Left Termination of the query pattern ss_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
ordered([]).
ordered(.(X1, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ss(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ss(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ss(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(perm(T5, T7), ordered(T7))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(perm(T5, T7), ordered(T7)), SCOPE: 2, CLAUSE: 1 | ','(perm(T5, T7), ordered(T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(perm(T5, T7), ordered(T7)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(perm(T5, T7), ordered(T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ordered([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ordered([]), SCOPE: 3, CLAUSE: 5 | ordered([]), SCOPE: 3, CLAUSE: 6 | ordered([]), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ordered([]), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ordered([]), SCOPE: 3, CLAUSE: 6 | ordered([]), SCOPE: 3, CLAUSE: 7\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: ordered([]), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(app(X23, .(T17, X24), T14), ','(app(X23, X24, X25), ','(perm(X25, T18), ordered(.(T17, T18)))))\n\nT14 is ground\nX23 is free; \nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(X23, .(T17, X24), T14)\n\nT14 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(T21, T22, X25), ','(perm(X25, T23), ordered(.(T24, T23))))\n\nT21 is ground\nT24 is ground\nT22 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(X23, .(T17, X24), T14), SCOPE: 4, CLAUSE: 3 | app(X23, .(T17, X24), T14), SCOPE: 4, CLAUSE: 4\n\nT14 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(X23, .(T17, X24), T14), SCOPE: 4, CLAUSE: 3\n\nT14 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(X23, .(T17, X24), T14), SCOPE: 4, CLAUSE: 4\n\nT14 is ground\nX23 is free; \nX24 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: app(X67, .(T38, X68), T37)\n\nT37 is ground\nX67 is free; \nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T21, T22, X25)\n\nT21 is ground\nT22 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(perm(T43, T23), ordered(.(T24, T23)))\n\nT24 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T21, T22, X25), SCOPE: 5, CLAUSE: 3 | app(T21, T22, X25), SCOPE: 5, CLAUSE: 4\n\nT21 is ground\nT22 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T21, T22, X25), SCOPE: 5, CLAUSE: 3\n\nT21 is ground\nT22 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T21, T22, X25), SCOPE: 5, CLAUSE: 4\n\nT21 is ground\nT22 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T58, T59, X101)\n\nT58 is ground\nT59 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: perm(T43, T23)\n\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ordered(.(T24, T62))\n\nT24 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: perm(T43, T23), SCOPE: 6, CLAUSE: 1 | perm(T43, T23), SCOPE: 6, CLAUSE: 2\n\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: perm(T43, T23), SCOPE: 6, CLAUSE: 1\n\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: perm(T43, T23), SCOPE: 6, CLAUSE: 2\n\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(app(X120, .(T72, X121), T69), ','(app(X120, X121, X122), perm(X122, T73)))\n\nT69 is ground\nX120 is free; \nX121 is free; \nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(X120, .(T72, X121), T69)\n\nT69 is ground\nX120 is free; \nX121 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(app(T76, T77, X122), perm(X122, T78))\n\nT76 is ground\nT77 is ground\nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(T76, T77, X122)\n\nT76 is ground\nT77 is ground\nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: perm(T83, T78)\n\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 5 | ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 6 | ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 7\n\nT24 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 6 | ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 7\n\nT24 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 6\n\nT24 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ordered(.(T24, T62)), SCOPE: 7, CLAUSE: 7\n\nT24 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(less(T99, s(T100)), ordered(.(T100, T101)))\n\nT99 is ground\nT100 is ground\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T99, s(T100))\n\nT99 is ground\nT100 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ordered(.(T100, T101))\n\nT100 is ground\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T99, s(T100)), SCOPE: 8, CLAUSE: 8 | less(T99, s(T100)), SCOPE: 8, CLAUSE: 9\n\nT99 is ground\nT100 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T99, s(T100)), SCOPE: 8, CLAUSE: 8\n\nT99 is ground\nT100 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(T99, s(T100)), SCOPE: 8, CLAUSE: 9\n\nT99 is ground\nT100 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: less(T115, T116)\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T115, T116), SCOPE: 9, CLAUSE: 8 | less(T115, T116), SCOPE: 9, CLAUSE: 9\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: less(T115, T116), SCOPE: 9, CLAUSE: 8\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T115, T116), SCOPE: 9, CLAUSE: 9\n\nT115 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(T128, T129)\n\nT128 is ground\nT129 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 2;

