Left Termination of the query pattern rev_in_2(a, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

rev([], []).
rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

rev(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rev(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | rev(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: rev(T1, []), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(rev(T7, X7), app(X7, .(T8, []), []))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(rev(T7, X7), app(X7, .(T8, []), [])), SCOPE: 2, CLAUSE: 0 | ','(rev(T7, X7), app(X7, .(T8, []), [])), SCOPE: 2, CLAUSE: 1\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rev(T7, X7), app(X7, .(T8, []), [])), SCOPE: 2, CLAUSE: 0\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(rev(T7, X7), app(X7, .(T8, []), [])), SCOPE: 2, CLAUSE: 1\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app([], .(T9, []), [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app([], .(T9, []), []), SCOPE: 3, CLAUSE: 2 | app([], .(T9, []), []), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], .(T9, []), []), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(rev(T19, X34), ','(app(X34, .(T20, []), X35), app(X35, .(T21, []), [])))\n\nX35 is free\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: rev(T19, X34)\n\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(T22, .(T23, []), X35), app(X35, .(T24, []), []))\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: rev(T19, X34), SCOPE: 4, CLAUSE: 0 | rev(T19, X34), SCOPE: 4, CLAUSE: 1\n\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: rev(T19, X34), SCOPE: 4, CLAUSE: 0\n\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: rev(T19, X34), SCOPE: 4, CLAUSE: 1\n\nX34 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: ','(rev(T31, X51), app(X51, .(T32, []), X52))\n\nX52 is free\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: rev(T31, X51)\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: app(T33, .(T34, []), X52)\n\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T33, .(T34, []), X52), SCOPE: 5, CLAUSE: 2 | app(T33, .(T34, []), X52), SCOPE: 5, CLAUSE: 3\n\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T33, .(T34, []), X52), SCOPE: 5, CLAUSE: 2\n\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T33, .(T34, []), X52), SCOPE: 5, CLAUSE: 3\n\nX52 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(T51, .(T52, []), X78)\n\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(T22, .(T23, []), X35)\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app(T57, .(T58, []), [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(T57, .(T58, []), []), SCOPE: 6, CLAUSE: 2 | app(T57, .(T58, []), []), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T57, .(T58, []), []), SCOPE: 6, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(rev(T76, X106), app(X106, .(T77, []), T75))\n\nT75 is ground\nX106 is free; \nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(rev(T76, X106), app(X106, .(T77, []), T75)), SCOPE: 7, CLAUSE: 0 | ','(rev(T76, X106), app(X106, .(T77, []), T75)), SCOPE: 7, CLAUSE: 1\n\nT75 is ground\nX106 is free; \nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(rev(T76, X106), app(X106, .(T77, []), T75)), SCOPE: 7, CLAUSE: 0\n\nT75 is ground\nX106 is free; \nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(rev(T76, X106), app(X106, .(T77, []), T75)), SCOPE: 7, CLAUSE: 1\n\nT75 is ground\nX106 is free; \nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app([], .(T78, []), T75)\n\nT75 is ground\nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app([], .(T78, []), T75), SCOPE: 8, CLAUSE: 2 | app([], .(T78, []), T75), SCOPE: 8, CLAUSE: 3\n\nT75 is ground\nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app([], .(T78, []), T75), SCOPE: 8, CLAUSE: 2\n\nT75 is ground\nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: app([], .(T78, []), T75), SCOPE: 8, CLAUSE: 3\n\nT75 is ground\nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(rev(T92, X134), ','(app(X134, .(T93, []), X135), app(X135, .(T94, []), T75)))\n\nT75 is ground\nX135 is free; \nX134 is free; \nrev(T1, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: rev(T92, X134)\n\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(app(T95, .(T96, []), X135), app(X135, .(T97, []), T75))\n\nT75 is ground\nX135 is free; \nrev(T98, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: app(T95, .(T96, []), X135)\n\nX135 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: app(T101, .(T102, []), T75)\n\nT75 is ground\nrev(T103, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: app(T101, .(T102, []), T75), SCOPE: 9, CLAUSE: 2 | app(T101, .(T102, []), T75), SCOPE: 9, CLAUSE: 3\n\nT75 is ground\nrev(T103, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: app(T101, .(T102, []), T75), SCOPE: 9, CLAUSE: 2\n\nT75 is ground\nrev(T103, T75) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: app(T101, .(T102, []), T75), SCOPE: 9, CLAUSE: 3\n\nT75 is ground\nrev(T103, T75) !=? rev([], [])", 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: app(T125, .(T126, []), T124)\n\nT124 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: app(T125, .(T126, []), T124), SCOPE: 10, CLAUSE: 2 | app(T125, .(T126, []), T124), SCOPE: 10, CLAUSE: 3\n\nT124 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: app(T125, .(T126, []), T124), SCOPE: 10, CLAUSE: 2\n\nT124 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: app(T125, .(T126, []), T124), SCOPE: 10, CLAUSE: 3\n\nT124 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: app(T146, .(T147, []), T145)\n\nT145 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="EVAL with clause\nrev([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="EVAL with clause\nrev(.(X103, X104), X105) :- ','(rev(X104, X106), app(X106, .(X103, []), X105)).\nand substitution\nX103 -> T77\nX104 -> T76\nT1 -> .(T77, T76)\nT2 -> T75\nX105 -> T75\nT74 -> T76\nT73 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 83 [arrowhead = none ];
83 -> 43;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 84 [arrowhead = none ];
84 -> 44;

