Left Termination of the query pattern
ss(g,g)
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).

Query: ss(g,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="G: ss(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ss(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(perm(T5, T6), ordered(T6))\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(perm(T5, T6), ordered(T6)), SCOPE: 2, CLAUSE: 1 | ','(perm(T5, T6), ordered(T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(perm(T5, T6), ordered(T6)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(perm(T5, T6), ordered(T6)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 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, .(T14, X24), T13), ','(app(X23, X24, X25), perm(X25, T15))), ordered(.(T14, T15)))\n\nT13 is ground\nT14 is ground\nT15 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="A: app(X23, .(T14, X24), T13)\n\nT13 is ground\nT14 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(app(T20, T21, X25), perm(X25, T15)), ordered(.(T14, T15)))\n\nT14 is ground\nT15 is ground\nT20 is ground\nT21 is ground\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(X23, .(T14, X24), T13), SCOPE: 4, CLAUSE: 3 | app(X23, .(T14, X24), T13), SCOPE: 4, CLAUSE: 4\n\nT13 is ground\nT14 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(X23, .(T14, X24), T13), SCOPE: 4, CLAUSE: 3\n\nT13 is ground\nT14 is ground\nX23 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(X23, .(T14, X24), T13), SCOPE: 4, CLAUSE: 4\n\nT13 is ground\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(X63, .(T42, X64), T44)\n\nT42 is ground\nT44 is ground\nX63 is free\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="B: app(T20, T21, X25)\n\nT20 is ground\nT21 is ground\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(perm(T51, T15), ordered(.(T14, T15)))\n\nT14 is ground\nT15 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T20, T21, X25), SCOPE: 5, CLAUSE: 3 | app(T20, T21, X25), SCOPE: 5, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T20, T21, X25), SCOPE: 5, CLAUSE: 3\n\nT20 is ground\nT21 is ground\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T20, T21, X25), SCOPE: 5, CLAUSE: 4\n\nT20 is ground\nT21 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(T66, T67, X92)\n\nT66 is ground\nT67 is ground\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="C: perm(T51, T15)\n\nT15 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="D: ordered(.(T14, T15))\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: perm(T51, T15), SCOPE: 6, CLAUSE: 1 | perm(T51, T15), SCOPE: 6, CLAUSE: 2\n\nT15 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: perm(T51, T15), SCOPE: 6, CLAUSE: 1\n\nT15 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: perm(T51, T15), SCOPE: 6, CLAUSE: 2\n\nT15 is ground\nT51 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(X107, .(T77, X108), T76), ','(app(X107, X108, X109), perm(X109, T78)))\n\nT76 is ground\nT77 is ground\nT78 is ground\nX107 is free\nX108 is free\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(X107, .(T77, X108), T76)\n\nT76 is ground\nT77 is ground\nX107 is free\nX108 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(app(T83, T84, X109), perm(X109, T78))\n\nT78 is ground\nT83 is ground\nT84 is ground\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(T83, T84, X109)\n\nT83 is ground\nT84 is ground\nX109 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: perm(T91, T78)\n\nT78 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 5 | ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 6 | ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 7\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 6 | ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 7\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 6\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ordered(.(T14, T15)), SCOPE: 7, CLAUSE: 7\n\nT14 is ground\nT15 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(T105, s(T106)), ordered(.(T106, T107)))\n\nT105 is ground\nT106 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="E: less(T105, s(T106))\n\nT105 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ordered(.(T106, T107))\n\nT106 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T105, s(T106)), SCOPE: 8, CLAUSE: 8 | less(T105, s(T106)), SCOPE: 8, CLAUSE: 9\n\nT105 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T105, s(T106)), SCOPE: 8, CLAUSE: 8\n\nT105 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(T105, s(T106)), SCOPE: 8, CLAUSE: 9\n\nT105 is ground\nT106 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="F: less(T121, T122)\n\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T121, T122), SCOPE: 9, CLAUSE: 8 | less(T121, T122), SCOPE: 9, CLAUSE: 9\n\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: less(T121, T122), SCOPE: 9, CLAUSE: 8\n\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T121, T122), SCOPE: 9, CLAUSE: 9\n\nT121 is ground\nT122 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(T134, T135)\n\nT134 is ground\nT135 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 substitutionT1 -> T5,\nX5 -> T5,\nT2 -> T6,\nX6 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT5 -> [],\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT5 -> T13,\nX20 -> T13,\nX21 -> T14,\nX22 -> T15,\nT6 -> .(T14, T15)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightblue];
10 -> 91 [arrowhead = none ];
91 -> 12;