80 [label="ONLY EVAL with clause\nss(X5, X6) :- ','(perm(X5, X6), ordered(X6)).\nand substitution\nT1 -> T5\nX5 -> T5\nT2 -> T7\nX6 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="EVAL with clause\nperm([], []).\nand substitution\nT5 -> []\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="EVAL with clause\nperm(X20, .(X21, X22)) :- ','(app(X23, .(X21, X24), X20), ','(app(X23, X24, X25), perm(X25, X22))).\nand substitution\nT5 -> T14\nX20 -> T14\nX21 -> T17\nX22 -> T18\nT7 -> .(T17, T18)\nT15 -> T17\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 89 [arrowhead = none ];
89 -> 10;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="ONLY EVAL with clause\nordered([]).\nand substitution", fontsize=14, style = filled, fillcolor = white];
10 -> 91 [arrowhead = none ];
91 -> 12;

92 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 92 [arrowhead = none ];
92 -> 14;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 93 [arrowhead = none ];
93 -> 13;

94 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 94 [arrowhead = none ];
94 -> 15;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 95 [arrowhead = none ];
95 -> 18;

96 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nT24 is ground\nT22 is ground\nreplacements:\nX23 -> T21\nX24 -> T22\nT18 -> T23\nT17 -> T24", fontsize=14, style = filled, fillcolor = violet];
16 -> 96 [arrowhead = none ];
96 -> 19;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 98 [arrowhead = none ];
98 -> 28;

99 [label="SPLIT 2\nnew knowledge:\nT43 is ground\nreplacements:\nX25 -> T43", fontsize=14, style = filled, fillcolor = violet];
19 -> 99 [arrowhead = none ];
99 -> 29;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 100 [arrowhead = none ];
100 -> 21;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 101 [arrowhead = none ];
101 -> 22;

102 [label="EVAL with clause\napp([], X45, X45).\nand substitution\nX23 -> []\nT17 -> T31\nX24 -> X46\nX45 -> .(T31, X46)\nT14 -> .(T31, X46)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 102 [arrowhead = none ];
102 -> 23;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 103 [arrowhead = none ];
103 -> 24;

104 [label="EVAL with clause\napp(.(X62, X63), X64, .(X62, X65)) :- app(X63, X64, X65).\nand substitution\nX62 -> X66\nX63 -> X67\nX23 -> .(X66, X67)\nT17 -> T38\nX24 -> X68\nX64 -> .(T38, X68)\nX65 -> T37\nT14 -> .(X66, T37)\nT36 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 106 [arrowhead = none ];
106 -> 25;

107 [label="INSTANCE with matching:\nX23 -> X67\nT17 -> T38\nX24 -> X68\nT14 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 107 [arrowhead = none , style=dashed];
107 -> 18[style=dashed];

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 108 [arrowhead = none ];
108 -> 30;

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 109 [arrowhead = none ];
109 -> 38;

110 [label="SPLIT 2\nnew knowledge:\nT62 is ground\nreplacements:\nT23 -> T62", fontsize=14, style = filled, fillcolor = violet];
29 -> 110 [arrowhead = none ];
110 -> 39;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 111 [arrowhead = none ];
111 -> 31;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 112 [arrowhead = none ];
112 -> 32;

113 [label="EVAL with clause\napp([], X85, X85).\nand substitution\nT21 -> []\nT22 -> T50\nX85 -> T50\nX25 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 113 [arrowhead = none ];
113 -> 33;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 114 [arrowhead = none ];
114 -> 34;

115 [label="EVAL with clause\napp(.(X97, X98), X99, .(X97, X100)) :- app(X98, X99, X100).\nand substitution\nX97 -> T57\nX98 -> T58\nT21 -> .(T57, T58)\nT22 -> T59\nX99 -> T59\nX100 -> X101\nX25 -> .(T57, X101)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 116 [arrowhead = none ];
116 -> 37;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 117 [arrowhead = none ];
117 -> 35;

118 [label="INSTANCE with matching:\nT21 -> T58\nT22 -> T59\nX25 -> X101", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 118 [arrowhead = none , style=dashed];
118 -> 28[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 119 [arrowhead = none ];
119 -> 40;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 120 [arrowhead = none ];
120 -> 52;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 122 [arrowhead = none ];
122 -> 42;

