Left Termination of the query pattern
goal(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)).

Query: goal(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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, X64)\n\nT19 is ground\nX64 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, X95)\n\nT25 is ground\nX95 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, X124)\n\nT31 is ground\nX124 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, X245, X246)\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 0 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 1 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 2 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 3 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 4\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 0\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 1 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 2 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 3 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 4\n\nX245 is free\nX246 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, X245, X246), SCOPE: 5, CLAUSE: 1\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 2 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 3 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 4\n\nX245 is free\nX246 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, X245, X246), SCOPE: 5, CLAUSE: 2\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 3 | tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 4\n\nX245 is free\nX246 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, X245, X246), SCOPE: 5, CLAUSE: 3\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: tappend(T50, X245, X246), SCOPE: 5, CLAUSE: 4\n\nX245 is free\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: tappend(T86, X334, X335)\n\nX334 is free\nX335 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: tappend(T96, X359, X360)\n\nX359 is free\nX360 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: tappend(T101, X391, X392)\n\nX391 is free\nX392 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, X421), tappend(node(nil, X420, X421), X4, X5))\n\nT106 is ground\nX4 is free\nX5 is free\nX420 is free\nX421 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: s2t(T106, X421)\n\nT106 is ground\nX421 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: tappend(node(nil, X420, T108), X4, X5)\n\nX4 is free\nX5 is free\nX420 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(T114, X457), tappend(node(X457, X458, nil), X4, X5))\n\nT114 is ground\nX4 is free\nX5 is free\nX457 is free\nX458 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: s2t(T114, X457)\n\nT114 is ground\nX457 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: tappend(node(T116, X458, nil), X4, X5)\n\nX4 is free\nX5 is free\nX458 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, X485, nil), X4, X5)\n\nX4 is free\nX5 is free\nX485 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 substitutionT4 -> T6,\nX2 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionX26 -> 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 = lightgreen];
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\nnew knowledge:\nT11 is ground\nreplacements:X29 -> 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(X61), node(X62, X63, X62)) :- s2t(X61, X62).\nand substitutionX61 -> T19,\nT11 -> s(T19),\nX62 -> X64,\nX63 -> X65,\nX29 -> node(X64, X65, X64)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 103 [arrowhead = none ];
103 -> 14;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 107 [arrowhead = none , style=dashed];
107 -> 9[style=dashed];

108 [label="EVAL with clause\ns2t(s(X91), node(nil, X92, X93)) :- s2t(X91, X93).\nand substitutionX91 -> T25,\nT11 -> s(T25),\nX92 -> X94,\nX93 -> X95,\nX29 -> node(nil, X94, X95)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 108 [arrowhead = none ];
108 -> 18;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 -> X95", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 112 [arrowhead = none , style=dashed];
112 -> 9[style=dashed];

113 [label="EVAL with clause\ns2t(s(X121), node(X122, X123, nil)) :- s2t(X121, X122).\nand substitutionX121 -> T31,\nT11 -> s(T31),\nX122 -> X124,\nX123 -> X125,\nX29 -> node(X124, X125, nil)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 113 [arrowhead = none ];
113 -> 22;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 -> X124", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 117 [arrowhead = none , style=dashed];
117 -> 9[style=dashed];

118 [label="EVAL with clause\ns2t(s(X143), node(nil, X144, nil)).\nand substitutionX143 -> T37,\nT11 -> s(T37),\nX144 -> X145,\nX29 -> node(nil, X145, nil)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 118 [arrowhead = none ];
118 -> 26;

