Left Termination of the query pattern goal_in_1(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).

Queries:

goal(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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; \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(X47, X48)) | append([], X10, X11), SCOPE: 4, CLAUSE: 2\n\nX10 is free\nX11 is free; \nX47 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(X47, X48)\n\nX47 is free\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: eq(X47, X48), SCOPE: 5, CLAUSE: 11\n\nX47 is free\nX48 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, X71), ','(s2l(X71, X73), append(.(X72, X73), X10, X11)))\n\nT6 is ground\nX10 is free; \nX11 is free; \nX72 is free; \nX73 is free; \nX71 is free; \ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T6, X71), ','(s2l(X71, X73), append(.(X72, X73), X10, X11))), SCOPE: 6, CLAUSE: 9 | ','(p(T6, X71), ','(s2l(X71, X73), append(.(X72, X73), X10, X11))), SCOPE: 6, CLAUSE: 10\n\nT6 is ground\nX10 is free; \nX11 is free; \nX72 is free; \nX73 is free; \nX71 is free; \ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T6, X71), ','(s2l(X71, X73), append(.(X72, X73), X10, X11))), SCOPE: 6, CLAUSE: 10\n\nT6 is ground\nX10 is free; \nX11 is free; \nX72 is free; \nX73 is free; \nX71 is free; \ns2l(T6, X9) !=? s2l(0, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(s2l(T9, X73), append(.(X72, X73), X10, X11))\n\nT9 is ground\nX10 is free; \nX11 is free; \nX72 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2l(T9, X73)\n\nT9 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append(.(X72, T10), X10, X11)\n\nX10 is free\nX11 is free; \nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: s2l(T9, X73), SCOPE: 7, CLAUSE: 3 | s2l(T9, X73), SCOPE: 7, CLAUSE: 4\n\nT9 is ground\nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_7, eq(X89, [])) | s2l(0, X73), SCOPE: 7, CLAUSE: 4\n\nX73 is free\nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: s2l(T9, X73), SCOPE: 7, CLAUSE: 4\n\nT9 is ground\nX73 is free; \ns2l(T9, X73) !=? s2l(0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: eq(X89, [])\n\nX89 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: eq(X89, []), SCOPE: 8, CLAUSE: 11\n\nX89 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, X113), s2l(X113, X115))\n\nT13 is ground\nX115 is free; \nX113 is free; \ns2l(T13, X73) !=? s2l(0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(T13, X113), s2l(X113, X115)), SCOPE: 9, CLAUSE: 9 | ','(p(T13, X113), s2l(X113, X115)), SCOPE: 9, CLAUSE: 10\n\nT13 is ground\nX115 is free; \nX113 is free; \ns2l(T13, X73) !=? s2l(0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T13, X113), s2l(X113, X115)), SCOPE: 9, CLAUSE: 10\n\nT13 is ground\nX115 is free; \nX113 is free; \ns2l(T13, X73) !=? s2l(0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: s2l(T16, X115)\n\nT16 is ground\nX115 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: append(.(X72, T10), X10, X11), SCOPE: 10, CLAUSE: 1 | append(.(X72, T10), X10, X11), SCOPE: 10, CLAUSE: 2\n\nX10 is free\nX11 is free; \nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: append(.(X72, T10), X10, X11), SCOPE: 10, CLAUSE: 2\n\nX10 is free\nX11 is free; \nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(head(.(X163, T22), X165), ','(tail(.(X163, T22), X162), append(X162, X164, X166)))\n\nX163 is free\nX164 is free; \nX165 is free; \nX166 is free; \nX162 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(.(X163, T22), X165), ','(tail(.(X163, T22), X162), append(X162, X164, X166))), SCOPE: 11, CLAUSE: 5 | ','(head(.(X163, T22), X165), ','(tail(.(X163, T22), X162), append(X162, X164, X166))), SCOPE: 11, CLAUSE: 6\n\nX163 is free\nX164 is free; \nX165 is free; \nX166 is free; \nX162 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(head(.(X163, T22), X165), ','(tail(.(X163, T22), X162), append(X162, X164, X166))), SCOPE: 11, CLAUSE: 6\n\nX163 is free\nX164 is free; \nX165 is free; \nX166 is free; \nX162 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(.(X184, T27), X162), append(X162, X164, X166))\n\nX164 is free\nX166 is free; \nX162 is free; \nX184 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(.