Left Termination of the query pattern slowsort_in_2(a, g) w.r.t. the given Prolog program could not be shown:

(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).

Queries:

slowsort(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: slowsort(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: slowsort(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(perm(T7, T6), sorted(T6))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(perm(T7, T6), sorted(T6)), SCOPE: 2, CLAUSE: 4 | ','(perm(T7, T6), sorted(T6)), SCOPE: 2, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(perm(T7, T6), sorted(T6)), SCOPE: 2, CLAUSE: 4\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(perm(T7, T6), sorted(T6)), SCOPE: 2, CLAUSE: 5\n\nT6 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(T18, .(T20, T21), X22), ','(perm(X22, T19), sorted(.(T18, .(T19, [])))))\n\nT18 is ground\nT19 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(T18, .(T20, T21), X22)\n\nT18 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(perm(T26, T19), sorted(.(T18, .(T19, []))))\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: delete(T18, .(T20, T21), X22), SCOPE: 4, CLAUSE: 6 | delete(T18, .(T20, T21), X22), SCOPE: 4, CLAUSE: 7\n\nT18 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: delete(T18, .(T20, T21), X22), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: delete(T18, .(T20, T21), X22), SCOPE: 4, CLAUSE: 7\n\nT18 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(T52, T55, X65)\n\nT52 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: delete(T52, T55, X65), SCOPE: 5, CLAUSE: 6 | delete(T52, T55, X65), SCOPE: 5, CLAUSE: 7\n\nT52 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: delete(T52, T55, X65), SCOPE: 5, CLAUSE: 6\n\nT52 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: delete(T52, T55, X65), SCOPE: 5, CLAUSE: 7\n\nT52 is ground\nX65 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(T76, T79, X96)\n\nT76 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: perm(T26, T19)\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: sorted(.(T18, .(T19, [])))\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: perm(T26, T19), SCOPE: 6, CLAUSE: 4 | perm(T26, T19), SCOPE: 6, CLAUSE: 5\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: perm(T26, T19), SCOPE: 6, CLAUSE: 4\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: perm(T26, T19), SCOPE: 6, CLAUSE: 5\n\nT19 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(T94, .(T96, T97), X118), perm(X118, T95))\n\nT94 is ground\nT95 is ground\nX118 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: delete(T94, .(T96, T97), X118)\n\nT94 is ground\nX118 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: perm(T102, T95)\n\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 1 | sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 3\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 3\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: sorted(.(T18, .(T19, []))), SCOPE: 7, CLAUSE: 3\n\nT18 is ground\nT19 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 substitution\nT1 -> T7\nX3 -> T7\nT2 -> T6\nX4 -> T6\nT5 -> T7", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT7 -> []\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nX18 -> T20\nX19 -> T21\nT7 -> .(T20, .(T21, []))\nX20 -> T18\nX21 -> T19\nT6 -> .(T18, .(T19, []))\nT16 -> T20\nT17 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 80 [arrowhead = none ];
80 -> 16;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
10 -> 85 [arrowhead = none ];
85 -> 12;

86 [label="BACKTRACK\nfor clause: sorted(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
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 = white];
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\nreplacements:\nX22 -> T26", 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", 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(X43, .(X43, X44), X44).\nand substitution\nT18 -> T39\nX43 -> T39\nT20 -> T39\nT21 -> T40\nX44 -> T40\nX22 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 96 [arrowhead = none ];
96 -> 23;

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

98 [label="ONLY EVAL with clause\ndelete(X61, .(X62, X63), X64) :- delete(X61, X63, X64).\nand substitution\nT18 -> T52\nX61 -> T52\nT20 -> T53\nX62 -> T53\nT21 -> T55\nX63 -> T55\nX22 -> X65\nX64 -> X65\nT54 -> T55", fontsize=14, style = filled, fillcolor = white];
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(X78, .(X78, X79), X79).\nand substitution\nT52 -> T68\nX78 -> T68\nX79 -> T69\nT55 -> .(T68, T69)\nX65 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 103 [arrowhead = none ];
103 -> 30;

