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), list(Xs)).
list([]).
list(X) :- ','(no(empty(X)), ','(tail(X, T), list(T))).
s2l(0, []).
s2l(X, .(X1, Xs)) :- ','(no(zero(X)), ','(p(X, P), s2l(P, Xs))).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
empty([]).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X3).
failure(b).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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(T2, X6), list(X6))\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2l(T2, X6)\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: list(T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2l(T2, X6), SCOPE: 2, CLAUSE: 3 | s2l(T2, X6), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: s2l(T2, X6), SCOPE: 2, CLAUSE: 3\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2l(T2, X6), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(no(zero(T4)), ','(p(T4, X13), s2l(X13, X15)))\n\nT4 is ground\nX15 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(zero(T4)), ','(p(T4, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 11 | ','(no(zero(T4)), ','(p(T4, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 12 | ?_3\n\nT4 is ground\nX15 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(zero(T5), ','(!_3, ','(failure(a), ','(p(T5, X13), s2l(X13, X15))))) | ','(no(zero(T5)), ','(p(T5, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 12 | ?_3\n\nT5 is ground\nX15 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(zero(T5), ','(!_3, ','(failure(a), ','(p(T5, X13), s2l(X13, X15))))), SCOPE: 4, CLAUSE: 10 | ','(no(zero(T5)), ','(p(T5, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 12 | ?_3\n\nT5 is ground\nX15 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(!_3, ','(failure(a), ','(p(0, X13), s2l(X13, X15)))) | ','(no(zero(0)), ','(p(0, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 12 | ?_3\n\nX15 is free\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(zero(T5)), ','(p(T5, X13), s2l(X13, X15))), SCOPE: 3, CLAUSE: 12\n\nT5 is ground\nX15 is free; \nX13 is free; \nzero(T5) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(failure(a), ','(p(0, X13), s2l(X13, X15)))\n\nX15 is free\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(failure(a), ','(p(0, X13), s2l(X13, X15))), SCOPE: 5, CLAUSE: 13\n\nX15 is free\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T6, X13), s2l(X13, X15))\n\nT6 is ground\nX15 is free; \nX13 is free; \nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T6, X13), s2l(X13, X15)), SCOPE: 6, CLAUSE: 7 | ','(p(T6, X13), s2l(X13, X15)), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free; \nX13 is free; \nzero(T6) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T6, X13), s2l(X13, X15)), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nX15 is free; \nX13 is free; \nzero(T6) !=? zero(0)\np(T6, X13) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: s2l(T7, X15)\n\nT7 is ground\nX15 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\nX13 is free\nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: list(T3), SCOPE: 7, CLAUSE: 1 | list(T3), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: list(T3), SCOPE: 7, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: list(T3), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(no(empty(T9)), ','(tail(T9, X24), list(X24)))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 11 | ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 12 | ?_8\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(empty(T11), ','(!_8, ','(failure(a), ','(tail(T11, X24), list(X24))))) | ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 12 | ?_8\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(empty(T11), ','(!_8, ','(failure(a), ','(tail(T11, X24), list(X24))))), SCOPE: 9, CLAUSE: 9 | ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 12 | ?_8\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(!_8, ','(failure(a), ','(tail([], X24), list(X24)))) | ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 12 | ?_8\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(no(empty(T9)), ','(tail(T9, X24), list(X24))), SCOPE: 8, CLAUSE: 12\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(failure(a), ','(tail([], X24), list(X24)))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(failure(a), ','(tail([], X24), list(X24))), SCOPE: 10, CLAUSE: 13\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(T13, X24), list(X24))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(T13, X24), list(X24)), SCOPE: 11, CLAUSE: 5 | ','(tail(T13, X24), list(X24)), SCOPE: 11, CLAUSE: 6\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: list([]) | ','(tail(T13, X24), list(X24)), SCOPE: 11, CLAUSE: 6\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(T13, X24), list(X24)), SCOPE: 11, CLAUSE: 6\n\nX24 is free\ntail(T13, X24) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: list([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail(T13, X24), list(X24)), SCOPE: 11, CLAUSE: 6\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: list([]), SCOPE: 12, CLAUSE: 1 | list([]), SCOPE: 12, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: list([]), SCOPE: 12, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: list([]), SCOPE: 12, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(no(empty([])), ','(tail([], X31), list(X31)))\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(no(empty([])), ','(tail([], X31), list(X31))), SCOPE: 13, CLAUSE: 11 | ','(no(empty([])), ','(tail([], X31), list(X31))), SCOPE: 13, CLAUSE: 12 | ?_13\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(empty([]), ','(!_13, ','(failure(a), ','(tail([], X31), list(X31))))) | ','(no(empty([])), ','(tail([], X31), list(X31))), SCOPE: 13, CLAUSE: 12 | ?_13\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(empty([]), ','(!_13, ','(failure(a), ','(tail([], X31), list(X31))))), SCOPE: 14, CLAUSE: 9 | ','(no(empty([])), ','(tail([], X31), list(X31))), SCOPE: 13, CLAUSE: 12 | ?_13\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(!_13, ','(failure(a), ','(tail([], X31), list(X31)))) | ','(no(empty([])), ','(tail([], X31), list(X31))), SCOPE: 13, CLAUSE: 12 | ?_13\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(failure(a), ','(tail([], X31), list(X31)))\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(failure(a), ','(tail([], X31), list(X31))), SCOPE: 15, CLAUSE: 13\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: list(T16)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\nX24 is free\nX36 is free; \nX37 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: list(T19)\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\nX24 is free\nX40 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 64 [arrowhead = none ];
64 -> 2;