(X184, T27), X162), append(X162, X164, X166)), SCOPE: 12, CLAUSE: 7 | ','(tail(.(X184, T27), X162), append(X162, X164, X166)), SCOPE: 12, CLAUSE: 8\n\nX164 is free\nX166 is free; \nX162 is free; \nX184 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(.(X184, T27), X162), append(X162, X164, X166)), SCOPE: 12, CLAUSE: 8\n\nX164 is free\nX166 is free; \nX162 is free; \nX184 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append(T32, X164, X166)\n\nX164 is free\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append(T32, X164, X166), SCOPE: 13, CLAUSE: 1 | append(T32, X164, X166), SCOPE: 13, CLAUSE: 2\n\nX164 is free\nX166 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_13, eq(X216, X217)) | append(T32, X164, X166), SCOPE: 13, CLAUSE: 2\n\nX164 is free\nX166 is free; \nX216 is free; \nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append(T32, X164, X166), SCOPE: 13, CLAUSE: 2\n\nX164 is free\nX166 is free; \nappend(T32, X164, X166) !=? append([], X214, X215)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(X216, X217)\n\nX216 is free\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: eq(X216, X217), SCOPE: 14, CLAUSE: 11\n\nX216 is free\nX217 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(T37, X251), ','(tail(T37, X249), append(X249, X250, X252)))\n\nX250 is free\nX251 is free; \nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T37, X251), ','(tail(T37, X249), append(X249, X250, X252))), SCOPE: 15, CLAUSE: 5 | ','(head(T37, X251), ','(tail(T37, X249), append(X249, X250, X252))), SCOPE: 15, CLAUSE: 6\n\nX250 is free\nX251 is free; \nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(head(T37, X251), ','(tail(T37, X249), append(X249, X250, X252))), SCOPE: 15, CLAUSE: 5\n\nX250 is free\nX251 is free; \nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T37, X251), ','(tail(T37, X249), append(X249, X250, X252))), SCOPE: 15, CLAUSE: 6\n\nX250 is free\nX251 is free; \nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail([], X249), append(X249, X250, X252))\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail([], X249), append(X249, X250, X252)), SCOPE: 16, CLAUSE: 7 | ','(tail([], X249), append(X249, X250, X252)), SCOPE: 16, CLAUSE: 8\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail([], X249), append(X249, X250, X252)), SCOPE: 16, CLAUSE: 7\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail([], X249), append(X249, X250, X252)), SCOPE: 16, CLAUSE: 8\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: append([], X250, X252)\n\nX250 is free\nX252 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(.(T44, T45), X249), append(X249, X250, X252))\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(.(T44, T45), X249), append(X249, X250, X252)), SCOPE: 17, CLAUSE: 7 | ','(tail(.(T44, T45), X249), append(X249, X250, X252)), SCOPE: 17, CLAUSE: 8\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(T44, T45), X249), append(X249, X250, X252)), SCOPE: 17, CLAUSE: 8\n\nX250 is free\nX252 is free; \nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: append(T53, X250, X252)\n\nX250 is free\nX252 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 substitution\nT1 -> T3\nX8 -> T3", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT3 -> 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 = lightblue];
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(X68, .(X69, X70)) :- ','(p(X68, X71), s2l(X71, X70)).\nand substitution\nT3 -> T6\nX68 -> T6\nX69 -> X72\nX70 -> X73\nX9 -> .(X72, X73)", fontsize=14, style = filled, fillcolor = white];
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(X31, X31).\nand substitution\nX17 -> []\nX31 -> []\nX32 -> []", fontsize=14, style = filled, fillcolor = white];
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([], X45, X46) :- ','(!_4, eq(X45, X46)).\nand substitution\nX10 -> X47\nX45 -> X47\nX11 -> X48\nX46 -> X48", fontsize=14, style = filled, fillcolor = white];
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(X53, X53).\nand substitution\nX47 -> X54\nX53 -> X54\nX48 -> X54", fontsize=14, style = filled, fillcolor = white];
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 = white];
17 -> 83 [arrowhead = none ];
83 -> 18;

