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

(0) Obligation:

Clauses:

tree(nil).
tree(node(L, X, R)) :- ','(tree(L), tree(R)).
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, T), tree(T)).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1), SCOPE: 1, CLAUSE: 7\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2t(T3, X3), tree(X3))\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 2 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 3 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 4 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 5 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 3 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 4 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 5 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(s2t(T8, X27), tree(node(X27, X28, X27)))\n\nT8 is ground\nX27 is free; \nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: s2t(T8, X27)\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: tree(node(T10, X28, T10))\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: s2t(T8, X27), SCOPE: 3, CLAUSE: 2 | s2t(T8, X27), SCOPE: 3, CLAUSE: 3 | s2t(T8, X27), SCOPE: 3, CLAUSE: 4 | s2t(T8, X27), SCOPE: 3, CLAUSE: 5 | s2t(T8, X27), SCOPE: 3, CLAUSE: 6\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: s2t(T8, X27), SCOPE: 3, CLAUSE: 2\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2t(T8, X27), SCOPE: 3, CLAUSE: 3 | s2t(T8, X27), SCOPE: 3, CLAUSE: 4 | s2t(T8, X27), SCOPE: 3, CLAUSE: 5 | s2t(T8, X27), SCOPE: 3, CLAUSE: 6\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: s2t(T16, X65)\n\nT16 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2t(T8, X27), SCOPE: 3, CLAUSE: 3\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: s2t(T8, X27), SCOPE: 3, CLAUSE: 4 | s2t(T8, X27), SCOPE: 3, CLAUSE: 5 | s2t(T8, X27), SCOPE: 3, CLAUSE: 6\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: s2t(T22, X99)\n\nT22 is ground\nX99 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: s2t(T8, X27), SCOPE: 3, CLAUSE: 4\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2t(T8, X27), SCOPE: 3, CLAUSE: 5 | s2t(T8, X27), SCOPE: 3, CLAUSE: 6\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: s2t(T28, X131)\n\nT28 is ground\nX131 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: s2t(T8, X27), SCOPE: 3, CLAUSE: 5\n\nT8 is ground\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2t(T8, X27), SCOPE: 3, CLAUSE: 6\n\nT8 is ground\nX27 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: tree(node(T10, X28, T10)), SCOPE: 4, CLAUSE: 0 | tree(node(T10, X28, T10)), SCOPE: 4, CLAUSE: 1\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: tree(node(T10, X28, T10)), SCOPE: 4, CLAUSE: 1\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tree(T40), tree(T40))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: tree(T40)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: tree(T41)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: tree(T41), SCOPE: 5, CLAUSE: 0 | tree(T41), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tree(T41), SCOPE: 5, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: tree(T41), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tree(T51), tree(T52))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: tree(T51)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: tree(T53)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 4 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 5 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(s2t(T58, X210), tree(node(nil, X209, X210)))\n\nT58 is ground\nX209 is free; \nX210 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: s2t(T58, X210)\n\nT58 is ground\nX210 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: tree(node(nil, X209, T60))\n\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 5 | ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(s2t(T67, X251), tree(node(X251, X252, nil)))\n\nT67 is ground\nX251 is free; \nX252 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: s2t(T67, X251)\n\nT67 is ground\nX251 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: tree(node(T69, X252, nil))\n\nX252 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(s2t(T3, X3), tree(X3)), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: tree(node(nil, X284, nil))\n\nX284 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: tree(node(nil, X284, nil)), SCOPE: 6, CLAUSE: 0 | tree(node(nil, X284, nil)), SCOPE: 6, CLAUSE: 1\n\nX284 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: tree(node(nil, X284, nil)), SCOPE: 6, CLAUSE: 1\n\nX284 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tree(nil), tree(nil))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: tree(nil), SCOPE: 7, CLAUSE: 0 | tree(nil), SCOPE: 7, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: tree(nil), SCOPE: 7, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: tree(nil), SCOPE: 7, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 76 [arrowhead = none ];
76 -> 2;

