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

(0) Obligation:

Clauses:

slowsort(X, Y) :- ','(perm(X, Y), sorted(Y)).
sorted([]).
sorted(.(X, [])).
sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z))).
perm([], []).
perm(.(X, .(Y, [])), .(U, .(V, []))) :- ','(delete(U, .(X, Y), Z), perm(Z, V)).
delete(X, .(X, Y), Y).
delete(X, .(Y, Z), W) :- delete(X, Z, W).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).

Query: slowsort(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: slowsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: slowsort(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(perm(T5, T7), sorted(T7))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 4 | ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: sorted([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: sorted([]), SCOPE: 3, CLAUSE: 1 | sorted([]), SCOPE: 3, CLAUSE: 2 | sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: sorted([]), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: sorted([]), SCOPE: 3, CLAUSE: 2 | sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(delete(T20, .(T16, T17), X22), perm(X22, T21)), sorted(.(T20, .(T21, []))))\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: delete(T20, .(T16, T17), X22)\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(perm(T26, T27), sorted(.(T28, .(T27, []))))\n\nT28 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 6 | delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 7\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 6\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 7\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: delete(T57, T56, X59)\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 6 | delete(T57, T56, X59), SCOPE: 5, CLAUSE: 7\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 6\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 7\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: delete(T81, T80, X88)\n\nT80 is ground\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: perm(T26, T27)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: sorted(.(T28, .(T86, [])))\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: perm(T26, T27), SCOPE: 6, CLAUSE: 4 | perm(T26, T27), SCOPE: 6, CLAUSE: 5\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: perm(T26, T27), SCOPE: 6, CLAUSE: 4\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: perm(T26, T27), SCOPE: 6, CLAUSE: 5\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(delete(T99, .(T95, T96), X106), perm(X106, T100))\n\nT95 is ground\nT96 is ground\nX106 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: delete(T99, .(T95, T96), X106)\n\nT95 is ground\nT96 is ground\nX106 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: perm(T105, T106)\n\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 1 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(le(T118, T119), sorted(.(T119, [])))\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: le(T118, T119)\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: sorted(.(T119, []))\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: le(T118, T119), SCOPE: 8, CLAUSE: 8 | le(T118, T119), SCOPE: 8, CLAUSE: 9 | le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: le(T118, T119), SCOPE: 8, CLAUSE: 8\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: le(T118, T119), SCOPE: 8, CLAUSE: 9 | le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: le(T132, T133)\n\nT132 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: le(T118, T119), SCOPE: 8, CLAUSE: 9\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", 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: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 1 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 73 [arrowhead = none ];
73 -> 2;

74 [label="ONLY EVAL with clause\nslowsort(X3, X4) :- ','(perm(X3, X4), sorted(X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 3;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 75 [arrowhead = none ];
75 -> 4;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 76 [arrowhead = none ];
76 -> 5;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="EVAL with clause\nperm([], []).\nand substitutionT5 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 79 [arrowhead = none ];
79 -> 8;

80 [label="EVAL with clause\nperm(.(X18, .(X19, [])), .(X20, .(X21, []))) :- ','(delete(X20, .(X18, X19), X22), perm(X22, X21)).\nand substitutionX18 -> T16,\nX19 -> T17,\nT5 -> .(T16, .(T17, [])),\nX20 -> T20,\nX21 -> T21,\nT7 -> .(T20, .(T21, [])),\nT18 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 80 [arrowhead = none ];
80 -> 16;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 81 [arrowhead = none ];
81 -> 17;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 82 [arrowhead = none ];
82 -> 9;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 83 [arrowhead = none ];
83 -> 10;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 84 [arrowhead = none ];
84 -> 11;

85 [label="ONLY EVAL with clause\nsorted([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 85 [arrowhead = none ];
85 -> 12;

86 [label="BACKTRACK\nfor clause: sorted(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 86 [arrowhead = none ];
86 -> 14;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 87 [arrowhead = none ];
87 -> 13;

88 [label="BACKTRACK\nfor clause: sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 88 [arrowhead = none ];
88 -> 15;

89 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 89 [arrowhead = none ];
89 -> 18;

90 [label="SPLIT 2\nnew knowledge:\nT28 is ground\nT16 is ground\nT17 is ground\nT26 is ground\nreplacements:X22 -> T26,\nT21 -> T27,\nT20 -> T28", fontsize=14, style = filled, fillcolor = violet];
16 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 91 [arrowhead = none ];
91 -> 20;