84 [label="EVAL with clause\np(s(X78), X78).\nand substitution\nX78 -> T9\nT6 -> s(T9)\nX71 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 84 [arrowhead = none ];
84 -> 19;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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\nreplacements:\nX73 -> 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, X88) :- ','(!_7, eq(X88, [])).\nand substitution\nT9 -> 0\nX73 -> X89\nX88 -> X89", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 90 [arrowhead = none ];
90 -> 24;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X110, .(X111, X112)) :- ','(p(X110, X113), s2l(X113, X112)).\nand substitution\nT9 -> T13\nX110 -> T13\nX111 -> X114\nX112 -> X115\nX73 -> .(X114, X115)", fontsize=14, style = filled, fillcolor = white];
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(X95, X95).\nand substitution\nX89 -> []\nX95 -> []\nX96 -> []", fontsize=14, style = filled, fillcolor = white];
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, X73), s2l(0, X88))", fontsize=14, style = filled, fillcolor = white];
31 -> 98 [arrowhead = none ];
98 -> 32;

99 [label="EVAL with clause\np(s(X120), X120).\nand substitution\nX120 -> T16\nT13 -> s(T16)\nX113 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 99 [arrowhead = none ];
99 -> 33;

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

101 [label="INSTANCE with matching:\nT9 -> T16\nX73 -> X115", 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 = white];
35 -> 102 [arrowhead = none ];
102 -> 36;

103 [label="ONLY EVAL with clause\nappend(X158, X159, .(X160, X161)) :- ','(head(X158, X160), ','(tail(X158, X162), append(X162, X159, X161))).\nand substitution\nX72 -> X163\nT10 -> T22\nX158 -> .(X163, T22)\nX10 -> X164\nX159 -> X164\nX160 -> X165\nX161 -> X166\nX11 -> .(X165, X166)\nT21 -> T22", fontsize=14, style = filled, fillcolor = white];
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 = white];
38 -> 105 [arrowhead = none ];
105 -> 39;

106 [label="ONLY EVAL with clause\nhead(.(X182, X183), X182).\nand substitution\nX163 -> X184\nX182 -> X184\nT22 -> T27\nX183 -> T27\nX165 -> X184\nT26 -> T27", fontsize=14, style = filled, fillcolor = white];
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 = white];
41 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="ONLY EVAL with clause\ntail(.(X199, X200), X200).\nand substitution\nX184 -> X201\nX199 -> X201\nT27 -> T32\nX200 -> T32\nX162 -> T32\nT31 -> T32", fontsize=14, style = filled, fillcolor = white];
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([], X214, X215) :- ','(!_13, eq(X214, X215)).\nand substitution\nT32 -> []\nX164 -> X216\nX214 -> X216\nX166 -> X217\nX215 -> X217", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X245, X246, .(X247, X248)) :- ','(head(X245, X247), ','(tail(X245, X249), append(X249, X246, X248))).\nand substitution\nT32 -> T37\nX245 -> T37\nX164 -> X250\nX246 -> X250\nX247 -> X251\nX248 -> X252\nX166 -> .(X251, X252)\nT36 -> T37", fontsize=14, style = filled, fillcolor = white];
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(X224, X224).\nand substitution\nX216 -> X225\nX224 -> X225\nX217 -> X225", fontsize=14, style = filled, fillcolor = white];
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([], X265).\nand substitution\nT37 -> []\nX251 -> X266\nX265 -> X266", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 121 [arrowhead = none ];
121 -> 55;

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

123 [label="EVAL with clause\nhead(.(X290, X291), X290).\nand substitution\nX290 -> T44\nX291 -> T45\nT37 -> .(T44, T45)\nX251 -> T44\nT42 -> T44\nT43 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 123 [arrowhead = none ];
123 -> 62;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nX249 -> []", fontsize=14, style = filled, fillcolor = white];
58 -> 128 [arrowhead = none ];
128 -> 60;

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

130 [label="INSTANCE with matching:\nX10 -> X250\nX11 -> X252", 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 = white];
64 -> 132 [arrowhead = none ];
132 -> 65;