85 [label="EVAL with clause\nrev(.(X4, X5), X6) :- ','(rev(X5, X7), app(X7, .(X4, []), X6)).\nand substitution\nX4 -> T8\nX5 -> T7\nT1 -> .(T8, T7)\nX6 -> []\nT6 -> T7\nT5 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 85 [arrowhead = none ];
85 -> 6;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 86 [arrowhead = none ];
86 -> 7;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 87 [arrowhead = none ];
87 -> 8;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 89 [arrowhead = none ];
89 -> 10;

90 [label="EVAL with clause\nrev([], []).\nand substitution\nT7 -> []\nX7 -> []\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 91 [arrowhead = none ];
91 -> 12;

92 [label="EVAL with clause\nrev(.(X31, X32), X33) :- ','(rev(X32, X34), app(X34, .(X31, []), X33)).\nand substitution\nX31 -> T20\nX32 -> T19\nT7 -> .(T20, T19)\nX7 -> X35\nX33 -> X35\nT18 -> T19\nT17 -> T20\nT8 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 92 [arrowhead = none ];
92 -> 16;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 93 [arrowhead = none ];
93 -> 17;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 94 [arrowhead = none ];
94 -> 13;

95 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 95 [arrowhead = none ];
95 -> 14;

96 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 96 [arrowhead = none ];
96 -> 15;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 97 [arrowhead = none ];
97 -> 18;

98 [label="SPLIT 2\nreplacements:\nX34 -> T22\nT20 -> T23\nT21 -> T24", fontsize=14, style = filled, fillcolor = violet];
16 -> 98 [arrowhead = none ];
98 -> 19;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 99 [arrowhead = none ];
99 -> 20;

100 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 100 [arrowhead = none ];
100 -> 38;

101 [label="SPLIT 2\nreplacements:\nX35 -> T57\nT24 -> T58", fontsize=14, style = filled, fillcolor = violet];
19 -> 101 [arrowhead = none ];
101 -> 39;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 102 [arrowhead = none ];
102 -> 21;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 103 [arrowhead = none ];
103 -> 22;

104 [label="EVAL with clause\nrev([], []).\nand substitution\nT19 -> []\nX34 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 104 [arrowhead = none ];
104 -> 23;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 105 [arrowhead = none ];
105 -> 24;

106 [label="EVAL with clause\nrev(.(X48, X49), X50) :- ','(rev(X49, X51), app(X51, .(X48, []), X50)).\nand substitution\nX48 -> T32\nX49 -> T31\nT19 -> .(T32, T31)\nX34 -> X52\nX50 -> X52\nT30 -> T31\nT29 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 106 [arrowhead = none ];
106 -> 26;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 107 [arrowhead = none ];
107 -> 27;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 108 [arrowhead = none ];
108 -> 25;

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 109 [arrowhead = none ];
109 -> 28;

110 [label="SPLIT 2\nreplacements:\nX51 -> T33\nT32 -> T34", fontsize=14, style = filled, fillcolor = violet];
26 -> 110 [arrowhead = none ];
110 -> 29;