92 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 92 [arrowhead = none ];
92 -> 35;

93 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nT86 is ground\nreplacements:T27 -> T86", fontsize=14, style = filled, fillcolor = violet];
19 -> 93 [arrowhead = none ];
93 -> 36;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 94 [arrowhead = none ];
94 -> 21;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 95 [arrowhead = none ];
95 -> 22;

96 [label="EVAL with clause\ndelete(X39, .(X39, X40), X40).\nand substitutionT20 -> T41,\nX39 -> T41,\nT16 -> T41,\nT17 -> T42,\nX40 -> T42,\nX22 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 96 [arrowhead = none ];
96 -> 23;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="ONLY EVAL with clause\ndelete(X55, .(X56, X57), X58) :- delete(X55, X57, X58).\nand substitutionT20 -> T57,\nX55 -> T57,\nT16 -> T55,\nX56 -> T55,\nT17 -> T56,\nX57 -> T56,\nX22 -> X59,\nX58 -> X59,\nT54 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 98 [arrowhead = none ];
98 -> 26;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 99 [arrowhead = none ];
99 -> 25;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 100 [arrowhead = none ];
100 -> 27;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 101 [arrowhead = none ];
101 -> 28;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 102 [arrowhead = none ];
102 -> 29;

103 [label="EVAL with clause\ndelete(X72, .(X72, X73), X73).\nand substitutionT57 -> T70,\nX72 -> T70,\nX73 -> T71,\nT56 -> .(T70, T71),\nX59 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 103 [arrowhead = none ];
103 -> 30;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 104 [arrowhead = none ];
104 -> 31;

105 [label="EVAL with clause\ndelete(X84, .(X85, X86), X87) :- delete(X84, X86, X87).\nand substitutionT57 -> T81,\nX84 -> T81,\nX85 -> T79,\nX86 -> T80,\nT56 -> .(T79, T80),\nX59 -> X88,\nX87 -> X88,\nT78 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 105 [arrowhead = none ];
105 -> 33;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 106 [arrowhead = none ];
106 -> 34;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="INSTANCE with matching:\nT57 -> T81\nT56 -> T80\nX59 -> X88", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 108 [arrowhead = none , style=dashed];
108 -> 26[style=dashed];

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 109 [arrowhead = none ];
109 -> 37;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 110 [arrowhead = none ];
110 -> 47;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 111 [arrowhead = none ];
111 -> 38;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 112 [arrowhead = none ];
112 -> 39;

113 [label="EVAL with clause\nperm([], []).\nand substitutionT26 -> [],\nT27 -> []", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 113 [arrowhead = none ];
113 -> 40;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 114 [arrowhead = none ];
114 -> 41;

115 [label="EVAL with clause\nperm(.(X102, .(X103, [])), .(X104, .(X105, []))) :- ','(delete(X104, .(X102, X103), X106), perm(X106, X105)).\nand substitutionX102 -> T95,\nX103 -> T96,\nT26 -> .(T95, .(T96, [])),\nX104 -> T99,\nX105 -> T100,\nT27 -> .(T99, .(T100, [])),\nT97 -> T99,\nT98 -> T100", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 115 [arrowhead = none ];
115 -> 43;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 116 [arrowhead = none ];
116 -> 44;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 117 [arrowhead = none ];
117 -> 42;

118 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 118 [arrowhead = none ];
118 -> 45;

119 [label="SPLIT 2\nnew knowledge:\nT99 is ground\nT95 is ground\nT96 is ground\nT105 is ground\nreplacements:X106 -> T105,\nT100 -> T106", fontsize=14, style = filled, fillcolor = violet];
43 -> 119 [arrowhead = none ];
119 -> 46;

120 [label="INSTANCE with matching:\nT20 -> T99\nT16 -> T95\nT17 -> T96\nX22 -> X106", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 120 [arrowhead = none , style=dashed];
120 -> 18[style=dashed];

121 [label="INSTANCE with matching:\nT26 -> T105\nT27 -> T106", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 121 [arrowhead = none , style=dashed];
121 -> 35[style=dashed];

122 [label="BACKTRACK\nfor clause: sorted([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 122 [arrowhead = none ];
122 -> 48;

123 [label="BACKTRACK\nfor clause: sorted(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 123 [arrowhead = none ];
123 -> 49;

124 [label="ONLY EVAL with clause\nsorted(.(X124, .(X125, X126))) :- ','(le(X124, X125), sorted(.(X125, X126))).\nand substitutionT28 -> T118,\nX124 -> T118,\nT86 -> T119,\nX125 -> T119,\nX126 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 125 [arrowhead = none ];
125 -> 51;

