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

(0) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tree(T)).
tree(nil) :- !.
tree(X) :- ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R)))).
s2t(0, L) :- ','(!, eq(L, nil)).
s2t(X, node(T, X1, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X2, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X3, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, nil)).
left(nil, nil).
left(node(L, X5, X6), L).
right(nil, nil).
right(node(X7, X8, R), R).
p(0, 0).
p(s(X), X).
eq(X, X).

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: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2t(T2, X11), tree(X11))\n\nT2 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2t(T2, X11)\n\nT2 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: tree(T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2t(T2, X11), SCOPE: 2, CLAUSE: 3 | s2t(T2, X11), SCOPE: 2, CLAUSE: 4 | s2t(T2, X11), SCOPE: 2, CLAUSE: 5 | s2t(T2, X11), SCOPE: 2, CLAUSE: 6 | s2t(T2, X11), SCOPE: 2, CLAUSE: 7 | ?_2\n\nT2 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, eq(X14, nil)) | s2t(0, X11), SCOPE: 2, CLAUSE: 4 | s2t(0, X11), SCOPE: 2, CLAUSE: 5 | s2t(0, X11), SCOPE: 2, CLAUSE: 6 | s2t(0, X11), SCOPE: 2, CLAUSE: 7 | ?_2\n\nX11 is free\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2t(T2, X11), SCOPE: 2, CLAUSE: 4 | s2t(T2, X11), SCOPE: 2, CLAUSE: 5 | s2t(T2, X11), SCOPE: 2, CLAUSE: 6 | s2t(T2, X11), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(X14, nil)\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(X14, nil), SCOPE: 3, CLAUSE: 14\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: s2t(T2, X11), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: s2t(T2, X11), SCOPE: 2, CLAUSE: 5 | s2t(T2, X11), SCOPE: 2, CLAUSE: 6 | s2t(T2, X11), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T4, X27), s2t(X27, X28))\n\nT4 is ground\nX11 is free; \nX13 is free; \nX28 is free; \nX27 is free; \ns2t(T4, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T4, X27), s2t(X27, X28)), SCOPE: 4, CLAUSE: 12 | ','(p(T4, X27), s2t(X27, X28)), SCOPE: 4, CLAUSE: 13\n\nT4 is ground\nX11 is free; \nX13 is free; \nX28 is free; \nX27 is free; \ns2t(T4, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T4, X27), s2t(X27, X28)), SCOPE: 4, CLAUSE: 13\n\nT4 is ground\nX11 is free; \nX13 is free; \nX28 is free; \nX27 is free; \ns2t(T4, X11) !=? s2t(0, X13)\np(T4, X27) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: s2t(T5, X28)\n\nT5 is ground\nX11 is free; \nX13 is free; \nX28 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\nX11 is free\nX13 is free; \nX27 is free; \nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: s2t(T2, X11), SCOPE: 2, CLAUSE: 5\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2t(T2, X11), SCOPE: 2, CLAUSE: 6 | s2t(T2, X11), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T6, X41), s2t(X41, X43))\n\nT6 is ground\nX11 is free; \nX13 is free; \nX43 is free; \nX41 is free; \ns2t(T6, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T6, X41), s2t(X41, X43)), SCOPE: 5, CLAUSE: 12 | ','(p(T6, X41), s2t(X41, X43)), SCOPE: 5, CLAUSE: 13\n\nT6 is ground\nX11 is free; \nX13 is free; \nX43 is free; \nX41 is free; \ns2t(T6, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T6, X41), s2t(X41, X43)), SCOPE: 5, CLAUSE: 13\n\nT6 is ground\nX11 is free; \nX13 is free; \nX43 is free; \nX41 is free; \ns2t(T6, X11) !=? s2t(0, X13)\np(T6, X41) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2t(T7, X43)\n\nT7 is ground\nX11 is free; \nX13 is free; \nX43 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\nX11 is free\nX13 is free; \nX41 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: s2t(T2, X11), SCOPE: 2, CLAUSE: 6\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: s2t(T2, X11), SCOPE: 2, CLAUSE: 7\n\nT2 is ground\nX11 is free; \nX13 is free; \ns2t(T2, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T8, X55), s2t(X55, X56))\n\nT8 is ground\nX11 is free; \nX13 is free; \nX56 is free; \nX55 is free; \ns2t(T8, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T8, X55), s2t(X55, X56)), SCOPE: 6, CLAUSE: 12 | ','(p(T8, X55), s2t(X55, X56)), SCOPE: 6, CLAUSE: 13\n\nT8 is ground\nX11 is free; \nX13 is free; \nX56 is free; \nX55 is free; \ns2t(T8, X11) !=? s2t(0, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(T8, X55), s2t(X55, X56)), SCOPE: 6, CLAUSE: 13\n\nT8 is ground\nX11 is free; \nX13 is free; \nX56 is free; \nX55 is free; \ns2t(T8, X11) !=? s2t(0, X13)\np(T8, X55) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: s2t(T9, X56)\n\nT9 is ground\nX11 is free; \nX13 is free; \nX56 is free; \nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\nX11 is free\nX13 is free; \nX55 is free; \nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\nX11 is free\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: tree(T3), SCOPE: 7, CLAUSE: 1 | tree(T3), SCOPE: 7, CLAUSE: 2 | ?_7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: !_7 | tree(T3), SCOPE: 7, CLAUSE: 2 | ?_7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: tree(T3), SCOPE: 7, CLAUSE: 2\n\ntree(T3) !=? tree(nil)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69))))\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 8 | ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(right(nil, X69), ','(tree(nil), tree(X69))) | ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(right(nil, X69), ','(tree(nil), tree(X69))), SCOPE: 9, CLAUSE: 10 | ','(right(nil, X69), ','(tree(nil), tree(X69))), SCOPE: 9, CLAUSE: 11 | ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tree(nil), tree(nil)) | ','(right(nil, X69), ','(tree(nil), tree(X69))), SCOPE: 9, CLAUSE: 11 | ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tree(nil), tree(nil))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(right(nil, X69), ','(tree(nil), tree(X69))), SCOPE: 9, CLAUSE: 11 | ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: tree(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: tree(nil), SCOPE: 10, CLAUSE: 1 | tree(nil), SCOPE: 10, CLAUSE: 2 | ?_10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: !_10 | tree(nil), SCOPE: 10, CLAUSE: 2 | ?_10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))), SCOPE: 8, CLAUSE: 9\n\nX67 is free\nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(right(node(X79, X80, X81), X69), ','(tree(X79), tree(X69)))\n\nX69 is free\nX79 is free; \nX80 is free; \nX81 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(right(node(X79, X80, X81), X69), ','(tree(X79), tree(X69))), SCOPE: 11, CLAUSE: 10 | ','(right(node(X79, X80, X81), X69), ','(tree(X79), tree(X69))), SCOPE: 11, CLAUSE: 11\n\nX69 is free\nX79 is free; \nX80 is free; \nX81 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(right(node(X79, X80, X81), X69), ','(tree(X79), tree(X69))), SCOPE: 11, CLAUSE: 11\n\nX69 is free\nX79 is free; \nX80 is free; \nX81 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tree(X88), tree(X90))\n\nX88 is free\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: tree(X88)\n\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: tree(X90)\n\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: tree(X90), SCOPE: 12, CLAUSE: 1 | tree(X90), SCOPE: 12, CLAUSE: 2 | ?_12\n\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: !_12 | tree(X90), SCOPE: 12, CLAUSE: 2 | ?_12\n\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 65 [arrowhead = none ];
65 -> 2;

