Left Termination of the query pattern front_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

front(void, []).
front(tree(X, void, void), .(X, [])).
front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

front(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: front(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: front(T1, T2), SCOPE: 1, CLAUSE: 0 | front(T1, T2), SCOPE: 1, CLAUSE: 1 | front(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | front(void, T2), SCOPE: 1, CLAUSE: 1 | front(void, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: front(T1, T2), SCOPE: 1, CLAUSE: 1 | front(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nfront(T1, T2) !=? front(void, [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: front(void, T2), SCOPE: 1, CLAUSE: 1 | front(void, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: front(void, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | front(tree(T4, void, void), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: front(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nfront(T1, T2) !=? front(void, [])\nfront(T1, T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: front(tree(T4, void, void), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9)))\n\nX18 is free\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 0 | ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 1 | ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 2\n\nX18 is free\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 0\n\nX18 is free\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 1 | ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 2\n\nX18 is free\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(front(void, X19), app([], X19, T9))\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: front(void, X19)\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app([], T13, T9)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: front(void, X19), SCOPE: 3, CLAUSE: 0 | front(void, X19), SCOPE: 3, CLAUSE: 1 | front(void, X19), SCOPE: 3, CLAUSE: 2\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: front(void, X19), SCOPE: 3, CLAUSE: 0\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: front(void, X19), SCOPE: 3, CLAUSE: 1 | front(void, X19), SCOPE: 3, CLAUSE: 2\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: front(void, X19), SCOPE: 3, CLAUSE: 2\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app([], T13, T9), SCOPE: 4, CLAUSE: 3 | app([], T13, T9), SCOPE: 4, CLAUSE: 4\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app([], T13, T9), SCOPE: 4, CLAUSE: 3\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app([], T13, T9), SCOPE: 4, CLAUSE: 4\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(front(void, X18), ','(front(void, X19), app(X18, X19, T9))), SCOPE: 2, CLAUSE: 2\n\nX18 is free\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29)))\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 0 | ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 1 | ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 0\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 1 | ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(front(T27, X57), app([], X57, T29))\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: front(T27, X57)\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app([], T30, T29)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: front(T27, X57), SCOPE: 6, CLAUSE: 0 | front(T27, X57), SCOPE: 6, CLAUSE: 1 | front(T27, X57), SCOPE: 6, CLAUSE: 2\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: front(T27, X57), SCOPE: 6, CLAUSE: 0\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: front(T27, X57), SCOPE: 6, CLAUSE: 1 | front(T27, X57), SCOPE: 6, CLAUSE: 2\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: front(T27, X57), SCOPE: 6, CLAUSE: 1\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: front(T27, X57), SCOPE: 6, CLAUSE: 2\n\nT27 is ground\nX57 is free; \nfront(tree(T25, void, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(front(T43, X84), ','(front(T44, X85), app(X84, X85, X86)))\n\nT43 is ground\nT44 is ground\nX86 is free; \nX84 is free; \nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: front(T43, X84)\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(front(T44, X85), app(T45, X85, X86))\n\nT44 is ground\nT45 is ground\nX86 is free; \nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: front(T43, X84), SCOPE: 7, CLAUSE: 0 | front(T43, X84), SCOPE: 7, CLAUSE: 1 | front(T43, X84), SCOPE: 7, CLAUSE: 2\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: front(T43, X84), SCOPE: 7, CLAUSE: 0\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: front(T43, X84), SCOPE: 7, CLAUSE: 1 | front(T43, X84), SCOPE: 7, CLAUSE: 2\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: front(T43, X84), SCOPE: 7, CLAUSE: 1\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: front(T43, X84), SCOPE: 7, CLAUSE: 2\n\nT43 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(front(T58, X113), ','(front(T59, X114), app(X113, X114, X115)))\n\nT58 is ground\nT59 is ground\nX115 is free; \nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: front(T44, X85)\n\nT44 is ground\nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T45, T60, X86)\n\nT45 is ground\nT60 is ground\nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T45, T60, X86), SCOPE: 8, CLAUSE: 3 | app(T45, T60, X86), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nT60 is ground\nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: app(T45, T60, X86), SCOPE: 8, CLAUSE: 3\n\nT45 is ground\nT60 is ground\nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: app(T45, T60, X86), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nT60 is ground\nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: app(T75, T76, X142)\n\nT75 is ground\nT76 is ground\nX142 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 1\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(front(T26, X56), ','(front(T27, X57), app(X56, X57, T29))), SCOPE: 5, CLAUSE: 2\n\nT26 is ground\nT27 is ground\nX56 is free; \nX57 is free; \nfront(tree(T25, T26, T27), T2) !=? front(tree(X8, void, void), .(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(front(T27, X57), app(.(T85, []), X57, T29))\n\nT27 is ground\nT85 is ground\nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: front(T27, X57)\n\nT27 is ground\nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: app(.(T85, []), T86, T29)\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: app(.(T85, []), T86, T29), SCOPE: 9, CLAUSE: 3 | app(.(T85, []), T86, T29), SCOPE: 9, CLAUSE: 4\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: app(.(T85, []), T86, T29), SCOPE: 9, CLAUSE: 4\n\nT85 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: app([], T97, T99)\n\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(front(T109, X198), ','(front(T110, X199), ','(app(X198, X199, X200), ','(front(T27, X57), app(X200, X57, T29)))))\n\nT27 is ground\nT109 is ground\nT110 is ground\nX57 is free; \nX200 is free; \nX198 is free; \nX199 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: front(T109, X198)\n\nT109 is ground\nX198 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(front(T110, X199), ','(app(T111, X199, X200), ','(front(T27, X57), app(X200, X57, T29))))\n\nT27 is ground\nT110 is ground\nT111 is ground\nX57 is free; \nX200 is free; \nX199 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: front(T110, X199)\n\nT110 is ground\nX199 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(app(T111, T112, X200), ','(front(T27, X57), app(X200, X57, T29)))\n\nT27 is ground\nT111 is ground\nT112 is ground\nX57 is free; \nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: app(T111, T112, X200)\n\nT111 is ground\nT112 is ground\nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(front(T27, X57), app(T115, X57, T29))\n\nT27 is ground\nT115 is ground\nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: front(T27, X57)\n\nT27 is ground\nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: app(T115, T118, T29)\n\nT115 is ground\nT118 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: app(T115, T118, T29), SCOPE: 10, CLAUSE: 3 | app(T115, T118, T29), SCOPE: 10, CLAUSE: 4\n\nT115 is ground\nT118 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: app(T115, T118, T29), SCOPE: 10, CLAUSE: 3\n\nT115 is ground\nT118 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: app(T115, T118, T29), SCOPE: 10, CLAUSE: 4\n\nT115 is ground\nT118 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: app(T135, T136, T138)\n\nT135 is ground\nT136 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 109 [arrowhead = none ];
109 -> 2;

