Left Termination proof of /tmp/tmpyAXpwa/search

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 8 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 218 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 8 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇒, 2 ms)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔, 0 ms)

        ↳29 YES

(0) Obligation:

Clauses:

search_tree(void).
search_tree(T) :- search_tree(T, X1, X2).
search_tree(tree(X, void, void), X, X).
search_tree(tree(X, void, Right), X, Max) :- ','(search_tree(Right, Min, Max), less(X, Min)).
search_tree(tree(X, Left, void), Min, X) :- ','(search_tree(Left, Min, Max), less(Max, X)).
search_tree(tree(X, Left, Right), Min1, Max2) :- ','(search_tree(Left, Min1, Max1), ','(less(Max1, X), ','(search_tree(Right, Min2, Max2), less(X, Min2)))).
less(0, s(X3)).
less(s(X), s(Y)) :- less(X, Y).

Query: search_tree(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: search_tree(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: search_tree(T1), SCOPE: 1, CLAUSE: 0 | search_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | search_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: search_tree(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nsearch_tree(T1) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: search_tree(void), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: search_tree(void, X6, X7)\n\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 2 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 3 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 4 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 5\n\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 3 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 4 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 5\n\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 4 | search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 5\n\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: search_tree(void, X6, X7), SCOPE: 2, CLAUSE: 5\n\nX6 is free\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: search_tree(T3, X22, X23)\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 2 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 3 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 4 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 2\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 3 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 4 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 3\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 4 | search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(search_tree(T18, X50, X51), less(T17, X50))\n\nT17 is ground\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: search_tree(T18, X50, X51)\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T17, T21)\n\nT17 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 2 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 3 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 4 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 2\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 3 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 4 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 3\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 4 | search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(search_tree(T39, X82, X83), less(T38, X82))\n\nT38 is ground\nT39 is ground\nX83 is free\nX82 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 4\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: search_tree(T18, X50, X51), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nX51 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(search_tree(T51, X108, X107), less(X107, T50))\n\nT50 is ground\nT51 is ground\nX108 is free\nX107 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: search_tree(T51, X108, X107)\n\nT51 is ground\nX108 is free\nX107 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T55, T50)\n\nT50 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(search_tree(T67, X138, X136), ','(less(X136, T66), ','(search_tree(T68, X137, X139), less(T66, X137))))\n\nT66 is ground\nT67 is ground\nT68 is ground\nX138 is free\nX139 is free\nX136 is free\nX137 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: search_tree(T67, X138, X136)\n\nT67 is ground\nX138 is free\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(less(T72, T66), ','(search_tree(T68, X137, X139), less(T66, X137)))\n\nT66 is ground\nT68 is ground\nT72 is ground\nX139 is free\nX137 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T72, T66)\n\nT66 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(search_tree(T68, X137, X139), less(T66, X137))\n\nT66 is ground\nT68 is ground\nX139 is free\nX137 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T17, T21), SCOPE: 5, CLAUSE: 6 | less(T17, T21), SCOPE: 5, CLAUSE: 7\n\nT17 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(T17, T21), SCOPE: 5, CLAUSE: 6\n\nT17 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: less(T17, T21), SCOPE: 5, CLAUSE: 7\n\nT17 is ground\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T92, T93)\n\nT92 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 4\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: search_tree(T3, X22, X23), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nX22 is free\nX23 is free\nsearch_tree(T3) !=? search_tree(void)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(search_tree(T105, X187, X186), less(X186, T104))\n\nT104 is ground\nT105 is ground\nX187 is free\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(search_tree(T115, X213, X211), ','(less(X211, T114), ','(search_tree(T116, X212, X214), less(T114, X212))))\n\nT114 is ground\nT115 is ground\nT116 is ground\nX213 is free\nX214 is free\nX211 is free\nX212 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 61 [arrowhead = none ];
61 -> 2;

62 [label="EVAL with clause\nsearch_tree(void).\nand substitutionT1 -> void", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 62 [arrowhead = none ];
62 -> 3;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 63 [arrowhead = none ];
63 -> 4;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 64 [arrowhead = none ];
64 -> 5;

65 [label="ONLY EVAL with clause\nsearch_tree(X21) :- search_tree(X21, X22, X23).\nand substitutionT1 -> T3,\nX21 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 65 [arrowhead = none ];
65 -> 12;

66 [label="ONLY EVAL with clause\nsearch_tree(X5) :- search_tree(X5, X6, X7).\nand substitutionX5 -> void", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 66 [arrowhead = none ];
66 -> 6;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 67 [arrowhead = none ];
67 -> 7;

68 [label="BACKTRACK\nfor clause: search_tree(tree(X, void, void), X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 68 [arrowhead = none ];
68 -> 8;

69 [label="BACKTRACK\nfor clause: search_tree(tree(X, void, Right), X, Max) :- ','(search_tree(Right, Min, Max), less(X, Min))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 69 [arrowhead = none ];
69 -> 9;

70 [label="BACKTRACK\nfor clause: search_tree(tree(X, Left, void), Min, X) :- ','(search_tree(Left, Min, Max), less(Max, X))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 70 [arrowhead = none ];
70 -> 10;

71 [label="BACKTRACK\nfor clause: search_tree(tree(X, Left, Right), Min1, Max2) :- ','(search_tree(Left, Min1, Max1), ','(less(Max1, X), ','(search_tree(Right, Min2, Max2), less(X, Min2))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 71 [arrowhead = none ];
71 -> 11;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 72 [arrowhead = none ];
72 -> 13;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 74 [arrowhead = none ];
74 -> 15;