111 [label="INSTANCE with matching:\nT19 -> T31\nX34 -> X51", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 111 [arrowhead = none , style=dashed];
111 -> 18[style=dashed];

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 112 [arrowhead = none ];
112 -> 30;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 113 [arrowhead = none ];
113 -> 31;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 114 [arrowhead = none ];
114 -> 32;

115 [label="EVAL with clause\napp([], X62, X62).\nand substitution\nT33 -> []\nT34 -> T41\nX62 -> .(T41, [])\nX52 -> .(T41, [])", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 115 [arrowhead = none ];
115 -> 33;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 116 [arrowhead = none ];
116 -> 34;

117 [label="EVAL with clause\napp(.(X74, X75), X76, .(X74, X77)) :- app(X75, X76, X77).\nand substitution\nX74 -> T48\nX75 -> T51\nT33 -> .(T48, T51)\nT34 -> T52\nX76 -> .(T52, [])\nX77 -> X78\nX52 -> .(T48, X78)\nT49 -> T51\nT50 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 117 [arrowhead = none ];
117 -> 36;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 118 [arrowhead = none ];
118 -> 37;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 119 [arrowhead = none ];
119 -> 35;

120 [label="INSTANCE with matching:\nT33 -> T51\nT34 -> T52\nX52 -> X78", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 120 [arrowhead = none , style=dashed];
120 -> 29[style=dashed];

121 [label="INSTANCE with matching:\nT33 -> T22\nT34 -> T23\nX52 -> X35", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 121 [arrowhead = none , style=dashed];
121 -> 29[style=dashed];

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 122 [arrowhead = none ];
122 -> 40;

123 [label="BACKTRACK\nfor clause: app([], X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 123 [arrowhead = none ];
123 -> 41;

124 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
41 -> 124 [arrowhead = none ];
124 -> 42;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 125 [arrowhead = none ];
125 -> 45;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 126 [arrowhead = none ];
126 -> 46;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 127 [arrowhead = none ];
127 -> 47;

128 [label="EVAL with clause\nrev([], []).\nand substitution\nT76 -> []\nX106 -> []\nT77 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 128 [arrowhead = none ];
128 -> 48;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="EVAL with clause\nrev(.(X131, X132), X133) :- ','(rev(X132, X134), app(X134, .(X131, []), X133)).\nand substitution\nX131 -> T93\nX132 -> T92\nT76 -> .(T93, T92)\nX106 -> X135\nX133 -> X135\nT91 -> T92\nT90 -> T93\nT77 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 130 [arrowhead = none ];
130 -> 57;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 131 [arrowhead = none ];
131 -> 58;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 132 [arrowhead = none ];
132 -> 50;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 133 [arrowhead = none ];
133 -> 51;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 134 [arrowhead = none ];
134 -> 52;

135 [label="EVAL with clause\napp([], X116, X116).\nand substitution\nT78 -> T85\nX116 -> .(T85, [])\nT75 -> .(T85, [])", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 135 [arrowhead = none ];
135 -> 53;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 136 [arrowhead = none ];
136 -> 54;

137 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 137 [arrowhead = none ];
137 -> 56;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 138 [arrowhead = none ];
138 -> 55;

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 139 [arrowhead = none ];
139 -> 59;

140 [label="SPLIT 2\nreplacements:\nX134 -> T95\nT93 -> T96\nT94 -> T97\nT1 -> T98", fontsize=14, style = filled, fillcolor = violet];
57 -> 140 [arrowhead = none ];
140 -> 60;

141 [label="INSTANCE with matching:\nT19 -> T92\nX34 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 141 [arrowhead = none , style=dashed];
141 -> 18[style=dashed];

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 142 [arrowhead = none ];
142 -> 61;

143 [label="SPLIT 2\nreplacements:\nX135 -> T101\nT97 -> T102\nT98 -> T103", fontsize=14, style = filled, fillcolor = violet];
60 -> 143 [arrowhead = none ];
143 -> 62;

144 [label="INSTANCE with matching:\nT22 -> T95\nT23 -> T96\nX35 -> X135", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 144 [arrowhead = none , style=dashed];
144 -> 38[style=dashed];

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 145 [arrowhead = none ];
145 -> 63;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 146 [arrowhead = none ];
146 -> 64;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 147 [arrowhead = none ];
147 -> 65;