92 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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:\nT20 is ground\nT14 is ground\nT21 is ground\nT13 is ground\nreplacements:X23 -> T20,\nX24 -> T21", 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:\nT20 is ground\nT21 is ground\nT51 is ground\nreplacements:X25 -> T51", 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([], X42, X42).\nand substitutionX23 -> [],\nT14 -> T34,\nX24 -> T35,\nX42 -> .(T34, T35),\nX43 -> T35,\nT13 -> .(T34, T35)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 102 [arrowhead = none ];
102 -> 23;

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

104 [label="EVAL with clause\napp(.(X58, X59), X60, .(X58, X61)) :- app(X59, X60, X61).\nand substitutionX58 -> T43,\nX59 -> X63,\nX23 -> .(T43, X63),\nT14 -> T42,\nX24 -> X64,\nX60 -> .(T42, X64),\nX62 -> T43,\nX61 -> T44,\nT13 -> .(T43, T44)", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 -> X63\nT14 -> T42\nX24 -> X64\nT13 -> T44", 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:\nT51 is ground\nT15 is ground", 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([], X77, X77).\nand substitutionT20 -> [],\nT21 -> T58,\nX77 -> T58,\nX25 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 113 [arrowhead = none ];
113 -> 33;

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

115 [label="EVAL with clause\napp(.(X88, X89), X90, .(X88, X91)) :- app(X89, X90, X91).\nand substitutionX88 -> T65,\nX89 -> T66,\nT20 -> .(T65, T66),\nT21 -> T67,\nX90 -> T67,\nX91 -> X92,\nX25 -> .(T65, X92)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT20 -> T66\nT21 -> T67\nX25 -> X92", 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 substitutionT51 -> [],\nT15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 123 [arrowhead = none ];
123 -> 43;

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

125 [label="EVAL with clause\nperm(X104, .(X105, X106)) :- ','(app(X107, .(X105, X108), X104), ','(app(X107, X108, X109), perm(X109, X106))).\nand substitutionT51 -> T76,\nX104 -> T76,\nX105 -> T77,\nX106 -> T78,\nT15 -> .(T77, T78)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT83 is ground\nT77 is ground\nT84 is ground\nT76 is ground\nreplacements:X107 -> T83,\nX108 -> T84", fontsize=14, style = filled, fillcolor = violet];
46 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="INSTANCE with matching:\nX23 -> X107\nT14 -> T77\nX24 -> X108\nT13 -> T76", 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\nT84 is ground\nT91 is ground\nreplacements:X109 -> T91", fontsize=14, style = filled, fillcolor = violet];
49 -> 132 [arrowhead = none ];
132 -> 51;

133 [label="INSTANCE with matching:\nT20 -> T83\nT21 -> T84\nX25 -> X109", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 133 [arrowhead = none , style=dashed];
133 -> 28[style=dashed];

134 [label="INSTANCE with matching:\nT51 -> T91\nT15 -> 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 = lightgreen];
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(.(X126, [])).\nand substitutionT14 -> T98,\nX126 -> T98,\nT15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 138 [arrowhead = none ];
138 -> 56;

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

140 [label="EVAL with clause\nordered(.(X133, .(X134, X135))) :- ','(less(X133, s(X134)), ordered(.(X134, X135))).\nand substitutionT14 -> T105,\nX133 -> T105,\nX134 -> T106,\nX135 -> T107,\nT15 -> .(T106, T107)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 140 [arrowhead = none ];
140 -> 59;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT105 is ground\nT106 is ground", 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:\nT14 -> T106\nT15 -> T107", 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(X144)).\nand substitutionT105 -> 0,\nT106 -> T116,\nX144 -> T116", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 149 [arrowhead = none ];
149 -> 66;

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

151 [label="EVAL with clause\nless(s(X149), s(X150)) :- less(X149, X150).\nand substitutionX149 -> T121,\nT105 -> s(T121),\nT106 -> T122,\nX150 -> T122", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X157)).\nand substitutionT121 -> 0,\nX157 -> T129,\nT122 -> s(T129)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 157 [arrowhead = none ];
157 -> 74;

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

159 [label="EVAL with clause\nless(s(X162), s(X163)) :- less(X162, X163).\nand substitutionX162 -> T134,\nT121 -> s(T134),\nX163 -> T135,\nT122 -> s(T135)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 159 [arrowhead = none ];
159 -> 77;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT121 -> T134\nT122 -> T135", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 162 [arrowhead = none , style=dashed];
162 -> 69[style=dashed];

