Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
perm([], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).

Queries:

perm(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: perm(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: perm(T1, T2), SCOPE: 1, CLAUSE: 0 | perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))) | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nperm(T1, T2) !=? perm(X4, .(X5, X6))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 2 | ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app(X46, .(T21, X47), T20), ','(app(.(X45, X46), X47, X9), perm(X9, T22)))\n\nT20 is ground\nX9 is free; \nX45 is free; \nX46 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app(X46, .(T21, X47), T20)\n\nT20 is ground\nX46 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(.(X45, T25), T26, X9), perm(X9, T27))\n\nT25 is ground\nT26 is ground\nX9 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app(X46, .(T21, X47), T20), SCOPE: 3, CLAUSE: 2 | app(X46, .(T21, X47), T20), SCOPE: 3, CLAUSE: 3\n\nT20 is ground\nX46 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(X46, .(T21, X47), T20), SCOPE: 3, CLAUSE: 2\n\nT20 is ground\nX46 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app(X46, .(T21, X47), T20), SCOPE: 3, CLAUSE: 3\n\nT20 is ground\nX46 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app(X99, .(T40, X100), T39)\n\nT39 is ground\nX99 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(.(X45, T25), T26, X9)\n\nT25 is ground\nT26 is ground\nX9 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: perm(T49, T27)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(.(X45, T25), T26, X9), SCOPE: 4, CLAUSE: 2 | app(.(X45, T25), T26, X9), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX9 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app(.(X45, T25), T26, X9), SCOPE: 4, CLAUSE: 2\n\nT25 is ground\nT26 is ground\nX9 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(.(X45, T25), T26, X9), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX9 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(T66, T67, X170)\n\nT66 is ground\nT67 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(T66, T67, X170), SCOPE: 5, CLAUSE: 2 | app(T66, T67, X170), SCOPE: 5, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T66, T67, X170), SCOPE: 5, CLAUSE: 2\n\nT66 is ground\nT67 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T66, T67, X170), SCOPE: 5, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T84, T85, X200)\n\nT84 is ground\nT85 is ground\nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: perm(T49, T27), SCOPE: 6, CLAUSE: 0 | perm(T49, T27), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: perm(T49, T27), SCOPE: 6, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: perm(T49, T27), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(app(X242, .(T111, X243), T110), ','(app(X242, X243, X244), perm(X244, T112)))\n\nX242 is free\nX243 is free; \nX244 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(X242, .(T111, X243), T110)\n\nX242 is free\nX243 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(app(T115, T116, X244), perm(X244, T117))\n\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app(X242, .(T111, X243), T110), SCOPE: 7, CLAUSE: 2 | app(X242, .(T111, X243), T110), SCOPE: 7, CLAUSE: 3\n\nX242 is free\nX243 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(X242, .(T111, X243), T110), SCOPE: 7, CLAUSE: 2\n\nX242 is free\nX243 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(X242, .(T111, X243), T110), SCOPE: 7, CLAUSE: 3\n\nX242 is free\nX243 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app(X295, .(T131, X296), T130)\n\nX295 is free\nX296 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(T115, T116, X244)\n\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: perm(T140, T141)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: app(T115, T116, X244), SCOPE: 8, CLAUSE: 2 | app(T115, T116, X244), SCOPE: 8, CLAUSE: 3\n\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: app(T115, T116, X244), SCOPE: 8, CLAUSE: 2\n\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: app(T115, T116, X244), SCOPE: 8, CLAUSE: 3\n\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: app(T160, T161, X349)\n\nX349 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(app(X7, .(T9, X8), T6), ','(app(X7, X8, X9), perm(X9, T10))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nX7 is free; \nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(app([], X378, X9), perm(X9, T176))\n\nX9 is free\nX378 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app([], X378, X9)\n\nX9 is free\nX378 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: perm(T180, T176)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: app([], X378, X9), SCOPE: 9, CLAUSE: 2 | app([], X378, X9), SCOPE: 9, CLAUSE: 3\n\nX9 is free\nX378 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app([], X378, X9), SCOPE: 9, CLAUSE: 3\n\nX9 is free\nX378 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: perm(T6, T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 80 [arrowhead = none ];
80 -> 2;

81 [label="EVAL with clause\nperm(X4, .(X5, X6)) :- ','(app(X7, .(X5, X8), X4), ','(app(X7, X8, X9), perm(X9, X6))).\nand substitution\nT1 -> T6\nX4 -> T6\nX5 -> T9\nX6 -> T10\nT2 -> .(T9, T10)\nT7 -> T9\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 81 [arrowhead = none ];
81 -> 3;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 82 [arrowhead = none ];
82 -> 4;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 83 [arrowhead = none ];
83 -> 5;

