Left Termination of the query pattern slowsort_in_2(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).

Queries:

slowsort(g,a).

(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\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, X65)\n\nT56 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: delete(T57, T56, X65), SCOPE: 5, CLAUSE: 6 | delete(T57, T56, X65), SCOPE: 5, CLAUSE: 7\n\nT56 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: delete(T57, T56, X65), SCOPE: 5, CLAUSE: 6\n\nT56 is ground\nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: delete(T57, T56, X65), SCOPE: 5, CLAUSE: 7\n\nT56 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(T81, T80, X96)\n\nT80 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, 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), X118), perm(X118, T100))\n\nT95 is ground\nT96 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(T99, .(T95, T96), X118)\n\nT95 is ground\nT96 is ground\nX118 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(T122, T123), sorted(.(T123, [])))\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: le(T122, T123)\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: sorted(.(T123, []))\n\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: le(T122, T123), SCOPE: 8, CLAUSE: 8 | le(T122, T123), SCOPE: 8, CLAUSE: 9 | le(T122, T123), SCOPE: 8, CLAUSE: 10\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: le(T122, T123), SCOPE: 8, CLAUSE: 8\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: le(T122, T123), SCOPE: 8, CLAUSE: 9 | le(T122, T123), SCOPE: 8, CLAUSE: 10\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: le(T136, T137)\n\nT136 is ground\nT137 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: le(T122, T123), SCOPE: 8, CLAUSE: 9\n\nT122 is ground\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: le(T122, T123), SCOPE: 8, CLAUSE: 10\n\nT122 is ground\nT123 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(.(T123, [])), SCOPE: 9, CLAUSE: 1 | sorted(.(T123, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T123, [])), SCOPE: 9, CLAUSE: 3\n\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: sorted(.(T123, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T123, [])), SCOPE: 9, CLAUSE: 3\n\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: sorted(.(T123, [])), SCOPE: 9, CLAUSE: 2\n\nT123 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: sorted(.(T123, [])), SCOPE: 9, CLAUSE: 3\n\nT123 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 -> T5\nX3 -> T5\nT2 -> T7\nX4 -> T7\nT6 -> 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\nT5 -> []\nT7 -> []", 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 -> 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 = 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\nnew knowledge:\nT28 is ground\nT26 is ground\nreplacements:\nX22 -> 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:\nT86 is ground\nreplacements:\nT27 -> 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(X43, .(X43, X44), X44).\nand substitution\nT20 -> T41\nX43 -> T41\nT16 -> T41\nT17 -> T42\nX44 -> T42\nX22 -> T42", 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\nT20 -> T57\nX61 -> T57\nT16 -> T55\nX62 -> T55\nT17 -> T56\nX63 -> T56\nX22 -> X65\nX64 -> X65\nT54 -> T57", 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\nT57 -> T70\nX78 -> T70\nX79 -> T71\nT56 -> .(T70, T71)\nX65 -> T71", 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\nT57 -> T81\nX92 -> T81\nX93 -> T79\nX94 -> T80\nT56 -> .(T79, T80)\nX65 -> X96\nX95 -> X96\nT78 -> T81", 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:\nT57 -> T81\nT56 -> T80\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 -> []\nT27 -> []", 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 -> T95\nX115 -> T96\nT26 -> .(T95, .(T96, []))\nX116 -> T99\nX117 -> 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 = 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\nnew knowledge:\nT99 is ground\nT105 is ground\nreplacements:\nX118 -> 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 -> X118", 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 = 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\nT28 -> T122\nX145 -> T122\nT86 -> T123\nX146 -> T123\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 -> T136\nT122 -> s(T136)\nX161 -> T137\nT123 -> s(T137)", 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:\nT122 -> T136\nT123 -> T137", 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\nT122 -> 0\nX170 -> T144\nT123 -> s(T144)", 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\nT122 -> 0\nT123 -> 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\nT123 -> T150\nX177 -> T150", 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(T81, .(T79, T80), X96) :- delete26(T81, T80, X96).
delete18(T57, T55, T56, X65) :- delete26(T57, T56, X65).
perm35(.(T95, .(T96, [])), .(T99, .(T100, []))) :- delete18(T99, T95, T96, X118).
perm35(.(T95, .(T96, [])), .(T99, .(T106, []))) :- ','(deletec18(T99, T95, T96, T105), perm35(T105, T106)).
le51(s(T136), s(T137)) :- le51(T136, T137).
slowsort1(.(T16, .(T17, [])), .(T20, .(T21, []))) :- delete18(T20, T16, T17, X22).
slowsort1(.(T16, .(T17, [])), .(T28, .(T27, []))) :- ','(deletec18(T28, T16, T17, T26), perm35(T26, T27)).
slowsort1(.(T16, .(T17, [])), .(T122, .(T123, []))) :- ','(deletec18(T122, T16, T17, T26), ','(permc35(T26, T123), le51(T122, T123))).

