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:

app1(.(X0, X), Y, .(X0, Z)) :- app1(X, Y, Z).
app1([], Y, Y).
app2(.(X0, X), Y, .(X0, Z)) :- app2(X, Y, Z).
app2([], Y, Y).
perm(X, .(X0, Y)) :- ','(app1(X1, .(X0, X2), X), ','(app2(X1, X2, Z), perm(Z, Y))).
perm([], []).

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: 4 | perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))) | perm(T6, T2), SCOPE: 1, CLAUSE: 5\n\nT6 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nperm(T1, T2) !=? perm(X6, .(X7, X8))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))), SCOPE: 2, CLAUSE: 0 | ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))), SCOPE: 2, CLAUSE: 1 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 5\n\nT6 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))), SCOPE: 2, CLAUSE: 0\n\nT6 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))), SCOPE: 2, CLAUSE: 1 | ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 5\n\nT6 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(app1(X48, .(T21, X49), T20), ','(app2(.(X47, X48), X49, X11), perm(X11, T22)))\n\nT20 is ground\nX11 is free; \nX47 is free; \nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: app1(X48, .(T21, X49), T20)\n\nT20 is ground\nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app2(.(X47, T25), T26, X11), perm(X11, T27))\n\nT25 is ground\nT26 is ground\nX11 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app1(X48, .(T21, X49), T20), SCOPE: 3, CLAUSE: 0 | app1(X48, .(T21, X49), T20), SCOPE: 3, CLAUSE: 1\n\nT20 is ground\nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app1(X48, .(T21, X49), T20), SCOPE: 3, CLAUSE: 0\n\nT20 is ground\nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app1(X48, .(T21, X49), T20), SCOPE: 3, CLAUSE: 1\n\nT20 is ground\nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app1(X101, .(T40, X102), T39)\n\nT39 is ground\nX101 is free; \nX102 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: app2(.(X47, T25), T26, X11)\n\nT25 is ground\nT26 is ground\nX11 is free; \nX47 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: app2(.(X47, T25), T26, X11), SCOPE: 4, CLAUSE: 2 | app2(.(X47, T25), T26, X11), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX11 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: app2(.(X47, T25), T26, X11), SCOPE: 4, CLAUSE: 2\n\nT25 is ground\nT26 is ground\nX11 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app2(.(X47, T25), T26, X11), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground\nX11 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app2(T66, T67, X172)\n\nT66 is ground\nT67 is ground\nX172 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app2(T66, T67, X172), SCOPE: 5, CLAUSE: 2 | app2(T66, T67, X172), SCOPE: 5, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nX172 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app2(T66, T67, X172), SCOPE: 5, CLAUSE: 2\n\nT66 is ground\nT67 is ground\nX172 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app2(T66, T67, X172), SCOPE: 5, CLAUSE: 3\n\nT66 is ground\nT67 is ground\nX172 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app2(T84, T85, X202)\n\nT84 is ground\nT85 is ground\nX202 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: 4 | perm(T49, T27), SCOPE: 6, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: perm(T49, T27), SCOPE: 6, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: perm(T49, T27), SCOPE: 6, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(app1(X244, .(T111, X245), T110), ','(app2(X244, X245, X246), perm(X246, T112)))\n\nX244 is free\nX245 is free; \nX246 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app1(X244, .(T111, X245), T110)\n\nX244 is free\nX245 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(app2(T115, T116, X246), perm(X246, T117))\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app1(X244, .(T111, X245), T110), SCOPE: 7, CLAUSE: 0 | app1(X244, .(T111, X245), T110), SCOPE: 7, CLAUSE: 1\n\nX244 is free\nX245 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app1(X244, .(T111, X245), T110), SCOPE: 7, CLAUSE: 0\n\nX244 is free\nX245 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app1(X244, .(T111, X245), T110), SCOPE: 7, CLAUSE: 1\n\nX244 is free\nX245 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app1(X297, .(T131, X298), T130)\n\nX297 is free\nX298 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: app2(T115, T116, X246)\n\nX246 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: app2(T115, T116, X246), SCOPE: 8, CLAUSE: 2 | app2(T115, T116, X246), SCOPE: 8, CLAUSE: 3\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: app2(T115, T116, X246), SCOPE: 8, CLAUSE: 2\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: app2(T115, T116, X246), SCOPE: 8, CLAUSE: 3\n\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: app2(T160, T161, X351)\n\nX351 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: ','(app1(X9, .(T9, X10), T6), ','(app2(X9, X10, X11), perm(X11, T10))), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nX9 is free; \nX10 is free; \nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ?_2 | perm(T6, T2), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(app2([], X380, X11), perm(X11, T176))\n\nX11 is free\nX380 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: app2([], X380, X11)\n\nX11 is free\nX380 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: app2([], X380, X11), SCOPE: 9, CLAUSE: 2 | app2([], X380, X11), SCOPE: 9, CLAUSE: 3\n\nX11 is free\nX380 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: app2([], X380, X11), SCOPE: 9, CLAUSE: 3\n\nX11 is free\nX380 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: 5\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(X6, .(X7, X8)) :- ','(app1(X9, .(X7, X10), X6), ','(app2(X9, X10, X11), perm(X11, X8))).\nand substitution\nT1 -> T6\nX6 -> T6\nX7 -> T9\nX8 -> 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\napp1(.(X43, X44), X45, .(X43, X46)) :- app1(X44, X45, X46).\nand substitution\nX43 -> X47\nX44 -> X48\nX9 -> .(X47, X48)\nT9 -> T21\nX10 -> X49\nX45 -> .(T21, X49)\nX46 -> T20\nT6 -> .(X47, 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:\nX48 -> T25\nX49 -> 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:\nX47 -> T48\nX11 -> 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\napp1(.(X96, X97), X98, .(X96, X99)) :- app1(X97, X98, X99).\nand substitution\nX96 -> X100\nX97 -> X101\nX48 -> .(X100, X101)\nT21 -> T40\nX49 -> X102\nX98 -> .(T40, X102)\nX99 -> T39\nT20 -> .(X100, 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\napp1([], X118, X118).\nand substitution\nX48 -> []\nT21 -> T45\nX49 -> X119\nX118 -> .(T45, X119)\nT20 -> .(T45, X119)", 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:\nX48 -> X101\nT21 -> T40\nX49 -> X102\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\napp2(.(X167, X168), X169, .(X167, X170)) :- app2(X168, X169, X170).\nand substitution\nX47 -> X171\nX167 -> X171\nT25 -> T66\nX168 -> T66\nT26 -> T67\nX169 -> T67\nX170 -> X172\nX11 -> .(X171, X172)", fontsize=14, style = filled, fillcolor = white];
23 -> 109 [arrowhead = none ];
109 -> 25;

