Left Termination of the query pattern in_order_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

in_order(void, []).
in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

in_order(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: in_order(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: in_order(T1, T2), SCOPE: 1, CLAUSE: 0 | in_order(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | in_order(void, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: in_order(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nin_order(T1, T2) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: in_order(void, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(in_order(T8, X13), ','(in_order(T9, X14), app(X13, .(T7, X14), T11)))\n\nT7 is ground\nT8 is ground\nT9 is ground\nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(in_order(T8, X13), ','(in_order(T9, X14), app(X13, .(T7, X14), T11))), SCOPE: 2, CLAUSE: 0 | ','(in_order(T8, X13), ','(in_order(T9, X14), app(X13, .(T7, X14), T11))), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nT9 is ground\nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(in_order(T8, X13), ','(in_order(T9, X14), app(X13, .(T7, X14), T11))), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground\nT9 is ground\nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(in_order(T8, X13), ','(in_order(T9, X14), app(X13, .(T7, X14), T11))), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nT9 is ground\nX13 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(in_order(T9, X14), app([], .(T7, X14), T11))\n\nT7 is ground\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: in_order(T9, X14)\n\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app([], .(T7, T12), T11)\n\nT7 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: in_order(T9, X14), SCOPE: 3, CLAUSE: 0 | in_order(T9, X14), SCOPE: 3, CLAUSE: 1\n\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: in_order(T9, X14), SCOPE: 3, CLAUSE: 0\n\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: in_order(T9, X14), SCOPE: 3, CLAUSE: 1\n\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(in_order(T20, X35), ','(in_order(T21, X36), app(X35, .(T19, X36), X37)))\n\nT19 is ground\nT20 is ground\nT21 is ground\nX37 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: in_order(T20, X35)\n\nT20 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(in_order(T21, X36), app(T22, .(T19, X36), X37))\n\nT19 is ground\nT21 is ground\nT22 is ground\nX37 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: in_order(T21, X36)\n\nT21 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T22, .(T19, T23), X37)\n\nT19 is ground\nT22 is ground\nT23 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T22, .(T19, T23), X37), SCOPE: 4, CLAUSE: 2 | app(T22, .(T19, T23), X37), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT22 is ground\nT23 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T22, .(T19, T23), X37), SCOPE: 4, CLAUSE: 2\n\nT19 is ground\nT22 is ground\nT23 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T22, .(T19, T23), X37), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT22 is ground\nT23 is ground\nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T47, .(T48, T49), X64)\n\nT47 is ground\nT48 is ground\nT49 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app([], .(T7, T12), T11), SCOPE: 5, CLAUSE: 2 | app([], .(T7, T12), T11), SCOPE: 5, CLAUSE: 3\n\nT7 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app([], .(T7, T12), T11), SCOPE: 5, CLAUSE: 2\n\nT7 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app([], .(T7, T12), T11), SCOPE: 5, CLAUSE: 3\n\nT7 is ground\nT12 is ground", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(in_order(T75, X99), ','(in_order(T76, X100), ','(app(X99, .(T74, X100), X101), ','(in_order(T9, X14), app(X101, .(T7, X14), T11)))))\n\nT7 is ground\nT9 is ground\nT74 is ground\nT75 is ground\nT76 is ground\nX14 is free; \nX101 is free; \nX99 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: in_order(T75, X99)\n\nT75 is ground\nX99 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(in_order(T76, X100), ','(app(T77, .(T74, X100), X101), ','(in_order(T9, X14), app(X101, .(T7, X14), T11))))\n\nT7 is ground\nT9 is ground\nT74 is ground\nT76 is ground\nT77 is ground\nX14 is free; \nX101 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: in_order(T76, X100)\n\nT76 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(app(T77, .(T74, T78), X101), ','(in_order(T9, X14), app(X101, .(T7, X14), T11)))\n\nT7 is ground\nT9 is ground\nT74 is ground\nT77 is ground\nT78 is ground\nX14 is free; \nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: app(T77, .(T74, T78), X101)\n\nT74 is ground\nT77 is ground\nT78 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(in_order(T9, X14), app(T83, .(T7, X14), T11))\n\nT7 is ground\nT9 is ground\nT83 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: in_order(T9, X14)\n\nT9 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: app(T83, .(T7, T88), T11)\n\nT7 is ground\nT83 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: app(T83, .(T7, T88), T11), SCOPE: 6, CLAUSE: 2 | app(T83, .(T7, T88), T11), SCOPE: 6, CLAUSE: 3\n\nT7 is ground\nT83 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: app(T83, .(T7, T88), T11), SCOPE: 6, CLAUSE: 2\n\nT7 is ground\nT83 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: app(T83, .(T7, T88), T11), SCOPE: 6, CLAUSE: 3\n\nT7 is ground\nT83 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: app(T114, .(T115, T116), T118)\n\nT114 is ground\nT115 is ground\nT116 is ground", 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\nin_order(void, []).\nand substitution\nT1 -> void\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 62 [arrowhead = none ];
62 -> 3;