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

105 [label="EVAL with clause\ndelete(X92, .(X93, X94), X95) :- delete(X92, X94, X95).\nand substitution\nT52 -> T76\nX92 -> T76\nX93 -> T77\nX94 -> T79\nT55 -> .(T77, T79)\nX65 -> X96\nX95 -> X96\nT78 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 105 [arrowhead = none ];
105 -> 33;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT52 -> T76\nT55 -> T79\nX65 -> X96", 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 substitution\nT26 -> []\nT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 113 [arrowhead = none ];
113 -> 40;

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

115 [label="EVAL with clause\nperm(.(X114, .(X115, [])), .(X116, .(X117, []))) :- ','(delete(X116, .(X114, X115), X118), perm(X118, X117)).\nand substitution\nX114 -> T96\nX115 -> T97\nT26 -> .(T96, .(T97, []))\nX116 -> T94\nX117 -> T95\nT19 -> .(T94, .(T95, []))\nT92 -> T96\nT93 -> T97", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 115 [arrowhead = none ];
115 -> 43;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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\nreplacements:\nX118 -> T102", fontsize=14, style = filled, fillcolor = violet];
43 -> 119 [arrowhead = none ];
119 -> 46;

120 [label="INSTANCE with matching:\nT18 -> T94\nT20 -> T96\nT21 -> T97\nX22 -> X118", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 120 [arrowhead = none , style=dashed];
120 -> 18[style=dashed];

121 [label="INSTANCE with matching:\nT26 -> T102\nT19 -> T95", 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 = white];
47 -> 122 [arrowhead = none ];
122 -> 48;

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

124 [label="ONLY EVAL with clause\nsorted(.(X145, .(X146, X147))) :- ','(le(X145, X146), sorted(.(X146, X147))).\nand substitution\nT18 -> T118\nX145 -> T118\nT19 -> T119\nX146 -> T119\nX147 -> []", fontsize=14, style = filled, fillcolor = white];
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", 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(X160), s(X161)) :- le(X160, X161).\nand substitution\nX160 -> T132\nT118 -> s(T132)\nX161 -> 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 = lightblue];
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(X170)).\nand substitution\nT118 -> 0\nX170 -> 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 = lightblue];
58 -> 137 [arrowhead = none ];
137 -> 61;

138 [label="EVAL with clause\nle(0, 0).\nand substitution\nT118 -> 0\nT119 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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(.(X177, [])).\nand substitution\nT119 -> T146\nX177 -> T146", fontsize=14, style = filled, fillcolor = white];
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 = white];
69 -> 146 [arrowhead = none ];
146 -> 72;

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

}

(2) Obligation:

Triples:

delete26(T76, .(T77, T79), X96) :- delete26(T76, T79, X96).
delete18(T52, T53, T55, X65) :- delete26(T52, T55, X65).
perm35(.(T96, .(T97, [])), .(T94, .(T95, []))) :- delete18(T94, T96, T97, X118).
perm35(.(T96, .(T97, [])), .(T94, .(T95, []))) :- ','(deletec18(T94, T96, T97, T102), perm35(T102, T95)).
le51(s(T132), s(T133)) :- le51(T132, T133).
slowsort1(.(T20, .(T21, [])), .(T18, .(T19, []))) :- delete18(T18, T20, T21, X22).
slowsort1(.(T20, .(T21, [])), .(T18, .(T19, []))) :- ','(deletec18(T18, T20, T21, T26), perm35(T26, T19)).
slowsort1(.(T20, .(T21, [])), .(T118, .(T119, []))) :- ','(deletec18(T118, T20, T21, T26), ','(permc35(T26, T119), le51(T118, T119))).