110 [label="BACKTRACK\nfor clause: app2([], Y, Y)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\napp2(.(X198, X199), X200, .(X198, X201)) :- app2(X199, X200, X201).\nand substitution\nX198 -> T83\nX199 -> T84\nT66 -> .(T83, T84)\nT67 -> T85\nX200 -> T85\nX201 -> X202\nX172 -> .(T83, X202)", 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\napp2([], X214, X214).\nand substitution\nT66 -> []\nT67 -> T91\nX214 -> T91\nX172 -> 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\nX172 -> X202", 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(X241, .(X242, X243)) :- ','(app1(X244, .(X242, X245), X241), ','(app2(X244, X245, X246), perm(X246, X243))).\nand substitution\nT49 -> T110\nX241 -> T110\nX242 -> T111\nX243 -> 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:\nX244 -> T115\nX245 -> 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:\nX246 -> 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\napp1(.(X292, X293), X294, .(X292, X295)) :- app1(X293, X294, X295).\nand substitution\nX292 -> X296\nX293 -> X297\nX244 -> .(X296, X297)\nT111 -> T131\nX245 -> X298\nX294 -> .(T131, X298)\nX295 -> T130\nT110 -> .(X296, 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\napp1([], X314, X314).\nand substitution\nX244 -> []\nT111 -> T136\nX245 -> X315\nX314 -> .(T136, X315)\nT110 -> .(T136, X315)", 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:\nX244 -> X297\nT111 -> T131\nX245 -> X298\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\napp2(.(X347, X348), X349, .(X347, X350)) :- app2(X348, X349, X350).\nand substitution\nX347 -> T157\nX348 -> T160\nT115 -> .(T157, T160)\nT116 -> T161\nX349 -> T161\nX350 -> X351\nX246 -> .(T157, X351)\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\napp2([], X363, X363).\nand substitution\nT115 -> []\nT116 -> T167\nX363 -> T167\nX246 -> 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\nX246 -> X351", 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\napp1([], X379, X379).\nand substitution\nX9 -> []\nT9 -> T175\nX10 -> X380\nX379 -> .(T175, X380)\nT6 -> .(T175, X380)\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:\nX380 -> T179\nX11 -> 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: app2(.(X0, X), Y, .(X0, Z)) :- app2(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
69 -> 157 [arrowhead = none ];
157 -> 70;