77 [label="ONLY EVAL with clause\ngoal(X2) :- ','(s2t(X2, X3), tree(X3)).\nand substitution\nT1 -> T3\nX2 -> T3", fontsize=14, style = filled, fillcolor = white];
2 -> 77 [arrowhead = none ];
77 -> 3;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 78 [arrowhead = none ];
78 -> 4;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 79 [arrowhead = none ];
79 -> 5;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 80 [arrowhead = none ];
80 -> 6;

81 [label="EVAL with clause\ns2t(s(X24), node(X25, X26, X25)) :- s2t(X24, X25).\nand substitution\nX24 -> T8\nT3 -> s(T8)\nX25 -> X27\nX26 -> X28\nX3 -> node(X27, X28, X27)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 81 [arrowhead = none ];
81 -> 7;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 8;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 83 [arrowhead = none ];
83 -> 47;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 84 [arrowhead = none ];
84 -> 48;

85 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 85 [arrowhead = none ];
85 -> 9;

86 [label="SPLIT 2\nreplacements:\nX27 -> T10", fontsize=14, style = filled, fillcolor = violet];
7 -> 86 [arrowhead = none ];
86 -> 10;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 87 [arrowhead = none ];
87 -> 11;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 88 [arrowhead = none ];
88 -> 32;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="EVAL with clause\ns2t(s(X62), node(X63, X64, X63)) :- s2t(X62, X63).\nand substitution\nX62 -> T16\nT8 -> s(T16)\nX63 -> X65\nX64 -> X66\nX27 -> node(X65, X66, X65)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 91 [arrowhead = none ];
91 -> 14;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 92 [arrowhead = none ];
92 -> 15;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 93 [arrowhead = none ];
93 -> 16;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="INSTANCE with matching:\nT8 -> T16\nX27 -> X65", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 95 [arrowhead = none , style=dashed];
95 -> 9[style=dashed];

96 [label="EVAL with clause\ns2t(s(X95), node(nil, X96, X97)) :- s2t(X95, X97).\nand substitution\nX95 -> T22\nT8 -> s(T22)\nX96 -> X98\nX97 -> X99\nX27 -> node(nil, X98, X99)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 96 [arrowhead = none ];
96 -> 18;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 97 [arrowhead = none ];
97 -> 19;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 98 [arrowhead = none ];
98 -> 20;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 99 [arrowhead = none ];
99 -> 21;

100 [label="INSTANCE with matching:\nT8 -> T22\nX27 -> X99", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 100 [arrowhead = none , style=dashed];
100 -> 9[style=dashed];

101 [label="EVAL with clause\ns2t(s(X128), node(X129, X130, nil)) :- s2t(X128, X129).\nand substitution\nX128 -> T28\nT8 -> s(T28)\nX129 -> X131\nX130 -> X132\nX27 -> node(X131, X132, nil)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 101 [arrowhead = none ];
101 -> 22;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 102 [arrowhead = none ];
102 -> 23;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 103 [arrowhead = none ];
103 -> 24;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 104 [arrowhead = none ];
104 -> 25;

105 [label="INSTANCE with matching:\nT8 -> T28\nX27 -> X131", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 105 [arrowhead = none , style=dashed];
105 -> 9[style=dashed];

106 [label="EVAL with clause\ns2t(s(X153), node(nil, X154, nil)).\nand substitution\nX153 -> T34\nT8 -> s(T34)\nX154 -> X155\nX27 -> node(nil, X155, nil)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 106 [arrowhead = none ];
106 -> 26;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 107 [arrowhead = none ];
107 -> 27;

108 [label="EVAL with clause\ns2t(0, nil).\nand substitution\nT8 -> 0\nX27 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 108 [arrowhead = none ];
108 -> 29;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 109 [arrowhead = none ];
109 -> 30;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 110 [arrowhead = none ];
110 -> 28;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 111 [arrowhead = none ];
111 -> 31;