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

120 [label="EVAL with clause\ns2t(0, nil).\nand substitutionT11 -> 0,\nX29 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 120 [arrowhead = none ];
120 -> 29;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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, X169, X170), X171, node(X171, X169, X170)).\nand substitutionT13 -> nil,\nX30 -> X172,\nX169 -> X172,\nX170 -> nil,\nX4 -> X173,\nX171 -> X173,\nX5 -> node(X173, X172, nil)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 127 [arrowhead = none ];
127 -> 36;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X194, X195, nil), X196, node(X194, X195, X196)).\nand substitutionT13 -> nil,\nX194 -> nil,\nX30 -> X197,\nX195 -> X197,\nT42 -> nil,\nX4 -> X198,\nX196 -> X198,\nX5 -> node(nil, X197, X198)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 132 [arrowhead = none ];
132 -> 41;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X239, X240, X241), X242, node(X243, X240, X241)) :- tappend(X239, X242, X243).\nand substitutionT13 -> T50,\nX239 -> T50,\nX30 -> X244,\nX240 -> X244,\nX241 -> T50,\nX4 -> X245,\nX242 -> X245,\nX243 -> X246,\nX5 -> node(X246, X244, T50),\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 137 [arrowhead = none ];
137 -> 46;

138 [label="ONLY EVAL with clause\ntappend(node(X385, X386, X387), X388, node(X385, X386, X389)) :- tappend(X387, X388, X389).\nand substitutionT13 -> T101,\nX385 -> T101,\nX30 -> X390,\nX386 -> X390,\nX387 -> T101,\nX4 -> X391,\nX388 -> X391,\nX389 -> X392,\nX5 -> node(T101, X390, X392),\nT100 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
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, X259, X259).\nand substitutionT50 -> nil,\nX245 -> X260,\nX259 -> X260,\nX246 -> X260", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 142 [arrowhead = none ];
142 -> 50;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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, X277, X278), X279, node(X279, X277, X278)).\nand substitutionX277 -> T59,\nX278 -> T60,\nT50 -> node(nil, T59, T60),\nX245 -> X280,\nX279 -> X280,\nX246 -> node(X280, T59, T60)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 147 [arrowhead = none ];
147 -> 55;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X297, X298, nil), X299, node(X297, X298, X299)).\nand substitutionX297 -> T69,\nX298 -> T70,\nT50 -> node(T69, T70, nil),\nX245 -> X300,\nX299 -> X300,\nX246 -> node(T69, T70, X300)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 152 [arrowhead = none ];
152 -> 60;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X329, X330, X331), X332, node(X333, X330, X331)) :- tappend(X329, X332, X333).\nand substitutionX329 -> T86,\nX330 -> T84,\nX331 -> T85,\nT50 -> node(T86, T84, T85),\nX245 -> X334,\nX332 -> X334,\nX333 -> X335,\nX246 -> node(X335, 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 = lightgreen];
63 -> 158 [arrowhead = none ];
158 -> 66;

159 [label="EVAL with clause\ntappend(node(X354, X355, X356), X357, node(X354, X355, X358)) :- tappend(X356, X357, X358).\nand substitutionX354 -> T93,\nX355 -> T94,\nX356 -> T96,\nT50 -> node(T93, T94, T96),\nX245 -> X359,\nX357 -> X359,\nX358 -> X360,\nX246 -> node(T93, T94, X360),\nT95 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 159 [arrowhead = none ];
159 -> 67;

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

161 [label="INSTANCE with matching:\nT50 -> T86\nX245 -> X334\nX246 -> X335", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 161 [arrowhead = none , style=dashed];
161 -> 46[style=dashed];

162 [label="INSTANCE with matching:\nT50 -> T96\nX245 -> X359\nX246 -> X360", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 162 [arrowhead = none , style=dashed];
162 -> 46[style=dashed];

163 [label="INSTANCE with matching:\nT50 -> T101\nX245 -> X391\nX246 -> X392", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 163 [arrowhead = none , style=dashed];
163 -> 46[style=dashed];

164 [label="EVAL with clause\ns2t(s(X417), node(nil, X418, X419)) :- s2t(X417, X419).\nand substitutionX417 -> T106,\nT6 -> s(T106),\nX418 -> X420,\nX419 -> X421,\nX3 -> node(nil, X420, X421)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 164 [arrowhead = none ];
164 -> 72;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT106 is ground\nreplacements:X421 -> T108", fontsize=14, style = filled, fillcolor = violet];
72 -> 169 [arrowhead = none ];
169 -> 75;

