Left Termination of the query pattern int_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

intlist([], []).
intlist(.(X, XS), .(s(X), YS)) :- intlist(XS, YS).
int(0, 0, .(0, [])).
int(0, s(Y), .(0, XS)) :- int(s(0), s(Y), XS).
int(s(X), 0, []).
int(s(X), s(Y), XS) :- ','(int(X, Y, ZS), intlist(ZS, XS)).

Queries:

int(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: int(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: int(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | int(0, 0, T3), SCOPE: 1, CLAUSE: 3 | int(0, 0, T3), SCOPE: 1, CLAUSE: 4 | int(0, 0, T3), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: int(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground\nint(T1, T2, T3) !=? int(0, 0, .(0, []))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: int(0, 0, T3), SCOPE: 1, CLAUSE: 3 | int(0, 0, T3), SCOPE: 1, CLAUSE: 4 | int(0, 0, T3), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: int(0, 0, T3), SCOPE: 1, CLAUSE: 4 | int(0, 0, T3), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: int(0, 0, T3), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: int(s(0), s(T6), T8) | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: int(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | int(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground\nint(T1, T2, T3) !=? int(0, 0, .(0, []))\nint(T1, T2, T3) !=? int(0, s(X9), .(0, X10))", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 2 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 3 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 4 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 5 | ?_2 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 3 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 4 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 5 | ?_2 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 4 | int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 5 | ?_2 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 5 | ?_2 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: int(s(0), s(T6), T8), SCOPE: 2, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_2 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(int(0, T19, X33), intlist(X33, T21))\n\nT19 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: int(0, T19, X33)\n\nT19 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: intlist(T22, T21)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: intlist(T22, T21), SCOPE: 3, CLAUSE: 0 | intlist(T22, T21), SCOPE: 3, CLAUSE: 1\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: intlist(T22, T21), SCOPE: 3, CLAUSE: 0\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: intlist(T22, T21), SCOPE: 3, CLAUSE: 1\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: intlist(T30, T32)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: int(0, s(T6), T3), SCOPE: 1, CLAUSE: 4 | int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: int(0, s(T6), T3), SCOPE: 1, CLAUSE: 5\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true | int(s(T34), 0, T3), SCOPE: 1, CLAUSE: 5\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: int(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground\nint(T1, T2, T3) !=? int(0, 0, .(0, []))\nint(T1, T2, T3) !=? int(0, s(X9), .(0, X10))\nint(T1, T2, T3) !=? int(s(X51), 0, [])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: int(s(T34), 0, T3), SCOPE: 1, CLAUSE: 5\n\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(int(T39, T40, X62), intlist(X62, T42))\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 2 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 3 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 4 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 2\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 3 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 4 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: intlist(.(0, []), T42)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: intlist(.(0, []), T42), SCOPE: 5, CLAUSE: 0 | intlist(.(0, []), T42), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: intlist(.(0, []), T42), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: intlist([], T47)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: intlist([], T47), SCOPE: 6, CLAUSE: 0 | intlist([], T47), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: intlist([], T47), SCOPE: 6, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: intlist([], T47), SCOPE: 6, CLAUSE: 1\n\n", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 3\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 4 | ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(int(s(0), s(T52), X98), intlist(.(0, X98), T42))\n\nT52 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: int(s(0), s(T52), X98)\n\nT52 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: intlist(.(0, T54), T42)\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 4\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(int(T39, T40, X62), intlist(X62, T42)), SCOPE: 4, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX62 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: intlist([], T42)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(int(T67, T68, X140), ','(intlist(X140, X141), intlist(X141, T42)))\n\nT67 is ground\nT68 is ground\nX141 is free; \nX140 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: int(T67, T68, X140)\n\nT67 is ground\nT68 is ground\nX140 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(intlist(T69, X141), intlist(X141, T42))\n\nT69 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: intlist(T69, X141)\n\nT69 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: intlist(T70, T42)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: intlist(T69, X141), SCOPE: 7, CLAUSE: 0 | intlist(T69, X141), SCOPE: 7, CLAUSE: 1\n\nT69 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: intlist(T69, X141), SCOPE: 7, CLAUSE: 0\n\nT69 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: intlist(T69, X141), SCOPE: 7, CLAUSE: 1\n\nT69 is ground\nX141 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: intlist(T76, X156)\n\nT76 is ground\nX156 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="EVAL with clause\nint(0, 0, .(0, [])).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> .(0, [])", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="EVAL with clause\nint(0, s(X9), .(0, X10)) :- int(s(0), s(X9), X10).\nand substitution\nT1 -> 0\nX9 -> T6\nT2 -> s(T6)\nX10 -> T8\nT3 -> .(0, T8)\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 81 [arrowhead = none ];
81 -> 9;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 82 [arrowhead = none ];
82 -> 10;