84 [label="EVAL with clause\nperm([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 84 [arrowhead = none ];
84 -> 77;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 85 [arrowhead = none ];
85 -> 78;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 86 [arrowhead = none ];
86 -> 6;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 87 [arrowhead = none ];
87 -> 7;

88 [label="EVAL with clause\napp(.(X41, X42), X43, .(X41, X44)) :- app(X42, X43, X44).\nand substitution\nX41 -> X45\nX42 -> X46\nX7 -> .(X45, X46)\nT9 -> T21\nX8 -> X47\nX43 -> .(T21, X47)\nX44 -> T20\nT6 -> .(X45, T20)\nT19 -> T21\nT10 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 88 [arrowhead = none ];
88 -> 8;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 89 [arrowhead = none ];
89 -> 9;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 90 [arrowhead = none ];
90 -> 63;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 91 [arrowhead = none ];
91 -> 64;

92 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 92 [arrowhead = none ];
92 -> 10;

93 [label="SPLIT 2\nnew knowledge:\nT25 is ground\nT21 is ground\nT26 is ground\nreplacements:\nX46 -> T25\nX47 -> T26\nT22 -> T27", fontsize=14, style = filled, fillcolor = violet];
8 -> 93 [arrowhead = none ];
93 -> 11;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 94 [arrowhead = none ];
94 -> 12;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 95 [arrowhead = none ];
95 -> 20;

96 [label="SPLIT 2\nreplacements:\nX45 -> T48\nX9 -> T49", fontsize=14, style = filled, fillcolor = violet];
11 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 97 [arrowhead = none ];
97 -> 13;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 98 [arrowhead = none ];
98 -> 14;

99 [label="EVAL with clause\napp(.(X94, X95), X96, .(X94, X97)) :- app(X95, X96, X97).\nand substitution\nX94 -> X98\nX95 -> X99\nX46 -> .(X98, X99)\nT21 -> T40\nX47 -> X100\nX96 -> .(T40, X100)\nX97 -> T39\nT20 -> .(X98, T39)\nT38 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 99 [arrowhead = none ];
99 -> 15;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 100 [arrowhead = none ];
100 -> 16;

101 [label="EVAL with clause\napp([], X116, X116).\nand substitution\nX46 -> []\nT21 -> T45\nX47 -> X117\nX116 -> .(T45, X117)\nT20 -> .(T45, X117)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 101 [arrowhead = none ];
101 -> 17;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 102 [arrowhead = none ];
102 -> 18;

103 [label="INSTANCE with matching:\nX46 -> X99\nT21 -> T40\nX47 -> X100\nT20 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 103 [arrowhead = none , style=dashed];
103 -> 10[style=dashed];

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 105 [arrowhead = none ];
105 -> 22;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 106 [arrowhead = none ];
106 -> 35;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 107 [arrowhead = none ];
107 -> 23;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 108 [arrowhead = none ];
108 -> 24;

109 [label="ONLY EVAL with clause\napp(.(X165, X166), X167, .(X165, X168)) :- app(X166, X167, X168).\nand substitution\nX45 -> X169\nX165 -> X169\nT25 -> T66\nX166 -> T66\nT26 -> T67\nX167 -> T67\nX168 -> X170\nX9 -> .(X169, X170)", fontsize=14, style = filled, fillcolor = white];
23 -> 109 [arrowhead = none ];
109 -> 25;

110 [label="BACKTRACK\nfor clause: app([], Ys, Ys)because of non-unification", fontsize=14, style = filled, fillcolor = white];
24 -> 110 [arrowhead = none ];
110 -> 34;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 113 [arrowhead = none ];
113 -> 28;

114 [label="EVAL with clause\napp(.(X196, X197), X198, .(X196, X199)) :- app(X197, X198, X199).\nand substitution\nX196 -> T83\nX197 -> T84\nT66 -> .(T83, T84)\nT67 -> T85\nX198 -> T85\nX199 -> X200\nX170 -> .(T83, X200)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 114 [arrowhead = none ];
114 -> 29;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 115 [arrowhead = none ];
115 -> 30;

116 [label="EVAL with clause\napp([], X212, X212).\nand substitution\nT66 -> []\nT67 -> T91\nX212 -> T91\nX170 -> T91", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 116 [arrowhead = none ];
116 -> 31;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 117 [arrowhead = none ];
117 -> 32;

