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), list(Xs)).
list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Query: goal(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="D: 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, X5), list(X5))\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(s2l(T3, X5), list(X5)), SCOPE: 2, CLAUSE: 3 | ','(s2l(T3, X5), list(X5)), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(','(!_2, eq(X11, [])), list(X11)) | ','(s2l(0, X5), list(X5)), SCOPE: 2, CLAUSE: 4\n\nX5 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(s2l(T3, X5), list(X5)), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX5 is free\nX10 is free\ns2l(T3, X5) !=? s2l(0, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X11, []), list(X11))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(X11, []), list(X11)), SCOPE: 3, CLAUSE: 9\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="B: list([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: list([]), SCOPE: 4, CLAUSE: 1 | list([]), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: !_4 | list([]), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(','(p(T6, X34), s2l(X34, X36)), list(.(X35, X36)))\n\nT6 is ground\nX5 is free\nX10 is free\nX35 is free\nX36 is free\nX34 is free\ns2l(T6, X5) !=? s2l(0, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(','(p(T6, X34), s2l(X34, X36)), list(.(X35, X36))), SCOPE: 5, CLAUSE: 7 | ','(','(p(T6, X34), s2l(X34, X36)), list(.(X35, X36))), SCOPE: 5, CLAUSE: 8\n\nT6 is ground\nX5 is free\nX10 is free\nX35 is free\nX36 is free\nX34 is free\ns2l(T6, X5) !=? s2l(0, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(p(T6, X34), s2l(X34, X36)), list(.(X35, X36))), SCOPE: 5, CLAUSE: 8\n\nT6 is ground\nX5 is free\nX10 is free\nX35 is free\nX36 is free\nX34 is free\ns2l(T6, X5) !=? s2l(0, X10)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(s2l(T9, X36), list(.(X35, X36)))\n\nT9 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="A: s2l(T9, X36)\n\nT9 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: list(.(X35, T10))\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: s2l(T9, X36), SCOPE: 6, CLAUSE: 3 | s2l(T9, X36), SCOPE: 6, CLAUSE: 4\n\nT9 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_6, eq(X51, [])) | s2l(0, X36), SCOPE: 6, CLAUSE: 4\n\nX36 is free\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: s2l(T9, X36), SCOPE: 6, CLAUSE: 4\n\nT9 is ground\nX36 is free\nX50 is free\ns2l(T9, X36) !=? s2l(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: eq(X51, [])\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq(X51, []), SCOPE: 7, CLAUSE: 9\n\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(p(T13, X74), s2l(X74, X76))\n\nT13 is ground\nX36 is free\nX50 is free\nX76 is free\nX74 is free\ns2l(T13, X36) !=? s2l(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T13, X74), s2l(X74, X76)), SCOPE: 8, CLAUSE: 7 | ','(p(T13, X74), s2l(X74, X76)), SCOPE: 8, CLAUSE: 8\n\nT13 is ground\nX36 is free\nX50 is free\nX76 is free\nX74 is free\ns2l(T13, X36) !=? s2l(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T13, X74), s2l(X74, X76)), SCOPE: 8, CLAUSE: 8\n\nT13 is ground\nX36 is free\nX50 is free\nX76 is free\nX74 is free\ns2l(T13, X36) !=? s2l(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: s2l(T16, X76)\n\nT16 is ground\nX76 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: list(.(X35, T10)), SCOPE: 9, CLAUSE: 1 | list(.(X35, T10)), SCOPE: 9, CLAUSE: 2\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: list(.(X35, T10)), SCOPE: 9, CLAUSE: 2\n\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(X91, T21), X90), list(X90))\n\nX91 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(tail(.(X91, T21), X90), list(X90)), SCOPE: 10, CLAUSE: 5 | ','(tail(.(X91, T21), X90), list(X90)), SCOPE: 10, CLAUSE: 6\n\nX91 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(tail(.(X91, T21), X90), list(X90)), SCOPE: 10, CLAUSE: 6\n\nX91 is free\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="C: list(T26)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: list(T26), SCOPE: 11, CLAUSE: 1 | list(T26), SCOPE: 11, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: !_11 | list(T26), SCOPE: 11, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: list(T26), SCOPE: 11, CLAUSE: 2\n\nlist(T26) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(T31, X105), list(X105))\n\nX105 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tail(T31, X105), list(X105)), SCOPE: 12, CLAUSE: 5 | ','(tail(T31, X105), list(X105)), SCOPE: 12, CLAUSE: 6\n\nX105 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail(T31, X105), list(X105)), SCOPE: 12, CLAUSE: 5\n\nX105 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail(T31, X105), list(X105)), SCOPE: 12, CLAUSE: 6\n\nX105 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: list([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: list(T38)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="ONLY EVAL with clause\ngoal(X4) :- ','(s2l(X4, X5), list(X5)).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="EVAL with clause\ns2l(0, X10) :- ','(!_2, eq(X10, [])).\nand substitutionT3 -> 0,\nX5 -> X11,\nX10 -> X11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="ONLY EVAL with clause\ns2l(X31, .(X32, X33)) :- ','(p(X31, X34), s2l(X34, X33)).\nand substitutionT3 -> T6,\nX31 -> T6,\nX32 -> X35,\nX33 -> X36,\nX5 -> .(X35, X36)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="ONLY EVAL with clause\neq(X16, X16).\nand substitutionX11 -> [],\nX16 -> [],\nX17 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="ONLY EVAL with clause\nlist([]) :- !_4.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 65 [arrowhead = none ];
65 -> 15;