75 [label="EVAL with clause\nsearch_tree(tree(X28, void, void), X28, X28).\nand substitutionX28 -> T8,\nT3 -> tree(T8, void, void),\nX22 -> T8,\nX23 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 75 [arrowhead = none ];
75 -> 16;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 76 [arrowhead = none ];
76 -> 17;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 77 [arrowhead = none ];
77 -> 19;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 78 [arrowhead = none ];
78 -> 20;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 79 [arrowhead = none ];
79 -> 18;

80 [label="EVAL with clause\nsearch_tree(tree(X47, void, X48), X47, X49) :- ','(search_tree(X48, X50, X49), less(X47, X50)).\nand substitutionX47 -> T17,\nX48 -> T18,\nT3 -> tree(T17, void, T18),\nX22 -> T17,\nX23 -> X51,\nX49 -> X51", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 82 [arrowhead = none ];
82 -> 55;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 83 [arrowhead = none ];
83 -> 56;

84 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 84 [arrowhead = none ];
84 -> 23;

85 [label="SPLIT 2\nnew knowledge:\nT18 is ground\nT21 is ground\nT22 is ground\nreplacements:X50 -> T21,\nX51 -> T22", fontsize=14, style = filled, fillcolor = violet];
21 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 87 [arrowhead = none ];
87 -> 47;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 89 [arrowhead = none ];
89 -> 27;

90 [label="EVAL with clause\nsearch_tree(tree(X60, void, void), X60, X60).\nand substitutionX60 -> T29,\nT18 -> tree(T29, void, void),\nX50 -> T29,\nX51 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 90 [arrowhead = none ];
90 -> 28;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 91 [arrowhead = none ];
91 -> 29;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="EVAL with clause\nsearch_tree(tree(X79, void, X80), X79, X81) :- ','(search_tree(X80, X82, X81), less(X79, X82)).\nand substitutionX79 -> T38,\nX80 -> T39,\nT18 -> tree(T38, void, T39),\nX50 -> T38,\nX51 -> X83,\nX81 -> X83", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 95 [arrowhead = none ];
95 -> 33;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 96 [arrowhead = none ];
96 -> 34;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 97 [arrowhead = none ];
97 -> 35;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 98 [arrowhead = none ];
98 -> 36;

99 [label="INSTANCE with matching:\nT18 -> T39\nX50 -> X82\nX51 -> X83\nT17 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 99 [arrowhead = none , style=dashed];
99 -> 21[style=dashed];

100 [label="EVAL with clause\nsearch_tree(tree(X104, X105, void), X106, X104) :- ','(search_tree(X105, X106, X107), less(X107, X104)).\nand substitutionX104 -> T50,\nX105 -> T51,\nT18 -> tree(T50, T51, void),\nX50 -> X108,\nX106 -> X108,\nX51 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 37;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="EVAL with clause\nsearch_tree(tree(X131, X132, X133), X134, X135) :- ','(search_tree(X132, X134, X136), ','(less(X136, X131), ','(search_tree(X133, X137, X135), less(X131, X137)))).\nand substitutionX131 -> T66,\nX132 -> T67,\nX133 -> T68,\nT18 -> tree(T66, T67, T68),\nX50 -> X138,\nX134 -> X138,\nX51 -> X139,\nX135 -> X139", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 102 [arrowhead = none ];
102 -> 41;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 103 [arrowhead = none ];
103 -> 42;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 104 [arrowhead = none ];
104 -> 39;

105 [label="SPLIT 2\nnew knowledge:\nT51 is ground\nT54 is ground\nT55 is ground\nreplacements:X108 -> T54,\nX107 -> T55", fontsize=14, style = filled, fillcolor = violet];
37 -> 105 [arrowhead = none ];
105 -> 40;

106 [label="INSTANCE with matching:\nT18 -> T51\nX50 -> X108\nX51 -> X107", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 106 [arrowhead = none , style=dashed];
106 -> 23[style=dashed];

107 [label="INSTANCE with matching:\nT17 -> T55\nT21 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 107 [arrowhead = none , style=dashed];
107 -> 24[style=dashed];

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 108 [arrowhead = none ];
108 -> 43;

109 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nT71 is ground\nT72 is ground\nreplacements:X138 -> T71,\nX136 -> T72", fontsize=14, style = filled, fillcolor = violet];
41 -> 109 [arrowhead = none ];
109 -> 44;

110 [label="INSTANCE with matching:\nT18 -> T67\nX50 -> X138\nX51 -> X136", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 110 [arrowhead = none , style=dashed];
110 -> 23[style=dashed];

111 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nT66 is ground", fontsize=14, style = filled, fillcolor = violet];
44 -> 112 [arrowhead = none ];
112 -> 46;

113 [label="INSTANCE with matching:\nT17 -> T72\nT21 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 113 [arrowhead = none , style=dashed];
113 -> 24[style=dashed];

114 [label="INSTANCE with matching:\nT18 -> T68\nX50 -> X137\nX51 -> X139\nT17 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 114 [arrowhead = none , style=dashed];
114 -> 21[style=dashed];

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 115 [arrowhead = none ];
115 -> 48;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 116 [arrowhead = none ];
116 -> 49;

117 [label="EVAL with clause\nless(0, s(X156)).\nand substitutionT17 -> 0,\nX156 -> T87,\nT21 -> s(T87)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 117 [arrowhead = none ];
117 -> 50;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 118 [arrowhead = none ];
118 -> 51;