170 [label="INSTANCE with matching:\nT11 -> T106\nX29 -> X421", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 170 [arrowhead = none , style=dashed];
170 -> 9[style=dashed];

171 [label="INSTANCE with matching:\nT50 -> node(nil, X420, T108)\nX245 -> X4\nX246 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 171 [arrowhead = none , style=dashed];
171 -> 46[style=dashed];

172 [label="EVAL with clause\ns2t(s(X454), node(X455, X456, nil)) :- s2t(X454, X455).\nand substitutionX454 -> T114,\nT6 -> s(T114),\nX455 -> X457,\nX456 -> X458,\nX3 -> node(X457, X458, nil)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 172 [arrowhead = none ];
172 -> 78;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT114 is ground\nreplacements:X457 -> T116", fontsize=14, style = filled, fillcolor = violet];
78 -> 177 [arrowhead = none ];
177 -> 81;

178 [label="INSTANCE with matching:\nT11 -> T114\nX29 -> X457", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 178 [arrowhead = none , style=dashed];
178 -> 9[style=dashed];

179 [label="INSTANCE with matching:\nT50 -> node(T116, X458, nil)\nX245 -> X4\nX246 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 179 [arrowhead = none , style=dashed];
179 -> 46[style=dashed];

180 [label="EVAL with clause\ns2t(s(X483), node(nil, X484, nil)).\nand substitutionX483 -> T122,\nT6 -> s(T122),\nX484 -> X485,\nX3 -> node(nil, X485, nil)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 180 [arrowhead = none ];
180 -> 84;

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

182 [label="EVAL with clause\ns2t(0, nil).\nand substitutionT6 -> 0,\nX3 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 182 [arrowhead = none ];
182 -> 86;

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

184 [label="INSTANCE with matching:\nT13 -> nil\nX30 -> X485", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 184 [arrowhead = none , style=dashed];
184 -> 10[style=dashed];

185 [label="INSTANCE with matching:\nT50 -> nil\nX245 -> X4\nX246 -> X5", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 185 [arrowhead = none , style=dashed];
185 -> 46[style=dashed];

}

(2) Obligation:

Clauses:

s2tA(s(T19), node(X64, X65, X64)) :- s2tA(T19, X64).
s2tA(s(T25), node(nil, X94, X95)) :- s2tA(T25, X95).
s2tA(s(T31), node(X124, X125, nil)) :- s2tA(T31, X124).
s2tA(s(T37), node(nil, X145, nil)).
s2tA(0, nil).
tappendB(nil, X260, X260).
tappendB(node(nil, T59, T60), X280, node(X280, T59, T60)).
tappendB(node(T69, T70, nil), X300, node(T69, T70, X300)).
tappendB(node(T86, T84, T85), X334, node(X335, T84, T85)) :- tappendB(T86, X334, X335).
tappendB(node(T93, T94, T96), X359, node(T93, T94, X360)) :- tappendB(T96, X359, X360).
tappendC(nil, X172, X173, node(X173, X172, nil)).
tappendC(nil, X197, X198, node(nil, X197, X198)).
tappendC(T50, X244, X245, node(X246, X244, T50)) :- tappendB(T50, X245, X246).
tappendC(T101, X390, X391, node(T101, X390, X392)) :- tappendB(T101, X391, X392).
goalD(s(T11)) :- s2tA(T11, X29).
goalD(s(T11)) :- ','(s2tA(T11, T13), tappendC(T13, X30, X4, X5)).
goalD(s(T106)) :- s2tA(T106, X421).
goalD(s(T106)) :- ','(s2tA(T106, T108), tappendB(node(nil, X420, T108), X4, X5)).
goalD(s(T114)) :- s2tA(T114, X457).
goalD(s(T114)) :- ','(s2tA(T114, T116), tappendB(node(T116, X458, nil), X4, X5)).
goalD(s(T122)) :- tappendC(nil, X485, X4, X5).
goalD(0) :- tappendB(nil, X4, X5).