110 [label="EVAL with clause\nfront(void, []).\nand substitution\nT1 -> void\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 110 [arrowhead = none ];
110 -> 3;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 111 [arrowhead = none ];
111 -> 4;

112 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 112 [arrowhead = none ];
112 -> 5;

113 [label="EVAL with clause\nfront(tree(X8, void, void), .(X8, [])).\nand substitution\nX8 -> T4\nT1 -> tree(T4, void, void)\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 113 [arrowhead = none ];
113 -> 8;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 114 [arrowhead = none ];
114 -> 9;

115 [label="BACKTRACK\nfor clause: front(tree(X, void, void), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 115 [arrowhead = none ];
115 -> 6;

116 [label="BACKTRACK\nfor clause: front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 116 [arrowhead = none ];
116 -> 7;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 117 [arrowhead = none ];
117 -> 10;

118 [label="EVAL with clause\nfront(tree(X52, X53, X54), X55) :- ','(front(X53, X56), ','(front(X54, X57), app(X56, X57, X55))).\nand substitution\nX52 -> T25\nX53 -> T26\nX54 -> T27\nT1 -> tree(T25, T26, T27)\nT2 -> T29\nX55 -> T29\nT28 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 118 [arrowhead = none ];
118 -> 34;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 119 [arrowhead = none ];
119 -> 35;

120 [label="ONLY EVAL with clause\nfront(tree(X14, X15, X16), X17) :- ','(front(X15, X18), ','(front(X16, X19), app(X18, X19, X17))).\nand substitution\nT4 -> T7\nX14 -> T7\nX15 -> void\nX16 -> void\nT2 -> T9\nX17 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = white];
10 -> 120 [arrowhead = none ];
120 -> 11;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 121 [arrowhead = none ];
121 -> 12;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 122 [arrowhead = none ];
122 -> 13;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 123 [arrowhead = none ];
123 -> 14;

124 [label="ONLY EVAL with clause\nfront(void, []).\nand substitution\nX18 -> []", fontsize=14, style = filled, fillcolor = white];
13 -> 124 [arrowhead = none ];
124 -> 15;

125 [label="BACKTRACK\nfor clause: front(tree(X, void, void), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 125 [arrowhead = none ];
125 -> 32;

126 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 126 [arrowhead = none ];
126 -> 16;

127 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nreplacements:\nX19 -> T13", fontsize=14, style = filled, fillcolor = violet];
15 -> 127 [arrowhead = none ];
127 -> 17;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 128 [arrowhead = none ];
128 -> 18;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 129 [arrowhead = none ];
129 -> 25;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 130 [arrowhead = none ];
130 -> 19;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 131 [arrowhead = none ];
131 -> 20;

132 [label="ONLY EVAL with clause\nfront(void, []).\nand substitution\nX19 -> []", fontsize=14, style = filled, fillcolor = white];
19 -> 132 [arrowhead = none ];
132 -> 21;

133 [label="BACKTRACK\nfor clause: front(tree(X, void, void), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 133 [arrowhead = none ];
133 -> 23;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 134 [arrowhead = none ];
134 -> 22;

135 [label="BACKTRACK\nfor clause: front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 135 [arrowhead = none ];
135 -> 24;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 136 [arrowhead = none ];
136 -> 26;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 137 [arrowhead = none ];
137 -> 27;

138 [label="EVAL with clause\napp([], X37, X37).\nand substitution\nT13 -> T20\nX37 -> T20\nT9 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 138 [arrowhead = none ];
138 -> 28;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 139 [arrowhead = none ];
139 -> 29;

140 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 140 [arrowhead = none ];
140 -> 31;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 141 [arrowhead = none ];
141 -> 30;

142 [label="BACKTRACK\nfor clause: front(tree(X1, L, R), Xs) :- ','(front(L, Ls), ','(front(R, Rs), app(Ls, Rs, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 142 [arrowhead = none ];
142 -> 33;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 143 [arrowhead = none ];
143 -> 36;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 144 [arrowhead = none ];
144 -> 37;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 145 [arrowhead = none ];
145 -> 38;

146 [label="EVAL with clause\nfront(void, []).\nand substitution\nT26 -> void\nX56 -> []", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 146 [arrowhead = none ];
146 -> 39;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 147 [arrowhead = none ];
147 -> 40;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 148 [arrowhead = none ];
148 -> 81;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 149 [arrowhead = none ];
149 -> 82;

150 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
39 -> 150 [arrowhead = none ];
150 -> 41;

151 [label="SPLIT 2\nnew knowledge:\nT30 is ground\nreplacements:\nX57 -> T30", fontsize=14, style = filled, fillcolor = violet];
39 -> 151 [arrowhead = none ];
151 -> 42;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 152 [arrowhead = none ];
152 -> 43;

153 [label="INSTANCE with matching:\nT13 -> T30\nT9 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 153 [arrowhead = none , style=dashed];
153 -> 17[style=dashed];

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 154 [arrowhead = none ];
154 -> 44;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 155 [arrowhead = none ];
155 -> 45;

156 [label="EVAL with clause\nfront(void, []).\nand substitution\nT27 -> void\nX57 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 156 [arrowhead = none ];
156 -> 46;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 157 [arrowhead = none ];
157 -> 47;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 158 [arrowhead = none ];
158 -> 49;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 159 [arrowhead = none ];
159 -> 50;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 160 [arrowhead = none ];
160 -> 48;

161 [label="EVAL with clause\nfront(tree(X66, void, void), .(X66, [])).\nand substitution\nX66 -> T35\nT27 -> tree(T35, void, void)\nX57 -> .(T35, [])", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 161 [arrowhead = none ];
161 -> 51;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 162 [arrowhead = none ];
162 -> 52;

163 [label="EVAL with clause\nfront(tree(X80, X81, X82), X83) :- ','(front(X81, X84), ','(front(X82, X85), app(X84, X85, X83))).\nand substitution\nX80 -> T42\nX81 -> T43\nX82 -> T44\nT27 -> tree(T42, T43, T44)\nX57 -> X86\nX83 -> X86", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 163 [arrowhead = none ];
163 -> 54;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 164 [arrowhead = none ];
164 -> 55;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 165 [arrowhead = none ];
165 -> 53;