Clauses:

deletec26(T68, .(T68, T69), T69).
deletec26(T76, .(T77, T79), X96) :- deletec26(T76, T79, X96).
deletec18(T39, T39, T40, T40).
deletec18(T52, T53, T55, X65) :- deletec26(T52, T55, X65).
permc35([], []).
permc35(.(T96, .(T97, [])), .(T94, .(T95, []))) :- ','(deletec18(T94, T96, T97, T102), permc35(T102, T95)).
lec51(s(T132), s(T133)) :- lec51(T132, T133).
lec51(0, s(T140)).
lec51(0, 0).

Afs:

slowsort1(x1, x2)  =  slowsort1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
slowsort1_in: (f,b)
delete18_in: (b,f,f,f)
delete26_in: (b,f,f)
deletec18_in: (b,f,f,f)
deletec26_in: (b,f,f)
perm35_in: (f,b)
permc35_in: (f,b)
le51_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → U7_AG(T20, T21, T18, T19, delete18_in_gaaa(T18, T20, T21, X22))
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → DELETE18_IN_GAAA(T18, T20, T21, X22)
DELETE18_IN_GAAA(T52, T53, T55, X65) → U2_GAAA(T52, T53, T55, X65, delete26_in_gaa(T52, T55, X65))
DELETE18_IN_GAAA(T52, T53, T55, X65) → DELETE26_IN_GAA(T52, T55, X65)
DELETE26_IN_GAA(T76, .(T77, T79), X96) → U1_GAA(T76, T77, T79, X96, delete26_in_gaa(T76, T79, X96))
DELETE26_IN_GAA(T76, .(T77, T79), X96) → DELETE26_IN_GAA(T76, T79, X96)
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → U8_AG(T20, T21, T18, T19, deletec18_in_gaaa(T18, T20, T21, T26))
U8_AG(T20, T21, T18, T19, deletec18_out_gaaa(T18, T20, T21, T26)) → U9_AG(T20, T21, T18, T19, perm35_in_ag(T26, T19))
U8_AG(T20, T21, T18, T19, deletec18_out_gaaa(T18, T20, T21, T26)) → PERM35_IN_AG(T26, T19)
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U3_AG(T96, T97, T94, T95, delete18_in_gaaa(T94, T96, T97, X118))
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → DELETE18_IN_GAAA(T94, T96, T97, X118)
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U4_AG(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U5_AG(T96, T97, T94, T95, perm35_in_ag(T102, T95))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → PERM35_IN_AG(T102, T95)
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T118, .(T119, []))) → U10_AG(T20, T21, T118, T119, deletec18_in_gaaa(T118, T20, T21, T26))
U10_AG(T20, T21, T118, T119, deletec18_out_gaaa(T118, T20, T21, T26)) → U11_AG(T20, T21, T118, T119, permc35_in_ag(T26, T119))
U11_AG(T20, T21, T118, T119, permc35_out_ag(T26, T119)) → U12_AG(T20, T21, T118, T119, le51_in_gg(T118, T119))
U11_AG(T20, T21, T118, T119, permc35_out_ag(T26, T119)) → LE51_IN_GG(T118, T119)
LE51_IN_GG(s(T132), s(T133)) → U6_GG(T132, T133, le51_in_gg(T132, T133))
LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
permc35_in_ag([], []) → permc35_out_ag([], [])
permc35_in_ag(.(T96, .(T97, [])), .(T94, .(T95, []))) → U16_ag(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U16_ag(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U17_ag(T96, T97, T94, T95, permc35_in_ag(T102, T95))
U17_ag(T96, T97, T94, T95, permc35_out_ag(T102, T95)) → permc35_out_ag(.(T96, .(T97, [])), .(T94, .(T95, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
delete18_in_gaaa(x1, x2, x3, x4)  =  delete18_in_gaaa(x1)
delete26_in_gaa(x1, x2, x3)  =  delete26_in_gaa(x1)
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
perm35_in_ag(x1, x2)  =  perm35_in_ag(x2)
permc35_in_ag(x1, x2)  =  permc35_in_ag(x2)
permc35_out_ag(x1, x2)  =  permc35_out_ag(x2)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x3, x4, x5)
U17_ag(x1, x2, x3, x4, x5)  =  U17_ag(x3, x4, x5)
le51_in_gg(x1, x2)  =  le51_in_gg(x1, x2)
s(x1)  =  s(x1)
SLOWSORT1_IN_AG(x1, x2)  =  SLOWSORT1_IN_AG(x2)
U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x3, x4, x5)
DELETE18_IN_GAAA(x1, x2, x3, x4)  =  DELETE18_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x1, x5)
DELETE26_IN_GAA(x1, x2, x3)  =  DELETE26_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)
U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x3, x4, x5)
PERM35_IN_AG(x1, x2)  =  PERM35_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x3, x4, x5)
U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
U12_AG(x1, x2, x3, x4, x5)  =  U12_AG(x3, x4, x5)
LE51_IN_GG(x1, x2)  =  LE51_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)