83 [label="BACKTRACK\nfor clause: int(0, s(Y), .(0, XS)) :- int(s(0), s(Y), XS)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="BACKTRACK\nfor clause: int(s(X), 0, [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="BACKTRACK\nfor clause: int(s(X), s(Y), XS) :- ','(int(X, Y, ZS), intlist(ZS, XS))because of non-unification", fontsize=14, style = filled, fillcolor = white];
7 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 86 [arrowhead = none ];
86 -> 11;

87 [label="EVAL with clause\nint(s(X51), 0, []).\nand substitution\nX51 -> T34\nT1 -> s(T34)\nT2 -> 0\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 87 [arrowhead = none ];
87 -> 31;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 88 [arrowhead = none ];
88 -> 32;

89 [label="BACKTRACK\nfor clause: int(0, 0, .(0, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="BACKTRACK\nfor clause: int(0, s(Y), .(0, XS)) :- int(s(0), s(Y), XS)because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="BACKTRACK\nfor clause: int(s(X), 0, [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 91 [arrowhead = none ];
91 -> 14;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 92 [arrowhead = none ];
92 -> 15;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 93 [arrowhead = none ];
93 -> 16;

94 [label="ONLY EVAL with clause\nint(s(X30), s(X31), X32) :- ','(int(X30, X31, X33), intlist(X33, X32)).\nand substitution\nX30 -> 0\nT6 -> T19\nX31 -> T19\nT8 -> T21\nX32 -> T21\nT20 -> T21", fontsize=14, style = filled, fillcolor = white];
15 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 95 [arrowhead = none ];
95 -> 28;

96 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 96 [arrowhead = none ];
96 -> 18;

97 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX33 -> T22", fontsize=14, style = filled, fillcolor = violet];
17 -> 97 [arrowhead = none ];
97 -> 19;

98 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> T19\nT3 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 98 [arrowhead = none , style=dashed];
98 -> 1[style=dashed];

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 99 [arrowhead = none ];
99 -> 20;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 100 [arrowhead = none ];
100 -> 21;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 101 [arrowhead = none ];
101 -> 22;

102 [label="EVAL with clause\nintlist([], []).\nand substitution\nT22 -> []\nT21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 102 [arrowhead = none ];
102 -> 23;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 103 [arrowhead = none ];
103 -> 24;

104 [label="EVAL with clause\nintlist(.(X40, X41), .(s(X40), X42)) :- intlist(X41, X42).\nand substitution\nX40 -> T29\nX41 -> T30\nT22 -> .(T29, T30)\nX42 -> T32\nT21 -> .(s(T29), T32)\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 106 [arrowhead = none ];
106 -> 25;

107 [label="INSTANCE with matching:\nT22 -> T30\nT21 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 107 [arrowhead = none , style=dashed];
107 -> 19[style=dashed];

108 [label="BACKTRACK\nfor clause: int(s(X), 0, [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
28 -> 108 [arrowhead = none ];
108 -> 29;