}

(2) Obligation:

Triples:

appA(.(X1, X2), X3, X4, .(X1, X5)) :- appA(X2, X3, X4, X5).
appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
permC(X1, .(X2, X3)) :- appA(X4, X2, X5, X1).
permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)).
permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))).
orderedD(s(X1), .(X2, X3)) :- lessF(X1, X2).
orderedD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedD(X2, X3)).
lessF(s(X1), s(X2)) :- lessF(X1, X2).
ssG(X1, .(X2, X3)) :- appA(X4, X2, X5, X1).
ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)).
ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))).
ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), ','(permcC(X6, X3), orderedD(X2, X3)))).

Clauses:

appcA([], X1, X2, .(X1, X2)).
appcA(.(X1, X2), X3, X4, .(X1, X5)) :- appcA(X2, X3, X4, X5).
appcB([], X1, X1).
appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
permcC([], []).
permcC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permcC(X6, X3))).
orderedcD(X1, []).
orderedcD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedcD(X2, X3)).
lesscF(0, s(X1)).
lesscF(s(X1), s(X2)) :- lesscF(X1, X2).
lesscE(0, X1).
lesscE(s(X1), X2) :- lesscF(X1, X2).

Afs:

ssG(x1, x2)  =  ssG(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
ssG_in: (b,b)
appA_in: (f,b,f,b)
appcA_in: (f,b,f,b)
appB_in: (b,b,f)
appcB_in: (b,b,f)
permC_in: (b,b)
permcC_in: (b,b)
orderedD_in: (b,b)
lessF_in: (b,b)
lesscE_in: (b,b)
lesscF_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SSG_IN_GG(X1, .(X2, X3)) → U12_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1))
SSG_IN_GG(X1, .(X2, X3)) → APPA_IN_AGAG(X4, X2, X5, X1)
APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → U1_AGAG(X1, X2, X3, X4, X5, appA_in_agag(X2, X3, X4, X5))
APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AGAG(X2, X3, X4, X5)
SSG_IN_GG(X1, .(X2, X3)) → U13_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U14_GG(X1, X2, X3, appB_in_gga(X4, X5, X6))
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → APPB_IN_GGA(X4, X5, X6)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U15_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U16_GG(X1, X2, X3, permC_in_gg(X6, X3))
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)
PERMC_IN_GG(X1, .(X2, X3)) → U3_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1))
PERMC_IN_GG(X1, .(X2, X3)) → APPA_IN_AGAG(X4, X2, X5, X1)
PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U5_GG(X1, X2, X3, appB_in_gga(X4, X5, X6))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → APPB_IN_GGA(X4, X5, X6)
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U7_GG(X1, X2, X3, permC_in_gg(X6, X3))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U17_GG(X1, X2, X3, permcC_in_gg(X6, X3))
U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) → U18_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) → ORDEREDD_IN_GG(X2, X3)
ORDEREDD_IN_GG(s(X1), .(X2, X3)) → U8_GG(X1, X2, X3, lessF_in_gg(X1, X2))
ORDEREDD_IN_GG(s(X1), .(X2, X3)) → LESSF_IN_GG(X1, X2)
LESSF_IN_GG(s(X1), s(X2)) → U11_GG(X1, X2, lessF_in_gg(X1, X2))
LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)
ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appA_in_agag(x1, x2, x3, x4)  =  appA_in_agag(x2, x4)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permC_in_gg(x1, x2)  =  permC_in_gg(x1, x2)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
orderedD_in_gg(x1, x2)  =  orderedD_in_gg(x1, x2)
s(x1)  =  s(x1)
lessF_in_gg(x1, x2)  =  lessF_in_gg(x1, x2)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
SSG_IN_GG(x1, x2)  =  SSG_IN_GG(x1, x2)
U12_GG(x1, x2, x3, x4)  =  U12_GG(x1, x2, x3, x4)
APPA_IN_AGAG(x1, x2, x3, x4)  =  APPA_IN_AGAG(x2, x4)
U1_AGAG(x1, x2, x3, x4, x5, x6)  =  U1_AGAG(x1, x3, x5, x6)
U13_GG(x1, x2, x3, x4)  =  U13_GG(x1, x2, x3, x4)
U14_GG(x1, x2, x3, x4)  =  U14_GG(x1, x2, x3, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U15_GG(x1, x2, x3, x4)  =  U15_GG(x1, x2, x3, x4)
U16_GG(x1, x2, x3, x4)  =  U16_GG(x1, x2, x3, x4)
PERMC_IN_GG(x1, x2)  =  PERMC_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x2, x3, x4)
U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x2, x3, x4)
U17_GG(x1, x2, x3, x4)  =  U17_GG(x1, x2, x3, x4)
U18_GG(x1, x2, x3, x4)  =  U18_GG(x1, x2, x3, x4)
ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
LESSF_IN_GG(x1, x2)  =  LESSF_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:

SSG_IN_GG(X1, .(X2, X3)) → U12_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1))
SSG_IN_GG(X1, .(X2, X3)) → APPA_IN_AGAG(X4, X2, X5, X1)
APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → U1_AGAG(X1, X2, X3, X4, X5, appA_in_agag(X2, X3, X4, X5))
APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AGAG(X2, X3, X4, X5)
SSG_IN_GG(X1, .(X2, X3)) → U13_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U14_GG(X1, X2, X3, appB_in_gga(X4, X5, X6))
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → APPB_IN_GGA(X4, X5, X6)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U15_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U16_GG(X1, X2, X3, permC_in_gg(X6, X3))
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)
PERMC_IN_GG(X1, .(X2, X3)) → U3_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1))
PERMC_IN_GG(X1, .(X2, X3)) → APPA_IN_AGAG(X4, X2, X5, X1)
PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U5_GG(X1, X2, X3, appB_in_gga(X4, X5, X6))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → APPB_IN_GGA(X4, X5, X6)
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U7_GG(X1, X2, X3, permC_in_gg(X6, X3))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)
U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U17_GG(X1, X2, X3, permcC_in_gg(X6, X3))
U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) → U18_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) → ORDEREDD_IN_GG(X2, X3)
ORDEREDD_IN_GG(s(X1), .(X2, X3)) → U8_GG(X1, X2, X3, lessF_in_gg(X1, X2))
ORDEREDD_IN_GG(s(X1), .(X2, X3)) → LESSF_IN_GG(X1, X2)
LESSF_IN_GG(s(X1), s(X2)) → U11_GG(X1, X2, lessF_in_gg(X1, X2))
LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)
ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appA_in_agag(x1, x2, x3, x4)  =  appA_in_agag(x2, x4)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appB_in_gga(x1, x2, x3)  =  appB_in_gga(x1, x2)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permC_in_gg(x1, x2)  =  permC_in_gg(x1, x2)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
orderedD_in_gg(x1, x2)  =  orderedD_in_gg(x1, x2)
s(x1)  =  s(x1)
lessF_in_gg(x1, x2)  =  lessF_in_gg(x1, x2)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
SSG_IN_GG(x1, x2)  =  SSG_IN_GG(x1, x2)
U12_GG(x1, x2, x3, x4)  =  U12_GG(x1, x2, x3, x4)
APPA_IN_AGAG(x1, x2, x3, x4)  =  APPA_IN_AGAG(x2, x4)
U1_AGAG(x1, x2, x3, x4, x5, x6)  =  U1_AGAG(x1, x3, x5, x6)
U13_GG(x1, x2, x3, x4)  =  U13_GG(x1, x2, x3, x4)
U14_GG(x1, x2, x3, x4)  =  U14_GG(x1, x2, x3, x4)
APPB_IN_GGA(x1, x2, x3)  =  APPB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U15_GG(x1, x2, x3, x4)  =  U15_GG(x1, x2, x3, x4)
U16_GG(x1, x2, x3, x4)  =  U16_GG(x1, x2, x3, x4)
PERMC_IN_GG(x1, x2)  =  PERMC_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x2, x3, x4)
U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x2, x3, x4)
U17_GG(x1, x2, x3, x4)  =  U17_GG(x1, x2, x3, x4)
U18_GG(x1, x2, x3, x4)  =  U18_GG(x1, x2, x3, x4)
ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
LESSF_IN_GG(x1, x2)  =  LESSF_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 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
s(x1)  =  s(x1)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
LESSF_IN_GG(x1, x2)  =  LESSF_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:

LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)

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:

LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)

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:

• LESSF_IN_GG(s(X1), s(X2)) → LESSF_IN_GG(X1, X2)
  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:

ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
s(x1)  =  s(x1)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_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:

ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)

The TRS R consists of the following rules:

lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))

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:

ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)

The TRS R consists of the following rules:

lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lesscE_in_gg(x0, x1)
U28_gg(x0, x1, x2)
lesscF_in_gg(x0, x1)
U27_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(X1, X2, X3, lesscE_out_gg(X1, X2)) → ORDEREDD_IN_GG(X2, X3)
  The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2