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:

SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → U7_AG(T20, T21, T18, T19, delete18_in_gaaa(T18, T20, T21, X22))
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → DELETE18_IN_GAAA(T18, T20, T21, X22)
DELETE18_IN_GAAA(T52, T53, T55, X65) → U2_GAAA(T52, T53, T55, X65, delete26_in_gaa(T52, T55, X65))
DELETE18_IN_GAAA(T52, T53, T55, X65) → DELETE26_IN_GAA(T52, T55, X65)
DELETE26_IN_GAA(T76, .(T77, T79), X96) → U1_GAA(T76, T77, T79, X96, delete26_in_gaa(T76, T79, X96))
DELETE26_IN_GAA(T76, .(T77, T79), X96) → DELETE26_IN_GAA(T76, T79, X96)
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T18, .(T19, []))) → U8_AG(T20, T21, T18, T19, deletec18_in_gaaa(T18, T20, T21, T26))
U8_AG(T20, T21, T18, T19, deletec18_out_gaaa(T18, T20, T21, T26)) → U9_AG(T20, T21, T18, T19, perm35_in_ag(T26, T19))
U8_AG(T20, T21, T18, T19, deletec18_out_gaaa(T18, T20, T21, T26)) → PERM35_IN_AG(T26, T19)
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U3_AG(T96, T97, T94, T95, delete18_in_gaaa(T94, T96, T97, X118))
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → DELETE18_IN_GAAA(T94, T96, T97, X118)
PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U4_AG(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U5_AG(T96, T97, T94, T95, perm35_in_ag(T102, T95))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → PERM35_IN_AG(T102, T95)
SLOWSORT1_IN_AG(.(T20, .(T21, [])), .(T118, .(T119, []))) → U10_AG(T20, T21, T118, T119, deletec18_in_gaaa(T118, T20, T21, T26))
U10_AG(T20, T21, T118, T119, deletec18_out_gaaa(T118, T20, T21, T26)) → U11_AG(T20, T21, T118, T119, permc35_in_ag(T26, T119))
U11_AG(T20, T21, T118, T119, permc35_out_ag(T26, T119)) → U12_AG(T20, T21, T118, T119, le51_in_gg(T118, T119))
U11_AG(T20, T21, T118, T119, permc35_out_ag(T26, T119)) → LE51_IN_GG(T118, T119)
LE51_IN_GG(s(T132), s(T133)) → U6_GG(T132, T133, le51_in_gg(T132, T133))
LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
permc35_in_ag([], []) → permc35_out_ag([], [])
permc35_in_ag(.(T96, .(T97, [])), .(T94, .(T95, []))) → U16_ag(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U16_ag(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U17_ag(T96, T97, T94, T95, permc35_in_ag(T102, T95))
U17_ag(T96, T97, T94, T95, permc35_out_ag(T102, T95)) → permc35_out_ag(.(T96, .(T97, [])), .(T94, .(T95, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
delete18_in_gaaa(x1, x2, x3, x4)  =  delete18_in_gaaa(x1)
delete26_in_gaa(x1, x2, x3)  =  delete26_in_gaa(x1)
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
perm35_in_ag(x1, x2)  =  perm35_in_ag(x2)
permc35_in_ag(x1, x2)  =  permc35_in_ag(x2)
permc35_out_ag(x1, x2)  =  permc35_out_ag(x2)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x3, x4, x5)
U17_ag(x1, x2, x3, x4, x5)  =  U17_ag(x3, x4, x5)
le51_in_gg(x1, x2)  =  le51_in_gg(x1, x2)
s(x1)  =  s(x1)
SLOWSORT1_IN_AG(x1, x2)  =  SLOWSORT1_IN_AG(x2)
U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x3, x4, x5)
DELETE18_IN_GAAA(x1, x2, x3, x4)  =  DELETE18_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5)  =  U2_GAAA(x1, x5)
DELETE26_IN_GAA(x1, x2, x3)  =  DELETE26_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)
U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x3, x4, x5)
PERM35_IN_AG(x1, x2)  =  PERM35_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x3, x4, x5)
U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
U12_AG(x1, x2, x3, x4, x5)  =  U12_AG(x3, x4, x5)
LE51_IN_GG(x1, x2)  =  LE51_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
permc35_in_ag([], []) → permc35_out_ag([], [])
permc35_in_ag(.(T96, .(T97, [])), .(T94, .(T95, []))) → U16_ag(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U16_ag(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U17_ag(T96, T97, T94, T95, permc35_in_ag(T102, T95))
U17_ag(T96, T97, T94, T95, permc35_out_ag(T102, T95)) → permc35_out_ag(.(T96, .(T97, [])), .(T94, .(T95, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
permc35_in_ag(x1, x2)  =  permc35_in_ag(x2)
permc35_out_ag(x1, x2)  =  permc35_out_ag(x2)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x3, x4, x5)
U17_ag(x1, x2, x3, x4, x5)  =  U17_ag(x3, x4, x5)
s(x1)  =  s(x1)
LE51_IN_GG(x1, x2)  =  LE51_IN_GG(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:

LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)

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

(10) PiDPToQDPProof (EQUIVALENT 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:

LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)

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:

• LE51_IN_GG(s(T132), s(T133)) → LE51_IN_GG(T132, T133)
  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:

DELETE26_IN_GAA(T76, .(T77, T79), X96) → DELETE26_IN_GAA(T76, T79, X96)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
permc35_in_ag([], []) → permc35_out_ag([], [])
permc35_in_ag(.(T96, .(T97, [])), .(T94, .(T95, []))) → U16_ag(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U16_ag(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U17_ag(T96, T97, T94, T95, permc35_in_ag(T102, T95))
U17_ag(T96, T97, T94, T95, permc35_out_ag(T102, T95)) → permc35_out_ag(.(T96, .(T97, [])), .(T94, .(T95, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
permc35_in_ag(x1, x2)  =  permc35_in_ag(x2)
permc35_out_ag(x1, x2)  =  permc35_out_ag(x2)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x3, x4, x5)
U17_ag(x1, x2, x3, x4, x5)  =  U17_ag(x3, x4, x5)
DELETE26_IN_GAA(x1, x2, x3)  =  DELETE26_IN_GAA(x1)

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:

DELETE26_IN_GAA(T76, .(T77, T79), X96) → DELETE26_IN_GAA(T76, T79, X96)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DELETE26_IN_GAA(x1, x2, x3)  =  DELETE26_IN_GAA(x1)

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:

DELETE26_IN_GAA(T76) → DELETE26_IN_GAA(T76)

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

(19) NonTerminationProof (EQUIVALENT transformation)

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

s = DELETE26_IN_GAA(T76) evaluates to t =DELETE26_IN_GAA(T76)

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DELETE26_IN_GAA(T76) to DELETE26_IN_GAA(T76).