123 [label="EVAL with clause\nperm([], []).\nand substitution\nT43 -> []\nT23 -> []", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 123 [arrowhead = none ];
123 -> 43;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 124 [arrowhead = none ];
124 -> 44;

125 [label="EVAL with clause\nperm(X117, .(X118, X119)) :- ','(app(X120, .(X118, X121), X117), ','(app(X120, X121, X122), perm(X122, X119))).\nand substitution\nT43 -> T69\nX117 -> T69\nX118 -> T72\nX119 -> T73\nT23 -> .(T72, T73)\nT70 -> T72\nT71 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 127 [arrowhead = none ];
127 -> 45;

128 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 128 [arrowhead = none ];
128 -> 48;

129 [label="SPLIT 2\nnew knowledge:\nT76 is ground\nT72 is ground\nT77 is ground\nreplacements:\nX120 -> T76\nX121 -> T77\nT73 -> T78", fontsize=14, style = filled, fillcolor = violet];
46 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="INSTANCE with matching:\nX23 -> X120\nT17 -> T72\nX24 -> X121\nT14 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 130 [arrowhead = none , style=dashed];
130 -> 18[style=dashed];

131 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 131 [arrowhead = none ];
131 -> 50;

132 [label="SPLIT 2\nnew knowledge:\nT83 is ground\nreplacements:\nX122 -> T83", fontsize=14, style = filled, fillcolor = violet];
49 -> 132 [arrowhead = none ];
132 -> 51;

133 [label="INSTANCE with matching:\nT21 -> T76\nT22 -> T77\nX25 -> X122", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 133 [arrowhead = none , style=dashed];
133 -> 28[style=dashed];

134 [label="INSTANCE with matching:\nT43 -> T83\nT23 -> T78", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 134 [arrowhead = none , style=dashed];
134 -> 38[style=dashed];

135 [label="BACKTRACK\nfor clause: ordered([])because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 135 [arrowhead = none ];
135 -> 53;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 136 [arrowhead = none ];
136 -> 54;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 137 [arrowhead = none ];
137 -> 55;

138 [label="EVAL with clause\nordered(.(X144, [])).\nand substitution\nT24 -> T92\nX144 -> T92\nT62 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 138 [arrowhead = none ];
138 -> 56;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 139 [arrowhead = none ];
139 -> 57;

140 [label="EVAL with clause\nordered(.(X152, .(X153, X154))) :- ','(less(X152, s(X153)), ordered(.(X153, X154))).\nand substitution\nT24 -> T99\nX152 -> T99\nX153 -> T100\nX154 -> T101\nT62 -> .(T100, T101)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 140 [arrowhead = none ];
140 -> 59;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 141 [arrowhead = none ];
141 -> 60;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 142 [arrowhead = none ];
142 -> 58;

143 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
59 -> 143 [arrowhead = none ];
143 -> 61;

144 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
59 -> 144 [arrowhead = none ];
144 -> 62;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 145 [arrowhead = none ];
145 -> 63;

146 [label="INSTANCE with matching:\nT24 -> T100\nT62 -> T101", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 146 [arrowhead = none , style=dashed];
146 -> 39[style=dashed];

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 147 [arrowhead = none ];
147 -> 64;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 148 [arrowhead = none ];
148 -> 65;

149 [label="EVAL with clause\nless(0, s(X166)).\nand substitution\nT99 -> 0\nT100 -> T110\nX166 -> T110", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 149 [arrowhead = none ];
149 -> 66;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 150 [arrowhead = none ];
150 -> 67;

151 [label="EVAL with clause\nless(s(X172), s(X173)) :- less(X172, X173).\nand substitution\nX172 -> T115\nT99 -> s(T115)\nT100 -> T116\nX173 -> T116", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 152 [arrowhead = none ];
152 -> 70;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 153 [arrowhead = none ];
153 -> 68;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 154 [arrowhead = none ];
154 -> 71;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 155 [arrowhead = none ];
155 -> 72;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 156 [arrowhead = none ];
156 -> 73;

157 [label="EVAL with clause\nless(0, s(X182)).\nand substitution\nT115 -> 0\nX182 -> T123\nT116 -> s(T123)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 157 [arrowhead = none ];
157 -> 74;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 158 [arrowhead = none ];
158 -> 75;