133 [label="ONLY EVAL with clause\ntail(.(X306, X307), X307).\nand substitution\nT44 -> T51\nX306 -> T51\nT45 -> T53\nX307 -> T53\nX249 -> T53\nT52 -> T53", fontsize=14, style = filled, fillcolor = white];
65 -> 133 [arrowhead = none ];
133 -> 66;

134 [label="INSTANCE with matching:\nT32 -> T53\nX164 -> X250\nX166 -> X252", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 134 [arrowhead = none , style=dashed];
134 -> 43[style=dashed];

}

(2) Obligation:

Triples:

s2l21(s(T16), .(X114, X115)) :- s2l21(T16, X115).
append43([], X250, .(X266, X252)) :- append9(X250, X252).
append43(.(T51, T53), X250, .(T51, X252)) :- append43(T53, X250, X252).
goal1(0) :- append9(X10, X11).
goal1(s(T9)) :- s2l21(T9, X73).
goal1(s(T9)) :- ','(s2lc21(T9, T32), append43(T32, X164, X166)).

Clauses:

s2lc21(0, []).
s2lc21(s(T16), .(X114, X115)) :- s2lc21(T16, X115).
appendc9(X54, X54).
appendc43([], X225, X225).
appendc43([], X250, .(X266, X252)) :- appendc9(X250, X252).
appendc43(.(T51, T53), X250, .(T51, X252)) :- appendc43(T53, X250, X252).

Afs:

goal1(x1)  =  goal1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

s2l21(s(T16), .(X114, X115)) :- s2l21(T16, X115).
append43(.(T51, T53), X250, .(T51, X252)) :- append43(T53, X250, X252).
goal1(s(T9)) :- s2l21(T9, X73).
goal1(s(T9)) :- ','(s2lc21(T9, T32), append43(T32, X164, X166)).

Clauses:

s2lc21(0, []).
s2lc21(s(T16), .(X114, X115)) :- s2lc21(T16, X115).
appendc9(X54, X54).
appendc43([], X225, X225).
appendc43([], X250, .(X266, X252)) :- appendc9(X250, X252).
appendc43(.(T51, T53), X250, .(T51, X252)) :- appendc43(T53, X250, X252).

Afs:

goal1(x1)  =  goal1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2l21_in: (b,f)
s2lc21_in: (b,f)
append43_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T9)) → U3_G(T9, s2l21_in_ga(T9, X73))
GOAL1_IN_G(s(T9)) → S2L21_IN_GA(T9, X73)
S2L21_IN_GA(s(T16), .(X114, X115)) → U1_GA(T16, X114, X115, s2l21_in_ga(T16, X115))
S2L21_IN_GA(s(T16), .(X114, X115)) → S2L21_IN_GA(T16, X115)
GOAL1_IN_G(s(T9)) → U4_G(T9, s2lc21_in_ga(T9, T32))
U4_G(T9, s2lc21_out_ga(T9, T32)) → U5_G(T9, append43_in_gaa(T32, X164, X166))
U4_G(T9, s2lc21_out_ga(T9, T32)) → APPEND43_IN_GAA(T32, X164, X166)
APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → U2_GAA(T51, T53, X250, X252, append43_in_gaa(T53, X250, X252))
APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → APPEND43_IN_GAA(T53, X250, X252)

The TRS R consists of the following rules:

s2lc21_in_ga(0, []) → s2lc21_out_ga(0, [])
s2lc21_in_ga(s(T16), .(X114, X115)) → U7_ga(T16, X114, X115, s2lc21_in_ga(T16, X115))
U7_ga(T16, X114, X115, s2lc21_out_ga(T16, X115)) → s2lc21_out_ga(s(T16), .(X114, X115))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l21_in_ga(x1, x2)  =  s2l21_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc21_in_ga(x1, x2)  =  s2lc21_in_ga(x1)
0  =  0
s2lc21_out_ga(x1, x2)  =  s2lc21_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append43_in_gaa(x1, x2, x3)  =  append43_in_gaa(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2L21_IN_GA(x1, x2)  =  S2L21_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
APPEND43_IN_GAA(x1, x2, x3)  =  APPEND43_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