126 [label="SPLIT 2\nnew knowledge:\nT118 is ground\nT119 is ground", fontsize=14, style = filled, fillcolor = violet];
50 -> 126 [arrowhead = none ];
126 -> 52;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 127 [arrowhead = none ];
127 -> 53;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 128 [arrowhead = none ];
128 -> 66;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 129 [arrowhead = none ];
129 -> 54;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="EVAL with clause\nle(s(X139), s(X140)) :- le(X139, X140).\nand substitutionX139 -> T132,\nT118 -> s(T132),\nX140 -> T133,\nT119 -> s(T133)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 132 [arrowhead = none ];
132 -> 57;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 133 [arrowhead = none ];
133 -> 58;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 134 [arrowhead = none ];
134 -> 59;

135 [label="INSTANCE with matching:\nT118 -> T132\nT119 -> T133", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 135 [arrowhead = none , style=dashed];
135 -> 51[style=dashed];

136 [label="EVAL with clause\nle(0, s(X147)).\nand substitutionT118 -> 0,\nX147 -> T140,\nT119 -> s(T140)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 136 [arrowhead = none ];
136 -> 60;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 137 [arrowhead = none ];
137 -> 61;

138 [label="EVAL with clause\nle(0, 0).\nand substitutionT118 -> 0,\nT119 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 140 [arrowhead = none ];
140 -> 62;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 141 [arrowhead = none ];
141 -> 65;

142 [label="BACKTRACK\nfor clause: sorted([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 142 [arrowhead = none ];
142 -> 67;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 143 [arrowhead = none ];
143 -> 68;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 144 [arrowhead = none ];
144 -> 69;

145 [label="ONLY EVAL with clause\nsorted(.(X152, [])).\nand substitutionT119 -> T145,\nX152 -> T145", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 145 [arrowhead = none ];
145 -> 70;

146 [label="BACKTRACK\nfor clause: sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 146 [arrowhead = none ];
146 -> 72;

147 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 147 [arrowhead = none ];
147 -> 71;

}

(2) Obligation:

Clauses:

deleteA(T70, .(T70, T71), T71).
deleteA(T81, .(T79, T80), X88) :- deleteA(T81, T80, X88).
deleteB(T41, T41, T42, T42).
deleteB(T57, T55, T56, X59) :- deleteA(T57, T56, X59).
permC([], []).
permC(.(T95, .(T96, [])), .(T99, .(T100, []))) :- deleteB(T99, T95, T96, X106).
permC(.(T95, .(T96, [])), .(T99, .(T106, []))) :- ','(deleteB(T99, T95, T96, T105), permC(T105, T106)).
leD(s(T132), s(T133)) :- leD(T132, T133).
leD(0, s(T140)).
leD(0, 0).
slowsortE([], []).
slowsortE(.(T16, .(T17, [])), .(T20, .(T21, []))) :- deleteB(T20, T16, T17, X22).
slowsortE(.(T16, .(T17, [])), .(T28, .(T27, []))) :- ','(deleteB(T28, T16, T17, T26), permC(T26, T27)).
slowsortE(.(T16, .(T17, [])), .(T118, .(T119, []))) :- ','(deleteB(T118, T16, T17, T26), ','(permC(T26, T119), leD(T118, T119))).
slowsortE(.(T16, .(T17, [])), .(T118, .(T145, []))) :- ','(deleteB(T118, T16, T17, T26), ','(permC(T26, T145), leD(T118, T145))).