66 [label="EVAL with clause\ngoal(X10) :- ','(s2t(X10, X11), tree(X11)).\nand substitution\nT1 -> T2\nX10 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 66 [arrowhead = none ];
66 -> 3;

67 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 67 [arrowhead = none ];
67 -> 4;

68 [label="SPLIT 2\nreplacements:\nX11 -> T3", fontsize=14, style = filled, fillcolor = violet];
3 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 69 [arrowhead = none ];
69 -> 6;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 70 [arrowhead = none ];
70 -> 36;

71 [label="EVAL with clause\ns2t(0, X13) :- ','(!, eq(X13, nil)).\nand substitution\nT2 -> 0\nX11 -> X14\nX13 -> X14", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 71 [arrowhead = none ];
71 -> 7;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 72 [arrowhead = none ];
72 -> 8;

73 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 73 [arrowhead = none ];
73 -> 9;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 74 [arrowhead = none ];
74 -> 13;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 75 [arrowhead = none ];
75 -> 14;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="EVAL with clause\neq(X16, X16).\nand substitution\nX14 -> nil\nX16 -> nil\nX17 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="EVAL with clause\ns2t(X24, node(X25, X26, X25)) :- ','(p(X24, X27), s2t(X27, X25)).\nand substitution\nT2 -> T4\nX24 -> T4\nX25 -> X28\nX26 -> X29\nX11 -> node(X28, X29, X28)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 79 [arrowhead = none ];
79 -> 15;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 80 [arrowhead = none ];
80 -> 20;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 81 [arrowhead = none ];
81 -> 21;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 82 [arrowhead = none ];
82 -> 16;