118 [label="INSTANCE with matching:\nT66 -> T84\nT67 -> T85\nX170 -> X200", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 118 [arrowhead = none , style=dashed];
118 -> 25[style=dashed];

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 119 [arrowhead = none ];
119 -> 33;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 120 [arrowhead = none ];
120 -> 36;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 121 [arrowhead = none ];
121 -> 37;

122 [label="EVAL with clause\nperm(X239, .(X240, X241)) :- ','(app(X242, .(X240, X243), X239), ','(app(X242, X243, X244), perm(X244, X241))).\nand substitution\nT49 -> T110\nX239 -> T110\nX240 -> T111\nX241 -> T112\nT27 -> .(T111, T112)\nT107 -> T110\nT108 -> T111\nT109 -> T112", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 122 [arrowhead = none ];
122 -> 38;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 123 [arrowhead = none ];
123 -> 39;

124 [label="EVAL with clause\nperm([], []).\nand substitution\nT49 -> []\nT27 -> []", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 125 [arrowhead = none ];
125 -> 61;

126 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
38 -> 126 [arrowhead = none ];
126 -> 40;

127 [label="SPLIT 2\nreplacements:\nX242 -> T115\nX243 -> T116\nT112 -> T117", fontsize=14, style = filled, fillcolor = violet];
38 -> 127 [arrowhead = none ];
127 -> 41;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 128 [arrowhead = none ];
128 -> 42;

129 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 129 [arrowhead = none ];
129 -> 50;

130 [label="SPLIT 2\nreplacements:\nX244 -> T140\nT117 -> T141", fontsize=14, style = filled, fillcolor = violet];
41 -> 130 [arrowhead = none ];
130 -> 51;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 131 [arrowhead = none ];
131 -> 43;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 132 [arrowhead = none ];
132 -> 44;

133 [label="EVAL with clause\napp(.(X290, X291), X292, .(X290, X293)) :- app(X291, X292, X293).\nand substitution\nX290 -> X294\nX291 -> X295\nX242 -> .(X294, X295)\nT111 -> T131\nX243 -> X296\nX292 -> .(T131, X296)\nX293 -> T130\nT110 -> .(X294, T130)\nT129 -> T130\nT128 -> T131", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 133 [arrowhead = none ];
133 -> 45;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 134 [arrowhead = none ];
134 -> 46;

135 [label="EVAL with clause\napp([], X312, X312).\nand substitution\nX242 -> []\nT111 -> T136\nX243 -> X313\nX312 -> .(T136, X313)\nT110 -> .(T136, X313)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 135 [arrowhead = none ];
135 -> 47;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 136 [arrowhead = none ];
136 -> 48;

137 [label="INSTANCE with matching:\nX242 -> X295\nT111 -> T131\nX243 -> X296\nT110 -> T130", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 137 [arrowhead = none , style=dashed];
137 -> 40[style=dashed];

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 138 [arrowhead = none ];
138 -> 49;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 139 [arrowhead = none ];
139 -> 52;

140 [label="INSTANCE with matching:\nT49 -> T140\nT27 -> T141", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 140 [arrowhead = none , style=dashed];
140 -> 21[style=dashed];

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 141 [arrowhead = none ];
141 -> 53;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 142 [arrowhead = none ];
142 -> 54;

143 [label="EVAL with clause\napp(.(X345, X346), X347, .(X345, X348)) :- app(X346, X347, X348).\nand substitution\nX345 -> T157\nX346 -> T160\nT115 -> .(T157, T160)\nT116 -> T161\nX347 -> T161\nX348 -> X349\nX244 -> .(T157, X349)\nT158 -> T160\nT159 -> T161", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 143 [arrowhead = none ];
143 -> 55;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 144 [arrowhead = none ];
144 -> 56;

145 [label="EVAL with clause\napp([], X361, X361).\nand substitution\nT115 -> []\nT116 -> T167\nX361 -> T167\nX244 -> T167", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 145 [arrowhead = none ];
145 -> 57;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 146 [arrowhead = none ];
146 -> 58;

147 [label="INSTANCE with matching:\nT115 -> T160\nT116 -> T161\nX244 -> X349", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 147 [arrowhead = none , style=dashed];
147 -> 50[style=dashed];

148 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 148 [arrowhead = none ];
148 -> 59;

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 149 [arrowhead = none ];
149 -> 62;

