Left Termination of the query pattern p_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

p(cons(X, nil)).
p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs)))).
p(cons(0, Xs)) :- p(Xs).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
mult(X1, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).

Queries:

p(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | p(cons(T3, nil)), SCOPE: 1, CLAUSE: 1 | p(cons(T3, nil)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\np(T1) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(cons(T3, nil)), SCOPE: 1, CLAUSE: 1 | p(cons(T3, nil)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(cons(T3, nil)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(nil), SCOPE: 2, CLAUSE: 0 | p(nil), SCOPE: 2, CLAUSE: 1 | p(nil), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(nil), SCOPE: 2, CLAUSE: 1 | p(nil), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(nil), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))) | p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\np(T1) !=? p(cons(X3, nil))\np(T1) !=? p(cons(s(s(X19)), cons(X20, X21)))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 0 | ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 1 | ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 1 | ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 1\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(cons(T24, cons(T25, T26))), ','(mult(T24, T25, X44), ','(p(cons(X44, T26)), ','(mult(s(s(T24)), T25, X22), p(cons(X22, T26))))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX22 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(cons(T24, cons(T25, T26)))\n\nT24 is ground\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(mult(T24, T25, X44), ','(p(cons(X44, T26)), ','(mult(s(s(T24)), T25, X22), p(cons(X22, T26)))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX22 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: mult(T24, T25, X44)\n\nT24 is ground\nT25 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(cons(T39, T26)), ','(mult(s(s(T24)), T25, X22), p(cons(X22, T26))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nT39 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: mult(T24, T25, X44), SCOPE: 4, CLAUSE: 5 | mult(T24, T25, X44), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: mult(T24, T25, X44), SCOPE: 4, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: mult(T24, T25, X44), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(mult(T51, T52, X82), sum(X82, T51, X83))\n\nT51 is ground\nT52 is ground\nX83 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: mult(T51, T52, X82)\n\nT51 is ground\nT52 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: sum(T55, T51, X83)\n\nT51 is ground\nT55 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: sum(T55, T51, X83), SCOPE: 5, CLAUSE: 3 | sum(T55, T51, X83), SCOPE: 5, CLAUSE: 4\n\nT51 is ground\nT55 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: sum(T55, T51, X83), SCOPE: 5, CLAUSE: 3\n\nT51 is ground\nT55 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: sum(T55, T51, X83), SCOPE: 5, CLAUSE: 4\n\nT51 is ground\nT55 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: sum(T69, T70, X110)\n\nT69 is ground\nT70 is ground\nX110 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: p(cons(T39, T26))\n\nT26 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(mult(s(s(T24)), T25, X22), p(cons(X22, T26)))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: mult(s(s(T24)), T25, X22)\n\nT24 is ground\nT25 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(cons(T79, T26))\n\nT26 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(cons(T8, cons(T9, T10))), ','(mult(T8, T9, X22), p(cons(X22, T10)))), SCOPE: 3, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ?_3 | p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(cons(T92, T93)), ','(mult(0, T92, X22), p(cons(X22, T93))))\n\nT92 is ground\nT93 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: p(cons(T92, T93))\n\nT92 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(mult(0, T92, X22), p(cons(X22, T93)))\n\nT92 is ground\nT93 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: mult(0, T92, X22)\n\nT92 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: p(cons(T98, T93))\n\nT93 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: mult(0, T92, X22), SCOPE: 6, CLAUSE: 5 | mult(0, T92, X22), SCOPE: 6, CLAUSE: 6\n\nT92 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: mult(0, T92, X22), SCOPE: 6, CLAUSE: 5\n\nT92 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: mult(0, T92, X22), SCOPE: 6, CLAUSE: 6\n\nT92 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(mult(0, T101, X158), sum(X158, 0, X159))\n\nT101 is ground\nX159 is free; \nX158 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: mult(0, T101, X158)\n\nT101 is ground\nX158 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: sum(T102, 0, X159)\n\nT102 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: sum(T102, 0, X159), SCOPE: 7, CLAUSE: 3 | sum(T102, 0, X159), SCOPE: 7, CLAUSE: 4\n\nT102 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: sum(T102, 0, X159), SCOPE: 7, CLAUSE: 3\n\nT102 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: sum(T102, 0, X159), SCOPE: 7, CLAUSE: 4\n\nT102 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(cons(s(s(T8)), cons(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: p(T113)\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: p(T113), SCOPE: 8, CLAUSE: 0 | p(T113), SCOPE: 8, CLAUSE: 1 | p(T113), SCOPE: 8, CLAUSE: 2\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: p(T113), SCOPE: 8, CLAUSE: 0\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: p(T113), SCOPE: 8, CLAUSE: 1 | p(T113), SCOPE: 8, CLAUSE: 2\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: p(T113), SCOPE: 8, CLAUSE: 1\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: p(T113), SCOPE: 8, CLAUSE: 2\n\nT113 is ground\np(cons(0, T113)) !=? p(cons(X3, nil))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(cons(T131, cons(T132, T133))), ','(mult(T131, T132, X205), p(cons(X205, T133))))\n\nT131 is ground\nT132 is ground\nT133 is ground\nX205 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: p(cons(T131, cons(T132, T133)))\n\nT131 is ground\nT132 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(mult(T131, T132, X205), p(cons(X205, T133)))\n\nT131 is ground\nT132 is ground\nT133 is ground\nX205 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: mult(T131, T132, X205)\n\nT131 is ground\nT132 is ground\nX205 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: p(cons(T146, T133))\n\nT133 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: p(T153)\n\nT153 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 91 [arrowhead = none ];
91 -> 2;