Query: slowsortE(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
slowsortE_in: (b,f)
deleteB_in: (f,b,b,f)
deleteA_in: (f,b,f)
permC_in: (b,f)
leD_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_GA(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → DELETEB_IN_AGGA(T20, T16, T17, X22)
DELETEB_IN_AGGA(T57, T55, T56, X59) → U2_AGGA(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
DELETEB_IN_AGGA(T57, T55, T56, X59) → DELETEA_IN_AGA(T57, T56, X59)
DELETEA_IN_AGA(T81, .(T79, T80), X88) → U1_AGA(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
DELETEA_IN_AGA(T81, .(T79, T80), X88) → DELETEA_IN_AGA(T81, T80, X88)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_GA(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_GA(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_GA(T16, T17, T28, T27, permC_in_ga(T26, T27))
U8_GA(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → PERMC_IN_GA(T26, T27)
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_GA(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → DELETEB_IN_AGGA(T99, T95, T96, X106)
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_GA(T95, T96, T99, T106, permC_in_ga(T105, T106))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → PERMC_IN_GA(T105, T106)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_GA(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_GA(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_GA(T16, T17, T118, T119, permC_in_ga(T26, T119))
U10_GA(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → PERMC_IN_GA(T26, T119)
U11_GA(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_GA(T16, T17, T118, T119, leD_in_ga(T118, T119))
U11_GA(T16, T17, T118, T119, permC_out_ga(T26, T119)) → LED_IN_GA(T118, T119)
LED_IN_GA(s(T132), s(T133)) → U6_GA(T132, T133, leD_in_ga(T132, T133))
LED_IN_GA(s(T132), s(T133)) → LED_IN_GA(T132, T133)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_GA(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_GA(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_GA(T16, T17, T118, T145, permC_in_ga(T26, T145))
U13_GA(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → PERMC_IN_GA(T26, T145)
U14_GA(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_GA(T16, T17, T118, T145, leD_in_ga(T118, T145))
U14_GA(T16, T17, T118, T145, permC_out_ga(T26, T145)) → LED_IN_GA(T118, T145)

The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
SLOWSORTE_IN_GA(x1, x2)  =  SLOWSORTE_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
DELETEB_IN_AGGA(x1, x2, x3, x4)  =  DELETEB_IN_AGGA(x2, x3)
U2_AGGA(x1, x2, x3, x4, x5)  =  U2_AGGA(x5)
DELETEA_IN_AGA(x1, x2, x3)  =  DELETEA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
PERMC_IN_GA(x1, x2)  =  PERMC_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
LED_IN_GA(x1, x2)  =  LED_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x3, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x5)

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

(6) Obligation:

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

SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_GA(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → DELETEB_IN_AGGA(T20, T16, T17, X22)
DELETEB_IN_AGGA(T57, T55, T56, X59) → U2_AGGA(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
DELETEB_IN_AGGA(T57, T55, T56, X59) → DELETEA_IN_AGA(T57, T56, X59)
DELETEA_IN_AGA(T81, .(T79, T80), X88) → U1_AGA(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
DELETEA_IN_AGA(T81, .(T79, T80), X88) → DELETEA_IN_AGA(T81, T80, X88)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_GA(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_GA(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_GA(T16, T17, T28, T27, permC_in_ga(T26, T27))
U8_GA(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → PERMC_IN_GA(T26, T27)
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_GA(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → DELETEB_IN_AGGA(T99, T95, T96, X106)
PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_GA(T95, T96, T99, T106, permC_in_ga(T105, T106))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → PERMC_IN_GA(T105, T106)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_GA(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_GA(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_GA(T16, T17, T118, T119, permC_in_ga(T26, T119))
U10_GA(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → PERMC_IN_GA(T26, T119)
U11_GA(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_GA(T16, T17, T118, T119, leD_in_ga(T118, T119))
U11_GA(T16, T17, T118, T119, permC_out_ga(T26, T119)) → LED_IN_GA(T118, T119)
LED_IN_GA(s(T132), s(T133)) → U6_GA(T132, T133, leD_in_ga(T132, T133))
LED_IN_GA(s(T132), s(T133)) → LED_IN_GA(T132, T133)
SLOWSORTE_IN_GA(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_GA(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_GA(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_GA(T16, T17, T118, T145, permC_in_ga(T26, T145))
U13_GA(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → PERMC_IN_GA(T26, T145)
U14_GA(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_GA(T16, T17, T118, T145, leD_in_ga(T118, T145))
U14_GA(T16, T17, T118, T145, permC_out_ga(T26, T145)) → LED_IN_GA(T118, T145)

The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
SLOWSORTE_IN_GA(x1, x2)  =  SLOWSORTE_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x5)
DELETEB_IN_AGGA(x1, x2, x3, x4)  =  DELETEB_IN_AGGA(x2, x3)
U2_AGGA(x1, x2, x3, x4, x5)  =  U2_AGGA(x5)
DELETEA_IN_AGA(x1, x2, x3)  =  DELETEA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x5)
PERMC_IN_GA(x1, x2)  =  PERMC_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x5)
LED_IN_GA(x1, x2)  =  LED_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x5)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x3, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 22 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LED_IN_GA(s(T132), s(T133)) → LED_IN_GA(T132, T133)