119 [label="EVAL with clause\nless(s(X161), s(X162)) :- less(X161, X162).\nand substitutionX161 -> T92,\nT17 -> s(T92),\nX162 -> T93,\nT21 -> s(T93)", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 119 [arrowhead = none ];
119 -> 53;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 120 [arrowhead = none ];
120 -> 54;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 121 [arrowhead = none ];
121 -> 52;

122 [label="INSTANCE with matching:\nT17 -> T92\nT21 -> T93", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 122 [arrowhead = none , style=dashed];
122 -> 24[style=dashed];

123 [label="EVAL with clause\nsearch_tree(tree(X183, X184, void), X185, X183) :- ','(search_tree(X184, X185, X186), less(X186, X183)).\nand substitutionX183 -> T104,\nX184 -> T105,\nT3 -> tree(T104, T105, void),\nX22 -> X187,\nX185 -> X187,\nX23 -> T104", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 123 [arrowhead = none ];
123 -> 57;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 124 [arrowhead = none ];
124 -> 58;

125 [label="EVAL with clause\nsearch_tree(tree(X206, X207, X208), X209, X210) :- ','(search_tree(X207, X209, X211), ','(less(X211, X206), ','(search_tree(X208, X212, X210), less(X206, X212)))).\nand substitutionX206 -> T114,\nX207 -> T115,\nX208 -> T116,\nT3 -> tree(T114, T115, T116),\nX22 -> X213,\nX209 -> X213,\nX23 -> X214,\nX210 -> X214", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 125 [arrowhead = none ];
125 -> 59;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 126 [arrowhead = none ];
126 -> 60;

127 [label="INSTANCE with matching:\nT51 -> T105\nX108 -> X187\nX107 -> X186\nT50 -> T104", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 127 [arrowhead = none , style=dashed];
127 -> 37[style=dashed];

128 [label="INSTANCE with matching:\nT67 -> T115\nX138 -> X213\nX136 -> X211\nT66 -> T114\nT68 -> T116\nX137 -> X212\nX139 -> X214", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 128 [arrowhead = none , style=dashed];
128 -> 41[style=dashed];

}

(2) Obligation:

Clauses:

pA(T18, X50, X51, T17) :- search_treeB(T18, X50, X51).
pA(T18, T21, T22, T17) :- ','(search_treeB(T18, T21, T22), lessC(T17, T21)).
search_treeB(tree(T29, void, void), T29, T29).
search_treeB(tree(T38, void, T39), T38, X83) :- pA(T39, X82, X83, T38).
search_treeB(tree(T50, T51, void), X108, T50) :- pD(T51, X108, X107, T50).
search_treeB(tree(T66, T67, T68), X138, X139) :- pE(T67, X138, X136, T66, T68, X137, X139).
lessC(0, s(T87)).
lessC(s(T92), s(T93)) :- lessC(T92, T93).
pD(T51, X108, X107, T50) :- search_treeB(T51, X108, X107).
pD(T51, T54, T55, T50) :- ','(search_treeB(T51, T54, T55), lessC(T55, T50)).
pE(T67, X138, X136, T66, T68, X137, X139) :- search_treeB(T67, X138, X136).
pE(T67, T71, T72, T66, T68, X137, X139) :- ','(search_treeB(T67, T71, T72), lessC(T72, T66)).
pE(T67, T71, T72, T66, T68, X137, X139) :- ','(search_treeB(T67, T71, T72), ','(lessC(T72, T66), pA(T68, X137, X139, T66))).
search_treeF(void).
search_treeF(tree(T8, void, void)).
search_treeF(tree(T17, void, T18)) :- pA(T18, X50, X51, T17).
search_treeF(tree(T104, T105, void)) :- pD(T105, X187, X186, T104).
search_treeF(tree(T114, T115, T116)) :- pE(T115, X213, X211, T114, T116, X212, X214).

Query: search_treeF(g)

(3) PrologToPiTRSProof (SOUND transformation)

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

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

(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:

SEARCH_TREEF_IN_G(tree(T17, void, T18)) → U15_G(T17, T18, pA_in_gaag(T18, X50, X51, T17))
SEARCH_TREEF_IN_G(tree(T17, void, T18)) → PA_IN_GAAG(T18, X50, X51, T17)
PA_IN_GAAG(T18, X50, X51, T17) → U1_GAAG(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
PA_IN_GAAG(T18, X50, X51, T17) → SEARCH_TREEB_IN_GAA(T18, X50, X51)
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → U4_GAA(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → PA_IN_GAAG(T39, X82, X83, T38)
PA_IN_GAAG(T18, T21, T22, T17) → U2_GAAG(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → U5_GAA(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → PD_IN_GAAG(T51, X108, X107, T50)
PD_IN_GAAG(T51, X108, X107, T50) → U8_GAAG(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
PD_IN_GAAG(T51, X108, X107, T50) → SEARCH_TREEB_IN_GAA(T51, X108, X107)
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → U6_GAA(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139)
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → U11_GAAGGAA(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → SEARCH_TREEB_IN_GAA(T67, X138, X136)
PE_IN_GAAGGAA(T67, T71, T72, T66, T68, X137, X139) → U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → LESSC_IN_AG(T72, T66)
LESSC_IN_AG(s(T92), s(T93)) → U7_AG(T92, T93, lessC_in_ag(T92, T93))
LESSC_IN_AG(s(T92), s(T93)) → LESSC_IN_AG(T92, T93)
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → PA_IN_GAAG(T68, X137, X139, T66)
PD_IN_GAAG(T51, T54, T55, T50) → U9_GAAG(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_GAAG(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_GAAG(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U9_GAAG(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → LESSC_IN_AG(T55, T50)
U2_GAAG(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_GAAG(T18, T21, T22, T17, lessC_in_gg(T17, T21))
U2_GAAG(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → LESSC_IN_GG(T17, T21)
LESSC_IN_GG(s(T92), s(T93)) → U7_GG(T92, T93, lessC_in_gg(T92, T93))
LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)
SEARCH_TREEF_IN_G(tree(T104, T105, void)) → U16_G(T104, T105, pD_in_gaag(T105, X187, X186, T104))
SEARCH_TREEF_IN_G(tree(T104, T105, void)) → PD_IN_GAAG(T105, X187, X186, T104)
SEARCH_TREEF_IN_G(tree(T114, T115, T116)) → U17_G(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
SEARCH_TREEF_IN_G(tree(T114, T115, T116)) → PE_IN_GAAGGAA(T115, X213, X211, T114, T116, X212, X214)