66 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T6, X5), s2l(0, X10))", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 66 [arrowhead = none ];
66 -> 16;

67 [label="EVAL with clause\np(s(X41), X41).\nand substitutionX41 -> T9,\nT6 -> s(T9),\nX34 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 67 [arrowhead = none ];
67 -> 17;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 68 [arrowhead = none ];
68 -> 18;

69 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 69 [arrowhead = none ];
69 -> 19;

70 [label="SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X36 -> T10", fontsize=14, style = filled, fillcolor = violet];
17 -> 70 [arrowhead = none ];
70 -> 20;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 71 [arrowhead = none ];
71 -> 21;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 72 [arrowhead = none ];
72 -> 33;

73 [label="EVAL with clause\ns2l(0, X50) :- ','(!_6, eq(X50, [])).\nand substitutionT9 -> 0,\nX36 -> X51,\nX50 -> X51", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 73 [arrowhead = none ];
73 -> 22;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 74 [arrowhead = none ];
74 -> 23;

75 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="ONLY EVAL with clause\ns2l(X71, .(X72, X73)) :- ','(p(X71, X74), s2l(X74, X73)).\nand substitutionT9 -> T13,\nX71 -> T13,\nX72 -> X75,\nX73 -> X76,\nX36 -> .(X75, X76)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 76 [arrowhead = none ];
76 -> 28;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 77 [arrowhead = none ];
77 -> 25;

78 [label="ONLY EVAL with clause\neq(X56, X56).\nand substitutionX51 -> [],\nX56 -> [],\nX57 -> []", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 78 [arrowhead = none ];
78 -> 26;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 80 [arrowhead = none ];
80 -> 29;

81 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T13, X36), s2l(0, X50))", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 81 [arrowhead = none ];
81 -> 30;

82 [label="EVAL with clause\np(s(X81), X81).\nand substitutionX81 -> T16,\nT13 -> s(T16),\nX74 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 82 [arrowhead = none ];
82 -> 31;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 83 [arrowhead = none ];
83 -> 32;

84 [label="INSTANCE with matching:\nT9 -> T16\nX36 -> X76", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 84 [arrowhead = none , style=dashed];
84 -> 19[style=dashed];

85 [label="BACKTRACK\nfor clause: list([]) :- !because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 85 [arrowhead = none ];
85 -> 34;

86 [label="ONLY EVAL with clause\nlist(X89) :- ','(tail(X89, X90), list(X90)).\nand substitutionX35 -> X91,\nT10 -> T21,\nX89 -> .(X91, T21),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 86 [arrowhead = none ];
86 -> 35;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 87 [arrowhead = none ];
87 -> 36;