Clauses:

deletec26(T70, .(T70, T71), T71).
deletec26(T81, .(T79, T80), X96) :- deletec26(T81, T80, X96).
deletec18(T41, T41, T42, T42).
deletec18(T57, T55, T56, X65) :- deletec26(T57, T56, X65).
permc35([], []).
permc35(.(T95, .(T96, [])), .(T99, .(T106, []))) :- ','(deletec18(T99, T95, T96, T105), permc35(T105, T106)).
lec51(s(T136), s(T137)) :- lec51(T136, T137).
lec51(0, s(T144)).
lec51(0, 0).

Afs:

slowsort1(x1, x2)  =  slowsort1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
slowsort1_in: (b,f)
delete18_in: (f,b,b,f)
delete26_in: (f,b,f)
deletec18_in: (f,b,b,f)
deletec26_in: (f,b,f)
perm35_in: (b,f)
permc35_in: (b,f)
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_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_GA(T16, T17, T20, T21, delete18_in_agga(T20, T16, T17, X22))
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → DELETE18_IN_AGGA(T20, T16, T17, X22)
DELETE18_IN_AGGA(T57, T55, T56, X65) → U2_AGGA(T57, T55, T56, X65, delete26_in_aga(T57, T56, X65))
DELETE18_IN_AGGA(T57, T55, T56, X65) → DELETE26_IN_AGA(T57, T56, X65)
DELETE26_IN_AGA(T81, .(T79, T80), X96) → U1_AGA(T81, T79, T80, X96, delete26_in_aga(T81, T80, X96))
DELETE26_IN_AGA(T81, .(T79, T80), X96) → DELETE26_IN_AGA(T81, T80, X96)
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_GA(T16, T17, T28, T27, deletec18_in_agga(T28, T16, T17, T26))
U8_GA(T16, T17, T28, T27, deletec18_out_agga(T28, T16, T17, T26)) → U9_GA(T16, T17, T28, T27, perm35_in_ga(T26, T27))
U8_GA(T16, T17, T28, T27, deletec18_out_agga(T28, T16, T17, T26)) → PERM35_IN_GA(T26, T27)
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_GA(T95, T96, T99, T100, delete18_in_agga(T99, T95, T96, X118))
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → DELETE18_IN_AGGA(T99, T95, T96, X118)
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U5_GA(T95, T96, T99, T106, perm35_in_ga(T105, T106))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105, T106)
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T122, .(T123, []))) → U10_GA(T16, T17, T122, T123, deletec18_in_agga(T122, T16, T17, T26))
U10_GA(T16, T17, T122, T123, deletec18_out_agga(T122, T16, T17, T26)) → U11_GA(T16, T17, T122, T123, permc35_in_ga(T26, T123))
U11_GA(T16, T17, T122, T123, permc35_out_ga(T26, T123)) → U12_GA(T16, T17, T122, T123, le51_in_gg(T122, T123))
U11_GA(T16, T17, T122, T123, permc35_out_ga(T26, T123)) → LE51_IN_GG(T122, T123)
LE51_IN_GG(s(T136), s(T137)) → U6_GG(T136, T137, le51_in_gg(T136, T137))
LE51_IN_GG(s(T136), s(T137)) → LE51_IN_GG(T136, T137)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
permc35_in_ga([], []) → permc35_out_ga([], [])
permc35_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U16_ga(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U16_ga(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U17_ga(T95, T96, T99, T106, permc35_in_ga(T105, T106))
U17_ga(T95, T96, T99, T106, permc35_out_ga(T105, T106)) → permc35_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
delete18_in_agga(x1, x2, x3, x4)  =  delete18_in_agga(x2, x3)
delete26_in_aga(x1, x2, x3)  =  delete26_in_aga(x2)
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
perm35_in_ga(x1, x2)  =  perm35_in_ga(x1)
permc35_in_ga(x1, x2)  =  permc35_in_ga(x1)
permc35_out_ga(x1, x2)  =  permc35_out_ga(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
le51_in_gg(x1, x2)  =  le51_in_gg(x1, x2)
s(x1)  =  s(x1)
SLOWSORT1_IN_GA(x1, x2)  =  SLOWSORT1_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
DELETE18_IN_AGGA(x1, x2, x3, x4)  =  DELETE18_IN_AGGA(x2, x3)
U2_AGGA(x1, x2, x3, x4, x5)  =  U2_AGGA(x2, x3, x5)
DELETE26_IN_AGA(x1, x2, x3)  =  DELETE26_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
PERM35_IN_GA(x1, x2)  =  PERM35_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, 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_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U7_GA(T16, T17, T20, T21, delete18_in_agga(T20, T16, T17, X22))
