Left Termination of the query pattern convert_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

convert([], B, 0).
convert(.(0, XS), B, X) :- ','(convert(XS, B, Y), times(Y, B, X)).
convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).

Queries:

convert(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: convert(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: convert(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | convert(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | convert(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | convert([], T5, T3), SCOPE: 1, CLAUSE: 1 | convert([], T5, T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: convert(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | convert(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nconvert(T1, T2, T3) !=? convert([], X2, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: convert([], T5, T3), SCOPE: 1, CLAUSE: 1 | convert([], T5, T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: convert([], T5, T3), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(convert(T9, T10, X17), times(X17, T10, T12)) | convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: convert(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nconvert(T1, T2, T3) !=? convert([], X2, 0)\nconvert(T1, T2, T3) !=? convert(.(0, X14), X15, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 2 | ?_2 | convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 0\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 2 | ?_2 | convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: times(0, T17, T12)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: times(0, T17, T12), SCOPE: 3, CLAUSE: 5 | times(0, T17, T12), SCOPE: 3, CLAUSE: 6\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: times(0, T17, T12), SCOPE: 3, CLAUSE: 5\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: times(0, T17, T12), SCOPE: 3, CLAUSE: 6\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 2 | ?_2 | convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(convert(T33, T34, X59), ','(times(X59, T34, X60), times(X60, T34, T12)))\n\nT33 is ground\nT34 is ground\nX60 is free; \nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: convert(T33, T34, X59)\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(times(T37, T34, X60), times(X60, T34, T12))\n\nT34 is ground\nT37 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: convert(T33, T34, X59), SCOPE: 4, CLAUSE: 0 | convert(T33, T34, X59), SCOPE: 4, CLAUSE: 1 | convert(T33, T34, X59), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: convert(T33, T34, X59), SCOPE: 4, CLAUSE: 0\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: convert(T33, T34, X59), SCOPE: 4, CLAUSE: 1 | convert(T33, T34, X59), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: convert(T33, T34, X59), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: convert(T33, T34, X59), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(convert(T53, T54, X95), times(X95, T54, X96))\n\nT53 is ground\nT54 is ground\nX96 is free; \nX95 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: convert(T53, T54, X95)\n\nT53 is ground\nT54 is ground\nX95 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: times(T57, T54, X96)\n\nT54 is ground\nT57 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: times(T57, T54, X96), SCOPE: 5, CLAUSE: 5 | times(T57, T54, X96), SCOPE: 5, CLAUSE: 6\n\nT54 is ground\nT57 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: times(T57, T54, X96), SCOPE: 5, CLAUSE: 5\n\nT54 is ground\nT57 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: times(T57, T54, X96), SCOPE: 5, CLAUSE: 6\n\nT54 is ground\nT57 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(times(T71, T72, X124), plus(T72, X124, X125))\n\nT71 is ground\nT72 is ground\nX125 is free; \nX124 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: times(T71, T72, X124)\n\nT71 is ground\nT72 is ground\nX124 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: plus(T72, T75, X125)\n\nT72 is ground\nT75 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: plus(T72, T75, X125), SCOPE: 6, CLAUSE: 3 | plus(T72, T75, X125), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT75 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: plus(T72, T75, X125), SCOPE: 6, CLAUSE: 3\n\nT72 is ground\nT75 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: plus(T72, T75, X125), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT75 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: plus(T89, T90, X152)\n\nT89 is ground\nT90 is ground\nX152 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: convert(.(T99, T100), T101, X172)\n\nT99 is ground\nT100 is ground\nT101 is ground\nX172 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: times(T37, T34, X60)\n\nT34 is ground\nT37 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: times(T106, T34, T12)\n\nT34 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: times(T106, T34, T12), SCOPE: 7, CLAUSE: 5 | times(T106, T34, T12), SCOPE: 7, CLAUSE: 6\n\nT34 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: times(T106, T34, T12), SCOPE: 7, CLAUSE: 5\n\nT34 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: times(T106, T34, T12), SCOPE: 7, CLAUSE: 6\n\nT34 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(times(T122, T123, X206), plus(T123, X206, T125))\n\nT122 is ground\nT123 is ground\nX206 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: times(T122, T123, X206)\n\nT122 is ground\nT123 is ground\nX206 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: plus(T123, T128, T125)\n\nT123 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: plus(T123, T128, T125), SCOPE: 8, CLAUSE: 3 | plus(T123, T128, T125), SCOPE: 8, CLAUSE: 4\n\nT123 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: plus(T123, T128, T125), SCOPE: 8, CLAUSE: 3\n\nT123 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: plus(T123, T128, T125), SCOPE: 8, CLAUSE: 4\n\nT123 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: plus(T144, T145, T147)\n\nT144 is ground\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(convert(T9, T10, X17), times(X17, T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ?_2 | convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(convert(.(T162, T163), T164, X260), times(s(X260), T164, T12))\n\nT162 is ground\nT163 is ground\nT164 is ground\nX260 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: convert(.(T162, T163), T164, X260)\n\nT162 is ground\nT163 is ground\nT164 is ground\nX260 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: times(s(T167), T164, T12)\n\nT164 is ground\nT167 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: convert(.(0, T9), T10, T3), SCOPE: 1, CLAUSE: 2\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: convert(.(T177, T178), T179, T181)\n\nT177 is ground\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 0 | convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 1 | convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 2\n\nT177 is ground\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 1 | convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 2\n\nT177 is ground\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 1\n\nT177 is ground\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: convert(.(T177, T178), T179, T181), SCOPE: 9, CLAUSE: 2\n\nT177 is ground\nT178 is ground\nT179 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(convert(T194, T195, X319), times(X319, T195, T197))\n\nT194 is ground\nT195 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: convert(T194, T195, X319)\n\nT194 is ground\nT195 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: times(T200, T195, T197)\n\nT195 is ground\nT200 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: convert(.(T213, T214), T215, T217)\n\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 100 [arrowhead = none ];
100 -> 2;