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

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

65 [label="EVAL with clause\nin_order(tree(X9, X10, X11), X12) :- ','(in_order(X10, X13), ','(in_order(X11, X14), app(X13, .(X9, X14), X12))).\nand substitution\nX9 -> T7\nX10 -> T8\nX11 -> T9\nT1 -> tree(T7, T8, T9)\nT2 -> T11\nX12 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 65 [arrowhead = none ];
65 -> 7;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 66 [arrowhead = none ];
66 -> 8;

67 [label="BACKTRACK\nfor clause: in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 67 [arrowhead = none ];
67 -> 6;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 68 [arrowhead = none ];
68 -> 9;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 69 [arrowhead = none ];
69 -> 10;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 70 [arrowhead = none ];
70 -> 11;

71 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT8 -> void\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 71 [arrowhead = none ];
71 -> 12;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 72 [arrowhead = none ];
72 -> 13;

73 [label="EVAL with clause\nin_order(tree(X95, X96, X97), X98) :- ','(in_order(X96, X99), ','(in_order(X97, X100), app(X99, .(X95, X100), X98))).\nand substitution\nX95 -> T74\nX96 -> T75\nX97 -> T76\nT8 -> tree(T74, T75, T76)\nX13 -> X101\nX98 -> X101", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 73 [arrowhead = none ];
73 -> 43;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 74 [arrowhead = none ];
74 -> 44;

75 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 75 [arrowhead = none ];
75 -> 14;

76 [label="SPLIT 2\nnew knowledge:\nT12 is ground\nreplacements:\nX14 -> T12", fontsize=14, style = filled, fillcolor = violet];
12 -> 76 [arrowhead = none ];
76 -> 15;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 77 [arrowhead = none ];
77 -> 16;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 78 [arrowhead = none ];
78 -> 36;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 80 [arrowhead = none ];
80 -> 18;

81 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT9 -> void\nX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 81 [arrowhead = none ];
81 -> 19;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 82 [arrowhead = none ];
82 -> 20;

83 [label="EVAL with clause\nin_order(tree(X31, X32, X33), X34) :- ','(in_order(X32, X35), ','(in_order(X33, X36), app(X35, .(X31, X36), X34))).\nand substitution\nX31 -> T19\nX32 -> T20\nX33 -> T21\nT9 -> tree(T19, T20, T21)\nX14 -> X37\nX34 -> X37", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 84 [arrowhead = none ];
84 -> 23;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 85 [arrowhead = none ];
85 -> 21;

86 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 86 [arrowhead = none ];
86 -> 24;

87 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX35 -> T22", fontsize=14, style = filled, fillcolor = violet];
22 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="INSTANCE with matching:\nT9 -> T20\nX14 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 88 [arrowhead = none , style=dashed];
88 -> 14[style=dashed];

89 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 89 [arrowhead = none ];
89 -> 26;