65 [label="EVAL with clause\ngoal(X5) :- ','(s2l(X5, X6), list(X6)).\nand substitution\nT1 -> T2\nX5 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 65 [arrowhead = none ];
65 -> 3;

66 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 66 [arrowhead = none ];
66 -> 4;

67 [label="SPLIT 2\nreplacements:\nX6 -> T3", fontsize=14, style = filled, fillcolor = violet];
3 -> 67 [arrowhead = none ];
67 -> 5;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 68 [arrowhead = none ];
68 -> 6;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 70 [arrowhead = none ];
70 -> 7;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 71 [arrowhead = none ];
71 -> 8;

72 [label="EVAL with clause\ns2l(0, []).\nand substitution\nT2 -> 0\nX6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 72 [arrowhead = none ];
72 -> 9;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 73 [arrowhead = none ];
73 -> 10;

74 [label="EVAL with clause\ns2l(X10, .(X11, X12)) :- ','(no(zero(X10)), ','(p(X10, X13), s2l(X13, X12))).\nand substitution\nT2 -> T4\nX10 -> T4\nX11 -> X14\nX12 -> X15\nX6 -> .(X14, X15)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 74 [arrowhead = none ];
74 -> 12;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 75 [arrowhead = none ];
75 -> 11;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="EVAL with clause\nno(X17) :- ','(X17, ','(!, failure(a))).\nand substitution\nT4 -> T5\nX17 -> zero(T5)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 78 [arrowhead = none ];
78 -> 15;

79 [label="EVAL with clause\nzero(0).\nand substitution\nT5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 79 [arrowhead = none ];
79 -> 16;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 80 [arrowhead = none ];
80 -> 17;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="EVAL with clause\nno(X19).\nand substitution\nT5 -> T6\nX19 -> zero(T6)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 83 [arrowhead = none ];
83 -> 19;

84 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 84 [arrowhead = none ];
84 -> 20;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 85 [arrowhead = none ];
85 -> 22;

86 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T6), zero(0))", fontsize=14, style = filled, fillcolor = white];
22 -> 86 [arrowhead = none ];
86 -> 23;

87 [label="EVAL with clause\np(s(X21), X21).\nand substitution\nX21 -> T7\nT6 -> s(T7)\nX13 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 87 [arrowhead = none ];
87 -> 24;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 88 [arrowhead = none ];
88 -> 25;

89 [label="INSTANCE with matching:\nT2 -> T7\nX6 -> X15", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 89 [arrowhead = none , style=dashed];
89 -> 4[style=dashed];