83 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T4, X11), s2t(0, X13))", fontsize=14, style = filled, fillcolor = white];
16 -> 83 [arrowhead = none ];
83 -> 17;

84 [label="EVAL with clause\np(s(X31), X31).\nand substitution\nX31 -> T5\nT4 -> s(T5)\nX27 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 84 [arrowhead = none ];
84 -> 18;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 85 [arrowhead = none ];
85 -> 19;

86 [label="INSTANCE with matching:\nT2 -> T5\nX11 -> X28", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 86 [arrowhead = none , style=dashed];
86 -> 4[style=dashed];

87 [label="EVAL with clause\ns2t(X38, node(nil, X39, X40)) :- ','(p(X38, X41), s2t(X41, X40)).\nand substitution\nT2 -> T6\nX38 -> T6\nX39 -> X42\nX40 -> X43\nX11 -> node(nil, X42, X43)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 87 [arrowhead = none ];
87 -> 22;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 88 [arrowhead = none ];
88 -> 27;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 89 [arrowhead = none ];
89 -> 28;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 90 [arrowhead = none ];
90 -> 23;

91 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T6, X11), s2t(0, X13))", fontsize=14, style = filled, fillcolor = white];
23 -> 91 [arrowhead = none ];
91 -> 24;

92 [label="EVAL with clause\np(s(X45), X45).\nand substitution\nX45 -> T7\nT6 -> s(T7)\nX41 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 92 [arrowhead = none ];
92 -> 25;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 93 [arrowhead = none ];
93 -> 26;

94 [label="INSTANCE with matching:\nT2 -> T7\nX11 -> X43", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 94 [arrowhead = none , style=dashed];
94 -> 4[style=dashed];

95 [label="EVAL with clause\ns2t(X52, node(X53, X54, nil)) :- ','(p(X52, X55), s2t(X55, X53)).\nand substitution\nT2 -> T8\nX52 -> T8\nX53 -> X56\nX54 -> X57\nX11 -> node(X56, X57, nil)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 95 [arrowhead = none ];
95 -> 29;

96 [label="EVAL with clause\ns2t(X62, node(nil, X63, nil)).\nand substitution\nT2 -> T10\nX62 -> T10\nX63 -> X64\nX11 -> node(nil, X64, nil)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 96 [arrowhead = none ];
96 -> 34;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 97 [arrowhead = none ];
97 -> 30;

98 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2t(T8, X11), s2t(0, X13))", fontsize=14, style = filled, fillcolor = white];
30 -> 98 [arrowhead = none ];
98 -> 31;

99 [label="EVAL with clause\np(s(X59), X59).\nand substitution\nX59 -> T9\nT8 -> s(T9)\nX55 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 99 [arrowhead = none ];
99 -> 32;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 100 [arrowhead = none ];
100 -> 33;

101 [label="INSTANCE with matching:\nT2 -> T9\nX11 -> X56", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 101 [arrowhead = none , style=dashed];
101 -> 4[style=dashed];

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 102 [arrowhead = none ];
102 -> 35;

103 [label="EVAL with clause\ntree(nil) :- !.\nand substitution\nT3 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 103 [arrowhead = none ];
103 -> 37;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 104 [arrowhead = none ];
104 -> 38;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 105 [arrowhead = none ];
105 -> 39;

106 [label="EVAL with clause\ntree(X66) :- ','(left(X67, X68), ','(right(X67, X69), ','(tree(X68), tree(X69)))).\nand substitution\nT3 -> T11\nX66 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 106 [arrowhead = none ];
106 -> 41;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 107 [arrowhead = none ];
107 -> 40;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="EVAL with clause\nleft(nil, nil).\nand substitution\nX67 -> nil\nX68 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 110 [arrowhead = none ];
110 -> 44;

111 [label="EVAL with clause\nright(nil, nil).\nand substitution\nX69 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 112 [arrowhead = none ];
112 -> 46;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 113 [arrowhead = none ];
113 -> 47;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 114 [arrowhead = none ];
114 -> 48;