92 [label="EVAL with clause\np(cons(X3, nil)).\nand substitution\nX3 -> T3\nT1 -> cons(T3, nil)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 92 [arrowhead = none ];
92 -> 3;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 93 [arrowhead = none ];
93 -> 4;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 94 [arrowhead = none ];
94 -> 5;

95 [label="EVAL with clause\np(cons(s(s(X19)), cons(X20, X21))) :- ','(p(cons(X19, cons(X20, X21))), ','(mult(X19, X20, X22), p(cons(X22, X21)))).\nand substitution\nX19 -> T8\nX20 -> T9\nX21 -> T10\nT1 -> cons(s(s(T8)), cons(T9, T10))", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 95 [arrowhead = none ];
95 -> 13;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 96 [arrowhead = none ];
96 -> 14;

97 [label="BACKTRACK\nfor clause: p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 97 [arrowhead = none ];
97 -> 6;

98 [label="EVAL with clause\np(cons(0, X9)) :- p(X9).\nand substitution\nT3 -> 0\nX9 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 99 [arrowhead = none ];
99 -> 8;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 100 [arrowhead = none ];
100 -> 9;

101 [label="BACKTRACK\nfor clause: p(cons(X, nil))because of non-unification", fontsize=14, style = filled, fillcolor = white];
9 -> 101 [arrowhead = none ];
101 -> 10;

102 [label="BACKTRACK\nfor clause: p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 102 [arrowhead = none ];
102 -> 11;

103 [label="BACKTRACK\nfor clause: p(cons(0, Xs)) :- p(Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 103 [arrowhead = none ];
103 -> 12;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 104 [arrowhead = none ];
104 -> 15;

105 [label="EVAL with clause\np(cons(0, X180)) :- p(X180).\nand substitution\nX180 -> T113\nT1 -> cons(0, T113)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 105 [arrowhead = none ];
105 -> 73;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 106 [arrowhead = none ];
106 -> 74;

107 [label="BACKTRACK\nfor clause: p(cons(X, nil))because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 107 [arrowhead = none ];
107 -> 16;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 108 [arrowhead = none ];
108 -> 17;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 109 [arrowhead = none ];
109 -> 18;

110 [label="EVAL with clause\np(cons(s(s(X41)), cons(X42, X43))) :- ','(p(cons(X41, cons(X42, X43))), ','(mult(X41, X42, X44), p(cons(X44, X43)))).\nand substitution\nX41 -> T24\nT8 -> s(s(T24))\nT9 -> T25\nX42 -> T25\nT10 -> T26\nX43 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 110 [arrowhead = none ];
110 -> 19;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 111 [arrowhead = none ];
111 -> 20;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 112 [arrowhead = none ];
112 -> 47;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 113 [arrowhead = none ];
113 -> 48;