The TRS R consists of the following rules:

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

The argument filtering Pi contains the following mapping:
search_treeF_in_g(x1) = search_treeF_in_g(x1)
void = void
search_treeF_out_g(x1) = search_treeF_out_g
tree(x1, x2, x3) = tree(x1, x2, x3)
U15_g(x1, x2, x3) = U15_g(x3)
pA_in_gaag(x1, x2, x3, x4) = pA_in_gaag(x1, x4)
U1_gaag(x1, x2, x3, x4, x5) = U1_gaag(x5)
search_treeB_in_gaa(x1, x2, x3) = search_treeB_in_gaa(x1)
search_treeB_out_gaa(x1, x2, x3) = search_treeB_out_gaa(x2)
U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4)
U2_gaag(x1, x2, x3, x4, x5) = U2_gaag(x4, x5)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x4)
pD_in_gaag(x1, x2, x3, x4) = pD_in_gaag(x1, x4)
U8_gaag(x1, x2, x3, x4, x5) = U8_gaag(x5)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
pE_in_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_in_gaaggaa(x1, x4, x5)
U11_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U11_gaaggaa(x8)
pE_out_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_out_gaaggaa(x2)
U12_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U12_gaaggaa(x4, x5, x8)
U13_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U13_gaaggaa(x2, x4, x5, x8)
lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessC_out_gg(x1, x2) = lessC_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U14_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U14_gaaggaa(x2, x8)
pA_out_gaag(x1, x2, x3, x4) = pA_out_gaag(x2)
lessC_in_ag(x1, x2) = lessC_in_ag(x2)
lessC_out_ag(x1, x2) = lessC_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
pD_out_gaag(x1, x2, x3, x4) = pD_out_gaag(x2)
U9_gaag(x1, x2, x3, x4, x5) = U9_gaag(x4, x5)
U10_gaag(x1, x2, x3, x4, x5) = U10_gaag(x2, x5)
U3_gaag(x1, x2, x3, x4, x5) = U3_gaag(x2, x5)
U16_g(x1, x2, x3) = U16_g(x3)
U17_g(x1, x2, x3, x4) = U17_g(x4)
SEARCH_TREEF_IN_G(x1) = SEARCH_TREEF_IN_G(x1)
U15_G(x1, x2, x3) = U15_G(x3)
PA_IN_GAAG(x1, x2, x3, x4) = PA_IN_GAAG(x1, x4)
U1_GAAG(x1, x2, x3, x4, x5) = U1_GAAG(x5)
SEARCH_TREEB_IN_GAA(x1, x2, x3) = SEARCH_TREEB_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4) = U4_GAA(x1, x4)
U2_GAAG(x1, x2, x3, x4, x5) = U2_GAAG(x4, x5)
U5_GAA(x1, x2, x3, x4) = U5_GAA(x4)
PD_IN_GAAG(x1, x2, x3, x4) = PD_IN_GAAG(x1, x4)
U8_GAAG(x1, x2, x3, x4, x5) = U8_GAAG(x5)
U6_GAA(x1, x2, x3, x4, x5, x6) = U6_GAA(x6)
PE_IN_GAAGGAA(x1, x2, x3, x4, x5, x6, x7) = PE_IN_GAAGGAA(x1, x4, x5)
U11_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GAAGGAA(x8)
U12_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GAAGGAA(x4, x5, x8)
U13_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GAAGGAA(x2, x4, x5, x8)
LESSC_IN_AG(x1, x2) = LESSC_IN_AG(x2)
U7_AG(x1, x2, x3) = U7_AG(x3)
U14_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U14_GAAGGAA(x2, x8)
U9_GAAG(x1, x2, x3, x4, x5) = U9_GAAG(x4, x5)
U10_GAAG(x1, x2, x3, x4, x5) = U10_GAAG(x2, x5)
U3_GAAG(x1, x2, x3, x4, x5) = U3_GAAG(x2, x5)
LESSC_IN_GG(x1, x2) = LESSC_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x3)
U16_G(x1, x2, x3) = U16_G(x3)
U17_G(x1, x2, x3, x4) = U17_G(x4)

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

(6) Obligation:

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

SEARCH_TREEF_IN_G(tree(T17, void, T18)) → U15_G(T17, T18, pA_in_gaag(T18, X50, X51, T17))
SEARCH_TREEF_IN_G(tree(T17, void, T18)) → PA_IN_GAAG(T18, X50, X51, T17)
PA_IN_GAAG(T18, X50, X51, T17) → U1_GAAG(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
PA_IN_GAAG(T18, X50, X51, T17) → SEARCH_TREEB_IN_GAA(T18, X50, X51)
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → U4_GAA(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → PA_IN_GAAG(T39, X82, X83, T38)
PA_IN_GAAG(T18, T21, T22, T17) → U2_GAAG(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → U5_GAA(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → PD_IN_GAAG(T51, X108, X107, T50)
PD_IN_GAAG(T51, X108, X107, T50) → U8_GAAG(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
PD_IN_GAAG(T51, X108, X107, T50) → SEARCH_TREEB_IN_GAA(T51, X108, X107)
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → U6_GAA(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139)
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → U11_GAAGGAA(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → SEARCH_TREEB_IN_GAA(T67, X138, X136)
PE_IN_GAAGGAA(T67, T71, T72, T66, T68, X137, X139) → U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → LESSC_IN_AG(T72, T66)
LESSC_IN_AG(s(T92), s(T93)) → U7_AG(T92, T93, lessC_in_ag(T92, T93))
LESSC_IN_AG(s(T92), s(T93)) → LESSC_IN_AG(T92, T93)
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → PA_IN_GAAG(T68, X137, X139, T66)
PD_IN_GAAG(T51, T54, T55, T50) → U9_GAAG(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_GAAG(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_GAAG(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U9_GAAG(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → LESSC_IN_AG(T55, T50)
U2_GAAG(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_GAAG(T18, T21, T22, T17, lessC_in_gg(T17, T21))
U2_GAAG(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → LESSC_IN_GG(T17, T21)
LESSC_IN_GG(s(T92), s(T93)) → U7_GG(T92, T93, lessC_in_gg(T92, T93))
LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)
SEARCH_TREEF_IN_G(tree(T104, T105, void)) → U16_G(T104, T105, pD_in_gaag(T105, X187, X186, T104))
SEARCH_TREEF_IN_G(tree(T104, T105, void)) → PD_IN_GAAG(T105, X187, X186, T104)
SEARCH_TREEF_IN_G(tree(T114, T115, T116)) → U17_G(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
SEARCH_TREEF_IN_G(tree(T114, T115, T116)) → PE_IN_GAAGGAA(T115, X213, X211, T114, T116, X212, X214)

The TRS R consists of the following rules:

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 22 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)

The TRS R consists of the following rules:

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

(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:

LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)

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:

LESSC_IN_GG(s(T92), s(T93)) → LESSC_IN_GG(T92, T93)
The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

LESSC_IN_AG(s(T92), s(T93)) → LESSC_IN_AG(T92, T93)

The TRS R consists of the following rules:

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

(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:

LESSC_IN_AG(s(T92), s(T93)) → LESSC_IN_AG(T92, T93)

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

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:

LESSC_IN_AG(s(T93)) → LESSC_IN_AG(T93)

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:

LESSC_IN_AG(s(T93)) → LESSC_IN_AG(T93)
The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

PA_IN_GAAG(T18, X50, X51, T17) → SEARCH_TREEB_IN_GAA(T18, X50, X51)
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → PA_IN_GAAG(T39, X82, X83, T38)
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → PD_IN_GAAG(T51, X108, X107, T50)
PD_IN_GAAG(T51, X108, X107, T50) → SEARCH_TREEB_IN_GAA(T51, X108, X107)
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139)
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → SEARCH_TREEB_IN_GAA(T67, X138, X136)
PE_IN_GAAGGAA(T67, T71, T72, T66, T68, X137, X139) → U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → PA_IN_GAAG(T68, X137, X139, T66)

The TRS R consists of the following rules:

search_treeF_in_g(void) → search_treeF_out_g(void)
search_treeF_in_g(tree(T8, void, void)) → search_treeF_out_g(tree(T8, void, void))
search_treeF_in_g(tree(T17, void, T18)) → U15_g(T17, T18, pA_in_gaag(T18, X50, X51, T17))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U15_g(T17, T18, pA_out_gaag(T18, X50, X51, T17)) → search_treeF_out_g(tree(T17, void, T18))
search_treeF_in_g(tree(T104, T105, void)) → U16_g(T104, T105, pD_in_gaag(T105, X187, X186, T104))
U16_g(T104, T105, pD_out_gaag(T105, X187, X186, T104)) → search_treeF_out_g(tree(T104, T105, void))
search_treeF_in_g(tree(T114, T115, T116)) → U17_g(T114, T115, T116, pE_in_gaaggaa(T115, X213, X211, T114, T116, X212, X214))
U17_g(T114, T115, T116, pE_out_gaaggaa(T115, X213, X211, T114, T116, X212, X214)) → search_treeF_out_g(tree(T114, T115, T116))