90 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nreplacements:\nX36 -> T23", fontsize=14, style = filled, fillcolor = violet];
25 -> 90 [arrowhead = none ];
90 -> 27;

91 [label="INSTANCE with matching:\nT9 -> T21\nX14 -> X36", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 91 [arrowhead = none , style=dashed];
91 -> 14[style=dashed];

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 92 [arrowhead = none ];
92 -> 28;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 93 [arrowhead = none ];
93 -> 29;

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

95 [label="EVAL with clause\napp([], X48, X48).\nand substitution\nT22 -> []\nT19 -> T36\nT23 -> T37\nX48 -> .(T36, T37)\nX37 -> .(T36, T37)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="EVAL with clause\napp(.(X60, X61), X62, .(X60, X63)) :- app(X61, X62, X63).\nand substitution\nX60 -> T46\nX61 -> T47\nT22 -> .(T46, T47)\nT19 -> T48\nT23 -> T49\nX62 -> .(T48, T49)\nX63 -> X64\nX37 -> .(T46, X64)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 99 [arrowhead = none ];
99 -> 33;

100 [label="INSTANCE with matching:\nT22 -> T47\nT19 -> T48\nT23 -> T49\nX37 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 100 [arrowhead = none , style=dashed];
100 -> 27[style=dashed];

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 101 [arrowhead = none ];
101 -> 37;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 102 [arrowhead = none ];
102 -> 38;

103 [label="EVAL with clause\napp([], X77, X77).\nand substitution\nT7 -> T66\nT12 -> T67\nX77 -> .(T66, T67)\nT11 -> .(T66, T67)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 103 [arrowhead = none ];
103 -> 39;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 104 [arrowhead = none ];
104 -> 40;

105 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 105 [arrowhead = none ];
105 -> 42;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 106 [arrowhead = none ];
106 -> 41;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 107 [arrowhead = none ];
107 -> 45;

108 [label="SPLIT 2\nnew knowledge:\nT77 is ground\nreplacements:\nX99 -> T77", fontsize=14, style = filled, fillcolor = violet];
43 -> 108 [arrowhead = none ];
108 -> 46;

109 [label="INSTANCE with matching:\nT9 -> T75\nX14 -> X99", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 109 [arrowhead = none , style=dashed];
109 -> 14[style=dashed];

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 110 [arrowhead = none ];
110 -> 47;

111 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nreplacements:\nX100 -> T78", fontsize=14, style = filled, fillcolor = violet];
46 -> 111 [arrowhead = none ];
111 -> 48;

112 [label="INSTANCE with matching:\nT9 -> T76\nX14 -> X100", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 112 [arrowhead = none , style=dashed];
112 -> 14[style=dashed];

113 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 113 [arrowhead = none ];
113 -> 49;

114 [label="SPLIT 2\nnew knowledge:\nT83 is ground\nreplacements:\nX101 -> T83", fontsize=14, style = filled, fillcolor = violet];
48 -> 114 [arrowhead = none ];
114 -> 50;

115 [label="INSTANCE with matching:\nT22 -> T77\nT19 -> T74\nT23 -> T78\nX37 -> X101", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 115 [arrowhead = none , style=dashed];
115 -> 27[style=dashed];

116 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 116 [arrowhead = none ];
116 -> 51;

117 [label="SPLIT 2\nnew knowledge:\nT88 is ground\nreplacements:\nX14 -> T88", fontsize=14, style = filled, fillcolor = violet];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 118 [arrowhead = none , style=dashed];
118 -> 14[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 119 [arrowhead = none ];
119 -> 53;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 120 [arrowhead = none ];
120 -> 54;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 121 [arrowhead = none ];
121 -> 55;

122 [label="EVAL with clause\napp([], X116, X116).\nand substitution\nT83 -> []\nT7 -> T101\nT88 -> T102\nX116 -> .(T101, T102)\nT11 -> .(T101, T102)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 122 [arrowhead = none ];
122 -> 56;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 123 [arrowhead = none ];
123 -> 57;