159 [label="EVAL with clause\nless(s(X188), s(X189)) :- less(X188, X189).\nand substitution\nX188 -> T128\nT115 -> s(T128)\nX189 -> T129\nT116 -> s(T129)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 159 [arrowhead = none ];
159 -> 77;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 160 [arrowhead = none ];
160 -> 78;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 161 [arrowhead = none ];
161 -> 76;

162 [label="INSTANCE with matching:\nT115 -> T128\nT116 -> T129", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 162 [arrowhead = none , style=dashed];
162 -> 69[style=dashed];

}

(2) Obligation:

Triples:

app18(.(X66, X67), T38, X68, .(X66, T37)) :- app18(X67, T38, X68, T37).
app28(.(T57, T58), T59, .(T57, X101)) :- app28(T58, T59, X101).
perm38(T69, .(T72, T73)) :- app18(X120, T72, X121, T69).
perm38(T69, .(T72, T78)) :- ','(appc18(T76, T72, T77, T69), app28(T76, T77, X122)).
perm38(T69, .(T72, T78)) :- ','(appc18(T76, T72, T77, T69), ','(appc28(T76, T77, T83), perm38(T83, T78))).
ordered39(s(T115), .(T116, T101)) :- less69(T115, T116).
ordered39(T99, .(T100, T101)) :- ','(lessc61(T99, T100), ordered39(T100, T101)).
less69(s(T128), s(T129)) :- less69(T128, T129).
ss1(T14, .(T17, T18)) :- app18(X23, T17, X24, T14).
ss1(T14, .(T24, T23)) :- ','(appc18(T21, T24, T22, T14), app28(T21, T22, X25)).
ss1(T14, .(T24, T23)) :- ','(appc18(T21, T24, T22, T14), ','(appc28(T21, T22, T43), perm38(T43, T23))).
ss1(T14, .(T24, T62)) :- ','(appc18(T21, T24, T22, T14), ','(appc28(T21, T22, T43), ','(permc38(T43, T62), ordered39(T24, T62)))).

Clauses:

appc18([], T31, X46, .(T31, X46)).
appc18(.(X66, X67), T38, X68, .(X66, T37)) :- appc18(X67, T38, X68, T37).
appc28([], T50, T50).
appc28(.(T57, T58), T59, .(T57, X101)) :- appc28(T58, T59, X101).
permc38([], []).
permc38(T69, .(T72, T78)) :- ','(appc18(T76, T72, T77, T69), ','(appc28(T76, T77, T83), permc38(T83, T78))).
orderedc39(T92, []).
orderedc39(T99, .(T100, T101)) :- ','(lessc61(T99, T100), orderedc39(T100, T101)).
lessc69(0, s(T123)).
lessc69(s(T128), s(T129)) :- lessc69(T128, T129).
lessc61(0, T110).
lessc61(s(T115), T116) :- lessc69(T115, T116).

Afs:

ss1(x1, x2)  =  ss1(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:
ss1_in: (b,f)
app18_in: (f,f,f,b)
appc18_in: (f,f,f,b)
app28_in: (b,b,f)
appc28_in: (b,b,f)
perm38_in: (b,f)
permc38_in: (b,f)
ordered39_in: (b,b)
less69_in: (b,b)
lessc61_in: (b,b)
lessc69_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SS1_IN_GA(T14, .(T17, T18)) → U12_GA(T14, T17, T18, app18_in_aaag(X23, T17, X24, T14))
SS1_IN_GA(T14, .(T17, T18)) → APP18_IN_AAAG(X23, T17, X24, T14)
APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → U1_AAAG(X66, X67, T38, X68, T37, app18_in_aaag(X67, T38, X68, T37))
APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → APP18_IN_AAAG(X67, T38, X68, T37)
SS1_IN_GA(T14, .(T24, T23)) → U13_GA(T14, T24, T23, appc18_in_aaag(T21, T24, T22, T14))
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → U14_GA(T14, T24, T23, app28_in_gga(T21, T22, X25))
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → APP28_IN_GGA(T21, T22, X25)
APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → U2_GGA(T57, T58, T59, X101, app28_in_gga(T58, T59, X101))
APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → APP28_IN_GGA(T58, T59, X101)
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → U15_GA(T14, T24, T23, appc28_in_gga(T21, T22, T43))
U15_GA(T14, T24, T23, appc28_out_gga(T21, T22, T43)) → U16_GA(T14, T24, T23, perm38_in_ga(T43, T23))
U15_GA(T14, T24, T23, appc28_out_gga(T21, T22, T43)) → PERM38_IN_GA(T43, T23)
PERM38_IN_GA(T69, .(T72, T73)) → U3_GA(T69, T72, T73, app18_in_aaag(X120, T72, X121, T69))
PERM38_IN_GA(T69, .(T72, T73)) → APP18_IN_AAAG(X120, T72, X121, T69)
PERM38_IN_GA(T69, .(T72, T78)) → U4_GA(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U5_GA(T69, T72, T78, app28_in_gga(T76, T77, X122))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → APP28_IN_GGA(T76, T77, X122)
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U7_GA(T69, T72, T78, perm38_in_ga(T83, T78))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83, T78)
SS1_IN_GA(T14, .(T24, T62)) → U17_GA(T14, T24, T62, appc18_in_aaag(T21, T24, T22, T14))
U17_GA(T14, T24, T62, appc18_out_aaag(T21, T24, T22, T14)) → U18_GA(T14, T24, T62, appc28_in_gga(T21, T22, T43))
U18_GA(T14, T24, T62, appc28_out_gga(T21, T22, T43)) → U19_GA(T14, T24, T62, permc38_in_ga(T43, T62))
U19_GA(T14, T24, T62, permc38_out_ga(T43, T62)) → U20_GA(T14, T24, T62, ordered39_in_gg(T24, T62))
U19_GA(T14, T24, T62, permc38_out_ga(T43, T62)) → ORDERED39_IN_GG(T24, T62)
ORDERED39_IN_GG(s(T115), .(T116, T101)) → U8_GG(T115, T116, T101, less69_in_gg(T115, T116))
ORDERED39_IN_GG(s(T115), .(T116, T101)) → LESS69_IN_GG(T115, T116)
LESS69_IN_GG(s(T128), s(T129)) → U11_GG(T128, T129, less69_in_gg(T128, T129))
LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)
ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → U10_GG(T99, T100, T101, ordered39_in_gg(T100, T101))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
app18_in_aaag(x1, x2, x3, x4)  =  app18_in_aaag(x4)
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app28_in_gga(x1, x2, x3)  =  app28_in_gga(x1, x2)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
perm38_in_ga(x1, x2)  =  perm38_in_ga(x1)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
ordered39_in_gg(x1, x2)  =  ordered39_in_gg(x1, x2)
s(x1)  =  s(x1)
less69_in_gg(x1, x2)  =  less69_in_gg(x1, x2)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
SS1_IN_GA(x1, x2)  =  SS1_IN_GA(x1)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x4)
APP18_IN_AAAG(x1, x2, x3, x4)  =  APP18_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
APP28_IN_GGA(x1, x2, x3)  =  APP28_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x4)
PERM38_IN_GA(x1, x2)  =  PERM38_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U17_GA(x1, x2, x3, x4)  =  U17_GA(x1, x4)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x2, x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
ORDERED39_IN_GG(x1, x2)  =  ORDERED39_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
LESS69_IN_GG(x1, x2)  =  LESS69_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)