112 [label="BACKTRACK\nfor clause: tree(nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="ONLY EVAL with clause\ntree(node(X170, X171, X172)) :- ','(tree(X170), tree(X172)).\nand substitution\nT10 -> T40\nX170 -> T40\nX28 -> X173\nX171 -> X173\nX172 -> T40\nT39 -> T40", fontsize=14, style = filled, fillcolor = white];
33 -> 113 [arrowhead = none ];
113 -> 34;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 114 [arrowhead = none ];
114 -> 35;

115 [label="SPLIT 2\nreplacements:\nT40 -> T41", fontsize=14, style = filled, fillcolor = violet];
34 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="INSTANCE with matching:\nT41 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 116 [arrowhead = none , style=dashed];
116 -> 36[style=dashed];

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 117 [arrowhead = none ];
117 -> 37;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 118 [arrowhead = none ];
118 -> 38;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 119 [arrowhead = none ];
119 -> 39;

120 [label="EVAL with clause\ntree(nil).\nand substitution\nT41 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 120 [arrowhead = none ];
120 -> 40;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="EVAL with clause\ntree(node(X180, X181, X182)) :- ','(tree(X180), tree(X182)).\nand substitution\nX180 -> T51\nX181 -> T49\nX182 -> T52\nT41 -> node(T51, T49, T52)\nT48 -> T51\nT50 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 122 [arrowhead = none ];
122 -> 43;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 123 [arrowhead = none ];
123 -> 44;

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 124 [arrowhead = none ];
124 -> 42;

125 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 125 [arrowhead = none ];
125 -> 45;

126 [label="SPLIT 2\nreplacements:\nT52 -> T53", fontsize=14, style = filled, fillcolor = violet];
43 -> 126 [arrowhead = none ];
126 -> 46;

127 [label="INSTANCE with matching:\nT40 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 127 [arrowhead = none , style=dashed];
127 -> 35[style=dashed];

128 [label="INSTANCE with matching:\nT40 -> T53", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 128 [arrowhead = none , style=dashed];
128 -> 35[style=dashed];

129 [label="EVAL with clause\ns2t(s(X206), node(nil, X207, X208)) :- s2t(X206, X208).\nand substitution\nX206 -> T58\nT3 -> s(T58)\nX207 -> X209\nX208 -> X210\nX3 -> node(nil, X209, X210)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 130 [arrowhead = none ];
130 -> 50;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 131 [arrowhead = none ];
131 -> 53;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 132 [arrowhead = none ];
132 -> 54;

133 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 133 [arrowhead = none ];
133 -> 51;

134 [label="SPLIT 2\nreplacements:\nX210 -> T60", fontsize=14, style = filled, fillcolor = violet];
49 -> 134 [arrowhead = none ];
134 -> 52;

135 [label="INSTANCE with matching:\nT8 -> T58\nX27 -> X210", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 135 [arrowhead = none , style=dashed];
135 -> 9[style=dashed];

136 [label="INSTANCE with matching:\nT40 -> node(nil, X209, T60)", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 136 [arrowhead = none , style=dashed];
136 -> 35[style=dashed];

137 [label="EVAL with clause\ns2t(s(X248), node(X249, X250, nil)) :- s2t(X248, X249).\nand substitution\nX248 -> T67\nT3 -> s(T67)\nX249 -> X251\nX250 -> X252\nX3 -> node(X251, X252, nil)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 137 [arrowhead = none ];
137 -> 55;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 138 [arrowhead = none ];
138 -> 56;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 139 [arrowhead = none ];
139 -> 59;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 140 [arrowhead = none ];
140 -> 60;

141 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 141 [arrowhead = none ];
141 -> 57;

142 [label="SPLIT 2\nreplacements:\nX251 -> T69", fontsize=14, style = filled, fillcolor = violet];
55 -> 142 [arrowhead = none ];
142 -> 58;

143 [label="INSTANCE with matching:\nT8 -> T67\nX27 -> X251", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 143 [arrowhead = none , style=dashed];
143 -> 9[style=dashed];