101 [label="EVAL with clause\nconvert([], X2, 0).\nand substitution\nT1 -> []\nT2 -> T5\nX2 -> T5\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 101 [arrowhead = none ];
101 -> 3;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 102 [arrowhead = none ];
102 -> 4;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 103 [arrowhead = none ];
103 -> 5;

104 [label="EVAL with clause\nconvert(.(0, X14), X15, X16) :- ','(convert(X14, X15, X17), times(X17, X15, X16)).\nand substitution\nX14 -> T9\nT1 -> .(0, T9)\nT2 -> T10\nX15 -> T10\nT3 -> T12\nX16 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 104 [arrowhead = none ];
104 -> 8;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 105 [arrowhead = none ];
105 -> 9;

106 [label="BACKTRACK\nfor clause: convert(.(0, XS), B, X) :- ','(convert(XS, B, Y), times(Y, B, X))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 106 [arrowhead = none ];
106 -> 6;

107 [label="BACKTRACK\nfor clause: convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 107 [arrowhead = none ];
107 -> 7;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 108 [arrowhead = none ];
108 -> 10;

109 [label="EVAL with clause\nconvert(.(s(X293), X294), X295, s(X296)) :- convert(.(X293, X294), X295, X296).\nand substitution\nX293 -> T177\nX294 -> T178\nT1 -> .(s(T177), T178)\nT2 -> T179\nX295 -> T179\nX296 -> T181\nT3 -> s(T181)\nT180 -> T181", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 109 [arrowhead = none ];
109 -> 88;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 110 [arrowhead = none ];
110 -> 89;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 111 [arrowhead = none ];
111 -> 11;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 112 [arrowhead = none ];
112 -> 12;

113 [label="EVAL with clause\nconvert([], X25, 0).\nand substitution\nT9 -> []\nT10 -> T17\nX25 -> T17\nX17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 113 [arrowhead = none ];
113 -> 13;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 114 [arrowhead = none ];
114 -> 14;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 115 [arrowhead = none ];
115 -> 22;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 116 [arrowhead = none ];
116 -> 23;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 117 [arrowhead = none ];
117 -> 15;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 118 [arrowhead = none ];
118 -> 16;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 119 [arrowhead = none ];
119 -> 17;

120 [label="EVAL with clause\ntimes(0, X33, 0).\nand substitution\nT17 -> T24\nX33 -> T24\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 120 [arrowhead = none ];
120 -> 18;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 121 [arrowhead = none ];
121 -> 19;

122 [label="BACKTRACK\nfor clause: times(s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
17 -> 122 [arrowhead = none ];
122 -> 21;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 123 [arrowhead = none ];
123 -> 20;

124 [label="EVAL with clause\nconvert(.(0, X56), X57, X58) :- ','(convert(X56, X57, X59), times(X59, X57, X58)).\nand substitution\nX56 -> T33\nT9 -> .(0, T33)\nT10 -> T34\nX57 -> T34\nX17 -> X60\nX58 -> X60", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 124 [arrowhead = none ];
124 -> 24;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 125 [arrowhead = none ];
125 -> 25;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 126 [arrowhead = none ];
126 -> 80;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 127 [arrowhead = none ];
127 -> 81;

128 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 128 [arrowhead = none ];
128 -> 26;

129 [label="SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:\nX59 -> T37", fontsize=14, style = filled, fillcolor = violet];
24 -> 129 [arrowhead = none ];
129 -> 27;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 130 [arrowhead = none ];
130 -> 28;

131 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
27 -> 131 [arrowhead = none ];
131 -> 60;

132 [label="SPLIT 2\nnew knowledge:\nT106 is ground\nreplacements:\nX60 -> T106", fontsize=14, style = filled, fillcolor = violet];
27 -> 132 [arrowhead = none ];
132 -> 61;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 133 [arrowhead = none ];
133 -> 29;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 134 [arrowhead = none ];
134 -> 30;

135 [label="EVAL with clause\nconvert([], X72, 0).\nand substitution\nT33 -> []\nT34 -> T44\nX72 -> T44\nX59 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 135 [arrowhead = none ];
135 -> 31;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 136 [arrowhead = none ];
136 -> 32;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 137 [arrowhead = none ];
137 -> 34;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 138 [arrowhead = none ];
138 -> 35;

139 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 139 [arrowhead = none ];
139 -> 33;

140 [label="EVAL with clause\nconvert(.(0, X92), X93, X94) :- ','(convert(X92, X93, X95), times(X95, X93, X94)).\nand substitution\nX92 -> T53\nT33 -> .(0, T53)\nT34 -> T54\nX93 -> T54\nX59 -> X96\nX94 -> X96", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 140 [arrowhead = none ];
140 -> 36;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 141 [arrowhead = none ];
141 -> 37;

142 [label="EVAL with clause\nconvert(.(s(X168), X169), X170, s(X171)) :- convert(.(X168, X169), X170, X171).\nand substitution\nX168 -> T99\nX169 -> T100\nT33 -> .(s(T99), T100)\nT34 -> T101\nX170 -> T101\nX171 -> X172\nX59 -> s(X172)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 142 [arrowhead = none ];
142 -> 58;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 143 [arrowhead = none ];
143 -> 59;

144 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 144 [arrowhead = none ];
144 -> 38;

145 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:\nX95 -> T57", fontsize=14, style = filled, fillcolor = violet];
36 -> 145 [arrowhead = none ];
145 -> 39;

146 [label="INSTANCE with matching:\nT33 -> T53\nT34 -> T54\nX59 -> X95", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 146 [arrowhead = none , style=dashed];
146 -> 26[style=dashed];

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 147 [arrowhead = none ];
147 -> 40;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 148 [arrowhead = none ];
148 -> 41;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 149 [arrowhead = none ];
149 -> 42;

150 [label="EVAL with clause\ntimes(0, X110, 0).\nand substitution\nT57 -> 0\nT54 -> T66\nX110 -> T66\nX96 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 150 [arrowhead = none ];
150 -> 43;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 151 [arrowhead = none ];
151 -> 44;