115 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
46 -> 115 [arrowhead = none ];
115 -> 49;

116 [label="BACKTRACK\nfor clause: right(node(X7, X8, R), R)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 116 [arrowhead = none ];
116 -> 54;

117 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 117 [arrowhead = none , style=dashed];
117 -> 49[style=dashed];

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 118 [arrowhead = none ];
118 -> 50;

119 [label="EVAL with clause\ntree(nil) :- !.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 119 [arrowhead = none ];
119 -> 51;

120 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 120 [arrowhead = none ];
120 -> 52;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 121 [arrowhead = none ];
121 -> 53;

122 [label="EVAL with clause\nleft(node(X76, X77, X78), X76).\nand substitution\nX76 -> X79\nX77 -> X80\nX78 -> X81\nX67 -> node(X79, X80, X81)\nX68 -> X79", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 122 [arrowhead = none ];
122 -> 55;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 123 [arrowhead = none ];
123 -> 56;

124 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
56 -> 124 [arrowhead = none ];
124 -> 57;

125 [label="EVAL with clause\nright(node(X85, X86, X87), X87).\nand substitution\nX79 -> X88\nX85 -> X88\nX80 -> X89\nX86 -> X89\nX81 -> X90\nX87 -> X90\nX69 -> X90", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 125 [arrowhead = none ];
125 -> 58;

126 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 126 [arrowhead = none ];
126 -> 59;

127 [label="SPLIT 2\nreplacements:\nX88 -> T12", fontsize=14, style = filled, fillcolor = violet];
58 -> 127 [arrowhead = none ];
127 -> 60;

128 [label="INSTANCE with matching:\nX90 -> X88", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 128 [arrowhead = none , style=dashed];
128 -> 60[style=dashed];

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 129 [arrowhead = none ];
129 -> 61;

130 [label="EVAL with clause\ntree(nil) :- !.\nand substitution\nX90 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 130 [arrowhead = none ];
130 -> 62;

131 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 131 [arrowhead = none ];
131 -> 63;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 132 [arrowhead = none ];
132 -> 64;

}

(2) Obligation:

Clauses:

s2t4(0, nil).
s2t4(s(T5), node(X28, X29, X28)) :- s2t4(T5, X28).
s2t4(s(T7), node(nil, X42, X43)) :- s2t4(T7, X43).
s2t4(s(T9), node(X56, X57, nil)) :- s2t4(T9, X56).
s2t4(T10, node(nil, X64, nil)).
tree49.
tree60(nil).
goal1(T2) :- s2t4(T2, X11).
goal1(T2) :- s2t4(T2, nil).
goal1(T2) :- ','(s2t4(T2, T11), tree49).
goal1(T2) :- ','(s2t4(T2, T11), tree49).
goal1(T2) :- ','(s2t4(T2, T11), tree60(X88)).
goal1(T2) :- ','(s2t4(T2, T11), tree60(T12)).

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)
s2t4_in: (b,f) (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)

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(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)

(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(T2) → U4_G(T2, s2t4_in_ga(T2, X11))
GOAL1_IN_G(T2) → S2T4_IN_GA(T2, X11)
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → U1_GA(T5, X28, X29, s2t4_in_ga(T5, X28))
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → S2T4_IN_GA(T5, X28)
S2T4_IN_GA(s(T7), node(nil, X42, X43)) → U2_GA(T7, X42, X43, s2t4_in_ga(T7, X43))
S2T4_IN_GA(s(T7), node(nil, X42, X43)) → S2T4_IN_GA(T7, X43)
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → U3_GA(T9, X56, X57, s2t4_in_ga(T9, X56))
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → S2T4_IN_GA(T9, X56)
GOAL1_IN_G(T2) → U5_G(T2, s2t4_in_gg(T2, nil))
GOAL1_IN_G(T2) → S2T4_IN_GG(T2, nil)
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → U1_GG(T5, X28, X29, s2t4_in_gg(T5, X28))
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → S2T4_IN_GG(T5, X28)
S2T4_IN_GG(s(T7), node(nil, X42, X43)) → U2_GG(T7, X42, X43, s2t4_in_gg(T7, X43))
S2T4_IN_GG(s(T7), node(nil, X42, X43)) → S2T4_IN_GG(T7, X43)
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → U3_GG(T9, X56, X57, s2t4_in_gg(T9, X56))
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → S2T4_IN_GG(T9, X56)
GOAL1_IN_G(T2) → U6_G(T2, s2t4_in_ga(T2, T11))
U6_G(T2, s2t4_out_ga(T2, T11)) → U7_G(T2, tree49_in_)
U6_G(T2, s2t4_out_ga(T2, T11)) → TREE49_IN_
U6_G(T2, s2t4_out_ga(T2, T11)) → U8_G(T2, tree60_in_a(X88))
U6_G(T2, s2t4_out_ga(T2, T11)) → TREE60_IN_A(X88)
U6_G(T2, s2t4_out_ga(T2, T11)) → U9_G(T2, tree60_in_a(T12))