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → DELETE18_IN_AGGA(T20, T16, T17, X22)
DELETE18_IN_AGGA(T57, T55, T56, X65) → U2_AGGA(T57, T55, T56, X65, delete26_in_aga(T57, T56, X65))
DELETE18_IN_AGGA(T57, T55, T56, X65) → DELETE26_IN_AGA(T57, T56, X65)
DELETE26_IN_AGA(T81, .(T79, T80), X96) → U1_AGA(T81, T79, T80, X96, delete26_in_aga(T81, T80, X96))
DELETE26_IN_AGA(T81, .(T79, T80), X96) → DELETE26_IN_AGA(T81, T80, X96)
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T28, .(T27, []))) → U8_GA(T16, T17, T28, T27, deletec18_in_agga(T28, T16, T17, T26))
U8_GA(T16, T17, T28, T27, deletec18_out_agga(T28, T16, T17, T26)) → U9_GA(T16, T17, T28, T27, perm35_in_ga(T26, T27))
U8_GA(T16, T17, T28, T27, deletec18_out_agga(T28, T16, T17, T26)) → PERM35_IN_GA(T26, T27)
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U3_GA(T95, T96, T99, T100, delete18_in_agga(T99, T95, T96, X118))
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → DELETE18_IN_AGGA(T99, T95, T96, X118)
PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U5_GA(T95, T96, T99, T106, perm35_in_ga(T105, T106))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105, T106)
SLOWSORT1_IN_GA(.(T16, .(T17, [])), .(T122, .(T123, []))) → U10_GA(T16, T17, T122, T123, deletec18_in_agga(T122, T16, T17, T26))
U10_GA(T16, T17, T122, T123, deletec18_out_agga(T122, T16, T17, T26)) → U11_GA(T16, T17, T122, T123, permc35_in_ga(T26, T123))
U11_GA(T16, T17, T122, T123, permc35_out_ga(T26, T123)) → U12_GA(T16, T17, T122, T123, le51_in_gg(T122, T123))
U11_GA(T16, T17, T122, T123, permc35_out_ga(T26, T123)) → LE51_IN_GG(T122, T123)
LE51_IN_GG(s(T136), s(T137)) → U6_GG(T136, T137, le51_in_gg(T136, T137))
LE51_IN_GG(s(T136), s(T137)) → LE51_IN_GG(T136, T137)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
permc35_in_ga([], []) → permc35_out_ga([], [])
permc35_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U16_ga(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U16_ga(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U17_ga(T95, T96, T99, T106, permc35_in_ga(T105, T106))
U17_ga(T95, T96, T99, T106, permc35_out_ga(T105, T106)) → permc35_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
delete18_in_agga(x1, x2, x3, x4)  =  delete18_in_agga(x2, x3)
delete26_in_aga(x1, x2, x3)  =  delete26_in_aga(x2)
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
perm35_in_ga(x1, x2)  =  perm35_in_ga(x1)
permc35_in_ga(x1, x2)  =  permc35_in_ga(x1)
permc35_out_ga(x1, x2)  =  permc35_out_ga(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
le51_in_gg(x1, x2)  =  le51_in_gg(x1, x2)
s(x1)  =  s(x1)
SLOWSORT1_IN_GA(x1, x2)  =  SLOWSORT1_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
DELETE18_IN_AGGA(x1, x2, x3, x4)  =  DELETE18_IN_AGGA(x2, x3)
U2_AGGA(x1, x2, x3, x4, x5)  =  U2_AGGA(x2, x3, x5)
DELETE26_IN_AGA(x1, x2, x3)  =  DELETE26_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
PERM35_IN_GA(x1, x2)  =  PERM35_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, 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(T136), s(T137)) → LE51_IN_GG(T136, T137)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
permc35_in_ga([], []) → permc35_out_ga([], [])
permc35_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U16_ga(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U16_ga(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U17_ga(T95, T96, T99, T106, permc35_in_ga(T105, T106))
U17_ga(T95, T96, T99, T106, permc35_out_ga(T105, T106)) → permc35_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
permc35_in_ga(x1, x2)  =  permc35_in_ga(x1)
permc35_out_ga(x1, x2)  =  permc35_out_ga(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, 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(T136), s(T137)) → LE51_IN_GG(T136, T137)