GOAL1_IN_G(s(T9)) → U3_G(T9, s2l21_in_ga(T9, X73))
GOAL1_IN_G(s(T9)) → S2L21_IN_GA(T9, X73)
S2L21_IN_GA(s(T16), .(X114, X115)) → U1_GA(T16, X114, X115, s2l21_in_ga(T16, X115))
S2L21_IN_GA(s(T16), .(X114, X115)) → S2L21_IN_GA(T16, X115)
GOAL1_IN_G(s(T9)) → U4_G(T9, s2lc21_in_ga(T9, T32))
U4_G(T9, s2lc21_out_ga(T9, T32)) → U5_G(T9, append43_in_gaa(T32, X164, X166))
U4_G(T9, s2lc21_out_ga(T9, T32)) → APPEND43_IN_GAA(T32, X164, X166)
APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → U2_GAA(T51, T53, X250, X252, append43_in_gaa(T53, X250, X252))
APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → APPEND43_IN_GAA(T53, X250, X252)

The TRS R consists of the following rules:

s2lc21_in_ga(0, []) → s2lc21_out_ga(0, [])
s2lc21_in_ga(s(T16), .(X114, X115)) → U7_ga(T16, X114, X115, s2lc21_in_ga(T16, X115))
U7_ga(T16, X114, X115, s2lc21_out_ga(T16, X115)) → s2lc21_out_ga(s(T16), .(X114, X115))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2l21_in_ga(x1, x2)  =  s2l21_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lc21_in_ga(x1, x2)  =  s2lc21_in_ga(x1)
0  =  0
s2lc21_out_ga(x1, x2)  =  s2lc21_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append43_in_gaa(x1, x2, x3)  =  append43_in_gaa(x1)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2L21_IN_GA(x1, x2)  =  S2L21_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
APPEND43_IN_GAA(x1, x2, x3)  =  APPEND43_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, 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 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → APPEND43_IN_GAA(T53, X250, X252)

The TRS R consists of the following rules:

s2lc21_in_ga(0, []) → s2lc21_out_ga(0, [])
s2lc21_in_ga(s(T16), .(X114, X115)) → U7_ga(T16, X114, X115, s2lc21_in_ga(T16, X115))
U7_ga(T16, X114, X115, s2lc21_out_ga(T16, X115)) → s2lc21_out_ga(s(T16), .(X114, X115))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc21_in_ga(x1, x2)  =  s2lc21_in_ga(x1)
0  =  0
s2lc21_out_ga(x1, x2)  =  s2lc21_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
APPEND43_IN_GAA(x1, x2, x3)  =  APPEND43_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:

APPEND43_IN_GAA(.(T51, T53), X250, .(T51, X252)) → APPEND43_IN_GAA(T53, X250, X252)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND43_IN_GAA(x1, x2, x3)  =  APPEND43_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:

APPEND43_IN_GAA(.(T53)) → APPEND43_IN_GAA(T53)

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:

• APPEND43_IN_GAA(.(T53)) → APPEND43_IN_GAA(T53)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2L21_IN_GA(s(T16), .(X114, X115)) → S2L21_IN_GA(T16, X115)

The TRS R consists of the following rules:

s2lc21_in_ga(0, []) → s2lc21_out_ga(0, [])
s2lc21_in_ga(s(T16), .(X114, X115)) → U7_ga(T16, X114, X115, s2lc21_in_ga(T16, X115))
U7_ga(T16, X114, X115, s2lc21_out_ga(T16, X115)) → s2lc21_out_ga(s(T16), .(X114, X115))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lc21_in_ga(x1, x2)  =  s2lc21_in_ga(x1)
0  =  0
s2lc21_out_ga(x1, x2)  =  s2lc21_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
S2L21_IN_GA(x1, x2)  =  S2L21_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:

S2L21_IN_GA(s(T16), .(X114, X115)) → S2L21_IN_GA(T16, X115)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L21_IN_GA(x1, x2)  =  S2L21_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:

S2L21_IN_GA(s(T16)) → S2L21_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:

• S2L21_IN_GA(s(T16)) → S2L21_IN_GA(T16)
  The graph contains the following edges 1 > 1

(22) YES