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 90 [arrowhead = none ];
90 -> 27;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 91 [arrowhead = none ];
91 -> 28;

92 [label="EVAL with clause\nlist([]).\nand substitution\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 92 [arrowhead = none ];
92 -> 29;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 93 [arrowhead = none ];
93 -> 30;

94 [label="EVAL with clause\nlist(X23) :- ','(no(empty(X23)), ','(tail(X23, X24), list(X24))).\nand substitution\nT3 -> T9\nX23 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 94 [arrowhead = none ];
94 -> 32;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 96 [arrowhead = none ];
96 -> 33;

97 [label="EVAL with clause\nno(X26) :- ','(X26, ','(!, failure(a))).\nand substitution\nT9 -> T11\nX26 -> empty(T11)\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="EVAL with clause\nempty([]).\nand substitution\nT11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 99 [arrowhead = none ];
99 -> 36;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 37;

101 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="EVAL with clause\nno(X28).\nand substitution\nT9 -> T13\nX28 -> empty(T13)\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 102 [arrowhead = none ];
102 -> 41;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 103 [arrowhead = none ];
103 -> 39;

104 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
39 -> 104 [arrowhead = none ];
104 -> 40;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 105 [arrowhead = none ];
105 -> 42;

106 [label="EVAL with clause\ntail([], []).\nand substitution\nT13 -> []\nX24 -> []", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 106 [arrowhead = none ];
106 -> 43;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 107 [arrowhead = none ];
107 -> 44;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 108 [arrowhead = none ];
108 -> 45;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 109 [arrowhead = none ];
109 -> 46;

110 [label="EVAL with clause\ntail(.(X40, X41), X41).\nand substitution\nX40 -> T17\nX41 -> T19\nT13 -> .(T17, T19)\nX24 -> T19\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 110 [arrowhead = none ];
110 -> 62;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 111 [arrowhead = none ];
111 -> 63;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 112 [arrowhead = none ];
112 -> 47;

113 [label="EVAL with clause\ntail(.(X36, X37), X37).\nand substitution\nX36 -> T14\nX37 -> T16\nT13 -> .(T14, T16)\nX24 -> T16\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 113 [arrowhead = none ];
113 -> 60;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 114 [arrowhead = none ];
114 -> 61;

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

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 116 [arrowhead = none ];
116 -> 49;

117 [label="EVAL with clause\nlist([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 117 [arrowhead = none ];
117 -> 50;

118 [label="EVAL with clause\nlist(X30) :- ','(no(empty(X30)), ','(tail(X30, X31), list(X31))).\nand substitution\nX30 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 118 [arrowhead = none ];
118 -> 52;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 119 [arrowhead = none ];
119 -> 51;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 120 [arrowhead = none ];
120 -> 53;

121 [label="EVAL with clause\nno(X33) :- ','(X33, ','(!, failure(a))).\nand substitution\nX33 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 121 [arrowhead = none ];
121 -> 54;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 122 [arrowhead = none ];
122 -> 55;

123 [label="EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 123 [arrowhead = none ];
123 -> 56;

124 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 124 [arrowhead = none ];
124 -> 57;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 125 [arrowhead = none ];
125 -> 58;

126 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
58 -> 126 [arrowhead = none ];
126 -> 59;

127 [label="INSTANCE with matching:\nT3 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 127 [arrowhead = none , style=dashed];
127 -> 5[style=dashed];

128 [label="INSTANCE with matching:\nT3 -> T19", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 128 [arrowhead = none , style=dashed];
128 -> 5[style=dashed];

}

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T7), .(X14, X15)) :- s2l4(T7, X15).
list5([]).
list5([]).
list5(.(T14, T16)) :- list5(T16).
list5(.(T17, T19)) :- list5(T19).
goal1(T2) :- s2l4(T2, X6).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- ','(s2l4(T2, .(T14, T16)), list5(T16)).
goal1(T2) :- ','(s2l4(T2, .(T17, T19)), list5(T19)).

Queries:

goal1(g).