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_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)
U5_G(x1, x2)  =  U5_G(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
TREE49_IN_  =  TREE49_IN_
U8_G(x1, x2)  =  U8_G(x2)
TREE60_IN_A(x1)  =  TREE60_IN_A
U9_G(x1, x2)  =  U9_G(x2)

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(T2) → U4_G(T2, s2t4_in_ga(T2, X11))
GOAL1_IN_G(T2) → S2T4_IN_GA(T2, X11)
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → U1_GA(T5, X28, X29, s2t4_in_ga(T5, X28))
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → S2T4_IN_GA(T5, X28)
S2T4_IN_GA(s(T7), node(nil, X42, X43)) → U2_GA(T7, X42, X43, s2t4_in_ga(T7, X43))
S2T4_IN_GA(s(T7), node(nil, X42, X43)) → S2T4_IN_GA(T7, X43)
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → U3_GA(T9, X56, X57, s2t4_in_ga(T9, X56))
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → S2T4_IN_GA(T9, X56)
GOAL1_IN_G(T2) → U5_G(T2, s2t4_in_gg(T2, nil))
GOAL1_IN_G(T2) → S2T4_IN_GG(T2, nil)
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → U1_GG(T5, X28, X29, s2t4_in_gg(T5, X28))
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → S2T4_IN_GG(T5, X28)
S2T4_IN_GG(s(T7), node(nil, X42, X43)) → U2_GG(T7, X42, X43, s2t4_in_gg(T7, X43))
S2T4_IN_GG(s(T7), node(nil, X42, X43)) → S2T4_IN_GG(T7, X43)
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → U3_GG(T9, X56, X57, s2t4_in_gg(T9, X56))
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → S2T4_IN_GG(T9, X56)
GOAL1_IN_G(T2) → U6_G(T2, s2t4_in_ga(T2, T11))
U6_G(T2, s2t4_out_ga(T2, T11)) → U7_G(T2, tree49_in_)
U6_G(T2, s2t4_out_ga(T2, T11)) → TREE49_IN_
U6_G(T2, s2t4_out_ga(T2, T11)) → U8_G(T2, tree60_in_a(X88))
U6_G(T2, s2t4_out_ga(T2, T11)) → TREE60_IN_A(X88)
U6_G(T2, s2t4_out_ga(T2, T11)) → U9_G(T2, tree60_in_a(T12))

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_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)
U5_G(x1, x2)  =  U5_G(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
TREE49_IN_  =  TREE49_IN_
U8_G(x1, x2)  =  U8_G(x2)
TREE60_IN_A(x1)  =  TREE60_IN_A
U9_G(x1, x2)  =  U9_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 16 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

S2T4_IN_GG(s(T7), node(nil, X42, X43)) → S2T4_IN_GG(T7, X43)
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → S2T4_IN_GG(T5, X28)
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → S2T4_IN_GG(T9, X56)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)

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:

S2T4_IN_GG(s(T7), node(nil, X42, X43)) → S2T4_IN_GG(T7, X43)
S2T4_IN_GG(s(T5), node(X28, X29, X28)) → S2T4_IN_GG(T5, X28)
S2T4_IN_GG(s(T9), node(X56, X57, nil)) → S2T4_IN_GG(T9, X56)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)

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:

S2T4_IN_GG(s(T7), node(nil, X43)) → S2T4_IN_GG(T7, X43)
S2T4_IN_GG(s(T5), node(X28, X28)) → S2T4_IN_GG(T5, X28)
S2T4_IN_GG(s(T9), node(X56, nil)) → S2T4_IN_GG(T9, X56)

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:

• S2T4_IN_GG(s(T7), node(nil, X43)) → S2T4_IN_GG(T7, X43)
  The graph contains the following edges 1 > 1, 2 > 2