144 [label="INSTANCE with matching:\nT40 -> node(T69, X252, nil)", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 144 [arrowhead = none , style=dashed];
144 -> 35[style=dashed];

145 [label="EVAL with clause\ns2t(s(X282), node(nil, X283, nil)).\nand substitution\nX282 -> T76\nT3 -> s(T76)\nX283 -> X284\nX3 -> node(nil, X284, nil)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 145 [arrowhead = none ];
145 -> 61;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 146 [arrowhead = none ];
146 -> 62;

147 [label="EVAL with clause\ns2t(0, nil).\nand substitution\nT3 -> 0\nX3 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 147 [arrowhead = none ];
147 -> 74;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 148 [arrowhead = none ];
148 -> 75;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 149 [arrowhead = none ];
149 -> 63;

150 [label="BACKTRACK\nfor clause: tree(nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
63 -> 150 [arrowhead = none ];
150 -> 64;

151 [label="ONLY EVAL with clause\ntree(node(X299, X300, X301)) :- ','(tree(X299), tree(X301)).\nand substitution\nX299 -> nil\nX284 -> X302\nX300 -> X302\nX301 -> nil", fontsize=14, style = filled, fillcolor = white];
64 -> 151 [arrowhead = none ];
151 -> 65;

152 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 152 [arrowhead = none ];
152 -> 66;

153 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
65 -> 153 [arrowhead = none ];
153 -> 67;

154 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 154 [arrowhead = none , style=dashed];
154 -> 67[style=dashed];

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 155 [arrowhead = none ];
155 -> 68;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 156 [arrowhead = none ];
156 -> 69;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 157 [arrowhead = none ];
157 -> 70;

158 [label="ONLY EVAL with clause\ntree(nil).\nand substitution", fontsize=14, style = filled, fillcolor = white];
69 -> 158 [arrowhead = none ];
158 -> 71;

159 [label="BACKTRACK\nfor clause: tree(node(L, X, R)) :- ','(tree(L), tree(R))because of non-unification", fontsize=14, style = filled, fillcolor = white];
70 -> 159 [arrowhead = none ];
159 -> 73;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 160 [arrowhead = none ];
160 -> 72;

161 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 161 [arrowhead = none , style=dashed];
161 -> 66[style=dashed];

}

(2) Obligation:

Clauses:

s2t9(s(T16), node(X65, X66, X65)) :- s2t9(T16, X65).
s2t9(s(T22), node(nil, X98, X99)) :- s2t9(T22, X99).
s2t9(s(T28), node(X131, X132, nil)) :- s2t9(T28, X131).
s2t9(s(T34), node(nil, X155, nil)).
s2t9(0, nil).
tree36(nil).
tree36(node(T51, T49, T52)) :- tree36(T51).
tree36(node(T51, T49, T53)) :- ','(tree36(T51), tree36(T53)).
tree67.
goal1(s(T8)) :- s2t9(T8, X27).
goal1(s(T8)) :- ','(s2t9(T8, T40), tree36(T40)).
goal1(s(T8)) :- ','(s2t9(T8, T41), ','(tree36(T41), tree36(T41))).
goal1(s(T58)) :- s2t9(T58, X210).
goal1(s(T58)) :- ','(s2t9(T58, T60), tree36(node(nil, X209, T60))).
goal1(s(T67)) :- s2t9(T67, X251).
goal1(s(T67)) :- ','(s2t9(T67, T69), tree36(node(T69, X252, nil))).
goal1(s(T76)) :- tree67.
goal1(s(T76)) :- ','(tree67, tree67).
goal1(0) :- tree67.

Queries:

goal1(g).