166 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 166 [arrowhead = none ];
166 -> 56;

167 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nreplacements:\nX84 -> T45", fontsize=14, style = filled, fillcolor = violet];
54 -> 167 [arrowhead = none ];
167 -> 57;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 168 [arrowhead = none ];
168 -> 58;

169 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 169 [arrowhead = none ];
169 -> 71;

170 [label="SPLIT 2\nnew knowledge:\nT60 is ground\nreplacements:\nX85 -> T60", fontsize=14, style = filled, fillcolor = violet];
57 -> 170 [arrowhead = none ];
170 -> 72;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 171 [arrowhead = none ];
171 -> 59;

172 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 172 [arrowhead = none ];
172 -> 60;

173 [label="EVAL with clause\nfront(void, []).\nand substitution\nT43 -> void\nX84 -> []", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 173 [arrowhead = none ];
173 -> 61;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 174 [arrowhead = none ];
174 -> 62;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 175 [arrowhead = none ];
175 -> 64;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 176 [arrowhead = none ];
176 -> 65;

177 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 177 [arrowhead = none ];
177 -> 63;

178 [label="EVAL with clause\nfront(tree(X95, void, void), .(X95, [])).\nand substitution\nX95 -> T50\nT43 -> tree(T50, void, void)\nX84 -> .(T50, [])", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 178 [arrowhead = none ];
178 -> 66;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 179 [arrowhead = none ];
179 -> 67;

180 [label="EVAL with clause\nfront(tree(X109, X110, X111), X112) :- ','(front(X110, X113), ','(front(X111, X114), app(X113, X114, X112))).\nand substitution\nX109 -> T57\nX110 -> T58\nX111 -> T59\nT43 -> tree(T57, T58, T59)\nX84 -> X115\nX112 -> X115", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 180 [arrowhead = none ];
180 -> 69;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 181 [arrowhead = none ];
181 -> 70;

182 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 182 [arrowhead = none ];
182 -> 68;

183 [label="INSTANCE with matching:\nT43 -> T58\nX84 -> X113\nT44 -> T59\nX85 -> X114\nX86 -> X115", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 183 [arrowhead = none , style=dashed];
183 -> 54[style=dashed];

184 [label="INSTANCE with matching:\nT43 -> T44\nX84 -> X85", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 184 [arrowhead = none , style=dashed];
184 -> 56[style=dashed];

185 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 185 [arrowhead = none ];
185 -> 73;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 186 [arrowhead = none ];
186 -> 74;

187 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 187 [arrowhead = none ];
187 -> 75;

188 [label="EVAL with clause\napp([], X126, X126).\nand substitution\nT45 -> []\nT60 -> T67\nX126 -> T67\nX86 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 188 [arrowhead = none ];
188 -> 76;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 189 [arrowhead = none ];
189 -> 77;

190 [label="EVAL with clause\napp(.(X138, X139), X140, .(X138, X141)) :- app(X139, X140, X141).\nand substitution\nX138 -> T74\nX139 -> T75\nT45 -> .(T74, T75)\nT60 -> T76\nX140 -> T76\nX141 -> X142\nX86 -> .(T74, X142)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 190 [arrowhead = none ];
190 -> 79;

191 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 191 [arrowhead = none ];
191 -> 80;

192 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 192 [arrowhead = none ];
192 -> 78;

193 [label="INSTANCE with matching:\nT45 -> T75\nT60 -> T76\nX86 -> X142", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 193 [arrowhead = none , style=dashed];
193 -> 72[style=dashed];

194 [label="EVAL with clause\nfront(tree(X155, void, void), .(X155, [])).\nand substitution\nX155 -> T85\nT26 -> tree(T85, void, void)\nX56 -> .(T85, [])", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 194 [arrowhead = none ];
194 -> 83;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 195 [arrowhead = none ];
195 -> 84;

196 [label="EVAL with clause\nfront(tree(X194, X195, X196), X197) :- ','(front(X195, X198), ','(front(X196, X199), app(X198, X199, X197))).\nand substitution\nX194 -> T108\nX195 -> T109\nX196 -> T110\nT26 -> tree(T108, T109, T110)\nX56 -> X200\nX197 -> X200", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 196 [arrowhead = none ];
196 -> 91;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 197 [arrowhead = none ];
197 -> 92;

198 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
83 -> 198 [arrowhead = none ];
198 -> 85;

199 [label="SPLIT 2\nnew knowledge:\nT86 is ground\nreplacements:\nX57 -> T86", fontsize=14, style = filled, fillcolor = violet];
83 -> 199 [arrowhead = none ];
199 -> 86;

200 [label="INSTANCE with matching:\nT43 -> T27\nX84 -> X57", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 200 [arrowhead = none , style=dashed];
200 -> 56[style=dashed];

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
86 -> 201 [arrowhead = none ];
201 -> 87;

202 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
87 -> 202 [arrowhead = none ];
202 -> 88;

203 [label="EVAL with clause\napp(.(X172, X173), X174, .(X172, X175)) :- app(X173, X174, X175).\nand substitution\nT85 -> T96\nX172 -> T96\nX173 -> []\nT86 -> T97\nX174 -> T97\nX175 -> T99\nT29 -> .(T96, T99)\nT98 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 203 [arrowhead = none ];
203 -> 89;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 204 [arrowhead = none ];
204 -> 90;

205 [label="INSTANCE with matching:\nT30 -> T97\nT29 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 205 [arrowhead = none , style=dashed];
205 -> 42[style=dashed];

206 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
91 -> 206 [arrowhead = none ];
206 -> 93;

207 [label="SPLIT 2\nnew knowledge:\nT111 is ground\nreplacements:\nX198 -> T111", fontsize=14, style = filled, fillcolor = violet];
91 -> 207 [arrowhead = none ];
207 -> 94;

208 [label="INSTANCE with matching:\nT27 -> T109\nX57 -> X198", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 208 [arrowhead = none , style=dashed];
208 -> 85[style=dashed];

209 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
94 -> 209 [arrowhead = none ];
209 -> 95;

210 [label="SPLIT 2\nnew knowledge:\nT112 is ground\nreplacements:\nX199 -> T112", fontsize=14, style = filled, fillcolor = violet];
94 -> 210 [arrowhead = none ];
210 -> 96;