109 [label="BACKTRACK\nfor clause: int(s(X), s(Y), XS) :- ','(int(X, Y, ZS), intlist(ZS, XS))because of non-unification", fontsize=14, style = filled, fillcolor = white];
29 -> 109 [arrowhead = none ];
109 -> 30;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="EVAL with clause\nint(s(X59), s(X60), X61) :- ','(int(X59, X60, X62), intlist(X62, X61)).\nand substitution\nX59 -> T39\nT1 -> s(T39)\nX60 -> T40\nT2 -> s(T40)\nT3 -> T42\nX61 -> T42\nT41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 111 [arrowhead = none ];
111 -> 35;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 112 [arrowhead = none ];
112 -> 36;

113 [label="BACKTRACK\nfor clause: int(s(X), s(Y), XS) :- ','(int(X, Y, ZS), intlist(ZS, XS))because of non-unification", fontsize=14, style = filled, fillcolor = white];
33 -> 113 [arrowhead = none ];
113 -> 34;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 114 [arrowhead = none ];
114 -> 37;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 115 [arrowhead = none ];
115 -> 38;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 116 [arrowhead = none ];
116 -> 39;

117 [label="EVAL with clause\nint(0, 0, .(0, [])).\nand substitution\nT39 -> 0\nT40 -> 0\nX62 -> .(0, [])", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 117 [arrowhead = none ];
117 -> 40;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 118 [arrowhead = none ];
118 -> 41;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 119 [arrowhead = none ];
119 -> 53;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 120 [arrowhead = none ];
120 -> 54;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 121 [arrowhead = none ];
121 -> 42;

122 [label="BACKTRACK\nfor clause: intlist([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
42 -> 122 [arrowhead = none ];
122 -> 43;

123 [label="EVAL with clause\nintlist(.(X75, X76), .(s(X75), X77)) :- intlist(X76, X77).\nand substitution\nX75 -> 0\nX76 -> []\nX77 -> T47\nT42 -> .(s(0), T47)\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 123 [arrowhead = none ];
123 -> 44;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 124 [arrowhead = none ];
124 -> 45;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="EVAL with clause\nintlist([], []).\nand substitution\nT47 -> []", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 128 [arrowhead = none ];
128 -> 49;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 129 [arrowhead = none ];
129 -> 50;

130 [label="BACKTRACK\nfor clause: intlist(.(X, XS), .(s(X), YS)) :- intlist(XS, YS)because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 130 [arrowhead = none ];
130 -> 52;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 131 [arrowhead = none ];
131 -> 51;

132 [label="EVAL with clause\nint(0, s(X96), .(0, X97)) :- int(s(0), s(X96), X97).\nand substitution\nT39 -> 0\nX96 -> T52\nT40 -> s(T52)\nX97 -> X98\nX62 -> .(0, X98)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 132 [arrowhead = none ];
132 -> 55;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 133 [arrowhead = none ];
133 -> 56;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 134 [arrowhead = none ];
134 -> 59;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 135 [arrowhead = none ];
135 -> 60;

136 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
55 -> 136 [arrowhead = none ];
136 -> 57;

137 [label="SPLIT 2\nnew knowledge:\nT54 is ground\nreplacements:\nX98 -> T54", fontsize=14, style = filled, fillcolor = violet];
55 -> 137 [arrowhead = none ];
137 -> 58;

138 [label="INSTANCE with matching:\nT1 -> s(0)\nT2 -> s(T52)\nT3 -> X98", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 138 [arrowhead = none , style=dashed];
138 -> 1[style=dashed];

139 [label="INSTANCE with matching:\nT22 -> .(0, T54)\nT21 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 139 [arrowhead = none , style=dashed];
139 -> 19[style=dashed];

140 [label="EVAL with clause\nint(s(X126), 0, []).\nand substitution\nX126 -> T62\nT39 -> s(T62)\nT40 -> 0\nX62 -> []", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 140 [arrowhead = none ];
140 -> 61;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 141 [arrowhead = none ];
141 -> 62;