The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
LED_IN_GA(x1, x2)  =  LED_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LED_IN_GA(s(T132), s(T133)) → LED_IN_GA(T132, T133)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LED_IN_GA(s(T132)) → LED_IN_GA(T132)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LED_IN_GA(s(T132)) → LED_IN_GA(T132)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

DELETEA_IN_AGA(T81, .(T79, T80), X88) → DELETEA_IN_AGA(T81, T80, X88)

The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
DELETEA_IN_AGA(x1, x2, x3)  =  DELETEA_IN_AGA(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

DELETEA_IN_AGA(T81, .(T79, T80), X88) → DELETEA_IN_AGA(T81, T80, X88)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

DELETEA_IN_AGA(.(T79, T80)) → DELETEA_IN_AGA(T80)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• DELETEA_IN_AGA(.(T79, T80)) → DELETEA_IN_AGA(T80)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → PERMC_IN_GA(T105, T106)

The TRS R consists of the following rules:

slowsortE_in_ga([], []) → slowsortE_out_ga([], [])
slowsortE_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_ga(T16, T17, T20, T21, deleteB_in_agga(T20, T16, T17, X22))
deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
U7_ga(T16, T17, T20, T21, deleteB_out_agga(T20, T16, T17, X22)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_ga(T16, T17, T28, T27, deleteB_in_agga(T28, T16, T17, T26))
U8_ga(T16, T17, T28, T27, deleteB_out_agga(T28, T16, T17, T26)) → U9_ga(T16, T17, T28, T27, permC_in_ga(T26, T27))
permC_in_ga([], []) → permC_out_ga([], [])
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_ga(T95, T96, T99, T100, deleteB_in_agga(T99, T95, T96, X106))
U3_ga(T95, T96, T99, T100, deleteB_out_agga(T99, T95, T96, X106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
permC_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_ga(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_ga(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → U5_ga(T95, T96, T99, T106, permC_in_ga(T105, T106))
U5_ga(T95, T96, T99, T106, permC_out_ga(T105, T106)) → permC_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))
U9_ga(T16, T17, T28, T27, permC_out_ga(T26, T27)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T28, .(T27, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T119, []))) → U10_ga(T16, T17, T118, T119, deleteB_in_agga(T118, T16, T17, T26))
U10_ga(T16, T17, T118, T119, deleteB_out_agga(T118, T16, T17, T26)) → U11_ga(T16, T17, T118, T119, permC_in_ga(T26, T119))
U11_ga(T16, T17, T118, T119, permC_out_ga(T26, T119)) → U12_ga(T16, T17, T118, T119, leD_in_ga(T118, T119))
leD_in_ga(s(T132), s(T133)) → U6_ga(T132, T133, leD_in_ga(T132, T133))
leD_in_ga(0, s(T140)) → leD_out_ga(0, s(T140))
leD_in_ga(0, 0) → leD_out_ga(0, 0)
U6_ga(T132, T133, leD_out_ga(T132, T133)) → leD_out_ga(s(T132), s(T133))
U12_ga(T16, T17, T118, T119, leD_out_ga(T118, T119)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T119, [])))
slowsortE_in_ga(.(T16, .(T17, [])), .(T118, .(T145, []))) → U13_ga(T16, T17, T118, T145, deleteB_in_agga(T118, T16, T17, T26))
U13_ga(T16, T17, T118, T145, deleteB_out_agga(T118, T16, T17, T26)) → U14_ga(T16, T17, T118, T145, permC_in_ga(T26, T145))
U14_ga(T16, T17, T118, T145, permC_out_ga(T26, T145)) → U15_ga(T16, T17, T118, T145, leD_in_ga(T118, T145))
U15_ga(T16, T17, T118, T145, leD_out_ga(T118, T145)) → slowsortE_out_ga(.(T16, .(T17, [])), .(T118, .(T145, [])))

The argument filtering Pi contains the following mapping:
slowsortE_in_ga(x1, x2)  =  slowsortE_in_ga(x1)
[]  =  []
slowsortE_out_ga(x1, x2)  =  slowsortE_out_ga
.(x1, x2)  =  .(x1, x2)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x5)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x5)
permC_in_ga(x1, x2)  =  permC_in_ga(x1)
permC_out_ga(x1, x2)  =  permC_out_ga
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
U10_ga(x1, x2, x3, x4, x5)  =  U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5)  =  U11_ga(x3, x5)
U12_ga(x1, x2, x3, x4, x5)  =  U12_ga(x5)
leD_in_ga(x1, x2)  =  leD_in_ga(x1)
s(x1)  =  s(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
0  =  0
leD_out_ga(x1, x2)  =  leD_out_ga
U13_ga(x1, x2, x3, x4, x5)  =  U13_ga(x5)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x3, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x5)
PERMC_IN_GA(x1, x2)  =  PERMC_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

