GOAL1_IN_G(s(T115)) → U15_G(T115, s2tc9_in_ga(T115, T117))
U15_G(T115, s2tc9_out_ga(T115, T117)) → U16_G(T115, tappend46_in_gaa(node(T117, X513, nil), X4, X5))
U15_G(T115, s2tc9_out_ga(T115, T117)) → TAPPEND46_IN_GAA(node(T117, X513, nil), X4, X5)
GOAL1_IN_G(s(T123)) → U17_G(T123, tappend10_in_gaaa(nil, X553, X4, X5))
GOAL1_IN_G(s(T123)) → TAPPEND10_IN_GAAA(nil, X553, X4, X5)
GOAL1_IN_G(0) → U18_G(tappend46_in_gaa(nil, X4, X5))
GOAL1_IN_G(0) → TAPPEND46_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2t9_in_ga(x1, x2)  =  s2t9_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
tappend10_in_gaaa(x1, x2, x3, x4)  =  tappend10_in_gaaa(x1)
tappend46_in_gaa(x1, x2, x3)  =  tappend46_in_gaa(x1)
nil  =  nil
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x1, x2)
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U9_G(x1, x2)  =  U9_G(x1, x2)
U10_G(x1, x2)  =  U10_G(x1, x2)
TAPPEND10_IN_GAAA(x1, x2, x3, x4)  =  TAPPEND10_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x1, x5)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x1, x3, x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x3, x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x1, x5)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1)  =  U18_G(x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 26 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
nil  =  nil
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)

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:

TAPPEND46_IN_GAA(node(T93, T94, T96), X401, node(T93, T94, X402)) → TAPPEND46_IN_GAA(T96, X401, X402)
TAPPEND46_IN_GAA(node(T86, T84, T85), X371, node(X372, T84, T85)) → TAPPEND46_IN_GAA(T86, X371, X372)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x3)
TAPPEND46_IN_GAA(x1, x2, x3)  =  TAPPEND46_IN_GAA(x1)

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:

TAPPEND46_IN_GAA(node(T93, T96)) → TAPPEND46_IN_GAA(T96)
TAPPEND46_IN_GAA(node(T86, T85)) → TAPPEND46_IN_GAA(T86)

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:

• TAPPEND46_IN_GAA(node(T93, T96)) → TAPPEND46_IN_GAA(T96)
  The graph contains the following edges 1 > 1

• TAPPEND46_IN_GAA(node(T86, T85)) → TAPPEND46_IN_GAA(T86)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)

The TRS R consists of the following rules:

s2tc9_in_ga(s(T19), node(X67, X68, X67)) → U20_ga(T19, X67, X68, s2tc9_in_ga(T19, X67))
s2tc9_in_ga(s(T25), node(nil, X100, X101)) → U21_ga(T25, X100, X101, s2tc9_in_ga(T25, X101))
s2tc9_in_ga(s(T31), node(X133, X134, nil)) → U22_ga(T31, X133, X134, s2tc9_in_ga(T31, X133))
s2tc9_in_ga(s(T37), node(nil, X157, nil)) → s2tc9_out_ga(s(T37), node(nil, X157, nil))
s2tc9_in_ga(0, nil) → s2tc9_out_ga(0, nil)
U22_ga(T31, X133, X134, s2tc9_out_ga(T31, X133)) → s2tc9_out_ga(s(T31), node(X133, X134, nil))
U21_ga(T25, X100, X101, s2tc9_out_ga(T25, X101)) → s2tc9_out_ga(s(T25), node(nil, X100, X101))
U20_ga(T19, X67, X68, s2tc9_out_ga(T19, X67)) → s2tc9_out_ga(s(T19), node(X67, X68, X67))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tc9_in_ga(x1, x2)  =  s2tc9_in_ga(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
s2tc9_out_ga(x1, x2)  =  s2tc9_out_ga(x1, x2)
0  =  0
nil  =  nil
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)

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:

S2T9_IN_GA(s(T25), node(nil, X100, X101)) → S2T9_IN_GA(T25, X101)
S2T9_IN_GA(s(T19), node(X67, X68, X67)) → S2T9_IN_GA(T19, X67)
S2T9_IN_GA(s(T31), node(X133, X134, nil)) → S2T9_IN_GA(T31, X133)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T9_IN_GA(x1, x2)  =  S2T9_IN_GA(x1)

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:

S2T9_IN_GA(s(T25)) → S2T9_IN_GA(T25)

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:

• S2T9_IN_GA(s(T25)) → S2T9_IN_GA(T25)
  The graph contains the following edges 1 > 1

(20) YES