• ORDEREDD_IN_GG(X1, .(X2, X3)) → U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
  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:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
s(x1)  =  s(x1)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
APPB_IN_GGA(x1, x2, x3)  =  APPB_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:

APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)

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

APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)

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:

• APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)
  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:

APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AGAG(X2, X3, X4, X5)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
s(x1)  =  s(x1)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
APPA_IN_AGAG(x1, x2, x3, x4)  =  APPA_IN_AGAG(x2, 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:

APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AGAG(X2, X3, X4, X5)

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

APPA_IN_AGAG(X3, .(X1, X5)) → APPA_IN_AGAG(X3, X5)

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:

• APPA_IN_AGAG(X3, .(X1, X5)) → APPA_IN_AGAG(X3, X5)
  The graph contains the following edges 1 >= 1, 2 > 2

(34) YES

(35) Obligation:

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

PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
permcC_in_gg([], []) → permcC_out_gg([], [])
permcC_in_gg(X1, .(X2, X3)) → U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → U24_gg(X1, X2, X3, permcC_in_gg(X6, X3))
U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) → permcC_out_gg(X1, .(X2, X3))
lesscE_in_gg(0, X1) → lesscE_out_gg(0, X1)
lesscE_in_gg(s(X1), X2) → U28_gg(X1, X2, lesscF_in_gg(X1, X2))
lesscF_in_gg(0, s(X1)) → lesscF_out_gg(0, s(X1))
lesscF_in_gg(s(X1), s(X2)) → U27_gg(X1, X2, lesscF_in_gg(X1, X2))
U27_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscF_out_gg(s(X1), s(X2))
U28_gg(X1, X2, lesscF_out_gg(X1, X2)) → lesscE_out_gg(s(X1), X2)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
permcC_in_gg(x1, x2)  =  permcC_in_gg(x1, x2)
permcC_out_gg(x1, x2)  =  permcC_out_gg(x1, x2)
U22_gg(x1, x2, x3, x4)  =  U22_gg(x1, x2, x3, x4)
U23_gg(x1, x2, x3, x4)  =  U23_gg(x1, x2, x3, x4)
U24_gg(x1, x2, x3, x4)  =  U24_gg(x1, x2, x3, x4)
s(x1)  =  s(x1)
lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
0  =  0
lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
PERMC_IN_GG(x1, x2)  =  PERMC_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x2, x3, 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:

PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)

The TRS R consists of the following rules:

appcA_in_agag([], X1, X2, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) → U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5))
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_agag(x1, x2, x3, x4)  =  appcA_in_agag(x2, x4)
appcA_out_agag(x1, x2, x3, x4)  =  appcA_out_agag(x1, x2, x3, x4)
U20_agag(x1, x2, x3, x4, x5, x6)  =  U20_agag(x1, x3, x5, x6)
appcB_in_gga(x1, x2, x3)  =  appcB_in_gga(x1, x2)
[]  =  []
appcB_out_gga(x1, x2, x3)  =  appcB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
PERMC_IN_GG(x1, x2)  =  PERMC_IN_GG(x1, x2)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x2, x3, 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:

PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X2, X1))
U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5))
U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)

The TRS R consists of the following rules:

appcA_in_agag(X1, .(X1, X2)) → appcA_out_agag([], X1, X2, .(X1, X2))
appcA_in_agag(X3, .(X1, X5)) → U20_agag(X1, X3, X5, appcA_in_agag(X3, X5))
appcB_in_gga([], X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3) → U21_gga(X1, X2, X3, appcB_in_gga(X2, X3))
U20_agag(X1, X3, X5, appcA_out_agag(X2, X3, X4, X5)) → appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5))
U21_gga(X1, X2, X3, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))

The set Q consists of the following terms:

appcA_in_agag(x0, x1)
appcB_in_gga(x0, x1)
U20_agag(x0, x1, x2, x3)
U21_gga(x0, x1, x2, x3)

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

(40) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) → U6_GG(X1, X2, X3, appcB_in_gga(X4, X5))
  The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 4 > 2, 3 >= 3

• U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) → PERMC_IN_GG(X6, X3)
  The graph contains the following edges 4 > 1, 3 >= 2

• PERMC_IN_GG(X1, .(X2, X3)) → U4_GG(X1, X2, X3, appcA_in_agag(X2, X1))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(41) YES