Left Termination proof of /tmp/tmpLBoMXa/inorder-fb.pl

Left Termination of the query pattern in_order_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 PiDPToQDPProof (⇐)

        ↳23 QDP

        ↳24 Narrowing (⇐)

        ↳25 QDP

        ↳26 NonTerminationProof (⇔)

        ↳27 NO

(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(a,g).

(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\nT2 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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | in_order(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: in_order(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nin_order(T1, T2) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: in_order(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(in_order(T9, X9), ','(in_order(T10, X10), app(X9, .(T11, X10), [])))\n\nX9 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(in_order(T9, X9), ','(in_order(T10, X10), app(X9, .(T11, X10), []))), SCOPE: 2, CLAUSE: 0 | ','(in_order(T9, X9), ','(in_order(T10, X10), app(X9, .(T11, X10), []))), SCOPE: 2, CLAUSE: 1\n\nX9 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(in_order(T9, X9), ','(in_order(T10, X10), app(X9, .(T11, X10), []))), SCOPE: 2, CLAUSE: 0\n\nX9 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(in_order(T9, X9), ','(in_order(T10, X10), app(X9, .(T11, X10), []))), SCOPE: 2, CLAUSE: 1\n\nX9 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(in_order(T12, X10), app([], .(T13, X10), []))\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: in_order(T12, X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], .(T15, T14), [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: in_order(T12, X10), SCOPE: 3, CLAUSE: 0 | in_order(T12, X10), SCOPE: 3, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: in_order(T12, X10), SCOPE: 3, CLAUSE: 0\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: in_order(T12, X10), SCOPE: 3, CLAUSE: 1\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(in_order(T25, X31), ','(in_order(T26, X32), app(X31, .(T27, X32), X33)))\n\nX33 is free\nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: in_order(T25, X31)\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(in_order(T29, X32), app(T28, .(T30, X32), X33))\n\nX33 is free\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: in_order(T29, X32)\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(T32, .(T33, T31), X33)\n\nX33 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T32, .(T33, T31), X33), SCOPE: 4, CLAUSE: 2 | app(T32, .(T33, T31), X33), SCOPE: 4, CLAUSE: 3\n\nX33 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T32, .(T33, T31), X33), SCOPE: 4, CLAUSE: 2\n\nX33 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T32, .(T33, T31), X33), SCOPE: 4, CLAUSE: 3\n\nX33 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T60, .(T61, T62), X60)\n\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app([], .(T15, T14), []), SCOPE: 5, CLAUSE: 2 | app([], .(T15, T14), []), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app([], .(T15, T14), []), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(in_order(T82, X94), ','(in_order(T83, X95), ','(app(X94, .(T84, X95), X96), ','(in_order(T85, X10), app(X96, .(T86, X10), [])))))\n\nX10 is free\nX96 is free; \nX94 is free; \nX95 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: in_order(T82, X94)\n\nX94 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(in_order(T88, X95), ','(app(T87, .(T89, X95), X96), ','(in_order(T90, X10), app(X96, .(T91, X10), []))))\n\nX10 is free\nX96 is free; \nX95 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: in_order(T88, X95)\n\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(app(T93, .(T94, T92), X96), ','(in_order(T95, X10), app(X96, .(T96, X10), [])))\n\nX10 is free\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(T93, .(T94, T92), X96)\n\nX96 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(in_order(T102, X10), app(T101, .(T103, X10), []))\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: in_order(T102, X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app(T109, .(T110, T108), [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(T109, .(T110, T108), []), SCOPE: 6, CLAUSE: 2 | app(T109, .(T110, T108), []), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: app(T109, .(T110, T108), []), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(in_order(T133, X124), ','(in_order(T134, X125), app(X124, .(T135, X125), T132)))\n\nT132 is ground\nX124 is free; \nX125 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(in_order(T133, X124), ','(in_order(T134, X125), app(X124, .(T135, X125), T132))), SCOPE: 7, CLAUSE: 0 | ','(in_order(T133, X124), ','(in_order(T134, X125), app(X124, .(T135, X125), T132))), SCOPE: 7, CLAUSE: 1\n\nT132 is ground\nX124 is free; \nX125 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(in_order(T133, X124), ','(in_order(T134, X125), app(X124, .(T135, X125), T132))), SCOPE: 7, CLAUSE: 0\n\nT132 is ground\nX124 is free; \nX125 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(in_order(T133, X124), ','(in_order(T134, X125), app(X124, .(T135, X125), T132))), SCOPE: 7, CLAUSE: 1\n\nT132 is ground\nX124 is free; \nX125 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(in_order(T136, X125), app([], .(T137, X125), T132))\n\nT132 is ground\nX125 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: in_order(T136, X125)\n\nX125 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: app([], .(T139, T138), T132)\n\nT132 is ground\nin_order(T140, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: app([], .(T139, T138), T132), SCOPE: 8, CLAUSE: 2 | app([], .(T139, T138), T132), SCOPE: 8, CLAUSE: 3\n\nT132 is ground\nin_order(T140, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: app([], .(T139, T138), T132), SCOPE: 8, CLAUSE: 2\n\nT132 is ground\nin_order(T140, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app([], .(T139, T138), T132), SCOPE: 8, CLAUSE: 3\n\nT132 is ground\nin_order(T140, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(in_order(T164, X158), ','(in_order(T165, X159), ','(app(X158, .(T166, X159), X160), ','(in_order(T167, X125), app(X160, .(T168, X125), T132)))))\n\nT132 is ground\nX125 is free; \nX160 is free; \nX158 is free; \nX159 is free; \nin_order(T1, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: in_order(T164, X158)\n\nX158 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(in_order(T170, X159), ','(app(T169, .(T171, X159), X160), ','(in_order(T172, X125), app(X160, .(T173, X125), T132))))\n\nT132 is ground\nX125 is free; \nX160 is free; \nX159 is free; \nin_order(T174, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: in_order(T170, X159)\n\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(app(T176, .(T177, T175), X160), ','(in_order(T178, X125), app(X160, .(T179, X125), T132)))\n\nT132 is ground\nX125 is free; \nX160 is free; \nin_order(T180, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T176, .(T177, T175), X160)\n\nX160 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(in_order(T186, X125), app(T185, .(T187, X125), T132))\n\nT132 is ground\nX125 is free; \nin_order(T188, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: in_order(T186, X125)\n\nX125 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: app(T194, .(T195, T193), T132)\n\nT132 is ground\nin_order(T196, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: app(T194, .(T195, T193), T132), SCOPE: 9, CLAUSE: 2 | app(T194, .(T195, T193), T132), SCOPE: 9, CLAUSE: 3\n\nT132 is ground\nin_order(T196, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: app(T194, .(T195, T193), T132), SCOPE: 9, CLAUSE: 2\n\nT132 is ground\nin_order(T196, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app(T194, .(T195, T193), T132), SCOPE: 9, CLAUSE: 3\n\nT132 is ground\nin_order(T196, T132) !=? in_order(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: app(T226, .(T227, T228), T225)\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: app(T226, .(T227, T228), T225), SCOPE: 10, CLAUSE: 2 | app(T226, .(T227, T228), T225), SCOPE: 10, CLAUSE: 3\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: app(T226, .(T227, T228), T225), SCOPE: 10, CLAUSE: 2\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: app(T226, .(T227, T228), T225), SCOPE: 10, CLAUSE: 3\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: app(T258, .(T259, T260), T257)\n\nT257 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 93 [arrowhead = none ];
93 -> 2;