152 [label="EVAL with clause\ntimes(s(X121), X122, X123) :- ','(times(X121, X122, X124), plus(X122, X124, X123)).\nand substitution\nX121 -> T71\nT57 -> s(T71)\nT54 -> T72\nX122 -> T72\nX96 -> X125\nX123 -> X125", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 152 [arrowhead = none ];
152 -> 46;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 153 [arrowhead = none ];
153 -> 47;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 154 [arrowhead = none ];
154 -> 45;

155 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 155 [arrowhead = none ];
155 -> 48;

156 [label="SPLIT 2\nnew knowledge:\nT75 is ground\nreplacements:\nX124 -> T75", fontsize=14, style = filled, fillcolor = violet];
46 -> 156 [arrowhead = none ];
156 -> 49;

157 [label="INSTANCE with matching:\nT57 -> T71\nT54 -> T72\nX96 -> X124", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 157 [arrowhead = none , style=dashed];
157 -> 39[style=dashed];

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 158 [arrowhead = none ];
158 -> 50;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 159 [arrowhead = none ];
159 -> 51;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 160 [arrowhead = none ];
160 -> 52;

161 [label="EVAL with clause\nplus(0, X139, X139).\nand substitution\nT72 -> 0\nT75 -> T84\nX139 -> T84\nX125 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 161 [arrowhead = none ];
161 -> 53;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 162 [arrowhead = none ];
162 -> 54;

163 [label="EVAL with clause\nplus(s(X149), X150, s(X151)) :- plus(X149, X150, X151).\nand substitution\nX149 -> T89\nT72 -> s(T89)\nT75 -> T90\nX150 -> T90\nX151 -> X152\nX125 -> s(X152)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 163 [arrowhead = none ];
163 -> 56;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 164 [arrowhead = none ];
164 -> 57;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 165 [arrowhead = none ];
165 -> 55;

166 [label="INSTANCE with matching:\nT72 -> T89\nT75 -> T90\nX125 -> X152", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 166 [arrowhead = none , style=dashed];
166 -> 49[style=dashed];

167 [label="INSTANCE with matching:\nT1 -> .(T99, T100)\nT2 -> T101\nT3 -> X172", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 167 [arrowhead = none , style=dashed];
167 -> 1[style=dashed];

168 [label="INSTANCE with matching:\nT57 -> T37\nT54 -> T34\nX96 -> X60", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 168 [arrowhead = none , style=dashed];
168 -> 39[style=dashed];

169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 169 [arrowhead = none ];
169 -> 62;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 170 [arrowhead = none ];
170 -> 63;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 171 [arrowhead = none ];
171 -> 64;

172 [label="EVAL with clause\ntimes(0, X194, 0).\nand substitution\nT106 -> 0\nT34 -> T115\nX194 -> T115\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 172 [arrowhead = none ];
172 -> 65;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 173 [arrowhead = none ];
173 -> 66;

174 [label="EVAL with clause\ntimes(s(X203), X204, X205) :- ','(times(X203, X204, X206), plus(X204, X206, X205)).\nand substitution\nX203 -> T122\nT106 -> s(T122)\nT34 -> T123\nX204 -> T123\nT12 -> T125\nX205 -> T125\nT124 -> T125", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 174 [arrowhead = none ];
174 -> 68;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 175 [arrowhead = none ];
175 -> 69;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 176 [arrowhead = none ];
176 -> 67;

177 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
68 -> 177 [arrowhead = none ];
177 -> 70;

178 [label="SPLIT 2\nnew knowledge:\nT128 is ground\nreplacements:\nX206 -> T128", fontsize=14, style = filled, fillcolor = violet];
68 -> 178 [arrowhead = none ];
178 -> 71;

179 [label="INSTANCE with matching:\nT37 -> T122\nT34 -> T123\nX60 -> X206", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 179 [arrowhead = none , style=dashed];
179 -> 60[style=dashed];

180 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 180 [arrowhead = none ];
180 -> 72;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 181 [arrowhead = none ];
181 -> 73;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 182 [arrowhead = none ];
182 -> 74;

183 [label="EVAL with clause\nplus(0, X220, X220).\nand substitution\nT123 -> 0\nT128 -> T137\nX220 -> T137\nT125 -> T137", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 183 [arrowhead = none ];
183 -> 75;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 184 [arrowhead = none ];
184 -> 76;

185 [label="EVAL with clause\nplus(s(X228), X229, s(X230)) :- plus(X228, X229, X230).\nand substitution\nX228 -> T144\nT123 -> s(T144)\nT128 -> T145\nX229 -> T145\nX230 -> T147\nT125 -> s(T147)\nT146 -> T147", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 185 [arrowhead = none ];
185 -> 78;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 186 [arrowhead = none ];
186 -> 79;

187 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 187 [arrowhead = none ];
187 -> 77;

188 [label="INSTANCE with matching:\nT123 -> T144\nT128 -> T145\nT125 -> T147", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 188 [arrowhead = none , style=dashed];
188 -> 71[style=dashed];

189 [label="EVAL with clause\nconvert(.(s(X256), X257), X258, s(X259)) :- convert(.(X256, X257), X258, X259).\nand substitution\nX256 -> T162\nX257 -> T163\nT9 -> .(s(T162), T163)\nT10 -> T164\nX258 -> T164\nX259 -> X260\nX17 -> s(X260)", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 189 [arrowhead = none ];
189 -> 82;

190 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 190 [arrowhead = none ];
190 -> 83;

191 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 191 [arrowhead = none ];
191 -> 86;

192 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 192 [arrowhead = none ];
192 -> 84;

193 [label="SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:\nX260 -> T167", fontsize=14, style = filled, fillcolor = violet];
82 -> 193 [arrowhead = none ];
193 -> 85;

194 [label="INSTANCE with matching:\nT1 -> .(T162, T163)\nT2 -> T164\nT3 -> X260", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 194 [arrowhead = none , style=dashed];
194 -> 1[style=dashed];

195 [label="INSTANCE with matching:\nT106 -> s(T167)\nT34 -> T164", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 195 [arrowhead = none , style=dashed];
195 -> 61[style=dashed];

196 [label="BACKTRACK\nfor clause: convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
86 -> 196 [arrowhead = none ];
196 -> 87;