(3) PrologToPiTRSProof (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)
tree36_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_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:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_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:

GOAL1_IN_G(s(T8)) → U7_G(T8, s2t9_in_ga(T8, X27))
GOAL1_IN_G(s(T8)) → S2T9_IN_GA(T8, X27)
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → U1_GA(T16, X65, X66, s2t9_in_ga(T16, X65))
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → S2T9_IN_GA(T16, X65)
S2T9_IN_GA(s(T22), node(nil, X98, X99)) → U2_GA(T22, X98, X99, s2t9_in_ga(T22, X99))
S2T9_IN_GA(s(T22), node(nil, X98, X99)) → S2T9_IN_GA(T22, X99)
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → U3_GA(T28, X131, X132, s2t9_in_ga(T28, X131))
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → S2T9_IN_GA(T28, X131)
GOAL1_IN_G(s(T8)) → U8_G(T8, s2t9_in_ga(T8, T40))
U8_G(T8, s2t9_out_ga(T8, T40)) → U9_G(T8, tree36_in_g(T40))
U8_G(T8, s2t9_out_ga(T8, T40)) → TREE36_IN_G(T40)
TREE36_IN_G(node(T51, T49, T52)) → U4_G(T51, T49, T52, tree36_in_g(T51))
TREE36_IN_G(node(T51, T49, T52)) → TREE36_IN_G(T51)
TREE36_IN_G(node(T51, T49, T53)) → U5_G(T51, T49, T53, tree36_in_g(T51))
U5_G(T51, T49, T53, tree36_out_g(T51)) → U6_G(T51, T49, T53, tree36_in_g(T53))
U5_G(T51, T49, T53, tree36_out_g(T51)) → TREE36_IN_G(T53)
GOAL1_IN_G(s(T8)) → U10_G(T8, s2t9_in_ga(T8, T41))
U10_G(T8, s2t9_out_ga(T8, T41)) → U11_G(T8, T41, tree36_in_g(T41))
U10_G(T8, s2t9_out_ga(T8, T41)) → TREE36_IN_G(T41)
U11_G(T8, T41, tree36_out_g(T41)) → U12_G(T8, tree36_in_g(T41))
U11_G(T8, T41, tree36_out_g(T41)) → TREE36_IN_G(T41)
GOAL1_IN_G(s(T58)) → U13_G(T58, s2t9_in_ga(T58, X210))
GOAL1_IN_G(s(T58)) → U14_G(T58, s2t9_in_ga(T58, T60))
U14_G(T58, s2t9_out_ga(T58, T60)) → U15_G(T58, tree36_in_g(node(nil, X209, T60)))
U14_G(T58, s2t9_out_ga(T58, T60)) → TREE36_IN_G(node(nil, X209, T60))
GOAL1_IN_G(s(T67)) → U16_G(T67, s2t9_in_ga(T67, X251))
GOAL1_IN_G(s(T67)) → U17_G(T67, s2t9_in_ga(T67, T69))
U17_G(T67, s2t9_out_ga(T67, T69)) → U18_G(T67, tree36_in_g(node(T69, X252, nil)))
U17_G(T67, s2t9_out_ga(T67, T69)) → TREE36_IN_G(node(T69, X252, nil))
GOAL1_IN_G(s(T76)) → U19_G(T76, tree67_in_)
GOAL1_IN_G(s(T76)) → TREE67_IN_
U19_G(T76, tree67_out_) → U20_G(T76, tree67_in_)
U19_G(T76, tree67_out_) → TREE67_IN_
GOAL1_IN_G(0) → U21_G(tree67_in_)
GOAL1_IN_G(0) → TREE67_IN_

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x2)
S2T9_IN_GA(x1, x2)  =  S2T9_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)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)
TREE36_IN_G(x1)  =  TREE36_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)
U5_G(x1, x2, x3, x4)  =  U5_G(x3, x4)
U6_G(x1, x2, x3, x4)  =  U6_G(x4)
U10_G(x1, x2)  =  U10_G(x2)
U11_G(x1, x2, x3)  =  U11_G(x2, x3)
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, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
TREE67_IN_  =  TREE67_IN_
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1)  =  U21_G(x1)

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

(6) Obligation:

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

GOAL1_IN_G(s(T8)) → U7_G(T8, s2t9_in_ga(T8, X27))
GOAL1_IN_G(s(T8)) → S2T9_IN_GA(T8, X27)
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → U1_GA(T16, X65, X66, s2t9_in_ga(T16, X65))
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → S2T9_IN_GA(T16, X65)
S2T9_IN_GA(s(T22), node(nil, X98, X99)) → U2_GA(T22, X98, X99, s2t9_in_ga(T22, X99))
S2T9_IN_GA(s(T22), node(nil, X98, X99)) → S2T9_IN_GA(T22, X99)
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → U3_GA(T28, X131, X132, s2t9_in_ga(T28, X131))
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → S2T9_IN_GA(T28, X131)
GOAL1_IN_G(s(T8)) → U8_G(T8, s2t9_in_ga(T8, T40))
U8_G(T8, s2t9_out_ga(T8, T40)) → U9_G(T8, tree36_in_g(T40))
U8_G(T8, s2t9_out_ga(T8, T40)) → TREE36_IN_G(T40)
TREE36_IN_G(node(T51, T49, T52)) → U4_G(T51, T49, T52, tree36_in_g(T51))
TREE36_IN_G(node(T51, T49, T52)) → TREE36_IN_G(T51)
TREE36_IN_G(node(T51, T49, T53)) → U5_G(T51, T49, T53, tree36_in_g(T51))
U5_G(T51, T49, T53, tree36_out_g(T51)) → U6_G(T51, T49, T53, tree36_in_g(T53))
U5_G(T51, T49, T53, tree36_out_g(T51)) → TREE36_IN_G(T53)
GOAL1_IN_G(s(T8)) → U10_G(T8, s2t9_in_ga(T8, T41))
U10_G(T8, s2t9_out_ga(T8, T41)) → U11_G(T8, T41, tree36_in_g(T41))
U10_G(T8, s2t9_out_ga(T8, T41)) → TREE36_IN_G(T41)
U11_G(T8, T41, tree36_out_g(T41)) → U12_G(T8, tree36_in_g(T41))
U11_G(T8, T41, tree36_out_g(T41)) → TREE36_IN_G(T41)
GOAL1_IN_G(s(T58)) → U13_G(T58, s2t9_in_ga(T58, X210))
GOAL1_IN_G(s(T58)) → U14_G(T58, s2t9_in_ga(T58, T60))
U14_G(T58, s2t9_out_ga(T58, T60)) → U15_G(T58, tree36_in_g(node(nil, X209, T60)))
U14_G(T58, s2t9_out_ga(T58, T60)) → TREE36_IN_G(node(nil, X209, T60))
GOAL1_IN_G(s(T67)) → U16_G(T67, s2t9_in_ga(T67, X251))
GOAL1_IN_G(s(T67)) → U17_G(T67, s2t9_in_ga(T67, T69))
U17_G(T67, s2t9_out_ga(T67, T69)) → U18_G(T67, tree36_in_g(node(T69, X252, nil)))
U17_G(T67, s2t9_out_ga(T67, T69)) → TREE36_IN_G(node(T69, X252, nil))
GOAL1_IN_G(s(T76)) → U19_G(T76, tree67_in_)
GOAL1_IN_G(s(T76)) → TREE67_IN_
U19_G(T76, tree67_out_) → U20_G(T76, tree67_in_)
U19_G(T76, tree67_out_) → TREE67_IN_
GOAL1_IN_G(0) → U21_G(tree67_in_)
GOAL1_IN_G(0) → TREE67_IN_