The argument filtering Pi contains the following mapping:
search_treeF_in_g(x1) = search_treeF_in_g(x1)
void = void
search_treeF_out_g(x1) = search_treeF_out_g
tree(x1, x2, x3) = tree(x1, x2, x3)
U15_g(x1, x2, x3) = U15_g(x3)
pA_in_gaag(x1, x2, x3, x4) = pA_in_gaag(x1, x4)
U1_gaag(x1, x2, x3, x4, x5) = U1_gaag(x5)
search_treeB_in_gaa(x1, x2, x3) = search_treeB_in_gaa(x1)
search_treeB_out_gaa(x1, x2, x3) = search_treeB_out_gaa(x2)
U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4)
U2_gaag(x1, x2, x3, x4, x5) = U2_gaag(x4, x5)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x4)
pD_in_gaag(x1, x2, x3, x4) = pD_in_gaag(x1, x4)
U8_gaag(x1, x2, x3, x4, x5) = U8_gaag(x5)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
pE_in_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_in_gaaggaa(x1, x4, x5)
U11_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U11_gaaggaa(x8)
pE_out_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_out_gaaggaa(x2)
U12_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U12_gaaggaa(x4, x5, x8)
U13_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U13_gaaggaa(x2, x4, x5, x8)
lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessC_out_gg(x1, x2) = lessC_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U14_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U14_gaaggaa(x2, x8)
pA_out_gaag(x1, x2, x3, x4) = pA_out_gaag(x2)
lessC_in_ag(x1, x2) = lessC_in_ag(x2)
lessC_out_ag(x1, x2) = lessC_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
pD_out_gaag(x1, x2, x3, x4) = pD_out_gaag(x2)
U9_gaag(x1, x2, x3, x4, x5) = U9_gaag(x4, x5)
U10_gaag(x1, x2, x3, x4, x5) = U10_gaag(x2, x5)
U3_gaag(x1, x2, x3, x4, x5) = U3_gaag(x2, x5)
U16_g(x1, x2, x3) = U16_g(x3)
U17_g(x1, x2, x3, x4) = U17_g(x4)
PA_IN_GAAG(x1, x2, x3, x4) = PA_IN_GAAG(x1, x4)
SEARCH_TREEB_IN_GAA(x1, x2, x3) = SEARCH_TREEB_IN_GAA(x1)
PD_IN_GAAG(x1, x2, x3, x4) = PD_IN_GAAG(x1, x4)
PE_IN_GAAGGAA(x1, x2, x3, x4, x5, x6, x7) = PE_IN_GAAGGAA(x1, x4, x5)
U12_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GAAGGAA(x4, x5, x8)
U13_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GAAGGAA(x2, x4, x5, x8)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

PA_IN_GAAG(T18, X50, X51, T17) → SEARCH_TREEB_IN_GAA(T18, X50, X51)
SEARCH_TREEB_IN_GAA(tree(T38, void, T39), T38, X83) → PA_IN_GAAG(T39, X82, X83, T38)
SEARCH_TREEB_IN_GAA(tree(T50, T51, void), X108, T50) → PD_IN_GAAG(T51, X108, X107, T50)
PD_IN_GAAG(T51, X108, X107, T50) → SEARCH_TREEB_IN_GAA(T51, X108, X107)
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68), X138, X139) → PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139)
PE_IN_GAAGGAA(T67, X138, X136, T66, T68, X137, X139) → SEARCH_TREEB_IN_GAA(T67, X138, X136)
PE_IN_GAAGGAA(T67, T71, T72, T66, T68, X137, X139) → U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U12_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
U13_GAAGGAA(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → PA_IN_GAAG(T68, X137, X139, T66)

The TRS R consists of the following rules:

search_treeB_in_gaa(tree(T29, void, void), T29, T29) → search_treeB_out_gaa(tree(T29, void, void), T29, T29)
search_treeB_in_gaa(tree(T38, void, T39), T38, X83) → U4_gaa(T38, T39, X83, pA_in_gaag(T39, X82, X83, T38))
search_treeB_in_gaa(tree(T50, T51, void), X108, T50) → U5_gaa(T50, T51, X108, pD_in_gaag(T51, X108, X107, T50))
search_treeB_in_gaa(tree(T66, T67, T68), X138, X139) → U6_gaa(T66, T67, T68, X138, X139, pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139))
lessC_in_ag(0, s(T87)) → lessC_out_ag(0, s(T87))
lessC_in_ag(s(T92), s(T93)) → U7_ag(T92, T93, lessC_in_ag(T92, T93))
U4_gaa(T38, T39, X83, pA_out_gaag(T39, X82, X83, T38)) → search_treeB_out_gaa(tree(T38, void, T39), T38, X83)
U5_gaa(T50, T51, X108, pD_out_gaag(T51, X108, X107, T50)) → search_treeB_out_gaa(tree(T50, T51, void), X108, T50)
U6_gaa(T66, T67, T68, X138, X139, pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)) → search_treeB_out_gaa(tree(T66, T67, T68), X138, X139)
U7_ag(T92, T93, lessC_out_ag(T92, T93)) → lessC_out_ag(s(T92), s(T93))
pA_in_gaag(T18, X50, X51, T17) → U1_gaag(T18, X50, X51, T17, search_treeB_in_gaa(T18, X50, X51))
pA_in_gaag(T18, T21, T22, T17) → U2_gaag(T18, T21, T22, T17, search_treeB_in_gaa(T18, T21, T22))
pD_in_gaag(T51, X108, X107, T50) → U8_gaag(T51, X108, X107, T50, search_treeB_in_gaa(T51, X108, X107))
pD_in_gaag(T51, T54, T55, T50) → U9_gaag(T51, T54, T55, T50, search_treeB_in_gaa(T51, T54, T55))
pE_in_gaaggaa(T67, X138, X136, T66, T68, X137, X139) → U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_in_gaa(T67, X138, X136))
pE_in_gaaggaa(T67, T71, T72, T66, T68, X137, X139) → U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_in_gaa(T67, T71, T72))
U1_gaag(T18, X50, X51, T17, search_treeB_out_gaa(T18, X50, X51)) → pA_out_gaag(T18, X50, X51, T17)
U2_gaag(T18, T21, T22, T17, search_treeB_out_gaa(T18, T21, T22)) → U3_gaag(T18, T21, T22, T17, lessC_in_gg(T17, T21))
U8_gaag(T51, X108, X107, T50, search_treeB_out_gaa(T51, X108, X107)) → pD_out_gaag(T51, X108, X107, T50)
U9_gaag(T51, T54, T55, T50, search_treeB_out_gaa(T51, T54, T55)) → U10_gaag(T51, T54, T55, T50, lessC_in_ag(T55, T50))
U11_gaaggaa(T67, X138, X136, T66, T68, X137, X139, search_treeB_out_gaa(T67, X138, X136)) → pE_out_gaaggaa(T67, X138, X136, T66, T68, X137, X139)
U12_gaaggaa(T67, T71, T72, T66, T68, X137, X139, search_treeB_out_gaa(T67, T71, T72)) → U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_in_ag(T72, T66))
U3_gaag(T18, T21, T22, T17, lessC_out_gg(T17, T21)) → pA_out_gaag(T18, T21, T22, T17)
U10_gaag(T51, T54, T55, T50, lessC_out_ag(T55, T50)) → pD_out_gaag(T51, T54, T55, T50)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U13_gaaggaa(T67, T71, T72, T66, T68, X137, X139, lessC_out_ag(T72, T66)) → U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_in_gaag(T68, X137, X139, T66))
lessC_in_gg(0, s(T87)) → lessC_out_gg(0, s(T87))
lessC_in_gg(s(T92), s(T93)) → U7_gg(T92, T93, lessC_in_gg(T92, T93))
U14_gaaggaa(T67, T71, T72, T66, T68, X137, X139, pA_out_gaag(T68, X137, X139, T66)) → pE_out_gaaggaa(T67, T71, T72, T66, T68, X137, X139)
U7_gg(T92, T93, lessC_out_gg(T92, T93)) → lessC_out_gg(s(T92), s(T93))

