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

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), append(Xs, X1, X2)).
append([], Y, Z) :- ','(!, eq(Y, Z)).
append(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), append(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Query: goal(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T3, X9), append(X9, X10, X11))\n\nT3 is ground\nX9 is free\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2l(T3, X9), append(X9, X10, X11)), SCOPE: 2, CLAUSE: 3 | ','(s2l(T3, X9), append(X9, X10, X11)), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX9 is free\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(','(!_2, eq(X17, [])), append(X17, X10, X11)) | ','(s2l(0, X9), append(X9, X10, X11)), SCOPE: 2, CLAUSE: 4\n\nX9 is free\nX10 is free\nX11 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2l(T3, X9), append(X9, X10, X11)), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX9 is free\nX10 is free\nX11 is free\nX16 is free\ns2l(T3, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X17, []), append(X17, X10, X11))\n\nX10 is free\nX11 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(X17, []), append(X17, X10, X11)), SCOPE: 3, CLAUSE: 11\n\nX10 is free\nX11 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: append([], X10, X11)\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: append([], X10, X11), SCOPE: 4, CLAUSE: 1 | append([], X10, X11), SCOPE: 4, CLAUSE: 2\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(!_4, eq(X46, X47)) | append([], X10, X11), SCOPE: 4, CLAUSE: 2\n\nX10 is free\nX11 is free\nX46 is free\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(X46, X47)\n\nX46 is free\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: eq(X46, X47), SCOPE: 5, CLAUSE: 11\n\nX46 is free\nX47 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(p(T6, X70), s2l(X70, X72)), append(.(X71, X72), X10, X11))\n\nT6 is ground\nX9 is free\nX10 is free\nX11 is free\nX16 is free\nX71 is free\nX72 is free\nX70 is free\ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(','(p(T6, X70), s2l(X70, X72)), append(.(X71, X72), X10, X11)), SCOPE: 6, CLAUSE: 9 | ','(','(p(T6, X70), s2l(X70, X72)), append(.(X71, X72), X10, X11)), SCOPE: 6, CLAUSE: 10\n\nT6 is ground\nX9 is free\nX10 is free\nX11 is free\nX16 is free\nX71 is free\nX72 is free\nX70 is free\ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(p(T6, X70), s2l(X70, X72)), append(.(X71, X72), X10, X11)), SCOPE: 6, CLAUSE: 10\n\nT6 is ground\nX9 is free\nX10 is free\nX11 is free\nX16 is free\nX71 is free\nX72 is free\nX70 is free\ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(s2l(T9, X72), append(.(X71, X72), X10, X11))\n\nT9 is ground\nX10 is free\nX11 is free\nX71 is free\nX72 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2l(T9, X72)\n\nT9 is ground\nX72 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append(.(X71, T10), X10, X11)\n\nX10 is free\nX11 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: s2l(T9, X72), SCOPE: 7, CLAUSE: 3 | s2l(T9, X72), SCOPE: 7, CLAUSE: 4\n\nT9 is ground\nX72 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_7, eq(X87, [])) | s2l(0, X72), SCOPE: 7, CLAUSE: 4\n\nX72 is free\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2l(T9, X72), SCOPE: 7, CLAUSE: 4\n\nT9 is ground\nX72 is free\nX86 is free\ns2l(T9, X72) !=? s2l(0, X86)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: eq(X87, [])\n\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: eq(X87, []), SCOPE: 8, CLAUSE: 11\n\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T13, X110), s2l(X110, X112))\n\nT13 is ground\nX72 is free\nX86 is free\nX112 is free\nX110 is free\ns2l(T13, X72) !=? s2l(0, X86)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(T13, X110), s2l(X110, X112)), SCOPE: 9, CLAUSE: 9 | ','(p(T13, X110), s2l(X110, X112)), SCOPE: 9, CLAUSE: 10\n\nT13 is ground\nX72 is free\nX86 is free\nX112 is free\nX110 is free\ns2l(T13, X72) !=? s2l(0, X86)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T13, X110), s2l(X110, X112)), SCOPE: 9, CLAUSE: 10\n\nT13 is ground\nX72 is free\nX86 is free\nX112 is free\nX110 is free\ns2l(T13, X72) !=? s2l(0, X86)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: s2l(T16, X112)\n\nT16 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: append(.(X71, T10), X10, X11), SCOPE: 10, CLAUSE: 1 | append(.(X71, T10), X10, X11), SCOPE: 10, CLAUSE: 2\n\nX10 is free\nX11 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: append(.(X71, T10), X10, X11), SCOPE: 10, CLAUSE: 2\n\nX10 is free\nX11 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(head(.(X150, T21), X152), ','(tail(.(X150, T21), X149), append(X149, X151, X153)))\n\nX150 is free\nX151 is free\nX152 is free\nX153 is free\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(.(X150, T21), X152), ','(tail(.(X150, T21), X149), append(X149, X151, X153))), SCOPE: 11, CLAUSE: 5 | ','(head(.(X150, T21), X152), ','(tail(.(X150, T21), X149), append(X149, X151, X153))), SCOPE: 11, CLAUSE: 6\n\nX150 is free\nX151 is free\nX152 is free\nX153 is free\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(head(.(X150, T21), X152), ','(tail(.(X150, T21), X149), append(X149, X151, X153))), SCOPE: 11, CLAUSE: 6\n\nX150 is free\nX151 is free\nX152 is free\nX153 is free\nX149 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(.(X171, T26), X149), append(X149, X151, X153))\n\nX151 is free\nX153 is free\nX149 is free\nX171 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(.(X171, T26), X149), append(X149, X151, X153)), SCOPE: 12, CLAUSE: 7 | ','(tail(.(X171, T26), X149), append(X149, X151, X153)), SCOPE: 12, CLAUSE: 8\n\nX151 is free\nX153 is free\nX149 is free\nX171 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(.(X171, T26), X149), append(X149, X151, X153)), SCOPE: 12, CLAUSE: 8\n\nX151 is free\nX153 is free\nX149 is free\nX171 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append(T31, X151, X153)\n\nX151 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append(T31, X151, X153), SCOPE: 13, CLAUSE: 1 | append(T31, X151, X153), SCOPE: 13, CLAUSE: 2\n\nX151 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_13, eq(X203, X204)) | append(T31, X151, X153), SCOPE: 13, CLAUSE: 2\n\nX151 is free\nX153 is free\nX203 is free\nX204 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append(T31, X151, X153), SCOPE: 13, CLAUSE: 2\n\nX151 is free\nX153 is free\nX201 is free\nX202 is free\nappend(T31, X151, X153) !=? append([], X201, X202)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(X203, X204)\n\nX203 is free\nX204 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(X203, X204), SCOPE: 14, CLAUSE: 11\n\nX203 is free\nX204 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(head(T36, X236), ','(tail(T36, X234), append(X234, X235, X237)))\n\nX235 is free\nX236 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T36, X236), ','(tail(T36, X234), append(X234, X235, X237))), SCOPE: 15, CLAUSE: 5 | ','(head(T36, X236), ','(tail(T36, X234), append(X234, X235, X237))), SCOPE: 15, CLAUSE: 6\n\nX235 is free\nX236 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(head(T36, X236), ','(tail(T36, X234), append(X234, X235, X237))), SCOPE: 15, CLAUSE: 5\n\nX235 is free\nX236 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T36, X236), ','(tail(T36, X234), append(X234, X235, X237))), SCOPE: 15, CLAUSE: 6\n\nX235 is free\nX236 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail([], X234), append(X234, X235, X237))\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail([], X234), append(X234, X235, X237)), SCOPE: 16, CLAUSE: 7 | ','(tail([], X234), append(X234, X235, X237)), SCOPE: 16, CLAUSE: 8\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail([], X234), append(X234, X235, X237)), SCOPE: 16, CLAUSE: 7\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail([], X234), append(X234, X235, X237)), SCOPE: 16, CLAUSE: 8\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: append([], X235, X237)\n\nX235 is free\nX237 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(.(T43, T44), X234), append(X234, X235, X237))\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(.(T43, T44), X234), append(X234, X235, X237)), SCOPE: 17, CLAUSE: 7 | ','(tail(.(T43, T44), X234), append(X234, X235, X237)), SCOPE: 17, CLAUSE: 8\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(T43, T44), X234), append(X234, X235, X237)), SCOPE: 17, CLAUSE: 8\n\nX235 is free\nX237 is free\nX234 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: append(T52, X235, X237)\n\nX235 is free\nX237 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 67 [arrowhead = none ];
67 -> 2;