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_g(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x2)
S2T9_IN_GA(x1, x2)  =  S2T9_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)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)
TREE36_IN_G(x1)  =  TREE36_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)
U5_G(x1, x2, x3, x4)  =  U5_G(x3, x4)
U6_G(x1, x2, x3, x4)  =  U6_G(x4)
U10_G(x1, x2)  =  U10_G(x2)
U11_G(x1, x2, x3)  =  U11_G(x2, x3)
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, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
TREE67_IN_  =  TREE67_IN_
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1)  =  U21_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 29 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TREE36_IN_G(node(T51, T49, T53)) → U5_G(T51, T49, T53, tree36_in_g(T51))
U5_G(T51, T49, T53, tree36_out_g(T51)) → TREE36_IN_G(T53)
TREE36_IN_G(node(T51, T49, T52)) → TREE36_IN_G(T51)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_g(x1)
TREE36_IN_G(x1)  =  TREE36_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x3, x4)

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:

TREE36_IN_G(node(T51, T49, T53)) → U5_G(T51, T49, T53, tree36_in_g(T51))
U5_G(T51, T49, T53, tree36_out_g(T51)) → TREE36_IN_G(T53)
TREE36_IN_G(node(T51, T49, T52)) → TREE36_IN_G(T51)

The TRS R consists of the following rules:

tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))

The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x3)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
TREE36_IN_G(x1)  =  TREE36_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x3, x4)

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:

TREE36_IN_G(node(T51, T53)) → U5_G(T53, tree36_in_g(T51))
U5_G(T53, tree36_out_g) → TREE36_IN_G(T53)
TREE36_IN_G(node(T51, T52)) → TREE36_IN_G(T51)

The TRS R consists of the following rules:

tree36_in_g(nil) → tree36_out_g
tree36_in_g(node(T51, T52)) → U4_g(tree36_in_g(T51))
tree36_in_g(node(T51, T53)) → U5_g(T53, tree36_in_g(T51))
U4_g(tree36_out_g) → tree36_out_g
U5_g(T53, tree36_out_g) → U6_g(tree36_in_g(T53))
U6_g(tree36_out_g) → tree36_out_g

The set Q consists of the following terms:

tree36_in_g(x0)
U4_g(x0)
U5_g(x0, x1)
U6_g(x0)

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:

• U5_G(T53, tree36_out_g) → TREE36_IN_G(T53)
  The graph contains the following edges 1 >= 1

• TREE36_IN_G(node(T51, T52)) → TREE36_IN_G(T51)
  The graph contains the following edges 1 > 1