PERMC_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deleteB_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deleteB_out_agga(T99, T95, T96, T105)) → PERMC_IN_GA(T105, T106)

The TRS R consists of the following rules:

deleteB_in_agga(T41, T41, T42, T42) → deleteB_out_agga(T41, T41, T42, T42)
deleteB_in_agga(T57, T55, T56, X59) → U2_agga(T57, T55, T56, X59, deleteA_in_aga(T57, T56, X59))
U2_agga(T57, T55, T56, X59, deleteA_out_aga(T57, T56, X59)) → deleteB_out_agga(T57, T55, T56, X59)
deleteA_in_aga(T70, .(T70, T71), T71) → deleteA_out_aga(T70, .(T70, T71), T71)
deleteA_in_aga(T81, .(T79, T80), X88) → U1_aga(T81, T79, T80, X88, deleteA_in_aga(T81, T80, X88))
U1_aga(T81, T79, T80, X88, deleteA_out_aga(T81, T80, X88)) → deleteA_out_aga(T81, .(T79, T80), X88)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
deleteB_in_agga(x1, x2, x3, x4)  =  deleteB_in_agga(x2, x3)
deleteB_out_agga(x1, x2, x3, x4)  =  deleteB_out_agga(x1, x4)
U2_agga(x1, x2, x3, x4, x5)  =  U2_agga(x5)
deleteA_in_aga(x1, x2, x3)  =  deleteA_in_aga(x2)
deleteA_out_aga(x1, x2, x3)  =  deleteA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
PERMC_IN_GA(x1, x2)  =  PERMC_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

PERMC_IN_GA(.(T95, .(T96, []))) → U4_GA(deleteB_in_agga(T95, T96))
U4_GA(deleteB_out_agga(T99, T105)) → PERMC_IN_GA(T105)

The TRS R consists of the following rules:

deleteB_in_agga(T41, T42) → deleteB_out_agga(T41, T42)
deleteB_in_agga(T55, T56) → U2_agga(deleteA_in_aga(T56))
U2_agga(deleteA_out_aga(T57, X59)) → deleteB_out_agga(T57, X59)
deleteA_in_aga(.(T70, T71)) → deleteA_out_aga(T70, T71)
deleteA_in_aga(.(T79, T80)) → U1_aga(deleteA_in_aga(T80))
U1_aga(deleteA_out_aga(T81, X88)) → deleteA_out_aga(T81, X88)

The set Q consists of the following terms:

deleteB_in_agga(x0, x1)
U2_agga(x0)
deleteA_in_aga(x0)
U1_aga(x0)

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

(28) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

PERMC_IN_GA(.(T95, .(T96, []))) → U4_GA(deleteB_in_agga(T95, T96))
U4_GA(deleteB_out_agga(T99, T105)) → PERMC_IN_GA(T105)

The following rules are removed from R:

deleteB_in_agga(T41, T42) → deleteB_out_agga(T41, T42)
U2_agga(deleteA_out_aga(T57, X59)) → deleteB_out_agga(T57, X59)
deleteA_in_aga(.(T70, T71)) → deleteA_out_aga(T70, T71)
deleteA_in_aga(.(T79, T80)) → U1_aga(deleteA_in_aga(T80))
U1_aga(deleteA_out_aga(T81, X88)) → deleteA_out_aga(T81, X88)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(PERMC_IN_GA(x₁)) = 1 + x₁   
POL(U1_aga(x₁)) = 1 + x₁   
POL(U2_agga(x₁)) = 2 + x₁   
POL(U4_GA(x₁)) = 2·x₁   
POL([]) = 0   
POL(deleteA_in_aga(x₁)) = x₁   
POL(deleteA_out_aga(x₁, x₂)) = x₁ + 2·x₂   
POL(deleteB_in_agga(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(deleteB_out_agga(x₁, x₂)) = 1 + x₁ + 2·x₂   

(29) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

deleteB_in_agga(T55, T56) → U2_agga(deleteA_in_aga(T56))

The set Q consists of the following terms:

deleteB_in_agga(x0, x1)
U2_agga(x0)
deleteA_in_aga(x0)
U1_aga(x0)

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

(30) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(31) YES