The argument filtering Pi contains the following mapping:
void = void
tree(x1, x2, x3) = tree(x1, x2, x3)
pA_in_gaag(x1, x2, x3, x4) = pA_in_gaag(x1, x4)
U1_gaag(x1, x2, x3, x4, x5) = U1_gaag(x5)
search_treeB_in_gaa(x1, x2, x3) = search_treeB_in_gaa(x1)
search_treeB_out_gaa(x1, x2, x3) = search_treeB_out_gaa(x2)
U4_gaa(x1, x2, x3, x4) = U4_gaa(x1, x4)
U2_gaag(x1, x2, x3, x4, x5) = U2_gaag(x4, x5)
U5_gaa(x1, x2, x3, x4) = U5_gaa(x4)
pD_in_gaag(x1, x2, x3, x4) = pD_in_gaag(x1, x4)
U8_gaag(x1, x2, x3, x4, x5) = U8_gaag(x5)
U6_gaa(x1, x2, x3, x4, x5, x6) = U6_gaa(x6)
pE_in_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_in_gaaggaa(x1, x4, x5)
U11_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U11_gaaggaa(x8)
pE_out_gaaggaa(x1, x2, x3, x4, x5, x6, x7) = pE_out_gaaggaa(x2)
U12_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U12_gaaggaa(x4, x5, x8)
U13_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U13_gaaggaa(x2, x4, x5, x8)
lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessC_out_gg(x1, x2) = lessC_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U14_gaaggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U14_gaaggaa(x2, x8)
pA_out_gaag(x1, x2, x3, x4) = pA_out_gaag(x2)
lessC_in_ag(x1, x2) = lessC_in_ag(x2)
lessC_out_ag(x1, x2) = lessC_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
pD_out_gaag(x1, x2, x3, x4) = pD_out_gaag(x2)
U9_gaag(x1, x2, x3, x4, x5) = U9_gaag(x4, x5)
U10_gaag(x1, x2, x3, x4, x5) = U10_gaag(x2, x5)
U3_gaag(x1, x2, x3, x4, x5) = U3_gaag(x2, x5)
PA_IN_GAAG(x1, x2, x3, x4) = PA_IN_GAAG(x1, x4)
SEARCH_TREEB_IN_GAA(x1, x2, x3) = SEARCH_TREEB_IN_GAA(x1)
PD_IN_GAAG(x1, x2, x3, x4) = PD_IN_GAAG(x1, x4)
PE_IN_GAAGGAA(x1, x2, x3, x4, x5, x6, x7) = PE_IN_GAAGGAA(x1, x4, x5)
U12_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GAAGGAA(x4, x5, x8)
U13_GAAGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U13_GAAGGAA(x2, x4, x5, x8)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

PA_IN_GAAG(T18, T17) → SEARCH_TREEB_IN_GAA(T18)
SEARCH_TREEB_IN_GAA(tree(T38, void, T39)) → PA_IN_GAAG(T39, T38)
SEARCH_TREEB_IN_GAA(tree(T50, T51, void)) → PD_IN_GAAG(T51, T50)
PD_IN_GAAG(T51, T50) → SEARCH_TREEB_IN_GAA(T51)
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68)) → PE_IN_GAAGGAA(T67, T66, T68)
PE_IN_GAAGGAA(T67, T66, T68) → SEARCH_TREEB_IN_GAA(T67)
PE_IN_GAAGGAA(T67, T66, T68) → U12_GAAGGAA(T66, T68, search_treeB_in_gaa(T67))
U12_GAAGGAA(T66, T68, search_treeB_out_gaa(T71)) → U13_GAAGGAA(T71, T66, T68, lessC_in_ag(T66))
U13_GAAGGAA(T71, T66, T68, lessC_out_ag(T72)) → PA_IN_GAAG(T68, T66)