148 [label="EVAL with clause\napp([], X149, X149).\nand substitution\nT101 -> []\nT102 -> T112\nX149 -> .(T112, [])\nT75 -> .(T112, [])", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 148 [arrowhead = none ];
148 -> 66;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 149 [arrowhead = none ];
149 -> 67;

150 [label="EVAL with clause\napp(.(X159, X160), X161, .(X159, X162)) :- app(X160, X161, X162).\nand substitution\nX159 -> T121\nX160 -> T125\nT101 -> .(T121, T125)\nT102 -> T126\nX161 -> .(T126, [])\nX162 -> T124\nT75 -> .(T121, T124)\nT122 -> T125\nT123 -> T126", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 150 [arrowhead = none ];
150 -> 69;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 151 [arrowhead = none ];
151 -> 70;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 152 [arrowhead = none ];
152 -> 68;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 153 [arrowhead = none ];
153 -> 71;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 154 [arrowhead = none ];
154 -> 72;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 155 [arrowhead = none ];
155 -> 73;

156 [label="EVAL with clause\napp([], X173, X173).\nand substitution\nT125 -> []\nT126 -> T133\nX173 -> .(T133, [])\nT124 -> .(T133, [])", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 156 [arrowhead = none ];
156 -> 74;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 157 [arrowhead = none ];
157 -> 75;

158 [label="EVAL with clause\napp(.(X183, X184), X185, .(X183, X186)) :- app(X184, X185, X186).\nand substitution\nX183 -> T142\nX184 -> T146\nT125 -> .(T142, T146)\nT126 -> T147\nX185 -> .(T147, [])\nX186 -> T145\nT124 -> .(T142, T145)\nT143 -> T146\nT144 -> T147", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 158 [arrowhead = none ];
158 -> 77;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 159 [arrowhead = none ];
159 -> 78;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 160 [arrowhead = none ];
160 -> 76;

161 [label="INSTANCE with matching:\nT125 -> T146\nT126 -> T147\nT124 -> T145", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 161 [arrowhead = none , style=dashed];
161 -> 69[style=dashed];

}

(2) Obligation:

Triples:

rev18(.(T32, T31), X52) :- rev18(T31, X51).
rev18(.(T34, T31), X52) :- ','(revc18(T31, T33), app29(T33, T34, X52)).
app29(.(T48, T51), T52, .(T48, X78)) :- app29(T51, T52, X78).
app69(.(T142, T146), T147, .(T142, T145)) :- app69(T146, T147, T145).
rev1(.(T21, .(T20, T19)), []) :- rev18(T19, X34).
rev1(.(T24, .(T23, T19)), []) :- ','(revc18(T19, T22), app29(T22, T23, X35)).
rev1(.(T94, .(T93, T92)), T75) :- rev18(T92, X134).
rev1(.(T97, .(T96, T92)), T75) :- ','(revc18(T92, T95), app29(T95, T96, X135)).
rev1(.(T126, .(T96, T92)), .(T121, T124)) :- ','(revc18(T92, T95), ','(appc29(T95, T96, .(T121, T125)), app69(T125, T126, T124))).

Clauses:

revc18([], []).
revc18(.(T34, T31), X52) :- ','(revc18(T31, T33), appc29(T33, T34, X52)).
appc29([], T41, .(T41, [])).
appc29(.(T48, T51), T52, .(T48, X78)) :- appc29(T51, T52, X78).
appc69([], T133, .(T133, [])).
appc69(.(T142, T146), T147, .(T142, T145)) :- appc69(T146, T147, T145).

Afs:

rev1(x1, x2)  =  rev1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev1_in: (f,b)
rev18_in: (f,f)
revc18_in: (f,f)
appc29_in: (b,f,f)
app29_in: (b,f,f)
app69_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REV1_IN_AG(.(T21, .(T20, T19)), []) → U6_AG(T21, T20, T19, rev18_in_aa(T19, X34))
REV1_IN_AG(.(T21, .(T20, T19)), []) → REV18_IN_AA(T19, X34)
REV18_IN_AA(.(T32, T31), X52) → U1_AA(T32, T31, X52, rev18_in_aa(T31, X51))
REV18_IN_AA(.(T32, T31), X52) → REV18_IN_AA(T31, X51)
REV18_IN_AA(.(T34, T31), X52) → U2_AA(T34, T31, X52, revc18_in_aa(T31, T33))
U2_AA(T34, T31, X52, revc18_out_aa(T31, T33)) → U3_AA(T34, T31, X52, app29_in_gaa(T33, T34, X52))
U2_AA(T34, T31, X52, revc18_out_aa(T31, T33)) → APP29_IN_GAA(T33, T34, X52)
APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → U4_GAA(T48, T51, T52, X78, app29_in_gaa(T51, T52, X78))
APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → APP29_IN_GAA(T51, T52, X78)
REV1_IN_AG(.(T24, .(T23, T19)), []) → U7_AG(T24, T23, T19, revc18_in_aa(T19, T22))
U7_AG(T24, T23, T19, revc18_out_aa(T19, T22)) → U8_AG(T24, T23, T19, app29_in_gaa(T22, T23, X35))
U7_AG(T24, T23, T19, revc18_out_aa(T19, T22)) → APP29_IN_GAA(T22, T23, X35)
REV1_IN_AG(.(T94, .(T93, T92)), T75) → U9_AG(T94, T93, T92, T75, rev18_in_aa(T92, X134))
REV1_IN_AG(.(T94, .(T93, T92)), T75) → REV18_IN_AA(T92, X134)
REV1_IN_AG(.(T97, .(T96, T92)), T75) → U10_AG(T97, T96, T92, T75, revc18_in_aa(T92, T95))
U10_AG(T97, T96, T92, T75, revc18_out_aa(T92, T95)) → U11_AG(T97, T96, T92, T75, app29_in_gaa(T95, T96, X135))
U10_AG(T97, T96, T92, T75, revc18_out_aa(T92, T95)) → APP29_IN_GAA(T95, T96, X135)
REV1_IN_AG(.(T126, .(T96, T92)), .(T121, T124)) → U12_AG(T126, T96, T92, T121, T124, revc18_in_aa(T92, T95))
U12_AG(T126, T96, T92, T121, T124, revc18_out_aa(T92, T95)) → U13_AG(T126, T96, T92, T121, T124, appc29_in_gaa(T95, T96, .(T121, T125)))
U13_AG(T126, T96, T92, T121, T124, appc29_out_gaa(T95, T96, .(T121, T125))) → U14_AG(T126, T96, T92, T121, T124, app69_in_gag(T125, T126, T124))
U13_AG(T126, T96, T92, T121, T124, appc29_out_gaa(T95, T96, .(T121, T125))) → APP69_IN_GAG(T125, T126, T124)
APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → U5_GAG(T142, T146, T147, T145, app69_in_gag(T146, T147, T145))
APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → APP69_IN_GAG(T146, T147, T145)

The TRS R consists of the following rules:

revc18_in_aa([], []) → revc18_out_aa([], [])
revc18_in_aa(.(T34, T31), X52) → U16_aa(T34, T31, X52, revc18_in_aa(T31, T33))
U16_aa(T34, T31, X52, revc18_out_aa(T31, T33)) → U17_aa(T34, T31, X52, appc29_in_gaa(T33, T34, X52))
appc29_in_gaa([], T41, .(T41, [])) → appc29_out_gaa([], T41, .(T41, []))
appc29_in_gaa(.(T48, T51), T52, .(T48, X78)) → U18_gaa(T48, T51, T52, X78, appc29_in_gaa(T51, T52, X78))
U18_gaa(T48, T51, T52, X78, appc29_out_gaa(T51, T52, X78)) → appc29_out_gaa(.(T48, T51), T52, .(T48, X78))
U17_aa(T34, T31, X52, appc29_out_gaa(T33, T34, X52)) → revc18_out_aa(.(T34, T31), X52)