150 [label="EVAL with clause\napp([], X377, X377).\nand substitution\nX7 -> []\nT9 -> T175\nX8 -> X378\nX377 -> .(T175, X378)\nT6 -> .(T175, X378)\nT10 -> T176", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 150 [arrowhead = none ];
150 -> 65;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 151 [arrowhead = none ];
151 -> 66;

152 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 152 [arrowhead = none ];
152 -> 73;

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 153 [arrowhead = none ];
153 -> 67;

154 [label="SPLIT 2\nreplacements:\nX378 -> T179\nX9 -> T180", fontsize=14, style = filled, fillcolor = violet];
65 -> 154 [arrowhead = none ];
154 -> 68;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 155 [arrowhead = none ];
155 -> 69;

156 [label="INSTANCE with matching:\nT49 -> T180\nT27 -> T176", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 156 [arrowhead = none , style=dashed];
156 -> 21[style=dashed];

157 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
69 -> 157 [arrowhead = none ];
157 -> 70;

158 [label="ONLY EVAL with clause\napp([], X407, X407).\nand substitution\nX378 -> X408\nX407 -> X408\nX9 -> X408", fontsize=14, style = filled, fillcolor = white];
70 -> 158 [arrowhead = none ];
158 -> 71;

159 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 159 [arrowhead = none ];
159 -> 72;

160 [label="EVAL with clause\nperm([], []).\nand substitution\nT6 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 160 [arrowhead = none ];
160 -> 74;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 161 [arrowhead = none ];
161 -> 75;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 162 [arrowhead = none ];
162 -> 76;

163 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 163 [arrowhead = none ];
163 -> 79;

}

(2) Obligation:

Triples:

app10(.(X98, X99), T40, X100, .(X98, T39)) :- app10(X99, T40, X100, T39).
app25(.(T83, T84), T85, .(T83, X200)) :- app25(T84, T85, X200).
app40(.(X294, X295), T131, X296, .(X294, T130)) :- app40(X295, T131, X296, T130).
perm21(T110, .(T111, T112)) :- app40(X242, T111, X243, T110).
perm21(T110, .(T111, T117)) :- ','(appc40(T115, T111, T116, T110), app50(T115, T116, X244)).
perm21(T110, .(T111, T141)) :- ','(appc40(T115, T111, T116, T110), ','(appc50(T115, T116, T140), perm21(T140, T141))).
app50(.(T157, T160), T161, .(T157, X349)) :- app50(T160, T161, X349).
perm1(.(X45, T20), .(T21, T22)) :- app10(X46, T21, X47, T20).
perm1(.(X169, T20), .(T21, T27)) :- ','(appc10(T66, T21, T67, T20), app25(T66, T67, X170)).
perm1(.(T48, T20), .(T21, T27)) :- ','(appc10(T25, T21, T26, T20), ','(appc20(T48, T25, T26, T49), perm21(T49, T27))).
perm1(.(T175, T179), .(T175, T176)) :- ','(appc67(T179, T180), perm21(T180, T176)).

Clauses:

appc10(.(X98, X99), T40, X100, .(X98, T39)) :- appc10(X99, T40, X100, T39).
appc10([], T45, X117, .(T45, X117)).
appc25(.(T83, T84), T85, .(T83, X200)) :- appc25(T84, T85, X200).
appc25([], T91, T91).
appc40(.(X294, X295), T131, X296, .(X294, T130)) :- appc40(X295, T131, X296, T130).
appc40([], T136, X313, .(T136, X313)).
permc21(T110, .(T111, T141)) :- ','(appc40(T115, T111, T116, T110), ','(appc50(T115, T116, T140), permc21(T140, T141))).
permc21([], []).
appc50(.(T157, T160), T161, .(T157, X349)) :- appc50(T160, T161, X349).
appc50([], T167, T167).
appc20(X169, T66, T67, .(X169, X170)) :- appc25(T66, T67, X170).
appc67(X408, X408).