• S2T4_IN_GG(s(T5), node(X28, X28)) → S2T4_IN_GG(T5, X28)
  The graph contains the following edges 1 > 1, 2 > 2

• S2T4_IN_GG(s(T9), node(X56, nil)) → S2T4_IN_GG(T9, X56)
  The graph contains the following edges 1 > 1, 2 > 2

(15) TRUE

(16) Obligation:

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

S2T4_IN_GA(s(T7), node(nil, X42, X43)) → S2T4_IN_GA(T7, X43)
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → S2T4_IN_GA(T5, X28)
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → S2T4_IN_GA(T9, X56)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2t4_in_ga(T2, X11))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T5), node(X28, X29, X28)) → U1_ga(T5, X28, X29, s2t4_in_ga(T5, X28))
s2t4_in_ga(s(T7), node(nil, X42, X43)) → U2_ga(T7, X42, X43, s2t4_in_ga(T7, X43))
s2t4_in_ga(s(T9), node(X56, X57, nil)) → U3_ga(T9, X56, X57, s2t4_in_ga(T9, X56))
s2t4_in_ga(T10, node(nil, X64, nil)) → s2t4_out_ga(T10, node(nil, X64, nil))
U3_ga(T9, X56, X57, s2t4_out_ga(T9, X56)) → s2t4_out_ga(s(T9), node(X56, X57, nil))
U2_ga(T7, X42, X43, s2t4_out_ga(T7, X43)) → s2t4_out_ga(s(T7), node(nil, X42, X43))
U1_ga(T5, X28, X29, s2t4_out_ga(T5, X28)) → s2t4_out_ga(s(T5), node(X28, X29, X28))
U4_g(T2, s2t4_out_ga(T2, X11)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2t4_in_gg(T2, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T5), node(X28, X29, X28)) → U1_gg(T5, X28, X29, s2t4_in_gg(T5, X28))
s2t4_in_gg(s(T7), node(nil, X42, X43)) → U2_gg(T7, X42, X43, s2t4_in_gg(T7, X43))
s2t4_in_gg(s(T9), node(X56, X57, nil)) → U3_gg(T9, X56, X57, s2t4_in_gg(T9, X56))
s2t4_in_gg(T10, node(nil, X64, nil)) → s2t4_out_gg(T10, node(nil, X64, nil))
U3_gg(T9, X56, X57, s2t4_out_gg(T9, X56)) → s2t4_out_gg(s(T9), node(X56, X57, nil))
U2_gg(T7, X42, X43, s2t4_out_gg(T7, X43)) → s2t4_out_gg(s(T7), node(nil, X42, X43))
U1_gg(T5, X28, X29, s2t4_out_gg(T5, X28)) → s2t4_out_gg(s(T5), node(X28, X29, X28))
U5_g(T2, s2t4_out_gg(T2, nil)) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2t4_in_ga(T2, T11))
U6_g(T2, s2t4_out_ga(T2, T11)) → U7_g(T2, tree49_in_)
tree49_in_ → tree49_out_
U7_g(T2, tree49_out_) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U8_g(T2, tree60_in_a(X88))
tree60_in_a(nil) → tree60_out_a(nil)
U8_g(T2, tree60_out_a(X88)) → goal1_out_g(T2)
U6_g(T2, s2t4_out_ga(T2, T11)) → U9_g(T2, tree60_in_a(T12))
U9_g(T2, tree60_out_a(T12)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x2)
s(x1)  =  s(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)
node(x1, x2, x3)  =  node(x1, x3)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
tree49_in_  =  tree49_in_
tree49_out_  =  tree49_out_
U8_g(x1, x2)  =  U8_g(x2)
tree60_in_a(x1)  =  tree60_in_a
tree60_out_a(x1)  =  tree60_out_a(x1)
U9_g(x1, x2)  =  U9_g(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_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:

S2T4_IN_GA(s(T7), node(nil, X42, X43)) → S2T4_IN_GA(T7, X43)
S2T4_IN_GA(s(T5), node(X28, X29, X28)) → S2T4_IN_GA(T5, X28)
S2T4_IN_GA(s(T9), node(X56, X57, nil)) → S2T4_IN_GA(T9, X56)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T4_IN_GA(x1, x2)  =  S2T4_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:

S2T4_IN_GA(s(T7)) → S2T4_IN_GA(T7)

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:

• S2T4_IN_GA(s(T7)) → S2T4_IN_GA(T7)
  The graph contains the following edges 1 > 1

(22) TRUE