(20) NO

(21) Obligation:

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

PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U4_AG(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → PERM35_IN_AG(T102, T95)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
permc35_in_ag([], []) → permc35_out_ag([], [])
permc35_in_ag(.(T96, .(T97, [])), .(T94, .(T95, []))) → U16_ag(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U16_ag(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → U17_ag(T96, T97, T94, T95, permc35_in_ag(T102, T95))
U17_ag(T96, T97, T94, T95, permc35_out_ag(T102, T95)) → permc35_out_ag(.(T96, .(T97, [])), .(T94, .(T95, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
permc35_in_ag(x1, x2)  =  permc35_in_ag(x2)
permc35_out_ag(x1, x2)  =  permc35_out_ag(x2)
U16_ag(x1, x2, x3, x4, x5)  =  U16_ag(x3, x4, x5)
U17_ag(x1, x2, x3, x4, x5)  =  U17_ag(x3, x4, x5)
PERM35_IN_AG(x1, x2)  =  PERM35_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)

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:

PERM35_IN_AG(.(T96, .(T97, [])), .(T94, .(T95, []))) → U4_AG(T96, T97, T94, T95, deletec18_in_gaaa(T94, T96, T97, T102))
U4_AG(T96, T97, T94, T95, deletec18_out_gaaa(T94, T96, T97, T102)) → PERM35_IN_AG(T102, T95)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39, T39, T40, T40) → deletec18_out_gaaa(T39, T39, T40, T40)
deletec18_in_gaaa(T52, T53, T55, X65) → U15_gaaa(T52, T53, T55, X65, deletec26_in_gaa(T52, T55, X65))
U15_gaaa(T52, T53, T55, X65, deletec26_out_gaa(T52, T55, X65)) → deletec18_out_gaaa(T52, T53, T55, X65)
deletec26_in_gaa(T68, .(T68, T69), T69) → deletec26_out_gaa(T68, .(T68, T69), T69)
deletec26_in_gaa(T76, .(T77, T79), X96) → U14_gaa(T76, T77, T79, X96, deletec26_in_gaa(T76, T79, X96))
U14_gaa(T76, T77, T79, X96, deletec26_out_gaa(T76, T79, X96)) → deletec26_out_gaa(T76, .(T77, T79), X96)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_gaaa(x1, x2, x3, x4)  =  deletec18_in_gaaa(x1)
deletec18_out_gaaa(x1, x2, x3, x4)  =  deletec18_out_gaaa(x1)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x5)
deletec26_in_gaa(x1, x2, x3)  =  deletec26_in_gaa(x1)
deletec26_out_gaa(x1, x2, x3)  =  deletec26_out_gaa(x1)
U14_gaa(x1, x2, x3, x4, x5)  =  U14_gaa(x1, x5)
PERM35_IN_AG(x1, x2)  =  PERM35_IN_AG(x2)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)

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:

PERM35_IN_AG(.(T94, .(T95, []))) → U4_AG(T94, T95, deletec18_in_gaaa(T94))
U4_AG(T94, T95, deletec18_out_gaaa(T94)) → PERM35_IN_AG(T95)

The TRS R consists of the following rules:

deletec18_in_gaaa(T39) → deletec18_out_gaaa(T39)
deletec18_in_gaaa(T52) → U15_gaaa(T52, deletec26_in_gaa(T52))
U15_gaaa(T52, deletec26_out_gaa(T52)) → deletec18_out_gaaa(T52)
deletec26_in_gaa(T68) → deletec26_out_gaa(T68)
deletec26_in_gaa(T76) → U14_gaa(T76, deletec26_in_gaa(T76))
U14_gaa(T76, deletec26_out_gaa(T76)) → deletec26_out_gaa(T76)

The set Q consists of the following terms:

deletec18_in_gaaa(x0)
U15_gaaa(x0, x1)
deletec26_in_gaa(x0)
U14_gaa(x0, x1)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U4_AG(T94, T95, deletec18_out_gaaa(T94)) → PERM35_IN_AG(T95)
  The graph contains the following edges 2 >= 1

• PERM35_IN_AG(.(T94, .(T95, []))) → U4_AG(T94, T95, deletec18_in_gaaa(T94))
  The graph contains the following edges 1 > 1, 1 > 2

(27) YES