158 [label="ONLY EVAL with clause\napp2([], X409, X409).\nand substitution\nX380 -> X410\nX409 -> X410\nX11 -> X410", 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:

app110(.(X100, X101), T40, X102, .(X100, T39)) :- app110(X101, T40, X102, T39).
app225(.(T83, T84), T85, .(T83, X202)) :- app225(T84, T85, X202).
app140(.(X296, X297), T131, X298, .(X296, T130)) :- app140(X297, T131, X298, T130).
perm21(T110, .(T111, T112)) :- app140(X244, T111, X245, T110).
perm21(T110, .(T111, T117)) :- ','(app1c40(T115, T111, T116, T110), app250(T115, T116, X246)).
perm21(T110, .(T111, T141)) :- ','(app1c40(T115, T111, T116, T110), ','(app2c50(T115, T116, T140), perm21(T140, T141))).
app250(.(T157, T160), T161, .(T157, X351)) :- app250(T160, T161, X351).
perm1(.(X47, T20), .(T21, T22)) :- app110(X48, T21, X49, T20).
perm1(.(X171, T20), .(T21, T27)) :- ','(app1c10(T66, T21, T67, T20), app225(T66, T67, X172)).
perm1(.(T48, T20), .(T21, T27)) :- ','(app1c10(T25, T21, T26, T20), ','(app2c20(T48, T25, T26, T49), perm21(T49, T27))).
perm1(.(T175, T179), .(T175, T176)) :- ','(app2c67(T179, T180), perm21(T180, T176)).

Clauses:

app1c10(.(X100, X101), T40, X102, .(X100, T39)) :- app1c10(X101, T40, X102, T39).
app1c10([], T45, X119, .(T45, X119)).
app2c25(.(T83, T84), T85, .(T83, X202)) :- app2c25(T84, T85, X202).
app2c25([], T91, T91).
app1c40(.(X296, X297), T131, X298, .(X296, T130)) :- app1c40(X297, T131, X298, T130).
app1c40([], T136, X315, .(T136, X315)).
permc21(T110, .(T111, T141)) :- ','(app1c40(T115, T111, T116, T110), ','(app2c50(T115, T116, T140), permc21(T140, T141))).
permc21([], []).
app2c50(.(T157, T160), T161, .(T157, X351)) :- app2c50(T160, T161, X351).
app2c50([], T167, T167).
app2c20(X171, T66, T67, .(X171, X172)) :- app2c25(T66, T67, X172).
app2c67(X410, X410).