Afs:

perm1(x1, x2)  =  perm1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm1_in: (b,f)
app10_in: (f,f,f,b)
appc10_in: (f,f,f,b)
app25_in: (b,b,f)
appc20_in: (b,b,b,f)
appc25_in: (b,b,f)
perm21_in: (b,f)
app40_in: (f,f,f,b)
appc40_in: (f,f,f,b)
app50_in: (b,b,f)
appc50_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERM1_IN_GA(.(X45, T20), .(T21, T22)) → U11_GA(X45, T20, T21, T22, app10_in_aaag(X46, T21, X47, T20))
PERM1_IN_GA(.(X45, T20), .(T21, T22)) → APP10_IN_AAAG(X46, T21, X47, T20)
APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → U1_AAAG(X98, X99, T40, X100, T39, app10_in_aaag(X99, T40, X100, T39))
APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → APP10_IN_AAAG(X99, T40, X100, T39)
PERM1_IN_GA(.(X169, T20), .(T21, T27)) → U12_GA(X169, T20, T21, T27, appc10_in_aaag(T66, T21, T67, T20))
U12_GA(X169, T20, T21, T27, appc10_out_aaag(T66, T21, T67, T20)) → U13_GA(X169, T20, T21, T27, app25_in_gga(T66, T67, X170))
U12_GA(X169, T20, T21, T27, appc10_out_aaag(T66, T21, T67, T20)) → APP25_IN_GGA(T66, T67, X170)
APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → U2_GGA(T83, T84, T85, X200, app25_in_gga(T84, T85, X200))
APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → APP25_IN_GGA(T84, T85, X200)
PERM1_IN_GA(.(T48, T20), .(T21, T27)) → U14_GA(T48, T20, T21, T27, appc10_in_aaag(T25, T21, T26, T20))
U14_GA(T48, T20, T21, T27, appc10_out_aaag(T25, T21, T26, T20)) → U15_GA(T48, T20, T21, T27, appc20_in_ggga(T48, T25, T26, T49))
U15_GA(T48, T20, T21, T27, appc20_out_ggga(T48, T25, T26, T49)) → U16_GA(T48, T20, T21, T27, perm21_in_ga(T49, T27))
U15_GA(T48, T20, T21, T27, appc20_out_ggga(T48, T25, T26, T49)) → PERM21_IN_GA(T49, T27)
PERM21_IN_GA(T110, .(T111, T112)) → U4_GA(T110, T111, T112, app40_in_aaag(X242, T111, X243, T110))
PERM21_IN_GA(T110, .(T111, T112)) → APP40_IN_AAAG(X242, T111, X243, T110)
APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → U3_AAAG(X294, X295, T131, X296, T130, app40_in_aaag(X295, T131, X296, T130))
APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → APP40_IN_AAAG(X295, T131, X296, T130)
PERM21_IN_GA(T110, .(T111, T117)) → U5_GA(T110, T111, T117, appc40_in_aaag(T115, T111, T116, T110))
U5_GA(T110, T111, T117, appc40_out_aaag(T115, T111, T116, T110)) → U6_GA(T110, T111, T117, app50_in_gga(T115, T116, X244))
U5_GA(T110, T111, T117, appc40_out_aaag(T115, T111, T116, T110)) → APP50_IN_GGA(T115, T116, X244)
APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → U10_GGA(T157, T160, T161, X349, app50_in_gga(T160, T161, X349))
APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → APP50_IN_GGA(T160, T161, X349)
PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, appc40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, appc50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → U9_GA(T110, T111, T141, perm21_in_ga(T140, T141))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)
PERM1_IN_GA(.(T175, T179), .(T175, T176)) → U17_GA(T175, T179, T176, appc67_in_ga(T179, T180))
U17_GA(T175, T179, T176, appc67_out_ga(T179, T180)) → U18_GA(T175, T179, T176, perm21_in_ga(T180, T176))
U17_GA(T175, T179, T176, appc67_out_ga(T179, T180)) → PERM21_IN_GA(T180, T176)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app10_in_aaag(x1, x2, x3, x4)  =  app10_in_aaag(x4)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
app25_in_gga(x1, x2, x3)  =  app25_in_gga(x1, x2)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
perm21_in_ga(x1, x2)  =  perm21_in_ga(x1)
app40_in_aaag(x1, x2, x3, x4)  =  app40_in_aaag(x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
app50_in_gga(x1, x2, x3)  =  app50_in_gga(x1, x2)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x5)
APP10_IN_AAAG(x1, x2, x3, x4)  =  APP10_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)
APP25_IN_GGA(x1, x2, x3)  =  APP25_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
PERM21_IN_GA(x1, x2)  =  PERM21_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
APP40_IN_AAAG(x1, x2, x3, x4)  =  APP40_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4, x5, x6)  =  U3_AAAG(x1, x5, x6)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
APP50_IN_GGA(x1, x2, x3)  =  APP50_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x4)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U17_GA(x1, x2, x3, x4)  =  U17_GA(x1, x2, x4)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)

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:

PERM1_IN_GA(.(X45, T20), .(T21, T22)) → U11_GA(X45, T20, T21, T22, app10_in_aaag(X46, T21, X47, T20))
PERM1_IN_GA(.(X45, T20), .(T21, T22)) → APP10_IN_AAAG(X46, T21, X47, T20)
APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → U1_AAAG(X98, X99, T40, X100, T39, app10_in_aaag(X99, T40, X100, T39))
APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → APP10_IN_AAAG(X99, T40, X100, T39)
PERM1_IN_GA(.(X169, T20), .(T21, T27)) → U12_GA(X169, T20, T21, T27, appc10_in_aaag(T66, T21, T67, T20))
U12_GA(X169, T20, T21, T27, appc10_out_aaag(T66, T21, T67, T20)) → U13_GA(X169, T20, T21, T27, app25_in_gga(T66, T67, X170))
U12_GA(X169, T20, T21, T27, appc10_out_aaag(T66, T21, T67, T20)) → APP25_IN_GGA(T66, T67, X170)
APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → U2_GGA(T83, T84, T85, X200, app25_in_gga(T84, T85, X200))
APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → APP25_IN_GGA(T84, T85, X200)
PERM1_IN_GA(.(T48, T20), .(T21, T27)) → U14_GA(T48, T20, T21, T27, appc10_in_aaag(T25, T21, T26, T20))
U14_GA(T48, T20, T21, T27, appc10_out_aaag(T25, T21, T26, T20)) → U15_GA(T48, T20, T21, T27, appc20_in_ggga(T48, T25, T26, T49))
U15_GA(T48, T20, T21, T27, appc20_out_ggga(T48, T25, T26, T49)) → U16_GA(T48, T20, T21, T27, perm21_in_ga(T49, T27))
U15_GA(T48, T20, T21, T27, appc20_out_ggga(T48, T25, T26, T49)) → PERM21_IN_GA(T49, T27)
PERM21_IN_GA(T110, .(T111, T112)) → U4_GA(T110, T111, T112, app40_in_aaag(X242, T111, X243, T110))
PERM21_IN_GA(T110, .(T111, T112)) → APP40_IN_AAAG(X242, T111, X243, T110)
APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → U3_AAAG(X294, X295, T131, X296, T130, app40_in_aaag(X295, T131, X296, T130))
APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → APP40_IN_AAAG(X295, T131, X296, T130)
PERM21_IN_GA(T110, .(T111, T117)) → U5_GA(T110, T111, T117, appc40_in_aaag(T115, T111, T116, T110))
U5_GA(T110, T111, T117, appc40_out_aaag(T115, T111, T116, T110)) → U6_GA(T110, T111, T117, app50_in_gga(T115, T116, X244))
U5_GA(T110, T111, T117, appc40_out_aaag(T115, T111, T116, T110)) → APP50_IN_GGA(T115, T116, X244)
APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → U10_GGA(T157, T160, T161, X349, app50_in_gga(T160, T161, X349))
APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → APP50_IN_GGA(T160, T161, X349)
PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, appc40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, appc50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → U9_GA(T110, T111, T141, perm21_in_ga(T140, T141))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)
PERM1_IN_GA(.(T175, T179), .(T175, T176)) → U17_GA(T175, T179, T176, appc67_in_ga(T179, T180))
U17_GA(T175, T179, T176, appc67_out_ga(T179, T180)) → U18_GA(T175, T179, T176, perm21_in_ga(T180, T176))
U17_GA(T175, T179, T176, appc67_out_ga(T179, T180)) → PERM21_IN_GA(T180, T176)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app10_in_aaag(x1, x2, x3, x4)  =  app10_in_aaag(x4)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
app25_in_gga(x1, x2, x3)  =  app25_in_gga(x1, x2)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
perm21_in_ga(x1, x2)  =  perm21_in_ga(x1)
app40_in_aaag(x1, x2, x3, x4)  =  app40_in_aaag(x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
app50_in_gga(x1, x2, x3)  =  app50_in_gga(x1, x2)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x5)
APP10_IN_AAAG(x1, x2, x3, x4)  =  APP10_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)
APP25_IN_GGA(x1, x2, x3)  =  APP25_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
PERM21_IN_GA(x1, x2)  =  PERM21_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
APP40_IN_AAAG(x1, x2, x3, x4)  =  APP40_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4, x5, x6)  =  U3_AAAG(x1, x5, x6)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
APP50_IN_GGA(x1, x2, x3)  =  APP50_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x4)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U17_GA(x1, x2, x3, x4)  =  U17_GA(x1, x2, x4)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → APP50_IN_GGA(T160, T161, X349)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
APP50_IN_GGA(x1, x2, x3)  =  APP50_IN_GGA(x1, x2)