211 [label="INSTANCE with matching:\nT27 -> T110\nX57 -> X199", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 211 [arrowhead = none , style=dashed];
211 -> 85[style=dashed];

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
96 -> 212 [arrowhead = none ];
212 -> 97;

213 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nreplacements:\nX200 -> T115", fontsize=14, style = filled, fillcolor = violet];
96 -> 213 [arrowhead = none ];
213 -> 98;

214 [label="INSTANCE with matching:\nT45 -> T111\nT60 -> T112\nX86 -> X200", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 214 [arrowhead = none , style=dashed];
214 -> 72[style=dashed];

215 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
98 -> 215 [arrowhead = none ];
215 -> 99;

216 [label="SPLIT 2\nnew knowledge:\nT118 is ground\nreplacements:\nX57 -> T118", fontsize=14, style = filled, fillcolor = violet];
98 -> 216 [arrowhead = none ];
216 -> 100;

217 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
99 -> 217 [arrowhead = none , style=dashed];
217 -> 85[style=dashed];

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 218 [arrowhead = none ];
218 -> 101;

219 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 219 [arrowhead = none ];
219 -> 102;

220 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 220 [arrowhead = none ];
220 -> 103;

221 [label="EVAL with clause\napp([], X215, X215).\nand substitution\nT115 -> []\nT118 -> T125\nX215 -> T125\nT29 -> T125", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 221 [arrowhead = none ];
221 -> 104;

222 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 222 [arrowhead = none ];
222 -> 105;

223 [label="EVAL with clause\napp(.(X225, X226), X227, .(X225, X228)) :- app(X226, X227, X228).\nand substitution\nX225 -> T134\nX226 -> T135\nT115 -> .(T134, T135)\nT118 -> T136\nX227 -> T136\nX228 -> T138\nT29 -> .(T134, T138)\nT137 -> T138", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 223 [arrowhead = none ];
223 -> 107;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 224 [arrowhead = none ];
224 -> 108;

225 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 225 [arrowhead = none ];
225 -> 106;

226 [label="INSTANCE with matching:\nT115 -> T135\nT118 -> T136\nT29 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
107 -> 226 [arrowhead = none , style=dashed];
226 -> 100[style=dashed];

}

(2) Obligation:

Triples:

p54(T43, X84, T44, X85, X86) :- front56(T43, X84).
p54(T43, T45, T44, X85, X86) :- ','(frontc56(T43, T45), front56(T44, X85)).
p54(T43, T45, T44, T60, X86) :- ','(frontc56(T43, T45), ','(frontc56(T44, T60), app72(T45, T60, X86))).
front56(tree(T57, T58, T59), X115) :- p54(T58, X113, T59, X114, X115).
app72(.(T74, T75), T76, .(T74, X142)) :- app72(T75, T76, X142).
app100(.(T134, T135), T136, .(T134, T138)) :- app100(T135, T136, T138).
front1(tree(T7, void, void), T9) :- ','(frontc16(T13), app17(T13, T9)).
front1(tree(T25, void, tree(T42, T43, T44)), T29) :- p54(T43, X84, T44, X85, X86).
front1(tree(T25, void, T27), T29) :- ','(frontc41(T27, T30), app17(T30, T29)).
front1(tree(T25, tree(T85, void, void), T27), T29) :- front56(T27, X57).
front1(tree(T25, tree(T96, void, void), T27), .(T96, T99)) :- ','(frontc56(T27, T97), app17(T97, T99)).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- front56(T109, X198).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), front56(T110, X199)).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), app72(T111, T112, X200))).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), ','(appc72(T111, T112, T115), front56(T27, X57)))).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), ','(appc72(T111, T112, T115), ','(frontc56(T27, T118), app100(T115, T118, T29))))).

Clauses:

appc17(T20, T20).
qc54(T43, T45, T44, T60, X86) :- ','(frontc56(T43, T45), ','(frontc56(T44, T60), appc72(T45, T60, X86))).
frontc56(void, []).
frontc56(tree(T50, void, void), .(T50, [])).
frontc56(tree(T57, T58, T59), X115) :- qc54(T58, X113, T59, X114, X115).
appc72([], T67, T67).
appc72(.(T74, T75), T76, .(T74, X142)) :- appc72(T75, T76, X142).
appc100([], T125, T125).
appc100(.(T134, T135), T136, .(T134, T138)) :- appc100(T135, T136, T138).
frontc16([]).
frontc41(void, []).
frontc41(tree(T35, void, void), .(T35, [])).
frontc41(tree(T42, T43, T44), X86) :- qc54(T43, X84, T44, X85, X86).

Afs:

front1(x1, x2)  =  front1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

p54(T43, X84, T44, X85, X86) :- front56(T43, X84).
p54(T43, T45, T44, X85, X86) :- ','(frontc56(T43, T45), front56(T44, X85)).
p54(T43, T45, T44, T60, X86) :- ','(frontc56(T43, T45), ','(frontc56(T44, T60), app72(T45, T60, X86))).
front56(tree(T57, T58, T59), X115) :- p54(T58, X113, T59, X114, X115).
app72(.(T74, T75), T76, .(T74, X142)) :- app72(T75, T76, X142).
app100(.(T134, T135), T136, .(T134, T138)) :- app100(T135, T136, T138).
front1(tree(T25, void, tree(T42, T43, T44)), T29) :- p54(T43, X84, T44, X85, X86).
front1(tree(T25, tree(T85, void, void), T27), T29) :- front56(T27, X57).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- front56(T109, X198).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), front56(T110, X199)).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), app72(T111, T112, X200))).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), ','(appc72(T111, T112, T115), front56(T27, X57)))).
front1(tree(T25, tree(T108, T109, T110), T27), T29) :- ','(frontc56(T109, T111), ','(frontc56(T110, T112), ','(appc72(T111, T112, T115), ','(frontc56(T27, T118), app100(T115, T118, T29))))).

Clauses:

appc17(T20, T20).
qc54(T43, T45, T44, T60, X86) :- ','(frontc56(T43, T45), ','(frontc56(T44, T60), appc72(T45, T60, X86))).
frontc56(void, []).
frontc56(tree(T50, void, void), .(T50, [])).
frontc56(tree(T57, T58, T59), X115) :- qc54(T58, X113, T59, X114, X115).
appc72([], T67, T67).
appc72(.(T74, T75), T76, .(T74, X142)) :- appc72(T75, T76, X142).
appc100([], T125, T125).
appc100(.(T134, T135), T136, .(T134, T138)) :- appc100(T135, T136, T138).
frontc16([]).
frontc41(void, []).
frontc41(tree(T35, void, void), .(T35, [])).
frontc41(tree(T42, T43, T44), X86) :- qc54(T43, X84, T44, X85, X86).