Query: goalD(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalD_in: (b)
s2tA_in: (b,f)
tappendC_in: (b,f,f,f)
tappendB_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

GOALD_IN_G(s(T11)) → U8_G(T11, s2tA_in_ga(T11, X29))
GOALD_IN_G(s(T11)) → S2TA_IN_GA(T11, X29)
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → U1_GA(T19, X64, X65, s2tA_in_ga(T19, X64))
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → S2TA_IN_GA(T19, X64)
S2TA_IN_GA(s(T25), node(nil, X94, X95)) → U2_GA(T25, X94, X95, s2tA_in_ga(T25, X95))
S2TA_IN_GA(s(T25), node(nil, X94, X95)) → S2TA_IN_GA(T25, X95)
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → U3_GA(T31, X124, X125, s2tA_in_ga(T31, X124))
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → S2TA_IN_GA(T31, X124)
GOALD_IN_G(s(T11)) → U9_G(T11, s2tA_in_ga(T11, T13))
U9_G(T11, s2tA_out_ga(T11, T13)) → U10_G(T11, tappendC_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tA_out_ga(T11, T13)) → TAPPENDC_IN_GAAA(T13, X30, X4, X5)
TAPPENDC_IN_GAAA(T50, X244, X245, node(X246, X244, T50)) → U6_GAAA(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
TAPPENDC_IN_GAAA(T50, X244, X245, node(X246, X244, T50)) → TAPPENDB_IN_GAA(T50, X245, X246)
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_GAA(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → TAPPENDB_IN_GAA(T86, X334, X335)
TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_GAA(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → TAPPENDB_IN_GAA(T96, X359, X360)
TAPPENDC_IN_GAAA(T101, X390, X391, node(T101, X390, X392)) → U7_GAAA(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
TAPPENDC_IN_GAAA(T101, X390, X391, node(T101, X390, X392)) → TAPPENDB_IN_GAA(T101, X391, X392)
GOALD_IN_G(s(T106)) → U11_G(T106, s2tA_in_ga(T106, X421))
GOALD_IN_G(s(T106)) → U12_G(T106, s2tA_in_ga(T106, T108))
U12_G(T106, s2tA_out_ga(T106, T108)) → U13_G(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U12_G(T106, s2tA_out_ga(T106, T108)) → TAPPENDB_IN_GAA(node(nil, X420, T108), X4, X5)
GOALD_IN_G(s(T114)) → U14_G(T114, s2tA_in_ga(T114, X457))
GOALD_IN_G(s(T114)) → U15_G(T114, s2tA_in_ga(T114, T116))
U15_G(T114, s2tA_out_ga(T114, T116)) → U16_G(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U15_G(T114, s2tA_out_ga(T114, T116)) → TAPPENDB_IN_GAA(node(T116, X458, nil), X4, X5)
GOALD_IN_G(s(T122)) → U17_G(T122, tappendC_in_gaaa(nil, X485, X4, X5))
GOALD_IN_G(s(T122)) → TAPPENDC_IN_GAAA(nil, X485, X4, X5)
GOALD_IN_G(0) → U18_G(tappendB_in_gaa(nil, X4, X5))
GOALD_IN_G(0) → TAPPENDB_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x2)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
TAPPENDC_IN_GAAA(x1, x2, x3, x4)  =  TAPPENDC_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x5)
TAPPENDB_IN_GAA(x1, x2, x3)  =  TAPPENDB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x5)
U11_G(x1, x2)  =  U11_G(x2)
U12_G(x1, x2)  =  U12_G(x2)
U13_G(x1, x2)  =  U13_G(x2)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1)  =  U18_G(x1)

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

(6) Obligation:

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

GOALD_IN_G(s(T11)) → U8_G(T11, s2tA_in_ga(T11, X29))
GOALD_IN_G(s(T11)) → S2TA_IN_GA(T11, X29)
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → U1_GA(T19, X64, X65, s2tA_in_ga(T19, X64))
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → S2TA_IN_GA(T19, X64)
S2TA_IN_GA(s(T25), node(nil, X94, X95)) → U2_GA(T25, X94, X95, s2tA_in_ga(T25, X95))
S2TA_IN_GA(s(T25), node(nil, X94, X95)) → S2TA_IN_GA(T25, X95)
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → U3_GA(T31, X124, X125, s2tA_in_ga(T31, X124))
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → S2TA_IN_GA(T31, X124)
GOALD_IN_G(s(T11)) → U9_G(T11, s2tA_in_ga(T11, T13))
U9_G(T11, s2tA_out_ga(T11, T13)) → U10_G(T11, tappendC_in_gaaa(T13, X30, X4, X5))
U9_G(T11, s2tA_out_ga(T11, T13)) → TAPPENDC_IN_GAAA(T13, X30, X4, X5)
TAPPENDC_IN_GAAA(T50, X244, X245, node(X246, X244, T50)) → U6_GAAA(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
TAPPENDC_IN_GAAA(T50, X244, X245, node(X246, X244, T50)) → TAPPENDB_IN_GAA(T50, X245, X246)
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_GAA(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → TAPPENDB_IN_GAA(T86, X334, X335)
TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_GAA(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → TAPPENDB_IN_GAA(T96, X359, X360)
TAPPENDC_IN_GAAA(T101, X390, X391, node(T101, X390, X392)) → U7_GAAA(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
TAPPENDC_IN_GAAA(T101, X390, X391, node(T101, X390, X392)) → TAPPENDB_IN_GAA(T101, X391, X392)
GOALD_IN_G(s(T106)) → U11_G(T106, s2tA_in_ga(T106, X421))
GOALD_IN_G(s(T106)) → U12_G(T106, s2tA_in_ga(T106, T108))
U12_G(T106, s2tA_out_ga(T106, T108)) → U13_G(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U12_G(T106, s2tA_out_ga(T106, T108)) → TAPPENDB_IN_GAA(node(nil, X420, T108), X4, X5)
GOALD_IN_G(s(T114)) → U14_G(T114, s2tA_in_ga(T114, X457))
GOALD_IN_G(s(T114)) → U15_G(T114, s2tA_in_ga(T114, T116))
U15_G(T114, s2tA_out_ga(T114, T116)) → U16_G(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U15_G(T114, s2tA_out_ga(T114, T116)) → TAPPENDB_IN_GAA(node(T116, X458, nil), X4, X5)
GOALD_IN_G(s(T122)) → U17_G(T122, tappendC_in_gaaa(nil, X485, X4, X5))
GOALD_IN_G(s(T122)) → TAPPENDC_IN_GAAA(nil, X485, X4, X5)
GOALD_IN_G(0) → U18_G(tappendB_in_gaa(nil, X4, X5))
GOALD_IN_G(0) → TAPPENDB_IN_GAA(nil, X4, X5)

The TRS R consists of the following rules:

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U8_G(x1, x2)  =  U8_G(x2)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U9_G(x1, x2)  =  U9_G(x2)
U10_G(x1, x2)  =  U10_G(x2)
TAPPENDC_IN_GAAA(x1, x2, x3, x4)  =  TAPPENDC_IN_GAAA(x1)
U6_GAAA(x1, x2, x3, x4, x5)  =  U6_GAAA(x5)
TAPPENDB_IN_GAA(x1, x2, x3)  =  TAPPENDB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x6)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U7_GAAA(x1, x2, x3, x4, x5)  =  U7_GAAA(x5)
U11_G(x1, x2)  =  U11_G(x2)
U12_G(x1, x2)  =  U12_G(x2)
U13_G(x1, x2)  =  U13_G(x2)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1)  =  U18_G(x1)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → TAPPENDB_IN_GAA(T96, X359, X360)
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → TAPPENDB_IN_GAA(T86, X334, X335)

The TRS R consists of the following rules:

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)
TAPPENDB_IN_GAA(x1, x2, x3)  =  TAPPENDB_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