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(T136), s(T137)) → LE51_IN_GG(T136, T137)

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(T136), s(T137)) → LE51_IN_GG(T136, T137)
  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_AGA(T81, .(T79, T80), X96) → DELETE26_IN_AGA(T81, T80, X96)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
permc35_in_ga([], []) → permc35_out_ga([], [])
permc35_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U16_ga(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U16_ga(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U17_ga(T95, T96, T99, T106, permc35_in_ga(T105, T106))
U17_ga(T95, T96, T99, T106, permc35_out_ga(T105, T106)) → permc35_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
permc35_in_ga(x1, x2)  =  permc35_in_ga(x1)
permc35_out_ga(x1, x2)  =  permc35_out_ga(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
DELETE26_IN_AGA(x1, x2, x3)  =  DELETE26_IN_AGA(x2)

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_AGA(T81, .(T79, T80), X96) → DELETE26_IN_AGA(T81, T80, X96)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DELETE26_IN_AGA(x1, x2, x3)  =  DELETE26_IN_AGA(x2)

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_AGA(.(T79, T80)) → DELETE26_IN_AGA(T80)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• DELETE26_IN_AGA(.(T79, T80)) → DELETE26_IN_AGA(T80)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

PERM35_IN_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105, T106)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
permc35_in_ga([], []) → permc35_out_ga([], [])
permc35_in_ga(.(T95, .(T96, [])), .(T99, .(T106, []))) → U16_ga(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U16_ga(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → U17_ga(T95, T96, T99, T106, permc35_in_ga(T105, T106))
U17_ga(T95, T96, T99, T106, permc35_out_ga(T105, T106)) → permc35_out_ga(.(T95, .(T96, [])), .(T99, .(T106, [])))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
permc35_in_ga(x1, x2)  =  permc35_in_ga(x1)
permc35_out_ga(x1, x2)  =  permc35_out_ga(x1, x2)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x3, x5)
PERM35_IN_GA(x1, x2)  =  PERM35_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, 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_GA(.(T95, .(T96, [])), .(T99, .(T106, []))) → U4_GA(T95, T96, T99, T106, deletec18_in_agga(T99, T95, T96, T105))
U4_GA(T95, T96, T99, T106, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105, T106)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T41, T42, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T57, T55, T56, X65) → U15_agga(T57, T55, T56, X65, deletec26_in_aga(T57, T56, X65))
U15_agga(T57, T55, T56, X65, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
deletec26_in_aga(T70, .(T70, T71), T71) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(T81, .(T79, T80), X96) → U14_aga(T81, T79, T80, X96, deletec26_in_aga(T81, T80, X96))
U14_aga(T81, T79, T80, X96, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