124 [label="EVAL with clause\napp(.(X126, X127), X128, .(X126, X129)) :- app(X127, X128, X129).\nand substitution\nX126 -> T113\nX127 -> T114\nT83 -> .(T113, T114)\nT7 -> T115\nT88 -> T116\nX128 -> .(T115, T116)\nX129 -> T118\nT11 -> .(T113, T118)\nT117 -> T118", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 124 [arrowhead = none ];
124 -> 59;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 125 [arrowhead = none ];
125 -> 60;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 126 [arrowhead = none ];
126 -> 58;

127 [label="INSTANCE with matching:\nT83 -> T114\nT7 -> T115\nT88 -> T116\nT11 -> T118", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 127 [arrowhead = none , style=dashed];
127 -> 52[style=dashed];

}

(2) Obligation:

Triples:

in_order14(tree(T19, T20, T21), X37) :- in_order14(T20, X35).
in_order14(tree(T19, T20, T21), X37) :- ','(in_orderc14(T20, T22), in_order14(T21, X36)).
in_order14(tree(T19, T20, T21), X37) :- ','(in_orderc14(T20, T22), ','(in_orderc14(T21, T23), app27(T22, T19, T23, X37))).
app27(.(T46, T47), T48, T49, .(T46, X64)) :- app27(T47, T48, T49, X64).
app52(.(T113, T114), T115, T116, .(T113, T118)) :- app52(T114, T115, T116, T118).
in_order1(tree(T7, void, T9), T11) :- in_order14(T9, X14).
in_order1(tree(T7, tree(T74, T75, T76), T9), T11) :- in_order14(T75, X99).
in_order1(tree(T7, tree(T74, T75, T76), T9), T11) :- ','(in_orderc14(T75, T77), in_order14(T76, X100)).
in_order1(tree(T7, tree(T74, T75, T76), T9), T11) :- ','(in_orderc14(T75, T77), ','(in_orderc14(T76, T78), app27(T77, T74, T78, X101))).
in_order1(tree(T7, tree(T74, T75, T76), T9), T11) :- ','(in_orderc14(T75, T77), ','(in_orderc14(T76, T78), ','(appc27(T77, T74, T78, T83), in_order14(T9, X14)))).
in_order1(tree(T7, tree(T74, T75, T76), T9), T11) :- ','(in_orderc14(T75, T77), ','(in_orderc14(T76, T78), ','(appc27(T77, T74, T78, T83), ','(in_orderc14(T9, T88), app52(T83, T7, T88, T11))))).

Clauses:

in_orderc14(void, []).
in_orderc14(tree(T19, T20, T21), X37) :- ','(in_orderc14(T20, T22), ','(in_orderc14(T21, T23), appc27(T22, T19, T23, X37))).
appc27([], T36, T37, .(T36, T37)).
appc27(.(T46, T47), T48, T49, .(T46, X64)) :- appc27(T47, T48, T49, X64).
appc52([], T101, T102, .(T101, T102)).
appc52(.(T113, T114), T115, T116, .(T113, T118)) :- appc52(T114, T115, T116, T118).

Afs:

in_order1(x1, x2)  =  in_order1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_order1_in: (b,f)
in_order14_in: (b,f)
in_orderc14_in: (b,f)
appc27_in: (b,b,b,f)
app27_in: (b,b,b,f)
app52_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