U10_GG(x1, x2, x3, x4)  =  U10_GG(x1, x2, x3, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SS1_IN_GA(T14, .(T17, T18)) → U12_GA(T14, T17, T18, app18_in_aaag(X23, T17, X24, T14))
SS1_IN_GA(T14, .(T17, T18)) → APP18_IN_AAAG(X23, T17, X24, T14)
APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → U1_AAAG(X66, X67, T38, X68, T37, app18_in_aaag(X67, T38, X68, T37))
APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → APP18_IN_AAAG(X67, T38, X68, T37)
SS1_IN_GA(T14, .(T24, T23)) → U13_GA(T14, T24, T23, appc18_in_aaag(T21, T24, T22, T14))
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → U14_GA(T14, T24, T23, app28_in_gga(T21, T22, X25))
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → APP28_IN_GGA(T21, T22, X25)
APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → U2_GGA(T57, T58, T59, X101, app28_in_gga(T58, T59, X101))
APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → APP28_IN_GGA(T58, T59, X101)
U13_GA(T14, T24, T23, appc18_out_aaag(T21, T24, T22, T14)) → U15_GA(T14, T24, T23, appc28_in_gga(T21, T22, T43))
U15_GA(T14, T24, T23, appc28_out_gga(T21, T22, T43)) → U16_GA(T14, T24, T23, perm38_in_ga(T43, T23))
U15_GA(T14, T24, T23, appc28_out_gga(T21, T22, T43)) → PERM38_IN_GA(T43, T23)
PERM38_IN_GA(T69, .(T72, T73)) → U3_GA(T69, T72, T73, app18_in_aaag(X120, T72, X121, T69))
PERM38_IN_GA(T69, .(T72, T73)) → APP18_IN_AAAG(X120, T72, X121, T69)
PERM38_IN_GA(T69, .(T72, T78)) → U4_GA(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U5_GA(T69, T72, T78, app28_in_gga(T76, T77, X122))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → APP28_IN_GGA(T76, T77, X122)
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U7_GA(T69, T72, T78, perm38_in_ga(T83, T78))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83, T78)
SS1_IN_GA(T14, .(T24, T62)) → U17_GA(T14, T24, T62, appc18_in_aaag(T21, T24, T22, T14))
U17_GA(T14, T24, T62, appc18_out_aaag(T21, T24, T22, T14)) → U18_GA(T14, T24, T62, appc28_in_gga(T21, T22, T43))
U18_GA(T14, T24, T62, appc28_out_gga(T21, T22, T43)) → U19_GA(T14, T24, T62, permc38_in_ga(T43, T62))
U19_GA(T14, T24, T62, permc38_out_ga(T43, T62)) → U20_GA(T14, T24, T62, ordered39_in_gg(T24, T62))
U19_GA(T14, T24, T62, permc38_out_ga(T43, T62)) → ORDERED39_IN_GG(T24, T62)
ORDERED39_IN_GG(s(T115), .(T116, T101)) → U8_GG(T115, T116, T101, less69_in_gg(T115, T116))
ORDERED39_IN_GG(s(T115), .(T116, T101)) → LESS69_IN_GG(T115, T116)
LESS69_IN_GG(s(T128), s(T129)) → U11_GG(T128, T129, less69_in_gg(T128, T129))
LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)
ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → U10_GG(T99, T100, T101, ordered39_in_gg(T100, T101))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
app18_in_aaag(x1, x2, x3, x4)  =  app18_in_aaag(x4)
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
app28_in_gga(x1, x2, x3)  =  app28_in_gga(x1, x2)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
perm38_in_ga(x1, x2)  =  perm38_in_ga(x1)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
ordered39_in_gg(x1, x2)  =  ordered39_in_gg(x1, x2)
s(x1)  =  s(x1)
less69_in_gg(x1, x2)  =  less69_in_gg(x1, x2)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
SS1_IN_GA(x1, x2)  =  SS1_IN_GA(x1)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x4)
APP18_IN_AAAG(x1, x2, x3, x4)  =  APP18_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
APP28_IN_GGA(x1, x2, x3)  =  APP28_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x4)
PERM38_IN_GA(x1, x2)  =  PERM38_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
U17_GA(x1, x2, x3, x4)  =  U17_GA(x1, x4)
U18_GA(x1, x2, x3, x4)  =  U18_GA(x1, x2, x4)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x2, x4)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
ORDERED39_IN_GG(x1, x2)  =  ORDERED39_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
LESS69_IN_GG(x1, x2)  =  LESS69_IN_GG(x1, x2)
U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)
U10_GG(x1, x2, x3, x4)  =  U10_GG(x1, x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 24 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
s(x1)  =  s(x1)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
LESS69_IN_GG(x1, x2)  =  LESS69_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:

LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)

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:

LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)

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:

• LESS69_IN_GG(s(T128), s(T129)) → LESS69_IN_GG(T128, T129)
  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:

ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
s(x1)  =  s(x1)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
ORDERED39_IN_GG(x1, x2)  =  ORDERED39_IN_GG(x1, x2)
U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)

The TRS R consists of the following rules:

lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))

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

(17) PiDPToQDPProof (EQUIVALENT 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:

ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)

The TRS R consists of the following rules:

lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))

The set Q consists of the following terms:

lessc61_in_gg(x0, x1)
U30_gg(x0, x1, x2)
lessc69_in_gg(x0, x1)
U29_gg(x0, x1, x2)

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:

• U9_GG(T99, T100, T101, lessc61_out_gg(T99, T100)) → ORDERED39_IN_GG(T100, T101)
  The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2

• ORDERED39_IN_GG(T99, .(T100, T101)) → U9_GG(T99, T100, T101, lessc61_in_gg(T99, T100))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(20) YES

(21) Obligation:

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

APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → APP28_IN_GGA(T58, T59, X101)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
s(x1)  =  s(x1)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
APP28_IN_GGA(x1, x2, x3)  =  APP28_IN_GGA(x1, x2)

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:

APP28_IN_GGA(.(T57, T58), T59, .(T57, X101)) → APP28_IN_GGA(T58, T59, X101)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP28_IN_GGA(x1, x2, x3)  =  APP28_IN_GGA(x1, x2)

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:

APP28_IN_GGA(.(T57, T58), T59) → APP28_IN_GGA(T58, T59)

R is empty.
Q is empty.
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:

• APP28_IN_GGA(.(T57, T58), T59) → APP28_IN_GGA(T58, T59)
  The graph contains the following edges 1 > 1, 2 >= 2

(27) YES

(28) Obligation:

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

APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → APP18_IN_AAAG(X67, T38, X68, T37)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
s(x1)  =  s(x1)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
APP18_IN_AAAG(x1, x2, x3, x4)  =  APP18_IN_AAAG(x4)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

APP18_IN_AAAG(.(X66, X67), T38, X68, .(X66, T37)) → APP18_IN_AAAG(X67, T38, X68, T37)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP18_IN_AAAG(x1, x2, x3, x4)  =  APP18_IN_AAAG(x4)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

APP18_IN_AAAG(.(X66, T37)) → APP18_IN_AAAG(T37)

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

(33) 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:

• APP18_IN_AAAG(.(X66, T37)) → APP18_IN_AAAG(T37)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

PERM38_IN_GA(T69, .(T72, T78)) → U4_GA(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83, T78)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))
permc38_in_ga([], []) → permc38_out_ga([], [])
permc38_in_ga(T69, .(T72, T78)) → U24_ga(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U24_ga(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U25_ga(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U25_ga(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → U26_ga(T69, T72, T78, permc38_in_ga(T83, T78))
U26_ga(T69, T72, T78, permc38_out_ga(T83, T78)) → permc38_out_ga(T69, .(T72, T78))
lessc61_in_gg(0, T110) → lessc61_out_gg(0, T110)
lessc61_in_gg(s(T115), T116) → U30_gg(T115, T116, lessc69_in_gg(T115, T116))
lessc69_in_gg(0, s(T123)) → lessc69_out_gg(0, s(T123))
lessc69_in_gg(s(T128), s(T129)) → U29_gg(T128, T129, lessc69_in_gg(T128, T129))
U29_gg(T128, T129, lessc69_out_gg(T128, T129)) → lessc69_out_gg(s(T128), s(T129))
U30_gg(T115, T116, lessc69_out_gg(T115, T116)) → lessc61_out_gg(s(T115), T116)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
permc38_in_ga(x1, x2)  =  permc38_in_ga(x1)
permc38_out_ga(x1, x2)  =  permc38_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x2, x4)
s(x1)  =  s(x1)
lessc61_in_gg(x1, x2)  =  lessc61_in_gg(x1, x2)
0  =  0
lessc61_out_gg(x1, x2)  =  lessc61_out_gg(x1, x2)
U30_gg(x1, x2, x3)  =  U30_gg(x1, x2, x3)
lessc69_in_gg(x1, x2)  =  lessc69_in_gg(x1, x2)
lessc69_out_gg(x1, x2)  =  lessc69_out_gg(x1, x2)
U29_gg(x1, x2, x3)  =  U29_gg(x1, x2, x3)
PERM38_IN_GA(x1, x2)  =  PERM38_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

PERM38_IN_GA(T69, .(T72, T78)) → U4_GA(T69, T72, T78, appc18_in_aaag(T76, T72, T77, T69))
U4_GA(T69, T72, T78, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, T72, T78, appc28_in_gga(T76, T77, T83))
U6_GA(T69, T72, T78, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83, T78)