142 [label="EVAL with clause\nint(s(X137), s(X138), X139) :- ','(int(X137, X138, X140), intlist(X140, X139)).\nand substitution\nX137 -> T67\nT39 -> s(T67)\nX138 -> T68\nT40 -> s(T68)\nX62 -> X141\nX139 -> X141", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 142 [arrowhead = none ];
142 -> 63;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 143 [arrowhead = none ];
143 -> 64;

144 [label="INSTANCE with matching:\nT47 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 144 [arrowhead = none , style=dashed];
144 -> 44[style=dashed];

145 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
63 -> 145 [arrowhead = none ];
145 -> 65;

146 [label="SPLIT 2\nnew knowledge:\nT69 is ground\nreplacements:\nX140 -> T69", fontsize=14, style = filled, fillcolor = violet];
63 -> 146 [arrowhead = none ];
146 -> 66;

147 [label="INSTANCE with matching:\nT1 -> T67\nT2 -> T68\nT3 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 147 [arrowhead = none , style=dashed];
147 -> 1[style=dashed];

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 148 [arrowhead = none ];
148 -> 67;

149 [label="SPLIT 2\nnew knowledge:\nT70 is ground\nreplacements:\nX141 -> T70", fontsize=14, style = filled, fillcolor = violet];
66 -> 149 [arrowhead = none ];
149 -> 68;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 150 [arrowhead = none ];
150 -> 69;

151 [label="INSTANCE with matching:\nT22 -> T70\nT21 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 151 [arrowhead = none , style=dashed];
151 -> 19[style=dashed];

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 152 [arrowhead = none ];
152 -> 70;

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

154 [label="EVAL with clause\nintlist([], []).\nand substitution\nT69 -> []\nX141 -> []", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 154 [arrowhead = none ];
154 -> 72;

155 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 155 [arrowhead = none ];
155 -> 73;

156 [label="EVAL with clause\nintlist(.(X153, X154), .(s(X153), X155)) :- intlist(X154, X155).\nand substitution\nX153 -> T75\nX154 -> T76\nT69 -> .(T75, T76)\nX155 -> X156\nX141 -> .(s(T75), X156)", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 156 [arrowhead = none ];
156 -> 75;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 157 [arrowhead = none ];
157 -> 76;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 158 [arrowhead = none ];
158 -> 74;

159 [label="INSTANCE with matching:\nT69 -> T76\nX141 -> X156", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 159 [arrowhead = none , style=dashed];
159 -> 67[style=dashed];

}

(2) Obligation:

Triples:

intlist19(.(T29, T30), .(s(T29), T32)) :- intlist19(T30, T32).
intlist67(.(T75, T76), .(s(T75), X156)) :- intlist67(T76, X156).
int1(0, s(T19), .(0, T21)) :- int1(0, T19, X33).
int1(0, s(T19), .(0, T21)) :- ','(intc1(0, T19, T22), intlist19(T22, T21)).
int1(s(0), s(0), .(s(0), T47)) :- intlist44(T47).
int1(s(0), s(s(T52)), T42) :- int1(s(0), s(T52), X98).
int1(s(0), s(s(T52)), T42) :- ','(intc1(s(0), s(T52), T54), intlist19(.(0, T54), T42)).
int1(s(s(T62)), s(0), T42) :- intlist44(T42).
int1(s(s(T67)), s(s(T68)), T42) :- int1(T67, T68, X140).
int1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), intlist67(T69, X141)).
int1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), ','(intlistc67(T69, T70), intlist19(T70, T42))).

Clauses:

intc1(0, 0, .(0, [])).
intc1(0, s(T19), .(0, T21)) :- ','(intc1(0, T19, T22), intlistc19(T22, T21)).
intc1(s(T34), 0, []).
intc1(s(0), s(0), .(s(0), T47)) :- intlistc44(T47).
intc1(s(0), s(s(T52)), T42) :- ','(intc1(s(0), s(T52), T54), intlistc19(.(0, T54), T42)).
intc1(s(s(T62)), s(0), T42) :- intlistc44(T42).
intc1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), ','(intlistc67(T69, T70), intlistc19(T70, T42))).
intlistc19([], []).
intlistc19(.(T29, T30), .(s(T29), T32)) :- intlistc19(T30, T32).
intlistc44([]).
intlistc67([], []).
intlistc67(.(T75, T76), .(s(T75), X156)) :- intlistc67(T76, X156).

Afs:

int1(x1, x2, x3)  =  int1(x1, x2)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

intlist19(.(T29, T30), .(s(T29), T32)) :- intlist19(T30, T32).
intlist67(.(T75, T76), .(s(T75), X156)) :- intlist67(T76, X156).
int1(0, s(T19), .(0, T21)) :- int1(0, T19, X33).
int1(0, s(T19), .(0, T21)) :- ','(intc1(0, T19, T22), intlist19(T22, T21)).
int1(s(0), s(s(T52)), T42) :- int1(s(0), s(T52), X98).
int1(s(0), s(s(T52)), T42) :- ','(intc1(s(0), s(T52), T54), intlist19(.(0, T54), T42)).
int1(s(s(T67)), s(s(T68)), T42) :- int1(T67, T68, X140).
int1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), intlist67(T69, X141)).
int1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), ','(intlistc67(T69, T70), intlist19(T70, T42))).

Clauses:

intc1(0, 0, .(0, [])).
intc1(0, s(T19), .(0, T21)) :- ','(intc1(0, T19, T22), intlistc19(T22, T21)).
intc1(s(T34), 0, []).
intc1(s(0), s(0), .(s(0), T47)) :- intlistc44(T47).
intc1(s(0), s(s(T52)), T42) :- ','(intc1(s(0), s(T52), T54), intlistc19(.(0, T54), T42)).
intc1(s(s(T62)), s(0), T42) :- intlistc44(T42).
intc1(s(s(T67)), s(s(T68)), T42) :- ','(intc1(T67, T68, T69), ','(intlistc67(T69, T70), intlistc19(T70, T42))).
intlistc19([], []).
intlistc19(.(T29, T30), .(s(T29), T32)) :- intlistc19(T30, T32).
intlistc44([]).
intlistc67([], []).
intlistc67(.(T75, T76), .(s(T75), X156)) :- intlistc67(T76, X156).

Afs:

int1(x1, x2, x3)  =  int1(x1, x2)

(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:
int1_in: (b,b,f)
intc1_in: (b,b,f)
intlistc67_in: (b,f)
intlistc19_in: (b,f)
intlist19_in: (b,f)
intlist67_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