114 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 114 [arrowhead = none ];
114 -> 21;

115 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
19 -> 115 [arrowhead = none ];
115 -> 22;

116 [label="INSTANCE with matching:\nT1 -> cons(T24, cons(T25, T26))", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 116 [arrowhead = none , style=dashed];
116 -> 1[style=dashed];

117 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 117 [arrowhead = none ];
117 -> 23;

118 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nreplacements:\nX44 -> T39", fontsize=14, style = filled, fillcolor = violet];
22 -> 118 [arrowhead = none ];
118 -> 24;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 119 [arrowhead = none ];
119 -> 25;

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 120 [arrowhead = none ];
120 -> 43;

121 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
24 -> 121 [arrowhead = none ];
121 -> 44;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 122 [arrowhead = none ];
122 -> 26;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 123 [arrowhead = none ];
123 -> 27;

124 [label="EVAL with clause\nmult(X68, 0, 0).\nand substitution\nT24 -> T46\nX68 -> T46\nT25 -> 0\nX44 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 124 [arrowhead = none ];
124 -> 28;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 125 [arrowhead = none ];
125 -> 29;

126 [label="EVAL with clause\nmult(X79, s(X80), X81) :- ','(mult(X79, X80, X82), sum(X82, X79, X81)).\nand substitution\nT24 -> T51\nX79 -> T51\nX80 -> T52\nT25 -> s(T52)\nX44 -> X83\nX81 -> X83", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 126 [arrowhead = none ];
126 -> 31;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 127 [arrowhead = none ];
127 -> 32;

128 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 128 [arrowhead = none ];
128 -> 30;

129 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 129 [arrowhead = none ];
129 -> 33;

130 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nreplacements:\nX82 -> T55", fontsize=14, style = filled, fillcolor = violet];
31 -> 130 [arrowhead = none ];
130 -> 34;

131 [label="INSTANCE with matching:\nT24 -> T51\nT25 -> T52\nX44 -> X82", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 131 [arrowhead = none , style=dashed];
131 -> 23[style=dashed];

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 132 [arrowhead = none ];
132 -> 35;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 133 [arrowhead = none ];
133 -> 36;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 134 [arrowhead = none ];
134 -> 37;

135 [label="EVAL with clause\nsum(X97, 0, X97).\nand substitution\nT55 -> T64\nX97 -> T64\nT51 -> 0\nX83 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 135 [arrowhead = none ];
135 -> 38;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 136 [arrowhead = none ];
136 -> 39;

137 [label="EVAL with clause\nsum(X107, s(X108), s(X109)) :- sum(X107, X108, X109).\nand substitution\nT55 -> T69\nX107 -> T69\nX108 -> T70\nT51 -> s(T70)\nX109 -> X110\nX83 -> s(X110)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 137 [arrowhead = none ];
137 -> 41;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 138 [arrowhead = none ];
138 -> 42;

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 139 [arrowhead = none ];
139 -> 40;

140 [label="INSTANCE with matching:\nT55 -> T69\nT51 -> T70\nX83 -> X110", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 140 [arrowhead = none , style=dashed];
140 -> 34[style=dashed];

141 [label="INSTANCE with matching:\nT1 -> cons(T39, T26)", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 141 [arrowhead = none , style=dashed];
141 -> 1[style=dashed];

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 142 [arrowhead = none ];
142 -> 45;

143 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nreplacements:\nX22 -> T79", fontsize=14, style = filled, fillcolor = violet];
44 -> 143 [arrowhead = none ];
143 -> 46;

144 [label="INSTANCE with matching:\nT24 -> s(s(T24))\nX44 -> X22", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 144 [arrowhead = none , style=dashed];
144 -> 23[style=dashed];

145 [label="INSTANCE with matching:\nT1 -> cons(T79, T26)", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 145 [arrowhead = none , style=dashed];
145 -> 1[style=dashed];