Afs:

front1(x1, x2)  =  front1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
front1_in: (b,f)
p54_in: (b,f,b,f,f)
front56_in: (b,f)
frontc56_in: (b,f)
qc54_in: (b,f,b,f,f)
appc72_in: (b,b,f)
app72_in: (b,b,f)
app100_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FRONT1_IN_GA(tree(T25, void, tree(T42, T43, T44)), T29) → U10_GA(T25, T42, T43, T44, T29, p54_in_gagaa(T43, X84, T44, X85, X86))
FRONT1_IN_GA(tree(T25, void, tree(T42, T43, T44)), T29) → P54_IN_GAGAA(T43, X84, T44, X85, X86)
P54_IN_GAGAA(T43, X84, T44, X85, X86) → U1_GAGAA(T43, X84, T44, X85, X86, front56_in_ga(T43, X84))
P54_IN_GAGAA(T43, X84, T44, X85, X86) → FRONT56_IN_GA(T43, X84)
FRONT56_IN_GA(tree(T57, T58, T59), X115) → U7_GA(T57, T58, T59, X115, p54_in_gagaa(T58, X113, T59, X114, X115))
FRONT56_IN_GA(tree(T57, T58, T59), X115) → P54_IN_GAGAA(T58, X113, T59, X114, X115)
P54_IN_GAGAA(T43, T45, T44, X85, X86) → U2_GAGAA(T43, T45, T44, X85, X86, frontc56_in_ga(T43, T45))
U2_GAGAA(T43, T45, T44, X85, X86, frontc56_out_ga(T43, T45)) → U3_GAGAA(T43, T45, T44, X85, X86, front56_in_ga(T44, X85))
U2_GAGAA(T43, T45, T44, X85, X86, frontc56_out_ga(T43, T45)) → FRONT56_IN_GA(T44, X85)
P54_IN_GAGAA(T43, T45, T44, T60, X86) → U4_GAGAA(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U4_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U5_GAGAA(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U5_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U6_GAGAA(T43, T45, T44, T60, X86, app72_in_gga(T45, T60, X86))
U5_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → APP72_IN_GGA(T45, T60, X86)
APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → U8_GGA(T74, T75, T76, X142, app72_in_gga(T75, T76, X142))
APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → APP72_IN_GGA(T75, T76, X142)
FRONT1_IN_GA(tree(T25, tree(T85, void, void), T27), T29) → U11_GA(T25, T85, T27, T29, front56_in_ga(T27, X57))
FRONT1_IN_GA(tree(T25, tree(T85, void, void), T27), T29) → FRONT56_IN_GA(T27, X57)
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → U12_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T109, X198))
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → FRONT56_IN_GA(T109, X198)
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → U13_GA(T25, T108, T109, T110, T27, T29, frontc56_in_ga(T109, T111))
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → U14_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T110, X199))
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → FRONT56_IN_GA(T110, X199)
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_in_ga(T110, T112))
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → U16_GA(T25, T108, T109, T110, T27, T29, app72_in_gga(T111, T112, X200))
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → APP72_IN_GGA(T111, T112, X200)
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → U17_GA(T25, T108, T109, T110, T27, T29, appc72_in_gga(T111, T112, T115))
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → U18_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T27, X57))
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → FRONT56_IN_GA(T27, X57)
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_in_ga(T27, T118))
U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_out_ga(T27, T118)) → U20_GA(T25, T108, T109, T110, T27, T29, app100_in_gga(T115, T118, T29))
U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_out_ga(T27, T118)) → APP100_IN_GGA(T115, T118, T29)
APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → U9_GGA(T134, T135, T136, T138, app100_in_gga(T135, T136, T138))
APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → APP100_IN_GGA(T135, T136, T138)

The TRS R consists of the following rules:

frontc56_in_ga(void, []) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void), .(T50, [])) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59), X115) → U25_ga(T57, T58, T59, X115, qc54_in_gagaa(T58, X113, T59, X114, X115))
qc54_in_gagaa(T43, T45, T44, T60, X86) → U22_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U22_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U23_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, X86, appc72_in_gga(T45, T60, X86))
appc72_in_gga([], T67, T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76, .(T74, X142)) → U26_gga(T74, T75, T76, X142, appc72_in_gga(T75, T76, X142))
U26_gga(T74, T75, T76, X142, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, X86, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, X115, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
p54_in_gagaa(x1, x2, x3, x4, x5)  =  p54_in_gagaa(x1, x3)
front56_in_ga(x1, x2)  =  front56_in_ga(x1)
frontc56_in_ga(x1, x2)  =  frontc56_in_ga(x1)
frontc56_out_ga(x1, x2)  =  frontc56_out_ga(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
qc54_in_gagaa(x1, x2, x3, x4, x5)  =  qc54_in_gagaa(x1, x3)
U22_gagaa(x1, x2, x3, x4, x5, x6)  =  U22_gagaa(x1, x3, x6)
U23_gagaa(x1, x2, x3, x4, x5, x6)  =  U23_gagaa(x1, x2, x3, x6)
U24_gagaa(x1, x2, x3, x4, x5, x6)  =  U24_gagaa(x1, x2, x3, x4, x6)
appc72_in_gga(x1, x2, x3)  =  appc72_in_gga(x1, x2)
[]  =  []
appc72_out_gga(x1, x2, x3)  =  appc72_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
qc54_out_gagaa(x1, x2, x3, x4, x5)  =  qc54_out_gagaa(x1, x2, x3, x4, x5)
app72_in_gga(x1, x2, x3)  =  app72_in_gga(x1, x2)
app100_in_gga(x1, x2, x3)  =  app100_in_gga(x1, x2)
FRONT1_IN_GA(x1, x2)  =  FRONT1_IN_GA(x1)
U10_GA(x1, x2, x3, x4, x5, x6)  =  U10_GA(x1, x2, x3, x4, x6)
P54_IN_GAGAA(x1, x2, x3, x4, x5)  =  P54_IN_GAGAA(x1, x3)
U1_GAGAA(x1, x2, x3, x4, x5, x6)  =  U1_GAGAA(x1, x3, x6)
FRONT56_IN_GA(x1, x2)  =  FRONT56_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U2_GAGAA(x1, x2, x3, x4, x5, x6)  =  U2_GAGAA(x1, x3, x6)
U3_GAGAA(x1, x2, x3, x4, x5, x6)  =  U3_GAGAA(x1, x3, x6)
U4_GAGAA(x1, x2, x3, x4, x5, x6)  =  U4_GAGAA(x1, x3, x6)
U5_GAGAA(x1, x2, x3, x4, x5, x6)  =  U5_GAGAA(x1, x2, x3, x6)
U6_GAGAA(x1, x2, x3, x4, x5, x6)  =  U6_GAGAA(x1, x3, x6)
APP72_IN_GGA(x1, x2, x3)  =  APP72_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GA(x1, x2, x3, x4, x5, x7)
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, x8)  =  U15_GA(x1, x2, x3, x4, x5, x7, x8)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x5, x7)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
U18_GA(x1, x2, x3, x4, x5, x6, x7)  =  U18_GA(x1, x2, x3, x4, x5, x7)
U19_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_GA(x1, x2, x3, x4, x5, x7, x8)
U20_GA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GA(x1, x2, x3, x4, x5, x7)
APP100_IN_GGA(x1, x2, x3)  =  APP100_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