IN_ORDER1_IN_GA(tree(T7, void, T9), T11) → U8_GA(T7, T9, T11, in_order14_in_ga(T9, X14))
IN_ORDER1_IN_GA(tree(T7, void, T9), T11) → IN_ORDER14_IN_GA(T9, X14)
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → U1_GA(T19, T20, T21, X37, in_order14_in_ga(T20, X35))
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → IN_ORDER14_IN_GA(T20, X35)
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → U2_GA(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U3_GA(T19, T20, T21, X37, in_order14_in_ga(T21, X36))
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → IN_ORDER14_IN_GA(T21, X36)
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U4_GA(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U4_GA(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U5_GA(T19, T20, T21, X37, app27_in_ggga(T22, T19, T23, X37))
U4_GA(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → APP27_IN_GGGA(T22, T19, T23, X37)
APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → U6_GGGA(T46, T47, T48, T49, X64, app27_in_ggga(T47, T48, T49, X64))
APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → APP27_IN_GGGA(T47, T48, T49, X64)
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → U9_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T75, X99))
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → IN_ORDER14_IN_GA(T75, X99)
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_in_ga(T75, T77))
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → U11_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T76, X100))
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → IN_ORDER14_IN_GA(T76, X100)
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_in_ga(T76, T78))
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → U13_GA(T7, T74, T75, T76, T9, T11, app27_in_ggga(T77, T74, T78, X101))
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → APP27_IN_GGGA(T77, T74, T78, X101)
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → U14_GA(T7, T74, T75, T76, T9, T11, appc27_in_ggga(T77, T74, T78, T83))
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → U15_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T9, X14))
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → IN_ORDER14_IN_GA(T9, X14)
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_in_ga(T9, T88))
U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_out_ga(T9, T88)) → U17_GA(T7, T74, T75, T76, T9, T11, app52_in_ggga(T83, T7, T88, T11))
U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_out_ga(T9, T88)) → APP52_IN_GGGA(T83, T7, T88, T11)
APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → U7_GGGA(T113, T114, T115, T116, T118, app52_in_ggga(T114, T115, T116, T118))
APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → APP52_IN_GGGA(T114, T115, T116, T118)

The TRS R consists of the following rules:

in_orderc14_in_ga(void, []) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21), X37) → U19_ga(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U19_ga(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U20_ga(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, X37, appc27_in_ggga(T22, T19, T23, X37))
appc27_in_ggga([], T36, T37, .(T36, T37)) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49, .(T46, X64)) → U22_ggga(T46, T47, T48, T49, X64, appc27_in_ggga(T47, T48, T49, X64))
U22_ggga(T46, T47, T48, T49, X64, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, X37, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_order14_in_ga(x1, x2)  =  in_order14_in_ga(x1)
in_orderc14_in_ga(x1, x2)  =  in_orderc14_in_ga(x1)
in_orderc14_out_ga(x1, x2)  =  in_orderc14_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appc27_in_ggga(x1, x2, x3, x4)  =  appc27_in_ggga(x1, x2, x3)
[]  =  []
appc27_out_ggga(x1, x2, x3, x4)  =  appc27_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
app27_in_ggga(x1, x2, x3, x4)  =  app27_in_ggga(x1, x2, x3)
app52_in_ggga(x1, x2, x3, x4)  =  app52_in_ggga(x1, x2, x3)
IN_ORDER1_IN_GA(x1, x2)  =  IN_ORDER1_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
IN_ORDER14_IN_GA(x1, x2)  =  IN_ORDER14_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x5, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
APP27_IN_GGGA(x1, x2, x3, x4)  =  APP27_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GA(x1, x2, x3, x4, x5, x7)
U10_GA(x1, x2, x3, x4, x5, x6, x7)  =  U10_GA(x1, x2, x3, x4, x5, x7)
U11_GA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GA(x1, x2, x3, x4, x5, x7)
U12_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GA(x1, x2, x3, x4, x5, x7, x8)
U13_GA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GA(x1, x2, x3, x4, x5, x7)
U14_GA(x1, x2, x3, x4, x5, x6, x7)  =  U14_GA(x1, x2, x3, x4, x5, x7)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x1, x2, x3, x4, x5, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GA(x1, x2, x3, x4, x5, x7, x8)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
APP52_IN_GGGA(x1, x2, x3, x4)  =  APP52_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