146 [label="EVAL with clause\np(cons(0, X130)) :- p(X130).\nand substitution\nT8 -> 0\nT9 -> T92\nT10 -> T93\nX130 -> cons(T92, T93)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 146 [arrowhead = none ];
146 -> 49;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 147 [arrowhead = none ];
147 -> 50;

148 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 148 [arrowhead = none ];
148 -> 71;

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 149 [arrowhead = none ];
149 -> 51;

150 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
49 -> 150 [arrowhead = none ];
150 -> 52;

151 [label="INSTANCE with matching:\nT1 -> cons(T92, T93)", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 151 [arrowhead = none , style=dashed];
151 -> 1[style=dashed];

152 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 152 [arrowhead = none ];
152 -> 53;

153 [label="SPLIT 2\nnew knowledge:\nT98 is ground\nreplacements:\nX22 -> T98", fontsize=14, style = filled, fillcolor = violet];
52 -> 153 [arrowhead = none ];
153 -> 54;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 154 [arrowhead = none ];
154 -> 55;

155 [label="INSTANCE with matching:\nT1 -> cons(T98, T93)", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 155 [arrowhead = none , style=dashed];
155 -> 1[style=dashed];

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 156 [arrowhead = none ];
156 -> 56;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 157 [arrowhead = none ];
157 -> 57;

158 [label="EVAL with clause\nmult(X144, 0, 0).\nand substitution\nX144 -> 0\nT92 -> 0\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 158 [arrowhead = none ];
158 -> 58;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 159 [arrowhead = none ];
159 -> 59;

160 [label="EVAL with clause\nmult(X155, s(X156), X157) :- ','(mult(X155, X156, X158), sum(X158, X155, X157)).\nand substitution\nX155 -> 0\nX156 -> T101\nT92 -> s(T101)\nX22 -> X159\nX157 -> X159", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 160 [arrowhead = none ];
160 -> 61;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 161 [arrowhead = none ];
161 -> 62;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 162 [arrowhead = none ];
162 -> 60;

163 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 163 [arrowhead = none ];
163 -> 63;

164 [label="SPLIT 2\nnew knowledge:\nT102 is ground\nreplacements:\nX158 -> T102", fontsize=14, style = filled, fillcolor = violet];
61 -> 164 [arrowhead = none ];
164 -> 64;

165 [label="INSTANCE with matching:\nT92 -> T101\nX22 -> X158", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 165 [arrowhead = none , style=dashed];
165 -> 53[style=dashed];

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 166 [arrowhead = none ];
166 -> 65;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 167 [arrowhead = none ];
167 -> 66;

168 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 168 [arrowhead = none ];
168 -> 67;

169 [label="ONLY EVAL with clause\nsum(X172, 0, X172).\nand substitution\nT102 -> T108\nX172 -> T108\nX159 -> T108", fontsize=14, style = filled, fillcolor = white];
66 -> 169 [arrowhead = none ];
169 -> 68;

170 [label="BACKTRACK\nfor clause: sum(X, s(Y), s(Z)) :- sum(X, Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = white];
67 -> 170 [arrowhead = none ];
170 -> 70;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 171 [arrowhead = none ];
171 -> 69;

172 [label="BACKTRACK\nfor clause: p(cons(0, Xs)) :- p(Xs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
71 -> 172 [arrowhead = none ];
172 -> 72;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 173 [arrowhead = none ];
173 -> 75;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 174 [arrowhead = none ];
174 -> 76;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 175 [arrowhead = none ];
175 -> 77;

176 [label="EVAL with clause\np(cons(X186, nil)).\nand substitution\nX186 -> T118\nT113 -> cons(T118, nil)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 176 [arrowhead = none ];
176 -> 78;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 177 [arrowhead = none ];
177 -> 79;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 178 [arrowhead = none ];
178 -> 81;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 179 [arrowhead = none ];
179 -> 82;

180 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 180 [arrowhead = none ];
180 -> 80;

181 [label="EVAL with clause\np(cons(s(s(X202)), cons(X203, X204))) :- ','(p(cons(X202, cons(X203, X204))), ','(mult(X202, X203, X205), p(cons(X205, X204)))).\nand substitution\nX202 -> T131\nX203 -> T132\nX204 -> T133\nT113 -> cons(s(s(T131)), cons(T132, T133))", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 181 [arrowhead = none ];
181 -> 83;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 182 [arrowhead = none ];
182 -> 84;