68 [label="ONLY EVAL with clause\ngoal(X8) :- ','(s2l(X8, X9), append(X9, X10, X11)).\nand substitutionT1 -> T3,\nX8 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 68 [arrowhead = none ];
68 -> 3;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 69 [arrowhead = none ];
69 -> 4;

70 [label="EVAL with clause\ns2l(0, X16) :- ','(!_2, eq(X16, [])).\nand substitutionT3 -> 0,\nX9 -> X17,\nX16 -> X17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 70 [arrowhead = none ];
70 -> 5;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 71 [arrowhead = none ];
71 -> 6;

72 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 72 [arrowhead = none ];
72 -> 7;

73 [label="ONLY EVAL with clause\ns2l(X67, .(X68, X69)) :- ','(p(X67, X70), s2l(X70, X69)).\nand substitutionT3 -> T6,\nX67 -> T6,\nX68 -> X71,\nX69 -> X72,\nX9 -> .(X71, X72)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 73 [arrowhead = none ];
73 -> 16;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 74 [arrowhead = none ];
74 -> 8;

75 [label="ONLY EVAL with clause\neq(X30, X30).\nand substitutionX17 -> [],\nX30 -> [],\nX31 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 75 [arrowhead = none ];
75 -> 9;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="ONLY EVAL with clause\nappend([], X44, X45) :- ','(!_4, eq(X44, X45)).\nand substitutionX10 -> X46,\nX44 -> X46,\nX11 -> X47,\nX45 -> X47", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="ONLY EVAL with clause\neq(X52, X52).\nand substitutionX46 -> X53,\nX52 -> X53,\nX47 -> X53", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 80 [arrowhead = none ];
80 -> 14;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 81 [arrowhead = none ];
81 -> 15;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 82 [arrowhead = none ];
82 -> 17;

83 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T6, X9), s2l(0, X16))", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 83 [arrowhead = none ];
83 -> 18;

84 [label="EVAL with clause\np(s(X77), X77).\nand substitutionX77 -> T9,\nT6 -> s(T9),\nX70 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 84 [arrowhead = none ];
84 -> 19;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 85 [arrowhead = none ];
85 -> 20;

86 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 86 [arrowhead = none ];
86 -> 21;

87 [label="SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X72 -> T10", fontsize=14, style = filled, fillcolor = violet];
19 -> 87 [arrowhead = none ];
87 -> 22;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 88 [arrowhead = none ];
88 -> 23;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="EVAL with clause\ns2l(0, X86) :- ','(!_7, eq(X86, [])).\nand substitutionT9 -> 0,\nX72 -> X87,\nX86 -> X87", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 90 [arrowhead = none ];
90 -> 24;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 91 [arrowhead = none ];
91 -> 25;

92 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 92 [arrowhead = none ];
92 -> 26;

93 [label="ONLY EVAL with clause\ns2l(X107, .(X108, X109)) :- ','(p(X107, X110), s2l(X110, X109)).\nand substitutionT9 -> T13,\nX107 -> T13,\nX108 -> X111,\nX109 -> X112,\nX72 -> .(X111, X112)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 93 [arrowhead = none ];
93 -> 30;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 94 [arrowhead = none ];
94 -> 27;

95 [label="ONLY EVAL with clause\neq(X92, X92).\nand substitutionX87 -> [],\nX92 -> [],\nX93 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 95 [arrowhead = none ];
95 -> 28;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 96 [arrowhead = none ];
96 -> 29;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 97 [arrowhead = none ];
97 -> 31;