88 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 88 [arrowhead = none ];
88 -> 37;

89 [label="ONLY EVAL with clause\ntail(.(X98, X99), X99).\nand substitutionX91 -> X100,\nX98 -> X100,\nT21 -> T26,\nX99 -> T26,\nX90 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 89 [arrowhead = none ];
89 -> 38;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 90 [arrowhead = none ];
90 -> 39;

91 [label="EVAL with clause\nlist([]) :- !_11.\nand substitutionT26 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="ONLY EVAL with clause\nlist(X104) :- ','(tail(X104, X105), list(X105)).\nand substitutionT26 -> T31,\nX104 -> T31,\nT30 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 94 [arrowhead = none ];
94 -> 44;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 95 [arrowhead = none ];
95 -> 43;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="EVAL with clause\ntail([], []).\nand substitutionT31 -> [],\nX105 -> []", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 99 [arrowhead = none ];
99 -> 48;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 100 [arrowhead = none ];
100 -> 49;

101 [label="EVAL with clause\ntail(.(X110, X111), X111).\nand substitutionX110 -> T36,\nX111 -> T38,\nT31 -> .(T36, T38),\nX105 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 102 [arrowhead = none ];
102 -> 51;

103 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 103 [arrowhead = none , style=dashed];
103 -> 9[style=dashed];

104 [label="INSTANCE with matching:\nT26 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 104 [arrowhead = none , style=dashed];
104 -> 38[style=dashed];

}

(2) Obligation:

Triples:

s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
listC([]) :- listB.
listC(.(X1, X2)) :- listC(X2).
goalD(0) :- listB.
goalD(s(X1)) :- s2lA(X1, X2).
goalD(s(X1)) :- ','(s2lcA(X1, X2), listC(X2)).

Clauses:

s2lcA(0, []).
s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
listcB.
listcC([]).
listcC([]) :- listcB.
listcC(.(X1, X2)) :- listcC(X2).

Afs:

goalD(x1)  =  goalD(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
listC(.(X1, X2)) :- listC(X2).
goalD(s(X1)) :- s2lA(X1, X2).
goalD(s(X1)) :- ','(s2lcA(X1, X2), listC(X2)).

Clauses:

s2lcA(0, []).
s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
listcB.
listcC([]).
listcC([]) :- listcB.
listcC(.(X1, X2)) :- listcC(X2).

Afs:

goalD(x1)  =  goalD(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalD_in: (b)
s2lA_in: (b,f)
s2lcA_in: (b,f)
listC_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOALD_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALD_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, listC_in_g(X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → LISTC_IN_G(X2)
LISTC_IN_G(.(X1, X2)) → U2_G(X1, X2, listC_in_g(X2))
LISTC_IN_G(.(X1, X2)) → LISTC_IN_G(X2)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
listC_in_g(x1)  =  listC_in_g(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2LA_IN_GA(x1, x2)  =  S2LA_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)
LISTC_IN_G(x1)  =  LISTC_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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:

GOALD_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALD_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, listC_in_g(X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → LISTC_IN_G(X2)
LISTC_IN_G(.(X1, X2)) → U2_G(X1, X2, listC_in_g(X2))
LISTC_IN_G(.(X1, X2)) → LISTC_IN_G(X2)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
listC_in_g(x1)  =  listC_in_g(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2LA_IN_GA(x1, x2)  =  S2LA_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)
LISTC_IN_G(x1)  =  LISTC_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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:

LISTC_IN_G(.(X1, X2)) → LISTC_IN_G(X2)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
LISTC_IN_G(x1)  =  LISTC_IN_G(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:

LISTC_IN_G(.(X1, X2)) → LISTC_IN_G(X2)

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

LISTC_IN_G(.(X2)) → LISTC_IN_G(X2)

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:

• LISTC_IN_G(.(X2)) → LISTC_IN_G(X2)
  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(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
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(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)

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(X1)) → S2LA_IN_GA(X1)

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(X1)) → S2LA_IN_GA(X1)
  The graph contains the following edges 1 > 1

(22) YES