183 [label="EVAL with clause\np(cons(0, X229)) :- p(X229).\nand substitution\nX229 -> T153\nT113 -> cons(0, T153)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 183 [arrowhead = none ];
183 -> 89;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 184 [arrowhead = none ];
184 -> 90;

185 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
83 -> 185 [arrowhead = none ];
185 -> 85;

186 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
83 -> 186 [arrowhead = none ];
186 -> 86;

187 [label="INSTANCE with matching:\nT1 -> cons(T131, cons(T132, T133))", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 187 [arrowhead = none , style=dashed];
187 -> 1[style=dashed];

188 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
86 -> 188 [arrowhead = none ];
188 -> 87;

189 [label="SPLIT 2\nnew knowledge:\nT146 is ground\nreplacements:\nX205 -> T146", fontsize=14, style = filled, fillcolor = violet];
86 -> 189 [arrowhead = none ];
189 -> 88;

190 [label="INSTANCE with matching:\nT24 -> T131\nT25 -> T132\nX44 -> X205", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 190 [arrowhead = none , style=dashed];
190 -> 23[style=dashed];

191 [label="INSTANCE with matching:\nT1 -> cons(T146, T133)", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 191 [arrowhead = none , style=dashed];
191 -> 1[style=dashed];

192 [label="INSTANCE with matching:\nT1 -> T153", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 192 [arrowhead = none , style=dashed];
192 -> 1[style=dashed];

}

(2) Obligation:

Triples:

mult23(T51, s(T52), X83) :- mult23(T51, T52, X82).
mult23(T51, s(T52), X83) :- ','(multc23(T51, T52, T55), sum34(T55, T51, X83)).
sum34(T69, s(T70), s(X110)) :- sum34(T69, T70, X110).
mult53(s(T101), X159) :- mult53(T101, X158).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- p1(cons(T24, cons(T25, T26))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(pc1(cons(T24, cons(T25, T26))), mult23(T24, T25, X44)).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(pc1(cons(T24, cons(T25, T26))), ','(multc23(T24, T25, T39), p1(cons(T39, T26)))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(pc1(cons(T24, cons(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(cons(T39, T26)), mult23(s(s(T24)), T25, X22)))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(pc1(cons(T24, cons(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(cons(T39, T26)), ','(multc23(s(s(T24)), T25, T79), p1(cons(T79, T26)))))).
p1(cons(s(s(0)), cons(T92, T93))) :- p1(cons(T92, T93)).
p1(cons(s(s(0)), cons(T92, T93))) :- ','(pc1(cons(T92, T93)), mult53(T92, X22)).
p1(cons(s(s(0)), cons(T92, T93))) :- ','(pc1(cons(T92, T93)), ','(multc53(T92, T98), p1(cons(T98, T93)))).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- p1(cons(T131, cons(T132, T133))).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- ','(pc1(cons(T131, cons(T132, T133))), mult23(T131, T132, X205)).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- ','(pc1(cons(T131, cons(T132, T133))), ','(multc23(T131, T132, T146), p1(cons(T146, T133)))).
p1(cons(0, cons(0, T153))) :- p1(T153).

Clauses:

pc1(cons(T3, nil)).
pc1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(pc1(cons(T24, cons(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(cons(T39, T26)), ','(multc23(s(s(T24)), T25, T79), pc1(cons(T79, T26)))))).
pc1(cons(s(s(0)), cons(T92, T93))) :- ','(pc1(cons(T92, T93)), ','(multc53(T92, T98), pc1(cons(T98, T93)))).
pc1(cons(0, cons(T118, nil))).
pc1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- ','(pc1(cons(T131, cons(T132, T133))), ','(multc23(T131, T132, T146), pc1(cons(T146, T133)))).
pc1(cons(0, cons(0, T153))) :- pc1(T153).
multc23(T46, 0, 0).
multc23(T51, s(T52), X83) :- ','(multc23(T51, T52, T55), sumc34(T55, T51, X83)).
sumc34(T64, 0, T64).
sumc34(T69, s(T70), s(X110)) :- sumc34(T69, T70, X110).
multc53(0, 0).
multc53(s(T101), T108) :- multc53(T101, T108).