94 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT1 -> void\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 94 [arrowhead = none ];
94 -> 3;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 95 [arrowhead = none ];
95 -> 4;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="EVAL with clause\nin_order(tree(X120, X121, X122), X123) :- ','(in_order(X121, X124), ','(in_order(X122, X125), app(X124, .(X120, X125), X123))).\nand substitution\nX120 -> T135\nX121 -> T133\nX122 -> T134\nT1 -> tree(T135, T133, T134)\nT2 -> T132\nX123 -> T132\nT130 -> T133\nT131 -> T134\nT129 -> T135", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 97 [arrowhead = none ];
97 -> 51;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 98 [arrowhead = none ];
98 -> 52;

99 [label="EVAL with clause\nin_order(tree(X5, X6, X7), X8) :- ','(in_order(X6, X9), ','(in_order(X7, X10), app(X9, .(X5, X10), X8))).\nand substitution\nX5 -> T11\nX6 -> T9\nX7 -> T10\nT1 -> tree(T11, T9, T10)\nX8 -> []\nT7 -> T9\nT8 -> T10\nT6 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 99 [arrowhead = none ];
99 -> 6;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 100 [arrowhead = none ];
100 -> 7;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 101 [arrowhead = none ];
101 -> 8;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 102 [arrowhead = none ];
102 -> 9;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 103 [arrowhead = none ];
103 -> 10;

104 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT9 -> void\nX9 -> []\nT10 -> T12\nT11 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 104 [arrowhead = none ];
104 -> 11;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 105 [arrowhead = none ];
105 -> 12;

106 [label="EVAL with clause\nin_order(tree(X90, X91, X92), X93) :- ','(in_order(X91, X94), ','(in_order(X92, X95), app(X94, .(X90, X95), X93))).\nand substitution\nX90 -> T84\nX91 -> T82\nX92 -> T83\nT9 -> tree(T84, T82, T83)\nX9 -> X96\nX93 -> X96\nT80 -> T82\nT81 -> T83\nT79 -> T84\nT10 -> T85\nT11 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 106 [arrowhead = none ];
106 -> 38;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 107 [arrowhead = none ];
107 -> 39;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 108 [arrowhead = none ];
108 -> 13;

109 [label="SPLIT 2\nreplacements:\nX10 -> T14\nT13 -> T15", fontsize=14, style = filled, fillcolor = violet];
11 -> 109 [arrowhead = none ];
109 -> 14;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 110 [arrowhead = none ];
110 -> 15;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 111 [arrowhead = none ];
111 -> 35;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 112 [arrowhead = none ];
112 -> 16;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 113 [arrowhead = none ];
113 -> 17;

114 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT12 -> void\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 114 [arrowhead = none ];
114 -> 18;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 115 [arrowhead = none ];
115 -> 19;

116 [label="EVAL with clause\nin_order(tree(X27, X28, X29), X30) :- ','(in_order(X28, X31), ','(in_order(X29, X32), app(X31, .(X27, X32), X30))).\nand substitution\nX27 -> T27\nX28 -> T25\nX29 -> T26\nT12 -> tree(T27, T25, T26)\nX10 -> X33\nX30 -> X33\nT23 -> T25\nT24 -> T26\nT22 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 116 [arrowhead = none ];
116 -> 21;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 118 [arrowhead = none ];
118 -> 20;

119 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 119 [arrowhead = none ];
119 -> 23;

120 [label="SPLIT 2\nreplacements:\nX31 -> T28\nT26 -> T29\nT27 -> T30", fontsize=14, style = filled, fillcolor = violet];
21 -> 120 [arrowhead = none ];
120 -> 24;

121 [label="INSTANCE with matching:\nT12 -> T25\nX10 -> X31", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 121 [arrowhead = none , style=dashed];
121 -> 13[style=dashed];

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 122 [arrowhead = none ];
122 -> 25;

123 [label="SPLIT 2\nreplacements:\nX32 -> T31\nT28 -> T32\nT30 -> T33", fontsize=14, style = filled, fillcolor = violet];
24 -> 123 [arrowhead = none ];
123 -> 26;

124 [label="INSTANCE with matching:\nT12 -> T29\nX10 -> X32", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 124 [arrowhead = none , style=dashed];
124 -> 13[style=dashed];

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 125 [arrowhead = none ];
125 -> 27;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 126 [arrowhead = none ];
126 -> 28;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 127 [arrowhead = none ];
127 -> 29;

128 [label="EVAL with clause\napp([], X44, X44).\nand substitution\nT32 -> []\nT33 -> T46\nT31 -> T47\nX44 -> .(T46, T47)\nX33 -> .(T46, T47)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 128 [arrowhead = none ];
128 -> 30;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 129 [arrowhead = none ];
129 -> 31;

130 [label="EVAL with clause\napp(.(X56, X57), X58, .(X56, X59)) :- app(X57, X58, X59).\nand substitution\nX56 -> T56\nX57 -> T60\nT32 -> .(T56, T60)\nT33 -> T61\nT31 -> T62\nX58 -> .(T61, T62)\nX59 -> X60\nX33 -> .(T56, X60)\nT57 -> T60\nT58 -> T61\nT59 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 130 [arrowhead = none ];
130 -> 33;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 131 [arrowhead = none ];
131 -> 34;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 132 [arrowhead = none ];
132 -> 32;

133 [label="INSTANCE with matching:\nT32 -> T60\nT33 -> T61\nT31 -> T62\nX33 -> X60", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 133 [arrowhead = none , style=dashed];
133 -> 26[style=dashed];

134 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
35 -> 134 [arrowhead = none ];
134 -> 36;

135 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
36 -> 135 [arrowhead = none ];
135 -> 37;

136 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
38 -> 136 [arrowhead = none ];
136 -> 40;

137 [label="SPLIT 2\nreplacements:\nX94 -> T87\nT83 -> T88\nT84 -> T89\nT85 -> T90\nT86 -> T91", fontsize=14, style = filled, fillcolor = violet];
38 -> 137 [arrowhead = none ];
137 -> 41;

138 [label="INSTANCE with matching:\nT12 -> T82\nX10 -> X94", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 138 [arrowhead = none , style=dashed];
138 -> 13[style=dashed];

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 139 [arrowhead = none ];
139 -> 42;

140 [label="SPLIT 2\nreplacements:\nX95 -> T92\nT87 -> T93\nT89 -> T94\nT90 -> T95\nT91 -> T96", fontsize=14, style = filled, fillcolor = violet];
41 -> 140 [arrowhead = none ];
140 -> 43;

141 [label="INSTANCE with matching:\nT12 -> T88\nX10 -> X95", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 141 [arrowhead = none , style=dashed];
141 -> 13[style=dashed];

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 142 [arrowhead = none ];
142 -> 44;

143 [label="SPLIT 2\nreplacements:\nX96 -> T101\nT95 -> T102\nT96 -> T103", fontsize=14, style = filled, fillcolor = violet];
43 -> 143 [arrowhead = none ];
143 -> 45;

144 [label="INSTANCE with matching:\nT32 -> T93\nT33 -> T94\nT31 -> T92\nX33 -> X96", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 144 [arrowhead = none , style=dashed];
144 -> 26[style=dashed];

145 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 145 [arrowhead = none ];
145 -> 46;

146 [label="SPLIT 2\nreplacements:\nX10 -> T108\nT101 -> T109\nT103 -> T110", fontsize=14, style = filled, fillcolor = violet];
45 -> 146 [arrowhead = none ];
146 -> 47;