IN_ORDER1_IN_GA(tree(T7, void, T9), T11) → U8_GA(T7, T9, T11, in_order14_in_ga(T9, X14))
IN_ORDER1_IN_GA(tree(T7, void, T9), T11) → IN_ORDER14_IN_GA(T9, X14)
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → U1_GA(T19, T20, T21, X37, in_order14_in_ga(T20, X35))
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → IN_ORDER14_IN_GA(T20, X35)
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → U2_GA(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U3_GA(T19, T20, T21, X37, in_order14_in_ga(T21, X36))
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → IN_ORDER14_IN_GA(T21, X36)
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U4_GA(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U4_GA(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U5_GA(T19, T20, T21, X37, app27_in_ggga(T22, T19, T23, X37))
U4_GA(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → APP27_IN_GGGA(T22, T19, T23, X37)
APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → U6_GGGA(T46, T47, T48, T49, X64, app27_in_ggga(T47, T48, T49, X64))
APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → APP27_IN_GGGA(T47, T48, T49, X64)
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → U9_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T75, X99))
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → IN_ORDER14_IN_GA(T75, X99)
IN_ORDER1_IN_GA(tree(T7, tree(T74, T75, T76), T9), T11) → U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_in_ga(T75, T77))
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → U11_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T76, X100))
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → IN_ORDER14_IN_GA(T76, X100)
U10_GA(T7, T74, T75, T76, T9, T11, in_orderc14_out_ga(T75, T77)) → U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_in_ga(T76, T78))
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → U13_GA(T7, T74, T75, T76, T9, T11, app27_in_ggga(T77, T74, T78, X101))
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → APP27_IN_GGGA(T77, T74, T78, X101)
U12_GA(T7, T74, T75, T76, T9, T11, T77, in_orderc14_out_ga(T76, T78)) → U14_GA(T7, T74, T75, T76, T9, T11, appc27_in_ggga(T77, T74, T78, T83))
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → U15_GA(T7, T74, T75, T76, T9, T11, in_order14_in_ga(T9, X14))
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → IN_ORDER14_IN_GA(T9, X14)
U14_GA(T7, T74, T75, T76, T9, T11, appc27_out_ggga(T77, T74, T78, T83)) → U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_in_ga(T9, T88))
U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_out_ga(T9, T88)) → U17_GA(T7, T74, T75, T76, T9, T11, app52_in_ggga(T83, T7, T88, T11))
U16_GA(T7, T74, T75, T76, T9, T11, T83, in_orderc14_out_ga(T9, T88)) → APP52_IN_GGGA(T83, T7, T88, T11)
APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → U7_GGGA(T113, T114, T115, T116, T118, app52_in_ggga(T114, T115, T116, T118))
APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → APP52_IN_GGGA(T114, T115, T116, T118)

The TRS R consists of the following rules:

in_orderc14_in_ga(void, []) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21), X37) → U19_ga(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U19_ga(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U20_ga(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, X37, appc27_in_ggga(T22, T19, T23, X37))
appc27_in_ggga([], T36, T37, .(T36, T37)) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49, .(T46, X64)) → U22_ggga(T46, T47, T48, T49, X64, appc27_in_ggga(T47, T48, T49, X64))
U22_ggga(T46, T47, T48, T49, X64, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, X37, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_order14_in_ga(x1, x2)  =  in_order14_in_ga(x1)
in_orderc14_in_ga(x1, x2)  =  in_orderc14_in_ga(x1)
in_orderc14_out_ga(x1, x2)  =  in_orderc14_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appc27_in_ggga(x1, x2, x3, x4)  =  appc27_in_ggga(x1, x2, x3)
[]  =  []
appc27_out_ggga(x1, x2, x3, x4)  =  appc27_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
app27_in_ggga(x1, x2, x3, x4)  =  app27_in_ggga(x1, x2, x3)
app52_in_ggga(x1, x2, x3, x4)  =  app52_in_ggga(x1, x2, x3)
IN_ORDER1_IN_GA(x1, x2)  =  IN_ORDER1_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
IN_ORDER14_IN_GA(x1, x2)  =  IN_ORDER14_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x5, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
APP27_IN_GGGA(x1, x2, x3, x4)  =  APP27_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GA(x1, x2, x3, x4, x5, x7)
U10_GA(x1, x2, x3, x4, x5, x6, x7)  =  U10_GA(x1, x2, x3, x4, x5, x7)
U11_GA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GA(x1, x2, x3, x4, x5, x7)
U12_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GA(x1, x2, x3, x4, x5, x7, x8)
U13_GA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GA(x1, x2, x3, x4, x5, x7)
U14_GA(x1, x2, x3, x4, x5, x6, x7)  =  U14_GA(x1, x2, x3, x4, x5, x7)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x1, x2, x3, x4, x5, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GA(x1, x2, x3, x4, x5, x7, x8)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
APP52_IN_GGGA(x1, x2, x3, x4)  =  APP52_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → APP52_IN_GGGA(T114, T115, T116, T118)