98 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T13, X72), s2l(0, X86))", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 98 [arrowhead = none ];
98 -> 32;

99 [label="EVAL with clause\np(s(X117), X117).\nand substitutionX117 -> T16,\nT13 -> s(T16),\nX110 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 99 [arrowhead = none ];
99 -> 33;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 100 [arrowhead = none ];
100 -> 34;

101 [label="INSTANCE with matching:\nT9 -> T16\nX72 -> X112", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 101 [arrowhead = none , style=dashed];
101 -> 21[style=dashed];

102 [label="BACKTRACK\nfor clause: append([], Y, Z) :- ','(!, eq(Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 102 [arrowhead = none ];
102 -> 36;

103 [label="ONLY EVAL with clause\nappend(X145, X146, .(X147, X148)) :- ','(head(X145, X147), ','(tail(X145, X149), append(X149, X146, X148))).\nand substitutionX71 -> X150,\nT10 -> T21,\nX145 -> .(X150, T21),\nX10 -> X151,\nX146 -> X151,\nX147 -> X152,\nX148 -> X153,\nX11 -> .(X152, X153),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 103 [arrowhead = none ];
103 -> 37;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 104 [arrowhead = none ];
104 -> 38;

105 [label="BACKTRACK\nfor clause: head([], X4)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 105 [arrowhead = none ];
105 -> 39;

106 [label="ONLY EVAL with clause\nhead(.(X169, X170), X169).\nand substitutionX150 -> X171,\nX169 -> X171,\nT21 -> T26,\nX170 -> T26,\nX152 -> X171,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 106 [arrowhead = none ];
106 -> 40;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 107 [arrowhead = none ];
107 -> 41;

108 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="ONLY EVAL with clause\ntail(.(X186, X187), X187).\nand substitutionX171 -> X188,\nX186 -> X188,\nT26 -> T31,\nX187 -> T31,\nX149 -> T31,\nT30 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 110 [arrowhead = none ];
110 -> 44;

111 [label="EVAL with clause\nappend([], X201, X202) :- ','(!_13, eq(X201, X202)).\nand substitutionT31 -> [],\nX151 -> X203,\nX201 -> X203,\nX153 -> X204,\nX202 -> X204", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 112 [arrowhead = none ];
112 -> 46;

113 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 113 [arrowhead = none ];
113 -> 47;

114 [label="ONLY EVAL with clause\nappend(X230, X231, .(X232, X233)) :- ','(head(X230, X232), ','(tail(X230, X234), append(X234, X231, X233))).\nand substitutionT31 -> T36,\nX230 -> T36,\nX151 -> X235,\nX231 -> X235,\nX232 -> X236,\nX233 -> X237,\nX153 -> .(X236, X237),\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 114 [arrowhead = none ];
114 -> 51;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 115 [arrowhead = none ];
115 -> 48;

116 [label="ONLY EVAL with clause\neq(X209, X209).\nand substitutionX203 -> X210,\nX209 -> X210,\nX204 -> X210", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 116 [arrowhead = none ];
116 -> 49;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 117 [arrowhead = none ];
117 -> 50;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 118 [arrowhead = none ];
118 -> 52;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 119 [arrowhead = none ];
119 -> 53;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 120 [arrowhead = none ];
120 -> 54;

121 [label="EVAL with clause\nhead([], X250).\nand substitutionT36 -> [],\nX236 -> X251,\nX250 -> X251", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 121 [arrowhead = none ];
121 -> 55;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 122 [arrowhead = none ];
122 -> 56;

123 [label="EVAL with clause\nhead(.(X274, X275), X274).\nand substitutionX274 -> T43,\nX275 -> T44,\nT36 -> .(T43, T44),\nX236 -> T43,\nT41 -> T43,\nT42 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 123 [arrowhead = none ];
123 -> 62;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 124 [arrowhead = none ];
124 -> 63;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 125 [arrowhead = none ];
125 -> 57;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 126 [arrowhead = none ];
126 -> 58;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 127 [arrowhead = none ];
127 -> 59;