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)
app110_in: (f,f,f,b)
app1c10_in: (f,f,f,b)
app225_in: (b,b,f)
app2c20_in: (b,b,b,f)
app2c25_in: (b,b,f)
perm21_in: (b,f)
app140_in: (f,f,f,b)
app1c40_in: (f,f,f,b)
app250_in: (b,b,f)
app2c50_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(.(X47, T20), .(T21, T22)) → U11_GA(X47, T20, T21, T22, app110_in_aaag(X48, T21, X49, T20))
PERM1_IN_GA(.(X47, T20), .(T21, T22)) → APP110_IN_AAAG(X48, T21, X49, T20)
APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → U1_AAAG(X100, X101, T40, X102, T39, app110_in_aaag(X101, T40, X102, T39))
APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → APP110_IN_AAAG(X101, T40, X102, T39)
PERM1_IN_GA(.(X171, T20), .(T21, T27)) → U12_GA(X171, T20, T21, T27, app1c10_in_aaag(T66, T21, T67, T20))
U12_GA(X171, T20, T21, T27, app1c10_out_aaag(T66, T21, T67, T20)) → U13_GA(X171, T20, T21, T27, app225_in_gga(T66, T67, X172))
U12_GA(X171, T20, T21, T27, app1c10_out_aaag(T66, T21, T67, T20)) → APP225_IN_GGA(T66, T67, X172)
APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → U2_GGA(T83, T84, T85, X202, app225_in_gga(T84, T85, X202))
APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → APP225_IN_GGA(T84, T85, X202)
PERM1_IN_GA(.(T48, T20), .(T21, T27)) → U14_GA(T48, T20, T21, T27, app1c10_in_aaag(T25, T21, T26, T20))
U14_GA(T48, T20, T21, T27, app1c10_out_aaag(T25, T21, T26, T20)) → U15_GA(T48, T20, T21, T27, app2c20_in_ggga(T48, T25, T26, T49))
U15_GA(T48, T20, T21, T27, app2c20_out_ggga(T48, T25, T26, T49)) → U16_GA(T48, T20, T21, T27, perm21_in_ga(T49, T27))
U15_GA(T48, T20, T21, T27, app2c20_out_ggga(T48, T25, T26, T49)) → PERM21_IN_GA(T49, T27)
PERM21_IN_GA(T110, .(T111, T112)) → U4_GA(T110, T111, T112, app140_in_aaag(X244, T111, X245, T110))
PERM21_IN_GA(T110, .(T111, T112)) → APP140_IN_AAAG(X244, T111, X245, T110)
APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → U3_AAAG(X296, X297, T131, X298, T130, app140_in_aaag(X297, T131, X298, T130))
APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → APP140_IN_AAAG(X297, T131, X298, T130)
PERM21_IN_GA(T110, .(T111, T117)) → U5_GA(T110, T111, T117, app1c40_in_aaag(T115, T111, T116, T110))
U5_GA(T110, T111, T117, app1c40_out_aaag(T115, T111, T116, T110)) → U6_GA(T110, T111, T117, app250_in_gga(T115, T116, X246))
U5_GA(T110, T111, T117, app1c40_out_aaag(T115, T111, T116, T110)) → APP250_IN_GGA(T115, T116, X246)
APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → U10_GGA(T157, T160, T161, X351, app250_in_gga(T160, T161, X351))
APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → APP250_IN_GGA(T160, T161, X351)
PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, app1c40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, app2c50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → U9_GA(T110, T111, T141, perm21_in_ga(T140, T141))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)
PERM1_IN_GA(.(T175, T179), .(T175, T176)) → U17_GA(T175, T179, T176, app2c67_in_ga(T179, T180))
U17_GA(T175, T179, T176, app2c67_out_ga(T179, T180)) → U18_GA(T175, T179, T176, perm21_in_ga(T180, T176))
U17_GA(T175, T179, T176, app2c67_out_ga(T179, T180)) → PERM21_IN_GA(T180, T176)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app110_in_aaag(x1, x2, x3, x4)  =  app110_in_aaag(x4)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app225_in_gga(x1, x2, x3)  =  app225_in_gga(x1, x2)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
perm21_in_ga(x1, x2)  =  perm21_in_ga(x1)
app140_in_aaag(x1, x2, x3, x4)  =  app140_in_aaag(x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app250_in_gga(x1, x2, x3)  =  app250_in_gga(x1, x2)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_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)
APP110_IN_AAAG(x1, x2, x3, x4)  =  APP110_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)
APP225_IN_GGA(x1, x2, x3)  =  APP225_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)