The argument filtering Pi contains the following mapping:
[]  =  []
rev18_in_aa(x1, x2)  =  rev18_in_aa
revc18_in_aa(x1, x2)  =  revc18_in_aa
revc18_out_aa(x1, x2)  =  revc18_out_aa(x1, x2)
U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
appc29_in_gaa(x1, x2, x3)  =  appc29_in_gaa(x1)
appc29_out_gaa(x1, x2, x3)  =  appc29_out_gaa(x1, x3)
.(x1, x2)  =  .(x2)
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
app29_in_gaa(x1, x2, x3)  =  app29_in_gaa(x1)
app69_in_gag(x1, x2, x3)  =  app69_in_gag(x1, x3)
REV1_IN_AG(x1, x2)  =  REV1_IN_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
REV18_IN_AA(x1, x2)  =  REV18_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
APP29_IN_GAA(x1, x2, x3)  =  APP29_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U8_AG(x1, x2, x3, x4)  =  U8_AG(x3, x4)
U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
U12_AG(x1, x2, x3, x4, x5, x6)  =  U12_AG(x5, x6)
U13_AG(x1, x2, x3, x4, x5, x6)  =  U13_AG(x3, x5, x6)
U14_AG(x1, x2, x3, x4, x5, x6)  =  U14_AG(x3, x5, x6)
APP69_IN_GAG(x1, x2, x3)  =  APP69_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)

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:

REV1_IN_AG(.(T21, .(T20, T19)), []) → U6_AG(T21, T20, T19, rev18_in_aa(T19, X34))
REV1_IN_AG(.(T21, .(T20, T19)), []) → REV18_IN_AA(T19, X34)
REV18_IN_AA(.(T32, T31), X52) → U1_AA(T32, T31, X52, rev18_in_aa(T31, X51))
REV18_IN_AA(.(T32, T31), X52) → REV18_IN_AA(T31, X51)
REV18_IN_AA(.(T34, T31), X52) → U2_AA(T34, T31, X52, revc18_in_aa(T31, T33))
U2_AA(T34, T31, X52, revc18_out_aa(T31, T33)) → U3_AA(T34, T31, X52, app29_in_gaa(T33, T34, X52))
U2_AA(T34, T31, X52, revc18_out_aa(T31, T33)) → APP29_IN_GAA(T33, T34, X52)
APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → U4_GAA(T48, T51, T52, X78, app29_in_gaa(T51, T52, X78))
APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → APP29_IN_GAA(T51, T52, X78)
REV1_IN_AG(.(T24, .(T23, T19)), []) → U7_AG(T24, T23, T19, revc18_in_aa(T19, T22))
U7_AG(T24, T23, T19, revc18_out_aa(T19, T22)) → U8_AG(T24, T23, T19, app29_in_gaa(T22, T23, X35))
U7_AG(T24, T23, T19, revc18_out_aa(T19, T22)) → APP29_IN_GAA(T22, T23, X35)
REV1_IN_AG(.(T94, .(T93, T92)), T75) → U9_AG(T94, T93, T92, T75, rev18_in_aa(T92, X134))
REV1_IN_AG(.(T94, .(T93, T92)), T75) → REV18_IN_AA(T92, X134)
REV1_IN_AG(.(T97, .(T96, T92)), T75) → U10_AG(T97, T96, T92, T75, revc18_in_aa(T92, T95))
U10_AG(T97, T96, T92, T75, revc18_out_aa(T92, T95)) → U11_AG(T97, T96, T92, T75, app29_in_gaa(T95, T96, X135))
U10_AG(T97, T96, T92, T75, revc18_out_aa(T92, T95)) → APP29_IN_GAA(T95, T96, X135)
REV1_IN_AG(.(T126, .(T96, T92)), .(T121, T124)) → U12_AG(T126, T96, T92, T121, T124, revc18_in_aa(T92, T95))
U12_AG(T126, T96, T92, T121, T124, revc18_out_aa(T92, T95)) → U13_AG(T126, T96, T92, T121, T124, appc29_in_gaa(T95, T96, .(T121, T125)))
U13_AG(T126, T96, T92, T121, T124, appc29_out_gaa(T95, T96, .(T121, T125))) → U14_AG(T126, T96, T92, T121, T124, app69_in_gag(T125, T126, T124))
U13_AG(T126, T96, T92, T121, T124, appc29_out_gaa(T95, T96, .(T121, T125))) → APP69_IN_GAG(T125, T126, T124)
APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → U5_GAG(T142, T146, T147, T145, app69_in_gag(T146, T147, T145))
APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → APP69_IN_GAG(T146, T147, T145)