The TRS R consists of the following rules:

in_orderc14_in_ga(void, []) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21), X37) → U19_ga(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U19_ga(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U20_ga(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, X37, appc27_in_ggga(T22, T19, T23, X37))
appc27_in_ggga([], T36, T37, .(T36, T37)) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49, .(T46, X64)) → U22_ggga(T46, T47, T48, T49, X64, appc27_in_ggga(T47, T48, T49, X64))
U22_ggga(T46, T47, T48, T49, X64, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, X37, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_orderc14_in_ga(x1, x2)  =  in_orderc14_in_ga(x1)
in_orderc14_out_ga(x1, x2)  =  in_orderc14_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appc27_in_ggga(x1, x2, x3, x4)  =  appc27_in_ggga(x1, x2, x3)
[]  =  []
appc27_out_ggga(x1, x2, x3, x4)  =  appc27_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
APP52_IN_GGGA(x1, x2, x3, x4)  =  APP52_IN_GGGA(x1, x2, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APP52_IN_GGGA(.(T113, T114), T115, T116, .(T113, T118)) → APP52_IN_GGGA(T114, T115, T116, T118)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP52_IN_GGGA(x1, x2, x3, x4)  =  APP52_IN_GGGA(x1, x2, x3)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP52_IN_GGGA(.(T113, T114), T115, T116) → APP52_IN_GGGA(T114, T115, T116)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• APP52_IN_GGGA(.(T113, T114), T115, T116) → APP52_IN_GGGA(T114, T115, T116)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(13) YES

(14) Obligation:

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

APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → APP27_IN_GGGA(T47, T48, T49, X64)

The TRS R consists of the following rules:

in_orderc14_in_ga(void, []) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21), X37) → U19_ga(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U19_ga(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U20_ga(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, X37, appc27_in_ggga(T22, T19, T23, X37))
appc27_in_ggga([], T36, T37, .(T36, T37)) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49, .(T46, X64)) → U22_ggga(T46, T47, T48, T49, X64, appc27_in_ggga(T47, T48, T49, X64))
U22_ggga(T46, T47, T48, T49, X64, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, X37, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_orderc14_in_ga(x1, x2)  =  in_orderc14_in_ga(x1)
in_orderc14_out_ga(x1, x2)  =  in_orderc14_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appc27_in_ggga(x1, x2, x3, x4)  =  appc27_in_ggga(x1, x2, x3)
[]  =  []
appc27_out_ggga(x1, x2, x3, x4)  =  appc27_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
APP27_IN_GGGA(x1, x2, x3, x4)  =  APP27_IN_GGGA(x1, x2, x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APP27_IN_GGGA(.(T46, T47), T48, T49, .(T46, X64)) → APP27_IN_GGGA(T47, T48, T49, X64)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP27_IN_GGGA(x1, x2, x3, x4)  =  APP27_IN_GGGA(x1, x2, x3)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APP27_IN_GGGA(.(T46, T47), T48, T49) → APP27_IN_GGGA(T47, T48, T49)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• APP27_IN_GGGA(.(T46, T47), T48, T49) → APP27_IN_GGGA(T47, T48, T49)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(20) YES