TAPPENDB_IN_GAA(node(T93, T94, T96), X359, node(T93, T94, X360)) → TAPPENDB_IN_GAA(T96, X359, X360)
TAPPENDB_IN_GAA(node(T86, T84, T85), X334, node(X335, T84, T85)) → TAPPENDB_IN_GAA(T86, X334, X335)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

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

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

(14) 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:

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

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

(15) YES

(16) Obligation:

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

S2TA_IN_GA(s(T25), node(nil, X94, X95)) → S2TA_IN_GA(T25, X95)
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → S2TA_IN_GA(T19, X64)
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → S2TA_IN_GA(T31, X124)

The TRS R consists of the following rules:

goalD_in_g(s(T11)) → U8_g(T11, s2tA_in_ga(T11, X29))
s2tA_in_ga(s(T19), node(X64, X65, X64)) → U1_ga(T19, X64, X65, s2tA_in_ga(T19, X64))
s2tA_in_ga(s(T25), node(nil, X94, X95)) → U2_ga(T25, X94, X95, s2tA_in_ga(T25, X95))
s2tA_in_ga(s(T31), node(X124, X125, nil)) → U3_ga(T31, X124, X125, s2tA_in_ga(T31, X124))
s2tA_in_ga(s(T37), node(nil, X145, nil)) → s2tA_out_ga(s(T37), node(nil, X145, nil))
s2tA_in_ga(0, nil) → s2tA_out_ga(0, nil)
U3_ga(T31, X124, X125, s2tA_out_ga(T31, X124)) → s2tA_out_ga(s(T31), node(X124, X125, nil))
U2_ga(T25, X94, X95, s2tA_out_ga(T25, X95)) → s2tA_out_ga(s(T25), node(nil, X94, X95))
U1_ga(T19, X64, X65, s2tA_out_ga(T19, X64)) → s2tA_out_ga(s(T19), node(X64, X65, X64))
U8_g(T11, s2tA_out_ga(T11, X29)) → goalD_out_g(s(T11))
goalD_in_g(s(T11)) → U9_g(T11, s2tA_in_ga(T11, T13))
U9_g(T11, s2tA_out_ga(T11, T13)) → U10_g(T11, tappendC_in_gaaa(T13, X30, X4, X5))
tappendC_in_gaaa(nil, X172, X173, node(X173, X172, nil)) → tappendC_out_gaaa(nil, X172, X173, node(X173, X172, nil))
tappendC_in_gaaa(nil, X197, X198, node(nil, X197, X198)) → tappendC_out_gaaa(nil, X197, X198, node(nil, X197, X198))
tappendC_in_gaaa(T50, X244, X245, node(X246, X244, T50)) → U6_gaaa(T50, X244, X245, X246, tappendB_in_gaa(T50, X245, X246))
tappendB_in_gaa(nil, X260, X260) → tappendB_out_gaa(nil, X260, X260)
tappendB_in_gaa(node(nil, T59, T60), X280, node(X280, T59, T60)) → tappendB_out_gaa(node(nil, T59, T60), X280, node(X280, T59, T60))
tappendB_in_gaa(node(T69, T70, nil), X300, node(T69, T70, X300)) → tappendB_out_gaa(node(T69, T70, nil), X300, node(T69, T70, X300))
tappendB_in_gaa(node(T86, T84, T85), X334, node(X335, T84, T85)) → U4_gaa(T86, T84, T85, X334, X335, tappendB_in_gaa(T86, X334, X335))
tappendB_in_gaa(node(T93, T94, T96), X359, node(T93, T94, X360)) → U5_gaa(T93, T94, T96, X359, X360, tappendB_in_gaa(T96, X359, X360))
U5_gaa(T93, T94, T96, X359, X360, tappendB_out_gaa(T96, X359, X360)) → tappendB_out_gaa(node(T93, T94, T96), X359, node(T93, T94, X360))
U4_gaa(T86, T84, T85, X334, X335, tappendB_out_gaa(T86, X334, X335)) → tappendB_out_gaa(node(T86, T84, T85), X334, node(X335, T84, T85))
U6_gaaa(T50, X244, X245, X246, tappendB_out_gaa(T50, X245, X246)) → tappendC_out_gaaa(T50, X244, X245, node(X246, X244, T50))
tappendC_in_gaaa(T101, X390, X391, node(T101, X390, X392)) → U7_gaaa(T101, X390, X391, X392, tappendB_in_gaa(T101, X391, X392))
U7_gaaa(T101, X390, X391, X392, tappendB_out_gaa(T101, X391, X392)) → tappendC_out_gaaa(T101, X390, X391, node(T101, X390, X392))
U10_g(T11, tappendC_out_gaaa(T13, X30, X4, X5)) → goalD_out_g(s(T11))
goalD_in_g(s(T106)) → U11_g(T106, s2tA_in_ga(T106, X421))
U11_g(T106, s2tA_out_ga(T106, X421)) → goalD_out_g(s(T106))
goalD_in_g(s(T106)) → U12_g(T106, s2tA_in_ga(T106, T108))
U12_g(T106, s2tA_out_ga(T106, T108)) → U13_g(T106, tappendB_in_gaa(node(nil, X420, T108), X4, X5))
U13_g(T106, tappendB_out_gaa(node(nil, X420, T108), X4, X5)) → goalD_out_g(s(T106))
goalD_in_g(s(T114)) → U14_g(T114, s2tA_in_ga(T114, X457))
U14_g(T114, s2tA_out_ga(T114, X457)) → goalD_out_g(s(T114))
goalD_in_g(s(T114)) → U15_g(T114, s2tA_in_ga(T114, T116))
U15_g(T114, s2tA_out_ga(T114, T116)) → U16_g(T114, tappendB_in_gaa(node(T116, X458, nil), X4, X5))
U16_g(T114, tappendB_out_gaa(node(T116, X458, nil), X4, X5)) → goalD_out_g(s(T114))
goalD_in_g(s(T122)) → U17_g(T122, tappendC_in_gaaa(nil, X485, X4, X5))
U17_g(T122, tappendC_out_gaaa(nil, X485, X4, X5)) → goalD_out_g(s(T122))
goalD_in_g(0) → U18_g(tappendB_in_gaa(nil, X4, X5))
U18_g(tappendB_out_gaa(nil, X4, X5)) → goalD_out_g(0)