APP140_IN_AAAG(x1, x2, x3, x4)  =  APP140_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)
APP250_IN_GGA(x1, x2, x3)  =  APP250_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(.(X47, T20), .(T21, T22)) → U11_GA(X47, T20, T21, T22, app110_in_aaag(X48, T21, X49, T20))
PERM1_IN_GA(.(X47, T20), .(T21, T22)) → APP110_IN_AAAG(X48, T21, X49, T20)
APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → U1_AAAG(X100, X101, T40, X102, T39, app110_in_aaag(X101, T40, X102, T39))
APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → APP110_IN_AAAG(X101, T40, X102, T39)
PERM1_IN_GA(.(X171, T20), .(T21, T27)) → U12_GA(X171, T20, T21, T27, app1c10_in_aaag(T66, T21, T67, T20))
U12_GA(X171, T20, T21, T27, app1c10_out_aaag(T66, T21, T67, T20)) → U13_GA(X171, T20, T21, T27, app225_in_gga(T66, T67, X172))
U12_GA(X171, T20, T21, T27, app1c10_out_aaag(T66, T21, T67, T20)) → APP225_IN_GGA(T66, T67, X172)
APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → U2_GGA(T83, T84, T85, X202, app225_in_gga(T84, T85, X202))
APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → APP225_IN_GGA(T84, T85, X202)
PERM1_IN_GA(.(T48, T20), .(T21, T27)) → U14_GA(T48, T20, T21, T27, app1c10_in_aaag(T25, T21, T26, T20))
U14_GA(T48, T20, T21, T27, app1c10_out_aaag(T25, T21, T26, T20)) → U15_GA(T48, T20, T21, T27, app2c20_in_ggga(T48, T25, T26, T49))
U15_GA(T48, T20, T21, T27, app2c20_out_ggga(T48, T25, T26, T49)) → U16_GA(T48, T20, T21, T27, perm21_in_ga(T49, T27))
U15_GA(T48, T20, T21, T27, app2c20_out_ggga(T48, T25, T26, T49)) → PERM21_IN_GA(T49, T27)
PERM21_IN_GA(T110, .(T111, T112)) → U4_GA(T110, T111, T112, app140_in_aaag(X244, T111, X245, T110))
PERM21_IN_GA(T110, .(T111, T112)) → APP140_IN_AAAG(X244, T111, X245, T110)
APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → U3_AAAG(X296, X297, T131, X298, T130, app140_in_aaag(X297, T131, X298, T130))
APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → APP140_IN_AAAG(X297, T131, X298, T130)
PERM21_IN_GA(T110, .(T111, T117)) → U5_GA(T110, T111, T117, app1c40_in_aaag(T115, T111, T116, T110))
U5_GA(T110, T111, T117, app1c40_out_aaag(T115, T111, T116, T110)) → U6_GA(T110, T111, T117, app250_in_gga(T115, T116, X246))
U5_GA(T110, T111, T117, app1c40_out_aaag(T115, T111, T116, T110)) → APP250_IN_GGA(T115, T116, X246)
APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → U10_GGA(T157, T160, T161, X351, app250_in_gga(T160, T161, X351))
APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → APP250_IN_GGA(T160, T161, X351)
PERM21_IN_GA(T110, .(T111, T141)) → U7_GA(T110, T111, T141, app1c40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, app2c50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → U9_GA(T110, T111, T141, perm21_in_ga(T140, T141))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)
PERM1_IN_GA(.(T175, T179), .(T175, T176)) → U17_GA(T175, T179, T176, app2c67_in_ga(T179, T180))
U17_GA(T175, T179, T176, app2c67_out_ga(T179, T180)) → U18_GA(T175, T179, T176, perm21_in_ga(T180, T176))
U17_GA(T175, T179, T176, app2c67_out_ga(T179, T180)) → PERM21_IN_GA(T180, T176)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app110_in_aaag(x1, x2, x3, x4)  =  app110_in_aaag(x4)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app225_in_gga(x1, x2, x3)  =  app225_in_gga(x1, x2)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
perm21_in_ga(x1, x2)  =  perm21_in_ga(x1)
app140_in_aaag(x1, x2, x3, x4)  =  app140_in_aaag(x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app250_in_gga(x1, x2, x3)  =  app250_in_gga(x1, x2)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_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)
APP110_IN_AAAG(x1, x2, x3, x4)  =  APP110_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)
APP225_IN_GGA(x1, x2, x3)  =  APP225_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)
APP140_IN_AAAG(x1, x2, x3, x4)  =  APP140_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)
APP250_IN_GGA(x1, x2, x3)  =  APP250_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:

APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → APP250_IN_GGA(T160, T161, X351)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_out_ga(x1, x2)
APP250_IN_GGA(x1, x2, x3)  =  APP250_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:

APP250_IN_GGA(.(T157, T160), T161, .(T157, X351)) → APP250_IN_GGA(T160, T161, X351)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP250_IN_GGA(x1, x2, x3)  =  APP250_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:

APP250_IN_GGA(.(T157, T160), T161) → APP250_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:

• APP250_IN_GGA(.(T157, T160), T161) → APP250_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:

APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → APP140_IN_AAAG(X297, T131, X298, T130)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_out_ga(x1, x2)
APP140_IN_AAAG(x1, x2, x3, x4)  =  APP140_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:

APP140_IN_AAAG(.(X296, X297), T131, X298, .(X296, T130)) → APP140_IN_AAAG(X297, T131, X298, T130)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP140_IN_AAAG(x1, x2, x3, x4)  =  APP140_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:

APP140_IN_AAAG(.(X296, T130)) → APP140_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:

• APP140_IN_AAAG(.(X296, T130)) → APP140_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, app1c40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, app2c50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_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, app1c40_in_aaag(T115, T111, T116, T110))
U7_GA(T110, T111, T141, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, T111, T141, app2c50_in_gga(T115, T116, T140))
U8_GA(T110, T111, T141, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140, T141)

The TRS R consists of the following rules:

app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_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, app1c40_in_aaag(T110))
U7_GA(T110, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, app2c50_in_gga(T115, T116))
U8_GA(T110, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140)

The TRS R consists of the following rules:

app1c40_in_aaag(.(X296, T130)) → U22_aaag(X296, T130, app1c40_in_aaag(T130))
app1c40_in_aaag(.(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
app2c50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, app2c50_in_gga(T160, T161))
app2c50_in_gga([], T167) → app2c50_out_gga([], T167, T167)
U22_aaag(X296, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
U26_gga(T157, T160, T161, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))

The set Q consists of the following terms:

app1c40_in_aaag(x0)
app2c50_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, app1c40_out_aaag(T115, T111, T116, T110)) → U8_GA(T110, app2c50_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(app1c40_in_aaag(x₁)) = x₁   
POL(app1c40_out_aaag(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃   
POL(app2c50_in_gga(x₁, x₂)) = x₁ + x₂   
POL(app2c50_out_gga(x₁, x₂, x₃)) = x₃   

The following usable rules [FROCOS05] were oriented:

app1c40_in_aaag(.(X296, T130)) → U22_aaag(X296, T130, app1c40_in_aaag(T130))
app1c40_in_aaag(.(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
app2c50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, app2c50_in_gga(T160, T161))
app2c50_in_gga([], T167) → app2c50_out_gga([], T167, T167)
U22_aaag(X296, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
U26_gga(T157, T160, T161, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))

(27) Obligation:

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

PERM21_IN_GA(T110) → U7_GA(T110, app1c40_in_aaag(T110))
U8_GA(T110, app2c50_out_gga(T115, T116, T140)) → PERM21_IN_GA(T140)

The TRS R consists of the following rules:

app1c40_in_aaag(.(X296, T130)) → U22_aaag(X296, T130, app1c40_in_aaag(T130))
app1c40_in_aaag(.(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
app2c50_in_gga(.(T157, T160), T161) → U26_gga(T157, T160, T161, app2c50_in_gga(T160, T161))
app2c50_in_gga([], T167) → app2c50_out_gga([], T167, T167)
U22_aaag(X296, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
U26_gga(T157, T160, T161, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))

The set Q consists of the following terms:

app1c40_in_aaag(x0)
app2c50_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:

APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → APP225_IN_GGA(T84, T85, X202)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_out_ga(x1, x2)
APP225_IN_GGA(x1, x2, x3)  =  APP225_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:

APP225_IN_GGA(.(T83, T84), T85, .(T83, X202)) → APP225_IN_GGA(T84, T85, X202)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP225_IN_GGA(x1, x2, x3)  =  APP225_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:

APP225_IN_GGA(.(T83, T84), T85) → APP225_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:

• APP225_IN_GGA(.(T83, T84), T85) → APP225_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:

APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → APP110_IN_AAAG(X101, T40, X102, T39)