The TRS R consists of the following rules:

revc18_in_aa([], []) → revc18_out_aa([], [])
revc18_in_aa(.(T34, T31), X52) → U16_aa(T34, T31, X52, revc18_in_aa(T31, T33))
U16_aa(T34, T31, X52, revc18_out_aa(T31, T33)) → U17_aa(T34, T31, X52, appc29_in_gaa(T33, T34, X52))
appc29_in_gaa([], T41, .(T41, [])) → appc29_out_gaa([], T41, .(T41, []))
appc29_in_gaa(.(T48, T51), T52, .(T48, X78)) → U18_gaa(T48, T51, T52, X78, appc29_in_gaa(T51, T52, X78))
U18_gaa(T48, T51, T52, X78, appc29_out_gaa(T51, T52, X78)) → appc29_out_gaa(.(T48, T51), T52, .(T48, X78))
U17_aa(T34, T31, X52, appc29_out_gaa(T33, T34, X52)) → revc18_out_aa(.(T34, T31), X52)

The argument filtering Pi contains the following mapping:
[]  =  []
rev18_in_aa(x1, x2)  =  rev18_in_aa
revc18_in_aa(x1, x2)  =  revc18_in_aa
revc18_out_aa(x1, x2)  =  revc18_out_aa(x1, x2)
U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
appc29_in_gaa(x1, x2, x3)  =  appc29_in_gaa(x1)
appc29_out_gaa(x1, x2, x3)  =  appc29_out_gaa(x1, x3)
.(x1, x2)  =  .(x2)
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
app29_in_gaa(x1, x2, x3)  =  app29_in_gaa(x1)
app69_in_gag(x1, x2, x3)  =  app69_in_gag(x1, x3)
REV1_IN_AG(x1, x2)  =  REV1_IN_AG(x2)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
REV18_IN_AA(x1, x2)  =  REV18_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
APP29_IN_GAA(x1, x2, x3)  =  APP29_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U8_AG(x1, x2, x3, x4)  =  U8_AG(x3, x4)
U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
U12_AG(x1, x2, x3, x4, x5, x6)  =  U12_AG(x5, x6)
U13_AG(x1, x2, x3, x4, x5, x6)  =  U13_AG(x3, x5, x6)
U14_AG(x1, x2, x3, x4, x5, x6)  =  U14_AG(x3, x5, x6)
APP69_IN_GAG(x1, x2, x3)  =  APP69_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 20 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → APP69_IN_GAG(T146, T147, T145)

The TRS R consists of the following rules:

revc18_in_aa([], []) → revc18_out_aa([], [])
revc18_in_aa(.(T34, T31), X52) → U16_aa(T34, T31, X52, revc18_in_aa(T31, T33))
U16_aa(T34, T31, X52, revc18_out_aa(T31, T33)) → U17_aa(T34, T31, X52, appc29_in_gaa(T33, T34, X52))
appc29_in_gaa([], T41, .(T41, [])) → appc29_out_gaa([], T41, .(T41, []))
appc29_in_gaa(.(T48, T51), T52, .(T48, X78)) → U18_gaa(T48, T51, T52, X78, appc29_in_gaa(T51, T52, X78))
U18_gaa(T48, T51, T52, X78, appc29_out_gaa(T51, T52, X78)) → appc29_out_gaa(.(T48, T51), T52, .(T48, X78))
U17_aa(T34, T31, X52, appc29_out_gaa(T33, T34, X52)) → revc18_out_aa(.(T34, T31), X52)

The argument filtering Pi contains the following mapping:
[]  =  []
revc18_in_aa(x1, x2)  =  revc18_in_aa
revc18_out_aa(x1, x2)  =  revc18_out_aa(x1, x2)
U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
appc29_in_gaa(x1, x2, x3)  =  appc29_in_gaa(x1)
appc29_out_gaa(x1, x2, x3)  =  appc29_out_gaa(x1, x3)
.(x1, x2)  =  .(x2)
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
APP69_IN_GAG(x1, x2, x3)  =  APP69_IN_GAG(x1, x3)

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:

APP69_IN_GAG(.(T142, T146), T147, .(T142, T145)) → APP69_IN_GAG(T146, T147, T145)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP69_IN_GAG(x1, x2, x3)  =  APP69_IN_GAG(x1, x3)

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

(10) PiDPToQDPProof (SOUND 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:

APP69_IN_GAG(.(T146), .(T145)) → APP69_IN_GAG(T146, T145)

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:

• APP69_IN_GAG(.(T146), .(T145)) → APP69_IN_GAG(T146, T145)
  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:

APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → APP29_IN_GAA(T51, T52, X78)

The TRS R consists of the following rules:

revc18_in_aa([], []) → revc18_out_aa([], [])
revc18_in_aa(.(T34, T31), X52) → U16_aa(T34, T31, X52, revc18_in_aa(T31, T33))
U16_aa(T34, T31, X52, revc18_out_aa(T31, T33)) → U17_aa(T34, T31, X52, appc29_in_gaa(T33, T34, X52))
appc29_in_gaa([], T41, .(T41, [])) → appc29_out_gaa([], T41, .(T41, []))
appc29_in_gaa(.(T48, T51), T52, .(T48, X78)) → U18_gaa(T48, T51, T52, X78, appc29_in_gaa(T51, T52, X78))
U18_gaa(T48, T51, T52, X78, appc29_out_gaa(T51, T52, X78)) → appc29_out_gaa(.(T48, T51), T52, .(T48, X78))
U17_aa(T34, T31, X52, appc29_out_gaa(T33, T34, X52)) → revc18_out_aa(.(T34, T31), X52)

The argument filtering Pi contains the following mapping:
[]  =  []
revc18_in_aa(x1, x2)  =  revc18_in_aa
revc18_out_aa(x1, x2)  =  revc18_out_aa(x1, x2)
U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
appc29_in_gaa(x1, x2, x3)  =  appc29_in_gaa(x1)
appc29_out_gaa(x1, x2, x3)  =  appc29_out_gaa(x1, x3)
.(x1, x2)  =  .(x2)
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
APP29_IN_GAA(x1, x2, x3)  =  APP29_IN_GAA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APP29_IN_GAA(.(T48, T51), T52, .(T48, X78)) → APP29_IN_GAA(T51, T52, X78)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP29_IN_GAA(x1, x2, x3)  =  APP29_IN_GAA(x1)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APP29_IN_GAA(.(T51)) → APP29_IN_GAA(T51)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APP29_IN_GAA(.(T51)) → APP29_IN_GAA(T51)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

REV18_IN_AA(.(T32, T31), X52) → REV18_IN_AA(T31, X51)

The TRS R consists of the following rules:

revc18_in_aa([], []) → revc18_out_aa([], [])
revc18_in_aa(.(T34, T31), X52) → U16_aa(T34, T31, X52, revc18_in_aa(T31, T33))
U16_aa(T34, T31, X52, revc18_out_aa(T31, T33)) → U17_aa(T34, T31, X52, appc29_in_gaa(T33, T34, X52))
appc29_in_gaa([], T41, .(T41, [])) → appc29_out_gaa([], T41, .(T41, []))
appc29_in_gaa(.(T48, T51), T52, .(T48, X78)) → U18_gaa(T48, T51, T52, X78, appc29_in_gaa(T51, T52, X78))
U18_gaa(T48, T51, T52, X78, appc29_out_gaa(T51, T52, X78)) → appc29_out_gaa(.(T48, T51), T52, .(T48, X78))
U17_aa(T34, T31, X52, appc29_out_gaa(T33, T34, X52)) → revc18_out_aa(.(T34, T31), X52)

The argument filtering Pi contains the following mapping:
[]  =  []
revc18_in_aa(x1, x2)  =  revc18_in_aa
revc18_out_aa(x1, x2)  =  revc18_out_aa(x1, x2)
U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
appc29_in_gaa(x1, x2, x3)  =  appc29_in_gaa(x1)
appc29_out_gaa(x1, x2, x3)  =  appc29_out_gaa(x1, x3)
.(x1, x2)  =  .(x2)
U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
REV18_IN_AA(x1, x2)  =  REV18_IN_AA

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:

REV18_IN_AA(.(T32, T31), X52) → REV18_IN_AA(T31, X51)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
REV18_IN_AA(x1, x2)  =  REV18_IN_AA

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:

REV18_IN_AA → REV18_IN_AA

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

(26) NonTerminationProof (EQUIVALENT transformation)

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

s = REV18_IN_AA evaluates to t =REV18_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV18_IN_AA to REV18_IN_AA.

(27) NO