197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 197 [arrowhead = none ];
197 -> 90;

198 [label="BACKTRACK\nfor clause: convert([], B, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
90 -> 198 [arrowhead = none ];
198 -> 91;

199 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 199 [arrowhead = none ];
199 -> 92;

200 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
91 -> 200 [arrowhead = none ];
200 -> 93;

201 [label="EVAL with clause\nconvert(.(0, X316), X317, X318) :- ','(convert(X316, X317, X319), times(X319, X317, X318)).\nand substitution\nT177 -> 0\nT178 -> T194\nX316 -> T194\nT179 -> T195\nX317 -> T195\nT181 -> T197\nX318 -> T197\nT196 -> T197", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 201 [arrowhead = none ];
201 -> 94;

202 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 202 [arrowhead = none ];
202 -> 95;

203 [label="EVAL with clause\nconvert(.(s(X337), X338), X339, s(X340)) :- convert(.(X337, X338), X339, X340).\nand substitution\nX337 -> T213\nT177 -> s(T213)\nT178 -> T214\nX338 -> T214\nT179 -> T215\nX339 -> T215\nX340 -> T217\nT181 -> s(T217)\nT216 -> T217", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 203 [arrowhead = none ];
203 -> 98;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 204 [arrowhead = none ];
204 -> 99;

205 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
94 -> 205 [arrowhead = none ];
205 -> 96;

206 [label="SPLIT 2\nnew knowledge:\nT200 is ground\nreplacements:\nX319 -> T200", fontsize=14, style = filled, fillcolor = violet];
94 -> 206 [arrowhead = none ];
206 -> 97;

207 [label="INSTANCE with matching:\nT33 -> T194\nT34 -> T195\nX59 -> X319", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 207 [arrowhead = none , style=dashed];
207 -> 26[style=dashed];

208 [label="INSTANCE with matching:\nT106 -> T200\nT34 -> T195\nT12 -> T197", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 208 [arrowhead = none , style=dashed];
208 -> 61[style=dashed];

209 [label="INSTANCE with matching:\nT1 -> .(T213, T214)\nT2 -> T215\nT3 -> T217", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 209 [arrowhead = none , style=dashed];
209 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

convert26([], T44, 0).
convert26(.(0, T53), T54, X96) :- convert26(T53, T54, X95).
convert26(.(0, T53), T54, X96) :- ','(convert26(T53, T54, T57), times39(T57, T54, X96)).
convert26(.(s(T99), T100), T101, s(X172)) :- convert1(.(T99, T100), T101, X172).
times39(0, T66, 0).
times39(s(T71), T72, X125) :- times39(T71, T72, X124).
times39(s(T71), T72, X125) :- ','(times39(T71, T72, T75), plus49(T72, T75, X125)).
plus49(0, T84, T84).
plus49(s(T89), T90, s(X152)) :- plus49(T89, T90, X152).
plus71(0, T137, T137).
plus71(s(T144), T145, s(T147)) :- plus71(T144, T145, T147).
times61(0, T115, 0).
times61(s(T122), T123, T125) :- times39(T122, T123, X206).
times61(s(T122), T123, T125) :- ','(times39(T122, T123, T128), plus71(T123, T128, T125)).
convert1([], T5, 0).
convert1(.(0, []), T24, 0).
convert1(.(0, .(0, T33)), T34, T12) :- convert26(T33, T34, X59).
convert1(.(0, .(0, T33)), T34, T12) :- ','(convert26(T33, T34, T37), times39(T37, T34, X60)).
convert1(.(0, .(0, T33)), T34, T12) :- ','(convert26(T33, T34, T37), ','(times39(T37, T34, T106), times61(T106, T34, T12))).
convert1(.(0, .(s(T162), T163)), T164, T12) :- convert1(.(T162, T163), T164, X260).
convert1(.(0, .(s(T162), T163)), T164, T12) :- ','(convert1(.(T162, T163), T164, T167), times61(s(T167), T164, T12)).
convert1(.(s(0), T194), T195, s(T197)) :- convert26(T194, T195, X319).
convert1(.(s(0), T194), T195, s(T197)) :- ','(convert26(T194, T195, T200), times61(T200, T195, T197)).
convert1(.(s(s(T213)), T214), T215, s(s(T217))) :- convert1(.(T213, T214), T215, T217).

Queries:

convert1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
convert1_in: (b,b,f)
convert26_in: (b,b,f)
times39_in: (f,b,f)
plus49_in: (b,f,f)
times61_in: (f,b,f)
plus71_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U13_GGA(T33, T34, T12, convert26_in_gga(T33, T34, X59))
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U1_GGA(T53, T54, X96, convert26_in_gga(T53, T54, X95))
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U2_GGA(T53, T54, X96, convert26_in_gga(T53, T54, T57))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → U4_GGA(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U14_GGA(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_GGA(T33, T34, T12, times39_in_aga(T37, T34, X60))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → TIMES39_IN_AGA(T37, T34, X60)
TIMES39_IN_AGA(s(T71), T72, X125) → U5_AGA(T71, T72, X125, times39_in_aga(T71, T72, X124))
TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)
TIMES39_IN_AGA(s(T71), T72, X125) → U6_AGA(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_AGA(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → PLUS49_IN_GAA(T72, T75, X125)
PLUS49_IN_GAA(s(T89), T90, s(X152)) → U8_GAA(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_GGA(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_GGA(T33, T34, T12, times61_in_aga(T106, T34, T12))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → TIMES61_IN_AGA(T106, T34, T12)
TIMES61_IN_AGA(s(T122), T123, T125) → U10_AGA(T122, T123, T125, times39_in_aga(T122, T123, X206))
TIMES61_IN_AGA(s(T122), T123, T125) → TIMES39_IN_AGA(T122, T123, X206)
TIMES61_IN_AGA(s(T122), T123, T125) → U11_AGA(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_AGA(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → PLUS71_IN_GAA(T123, T128, T125)
PLUS71_IN_GAA(s(T144), T145, s(T147)) → U9_GAA(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U18_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U19_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U21_GGA(T194, T195, T197, convert26_in_gga(T194, T195, X319))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U22_GGA(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_GGA(T194, T195, T197, times61_in_aga(T200, T195, T197))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → TIMES61_IN_AGA(T200, T195, T197)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → U24_GGA(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_GGA(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → TIMES61_IN_AGA(s(T167), T164, T12)
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_GGA(T53, T54, X96, times39_in_aga(T57, T54, X96))
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → TIMES39_IN_AGA(T57, T54, X96)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4)  =  U5_AGA(x4)
U6_AGA(x1, x2, x3, x4)  =  U6_AGA(x2, x4)
U7_AGA(x1, x2, x3, x4)  =  U7_AGA(x1, x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x4)
TIMES61_IN_AGA(x1, x2, x3)  =  TIMES61_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(6) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U13_GGA(T33, T34, T12, convert26_in_gga(T33, T34, X59))
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U1_GGA(T53, T54, X96, convert26_in_gga(T53, T54, X95))
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U2_GGA(T53, T54, X96, convert26_in_gga(T53, T54, T57))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → U4_GGA(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U14_GGA(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_GGA(T33, T34, T12, times39_in_aga(T37, T34, X60))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → TIMES39_IN_AGA(T37, T34, X60)
TIMES39_IN_AGA(s(T71), T72, X125) → U5_AGA(T71, T72, X125, times39_in_aga(T71, T72, X124))
TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)
TIMES39_IN_AGA(s(T71), T72, X125) → U6_AGA(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_AGA(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → PLUS49_IN_GAA(T72, T75, X125)
PLUS49_IN_GAA(s(T89), T90, s(X152)) → U8_GAA(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_GGA(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_GGA(T33, T34, T12, times61_in_aga(T106, T34, T12))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → TIMES61_IN_AGA(T106, T34, T12)
TIMES61_IN_AGA(s(T122), T123, T125) → U10_AGA(T122, T123, T125, times39_in_aga(T122, T123, X206))
TIMES61_IN_AGA(s(T122), T123, T125) → TIMES39_IN_AGA(T122, T123, X206)
TIMES61_IN_AGA(s(T122), T123, T125) → U11_AGA(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_AGA(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → PLUS71_IN_GAA(T123, T128, T125)
PLUS71_IN_GAA(s(T144), T145, s(T147)) → U9_GAA(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U18_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U19_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U21_GGA(T194, T195, T197, convert26_in_gga(T194, T195, X319))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U22_GGA(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_GGA(T194, T195, T197, times61_in_aga(T200, T195, T197))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → TIMES61_IN_AGA(T200, T195, T197)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → U24_GGA(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_GGA(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → TIMES61_IN_AGA(s(T167), T164, T12)
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_GGA(T53, T54, X96, times39_in_aga(T57, T54, X96))
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → TIMES39_IN_AGA(T57, T54, X96)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x2, x4)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4)  =  U5_AGA(x4)
U6_AGA(x1, x2, x3, x4)  =  U6_AGA(x2, x4)
U7_AGA(x1, x2, x3, x4)  =  U7_AGA(x1, x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x4)
TIMES61_IN_AGA(x1, x2, x3)  =  TIMES61_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x5)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 32 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

PLUS71_IN_GAA(s(T144)) → PLUS71_IN_GAA(T144)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• PLUS71_IN_GAA(s(T144)) → PLUS71_IN_GAA(T144)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

PLUS49_IN_GAA(s(T89)) → PLUS49_IN_GAA(T89)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• PLUS49_IN_GAA(s(T89)) → PLUS49_IN_GAA(T89)
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

TIMES39_IN_AGA(T72) → TIMES39_IN_AGA(T72)

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

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = TIMES39_IN_AGA(T72) evaluates to t =TIMES39_IN_AGA(T72)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TIMES39_IN_AGA(T72) to TIMES39_IN_AGA(T72).

(29) NO

(30) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x2, x4)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34) → CONVERT26_IN_GGA(T33, T34)
CONVERT26_IN_GGA(.(0, T53), T54) → CONVERT26_IN_GGA(T53, T54)
CONVERT26_IN_GGA(.(s(T99), T100), T101) → CONVERT1_IN_GGA(.(T99, T100), T101)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164) → CONVERT1_IN_GGA(.(T162, T163), T164)
CONVERT1_IN_GGA(.(s(0), T194), T195) → CONVERT26_IN_GGA(T194, T195)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215) → CONVERT1_IN_GGA(.(T213, T214), T215)

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

(35) 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:

CONVERT1_IN_GGA(.(0, .(0, T33)), T34) → CONVERT26_IN_GGA(T33, T34)
CONVERT26_IN_GGA(.(0, T53), T54) → CONVERT26_IN_GGA(T53, T54)
CONVERT26_IN_GGA(.(s(T99), T100), T101) → CONVERT1_IN_GGA(.(T99, T100), T101)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164) → CONVERT1_IN_GGA(.(T162, T163), T164)
CONVERT1_IN_GGA(.(s(0), T194), T195) → CONVERT26_IN_GGA(T194, T195)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215) → CONVERT1_IN_GGA(.(T213, T214), T215)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + 2·x₂   
POL(0) = 0   
POL(CONVERT1_IN_GGA(x₁, x₂)) = x₁ + x₂   
POL(CONVERT26_IN_GGA(x₁, x₂)) = 2·x₁ + x₂   
POL(s(x₁)) = 2·x₁   

(36) Obligation:

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

(37) PisEmptyProof (EQUIVALENT transformation)

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

(38) YES

(39) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
convert1_in: (b,b,f)
convert26_in: (b,b,f)
times39_in: (f,b,f)
plus49_in: (b,f,f)
times61_in: (f,b,f)
plus71_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(40) Obligation:

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

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)

(41) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U13_GGA(T33, T34, T12, convert26_in_gga(T33, T34, X59))
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U1_GGA(T53, T54, X96, convert26_in_gga(T53, T54, X95))
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U2_GGA(T53, T54, X96, convert26_in_gga(T53, T54, T57))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → U4_GGA(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U14_GGA(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_GGA(T33, T34, T12, times39_in_aga(T37, T34, X60))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → TIMES39_IN_AGA(T37, T34, X60)
TIMES39_IN_AGA(s(T71), T72, X125) → U5_AGA(T71, T72, X125, times39_in_aga(T71, T72, X124))
TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)
TIMES39_IN_AGA(s(T71), T72, X125) → U6_AGA(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_AGA(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → PLUS49_IN_GAA(T72, T75, X125)
PLUS49_IN_GAA(s(T89), T90, s(X152)) → U8_GAA(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_GGA(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_GGA(T33, T34, T12, times61_in_aga(T106, T34, T12))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → TIMES61_IN_AGA(T106, T34, T12)
TIMES61_IN_AGA(s(T122), T123, T125) → U10_AGA(T122, T123, T125, times39_in_aga(T122, T123, X206))
TIMES61_IN_AGA(s(T122), T123, T125) → TIMES39_IN_AGA(T122, T123, X206)
TIMES61_IN_AGA(s(T122), T123, T125) → U11_AGA(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_AGA(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → PLUS71_IN_GAA(T123, T128, T125)
PLUS71_IN_GAA(s(T144), T145, s(T147)) → U9_GAA(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U18_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U19_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U21_GGA(T194, T195, T197, convert26_in_gga(T194, T195, X319))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U22_GGA(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_GGA(T194, T195, T197, times61_in_aga(T200, T195, T197))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → TIMES61_IN_AGA(T200, T195, T197)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → U24_GGA(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_GGA(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → TIMES61_IN_AGA(s(T167), T164, T12)
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_GGA(T53, T54, X96, times39_in_aga(T57, T54, X96))
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → TIMES39_IN_AGA(T57, T54, X96)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_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)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4)  =  U5_AGA(x2, x4)
U6_AGA(x1, x2, x3, x4)  =  U6_AGA(x2, x4)
U7_AGA(x1, x2, x3, x4)  =  U7_AGA(x1, x2, x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x1, x2, x4)
TIMES61_IN_AGA(x1, x2, x3)  =  TIMES61_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x2, x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x1, x2, x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x1, x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x1, x2, x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(42) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U13_GGA(T33, T34, T12, convert26_in_gga(T33, T34, X59))
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U1_GGA(T53, T54, X96, convert26_in_gga(T53, T54, X95))
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → U2_GGA(T53, T54, X96, convert26_in_gga(T53, T54, T57))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → U4_GGA(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → U14_GGA(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_GGA(T33, T34, T12, times39_in_aga(T37, T34, X60))
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → TIMES39_IN_AGA(T37, T34, X60)
TIMES39_IN_AGA(s(T71), T72, X125) → U5_AGA(T71, T72, X125, times39_in_aga(T71, T72, X124))
TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)
TIMES39_IN_AGA(s(T71), T72, X125) → U6_AGA(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_AGA(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
U6_AGA(T71, T72, X125, times39_out_aga(T71, T72, T75)) → PLUS49_IN_GAA(T72, T75, X125)
PLUS49_IN_GAA(s(T89), T90, s(X152)) → U8_GAA(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)
U14_GGA(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_GGA(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_GGA(T33, T34, T12, times61_in_aga(T106, T34, T12))
U16_GGA(T33, T34, T12, times39_out_aga(T37, T34, T106)) → TIMES61_IN_AGA(T106, T34, T12)
TIMES61_IN_AGA(s(T122), T123, T125) → U10_AGA(T122, T123, T125, times39_in_aga(T122, T123, X206))
TIMES61_IN_AGA(s(T122), T123, T125) → TIMES39_IN_AGA(T122, T123, X206)
TIMES61_IN_AGA(s(T122), T123, T125) → U11_AGA(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_AGA(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
U11_AGA(T122, T123, T125, times39_out_aga(T122, T123, T128)) → PLUS71_IN_GAA(T123, T128, T125)
PLUS71_IN_GAA(s(T144), T145, s(T147)) → U9_GAA(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U18_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → U19_GGA(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U21_GGA(T194, T195, T197, convert26_in_gga(T194, T195, X319))
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → U22_GGA(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_GGA(T194, T195, T197, times61_in_aga(T200, T195, T197))
U22_GGA(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → TIMES61_IN_AGA(T200, T195, T197)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → U24_GGA(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_GGA(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U19_GGA(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → TIMES61_IN_AGA(s(T167), T164, T12)
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_GGA(T53, T54, X96, times39_in_aga(T57, T54, X96))
U2_GGA(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → TIMES39_IN_AGA(T57, T54, X96)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_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)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)
U5_AGA(x1, x2, x3, x4)  =  U5_AGA(x2, x4)
U6_AGA(x1, x2, x3, x4)  =  U6_AGA(x2, x4)
U7_AGA(x1, x2, x3, x4)  =  U7_AGA(x1, x2, x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x1, x2, x4)
TIMES61_IN_AGA(x1, x2, x3)  =  TIMES61_IN_AGA(x2)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
U11_AGA(x1, x2, x3, x4)  =  U11_AGA(x2, x4)
U12_AGA(x1, x2, x3, x4)  =  U12_AGA(x1, x2, x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x1, x2, x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x1, x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x1, x2, x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 32 less nodes.

(44) Complex Obligation (AND)

(45) Obligation:

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

PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

PLUS71_IN_GAA(s(T144), T145, s(T147)) → PLUS71_IN_GAA(T144, T145, T147)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS71_IN_GAA(x1, x2, x3)  =  PLUS71_IN_GAA(x1)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

PLUS71_IN_GAA(s(T144)) → PLUS71_IN_GAA(T144)

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

(50) 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:

• PLUS71_IN_GAA(s(T144)) → PLUS71_IN_GAA(T144)
  The graph contains the following edges 1 > 1

(51) YES

(52) Obligation:

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

PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)

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

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) Obligation:

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

PLUS49_IN_GAA(s(T89), T90, s(X152)) → PLUS49_IN_GAA(T89, T90, X152)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PLUS49_IN_GAA(x1, x2, x3)  =  PLUS49_IN_GAA(x1)

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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

PLUS49_IN_GAA(s(T89)) → PLUS49_IN_GAA(T89)

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

(57) 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:

• PLUS49_IN_GAA(s(T89)) → PLUS49_IN_GAA(T89)
  The graph contains the following edges 1 > 1

(58) YES

(59) Obligation:

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

TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

TIMES39_IN_AGA(s(T71), T72, X125) → TIMES39_IN_AGA(T71, T72, X124)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
TIMES39_IN_AGA(x1, x2, x3)  =  TIMES39_IN_AGA(x2)

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

(62) PiDPToQDPProof (SOUND transformation)

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

(63) Obligation:

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

TIMES39_IN_AGA(T72) → TIMES39_IN_AGA(T72)

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

(64) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = TIMES39_IN_AGA(T72) evaluates to t =TIMES39_IN_AGA(T72)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TIMES39_IN_AGA(T72) to TIMES39_IN_AGA(T72).

(65) NO

(66) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)

The TRS R consists of the following rules:

convert1_in_gga([], T5, 0) → convert1_out_gga([], T5, 0)
convert1_in_gga(.(0, []), T24, 0) → convert1_out_gga(.(0, []), T24, 0)
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U13_gga(T33, T34, T12, convert26_in_gga(T33, T34, X59))
convert26_in_gga([], T44, 0) → convert26_out_gga([], T44, 0)
convert26_in_gga(.(0, T53), T54, X96) → U1_gga(T53, T54, X96, convert26_in_gga(T53, T54, X95))
convert26_in_gga(.(0, T53), T54, X96) → U2_gga(T53, T54, X96, convert26_in_gga(T53, T54, T57))
convert26_in_gga(.(s(T99), T100), T101, s(X172)) → U4_gga(T99, T100, T101, X172, convert1_in_gga(.(T99, T100), T101, X172))
convert1_in_gga(.(0, .(0, T33)), T34, T12) → U14_gga(T33, T34, T12, convert26_in_gga(T33, T34, T37))
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U15_gga(T33, T34, T12, times39_in_aga(T37, T34, X60))
times39_in_aga(0, T66, 0) → times39_out_aga(0, T66, 0)
times39_in_aga(s(T71), T72, X125) → U5_aga(T71, T72, X125, times39_in_aga(T71, T72, X124))
times39_in_aga(s(T71), T72, X125) → U6_aga(T71, T72, X125, times39_in_aga(T71, T72, T75))
U6_aga(T71, T72, X125, times39_out_aga(T71, T72, T75)) → U7_aga(T71, T72, X125, plus49_in_gaa(T72, T75, X125))
plus49_in_gaa(0, T84, T84) → plus49_out_gaa(0, T84, T84)
plus49_in_gaa(s(T89), T90, s(X152)) → U8_gaa(T89, T90, X152, plus49_in_gaa(T89, T90, X152))
U8_gaa(T89, T90, X152, plus49_out_gaa(T89, T90, X152)) → plus49_out_gaa(s(T89), T90, s(X152))
U7_aga(T71, T72, X125, plus49_out_gaa(T72, T75, X125)) → times39_out_aga(s(T71), T72, X125)
U5_aga(T71, T72, X125, times39_out_aga(T71, T72, X124)) → times39_out_aga(s(T71), T72, X125)
U15_gga(T33, T34, T12, times39_out_aga(T37, T34, X60)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
U14_gga(T33, T34, T12, convert26_out_gga(T33, T34, T37)) → U16_gga(T33, T34, T12, times39_in_aga(T37, T34, T106))
U16_gga(T33, T34, T12, times39_out_aga(T37, T34, T106)) → U17_gga(T33, T34, T12, times61_in_aga(T106, T34, T12))
times61_in_aga(0, T115, 0) → times61_out_aga(0, T115, 0)
times61_in_aga(s(T122), T123, T125) → U10_aga(T122, T123, T125, times39_in_aga(T122, T123, X206))
U10_aga(T122, T123, T125, times39_out_aga(T122, T123, X206)) → times61_out_aga(s(T122), T123, T125)
times61_in_aga(s(T122), T123, T125) → U11_aga(T122, T123, T125, times39_in_aga(T122, T123, T128))
U11_aga(T122, T123, T125, times39_out_aga(T122, T123, T128)) → U12_aga(T122, T123, T125, plus71_in_gaa(T123, T128, T125))
plus71_in_gaa(0, T137, T137) → plus71_out_gaa(0, T137, T137)
plus71_in_gaa(s(T144), T145, s(T147)) → U9_gaa(T144, T145, T147, plus71_in_gaa(T144, T145, T147))
U9_gaa(T144, T145, T147, plus71_out_gaa(T144, T145, T147)) → plus71_out_gaa(s(T144), T145, s(T147))
U12_aga(T122, T123, T125, plus71_out_gaa(T123, T128, T125)) → times61_out_aga(s(T122), T123, T125)
U17_gga(T33, T34, T12, times61_out_aga(T106, T34, T12)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U18_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, X260))
convert1_in_gga(.(0, .(s(T162), T163)), T164, T12) → U19_gga(T162, T163, T164, T12, convert1_in_gga(.(T162, T163), T164, T167))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U21_gga(T194, T195, T197, convert26_in_gga(T194, T195, X319))
U21_gga(T194, T195, T197, convert26_out_gga(T194, T195, X319)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(0), T194), T195, s(T197)) → U22_gga(T194, T195, T197, convert26_in_gga(T194, T195, T200))
U22_gga(T194, T195, T197, convert26_out_gga(T194, T195, T200)) → U23_gga(T194, T195, T197, times61_in_aga(T200, T195, T197))
U23_gga(T194, T195, T197, times61_out_aga(T200, T195, T197)) → convert1_out_gga(.(s(0), T194), T195, s(T197))
convert1_in_gga(.(s(s(T213)), T214), T215, s(s(T217))) → U24_gga(T213, T214, T215, T217, convert1_in_gga(.(T213, T214), T215, T217))
U24_gga(T213, T214, T215, T217, convert1_out_gga(.(T213, T214), T215, T217)) → convert1_out_gga(.(s(s(T213)), T214), T215, s(s(T217)))
U19_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, T167)) → U20_gga(T162, T163, T164, T12, times61_in_aga(s(T167), T164, T12))
U20_gga(T162, T163, T164, T12, times61_out_aga(s(T167), T164, T12)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U18_gga(T162, T163, T164, T12, convert1_out_gga(.(T162, T163), T164, X260)) → convert1_out_gga(.(0, .(s(T162), T163)), T164, T12)
U4_gga(T99, T100, T101, X172, convert1_out_gga(.(T99, T100), T101, X172)) → convert26_out_gga(.(s(T99), T100), T101, s(X172))
U2_gga(T53, T54, X96, convert26_out_gga(T53, T54, T57)) → U3_gga(T53, T54, X96, times39_in_aga(T57, T54, X96))
U3_gga(T53, T54, X96, times39_out_aga(T57, T54, X96)) → convert26_out_gga(.(0, T53), T54, X96)
U1_gga(T53, T54, X96, convert26_out_gga(T53, T54, X95)) → convert26_out_gga(.(0, T53), T54, X96)
U13_gga(T33, T34, T12, convert26_out_gga(T33, T34, X59)) → convert1_out_gga(.(0, .(0, T33)), T34, T12)

The argument filtering Pi contains the following mapping:
convert1_in_gga(x1, x2, x3)  =  convert1_in_gga(x1, x2)
[]  =  []
convert1_out_gga(x1, x2, x3)  =  convert1_out_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
convert26_in_gga(x1, x2, x3)  =  convert26_in_gga(x1, x2)
convert26_out_gga(x1, x2, x3)  =  convert26_out_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)
s(x1)  =  s(x1)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x1, x2, x4)
plus49_in_gaa(x1, x2, x3)  =  plus49_in_gaa(x1)
plus49_out_gaa(x1, x2, x3)  =  plus49_out_gaa(x1)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x4)
U16_gga(x1, x2, x3, x4)  =  U16_gga(x1, x2, x4)
U17_gga(x1, x2, x3, x4)  =  U17_gga(x1, x2, x4)
times61_in_aga(x1, x2, x3)  =  times61_in_aga(x2)
times61_out_aga(x1, x2, x3)  =  times61_out_aga(x1, x2)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
times39_in_aga(x1, x2, x3)  =  times39_in_aga(x2)
times39_out_aga(x1, x2, x3)  =  times39_out_aga(x1, x2)
U5_aga(x1, x2, x3, x4)  =  U5_aga(x2, x4)
U6_aga(x1, x2, x3, x4)  =  U6_aga(x2, x4)
U7_aga(x1, x2, x3, x4)  =  U7_aga(x1, x2, x4)
U11_aga(x1, x2, x3, x4)  =  U11_aga(x2, x4)
U12_aga(x1, x2, x3, x4)  =  U12_aga(x1, x2, x4)
plus71_in_gaa(x1, x2, x3)  =  plus71_in_gaa(x1)
plus71_out_gaa(x1, x2, x3)  =  plus71_out_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4)  =  U21_gga(x1, x2, x4)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
U24_gga(x1, x2, x3, x4, x5)  =  U24_gga(x1, x2, x3, x5)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)

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

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34, T12) → CONVERT26_IN_GGA(T33, T34, X59)
CONVERT26_IN_GGA(.(0, T53), T54, X96) → CONVERT26_IN_GGA(T53, T54, X95)
CONVERT26_IN_GGA(.(s(T99), T100), T101, s(X172)) → CONVERT1_IN_GGA(.(T99, T100), T101, X172)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164, T12) → CONVERT1_IN_GGA(.(T162, T163), T164, X260)
CONVERT1_IN_GGA(.(s(0), T194), T195, s(T197)) → CONVERT26_IN_GGA(T194, T195, X319)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215, s(s(T217))) → CONVERT1_IN_GGA(.(T213, T214), T215, T217)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
CONVERT1_IN_GGA(x1, x2, x3)  =  CONVERT1_IN_GGA(x1, x2)
CONVERT26_IN_GGA(x1, x2, x3)  =  CONVERT26_IN_GGA(x1, x2)

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

(69) PiDPToQDPProof (SOUND transformation)

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

(70) Obligation:

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

CONVERT1_IN_GGA(.(0, .(0, T33)), T34) → CONVERT26_IN_GGA(T33, T34)
CONVERT26_IN_GGA(.(0, T53), T54) → CONVERT26_IN_GGA(T53, T54)
CONVERT26_IN_GGA(.(s(T99), T100), T101) → CONVERT1_IN_GGA(.(T99, T100), T101)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164) → CONVERT1_IN_GGA(.(T162, T163), T164)
CONVERT1_IN_GGA(.(s(0), T194), T195) → CONVERT26_IN_GGA(T194, T195)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215) → CONVERT1_IN_GGA(.(T213, T214), T215)

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

(71) 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:

CONVERT1_IN_GGA(.(0, .(0, T33)), T34) → CONVERT26_IN_GGA(T33, T34)
CONVERT26_IN_GGA(.(0, T53), T54) → CONVERT26_IN_GGA(T53, T54)
CONVERT26_IN_GGA(.(s(T99), T100), T101) → CONVERT1_IN_GGA(.(T99, T100), T101)
CONVERT1_IN_GGA(.(0, .(s(T162), T163)), T164) → CONVERT1_IN_GGA(.(T162, T163), T164)
CONVERT1_IN_GGA(.(s(0), T194), T195) → CONVERT26_IN_GGA(T194, T195)
CONVERT1_IN_GGA(.(s(s(T213)), T214), T215) → CONVERT1_IN_GGA(.(T213, T214), T215)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + 2·x₂   
POL(0) = 0   
POL(CONVERT1_IN_GGA(x₁, x₂)) = x₁ + x₂   
POL(CONVERT26_IN_GGA(x₁, x₂)) = 2·x₁ + x₂   
POL(s(x₁)) = 2·x₁   

(72) Obligation:

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

(73) PisEmptyProof (EQUIVALENT transformation)

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

(74) YES