147 [label="INSTANCE with matching:\nT12 -> T102", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 147 [arrowhead = none , style=dashed];
147 -> 13[style=dashed];

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 148 [arrowhead = none ];
148 -> 48;

149 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 149 [arrowhead = none ];
149 -> 49;

150 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 150 [arrowhead = none ];
150 -> 50;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 151 [arrowhead = none ];
151 -> 53;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 152 [arrowhead = none ];
152 -> 54;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 153 [arrowhead = none ];
153 -> 55;

154 [label="EVAL with clause\nin_order(void, []).\nand substitution\nT133 -> void\nX124 -> []\nT134 -> T136\nT135 -> T137", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 154 [arrowhead = none ];
154 -> 56;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 155 [arrowhead = none ];
155 -> 57;

156 [label="EVAL with clause\nin_order(tree(X154, X155, X156), X157) :- ','(in_order(X155, X158), ','(in_order(X156, X159), app(X158, .(X154, X159), X157))).\nand substitution\nX154 -> T166\nX155 -> T164\nX156 -> T165\nT133 -> tree(T166, T164, T165)\nX124 -> X160\nX157 -> X160\nT162 -> T164\nT163 -> T165\nT161 -> T166\nT134 -> T167\nT135 -> T168", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 156 [arrowhead = none ];
156 -> 67;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 157 [arrowhead = none ];
157 -> 68;

158 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 158 [arrowhead = none ];
158 -> 58;

159 [label="SPLIT 2\nreplacements:\nX125 -> T138\nT137 -> T139\nT1 -> T140", fontsize=14, style = filled, fillcolor = violet];
56 -> 159 [arrowhead = none ];
159 -> 59;

160 [label="INSTANCE with matching:\nT12 -> T136\nX10 -> X125", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 160 [arrowhead = none , style=dashed];
160 -> 13[style=dashed];

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 161 [arrowhead = none ];
161 -> 60;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 162 [arrowhead = none ];
162 -> 61;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 163 [arrowhead = none ];
163 -> 62;

164 [label="EVAL with clause\napp([], X136, X136).\nand substitution\nT139 -> T153\nT138 -> T154\nX136 -> .(T153, T154)\nT132 -> .(T153, T154)", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 164 [arrowhead = none ];
164 -> 63;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 165 [arrowhead = none ];
165 -> 64;

166 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 166 [arrowhead = none ];
166 -> 66;

167 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 167 [arrowhead = none ];
167 -> 65;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 168 [arrowhead = none ];
168 -> 69;

169 [label="SPLIT 2\nreplacements:\nX158 -> T169\nT165 -> T170\nT166 -> T171\nT167 -> T172\nT168 -> T173\nT1 -> T174", fontsize=14, style = filled, fillcolor = violet];
67 -> 169 [arrowhead = none ];
169 -> 70;

170 [label="INSTANCE with matching:\nT136 -> T164\nX125 -> X158", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 170 [arrowhead = none , style=dashed];
170 -> 58[style=dashed];

171 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
70 -> 171 [arrowhead = none ];
171 -> 71;

172 [label="SPLIT 2\nreplacements:\nX159 -> T175\nT169 -> T176\nT171 -> T177\nT172 -> T178\nT173 -> T179\nT174 -> T180", fontsize=14, style = filled, fillcolor = violet];
70 -> 172 [arrowhead = none ];
172 -> 72;

173 [label="INSTANCE with matching:\nT136 -> T170\nX125 -> X159", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 173 [arrowhead = none , style=dashed];
173 -> 58[style=dashed];

174 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 174 [arrowhead = none ];
174 -> 73;

175 [label="SPLIT 2\nreplacements:\nX160 -> T185\nT178 -> T186\nT179 -> T187\nT180 -> T188", fontsize=14, style = filled, fillcolor = violet];
72 -> 175 [arrowhead = none ];
175 -> 74;

176 [label="INSTANCE with matching:\nT93 -> T176\nT94 -> T177\nT92 -> T175\nX96 -> X160", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 176 [arrowhead = none , style=dashed];
176 -> 44[style=dashed];

177 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
74 -> 177 [arrowhead = none ];
177 -> 75;

178 [label="SPLIT 2\nreplacements:\nX125 -> T193\nT185 -> T194\nT187 -> T195\nT188 -> T196", fontsize=14, style = filled, fillcolor = violet];
74 -> 178 [arrowhead = none ];
178 -> 76;

179 [label="INSTANCE with matching:\nT136 -> T186", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 179 [arrowhead = none , style=dashed];
179 -> 58[style=dashed];

180 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 180 [arrowhead = none ];
180 -> 77;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 181 [arrowhead = none ];
181 -> 78;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 182 [arrowhead = none ];
182 -> 79;

183 [label="EVAL with clause\napp([], X175, X175).\nand substitution\nT194 -> []\nT195 -> T209\nT193 -> T210\nX175 -> .(T209, T210)\nT132 -> .(T209, T210)", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 183 [arrowhead = none ];
183 -> 80;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 184 [arrowhead = none ];
184 -> 81;

185 [label="EVAL with clause\napp(.(X185, X186), X187, .(X185, X188)) :- app(X186, X187, X188).\nand substitution\nX185 -> T221\nX186 -> T226\nT194 -> .(T221, T226)\nT195 -> T227\nT193 -> T228\nX187 -> .(T227, T228)\nX188 -> T225\nT132 -> .(T221, T225)\nT222 -> T226\nT223 -> T227\nT224 -> T228", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 185 [arrowhead = none ];
185 -> 83;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 186 [arrowhead = none ];
186 -> 84;

187 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 187 [arrowhead = none ];
187 -> 82;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 188 [arrowhead = none ];
188 -> 85;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 189 [arrowhead = none ];
189 -> 86;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
85 -> 190 [arrowhead = none ];
190 -> 87;

191 [label="EVAL with clause\napp([], X199, X199).\nand substitution\nT226 -> []\nT227 -> T241\nT228 -> T242\nX199 -> .(T241, T242)\nT225 -> .(T241, T242)", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 191 [arrowhead = none ];
191 -> 88;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 192 [arrowhead = none ];
192 -> 89;

193 [label="EVAL with clause\napp(.(X209, X210), X211, .(X209, X212)) :- app(X210, X211, X212).\nand substitution\nX209 -> T253\nX210 -> T258\nT226 -> .(T253, T258)\nT227 -> T259\nT228 -> T260\nX211 -> .(T259, T260)\nX212 -> T257\nT225 -> .(T253, T257)\nT254 -> T258\nT255 -> T259\nT256 -> T260", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 193 [arrowhead = none ];
193 -> 91;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 194 [arrowhead = none ];
194 -> 92;

195 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 195 [arrowhead = none ];
195 -> 90;

196 [label="INSTANCE with matching:\nT226 -> T258\nT227 -> T259\nT228 -> T260\nT225 -> T257", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 196 [arrowhead = none , style=dashed];
196 -> 83[style=dashed];

}

(2) Obligation:

Triples:

in_order13(tree(T27, T25, T26), X33) :- in_order13(T25, X31).
in_order13(tree(T30, T25, T29), X33) :- ','(in_orderc13(T25, T28), in_order13(T29, X32)).
in_order13(tree(T33, T25, T29), X33) :- ','(in_orderc13(T25, T32), ','(in_orderc13(T29, T31), app26(T32, T33, T31, X33))).
app26(.(T56, T60), T61, T62, .(T56, X60)) :- app26(T60, T61, T62, X60).
app83(.(T253, T258), T259, T260, .(T253, T257)) :- app83(T258, T259, T260, T257).
in_order1(tree(T13, void, T12), []) :- in_order13(T12, X10).
in_order1(tree(T86, tree(T84, T82, T83), T85), []) :- in_order13(T82, X94).
in_order1(tree(T91, tree(T89, T82, T88), T90), []) :- ','(in_orderc13(T82, T87), in_order13(T88, X95)).
in_order1(tree(T96, tree(T94, T82, T88), T95), []) :- ','(in_orderc13(T82, T93), ','(in_orderc13(T88, T92), app26(T93, T94, T92, X96))).
in_order1(tree(T103, tree(T94, T82, T88), T102), []) :- ','(in_orderc13(T82, T93), ','(in_orderc13(T88, T92), ','(appc26(T93, T94, T92, T101), in_order13(T102, X10)))).
in_order1(tree(T137, void, T136), T132) :- in_order13(T136, X125).
in_order1(tree(T168, tree(T166, T164, T165), T167), T132) :- in_order13(T164, X158).
in_order1(tree(T173, tree(T171, T164, T170), T172), T132) :- ','(in_orderc13(T164, T169), in_order13(T170, X159)).
in_order1(tree(T179, tree(T177, T164, T170), T178), T132) :- ','(in_orderc13(T164, T176), ','(in_orderc13(T170, T175), app26(T176, T177, T175, X160))).
in_order1(tree(T187, tree(T177, T164, T170), T186), T132) :- ','(in_orderc13(T164, T176), ','(in_orderc13(T170, T175), ','(appc26(T176, T177, T175, T185), in_order13(T186, X125)))).
in_order1(tree(T227, tree(T177, T164, T170), T186), .(T221, T225)) :- ','(in_orderc13(T164, T176), ','(in_orderc13(T170, T175), ','(appc26(T176, T177, T175, .(T221, T226)), ','(in_orderc13(T186, T228), app83(T226, T227, T228, T225))))).

Clauses:

in_orderc13(void, []).
in_orderc13(tree(T33, T25, T29), X33) :- ','(in_orderc13(T25, T32), ','(in_orderc13(T29, T31), appc26(T32, T33, T31, X33))).
appc26([], T46, T47, .(T46, T47)).
appc26(.(T56, T60), T61, T62, .(T56, X60)) :- appc26(T60, T61, T62, X60).
appc83([], T241, T242, .(T241, T242)).
appc83(.(T253, T258), T259, T260, .(T253, T257)) :- appc83(T258, T259, T260, T257).

Afs:

in_order1(x1, x2) = in_order1(x2)

(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: (f,b)
in_order13_in: (f,f)
in_orderc13_in: (f,f)
appc26_in: (b,f,b,f)
app26_in: (b,f,b,f)
app83_in: (b,f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

IN_ORDER1_IN_AG(tree(T13, void, T12), []) → U9_AG(T13, T12, in_order13_in_aa(T12, X10))
IN_ORDER1_IN_AG(tree(T13, void, T12), []) → IN_ORDER13_IN_AA(T12, X10)
IN_ORDER13_IN_AA(tree(T27, T25, T26), X33) → U1_AA(T27, T25, T26, X33, in_order13_in_aa(T25, X31))
IN_ORDER13_IN_AA(tree(T27, T25, T26), X33) → IN_ORDER13_IN_AA(T25, X31)
IN_ORDER13_IN_AA(tree(T30, T25, T29), X33) → U2_AA(T30, T25, T29, X33, in_orderc13_in_aa(T25, T28))
U2_AA(T30, T25, T29, X33, in_orderc13_out_aa(T25, T28)) → U3_AA(T30, T25, T29, X33, in_order13_in_aa(T29, X32))
U2_AA(T30, T25, T29, X33, in_orderc13_out_aa(T25, T28)) → IN_ORDER13_IN_AA(T29, X32)
IN_ORDER13_IN_AA(tree(T33, T25, T29), X33) → U4_AA(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U4_AA(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U5_AA(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U5_AA(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U6_AA(T33, T25, T29, X33, app26_in_gaga(T32, T33, T31, X33))
U5_AA(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → APP26_IN_GAGA(T32, T33, T31, X33)
APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → U7_GAGA(T56, T60, T61, T62, X60, app26_in_gaga(T60, T61, T62, X60))
APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → APP26_IN_GAGA(T60, T61, T62, X60)
IN_ORDER1_IN_AG(tree(T86, tree(T84, T82, T83), T85), []) → U10_AG(T86, T84, T82, T83, T85, in_order13_in_aa(T82, X94))
IN_ORDER1_IN_AG(tree(T86, tree(T84, T82, T83), T85), []) → IN_ORDER13_IN_AA(T82, X94)
IN_ORDER1_IN_AG(tree(T91, tree(T89, T82, T88), T90), []) → U11_AG(T91, T89, T82, T88, T90, in_orderc13_in_aa(T82, T87))
U11_AG(T91, T89, T82, T88, T90, in_orderc13_out_aa(T82, T87)) → U12_AG(T91, T89, T82, T88, T90, in_order13_in_aa(T88, X95))
U11_AG(T91, T89, T82, T88, T90, in_orderc13_out_aa(T82, T87)) → IN_ORDER13_IN_AA(T88, X95)
IN_ORDER1_IN_AG(tree(T96, tree(T94, T82, T88), T95), []) → U13_AG(T96, T94, T82, T88, T95, in_orderc13_in_aa(T82, T93))
U13_AG(T96, T94, T82, T88, T95, in_orderc13_out_aa(T82, T93)) → U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_in_aa(T88, T92))
U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_out_aa(T88, T92)) → U15_AG(T96, T94, T82, T88, T95, app26_in_gaga(T93, T94, T92, X96))
U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_out_aa(T88, T92)) → APP26_IN_GAGA(T93, T94, T92, X96)
IN_ORDER1_IN_AG(tree(T103, tree(T94, T82, T88), T102), []) → U16_AG(T103, T94, T82, T88, T102, in_orderc13_in_aa(T82, T93))
U16_AG(T103, T94, T82, T88, T102, in_orderc13_out_aa(T82, T93)) → U17_AG(T103, T94, T82, T88, T102, T93, in_orderc13_in_aa(T88, T92))
U17_AG(T103, T94, T82, T88, T102, T93, in_orderc13_out_aa(T88, T92)) → U18_AG(T103, T94, T82, T88, T102, appc26_in_gaga(T93, T94, T92, T101))
U18_AG(T103, T94, T82, T88, T102, appc26_out_gaga(T93, T94, T92, T101)) → U19_AG(T103, T94, T82, T88, T102, in_order13_in_aa(T102, X10))
U18_AG(T103, T94, T82, T88, T102, appc26_out_gaga(T93, T94, T92, T101)) → IN_ORDER13_IN_AA(T102, X10)
IN_ORDER1_IN_AG(tree(T137, void, T136), T132) → U20_AG(T137, T136, T132, in_order13_in_aa(T136, X125))
IN_ORDER1_IN_AG(tree(T137, void, T136), T132) → IN_ORDER13_IN_AA(T136, X125)
IN_ORDER1_IN_AG(tree(T168, tree(T166, T164, T165), T167), T132) → U21_AG(T168, T166, T164, T165, T167, T132, in_order13_in_aa(T164, X158))
IN_ORDER1_IN_AG(tree(T168, tree(T166, T164, T165), T167), T132) → IN_ORDER13_IN_AA(T164, X158)
IN_ORDER1_IN_AG(tree(T173, tree(T171, T164, T170), T172), T132) → U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_in_aa(T164, T169))
U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_out_aa(T164, T169)) → U23_AG(T173, T171, T164, T170, T172, T132, in_order13_in_aa(T170, X159))
U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_out_aa(T164, T169)) → IN_ORDER13_IN_AA(T170, X159)
IN_ORDER1_IN_AG(tree(T179, tree(T177, T164, T170), T178), T132) → U24_AG(T179, T177, T164, T170, T178, T132, in_orderc13_in_aa(T164, T176))
U24_AG(T179, T177, T164, T170, T178, T132, in_orderc13_out_aa(T164, T176)) → U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_in_aa(T170, T175))
U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_out_aa(T170, T175)) → U26_AG(T179, T177, T164, T170, T178, T132, app26_in_gaga(T176, T177, T175, X160))
U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_out_aa(T170, T175)) → APP26_IN_GAGA(T176, T177, T175, X160)
IN_ORDER1_IN_AG(tree(T187, tree(T177, T164, T170), T186), T132) → U27_AG(T187, T177, T164, T170, T186, T132, in_orderc13_in_aa(T164, T176))
U27_AG(T187, T177, T164, T170, T186, T132, in_orderc13_out_aa(T164, T176)) → U28_AG(T187, T177, T164, T170, T186, T132, T176, in_orderc13_in_aa(T170, T175))
U28_AG(T187, T177, T164, T170, T186, T132, T176, in_orderc13_out_aa(T170, T175)) → U29_AG(T187, T177, T164, T170, T186, T132, appc26_in_gaga(T176, T177, T175, T185))
U29_AG(T187, T177, T164, T170, T186, T132, appc26_out_gaga(T176, T177, T175, T185)) → U30_AG(T187, T177, T164, T170, T186, T132, in_order13_in_aa(T186, X125))
U29_AG(T187, T177, T164, T170, T186, T132, appc26_out_gaga(T176, T177, T175, T185)) → IN_ORDER13_IN_AA(T186, X125)
IN_ORDER1_IN_AG(tree(T227, tree(T177, T164, T170), T186), .(T221, T225)) → U31_AG(T227, T177, T164, T170, T186, T221, T225, in_orderc13_in_aa(T164, T176))
U31_AG(T227, T177, T164, T170, T186, T221, T225, in_orderc13_out_aa(T164, T176)) → U32_AG(T227, T177, T164, T170, T186, T221, T225, T176, in_orderc13_in_aa(T170, T175))
U32_AG(T227, T177, T164, T170, T186, T221, T225, T176, in_orderc13_out_aa(T170, T175)) → U33_AG(T227, T177, T164, T170, T186, T221, T225, appc26_in_gaga(T176, T177, T175, .(T221, T226)))
U33_AG(T227, T177, T164, T170, T186, T221, T225, appc26_out_gaga(T176, T177, T175, .(T221, T226))) → U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_in_aa(T186, T228))
U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_out_aa(T186, T228)) → U35_AG(T227, T177, T164, T170, T186, T221, T225, app83_in_gagg(T226, T227, T228, T225))
U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_out_aa(T186, T228)) → APP83_IN_GAGG(T226, T227, T228, T225)
APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → U8_GAGG(T253, T258, T259, T260, T257, app83_in_gagg(T258, T259, T260, T257))
APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → APP83_IN_GAGG(T258, T259, T260, T257)