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:

APP50_IN_GGA(.(T157, T160), T161, .(T157, X349)) → APP50_IN_GGA(T160, T161, X349)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP50_IN_GGA(x1, x2, x3)  =  APP50_IN_GGA(x1, x2)

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:

APP50_IN_GGA(.(T157, T160), T161) → APP50_IN_GGA(T160, T161)

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:

• APP50_IN_GGA(.(T157, T160), T161) → APP50_IN_GGA(T160, T161)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → APP40_IN_AAAG(X295, T131, X296, T130)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
APP40_IN_AAAG(x1, x2, x3, x4)  =  APP40_IN_AAAG(x4)

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:

APP40_IN_AAAG(.(X294, X295), T131, X296, .(X294, T130)) → APP40_IN_AAAG(X295, T131, X296, T130)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP40_IN_AAAG(x1, x2, x3, x4)  =  APP40_IN_AAAG(x4)

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:

APP40_IN_AAAG(.(X294, T130)) → APP40_IN_AAAG(T130)

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:

• APP40_IN_AAAG(.(X294, T130)) → APP40_IN_AAAG(T130)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, appc40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, appc50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
PERM21_IN_GA(x1, x2)  =  PERM21_IN_GA(x1)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, appc40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, appc50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)

The TRS R consists of the following rules:

appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
PERM21_IN_GA(x1, x2)  =  PERM21_IN_GA(x1)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x4)

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:

PERM21_IN_GA(T110) → U7_GA(T110, appc40_in_aaag(T110))
U7_GA(T110, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, appc50_in_gga(T115, T116))
U8_GA(T110, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140)

The TRS R consists of the following rules:

appc40_in_aaag(.(X294, T130)) → U22_aaag(X294, T130, appc40_in_aaag(T130))
appc40_in_aaag(.(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
appc50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, appc50_in_gga(T160, T161))
appc50_in_gga([], T167) → appc50_out_gga([], T167, T167)
U22_aaag(X294, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
U26_gga(T157, T160, T161, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))

The set Q consists of the following terms:

appc40_in_aaag(x0)
appc50_in_gga(x0, x1)
U22_aaag(x0, x1, x2)
U26_gga(x0, x1, x2, x3)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_GA(T110, appc40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, appc50_in_gga(T115, T116))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(PERM21_IN_GA(x₁)) = 1 + x₁   
POL(U22_aaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U26_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U7_GA(x₁, x₂)) = 1 + x₂   
POL(U8_GA(x₁, x₂)) = 1 + x₂   
POL([]) = 0   
POL(appc40_in_aaag(x₁)) = x₁   
POL(appc40_out_aaag(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃   
POL(appc50_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appc50_out_gga(x₁, x₂, x₃)) = x₃   

The following usable rules [FROCOS05] were oriented:

appc40_in_aaag(.(X294, T130)) → U22_aaag(X294, T130, appc40_in_aaag(T130))
appc40_in_aaag(.(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
appc50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, appc50_in_gga(T160, T161))
appc50_in_gga([], T167) → appc50_out_gga([], T167, T167)
U22_aaag(X294, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
U26_gga(T157, T160, T161, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))

(27) Obligation:

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

PERM21_IN_GA(T110) → U7_GA(T110, appc40_in_aaag(T110))
U8_GA(T110, appc50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140)

The TRS R consists of the following rules:

appc40_in_aaag(.(X294, T130)) → U22_aaag(X294, T130, appc40_in_aaag(T130))
appc40_in_aaag(.(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
appc50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, appc50_in_gga(T160, T161))
appc50_in_gga([], T167) → appc50_out_gga([], T167, T167)
U22_aaag(X294, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
U26_gga(T157, T160, T161, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))

The set Q consists of the following terms:

appc40_in_aaag(x0)
appc50_in_gga(x0, x1)
U22_aaag(x0, x1, x2)
U26_gga(x0, x1, x2, x3)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(29) TRUE

(30) Obligation:

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

APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → APP25_IN_GGA(T84, T85, X200)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
APP25_IN_GGA(x1, x2, x3)  =  APP25_IN_GGA(x1, x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APP25_IN_GGA(.(T83, T84), T85, .(T83, X200)) → APP25_IN_GGA(T84, T85, X200)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP25_IN_GGA(x1, x2, x3)  =  APP25_IN_GGA(x1, x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP25_IN_GGA(.(T83, T84), T85) → APP25_IN_GGA(T84, T85)