The TRS R consists of the following rules:

appc18_in_aaag([], T31, X46, .(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, X67), T38, X68, .(X66, T37)) → U22_aaag(X66, X67, T38, X68, T37, appc18_in_aaag(X67, T38, X68, T37))
appc28_in_gga([], T50, T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59, .(T57, X101)) → U23_gga(T57, T58, T59, X101, appc28_in_gga(T58, T59, X101))
U22_aaag(X66, X67, T38, X68, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
U23_gga(T57, T58, T59, X101, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appc18_in_aaag(x1, x2, x3, x4)  =  appc18_in_aaag(x4)
appc18_out_aaag(x1, x2, x3, x4)  =  appc18_out_aaag(x1, x2, x3, x4)
U22_aaag(x1, x2, x3, x4, x5, x6)  =  U22_aaag(x1, x5, x6)
appc28_in_gga(x1, x2, x3)  =  appc28_in_gga(x1, x2)
[]  =  []
appc28_out_gga(x1, x2, x3)  =  appc28_out_gga(x1, x2, x3)
U23_gga(x1, x2, x3, x4, x5)  =  U23_gga(x1, x2, x3, x5)
PERM38_IN_GA(x1, x2)  =  PERM38_IN_GA(x1)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

PERM38_IN_GA(T69) → U4_GA(T69, appc18_in_aaag(T69))
U4_GA(T69, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, appc28_in_gga(T76, T77))
U6_GA(T69, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83)

The TRS R consists of the following rules:

appc18_in_aaag(.(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, T37)) → U22_aaag(X66, T37, appc18_in_aaag(T37))
appc28_in_gga([], T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59) → U23_gga(T57, T58, T59, appc28_in_gga(T58, T59))
U22_aaag(X66, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
U23_gga(T57, T58, T59, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))

The set Q consists of the following terms:

appc18_in_aaag(x0)
appc28_in_gga(x0, x1)
U22_aaag(x0, x1, x2)
U23_gga(x0, x1, x2, x3)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PERM38_IN_GA(T69) → U4_GA(T69, appc18_in_aaag(T69))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(PERM38_IN_GA(x₁)) = 1 + x₁   
POL(U22_aaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U23_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U4_GA(x₁, x₂)) = x₂   
POL(U6_GA(x₁, x₂)) = x₂   
POL([]) = 1   
POL(appc18_in_aaag(x₁)) = x₁   
POL(appc18_out_aaag(x₁, x₂, x₃, x₄)) = x₁ + x₃   
POL(appc28_in_gga(x₁, x₂)) = x₁ + x₂   
POL(appc28_out_gga(x₁, x₂, x₃)) = 1 + x₃   

The following usable rules [FROCOS05] were oriented:

appc18_in_aaag(.(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, T37)) → U22_aaag(X66, T37, appc18_in_aaag(T37))
appc28_in_gga([], T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59) → U23_gga(T57, T58, T59, appc28_in_gga(T58, T59))
U22_aaag(X66, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
U23_gga(T57, T58, T59, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))

(41) Obligation:

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

U4_GA(T69, appc18_out_aaag(T76, T72, T77, T69)) → U6_GA(T69, appc28_in_gga(T76, T77))
U6_GA(T69, appc28_out_gga(T76, T77, T83)) → PERM38_IN_GA(T83)

The TRS R consists of the following rules:

appc18_in_aaag(.(T31, X46)) → appc18_out_aaag([], T31, X46, .(T31, X46))
appc18_in_aaag(.(X66, T37)) → U22_aaag(X66, T37, appc18_in_aaag(T37))
appc28_in_gga([], T50) → appc28_out_gga([], T50, T50)
appc28_in_gga(.(T57, T58), T59) → U23_gga(T57, T58, T59, appc28_in_gga(T58, T59))
U22_aaag(X66, T37, appc18_out_aaag(X67, T38, X68, T37)) → appc18_out_aaag(.(X66, X67), T38, X68, .(X66, T37))
U23_gga(T57, T58, T59, appc28_out_gga(T58, T59, X101)) → appc28_out_gga(.(T57, T58), T59, .(T57, X101))

The set Q consists of the following terms:

appc18_in_aaag(x0)
appc28_in_gga(x0, x1)
U22_aaag(x0, x1, x2)
U23_gga(x0, x1, x2, x3)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) TRUE