• TREE36_IN_G(node(T51, T53)) → U5_G(T53, tree36_in_g(T51))
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2T9_IN_GA(s(T22), node(nil, X98, X99)) → S2T9_IN_GA(T22, X99)
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → S2T9_IN_GA(T16, X65)
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → S2T9_IN_GA(T28, X131)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U7_g(T8, s2t9_in_ga(T8, X27))
s2t9_in_ga(s(T16), node(X65, X66, X65)) → U1_ga(T16, X65, X66, s2t9_in_ga(T16, X65))
s2t9_in_ga(s(T22), node(nil, X98, X99)) → U2_ga(T22, X98, X99, s2t9_in_ga(T22, X99))
s2t9_in_ga(s(T28), node(X131, X132, nil)) → U3_ga(T28, X131, X132, s2t9_in_ga(T28, X131))
s2t9_in_ga(s(T34), node(nil, X155, nil)) → s2t9_out_ga(s(T34), node(nil, X155, nil))
s2t9_in_ga(0, nil) → s2t9_out_ga(0, nil)
U3_ga(T28, X131, X132, s2t9_out_ga(T28, X131)) → s2t9_out_ga(s(T28), node(X131, X132, nil))
U2_ga(T22, X98, X99, s2t9_out_ga(T22, X99)) → s2t9_out_ga(s(T22), node(nil, X98, X99))
U1_ga(T16, X65, X66, s2t9_out_ga(T16, X65)) → s2t9_out_ga(s(T16), node(X65, X66, X65))
U7_g(T8, s2t9_out_ga(T8, X27)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U8_g(T8, s2t9_in_ga(T8, T40))
U8_g(T8, s2t9_out_ga(T8, T40)) → U9_g(T8, tree36_in_g(T40))
tree36_in_g(nil) → tree36_out_g(nil)
tree36_in_g(node(T51, T49, T52)) → U4_g(T51, T49, T52, tree36_in_g(T51))
tree36_in_g(node(T51, T49, T53)) → U5_g(T51, T49, T53, tree36_in_g(T51))
U5_g(T51, T49, T53, tree36_out_g(T51)) → U6_g(T51, T49, T53, tree36_in_g(T53))
U6_g(T51, T49, T53, tree36_out_g(T53)) → tree36_out_g(node(T51, T49, T53))
U4_g(T51, T49, T52, tree36_out_g(T51)) → tree36_out_g(node(T51, T49, T52))
U9_g(T8, tree36_out_g(T40)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U10_g(T8, s2t9_in_ga(T8, T41))
U10_g(T8, s2t9_out_ga(T8, T41)) → U11_g(T8, T41, tree36_in_g(T41))
U11_g(T8, T41, tree36_out_g(T41)) → U12_g(T8, tree36_in_g(T41))
U12_g(T8, tree36_out_g(T41)) → goal1_out_g(s(T8))
goal1_in_g(s(T58)) → U13_g(T58, s2t9_in_ga(T58, X210))
U13_g(T58, s2t9_out_ga(T58, X210)) → goal1_out_g(s(T58))
goal1_in_g(s(T58)) → U14_g(T58, s2t9_in_ga(T58, T60))
U14_g(T58, s2t9_out_ga(T58, T60)) → U15_g(T58, tree36_in_g(node(nil, X209, T60)))
U15_g(T58, tree36_out_g(node(nil, X209, T60))) → goal1_out_g(s(T58))
goal1_in_g(s(T67)) → U16_g(T67, s2t9_in_ga(T67, X251))
U16_g(T67, s2t9_out_ga(T67, X251)) → goal1_out_g(s(T67))
goal1_in_g(s(T67)) → U17_g(T67, s2t9_in_ga(T67, T69))
U17_g(T67, s2t9_out_ga(T67, T69)) → U18_g(T67, tree36_in_g(node(T69, X252, nil)))
U18_g(T67, tree36_out_g(node(T69, X252, nil))) → goal1_out_g(s(T67))
goal1_in_g(s(T76)) → U19_g(T76, tree67_in_)
tree67_in_ → tree67_out_
U19_g(T76, tree67_out_) → goal1_out_g(s(T76))
U19_g(T76, tree67_out_) → U20_g(T76, tree67_in_)
U20_g(T76, tree67_out_) → goal1_out_g(s(T76))
goal1_in_g(0) → U21_g(tree67_in_)
U21_g(tree67_out_) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2)  =  U7_g(x2)
s2t9_in_ga(x1, x2)  =  s2t9_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)
s2t9_out_ga(x1, x2)  =  s2t9_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
goal1_out_g(x1)  =  goal1_out_g
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
tree36_in_g(x1)  =  tree36_in_g(x1)
nil  =  nil
tree36_out_g(x1)  =  tree36_out_g
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
U5_g(x1, x2, x3, x4)  =  U5_g(x3, x4)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
U10_g(x1, x2)  =  U10_g(x2)
U11_g(x1, x2, x3)  =  U11_g(x2, x3)
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, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tree67_in_  =  tree67_in_
tree67_out_  =  tree67_out_
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1)  =  U21_g(x1)
S2T9_IN_GA(x1, x2)  =  S2T9_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:

S2T9_IN_GA(s(T22), node(nil, X98, X99)) → S2T9_IN_GA(T22, X99)
S2T9_IN_GA(s(T16), node(X65, X66, X65)) → S2T9_IN_GA(T16, X65)
S2T9_IN_GA(s(T28), node(X131, X132, nil)) → S2T9_IN_GA(T28, X131)

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

(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:

S2T9_IN_GA(s(T22)) → S2T9_IN_GA(T22)

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:

• S2T9_IN_GA(s(T22)) → S2T9_IN_GA(T22)
  The graph contains the following edges 1 > 1

(22) YES