128 [label="ONLY EVAL with clause\ntail([], []).\nand substitutionX234 -> []", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 128 [arrowhead = none ];
128 -> 60;

129 [label="BACKTRACK\nfor clause: tail(.(X6, Xs), Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 129 [arrowhead = none ];
129 -> 61;

130 [label="INSTANCE with matching:\nX10 -> X235\nX11 -> X237", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 130 [arrowhead = none , style=dashed];
130 -> 9[style=dashed];

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 131 [arrowhead = none ];
131 -> 64;

132 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 132 [arrowhead = none ];
132 -> 65;

133 [label="ONLY EVAL with clause\ntail(.(X288, X289), X289).\nand substitutionT43 -> T50,\nX288 -> T50,\nT44 -> T52,\nX289 -> T52,\nX234 -> T52,\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 133 [arrowhead = none ];
133 -> 66;

134 [label="INSTANCE with matching:\nT31 -> T52\nX151 -> X235\nX153 -> X237", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 134 [arrowhead = none , style=dashed];
134 -> 43[style=dashed];

}

(2) Obligation:

Clauses:

s2lA(0, []).
s2lA(s(T16), .(X111, X112)) :- s2lA(T16, X112).
appendB(X53, X53).
appendC([], X210, X210).
appendC([], X235, .(X251, X237)) :- appendB(X235, X237).
appendC(.(T50, T52), X235, .(T50, X237)) :- appendC(T52, X235, X237).
goalD(0) :- appendB(X10, X11).
goalD(s(T9)) :- s2lA(T9, X72).
goalD(s(T9)) :- ','(s2lA(T9, T31), appendC(T31, X151, X153)).

Query: goalD(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalD_in: (b)
s2lA_in: (b,f)
appendC_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOALD_IN_G(0) → U4_G(appendB_in_aa(X10, X11))
GOALD_IN_G(0) → APPENDB_IN_AA(X10, X11)
GOALD_IN_G(s(T9)) → U5_G(T9, s2lA_in_ga(T9, X72))
GOALD_IN_G(s(T9)) → S2LA_IN_GA(T9, X72)
S2LA_IN_GA(s(T16), .(X111, X112)) → U1_GA(T16, X111, X112, s2lA_in_ga(T16, X112))
S2LA_IN_GA(s(T16), .(X111, X112)) → S2LA_IN_GA(T16, X112)
GOALD_IN_G(s(T9)) → U6_G(T9, s2lA_in_ga(T9, T31))
U6_G(T9, s2lA_out_ga(T9, T31)) → U7_G(T9, appendC_in_gaa(T31, X151, X153))
U6_G(T9, s2lA_out_ga(T9, T31)) → APPENDC_IN_GAA(T31, X151, X153)
APPENDC_IN_GAA([], X235, .(X251, X237)) → U2_GAA(X235, X251, X237, appendB_in_aa(X235, X237))
APPENDC_IN_GAA([], X235, .(X251, X237)) → APPENDB_IN_AA(X235, X237)
APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → U3_GAA(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → APPENDC_IN_GAA(T52, X235, X237)

The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U4_G(x1)  =  U4_G(x1)
APPENDB_IN_AA(x1, x2)  =  APPENDB_IN_AA
U5_G(x1, x2)  =  U5_G(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(6) Obligation:

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

GOALD_IN_G(0) → U4_G(appendB_in_aa(X10, X11))
GOALD_IN_G(0) → APPENDB_IN_AA(X10, X11)
GOALD_IN_G(s(T9)) → U5_G(T9, s2lA_in_ga(T9, X72))
GOALD_IN_G(s(T9)) → S2LA_IN_GA(T9, X72)
S2LA_IN_GA(s(T16), .(X111, X112)) → U1_GA(T16, X111, X112, s2lA_in_ga(T16, X112))
S2LA_IN_GA(s(T16), .(X111, X112)) → S2LA_IN_GA(T16, X112)
GOALD_IN_G(s(T9)) → U6_G(T9, s2lA_in_ga(T9, T31))
U6_G(T9, s2lA_out_ga(T9, T31)) → U7_G(T9, appendC_in_gaa(T31, X151, X153))
U6_G(T9, s2lA_out_ga(T9, T31)) → APPENDC_IN_GAA(T31, X151, X153)
APPENDC_IN_GAA([], X235, .(X251, X237)) → U2_GAA(X235, X251, X237, appendB_in_aa(X235, X237))
APPENDC_IN_GAA([], X235, .(X251, X237)) → APPENDB_IN_AA(X235, X237)
APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → U3_GAA(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → APPENDC_IN_GAA(T52, X235, X237)