Afs:

p1(x1)  =  p1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b)
pc1_in: (b)
multc23_in: (b,b,f)
sumc34_in: (b,b,f)
multc53_in: (b,f)
mult23_in: (b,b,f)
sum34_in: (b,b,f)
mult53_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U6_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_gga(T55, T51, X83))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → SUM34_IN_GGA(T55, T51, X83)
SUM34_IN_GGA(T69, s(T70), s(X110)) → U4_GGA(T69, T70, X110, sum34_in_gga(T69, T70, X110))
SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_g(cons(T39, T26)))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(cons(T39, T26))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U15_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U16_G(T92, T93, pc1_in_g(cons(T92, T93)))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U17_G(T92, T93, mult53_in_ga(T92, X22))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → U19_G(T92, T93, p1_in_g(cons(T98, T93)))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(cons(T98, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U20_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U22_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U24_G(T131, T132, T133, p1_in_g(cons(T146, T133)))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(cons(T146, T133))
P1_IN_G(cons(0, cons(0, T153))) → U25_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U14_G(T24, T25, T26, p1_in_g(cons(T79, T26)))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(cons(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
sum34_in_gga(x1, x2, x3)  =  sum34_in_gga(x1, x2)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
P1_IN_G(x1)  =  P1_IN_G(x1)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x2, x3, x4)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x3, x4)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U19_G(x1, x2, x3)  =  U19_G(x1, x2, x3)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U22_G(x1, x2, x3, x4)  =  U22_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U24_G(x1, x2, x3, x4)  =  U24_G(x1, x2, x3, x4)
U25_G(x1, x2)  =  U25_G(x1, x2)
U11_G(x1, x2, x3, x4)  =  U11_G(x1, x2, x3, x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x1, x2, x3, x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x1, x2, x3, x4)
U14_G(x1, x2, x3, x4)  =  U14_G(x1, x2, x3, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U6_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_gga(T55, T51, X83))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → SUM34_IN_GGA(T55, T51, X83)
SUM34_IN_GGA(T69, s(T70), s(X110)) → U4_GGA(T69, T70, X110, sum34_in_gga(T69, T70, X110))
SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_g(cons(T39, T26)))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(cons(T39, T26))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U15_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U16_G(T92, T93, pc1_in_g(cons(T92, T93)))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U17_G(T92, T93, mult53_in_ga(T92, X22))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → U19_G(T92, T93, p1_in_g(cons(T98, T93)))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(cons(T98, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U20_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U22_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U24_G(T131, T132, T133, p1_in_g(cons(T146, T133)))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(cons(T146, T133))
P1_IN_G(cons(0, cons(0, T153))) → U25_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U14_G(T24, T25, T26, p1_in_g(cons(T79, T26)))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(cons(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
sum34_in_gga(x1, x2, x3)  =  sum34_in_gga(x1, x2)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
P1_IN_G(x1)  =  P1_IN_G(x1)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x2, x3, x4)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x3, x4)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U19_G(x1, x2, x3)  =  U19_G(x1, x2, x3)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U22_G(x1, x2, x3, x4)  =  U22_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U24_G(x1, x2, x3, x4)  =  U24_G(x1, x2, x3, x4)
U25_G(x1, x2)  =  U25_G(x1, x2)
U11_G(x1, x2, x3, x4)  =  U11_G(x1, x2, x3, x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x1, x2, x3, x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x1, x2, x3, x4)
U14_G(x1, x2, x3, x4)  =  U14_G(x1, x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)

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

(12) 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:

• MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

SUM34_IN_GGA(T69, s(T70)) → SUM34_IN_GGA(T69, T70)

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

(19) 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:

• SUM34_IN_GGA(T69, s(T70)) → SUM34_IN_GGA(T69, T70)
  The graph contains the following edges 1 >= 1, 2 > 2

(20) YES

(21) Obligation:

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

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)

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

(26) 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:

• MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)
  The graph contains the following edges 1 >= 1, 2 > 2

(27) YES

(28) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(cons(T39, T26))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U16_G(T92, T93, pc1_in_g(cons(T92, T93)))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(cons(T98, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(cons(T146, T133))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(cons(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
nil  =  nil
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
P1_IN_G(x1)  =  P1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U11_G(x1, x2, x3, x4)  =  U11_G(x1, x2, x3, x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x1, x2, x3, x4)

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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(cons(T39, T26))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U16_G(T92, T93, pc1_in_g(cons(T92, T93)))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(cons(T98, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(cons(T146, T133))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(cons(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U16_G(T92, T93, pc1_in_g(cons(T92, T93)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P1_IN_G(x₁)) = x₁   
POL(U11_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U13_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U16_G(x₁, x₂, x₃)) = 1 + x₂   
POL(U18_G(x₁, x₂, x₃)) = 1 + x₂   
POL(U21_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U23_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U27_g(x₁, x₂, x₃, x₄)) = 0   
POL(U28_g(x₁, x₂, x₃, x₄)) = 0   
POL(U29_g(x₁, x₂, x₃, x₄)) = 0   
POL(U30_g(x₁, x₂, x₃, x₄)) = 0   
POL(U31_g(x₁, x₂, x₃, x₄)) = 0   
POL(U32_g(x₁, x₂, x₃)) = 0   
POL(U33_g(x₁, x₂, x₃)) = 0   
POL(U34_g(x₁, x₂, x₃)) = 0   
POL(U35_g(x₁, x₂, x₃, x₄)) = 0   
POL(U36_g(x₁, x₂, x₃, x₄)) = 0   
POL(U37_g(x₁, x₂, x₃, x₄)) = 0   
POL(U38_g(x₁, x₂)) = 0   
POL(U39_gga(x₁, x₂, x₃)) = 0   
POL(U40_gga(x₁, x₂, x₃)) = 0   
POL(U41_gga(x₁, x₂, x₃)) = 0   
POL(U42_ga(x₁, x₂)) = 0   
POL(U7_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U9_G(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(cons(x₁, x₂)) = 1 + x₂   
POL(multc23_in_gga(x₁, x₂)) = 0   
POL(multc23_out_gga(x₁, x₂, x₃)) = 0   
POL(multc53_in_ga(x₁)) = 0   
POL(multc53_out_ga(x₁, x₂)) = 0   
POL(nil) = 0   
POL(pc1_in_g(x₁)) = 0   
POL(pc1_out_g(x₁)) = 0   
POL(s(x₁)) = 0   
POL(sumc34_in_gga(x₁, x₂)) = 0   
POL(sumc34_out_gga(x₁, x₂, x₃)) = 0   

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

U7_G(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(cons(T39, T26))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
U16_G(T92, T93, pc1_out_g(cons(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(cons(T98, T93))
U21_G(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(cons(T146, T133))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(cons(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(33) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.

(34) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))

The TRS R consists of the following rules:

pc1_in_g(cons(T3, nil)) → pc1_out_g(cons(T3, nil))
pc1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(cons(T24, cons(T25, T26))))
pc1_in_g(cons(s(s(0)), cons(T92, T93))) → U32_g(T92, T93, pc1_in_g(cons(T92, T93)))
pc1_in_g(cons(0, cons(T118, nil))) → pc1_out_g(cons(0, cons(T118, nil)))
pc1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(cons(T131, cons(T132, T133))))
pc1_in_g(cons(0, cons(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(cons(0, cons(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(cons(T131, cons(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(cons(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(cons(T146, T133))) → pc1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U32_g(T92, T93, pc1_out_g(cons(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(cons(T98, T93)))
U34_g(T92, T93, pc1_out_g(cons(T98, T93))) → pc1_out_g(cons(s(s(0)), cons(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(cons(T24, cons(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(cons(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(cons(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(cons(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(cons(T79, T26))) → pc1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(35) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(36) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))

R is empty.
The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(37) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

(38) Obligation:

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

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))

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

(39) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(P1_IN_G(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = x₁   

(40) Obligation:

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

(41) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(42) YES