(21) Obligation:

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

IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → U2_GA(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U2_GA(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → IN_ORDER14_IN_GA(T21, X36)
IN_ORDER14_IN_GA(tree(T19, T20, T21), X37) → IN_ORDER14_IN_GA(T20, X35)

The TRS R consists of the following rules:

in_orderc14_in_ga(void, []) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21), X37) → U19_ga(T19, T20, T21, X37, in_orderc14_in_ga(T20, T22))
U19_ga(T19, T20, T21, X37, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, X37, T22, in_orderc14_in_ga(T21, T23))
U20_ga(T19, T20, T21, X37, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, X37, appc27_in_ggga(T22, T19, T23, X37))
appc27_in_ggga([], T36, T37, .(T36, T37)) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49, .(T46, X64)) → U22_ggga(T46, T47, T48, T49, X64, appc27_in_ggga(T47, T48, T49, X64))
U22_ggga(T46, T47, T48, T49, X64, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, X37, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_orderc14_in_ga(x1, x2)  =  in_orderc14_in_ga(x1)
in_orderc14_out_ga(x1, x2)  =  in_orderc14_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appc27_in_ggga(x1, x2, x3, x4)  =  appc27_in_ggga(x1, x2, x3)
[]  =  []
appc27_out_ggga(x1, x2, x3, x4)  =  appc27_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
IN_ORDER14_IN_GA(x1, x2)  =  IN_ORDER14_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

IN_ORDER14_IN_GA(tree(T19, T20, T21)) → U2_GA(T19, T20, T21, in_orderc14_in_ga(T20))
U2_GA(T19, T20, T21, in_orderc14_out_ga(T20, T22)) → IN_ORDER14_IN_GA(T21)
IN_ORDER14_IN_GA(tree(T19, T20, T21)) → IN_ORDER14_IN_GA(T20)

The TRS R consists of the following rules:

in_orderc14_in_ga(void) → in_orderc14_out_ga(void, [])
in_orderc14_in_ga(tree(T19, T20, T21)) → U19_ga(T19, T20, T21, in_orderc14_in_ga(T20))
U19_ga(T19, T20, T21, in_orderc14_out_ga(T20, T22)) → U20_ga(T19, T20, T21, T22, in_orderc14_in_ga(T21))
U20_ga(T19, T20, T21, T22, in_orderc14_out_ga(T21, T23)) → U21_ga(T19, T20, T21, appc27_in_ggga(T22, T19, T23))
appc27_in_ggga([], T36, T37) → appc27_out_ggga([], T36, T37, .(T36, T37))
appc27_in_ggga(.(T46, T47), T48, T49) → U22_ggga(T46, T47, T48, T49, appc27_in_ggga(T47, T48, T49))
U22_ggga(T46, T47, T48, T49, appc27_out_ggga(T47, T48, T49, X64)) → appc27_out_ggga(.(T46, T47), T48, T49, .(T46, X64))
U21_ga(T19, T20, T21, appc27_out_ggga(T22, T19, T23, X37)) → in_orderc14_out_ga(tree(T19, T20, T21), X37)

The set Q consists of the following terms:

in_orderc14_in_ga(x0)
U19_ga(x0, x1, x2, x3)
U20_ga(x0, x1, x2, x3, x4)
appc27_in_ggga(x0, x1, x2)
U22_ggga(x0, x1, x2, x3, x4)
U21_ga(x0, x1, x2, x3)

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

(24) 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:

• U2_GA(T19, T20, T21, in_orderc14_out_ga(T20, T22)) → IN_ORDER14_IN_GA(T21)
  The graph contains the following edges 3 >= 1

• IN_ORDER14_IN_GA(tree(T19, T20, T21)) → IN_ORDER14_IN_GA(T20)
  The graph contains the following edges 1 > 1

• IN_ORDER14_IN_GA(tree(T19, T20, T21)) → U2_GA(T19, T20, T21, in_orderc14_in_ga(T20))
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

(25) YES