The TRS R consists of the following rules:

in_orderc13_in_aa(void, []) → in_orderc13_out_aa(void, [])
in_orderc13_in_aa(tree(T33, T25, T29), X33) → U37_aa(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U37_aa(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U38_aa(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U38_aa(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T33, T25, T29, X33, appc26_in_gaga(T32, T33, T31, X33))
appc26_in_gaga([], T46, T47, .(T46, T47)) → appc26_out_gaga([], T46, T47, .(T46, T47))
appc26_in_gaga(.(T56, T60), T61, T62, .(T56, X60)) → U40_gaga(T56, T60, T61, T62, X60, appc26_in_gaga(T60, T61, T62, X60))
U40_gaga(T56, T60, T61, T62, X60, appc26_out_gaga(T60, T61, T62, X60)) → appc26_out_gaga(.(T56, T60), T61, T62, .(T56, X60))
U39_aa(T33, T25, T29, X33, appc26_out_gaga(T32, T33, T31, X33)) → in_orderc13_out_aa(tree(T33, T25, T29), X33)

The argument filtering Pi contains the following mapping:
[] = []
in_order13_in_aa(x1, x2) = in_order13_in_aa
in_orderc13_in_aa(x1, x2) = in_orderc13_in_aa
in_orderc13_out_aa(x1, x2) = in_orderc13_out_aa(x1, x2)
U37_aa(x1, x2, x3, x4, x5) = U37_aa(x5)
U38_aa(x1, x2, x3, x4, x5, x6) = U38_aa(x2, x5, x6)
U39_aa(x1, x2, x3, x4, x5) = U39_aa(x2, x3, x5)
appc26_in_gaga(x1, x2, x3, x4) = appc26_in_gaga(x1, x3)
appc26_out_gaga(x1, x2, x3, x4) = appc26_out_gaga(x1, x3, x4)
.(x1, x2) = .(x2)
U40_gaga(x1, x2, x3, x4, x5, x6) = U40_gaga(x2, x4, x6)
tree(x1, x2, x3) = tree(x2, x3)
app26_in_gaga(x1, x2, x3, x4) = app26_in_gaga(x1, x3)
app83_in_gagg(x1, x2, x3, x4) = app83_in_gagg(x1, x3, x4)
IN_ORDER1_IN_AG(x1, x2) = IN_ORDER1_IN_AG(x2)
U9_AG(x1, x2, x3) = U9_AG(x3)
IN_ORDER13_IN_AA(x1, x2) = IN_ORDER13_IN_AA
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U4_AA(x1, x2, x3, x4, x5) = U4_AA(x5)
U5_AA(x1, x2, x3, x4, x5, x6) = U5_AA(x5, x6)
U6_AA(x1, x2, x3, x4, x5) = U6_AA(x5)
APP26_IN_GAGA(x1, x2, x3, x4) = APP26_IN_GAGA(x1, x3)
U7_GAGA(x1, x2, x3, x4, x5, x6) = U7_GAGA(x2, x4, x6)
U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x6)
U11_AG(x1, x2, x3, x4, x5, x6) = U11_AG(x6)
U12_AG(x1, x2, x3, x4, x5, x6) = U12_AG(x6)
U13_AG(x1, x2, x3, x4, x5, x6) = U13_AG(x6)
U14_AG(x1, x2, x3, x4, x5, x6, x7) = U14_AG(x6, x7)
U15_AG(x1, x2, x3, x4, x5, x6) = U15_AG(x6)
U16_AG(x1, x2, x3, x4, x5, x6) = U16_AG(x6)
U17_AG(x1, x2, x3, x4, x5, x6, x7) = U17_AG(x6, x7)
U18_AG(x1, x2, x3, x4, x5, x6) = U18_AG(x6)
U19_AG(x1, x2, x3, x4, x5, x6) = U19_AG(x6)
U20_AG(x1, x2, x3, x4) = U20_AG(x3, x4)
U21_AG(x1, x2, x3, x4, x5, x6, x7) = U21_AG(x6, x7)
U22_AG(x1, x2, x3, x4, x5, x6, x7) = U22_AG(x6, x7)
U23_AG(x1, x2, x3, x4, x5, x6, x7) = U23_AG(x6, x7)
U24_AG(x1, x2, x3, x4, x5, x6, x7) = U24_AG(x6, x7)
U25_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U25_AG(x6, x7, x8)
U26_AG(x1, x2, x3, x4, x5, x6, x7) = U26_AG(x6, x7)
U27_AG(x1, x2, x3, x4, x5, x6, x7) = U27_AG(x6, x7)
U28_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U28_AG(x6, x7, x8)
U29_AG(x1, x2, x3, x4, x5, x6, x7) = U29_AG(x6, x7)
U30_AG(x1, x2, x3, x4, x5, x6, x7) = U30_AG(x6, x7)
U31_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U31_AG(x7, x8)
U32_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U32_AG(x7, x8, x9)
U33_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U33_AG(x7, x8)
U34_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U34_AG(x7, x8, x9)
U35_AG(x1, x2, x3, x4, x5, x6, x7, x8) = U35_AG(x7, x8)
APP83_IN_GAGG(x1, x2, x3, x4) = APP83_IN_GAGG(x1, x3, x4)
U8_GAGG(x1, x2, x3, x4, x5, x6) = U8_GAGG(x2, x4, x5, 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_AG(tree(T13, void, T12), []) → U9_AG(T13, T12, in_order13_in_aa(T12, X10))
IN_ORDER1_IN_AG(tree(T13, void, T12), []) → IN_ORDER13_IN_AA(T12, X10)
IN_ORDER13_IN_AA(tree(T27, T25, T26), X33) → U1_AA(T27, T25, T26, X33, in_order13_in_aa(T25, X31))
IN_ORDER13_IN_AA(tree(T27, T25, T26), X33) → IN_ORDER13_IN_AA(T25, X31)
IN_ORDER13_IN_AA(tree(T30, T25, T29), X33) → U2_AA(T30, T25, T29, X33, in_orderc13_in_aa(T25, T28))
U2_AA(T30, T25, T29, X33, in_orderc13_out_aa(T25, T28)) → U3_AA(T30, T25, T29, X33, in_order13_in_aa(T29, X32))
U2_AA(T30, T25, T29, X33, in_orderc13_out_aa(T25, T28)) → IN_ORDER13_IN_AA(T29, X32)
IN_ORDER13_IN_AA(tree(T33, T25, T29), X33) → U4_AA(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U4_AA(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U5_AA(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U5_AA(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U6_AA(T33, T25, T29, X33, app26_in_gaga(T32, T33, T31, X33))
U5_AA(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → APP26_IN_GAGA(T32, T33, T31, X33)
APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → U7_GAGA(T56, T60, T61, T62, X60, app26_in_gaga(T60, T61, T62, X60))
APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → APP26_IN_GAGA(T60, T61, T62, X60)
IN_ORDER1_IN_AG(tree(T86, tree(T84, T82, T83), T85), []) → U10_AG(T86, T84, T82, T83, T85, in_order13_in_aa(T82, X94))