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

(35) 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:

• APP25_IN_GGA(.(T83, T84), T85) → APP25_IN_GGA(T84, T85)
  The graph contains the following edges 1 > 1, 2 >= 2

(36) YES

(37) Obligation:

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

APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → APP10_IN_AAAG(X99, T40, X100, T39)

The TRS R consists of the following rules:

appc10_in_aaag(.(X98, X99), T40, X100, .(X98, T39)) → U20_aaag(X98, X99, T40, X100, T39, appc10_in_aaag(X99, T40, X100, T39))
appc10_in_aaag([], T45, X117, .(T45, X117)) → appc10_out_aaag([], T45, X117, .(T45, X117))
U20_aaag(X98, X99, T40, X100, T39, appc10_out_aaag(X99, T40, X100, T39)) → appc10_out_aaag(.(X98, X99), T40, X100, .(X98, T39))
appc20_in_ggga(X169, T66, T67, .(X169, X170)) → U27_ggga(X169, T66, T67, X170, appc25_in_gga(T66, T67, X170))
appc25_in_gga(.(T83, T84), T85, .(T83, X200)) → U21_gga(T83, T84, T85, X200, appc25_in_gga(T84, T85, X200))
appc25_in_gga([], T91, T91) → appc25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X200, appc25_out_gga(T84, T85, X200)) → appc25_out_gga(.(T83, T84), T85, .(T83, X200))
U27_ggga(X169, T66, T67, X170, appc25_out_gga(T66, T67, X170)) → appc20_out_ggga(X169, T66, T67, .(X169, X170))
appc40_in_aaag(.(X294, X295), T131, X296, .(X294, T130)) → U22_aaag(X294, X295, T131, X296, T130, appc40_in_aaag(X295, T131, X296, T130))
appc40_in_aaag([], T136, X313, .(T136, X313)) → appc40_out_aaag([], T136, X313, .(T136, X313))
U22_aaag(X294, X295, T131, X296, T130, appc40_out_aaag(X295, T131, X296, T130)) → appc40_out_aaag(.(X294, X295), T131, X296, .(X294, T130))
appc50_in_gga(.(T157, T160), T161, .(T157, X349)) → U26_gga(T157, T160, T161, X349, appc50_in_gga(T160, T161, X349))
appc50_in_gga([], T167, T167) → appc50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X349, appc50_out_gga(T160, T161, X349)) → appc50_out_gga(.(T157, T160), T161, .(T157, X349))
appc67_in_ga(X408, X408) → appc67_out_ga(X408, X408)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc10_in_aaag(x1, x2, x3, x4)  =  appc10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
appc10_out_aaag(x1, x2, x3, x4)  =  appc10_out_aaag(x1, x2, x3, x4)
appc20_in_ggga(x1, x2, x3, x4)  =  appc20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
appc25_in_gga(x1, x2, x3)  =  appc25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
appc25_out_gga(x1, x2, x3)  =  appc25_out_gga(x1, x2, x3)
appc20_out_ggga(x1, x2, x3, x4)  =  appc20_out_ggga(x1, x2, x3, x4)
appc40_in_aaag(x1, x2, x3, x4)  =  appc40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc40_out_aaag(x1, x2, x3, x4)  =  appc40_out_aaag(x1, x2, x3, x4)
appc50_in_gga(x1, x2, x3)  =  appc50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
appc50_out_gga(x1, x2, x3)  =  appc50_out_gga(x1, x2, x3)
appc67_in_ga(x1, x2)  =  appc67_in_ga(x1)
appc67_out_ga(x1, x2)  =  appc67_out_ga(x1, x2)
APP10_IN_AAAG(x1, x2, x3, x4)  =  APP10_IN_AAAG(x4)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

APP10_IN_AAAG(.(X98, X99), T40, X100, .(X98, T39)) → APP10_IN_AAAG(X99, T40, X100, T39)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP10_IN_AAAG(x1, x2, x3, x4)  =  APP10_IN_AAAG(x4)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

APP10_IN_AAAG(.(X98, T39)) → APP10_IN_AAAG(T39)

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

(42) 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:

• APP10_IN_AAAG(.(X98, T39)) → APP10_IN_AAAG(T39)
  The graph contains the following edges 1 > 1

(43) YES