FRONT1_IN_GA(tree(T25, void, tree(T42, T43, T44)), T29) → U10_GA(T25, T42, T43, T44, T29, p54_in_gagaa(T43, X84, T44, X85, X86))
FRONT1_IN_GA(tree(T25, void, tree(T42, T43, T44)), T29) → P54_IN_GAGAA(T43, X84, T44, X85, X86)
P54_IN_GAGAA(T43, X84, T44, X85, X86) → U1_GAGAA(T43, X84, T44, X85, X86, front56_in_ga(T43, X84))
P54_IN_GAGAA(T43, X84, T44, X85, X86) → FRONT56_IN_GA(T43, X84)
FRONT56_IN_GA(tree(T57, T58, T59), X115) → U7_GA(T57, T58, T59, X115, p54_in_gagaa(T58, X113, T59, X114, X115))
FRONT56_IN_GA(tree(T57, T58, T59), X115) → P54_IN_GAGAA(T58, X113, T59, X114, X115)
P54_IN_GAGAA(T43, T45, T44, X85, X86) → U2_GAGAA(T43, T45, T44, X85, X86, frontc56_in_ga(T43, T45))
U2_GAGAA(T43, T45, T44, X85, X86, frontc56_out_ga(T43, T45)) → U3_GAGAA(T43, T45, T44, X85, X86, front56_in_ga(T44, X85))
U2_GAGAA(T43, T45, T44, X85, X86, frontc56_out_ga(T43, T45)) → FRONT56_IN_GA(T44, X85)
P54_IN_GAGAA(T43, T45, T44, T60, X86) → U4_GAGAA(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U4_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U5_GAGAA(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U5_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U6_GAGAA(T43, T45, T44, T60, X86, app72_in_gga(T45, T60, X86))
U5_GAGAA(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → APP72_IN_GGA(T45, T60, X86)
APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → U8_GGA(T74, T75, T76, X142, app72_in_gga(T75, T76, X142))
APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → APP72_IN_GGA(T75, T76, X142)
FRONT1_IN_GA(tree(T25, tree(T85, void, void), T27), T29) → U11_GA(T25, T85, T27, T29, front56_in_ga(T27, X57))
FRONT1_IN_GA(tree(T25, tree(T85, void, void), T27), T29) → FRONT56_IN_GA(T27, X57)
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → U12_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T109, X198))
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → FRONT56_IN_GA(T109, X198)
FRONT1_IN_GA(tree(T25, tree(T108, T109, T110), T27), T29) → U13_GA(T25, T108, T109, T110, T27, T29, frontc56_in_ga(T109, T111))
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → U14_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T110, X199))
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → FRONT56_IN_GA(T110, X199)
U13_GA(T25, T108, T109, T110, T27, T29, frontc56_out_ga(T109, T111)) → U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_in_ga(T110, T112))
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → U16_GA(T25, T108, T109, T110, T27, T29, app72_in_gga(T111, T112, X200))
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → APP72_IN_GGA(T111, T112, X200)
U15_GA(T25, T108, T109, T110, T27, T29, T111, frontc56_out_ga(T110, T112)) → U17_GA(T25, T108, T109, T110, T27, T29, appc72_in_gga(T111, T112, T115))
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → U18_GA(T25, T108, T109, T110, T27, T29, front56_in_ga(T27, X57))
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → FRONT56_IN_GA(T27, X57)
U17_GA(T25, T108, T109, T110, T27, T29, appc72_out_gga(T111, T112, T115)) → U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_in_ga(T27, T118))
U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_out_ga(T27, T118)) → U20_GA(T25, T108, T109, T110, T27, T29, app100_in_gga(T115, T118, T29))
U19_GA(T25, T108, T109, T110, T27, T29, T115, frontc56_out_ga(T27, T118)) → APP100_IN_GGA(T115, T118, T29)
APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → U9_GGA(T134, T135, T136, T138, app100_in_gga(T135, T136, T138))
APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → APP100_IN_GGA(T135, T136, T138)

The TRS R consists of the following rules:

frontc56_in_ga(void, []) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void), .(T50, [])) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59), X115) → U25_ga(T57, T58, T59, X115, qc54_in_gagaa(T58, X113, T59, X114, X115))
qc54_in_gagaa(T43, T45, T44, T60, X86) → U22_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U22_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U23_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, X86, appc72_in_gga(T45, T60, X86))
appc72_in_gga([], T67, T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76, .(T74, X142)) → U26_gga(T74, T75, T76, X142, appc72_in_gga(T75, T76, X142))
U26_gga(T74, T75, T76, X142, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, X86, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, X115, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
p54_in_gagaa(x1, x2, x3, x4, x5)  =  p54_in_gagaa(x1, x3)
front56_in_ga(x1, x2)  =  front56_in_ga(x1)
frontc56_in_ga(x1, x2)  =  frontc56_in_ga(x1)
frontc56_out_ga(x1, x2)  =  frontc56_out_ga(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
qc54_in_gagaa(x1, x2, x3, x4, x5)  =  qc54_in_gagaa(x1, x3)
U22_gagaa(x1, x2, x3, x4, x5, x6)  =  U22_gagaa(x1, x3, x6)
U23_gagaa(x1, x2, x3, x4, x5, x6)  =  U23_gagaa(x1, x2, x3, x6)
U24_gagaa(x1, x2, x3, x4, x5, x6)  =  U24_gagaa(x1, x2, x3, x4, x6)
appc72_in_gga(x1, x2, x3)  =  appc72_in_gga(x1, x2)
[]  =  []
appc72_out_gga(x1, x2, x3)  =  appc72_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
qc54_out_gagaa(x1, x2, x3, x4, x5)  =  qc54_out_gagaa(x1, x2, x3, x4, x5)
app72_in_gga(x1, x2, x3)  =  app72_in_gga(x1, x2)
app100_in_gga(x1, x2, x3)  =  app100_in_gga(x1, x2)
FRONT1_IN_GA(x1, x2)  =  FRONT1_IN_GA(x1)
U10_GA(x1, x2, x3, x4, x5, x6)  =  U10_GA(x1, x2, x3, x4, x6)
P54_IN_GAGAA(x1, x2, x3, x4, x5)  =  P54_IN_GAGAA(x1, x3)
U1_GAGAA(x1, x2, x3, x4, x5, x6)  =  U1_GAGAA(x1, x3, x6)
FRONT56_IN_GA(x1, x2)  =  FRONT56_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U2_GAGAA(x1, x2, x3, x4, x5, x6)  =  U2_GAGAA(x1, x3, x6)
U3_GAGAA(x1, x2, x3, x4, x5, x6)  =  U3_GAGAA(x1, x3, x6)
U4_GAGAA(x1, x2, x3, x4, x5, x6)  =  U4_GAGAA(x1, x3, x6)
U5_GAGAA(x1, x2, x3, x4, x5, x6)  =  U5_GAGAA(x1, x2, x3, x6)
U6_GAGAA(x1, x2, x3, x4, x5, x6)  =  U6_GAGAA(x1, x3, x6)
APP72_IN_GGA(x1, x2, x3)  =  APP72_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GA(x1, x2, x3, x4, x5, x7)
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, x8)  =  U15_GA(x1, x2, x3, x4, x5, x7, x8)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x5, x7)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
U18_GA(x1, x2, x3, x4, x5, x6, x7)  =  U18_GA(x1, x2, x3, x4, x5, x7)
U19_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_GA(x1, x2, x3, x4, x5, x7, x8)
U20_GA(x1, x2, x3, x4, x5, x6, x7)  =  U20_GA(x1, x2, x3, x4, x5, x7)
APP100_IN_GGA(x1, x2, x3)  =  APP100_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 27 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → APP100_IN_GGA(T135, T136, T138)

The TRS R consists of the following rules:

frontc56_in_ga(void, []) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void), .(T50, [])) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59), X115) → U25_ga(T57, T58, T59, X115, qc54_in_gagaa(T58, X113, T59, X114, X115))
qc54_in_gagaa(T43, T45, T44, T60, X86) → U22_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U22_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U23_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, X86, appc72_in_gga(T45, T60, X86))
appc72_in_gga([], T67, T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76, .(T74, X142)) → U26_gga(T74, T75, T76, X142, appc72_in_gga(T75, T76, X142))
U26_gga(T74, T75, T76, X142, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, X86, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, X115, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
frontc56_in_ga(x1, x2)  =  frontc56_in_ga(x1)
frontc56_out_ga(x1, x2)  =  frontc56_out_ga(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
qc54_in_gagaa(x1, x2, x3, x4, x5)  =  qc54_in_gagaa(x1, x3)
U22_gagaa(x1, x2, x3, x4, x5, x6)  =  U22_gagaa(x1, x3, x6)
U23_gagaa(x1, x2, x3, x4, x5, x6)  =  U23_gagaa(x1, x2, x3, x6)
U24_gagaa(x1, x2, x3, x4, x5, x6)  =  U24_gagaa(x1, x2, x3, x4, x6)
appc72_in_gga(x1, x2, x3)  =  appc72_in_gga(x1, x2)
[]  =  []
appc72_out_gga(x1, x2, x3)  =  appc72_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
qc54_out_gagaa(x1, x2, x3, x4, x5)  =  qc54_out_gagaa(x1, x2, x3, x4, x5)
APP100_IN_GGA(x1, x2, x3)  =  APP100_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP100_IN_GGA(.(T134, T135), T136, .(T134, T138)) → APP100_IN_GGA(T135, T136, T138)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP100_IN_GGA(x1, x2, x3)  =  APP100_IN_GGA(x1, x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP100_IN_GGA(.(T134, T135), T136) → APP100_IN_GGA(T135, T136)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP100_IN_GGA(.(T134, T135), T136) → APP100_IN_GGA(T135, T136)
  The graph contains the following edges 1 > 1, 2 >= 2

(15) YES

(16) Obligation:

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

APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → APP72_IN_GGA(T75, T76, X142)

The TRS R consists of the following rules:

frontc56_in_ga(void, []) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void), .(T50, [])) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59), X115) → U25_ga(T57, T58, T59, X115, qc54_in_gagaa(T58, X113, T59, X114, X115))
qc54_in_gagaa(T43, T45, T44, T60, X86) → U22_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U22_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U23_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, X86, appc72_in_gga(T45, T60, X86))
appc72_in_gga([], T67, T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76, .(T74, X142)) → U26_gga(T74, T75, T76, X142, appc72_in_gga(T75, T76, X142))
U26_gga(T74, T75, T76, X142, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, X86, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, X115, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
frontc56_in_ga(x1, x2)  =  frontc56_in_ga(x1)
frontc56_out_ga(x1, x2)  =  frontc56_out_ga(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
qc54_in_gagaa(x1, x2, x3, x4, x5)  =  qc54_in_gagaa(x1, x3)
U22_gagaa(x1, x2, x3, x4, x5, x6)  =  U22_gagaa(x1, x3, x6)
U23_gagaa(x1, x2, x3, x4, x5, x6)  =  U23_gagaa(x1, x2, x3, x6)
U24_gagaa(x1, x2, x3, x4, x5, x6)  =  U24_gagaa(x1, x2, x3, x4, x6)
appc72_in_gga(x1, x2, x3)  =  appc72_in_gga(x1, x2)
[]  =  []
appc72_out_gga(x1, x2, x3)  =  appc72_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
qc54_out_gagaa(x1, x2, x3, x4, x5)  =  qc54_out_gagaa(x1, x2, x3, x4, x5)
APP72_IN_GGA(x1, x2, x3)  =  APP72_IN_GGA(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP72_IN_GGA(.(T74, T75), T76, .(T74, X142)) → APP72_IN_GGA(T75, T76, X142)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP72_IN_GGA(x1, x2, x3)  =  APP72_IN_GGA(x1, x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP72_IN_GGA(.(T74, T75), T76) → APP72_IN_GGA(T75, T76)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP72_IN_GGA(.(T74, T75), T76) → APP72_IN_GGA(T75, T76)
  The graph contains the following edges 1 > 1, 2 >= 2