The TRS R consists of the following rules:

app1c10_in_aaag(.(X100, X101), T40, X102, .(X100, T39)) → U20_aaag(X100, X101, T40, X102, T39, app1c10_in_aaag(X101, T40, X102, T39))
app1c10_in_aaag([], T45, X119, .(T45, X119)) → app1c10_out_aaag([], T45, X119, .(T45, X119))
U20_aaag(X100, X101, T40, X102, T39, app1c10_out_aaag(X101, T40, X102, T39)) → app1c10_out_aaag(.(X100, X101), T40, X102, .(X100, T39))
app2c20_in_ggga(X171, T66, T67, .(X171, X172)) → U27_ggga(X171, T66, T67, X172, app2c25_in_gga(T66, T67, X172))
app2c25_in_gga(.(T83, T84), T85, .(T83, X202)) → U21_gga(T83, T84, T85, X202, app2c25_in_gga(T84, T85, X202))
app2c25_in_gga([], T91, T91) → app2c25_out_gga([], T91, T91)
U21_gga(T83, T84, T85, X202, app2c25_out_gga(T84, T85, X202)) → app2c25_out_gga(.(T83, T84), T85, .(T83, X202))
U27_ggga(X171, T66, T67, X172, app2c25_out_gga(T66, T67, X172)) → app2c20_out_ggga(X171, T66, T67, .(X171, X172))
app1c40_in_aaag(.(X296, X297), T131, X298, .(X296, T130)) → U22_aaag(X296, X297, T131, X298, T130, app1c40_in_aaag(X297, T131, X298, T130))
app1c40_in_aaag([], T136, X315, .(T136, X315)) → app1c40_out_aaag([], T136, X315, .(T136, X315))
U22_aaag(X296, X297, T131, X298, T130, app1c40_out_aaag(X297, T131, X298, T130)) → app1c40_out_aaag(.(X296, X297), T131, X298, .(X296, T130))
app2c50_in_gga(.(T157, T160), T161, .(T157, X351)) → U26_gga(T157, T160, T161, X351, app2c50_in_gga(T160, T161, X351))
app2c50_in_gga([], T167, T167) → app2c50_out_gga([], T167, T167)
U26_gga(T157, T160, T161, X351, app2c50_out_gga(T160, T161, X351)) → app2c50_out_gga(.(T157, T160), T161, .(T157, X351))
app2c67_in_ga(X410, X410) → app2c67_out_ga(X410, X410)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
app1c10_in_aaag(x1, x2, x3, x4)  =  app1c10_in_aaag(x4)
U20_aaag(x1, x2, x3, x4, x5, x6)  =  U20_aaag(x1, x5, x6)
app1c10_out_aaag(x1, x2, x3, x4)  =  app1c10_out_aaag(x1, x2, x3, x4)
app2c20_in_ggga(x1, x2, x3, x4)  =  app2c20_in_ggga(x1, x2, x3)
U27_ggga(x1, x2, x3, x4, x5)  =  U27_ggga(x1, x2, x3, x5)
app2c25_in_gga(x1, x2, x3)  =  app2c25_in_gga(x1, x2)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
[]  =  []
app2c25_out_gga(x1, x2, x3)  =  app2c25_out_gga(x1, x2, x3)
app2c20_out_ggga(x1, x2, x3, x4)  =  app2c20_out_ggga(x1, x2, x3, x4)
app1c40_in_aaag(x1, x2, x3, x4)  =  app1c40_in_aaag(x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app1c40_out_aaag(x1, x2, x3, x4)  =  app1c40_out_aaag(x1, x2, x3, x4)
app2c50_in_gga(x1, x2, x3)  =  app2c50_in_gga(x1, x2)
U26_gga(x1, x2, x3, x4, x5)  =  U26_gga(x1, x2, x3, x5)
app2c50_out_gga(x1, x2, x3)  =  app2c50_out_gga(x1, x2, x3)
app2c67_in_ga(x1, x2)  =  app2c67_in_ga(x1)
app2c67_out_ga(x1, x2)  =  app2c67_out_ga(x1, x2)
APP110_IN_AAAG(x1, x2, x3, x4)  =  APP110_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:

APP110_IN_AAAG(.(X100, X101), T40, X102, .(X100, T39)) → APP110_IN_AAAG(X101, T40, X102, T39)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP110_IN_AAAG(x1, x2, x3, x4)  =  APP110_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:

APP110_IN_AAAG(.(X100, T39)) → APP110_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:

• APP110_IN_AAAG(.(X100, T39)) → APP110_IN_AAAG(T39)
  The graph contains the following edges 1 > 1

(43) YES