The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U4_G(x1)  =  U4_G(x1)
APPENDB_IN_AA(x1, x2)  =  APPENDB_IN_AA
U5_G(x1, x2)  =  U5_G(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → APPENDC_IN_GAA(T52, X235, X237)

The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPENDC_IN_GAA(.(T50, T52), X235, .(T50, X237)) → APPENDC_IN_GAA(T52, X235, X237)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPENDC_IN_GAA(.(T52)) → APPENDC_IN_GAA(T52)

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

(14) 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:

• APPENDC_IN_GAA(.(T52)) → APPENDC_IN_GAA(T52)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2LA_IN_GA(s(T16), .(X111, X112)) → S2LA_IN_GA(T16, X112)

The TRS R consists of the following rules:

goalD_in_g(0) → U4_g(appendB_in_aa(X10, X11))
appendB_in_aa(X53, X53) → appendB_out_aa(X53, X53)
U4_g(appendB_out_aa(X10, X11)) → goalD_out_g(0)
goalD_in_g(s(T9)) → U5_g(T9, s2lA_in_ga(T9, X72))
s2lA_in_ga(0, []) → s2lA_out_ga(0, [])
s2lA_in_ga(s(T16), .(X111, X112)) → U1_ga(T16, X111, X112, s2lA_in_ga(T16, X112))
U1_ga(T16, X111, X112, s2lA_out_ga(T16, X112)) → s2lA_out_ga(s(T16), .(X111, X112))
U5_g(T9, s2lA_out_ga(T9, X72)) → goalD_out_g(s(T9))
goalD_in_g(s(T9)) → U6_g(T9, s2lA_in_ga(T9, T31))
U6_g(T9, s2lA_out_ga(T9, T31)) → U7_g(T9, appendC_in_gaa(T31, X151, X153))
appendC_in_gaa([], X210, X210) → appendC_out_gaa([], X210, X210)
appendC_in_gaa([], X235, .(X251, X237)) → U2_gaa(X235, X251, X237, appendB_in_aa(X235, X237))
U2_gaa(X235, X251, X237, appendB_out_aa(X235, X237)) → appendC_out_gaa([], X235, .(X251, X237))
appendC_in_gaa(.(T50, T52), X235, .(T50, X237)) → U3_gaa(T50, T52, X235, X237, appendC_in_gaa(T52, X235, X237))
U3_gaa(T50, T52, X235, X237, appendC_out_gaa(T52, X235, X237)) → appendC_out_gaa(.(T50, T52), X235, .(T50, X237))
U7_g(T9, appendC_out_gaa(T31, X151, X153)) → goalD_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalD_in_g(x1)  =  goalD_in_g(x1)
0  =  0
U4_g(x1)  =  U4_g(x1)
appendB_in_aa(x1, x2)  =  appendB_in_aa
appendB_out_aa(x1, x2)  =  appendB_out_aa
goalD_out_g(x1)  =  goalD_out_g
s(x1)  =  s(x1)
U5_g(x1, x2)  =  U5_g(x2)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
s2lA_out_ga(x1, x2)  =  s2lA_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
[]  =  []
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

S2LA_IN_GA(s(T16), .(X111, X112)) → S2LA_IN_GA(T16, X112)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

S2LA_IN_GA(s(T16)) → S2LA_IN_GA(T16)

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

(21) 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:

• S2LA_IN_GA(s(T16)) → S2LA_IN_GA(T16)
  The graph contains the following edges 1 > 1

(22) YES