IN_ORDER1_IN_AG(tree(T86, tree(T84, T82, T83), T85), []) → IN_ORDER13_IN_AA(T82, X94)
IN_ORDER1_IN_AG(tree(T91, tree(T89, T82, T88), T90), []) → U11_AG(T91, T89, T82, T88, T90, in_orderc13_in_aa(T82, T87))
U11_AG(T91, T89, T82, T88, T90, in_orderc13_out_aa(T82, T87)) → U12_AG(T91, T89, T82, T88, T90, in_order13_in_aa(T88, X95))
U11_AG(T91, T89, T82, T88, T90, in_orderc13_out_aa(T82, T87)) → IN_ORDER13_IN_AA(T88, X95)
IN_ORDER1_IN_AG(tree(T96, tree(T94, T82, T88), T95), []) → U13_AG(T96, T94, T82, T88, T95, in_orderc13_in_aa(T82, T93))
U13_AG(T96, T94, T82, T88, T95, in_orderc13_out_aa(T82, T93)) → U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_in_aa(T88, T92))
U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_out_aa(T88, T92)) → U15_AG(T96, T94, T82, T88, T95, app26_in_gaga(T93, T94, T92, X96))
U14_AG(T96, T94, T82, T88, T95, T93, in_orderc13_out_aa(T88, T92)) → APP26_IN_GAGA(T93, T94, T92, X96)
IN_ORDER1_IN_AG(tree(T103, tree(T94, T82, T88), T102), []) → U16_AG(T103, T94, T82, T88, T102, in_orderc13_in_aa(T82, T93))
U16_AG(T103, T94, T82, T88, T102, in_orderc13_out_aa(T82, T93)) → U17_AG(T103, T94, T82, T88, T102, T93, in_orderc13_in_aa(T88, T92))
U17_AG(T103, T94, T82, T88, T102, T93, in_orderc13_out_aa(T88, T92)) → U18_AG(T103, T94, T82, T88, T102, appc26_in_gaga(T93, T94, T92, T101))
U18_AG(T103, T94, T82, T88, T102, appc26_out_gaga(T93, T94, T92, T101)) → U19_AG(T103, T94, T82, T88, T102, in_order13_in_aa(T102, X10))
U18_AG(T103, T94, T82, T88, T102, appc26_out_gaga(T93, T94, T92, T101)) → IN_ORDER13_IN_AA(T102, X10)
IN_ORDER1_IN_AG(tree(T137, void, T136), T132) → U20_AG(T137, T136, T132, in_order13_in_aa(T136, X125))
IN_ORDER1_IN_AG(tree(T137, void, T136), T132) → IN_ORDER13_IN_AA(T136, X125)
IN_ORDER1_IN_AG(tree(T168, tree(T166, T164, T165), T167), T132) → U21_AG(T168, T166, T164, T165, T167, T132, in_order13_in_aa(T164, X158))
IN_ORDER1_IN_AG(tree(T168, tree(T166, T164, T165), T167), T132) → IN_ORDER13_IN_AA(T164, X158)
IN_ORDER1_IN_AG(tree(T173, tree(T171, T164, T170), T172), T132) → U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_in_aa(T164, T169))
U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_out_aa(T164, T169)) → U23_AG(T173, T171, T164, T170, T172, T132, in_order13_in_aa(T170, X159))
U22_AG(T173, T171, T164, T170, T172, T132, in_orderc13_out_aa(T164, T169)) → IN_ORDER13_IN_AA(T170, X159)
IN_ORDER1_IN_AG(tree(T179, tree(T177, T164, T170), T178), T132) → U24_AG(T179, T177, T164, T170, T178, T132, in_orderc13_in_aa(T164, T176))
U24_AG(T179, T177, T164, T170, T178, T132, in_orderc13_out_aa(T164, T176)) → U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_in_aa(T170, T175))
U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_out_aa(T170, T175)) → U26_AG(T179, T177, T164, T170, T178, T132, app26_in_gaga(T176, T177, T175, X160))
U25_AG(T179, T177, T164, T170, T178, T132, T176, in_orderc13_out_aa(T170, T175)) → APP26_IN_GAGA(T176, T177, T175, X160)
IN_ORDER1_IN_AG(tree(T187, tree(T177, T164, T170), T186), T132) → U27_AG(T187, T177, T164, T170, T186, T132, in_orderc13_in_aa(T164, T176))
U27_AG(T187, T177, T164, T170, T186, T132, in_orderc13_out_aa(T164, T176)) → U28_AG(T187, T177, T164, T170, T186, T132, T176, in_orderc13_in_aa(T170, T175))
U28_AG(T187, T177, T164, T170, T186, T132, T176, in_orderc13_out_aa(T170, T175)) → U29_AG(T187, T177, T164, T170, T186, T132, appc26_in_gaga(T176, T177, T175, T185))
U29_AG(T187, T177, T164, T170, T186, T132, appc26_out_gaga(T176, T177, T175, T185)) → U30_AG(T187, T177, T164, T170, T186, T132, in_order13_in_aa(T186, X125))
U29_AG(T187, T177, T164, T170, T186, T132, appc26_out_gaga(T176, T177, T175, T185)) → IN_ORDER13_IN_AA(T186, X125)
IN_ORDER1_IN_AG(tree(T227, tree(T177, T164, T170), T186), .(T221, T225)) → U31_AG(T227, T177, T164, T170, T186, T221, T225, in_orderc13_in_aa(T164, T176))
U31_AG(T227, T177, T164, T170, T186, T221, T225, in_orderc13_out_aa(T164, T176)) → U32_AG(T227, T177, T164, T170, T186, T221, T225, T176, in_orderc13_in_aa(T170, T175))
U32_AG(T227, T177, T164, T170, T186, T221, T225, T176, in_orderc13_out_aa(T170, T175)) → U33_AG(T227, T177, T164, T170, T186, T221, T225, appc26_in_gaga(T176, T177, T175, .(T221, T226)))
U33_AG(T227, T177, T164, T170, T186, T221, T225, appc26_out_gaga(T176, T177, T175, .(T221, T226))) → U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_in_aa(T186, T228))
U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_out_aa(T186, T228)) → U35_AG(T227, T177, T164, T170, T186, T221, T225, app83_in_gagg(T226, T227, T228, T225))
U34_AG(T227, T177, T164, T170, T186, T221, T225, T226, in_orderc13_out_aa(T186, T228)) → APP83_IN_GAGG(T226, T227, T228, T225)
APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → U8_GAGG(T253, T258, T259, T260, T257, app83_in_gagg(T258, T259, T260, T257))
APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → APP83_IN_GAGG(T258, T259, T260, T257)