deletec18_in_agga(x1, x2, x3, x4)  =  deletec18_in_agga(x2, x3)
deletec18_out_agga(x1, x2, x3, x4)  =  deletec18_out_agga(x1, x2, x3, x4)
U15_agga(x1, x2, x3, x4, x5)  =  U15_agga(x2, x3, x5)
deletec26_in_aga(x1, x2, x3)  =  deletec26_in_aga(x2)
deletec26_out_aga(x1, x2, x3)  =  deletec26_out_aga(x1, x2, x3)
U14_aga(x1, x2, x3, x4, x5)  =  U14_aga(x2, x3, x5)
PERM35_IN_GA(x1, x2)  =  PERM35_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, 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_GA(.(T95, .(T96, []))) → U4_GA(T95, T96, deletec18_in_agga(T95, T96))
U4_GA(T95, T96, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105)

The TRS R consists of the following rules:

deletec18_in_agga(T41, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T55, T56) → U15_agga(T55, T56, deletec26_in_aga(T56))
U15_agga(T55, T56, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
deletec26_in_aga(.(T70, T71)) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(.(T79, T80)) → U14_aga(T79, T80, deletec26_in_aga(T80))
U14_aga(T79, T80, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)

The set Q consists of the following terms:

deletec18_in_agga(x0, x1)
U15_agga(x0, x1, x2)
deletec26_in_aga(x0)
U14_aga(x0, x1, x2)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PERM35_IN_GA(.(T95, .(T96, []))) → U4_GA(T95, T96, deletec18_in_agga(T95, T96))
U4_GA(T95, T96, deletec18_out_agga(T99, T95, T96, T105)) → PERM35_IN_GA(T105)

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

POL(.(x₁, x₂)) = 1 + x₁ + x₂   
POL(PERM35_IN_GA(x₁)) = x₁   
POL(U14_aga(x₁, x₂, x₃)) = 1 + x₁ + x₃   
POL(U15_agga(x₁, x₂, x₃)) = x₃   
POL(U4_GA(x₁, x₂, x₃)) = x₃   
POL([]) = 1   
POL(deletec18_in_agga(x₁, x₂)) = 1 + x₁ + x₂   
POL(deletec18_out_agga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(deletec26_in_aga(x₁)) = 1 + x₁   
POL(deletec26_out_aga(x₁, x₂, x₃)) = 1 + x₁ + x₃   

The following usable rules [FROCOS05] were oriented:

deletec18_in_agga(T41, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T55, T56) → U15_agga(T55, T56, deletec26_in_aga(T56))
deletec26_in_aga(.(T70, T71)) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(.(T79, T80)) → U14_aga(T79, T80, deletec26_in_aga(T80))
U15_agga(T55, T56, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
U14_aga(T79, T80, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)

(27) Obligation:

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

deletec18_in_agga(T41, T42) → deletec18_out_agga(T41, T41, T42, T42)
deletec18_in_agga(T55, T56) → U15_agga(T55, T56, deletec26_in_aga(T56))
U15_agga(T55, T56, deletec26_out_aga(T57, T56, X65)) → deletec18_out_agga(T57, T55, T56, X65)
deletec26_in_aga(.(T70, T71)) → deletec26_out_aga(T70, .(T70, T71), T71)
deletec26_in_aga(.(T79, T80)) → U14_aga(T79, T80, deletec26_in_aga(T80))
U14_aga(T79, T80, deletec26_out_aga(T81, T80, X96)) → deletec26_out_aga(T81, .(T79, T80), X96)

The set Q consists of the following terms:

deletec18_in_agga(x0, x1)
U15_agga(x0, x1, x2)
deletec26_in_aga(x0)
U14_aga(x0, x1, x2)

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) YES