(3) PrologToPiTRSProof (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)
s2l4_in: (b,f) (b,b)
list5_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

(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:

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X6))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X6)
S2L4_IN_GA(s(T7), .(X14, X15)) → U1_GA(T7, X14, X15, s2l4_in_ga(T7, X15))
S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T7), .(X14, X15)) → U1_GG(T7, X14, X15, s2l4_in_gg(T7, X15))
S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T14, T16)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T14, T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_G(T2, list5_in_g(T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → LIST5_IN_G(T16)
LIST5_IN_G(.(T14, T16)) → U2_G(T14, T16, list5_in_g(T16))
LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)
LIST5_IN_G(.(T17, T19)) → U3_G(T17, T19, list5_in_g(T19))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_G(T2, list5_in_g(T19))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → LIST5_IN_G(T19)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X6))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X6)
S2L4_IN_GA(s(T7), .(X14, X15)) → U1_GA(T7, X14, X15, s2l4_in_ga(T7, X15))
S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T7), .(X14, X15)) → U1_GG(T7, X14, X15, s2l4_in_gg(T7, X15))
S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T14, T16)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T14, T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_G(T2, list5_in_g(T16))
U6_G(T2, s2l4_out_ga(T2, .(T14, T16))) → LIST5_IN_G(T16)
LIST5_IN_G(.(T14, T16)) → U2_G(T14, T16, list5_in_g(T16))
LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)
LIST5_IN_G(.(T17, T19)) → U3_G(T17, T19, list5_in_g(T19))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_G(T2, list5_in_g(T19))
U8_G(T2, s2l4_out_ga(T2, .(T17, T19))) → LIST5_IN_G(T19)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
LIST5_IN_G(x1)  =  LIST5_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:

LIST5_IN_G(.(T14, T16)) → LIST5_IN_G(T16)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LIST5_IN_G(x1)  =  LIST5_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:

LIST5_IN_G(.(T16)) → LIST5_IN_G(T16)

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:

• LIST5_IN_G(.(T16)) → LIST5_IN_G(T16)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T7), .(X14, X15)) → S2L4_IN_GG(T7, X15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T7), .(X15)) → S2L4_IN_GG(T7, X15)

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:

• S2L4_IN_GG(s(T7), .(X15)) → S2L4_IN_GG(T7, X15)
  The graph contains the following edges 1 > 1, 2 > 2

(22) TRUE

(23) Obligation:

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

S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X6))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T7), .(X14, X15)) → U1_ga(T7, X14, X15, s2l4_in_ga(T7, X15))
U1_ga(T7, X14, X15, s2l4_out_ga(T7, X15)) → s2l4_out_ga(s(T7), .(X14, X15))
U4_g(T2, s2l4_out_ga(T2, X6)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T7), .(X14, X15)) → U1_gg(T7, X14, X15, s2l4_in_gg(T7, X15))
U1_gg(T7, X14, X15, s2l4_out_gg(T7, X15)) → s2l4_out_gg(s(T7), .(X14, X15))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T14, T16)))
U6_g(T2, s2l4_out_ga(T2, .(T14, T16))) → U7_g(T2, list5_in_g(T16))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T14, T16)) → U2_g(T14, T16, list5_in_g(T16))
list5_in_g(.(T17, T19)) → U3_g(T17, T19, list5_in_g(T19))
U3_g(T17, T19, list5_out_g(T19)) → list5_out_g(.(T17, T19))
U2_g(T14, T16, list5_out_g(T16)) → list5_out_g(.(T14, T16))
U7_g(T2, list5_out_g(T16)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T17, T19)))
U8_g(T2, s2l4_out_ga(T2, .(T17, T19))) → U9_g(T2, list5_in_g(T19))
U9_g(T2, list5_out_g(T19)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

S2L4_IN_GA(s(T7), .(X14, X15)) → S2L4_IN_GA(T7, X15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)

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

(28) 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:

• S2L4_IN_GA(s(T7)) → S2L4_IN_GA(T7)
  The graph contains the following edges 1 > 1

(29) TRUE