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

(0) Obligation:

Clauses:

p(.(X, [])).
p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), ','(mult(X, Y, Z), p(.(Z, Xs)))).
p(.(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)).

Query: p(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: 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(.(T3, [])), SCOPE: 1, CLAUSE: 1 | p(.(T3, [])), 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\nX3 is free\np(T1) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(.(T3, [])), SCOPE: 1, CLAUSE: 1 | p(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p([]), SCOPE: 2, CLAUSE: 0 | p([]), SCOPE: 2, CLAUSE: 1 | p([]), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p([]), SCOPE: 2, CLAUSE: 1 | p([]), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p([]), 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(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))) | p(.(s(s(T8)), .(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX3 is free\nX17 is free\nX18 is free\nX19 is free\np(T1) !=? p(.(X3, []))\np(T1) !=? p(.(s(s(X17)), .(X18, X19)))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 0 | ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 1 | ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(.(s(s(T8)), .(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 1 | ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(.(s(s(T8)), .(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 1\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 2 | ?_3 | p(.(s(s(T8)), .(T9, T10))), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(p(.(T24, .(T25, T26))), ','(mult(T24, T25, X39), p(.(X39, T26)))), ','(mult(s(s(T24)), T25, X20), p(.(X20, T26))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX20 is free\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(.(T24, .(T25, T26)))\n\nT24 is ground\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(mult(T24, T25, X39), p(.(X39, T26))), ','(mult(s(s(T24)), T25, X20), p(.(X20, T26))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX20 is free\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="B: mult(T24, T25, X39)\n\nT24 is ground\nT25 is ground\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(.(T33, T26)), ','(mult(s(s(T24)), T25, X20), p(.(X20, T26))))\n\nT24 is ground\nT25 is ground\nT26 is ground\nT33 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: mult(T24, T25, X39), SCOPE: 4, CLAUSE: 5 | mult(T24, T25, X39), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: mult(T24, T25, X39), SCOPE: 4, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nX39 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: mult(T24, T25, X39), SCOPE: 4, CLAUSE: 6\n\nT24 is ground\nT25 is ground\nX39 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(T45, T46, X65), sum(X65, T45, X66))\n\nT45 is ground\nT46 is ground\nX66 is free\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: mult(T45, T46, X65)\n\nT45 is ground\nT46 is ground\nX65 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="D: sum(T49, T45, X66)\n\nT45 is ground\nT49 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: sum(T49, T45, X66), SCOPE: 5, CLAUSE: 3 | sum(T49, T45, X66), SCOPE: 5, CLAUSE: 4\n\nT45 is ground\nT49 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: sum(T49, T45, X66), SCOPE: 5, CLAUSE: 3\n\nT45 is ground\nT49 is ground\nX66 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: sum(T49, T45, X66), SCOPE: 5, CLAUSE: 4\n\nT45 is ground\nT49 is ground\nX66 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(T63, T64, X89)\n\nT63 is ground\nT64 is ground\nX89 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: p(.(T33, T26))\n\nT26 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(mult(s(s(T24)), T25, X20), p(.(X20, T26)))\n\nT24 is ground\nT25 is ground\nT26 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: mult(s(s(T24)), T25, X20)\n\nT24 is ground\nT25 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: p(.(T73, T26))\n\nT26 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(.(T8, .(T9, T10))), ','(mult(T8, T9, X20), p(.(X20, T10)))), SCOPE: 3, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ?_3 | p(.(s(s(T8)), .(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(.(T86, T87)), ','(mult(0, T86, X20), p(.(X20, T87))))\n\nT86 is ground\nT87 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: p(.(T86, T87))\n\nT86 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(mult(0, T86, X20), p(.(X20, T87)))\n\nT86 is ground\nT87 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="C: mult(0, T86, X20)\n\nT86 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: p(.(T92, T87))\n\nT87 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: mult(0, T86, X20), SCOPE: 6, CLAUSE: 5 | mult(0, T86, X20), SCOPE: 6, CLAUSE: 6\n\nT86 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: mult(0, T86, X20), SCOPE: 6, CLAUSE: 5\n\nT86 is ground\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: mult(0, T86, X20), SCOPE: 6, CLAUSE: 6\n\nT86 is ground\nX20 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, T95, X132), sum(X132, 0, X133))\n\nT95 is ground\nX133 is free\nX132 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: mult(0, T95, X132)\n\nT95 is ground\nX132 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: sum(T96, 0, X133)\n\nT96 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: sum(T96, 0, X133), SCOPE: 7, CLAUSE: 3 | sum(T96, 0, X133), SCOPE: 7, CLAUSE: 4\n\nT96 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: sum(T96, 0, X133), SCOPE: 7, CLAUSE: 3\n\nT96 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: sum(T96, 0, X133), SCOPE: 7, CLAUSE: 4\n\nT96 is ground\nX133 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(.(s(s(T8)), .(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(T107)\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: p(T107), SCOPE: 8, CLAUSE: 0 | p(T107), SCOPE: 8, CLAUSE: 1 | p(T107), SCOPE: 8, CLAUSE: 2\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: p(T107), SCOPE: 8, CLAUSE: 0\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: p(T107), SCOPE: 8, CLAUSE: 1 | p(T107), SCOPE: 8, CLAUSE: 2\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", 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(T107), SCOPE: 8, CLAUSE: 1\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: p(T107), SCOPE: 8, CLAUSE: 2\n\nT107 is ground\nX3 is free\np(.(0, T107)) !=? p(.(X3, []))", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(.(T125, .(T126, T127))), ','(mult(T125, T126, X174), p(.(X174, T127))))\n\nT125 is ground\nT126 is ground\nT127 is ground\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: p(.(T125, .(T126, T127)))\n\nT125 is ground\nT126 is ground\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(mult(T125, T126, X174), p(.(X174, T127)))\n\nT125 is ground\nT126 is ground\nT127 is ground\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: mult(T125, T126, X174)\n\nT125 is ground\nT126 is ground\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: p(.(T134, T127))\n\nT127 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: p(T141)\n\nT141 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(.(X3, [])).\nand substitutionX3 -> T3,\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 92 [arrowhead = none ];
92 -> 3;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(s(s(X17)), .(X18, X19))) :- ','(p(.(X17, .(X18, X19))), ','(mult(X17, X18, X20), p(.(X20, X19)))).\nand substitutionX17 -> T8,\nX18 -> T9,\nX19 -> T10,\nT1 -> .(s(s(T8)), .(T9, T10))", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 95 [arrowhead = none ];
95 -> 13;

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

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

98 [label="EVAL with clause\np(.(0, X8)) :- p(X8).\nand substitutionT3 -> 0,\nX8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 101 [arrowhead = none ];
101 -> 10;

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

103 [label="BACKTRACK\nfor clause: p(.(0, Xs)) :- p(Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(0, X151)) :- p(X151).\nand substitutionX151 -> T107,\nT1 -> .(0, T107)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 105 [arrowhead = none ];
105 -> 73;

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

107 [label="BACKTRACK\nfor clause: p(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(s(s(X36)), .(X37, X38))) :- ','(p(.(X36, .(X37, X38))), ','(mult(X36, X37, X39), p(.(X39, X38)))).\nand substitutionX36 -> T24,\nT8 -> s(s(T24)),\nT9 -> T25,\nX37 -> T25,\nT10 -> T26,\nX38 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 110 [arrowhead = none ];
110 -> 19;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT24 is ground\nT25 is ground\nT26 is ground", fontsize=14, style = filled, fillcolor = violet];
19 -> 115 [arrowhead = none ];
115 -> 22;

116 [label="INSTANCE with matching:\nT1 -> .(T24, .(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:\nT24 is ground\nT25 is ground\nT33 is ground\nreplacements:X39 -> T33", 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\nnew knowledge:\nT33 is ground\nT26 is ground", 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(X52, 0, 0).\nand substitutionT24 -> T40,\nX52 -> T40,\nT25 -> 0,\nX39 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 124 [arrowhead = none ];
124 -> 28;

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

126 [label="EVAL with clause\nmult(X62, s(X63), X64) :- ','(mult(X62, X63, X65), sum(X65, X62, X64)).\nand substitutionT24 -> T45,\nX62 -> T45,\nX63 -> T46,\nT25 -> s(T46),\nX39 -> X66,\nX64 -> X66", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 126 [arrowhead = none ];
126 -> 31;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X65 -> T49", fontsize=14, style = filled, fillcolor = violet];
31 -> 130 [arrowhead = none ];
130 -> 34;

131 [label="INSTANCE with matching:\nT24 -> T45\nT25 -> T46\nX39 -> X65", 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(X77, 0, X77).\nand substitutionT49 -> T58,\nX77 -> T58,\nT45 -> 0,\nX66 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 135 [arrowhead = none ];
135 -> 38;

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

137 [label="EVAL with clause\nsum(X86, s(X87), s(X88)) :- sum(X86, X87, X88).\nand substitutionT49 -> T63,\nX86 -> T63,\nX87 -> T64,\nT45 -> s(T64),\nX88 -> X89,\nX66 -> s(X89)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 137 [arrowhead = none ];
137 -> 41;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT49 -> T63\nT45 -> T64\nX66 -> X89", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 140 [arrowhead = none , style=dashed];
140 -> 34[style=dashed];

141 [label="INSTANCE with matching:\nT1 -> .(T33, 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:\nT24 is ground\nT25 is ground\nT73 is ground\nreplacements:X20 -> T73", fontsize=14, style = filled, fillcolor = violet];
44 -> 143 [arrowhead = none ];
143 -> 46;

144 [label="INSTANCE with matching:\nT24 -> s(s(T24))\nX39 -> X20", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 144 [arrowhead = none , style=dashed];
144 -> 23[style=dashed];

145 [label="INSTANCE with matching:\nT1 -> .(T73, T26)", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 145 [arrowhead = none , style=dashed];
145 -> 1[style=dashed];

146 [label="EVAL with clause\np(.(0, X106)) :- p(X106).\nand substitutionT8 -> 0,\nT9 -> T86,\nT10 -> T87,\nX106 -> .(T86, T87)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 146 [arrowhead = none ];
146 -> 49;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT86 is ground\nT87 is ground", fontsize=14, style = filled, fillcolor = violet];
49 -> 150 [arrowhead = none ];
150 -> 52;

151 [label="INSTANCE with matching:\nT1 -> .(T86, T87)", 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:\nT86 is ground\nT92 is ground\nreplacements:X20 -> T92", 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 -> .(T92, T87)", 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(X119, 0, 0).\nand substitutionX119 -> 0,\nT86 -> 0,\nX20 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 158 [arrowhead = none ];
158 -> 58;

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

160 [label="EVAL with clause\nmult(X129, s(X130), X131) :- ','(mult(X129, X130, X132), sum(X132, X129, X131)).\nand substitutionX129 -> 0,\nX130 -> T95,\nT86 -> s(T95),\nX20 -> X133,\nX131 -> X133", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 160 [arrowhead = none ];
160 -> 61;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT95 is ground\nT96 is ground\nreplacements:X132 -> T96", fontsize=14, style = filled, fillcolor = violet];
61 -> 164 [arrowhead = none ];
164 -> 64;

165 [label="INSTANCE with matching:\nT86 -> T95\nX20 -> X132", 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(X143, 0, X143).\nand substitutionT96 -> T102,\nX143 -> T102,\nX133 -> T102", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
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(.(0, Xs)) :- p(Xs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X156, [])).\nand substitutionX156 -> T112,\nT107 -> .(T112, [])", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 176 [arrowhead = none ];
176 -> 78;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(s(s(X171)), .(X172, X173))) :- ','(p(.(X171, .(X172, X173))), ','(mult(X171, X172, X174), p(.(X174, X173)))).\nand substitutionX171 -> T125,\nX172 -> T126,\nX173 -> T127,\nT107 -> .(s(s(T125)), .(T126, T127))", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 181 [arrowhead = none ];
181 -> 83;

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

183 [label="EVAL with clause\np(.(0, X187)) :- p(X187).\nand substitutionX187 -> T141,\nT107 -> .(0, T141)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 183 [arrowhead = none ];
183 -> 89;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT125 is ground\nT126 is ground\nT127 is ground", fontsize=14, style = filled, fillcolor = violet];
83 -> 186 [arrowhead = none ];
186 -> 86;

187 [label="INSTANCE with matching:\nT1 -> .(T125, .(T126, T127))", 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:\nT125 is ground\nT126 is ground\nT134 is ground\nreplacements:X174 -> T134", fontsize=14, style = filled, fillcolor = violet];
86 -> 189 [arrowhead = none ];
189 -> 88;

190 [label="INSTANCE with matching:\nT24 -> T125\nT25 -> T126\nX39 -> X174", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 190 [arrowhead = none , style=dashed];
190 -> 23[style=dashed];

191 [label="INSTANCE with matching:\nT1 -> .(T134, T127)", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 191 [arrowhead = none , style=dashed];
191 -> 1[style=dashed];

192 [label="INSTANCE with matching:\nT1 -> T141", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 192 [arrowhead = none , style=dashed];
192 -> 1[style=dashed];

}

(2) Obligation:

Triples:

multB(X1, s(X2), X3) :- multB(X1, X2, X4).
multB(X1, s(X2), X3) :- ','(multcB(X1, X2, X4), sumD(X4, X1, X3)).
sumD(X1, s(X2), s(X3)) :- sumD(X1, X2, X3).
multC(s(X1), X2) :- multC(X1, X3).
pA(.(s(s(s(s(X1)))), .(X2, X3))) :- pA(.(X1, .(X2, X3))).
pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), multB(X1, X2, X4)).
pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), pA(.(X4, X3)))).
pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), ','(pcA(.(X4, X3)), multB(s(s(X1)), X2, X5)))).
pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), ','(pcA(.(X4, X3)), ','(multcB(s(s(X1)), X2, X5), pA(.(X5, X3)))))).
pA(.(s(s(0)), .(X1, X2))) :- pA(.(X1, X2)).
pA(.(s(s(0)), .(X1, X2))) :- ','(pcA(.(X1, X2)), multC(X1, X3)).
pA(.(s(s(0)), .(X1, X2))) :- ','(pcA(.(X1, X2)), ','(multcC(X1, X3), pA(.(X3, X2)))).
pA(.(0, .(s(s(X1)), .(X2, X3)))) :- pA(.(X1, .(X2, X3))).
pA(.(0, .(s(s(X1)), .(X2, X3)))) :- ','(pcA(.(X1, .(X2, X3))), multB(X1, X2, X4)).
pA(.(0, .(s(s(X1)), .(X2, X3)))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), pA(.(X4, X3)))).
pA(.(0, .(0, X1))) :- pA(X1).

Clauses:

pcA(.(X1, [])).
pcA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), ','(pcA(.(X4, X3)), ','(multcB(s(s(X1)), X2, X5), pcA(.(X5, X3)))))).
pcA(.(s(s(0)), .(X1, X2))) :- ','(pcA(.(X1, X2)), ','(multcC(X1, X3), pcA(.(X3, X2)))).
pcA(.(0, .(X1, []))).
pcA(.(0, .(s(s(X1)), .(X2, X3)))) :- ','(pcA(.(X1, .(X2, X3))), ','(multcB(X1, X2, X4), pcA(.(X4, X3)))).
pcA(.(0, .(0, X1))) :- pcA(X1).
multcB(X1, 0, 0).
multcB(X1, s(X2), X3) :- ','(multcB(X1, X2, X4), sumcD(X4, X1, X3)).
sumcD(X1, 0, X1).
sumcD(X1, s(X2), s(X3)) :- sumcD(X1, X2, X3).
multcC(0, 0).
multcC(s(X1), X2) :- multcC(X1, X2).

Afs:

pA(x1)  =  pA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
pA_in: (b)
pcA_in: (b)
multcB_in: (b,b,f)
sumcD_in: (b,b,f)
multcC_in: (b,f)
multB_in: (b,b,f)
sumD_in: (b,b,f)
multC_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U6_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U7_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U8_G(X1, X2, X3, multB_in_gga(X1, X2, X4))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → MULTB_IN_GGA(X1, X2, X4)
MULTB_IN_GGA(X1, s(X2), X3) → U1_GGA(X1, X2, X3, multB_in_gga(X1, X2, X4))
MULTB_IN_GGA(X1, s(X2), X3) → MULTB_IN_GGA(X1, X2, X4)
MULTB_IN_GGA(X1, s(X2), X3) → U2_GGA(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, sumD_in_gga(X4, X1, X3))
U2_GGA(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → SUMD_IN_GGA(X4, X1, X3)
SUMD_IN_GGA(X1, s(X2), s(X3)) → U4_GGA(X1, X2, X3, sumD_in_gga(X1, X2, X3))
SUMD_IN_GGA(X1, s(X2), s(X3)) → SUMD_IN_GGA(X1, X2, X3)
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U9_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U10_G(X1, X2, X3, pA_in_g(.(X4, X3)))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U15_G(X1, X2, pA_in_g(.(X1, X2)))
PA_IN_G(.(s(s(0)), .(X1, X2))) → PA_IN_G(.(X1, X2))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U16_G(X1, X2, pcA_in_g(.(X1, X2)))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U17_G(X1, X2, multC_in_ga(X1, X3))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → MULTC_IN_GA(X1, X3)
MULTC_IN_GA(s(X1), X2) → U5_GA(X1, X2, multC_in_ga(X1, X3))
MULTC_IN_GA(s(X1), X2) → MULTC_IN_GA(X1, X3)
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U18_G(X1, X2, multcC_in_ga(X1, X3))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → U19_G(X1, X2, pA_in_g(.(X3, X2)))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → PA_IN_G(.(X3, X2))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U20_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U21_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U22_G(X1, X2, X3, multB_in_gga(X1, X2, X4))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → MULTB_IN_GGA(X1, X2, X4)
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U23_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U24_G(X1, X2, X3, pA_in_g(.(X4, X3)))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(0, .(0, X1))) → U25_G(X1, pA_in_g(X1))
PA_IN_G(.(0, .(0, X1))) → PA_IN_G(X1)
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U11_G(X1, X2, X3, pcA_in_g(.(X4, X3)))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U12_G(X1, X2, X3, multB_in_gga(s(s(X1)), X2, X5))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → MULTB_IN_GGA(s(s(X1)), X2, X5)
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U13_G(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U14_G(X1, X2, X3, pA_in_g(.(X5, X3)))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → PA_IN_G(.(X5, X3))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
multB_in_gga(x1, x2, x3)  =  multB_in_gga(x1, x2)
sumD_in_gga(x1, x2, x3)  =  sumD_in_gga(x1, x2)
multC_in_ga(x1, x2)  =  multC_in_ga(x1)
PA_IN_G(x1)  =  PA_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)
MULTB_IN_GGA(x1, x2, x3)  =  MULTB_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)
SUMD_IN_GGA(x1, x2, x3)  =  SUMD_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)
MULTC_IN_GA(x1, x2)  =  MULTC_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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U6_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U7_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U8_G(X1, X2, X3, multB_in_gga(X1, X2, X4))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → MULTB_IN_GGA(X1, X2, X4)
MULTB_IN_GGA(X1, s(X2), X3) → U1_GGA(X1, X2, X3, multB_in_gga(X1, X2, X4))
MULTB_IN_GGA(X1, s(X2), X3) → MULTB_IN_GGA(X1, X2, X4)
MULTB_IN_GGA(X1, s(X2), X3) → U2_GGA(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, sumD_in_gga(X4, X1, X3))
U2_GGA(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → SUMD_IN_GGA(X4, X1, X3)
SUMD_IN_GGA(X1, s(X2), s(X3)) → U4_GGA(X1, X2, X3, sumD_in_gga(X1, X2, X3))
SUMD_IN_GGA(X1, s(X2), s(X3)) → SUMD_IN_GGA(X1, X2, X3)
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U9_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U10_G(X1, X2, X3, pA_in_g(.(X4, X3)))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U15_G(X1, X2, pA_in_g(.(X1, X2)))
PA_IN_G(.(s(s(0)), .(X1, X2))) → PA_IN_G(.(X1, X2))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U16_G(X1, X2, pcA_in_g(.(X1, X2)))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U17_G(X1, X2, multC_in_ga(X1, X3))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → MULTC_IN_GA(X1, X3)
MULTC_IN_GA(s(X1), X2) → U5_GA(X1, X2, multC_in_ga(X1, X3))
MULTC_IN_GA(s(X1), X2) → MULTC_IN_GA(X1, X3)
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U18_G(X1, X2, multcC_in_ga(X1, X3))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → U19_G(X1, X2, pA_in_g(.(X3, X2)))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → PA_IN_G(.(X3, X2))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U20_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U21_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U22_G(X1, X2, X3, multB_in_gga(X1, X2, X4))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → MULTB_IN_GGA(X1, X2, X4)
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U23_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U24_G(X1, X2, X3, pA_in_g(.(X4, X3)))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(0, .(0, X1))) → U25_G(X1, pA_in_g(X1))
PA_IN_G(.(0, .(0, X1))) → PA_IN_G(X1)
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U11_G(X1, X2, X3, pcA_in_g(.(X4, X3)))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U12_G(X1, X2, X3, multB_in_gga(s(s(X1)), X2, X5))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → MULTB_IN_GGA(s(s(X1)), X2, X5)
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U13_G(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U14_G(X1, X2, X3, pA_in_g(.(X5, X3)))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → PA_IN_G(.(X5, X3))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
pA_in_g(x1)  =  pA_in_g(x1)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
multB_in_gga(x1, x2, x3)  =  multB_in_gga(x1, x2)
sumD_in_gga(x1, x2, x3)  =  sumD_in_gga(x1, x2)
multC_in_ga(x1, x2)  =  multC_in_ga(x1)
PA_IN_G(x1)  =  PA_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)
MULTB_IN_GGA(x1, x2, x3)  =  MULTB_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)
SUMD_IN_GGA(x1, x2, x3)  =  SUMD_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)
MULTC_IN_GA(x1, x2)  =  MULTC_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:

MULTC_IN_GA(s(X1), X2) → MULTC_IN_GA(X1, X3)

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
MULTC_IN_GA(x1, x2)  =  MULTC_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:

MULTC_IN_GA(s(X1), X2) → MULTC_IN_GA(X1, X3)

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

MULTC_IN_GA(s(X1)) → MULTC_IN_GA(X1)

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:

• MULTC_IN_GA(s(X1)) → MULTC_IN_GA(X1)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

SUMD_IN_GGA(X1, s(X2), s(X3)) → SUMD_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
SUMD_IN_GGA(x1, x2, x3)  =  SUMD_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:

SUMD_IN_GGA(X1, s(X2), s(X3)) → SUMD_IN_GGA(X1, X2, X3)

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

SUMD_IN_GGA(X1, s(X2)) → SUMD_IN_GGA(X1, X2)

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:

• SUMD_IN_GGA(X1, s(X2)) → SUMD_IN_GGA(X1, X2)
  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:

MULTB_IN_GGA(X1, s(X2), X3) → MULTB_IN_GGA(X1, X2, X4)

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
MULTB_IN_GGA(x1, x2, x3)  =  MULTB_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:

MULTB_IN_GGA(X1, s(X2), X3) → MULTB_IN_GGA(X1, X2, X4)

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

MULTB_IN_GGA(X1, s(X2)) → MULTB_IN_GGA(X1, X2)

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:

• MULTB_IN_GGA(X1, s(X2)) → MULTB_IN_GGA(X1, X2)
  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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U7_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U9_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(s(s(0)), .(X1, X2))) → PA_IN_G(.(X1, X2))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U16_G(X1, X2, pcA_in_g(.(X1, X2)))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U18_G(X1, X2, multcC_in_ga(X1, X3))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → PA_IN_G(.(X3, X2))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U21_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U23_G(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(0, .(0, X1))) → PA_IN_G(X1)
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U11_G(X1, X2, X3, pcA_in_g(.(X4, X3)))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U13_G(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → PA_IN_G(.(X5, X3))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
multcB_in_gga(X1, 0, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2), X3) → U39_gga(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U39_gga(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, X3, sumcD_in_gga(X4, X1, X3))
sumcD_in_gga(X1, 0, X1) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2), s(X3)) → U41_gga(X1, X2, X3, sumcD_in_gga(X1, X2, X3))
U41_gga(X1, X2, X3, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, X3, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1, X3))
multcC_in_ga(0, 0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1), X2) → U42_ga(X1, X2, multcC_in_ga(X1, X2))
U42_ga(X1, X2, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2, X4))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2, X5))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pcA_in_g(x1)  =  pcA_in_g(x1)
[]  =  []
pcA_out_g(x1)  =  pcA_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)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
multcB_out_gga(x1, x2, x3)  =  multcB_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)
sumcD_in_gga(x1, x2, x3)  =  sumcD_in_gga(x1, x2)
sumcD_out_gga(x1, x2, x3)  =  sumcD_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)
multcC_in_ga(x1, x2)  =  multcC_in_ga(x1)
multcC_out_ga(x1, x2)  =  multcC_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)
PA_IN_G(x1)  =  PA_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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U7_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U9_G(X1, X2, X3, multcB_in_gga(X1, X2))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(s(s(0)), .(X1, X2))) → PA_IN_G(.(X1, X2))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U16_G(X1, X2, pcA_in_g(.(X1, X2)))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U18_G(X1, X2, multcC_in_ga(X1))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → PA_IN_G(.(X3, X2))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U21_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U23_G(X1, X2, X3, multcB_in_gga(X1, X2))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(0, .(0, X1))) → PA_IN_G(X1)
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U11_G(X1, X2, X3, pcA_in_g(.(X4, X3)))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U13_G(X1, X2, X3, multcB_in_gga(s(s(X1)), X2))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → PA_IN_G(.(X5, X3))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2))
multcB_in_gga(X1, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2)) → U39_gga(X1, X2, multcB_in_gga(X1, X2))
U39_gga(X1, X2, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, sumcD_in_gga(X4, X1))
sumcD_in_gga(X1, 0) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2)) → U41_gga(X1, X2, sumcD_in_gga(X1, X2))
U41_gga(X1, X2, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1))
multcC_in_ga(0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1)) → U42_ga(X1, multcC_in_ga(X1))
U42_ga(X1, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The set Q consists of the following terms:

pcA_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multcB_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumcD_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)
multcC_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,JAR06].

The following pairs can be oriented strictly and are deleted.

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → U7_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(s(s(0)), .(X1, X2))) → PA_IN_G(.(X1, X2))
PA_IN_G(.(s(s(0)), .(X1, X2))) → U16_G(X1, X2, pcA_in_g(.(X1, X2)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → PA_IN_G(.(X1, .(X2, X3)))
PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) → U21_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
PA_IN_G(.(0, .(0, X1))) → PA_IN_G(X1)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(PA_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([]) = 0   
POL(multcB_in_gga(x₁, x₂)) = 0   
POL(multcB_out_gga(x₁, x₂, x₃)) = 0   
POL(multcC_in_ga(x₁)) = 0   
POL(multcC_out_ga(x₁, x₂)) = 0   
POL(pcA_in_g(x₁)) = 0   
POL(pcA_out_g(x₁)) = 0   
POL(s(x₁)) = 0   
POL(sumcD_in_gga(x₁, x₂)) = 0   
POL(sumcD_out_gga(x₁, x₂, x₃)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(32) Obligation:

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

U7_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U9_G(X1, X2, X3, multcB_in_gga(X1, X2))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))
U16_G(X1, X2, pcA_out_g(.(X1, X2))) → U18_G(X1, X2, multcC_in_ga(X1))
U18_G(X1, X2, multcC_out_ga(X1, X3)) → PA_IN_G(.(X3, X2))
U21_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U23_G(X1, X2, X3, multcB_in_gga(X1, X2))
U23_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → PA_IN_G(.(X4, X3))
U9_G(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U11_G(X1, X2, X3, pcA_in_g(.(X4, X3)))
U11_G(X1, X2, X3, pcA_out_g(.(X4, X3))) → U13_G(X1, X2, X3, multcB_in_gga(s(s(X1)), X2))
U13_G(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → PA_IN_G(.(X5, X3))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2))
multcB_in_gga(X1, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2)) → U39_gga(X1, X2, multcB_in_gga(X1, X2))
U39_gga(X1, X2, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, sumcD_in_gga(X4, X1))
sumcD_in_gga(X1, 0) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2)) → U41_gga(X1, X2, sumcD_in_gga(X1, X2))
U41_gga(X1, X2, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1))
multcC_in_ga(0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1)) → U42_ga(X1, multcC_in_ga(X1))
U42_ga(X1, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The set Q consists of the following terms:

pcA_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multcB_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumcD_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)
multcC_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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))

The TRS R consists of the following rules:

pcA_in_g(.(X1, [])) → pcA_out_g(.(X1, []))
pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) → U27_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(s(s(0)), .(X1, X2))) → U32_g(X1, X2, pcA_in_g(.(X1, X2)))
pcA_in_g(.(0, .(X1, []))) → pcA_out_g(.(0, .(X1, [])))
pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) → U35_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3))))
pcA_in_g(.(0, .(0, X1))) → U38_g(X1, pcA_in_g(X1))
U38_g(X1, pcA_out_g(X1)) → pcA_out_g(.(0, .(0, X1)))
U35_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U36_g(X1, X2, X3, multcB_in_gga(X1, X2))
multcB_in_gga(X1, 0) → multcB_out_gga(X1, 0, 0)
multcB_in_gga(X1, s(X2)) → U39_gga(X1, X2, multcB_in_gga(X1, X2))
U39_gga(X1, X2, multcB_out_gga(X1, X2, X4)) → U40_gga(X1, X2, sumcD_in_gga(X4, X1))
sumcD_in_gga(X1, 0) → sumcD_out_gga(X1, 0, X1)
sumcD_in_gga(X1, s(X2)) → U41_gga(X1, X2, sumcD_in_gga(X1, X2))
U41_gga(X1, X2, sumcD_out_gga(X1, X2, X3)) → sumcD_out_gga(X1, s(X2), s(X3))
U40_gga(X1, X2, sumcD_out_gga(X4, X1, X3)) → multcB_out_gga(X1, s(X2), X3)
U36_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U37_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U37_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → pcA_out_g(.(0, .(s(s(X1)), .(X2, X3))))
U32_g(X1, X2, pcA_out_g(.(X1, X2))) → U33_g(X1, X2, multcC_in_ga(X1))
multcC_in_ga(0) → multcC_out_ga(0, 0)
multcC_in_ga(s(X1)) → U42_ga(X1, multcC_in_ga(X1))
U42_ga(X1, multcC_out_ga(X1, X2)) → multcC_out_ga(s(X1), X2)
U33_g(X1, X2, multcC_out_ga(X1, X3)) → U34_g(X1, X2, pcA_in_g(.(X3, X2)))
U34_g(X1, X2, pcA_out_g(.(X3, X2))) → pcA_out_g(.(s(s(0)), .(X1, X2)))
U27_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) → U28_g(X1, X2, X3, multcB_in_gga(X1, X2))
U28_g(X1, X2, X3, multcB_out_gga(X1, X2, X4)) → U29_g(X1, X2, X3, pcA_in_g(.(X4, X3)))
U29_g(X1, X2, X3, pcA_out_g(.(X4, X3))) → U30_g(X1, X2, X3, multcB_in_gga(s(s(X1)), X2))
U30_g(X1, X2, X3, multcB_out_gga(s(s(X1)), X2, X5)) → U31_g(X1, X2, X3, pcA_in_g(.(X5, X3)))
U31_g(X1, X2, X3, pcA_out_g(.(X5, X3))) → pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3)))

The set Q consists of the following terms:

pcA_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multcB_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumcD_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)
multcC_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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))

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

pcA_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multcB_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumcD_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)
multcC_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].

pcA_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multcB_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumcD_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)
multcC_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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))

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:

PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) → PA_IN_G(.(X1, .(X2, X3)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂   
POL(PA_IN_G(x₁)) = x₁   
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