The TRS R consists of the following rules:

in_orderc13_in_aa(void, []) → in_orderc13_out_aa(void, [])
in_orderc13_in_aa(tree(T33, T25, T29), X33) → U37_aa(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U37_aa(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U38_aa(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U38_aa(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T33, T25, T29, X33, appc26_in_gaga(T32, T33, T31, X33))
appc26_in_gaga([], T46, T47, .(T46, T47)) → appc26_out_gaga([], T46, T47, .(T46, T47))
appc26_in_gaga(.(T56, T60), T61, T62, .(T56, X60)) → U40_gaga(T56, T60, T61, T62, X60, appc26_in_gaga(T60, T61, T62, X60))
U40_gaga(T56, T60, T61, T62, X60, appc26_out_gaga(T60, T61, T62, X60)) → appc26_out_gaga(.(T56, T60), T61, T62, .(T56, X60))
U39_aa(T33, T25, T29, X33, appc26_out_gaga(T32, T33, T31, X33)) → in_orderc13_out_aa(tree(T33, T25, T29), X33)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 46 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → APP83_IN_GAGG(T258, T259, T260, T257)

The TRS R consists of the following rules:

in_orderc13_in_aa(void, []) → in_orderc13_out_aa(void, [])
in_orderc13_in_aa(tree(T33, T25, T29), X33) → U37_aa(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U37_aa(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U38_aa(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U38_aa(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T33, T25, T29, X33, appc26_in_gaga(T32, T33, T31, X33))
appc26_in_gaga([], T46, T47, .(T46, T47)) → appc26_out_gaga([], T46, T47, .(T46, T47))
appc26_in_gaga(.(T56, T60), T61, T62, .(T56, X60)) → U40_gaga(T56, T60, T61, T62, X60, appc26_in_gaga(T60, T61, T62, X60))
U40_gaga(T56, T60, T61, T62, X60, appc26_out_gaga(T60, T61, T62, X60)) → appc26_out_gaga(.(T56, T60), T61, T62, .(T56, X60))
U39_aa(T33, T25, T29, X33, appc26_out_gaga(T32, T33, T31, X33)) → in_orderc13_out_aa(tree(T33, T25, T29), X33)

The argument filtering Pi contains the following mapping:
[] = []
in_orderc13_in_aa(x1, x2) = in_orderc13_in_aa
in_orderc13_out_aa(x1, x2) = in_orderc13_out_aa(x1, x2)
U37_aa(x1, x2, x3, x4, x5) = U37_aa(x5)
U38_aa(x1, x2, x3, x4, x5, x6) = U38_aa(x2, x5, x6)
U39_aa(x1, x2, x3, x4, x5) = U39_aa(x2, x3, x5)
appc26_in_gaga(x1, x2, x3, x4) = appc26_in_gaga(x1, x3)
appc26_out_gaga(x1, x2, x3, x4) = appc26_out_gaga(x1, x3, x4)
.(x1, x2) = .(x2)
U40_gaga(x1, x2, x3, x4, x5, x6) = U40_gaga(x2, x4, x6)
tree(x1, x2, x3) = tree(x2, x3)
APP83_IN_GAGG(x1, x2, x3, x4) = APP83_IN_GAGG(x1, x3, x4)

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:

APP83_IN_GAGG(.(T253, T258), T259, T260, .(T253, T257)) → APP83_IN_GAGG(T258, T259, T260, T257)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP83_IN_GAGG(x1, x2, x3, x4) = APP83_IN_GAGG(x1, x3, x4)

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:

APP83_IN_GAGG(.(T258), T260, .(T257)) → APP83_IN_GAGG(T258, T260, T257)

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:

APP83_IN_GAGG(.(T258), T260, .(T257)) → APP83_IN_GAGG(T258, T260, T257)
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:

APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → APP26_IN_GAGA(T60, T61, T62, X60)

The TRS R consists of the following rules:

in_orderc13_in_aa(void, []) → in_orderc13_out_aa(void, [])
in_orderc13_in_aa(tree(T33, T25, T29), X33) → U37_aa(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U37_aa(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U38_aa(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U38_aa(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T33, T25, T29, X33, appc26_in_gaga(T32, T33, T31, X33))
appc26_in_gaga([], T46, T47, .(T46, T47)) → appc26_out_gaga([], T46, T47, .(T46, T47))
appc26_in_gaga(.(T56, T60), T61, T62, .(T56, X60)) → U40_gaga(T56, T60, T61, T62, X60, appc26_in_gaga(T60, T61, T62, X60))
U40_gaga(T56, T60, T61, T62, X60, appc26_out_gaga(T60, T61, T62, X60)) → appc26_out_gaga(.(T56, T60), T61, T62, .(T56, X60))
U39_aa(T33, T25, T29, X33, appc26_out_gaga(T32, T33, T31, X33)) → in_orderc13_out_aa(tree(T33, T25, T29), X33)

The argument filtering Pi contains the following mapping:
[] = []
in_orderc13_in_aa(x1, x2) = in_orderc13_in_aa
in_orderc13_out_aa(x1, x2) = in_orderc13_out_aa(x1, x2)
U37_aa(x1, x2, x3, x4, x5) = U37_aa(x5)
U38_aa(x1, x2, x3, x4, x5, x6) = U38_aa(x2, x5, x6)
U39_aa(x1, x2, x3, x4, x5) = U39_aa(x2, x3, x5)
appc26_in_gaga(x1, x2, x3, x4) = appc26_in_gaga(x1, x3)
appc26_out_gaga(x1, x2, x3, x4) = appc26_out_gaga(x1, x3, x4)
.(x1, x2) = .(x2)
U40_gaga(x1, x2, x3, x4, x5, x6) = U40_gaga(x2, x4, x6)
tree(x1, x2, x3) = tree(x2, x3)
APP26_IN_GAGA(x1, x2, x3, x4) = APP26_IN_GAGA(x1, 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:

APP26_IN_GAGA(.(T56, T60), T61, T62, .(T56, X60)) → APP26_IN_GAGA(T60, T61, T62, X60)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP26_IN_GAGA(x1, x2, x3, x4) = APP26_IN_GAGA(x1, 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:

APP26_IN_GAGA(.(T60), T62) → APP26_IN_GAGA(T60, T62)

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:

APP26_IN_GAGA(.(T60), T62) → APP26_IN_GAGA(T60, T62)
The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

IN_ORDER13_IN_AA(tree(T30, T25, T29), X33) → U2_AA(T30, T25, T29, X33, in_orderc13_in_aa(T25, T28))
U2_AA(T30, T25, T29, X33, in_orderc13_out_aa(T25, T28)) → IN_ORDER13_IN_AA(T29, X32)
IN_ORDER13_IN_AA(tree(T27, T25, T26), X33) → IN_ORDER13_IN_AA(T25, X31)

The TRS R consists of the following rules:

in_orderc13_in_aa(void, []) → in_orderc13_out_aa(void, [])
in_orderc13_in_aa(tree(T33, T25, T29), X33) → U37_aa(T33, T25, T29, X33, in_orderc13_in_aa(T25, T32))
U37_aa(T33, T25, T29, X33, in_orderc13_out_aa(T25, T32)) → U38_aa(T33, T25, T29, X33, T32, in_orderc13_in_aa(T29, T31))
U38_aa(T33, T25, T29, X33, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T33, T25, T29, X33, appc26_in_gaga(T32, T33, T31, X33))
appc26_in_gaga([], T46, T47, .(T46, T47)) → appc26_out_gaga([], T46, T47, .(T46, T47))
appc26_in_gaga(.(T56, T60), T61, T62, .(T56, X60)) → U40_gaga(T56, T60, T61, T62, X60, appc26_in_gaga(T60, T61, T62, X60))
U40_gaga(T56, T60, T61, T62, X60, appc26_out_gaga(T60, T61, T62, X60)) → appc26_out_gaga(.(T56, T60), T61, T62, .(T56, X60))
U39_aa(T33, T25, T29, X33, appc26_out_gaga(T32, T33, T31, X33)) → in_orderc13_out_aa(tree(T33, T25, T29), X33)

The argument filtering Pi contains the following mapping:
[] = []
in_orderc13_in_aa(x1, x2) = in_orderc13_in_aa
in_orderc13_out_aa(x1, x2) = in_orderc13_out_aa(x1, x2)
U37_aa(x1, x2, x3, x4, x5) = U37_aa(x5)
U38_aa(x1, x2, x3, x4, x5, x6) = U38_aa(x2, x5, x6)
U39_aa(x1, x2, x3, x4, x5) = U39_aa(x2, x3, x5)
appc26_in_gaga(x1, x2, x3, x4) = appc26_in_gaga(x1, x3)
appc26_out_gaga(x1, x2, x3, x4) = appc26_out_gaga(x1, x3, x4)
.(x1, x2) = .(x2)
U40_gaga(x1, x2, x3, x4, x5, x6) = U40_gaga(x2, x4, x6)
tree(x1, x2, x3) = tree(x2, x3)
IN_ORDER13_IN_AA(x1, x2) = IN_ORDER13_IN_AA
U2_AA(x1, x2, x3, x4, x5) = U2_AA(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_ORDER13_IN_AA → U2_AA(in_orderc13_in_aa)
U2_AA(in_orderc13_out_aa(T25, T28)) → IN_ORDER13_IN_AA
IN_ORDER13_IN_AA → IN_ORDER13_IN_AA

The TRS R consists of the following rules:

in_orderc13_in_aa → in_orderc13_out_aa(void, [])
in_orderc13_in_aa → U37_aa(in_orderc13_in_aa)
U37_aa(in_orderc13_out_aa(T25, T32)) → U38_aa(T25, T32, in_orderc13_in_aa)
U38_aa(T25, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T25, T29, appc26_in_gaga(T32, T31))
appc26_in_gaga([], T47) → appc26_out_gaga([], T47, .(T47))
appc26_in_gaga(.(T60), T62) → U40_gaga(T60, T62, appc26_in_gaga(T60, T62))
U40_gaga(T60, T62, appc26_out_gaga(T60, T62, X60)) → appc26_out_gaga(.(T60), T62, .(X60))
U39_aa(T25, T29, appc26_out_gaga(T32, T31, X33)) → in_orderc13_out_aa(tree(T25, T29), X33)

The set Q consists of the following terms:

in_orderc13_in_aa
U37_aa(x0)
U38_aa(x0, x1, x2)
appc26_in_gaga(x0, x1)
U40_gaga(x0, x1, x2)
U39_aa(x0, x1, x2)

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

(24) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule IN_ORDER13_IN_AA → U2_AA(in_orderc13_in_aa) at position [0] we obtained the following new rules [LPAR04]:

IN_ORDER13_IN_AA → U2_AA(in_orderc13_out_aa(void, []))
IN_ORDER13_IN_AA → U2_AA(U37_aa(in_orderc13_in_aa))

(25) Obligation:

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

U2_AA(in_orderc13_out_aa(T25, T28)) → IN_ORDER13_IN_AA
IN_ORDER13_IN_AA → IN_ORDER13_IN_AA
IN_ORDER13_IN_AA → U2_AA(in_orderc13_out_aa(void, []))
IN_ORDER13_IN_AA → U2_AA(U37_aa(in_orderc13_in_aa))

The TRS R consists of the following rules:

in_orderc13_in_aa → in_orderc13_out_aa(void, [])
in_orderc13_in_aa → U37_aa(in_orderc13_in_aa)
U37_aa(in_orderc13_out_aa(T25, T32)) → U38_aa(T25, T32, in_orderc13_in_aa)
U38_aa(T25, T32, in_orderc13_out_aa(T29, T31)) → U39_aa(T25, T29, appc26_in_gaga(T32, T31))
appc26_in_gaga([], T47) → appc26_out_gaga([], T47, .(T47))
appc26_in_gaga(.(T60), T62) → U40_gaga(T60, T62, appc26_in_gaga(T60, T62))
U40_gaga(T60, T62, appc26_out_gaga(T60, T62, X60)) → appc26_out_gaga(.(T60), T62, .(X60))
U39_aa(T25, T29, appc26_out_gaga(T32, T31, X33)) → in_orderc13_out_aa(tree(T25, T29), X33)

The set Q consists of the following terms:

in_orderc13_in_aa
U37_aa(x0)
U38_aa(x0, x1, x2)
appc26_in_gaga(x0, x1)
U40_gaga(x0, x1, x2)
U39_aa(x0, x1, x2)

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

(26) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = IN_ORDER13_IN_AA evaluates to t =IN_ORDER13_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from IN_ORDER13_IN_AA to IN_ORDER13_IN_AA.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) PiDPToQDPProof (SOUND transformation)

(23) Obligation:

(24) Narrowing (SOUND transformation)

(25) Obligation:

(26) NonTerminationProof (EQUIVALENT transformation)

(27) NO