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
s(x1)  =  s(x1)
U8_g(x1, x2)  =  U8_g(x2)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
s2tA_out_ga(x1, x2)  =  s2tA_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goalD_out_g(x1)  =  goalD_out_g
U9_g(x1, x2)  =  U9_g(x2)
U10_g(x1, x2)  =  U10_g(x2)
tappendC_in_gaaa(x1, x2, x3, x4)  =  tappendC_in_gaaa(x1)
nil  =  nil
tappendC_out_gaaa(x1, x2, x3, x4)  =  tappendC_out_gaaa
U6_gaaa(x1, x2, x3, x4, x5)  =  U6_gaaa(x5)
tappendB_in_gaa(x1, x2, x3)  =  tappendB_in_gaa(x1)
tappendB_out_gaa(x1, x2, x3)  =  tappendB_out_gaa
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x6)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U7_gaaa(x1, x2, x3, x4, x5)  =  U7_gaaa(x5)
U11_g(x1, x2)  =  U11_g(x2)
U12_g(x1, x2)  =  U12_g(x2)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1)  =  U18_g(x1)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

S2TA_IN_GA(s(T25), node(nil, X94, X95)) → S2TA_IN_GA(T25, X95)
S2TA_IN_GA(s(T19), node(X64, X65, X64)) → S2TA_IN_GA(T19, X64)
S2TA_IN_GA(s(T31), node(X124, X125, nil)) → S2TA_IN_GA(T31, X124)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

S2TA_IN_GA(s(T25)) → S2TA_IN_GA(T25)

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

(21) 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:

• S2TA_IN_GA(s(T25)) → S2TA_IN_GA(T25)
  The graph contains the following edges 1 > 1

(22) YES