(22) YES

(23) Obligation:

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

P54_IN_GAGAA(T43, X84, T44, X85, X86) → FRONT56_IN_GA(T43, X84)
FRONT56_IN_GA(tree(T57, T58, T59), X115) → P54_IN_GAGAA(T58, X113, T59, X114, X115)
P54_IN_GAGAA(T43, T45, T44, X85, X86) → U2_GAGAA(T43, T45, T44, X85, X86, frontc56_in_ga(T43, T45))
U2_GAGAA(T43, T45, T44, X85, X86, frontc56_out_ga(T43, T45)) → FRONT56_IN_GA(T44, X85)

The TRS R consists of the following rules:

frontc56_in_ga(void, []) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void), .(T50, [])) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59), X115) → U25_ga(T57, T58, T59, X115, qc54_in_gagaa(T58, X113, T59, X114, X115))
qc54_in_gagaa(T43, T45, T44, T60, X86) → U22_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T43, T45))
U22_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, T60, X86, frontc56_in_ga(T44, T60))
U23_gagaa(T43, T45, T44, T60, X86, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, X86, appc72_in_gga(T45, T60, X86))
appc72_in_gga([], T67, T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76, .(T74, X142)) → U26_gga(T74, T75, T76, X142, appc72_in_gga(T75, T76, X142))
U26_gga(T74, T75, T76, X142, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, X86, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, X115, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
frontc56_in_ga(x1, x2)  =  frontc56_in_ga(x1)
frontc56_out_ga(x1, x2)  =  frontc56_out_ga(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
qc54_in_gagaa(x1, x2, x3, x4, x5)  =  qc54_in_gagaa(x1, x3)
U22_gagaa(x1, x2, x3, x4, x5, x6)  =  U22_gagaa(x1, x3, x6)
U23_gagaa(x1, x2, x3, x4, x5, x6)  =  U23_gagaa(x1, x2, x3, x6)
U24_gagaa(x1, x2, x3, x4, x5, x6)  =  U24_gagaa(x1, x2, x3, x4, x6)
appc72_in_gga(x1, x2, x3)  =  appc72_in_gga(x1, x2)
[]  =  []
appc72_out_gga(x1, x2, x3)  =  appc72_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
qc54_out_gagaa(x1, x2, x3, x4, x5)  =  qc54_out_gagaa(x1, x2, x3, x4, x5)
P54_IN_GAGAA(x1, x2, x3, x4, x5)  =  P54_IN_GAGAA(x1, x3)
FRONT56_IN_GA(x1, x2)  =  FRONT56_IN_GA(x1)
U2_GAGAA(x1, x2, x3, x4, x5, x6)  =  U2_GAGAA(x1, x3, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

P54_IN_GAGAA(T43, T44) → FRONT56_IN_GA(T43)
FRONT56_IN_GA(tree(T57, T58, T59)) → P54_IN_GAGAA(T58, T59)
P54_IN_GAGAA(T43, T44) → U2_GAGAA(T43, T44, frontc56_in_ga(T43))
U2_GAGAA(T43, T44, frontc56_out_ga(T43, T45)) → FRONT56_IN_GA(T44)

The TRS R consists of the following rules:

frontc56_in_ga(void) → frontc56_out_ga(void, [])
frontc56_in_ga(tree(T50, void, void)) → frontc56_out_ga(tree(T50, void, void), .(T50, []))
frontc56_in_ga(tree(T57, T58, T59)) → U25_ga(T57, T58, T59, qc54_in_gagaa(T58, T59))
qc54_in_gagaa(T43, T44) → U22_gagaa(T43, T44, frontc56_in_ga(T43))
U22_gagaa(T43, T44, frontc56_out_ga(T43, T45)) → U23_gagaa(T43, T45, T44, frontc56_in_ga(T44))
U23_gagaa(T43, T45, T44, frontc56_out_ga(T44, T60)) → U24_gagaa(T43, T45, T44, T60, appc72_in_gga(T45, T60))
appc72_in_gga([], T67) → appc72_out_gga([], T67, T67)
appc72_in_gga(.(T74, T75), T76) → U26_gga(T74, T75, T76, appc72_in_gga(T75, T76))
U26_gga(T74, T75, T76, appc72_out_gga(T75, T76, X142)) → appc72_out_gga(.(T74, T75), T76, .(T74, X142))
U24_gagaa(T43, T45, T44, T60, appc72_out_gga(T45, T60, X86)) → qc54_out_gagaa(T43, T45, T44, T60, X86)
U25_ga(T57, T58, T59, qc54_out_gagaa(T58, X113, T59, X114, X115)) → frontc56_out_ga(tree(T57, T58, T59), X115)

The set Q consists of the following terms:

frontc56_in_ga(x0)
qc54_in_gagaa(x0, x1)
U22_gagaa(x0, x1, x2)
U23_gagaa(x0, x1, x2, x3)
appc72_in_gga(x0, x1)
U26_gga(x0, x1, x2, x3)
U24_gagaa(x0, x1, x2, x3, x4)
U25_ga(x0, x1, x2, x3)

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

(26) 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:

• FRONT56_IN_GA(tree(T57, T58, T59)) → P54_IN_GAGAA(T58, T59)
  The graph contains the following edges 1 > 1, 1 > 2

• P54_IN_GAGAA(T43, T44) → FRONT56_IN_GA(T43)
  The graph contains the following edges 1 >= 1

• P54_IN_GAGAA(T43, T44) → U2_GAGAA(T43, T44, frontc56_in_ga(T43))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U2_GAGAA(T43, T44, frontc56_out_ga(T43, T45)) → FRONT56_IN_GA(T44)
  The graph contains the following edges 2 >= 1

(27) YES