INT1_IN_GGA(0, s(T19), .(0, T21)) → U3_GGA(T19, T21, int1_in_gga(0, T19, X33))
INT1_IN_GGA(0, s(T19), .(0, T21)) → INT1_IN_GGA(0, T19, X33)
INT1_IN_GGA(0, s(T19), .(0, T21)) → U4_GGA(T19, T21, intc1_in_gga(0, T19, T22))
U4_GGA(T19, T21, intc1_out_gga(0, T19, T22)) → U5_GGA(T19, T21, intlist19_in_ga(T22, T21))
U4_GGA(T19, T21, intc1_out_gga(0, T19, T22)) → INTLIST19_IN_GA(T22, T21)
INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → U1_GA(T29, T30, T32, intlist19_in_ga(T30, T32))
INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → INTLIST19_IN_GA(T30, T32)
INT1_IN_GGA(s(0), s(s(T52)), T42) → U6_GGA(T52, T42, int1_in_gga(s(0), s(T52), X98))
INT1_IN_GGA(s(0), s(s(T52)), T42) → INT1_IN_GGA(s(0), s(T52), X98)
INT1_IN_GGA(s(0), s(s(T52)), T42) → U7_GGA(T52, T42, intc1_in_gga(s(0), s(T52), T54))
U7_GGA(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U8_GGA(T52, T42, intlist19_in_ga(.(0, T54), T42))
U7_GGA(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → INTLIST19_IN_GA(.(0, T54), T42)
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → U9_GGA(T67, T68, T42, int1_in_gga(T67, T68, X140))
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → INT1_IN_GGA(T67, T68, X140)
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → U10_GGA(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U11_GGA(T67, T68, T42, intlist67_in_ga(T69, X141))
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → INTLIST67_IN_GA(T69, X141)
INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → U2_GA(T75, T76, X156, intlist67_in_ga(T76, X156))
INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → INTLIST67_IN_GA(T76, X156)
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U12_GGA(T67, T68, T42, intlistc67_in_ga(T69, T70))
U12_GGA(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U13_GGA(T67, T68, T42, intlist19_in_ga(T70, T42))
U12_GGA(T67, T68, T42, intlistc67_out_ga(T69, T70)) → INTLIST19_IN_GA(T70, T42)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
int1_in_gga(x1, x2, x3)  =  int1_in_gga(x1, x2)
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
intlist19_in_ga(x1, x2)  =  intlist19_in_ga(x1)
intlist67_in_ga(x1, x2)  =  intlist67_in_ga(x1)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3)  =  U3_GGA(x1, x3)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x1, x3)
INTLIST19_IN_GA(x1, x2)  =  INTLIST19_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U6_GGA(x1, x2, x3)  =  U6_GGA(x1, x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x3)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
INTLIST67_IN_GA(x1, x2)  =  INTLIST67_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)

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:

INT1_IN_GGA(0, s(T19), .(0, T21)) → U3_GGA(T19, T21, int1_in_gga(0, T19, X33))
INT1_IN_GGA(0, s(T19), .(0, T21)) → INT1_IN_GGA(0, T19, X33)
INT1_IN_GGA(0, s(T19), .(0, T21)) → U4_GGA(T19, T21, intc1_in_gga(0, T19, T22))
U4_GGA(T19, T21, intc1_out_gga(0, T19, T22)) → U5_GGA(T19, T21, intlist19_in_ga(T22, T21))
U4_GGA(T19, T21, intc1_out_gga(0, T19, T22)) → INTLIST19_IN_GA(T22, T21)
INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → U1_GA(T29, T30, T32, intlist19_in_ga(T30, T32))
INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → INTLIST19_IN_GA(T30, T32)
INT1_IN_GGA(s(0), s(s(T52)), T42) → U6_GGA(T52, T42, int1_in_gga(s(0), s(T52), X98))
INT1_IN_GGA(s(0), s(s(T52)), T42) → INT1_IN_GGA(s(0), s(T52), X98)
INT1_IN_GGA(s(0), s(s(T52)), T42) → U7_GGA(T52, T42, intc1_in_gga(s(0), s(T52), T54))
U7_GGA(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U8_GGA(T52, T42, intlist19_in_ga(.(0, T54), T42))
U7_GGA(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → INTLIST19_IN_GA(.(0, T54), T42)
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → U9_GGA(T67, T68, T42, int1_in_gga(T67, T68, X140))
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → INT1_IN_GGA(T67, T68, X140)
INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → U10_GGA(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U11_GGA(T67, T68, T42, intlist67_in_ga(T69, X141))
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → INTLIST67_IN_GA(T69, X141)
INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → U2_GA(T75, T76, X156, intlist67_in_ga(T76, X156))
INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → INTLIST67_IN_GA(T76, X156)
U10_GGA(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U12_GGA(T67, T68, T42, intlistc67_in_ga(T69, T70))
U12_GGA(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U13_GGA(T67, T68, T42, intlist19_in_ga(T70, T42))
U12_GGA(T67, T68, T42, intlistc67_out_ga(T69, T70)) → INTLIST19_IN_GA(T70, T42)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
int1_in_gga(x1, x2, x3)  =  int1_in_gga(x1, x2)
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
intlist19_in_ga(x1, x2)  =  intlist19_in_ga(x1)
intlist67_in_ga(x1, x2)  =  intlist67_in_ga(x1)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3)  =  U3_GGA(x1, x3)
U4_GGA(x1, x2, x3)  =  U4_GGA(x1, x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x1, x3)
INTLIST19_IN_GA(x1, x2)  =  INTLIST19_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U6_GGA(x1, x2, x3)  =  U6_GGA(x1, x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x1, x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x3)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
INTLIST67_IN_GA(x1, x2)  =  INTLIST67_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 17 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → INTLIST67_IN_GA(T76, X156)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
INTLIST67_IN_GA(x1, x2)  =  INTLIST67_IN_GA(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:

INTLIST67_IN_GA(.(T75, T76), .(s(T75), X156)) → INTLIST67_IN_GA(T76, X156)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
INTLIST67_IN_GA(x1, x2)  =  INTLIST67_IN_GA(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:

INTLIST67_IN_GA(.(T75, T76)) → INTLIST67_IN_GA(T76)

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:

• INTLIST67_IN_GA(.(T75, T76)) → INTLIST67_IN_GA(T76)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → INTLIST19_IN_GA(T30, T32)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
INTLIST19_IN_GA(x1, x2)  =  INTLIST19_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:

INTLIST19_IN_GA(.(T29, T30), .(s(T29), T32)) → INTLIST19_IN_GA(T30, T32)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
INTLIST19_IN_GA(x1, x2)  =  INTLIST19_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:

INTLIST19_IN_GA(.(T29, T30)) → INTLIST19_IN_GA(T30)

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:

• INTLIST19_IN_GA(.(T29, T30)) → INTLIST19_IN_GA(T30)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

INT1_IN_GGA(s(0), s(s(T52)), T42) → INT1_IN_GGA(s(0), s(T52), X98)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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:

INT1_IN_GGA(s(0), s(s(T52)), T42) → INT1_IN_GGA(s(0), s(T52), X98)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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:

INT1_IN_GGA(s(0), s(s(T52))) → INT1_IN_GGA(s(0), s(T52))

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:

• INT1_IN_GGA(s(0), s(s(T52))) → INT1_IN_GGA(s(0), s(T52))
  The graph contains the following edges 1 >= 1, 2 > 2

(29) YES

(30) Obligation:

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

INT1_IN_GGA(0, s(T19), .(0, T21)) → INT1_IN_GGA(0, T19, X33)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

INT1_IN_GGA(0, s(T19), .(0, T21)) → INT1_IN_GGA(0, T19, X33)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
.(x1, x2)  =  .(x1, x2)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

INT1_IN_GGA(0, s(T19)) → INT1_IN_GGA(0, T19)

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

(35) 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:

• INT1_IN_GGA(0, s(T19)) → INT1_IN_GGA(0, T19)
  The graph contains the following edges 1 >= 1, 2 > 2

(36) YES

(37) Obligation:

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

INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → INT1_IN_GGA(T67, T68, X140)

The TRS R consists of the following rules:

intc1_in_gga(0, 0, .(0, [])) → intc1_out_gga(0, 0, .(0, []))
intc1_in_gga(0, s(T19), .(0, T21)) → U15_gga(T19, T21, intc1_in_gga(0, T19, T22))
intc1_in_gga(s(T34), 0, []) → intc1_out_gga(s(T34), 0, [])
intc1_in_gga(s(0), s(0), .(s(0), T47)) → U17_gga(T47, intlistc44_in_a(T47))
intlistc44_in_a([]) → intlistc44_out_a([])
U17_gga(T47, intlistc44_out_a(T47)) → intc1_out_gga(s(0), s(0), .(s(0), T47))
intc1_in_gga(s(0), s(s(T52)), T42) → U18_gga(T52, T42, intc1_in_gga(s(0), s(T52), T54))
intc1_in_gga(s(s(T62)), s(0), T42) → U20_gga(T62, T42, intlistc44_in_a(T42))
U20_gga(T62, T42, intlistc44_out_a(T42)) → intc1_out_gga(s(s(T62)), s(0), T42)
intc1_in_gga(s(s(T67)), s(s(T68)), T42) → U21_gga(T67, T68, T42, intc1_in_gga(T67, T68, T69))
U21_gga(T67, T68, T42, intc1_out_gga(T67, T68, T69)) → U22_gga(T67, T68, T42, intlistc67_in_ga(T69, T70))
intlistc67_in_ga([], []) → intlistc67_out_ga([], [])
intlistc67_in_ga(.(T75, T76), .(s(T75), X156)) → U25_ga(T75, T76, X156, intlistc67_in_ga(T76, X156))
U25_ga(T75, T76, X156, intlistc67_out_ga(T76, X156)) → intlistc67_out_ga(.(T75, T76), .(s(T75), X156))
U22_gga(T67, T68, T42, intlistc67_out_ga(T69, T70)) → U23_gga(T67, T68, T42, intlistc19_in_ga(T70, T42))
intlistc19_in_ga([], []) → intlistc19_out_ga([], [])
intlistc19_in_ga(.(T29, T30), .(s(T29), T32)) → U24_ga(T29, T30, T32, intlistc19_in_ga(T30, T32))
U24_ga(T29, T30, T32, intlistc19_out_ga(T30, T32)) → intlistc19_out_ga(.(T29, T30), .(s(T29), T32))
U23_gga(T67, T68, T42, intlistc19_out_ga(T70, T42)) → intc1_out_gga(s(s(T67)), s(s(T68)), T42)
U18_gga(T52, T42, intc1_out_gga(s(0), s(T52), T54)) → U19_gga(T52, T42, intlistc19_in_ga(.(0, T54), T42))
U19_gga(T52, T42, intlistc19_out_ga(.(0, T54), T42)) → intc1_out_gga(s(0), s(s(T52)), T42)
U15_gga(T19, T21, intc1_out_gga(0, T19, T22)) → U16_gga(T19, T21, intlistc19_in_ga(T22, T21))
U16_gga(T19, T21, intlistc19_out_ga(T22, T21)) → intc1_out_gga(0, s(T19), .(0, T21))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
intc1_in_gga(x1, x2, x3)  =  intc1_in_gga(x1, x2)
intc1_out_gga(x1, x2, x3)  =  intc1_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3)  =  U15_gga(x1, x3)
U17_gga(x1, x2)  =  U17_gga(x2)
intlistc44_in_a(x1)  =  intlistc44_in_a
intlistc44_out_a(x1)  =  intlistc44_out_a(x1)
U18_gga(x1, x2, x3)  =  U18_gga(x1, x3)
U20_gga(x1, x2, x3)  =  U20_gga(x1, x3)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
intlistc67_in_ga(x1, x2)  =  intlistc67_in_ga(x1)
[]  =  []
intlistc67_out_ga(x1, x2)  =  intlistc67_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
intlistc19_in_ga(x1, x2)  =  intlistc19_in_ga(x1)
intlistc19_out_ga(x1, x2)  =  intlistc19_out_ga(x1, x2)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x2, x4)
U19_gga(x1, x2, x3)  =  U19_gga(x1, x3)
U16_gga(x1, x2, x3)  =  U16_gga(x1, x3)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

INT1_IN_GGA(s(s(T67)), s(s(T68)), T42) → INT1_IN_GGA(T67, T68, X140)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
INT1_IN_GGA(x1, x2, x3)  =  INT1_IN_GGA(x1, x2)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

INT1_IN_GGA(s(s(T67)), s(s(T68))) → INT1_IN_GGA(T67, T68)

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

(42) 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:

• INT1_IN_GGA(s(s(T67)), s(s(T68))) → INT1_IN_GGA(T67, T68)
  The graph contains the following edges 1 > 1, 2 > 2

(43) YES