The TRS R consists of the following rules:

search_treeB_in_gaa(tree(T29, void, void)) → search_treeB_out_gaa(T29)
search_treeB_in_gaa(tree(T38, void, T39)) → U4_gaa(T38, pA_in_gaag(T39, T38))
search_treeB_in_gaa(tree(T50, T51, void)) → U5_gaa(pD_in_gaag(T51, T50))
search_treeB_in_gaa(tree(T66, T67, T68)) → U6_gaa(pE_in_gaaggaa(T67, T66, T68))
lessC_in_ag(s(T87)) → lessC_out_ag(0)
lessC_in_ag(s(T93)) → U7_ag(lessC_in_ag(T93))
U4_gaa(T38, pA_out_gaag(X82)) → search_treeB_out_gaa(T38)
U5_gaa(pD_out_gaag(X108)) → search_treeB_out_gaa(X108)
U6_gaa(pE_out_gaaggaa(X138)) → search_treeB_out_gaa(X138)
U7_ag(lessC_out_ag(T92)) → lessC_out_ag(s(T92))
pA_in_gaag(T18, T17) → U1_gaag(search_treeB_in_gaa(T18))
pA_in_gaag(T18, T17) → U2_gaag(T17, search_treeB_in_gaa(T18))
pD_in_gaag(T51, T50) → U8_gaag(search_treeB_in_gaa(T51))
pD_in_gaag(T51, T50) → U9_gaag(T50, search_treeB_in_gaa(T51))
pE_in_gaaggaa(T67, T66, T68) → U11_gaaggaa(search_treeB_in_gaa(T67))
pE_in_gaaggaa(T67, T66, T68) → U12_gaaggaa(T66, T68, search_treeB_in_gaa(T67))
U1_gaag(search_treeB_out_gaa(X50)) → pA_out_gaag(X50)
U2_gaag(T17, search_treeB_out_gaa(T21)) → U3_gaag(T21, lessC_in_gg(T17, T21))
U8_gaag(search_treeB_out_gaa(X108)) → pD_out_gaag(X108)
U9_gaag(T50, search_treeB_out_gaa(T54)) → U10_gaag(T54, lessC_in_ag(T50))
U11_gaaggaa(search_treeB_out_gaa(X138)) → pE_out_gaaggaa(X138)
U12_gaaggaa(T66, T68, search_treeB_out_gaa(T71)) → U13_gaaggaa(T71, T66, T68, lessC_in_ag(T66))
U3_gaag(T21, lessC_out_gg) → pA_out_gaag(T21)
U10_gaag(T54, lessC_out_ag(T55)) → pD_out_gaag(T54)
U13_gaaggaa(T71, T66, T68, lessC_out_ag(T72)) → pE_out_gaaggaa(T71)
U13_gaaggaa(T71, T66, T68, lessC_out_ag(T72)) → U14_gaaggaa(T71, pA_in_gaag(T68, T66))
lessC_in_gg(0, s(T87)) → lessC_out_gg
lessC_in_gg(s(T92), s(T93)) → U7_gg(lessC_in_gg(T92, T93))
U14_gaaggaa(T71, pA_out_gaag(X137)) → pE_out_gaaggaa(T71)
U7_gg(lessC_out_gg) → lessC_out_gg

The set Q consists of the following terms:

search_treeB_in_gaa(x0)
lessC_in_ag(x0)
U4_gaa(x0, x1)
U5_gaa(x0)
U6_gaa(x0)
U7_ag(x0)
pA_in_gaag(x0, x1)
pD_in_gaag(x0, x1)
pE_in_gaaggaa(x0, x1, x2)
U1_gaag(x0)
U2_gaag(x0, x1)
U8_gaag(x0)
U9_gaag(x0, x1)
U11_gaaggaa(x0)
U12_gaaggaa(x0, x1, x2)
U3_gaag(x0, x1)
U10_gaag(x0, x1)
U13_gaaggaa(x0, x1, x2, x3)
lessC_in_gg(x0, x1)
U14_gaaggaa(x0, x1)
U7_gg(x0)

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

(28) 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:

SEARCH_TREEB_IN_GAA(tree(T38, void, T39)) → PA_IN_GAAG(T39, T38)
The graph contains the following edges 1 > 1, 1 > 2
U13_GAAGGAA(T71, T66, T68, lessC_out_ag(T72)) → PA_IN_GAAG(T68, T66)
The graph contains the following edges 3 >= 1, 2 >= 2
PA_IN_GAAG(T18, T17) → SEARCH_TREEB_IN_GAA(T18)
The graph contains the following edges 1 >= 1
PD_IN_GAAG(T51, T50) → SEARCH_TREEB_IN_GAA(T51)
The graph contains the following edges 1 >= 1
PE_IN_GAAGGAA(T67, T66, T68) → SEARCH_TREEB_IN_GAA(T67)
The graph contains the following edges 1 >= 1
SEARCH_TREEB_IN_GAA(tree(T50, T51, void)) → PD_IN_GAAG(T51, T50)
The graph contains the following edges 1 > 1, 1 > 2
SEARCH_TREEB_IN_GAA(tree(T66, T67, T68)) → PE_IN_GAAGGAA(T67, T66, T68)
The graph contains the following edges 1 > 1, 1 > 2, 1 > 3
PE_IN_GAAGGAA(T67, T66, T68) → U12_GAAGGAA(T66, T68, search_treeB_in_gaa(T67))
The graph contains the following edges 2 >= 1, 3 >= 2
U12_GAAGGAA(T66, T68, search_treeB_out_gaa(T71)) → U13_GAAGGAA(T71, T66, T68, lessC_in_ag(T66))
The graph contains the following edges 3 > 1, 1 >= 2, 2 >= 3

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES