Left Termination of the query pattern f_in_1(g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

pred(0, 0).
pred(s(0), 0).
pred(s(s(X)), s(Y)) :- pred(s(X), Y).
double(0, 0).
double(s(X), s(s(Y))) :- ','(pred(s(X), Z), double(Z, Y)).
half(0, 0).
half(s(s(X)), s(U)) :- ','(pred(s(s(X)), Y), ','(pred(Y, Z), half(Z, U))).
f(s(X)) :- ','(half(s(X), Y), ','(double(Y, Z), f(Z))).

Queries:

f(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1), SCOPE: 1, CLAUSE: 7\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(half(s(T3), X3), ','(double(X3, X4), f(X4)))\n\nT3 is ground\nX3 is free; \nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(half(s(T3), X3), ','(double(X3, X4), f(X4))), SCOPE: 2, CLAUSE: 5 | ','(half(s(T3), X3), ','(double(X3, X4), f(X4))), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; \nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(half(s(T3), X3), ','(double(X3, X4), f(X4))), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free; \nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(pred(s(s(T6)), X16), ','(pred(X16, X17), ','(half(X17, X18), ','(double(s(X18), X4), f(X4)))))\n\nT6 is ground\nX4 is free; \nX18 is free; \nX16 is free; \nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: pred(s(s(T6)), X16)\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(pred(T8, X17), ','(half(X17, X18), ','(double(s(X18), X4), f(X4))))\n\nT8 is ground\nX4 is free; \nX18 is free; \nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 0 | pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 1 | pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 2\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 1 | pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 2\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: pred(s(s(T6)), X16), SCOPE: 3, CLAUSE: 2\n\nT6 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: pred(s(T12), X35)\n\nT12 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: pred(s(T12), X35), SCOPE: 4, CLAUSE: 0 | pred(s(T12), X35), SCOPE: 4, CLAUSE: 1 | pred(s(T12), X35), SCOPE: 4, CLAUSE: 2\n\nT12 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: pred(s(T12), X35), SCOPE: 4, CLAUSE: 1 | pred(s(T12), X35), SCOPE: 4, CLAUSE: 2\n\nT12 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: pred(s(T12), X35), SCOPE: 4, CLAUSE: 1\n\nT12 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: pred(s(T12), X35), SCOPE: 4, CLAUSE: 2\n\nT12 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\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: pred(s(T15), X44)\n\nT15 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: pred(T8, X17)\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(half(T16, X18), ','(double(s(X18), X4), f(X4)))\n\nT16 is ground\nX4 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: pred(T8, X17), SCOPE: 5, CLAUSE: 0 | pred(T8, X17), SCOPE: 5, CLAUSE: 1 | pred(T8, X17), SCOPE: 5, CLAUSE: 2\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: pred(T8, X17), SCOPE: 5, CLAUSE: 0\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: pred(T8, X17), SCOPE: 5, CLAUSE: 1 | pred(T8, X17), SCOPE: 5, CLAUSE: 2\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: pred(T8, X17), SCOPE: 5, CLAUSE: 1\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: pred(T8, X17), SCOPE: 5, CLAUSE: 2\n\nT8 is ground\nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: pred(s(T19), X55)\n\nT19 is ground\nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: half(T16, X18)\n\nT16 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(double(s(T20), X4), f(X4))\n\nT20 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: half(T16, X18), SCOPE: 6, CLAUSE: 5 | half(T16, X18), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: half(T16, X18), SCOPE: 6, CLAUSE: 5\n\nT16 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: half(T16, X18), SCOPE: 6, CLAUSE: 6\n\nT16 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(pred(s(s(T23)), X68), ','(pred(X68, X69), half(X69, X70)))\n\nT23 is ground\nX70 is free; \nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: pred(s(s(T23)), X68)\n\nT23 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(pred(T25, X69), half(X69, X70))\n\nT25 is ground\nX70 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: pred(T25, X69)\n\nT25 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: half(T27, X70)\n\nT27 is ground\nX70 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: double(s(T20), X4)\n\nT20 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: f(T29)\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: double(s(T20), X4), SCOPE: 7, CLAUSE: 3 | double(s(T20), X4), SCOPE: 7, CLAUSE: 4\n\nT20 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: double(s(T20), X4), SCOPE: 7, CLAUSE: 4\n\nT20 is ground\nX4 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(pred(s(T33), X96), double(X96, X97))\n\nT33 is ground\nX97 is free; \nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: pred(s(T33), X96)\n\nT33 is ground\nX96 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: double(T34, X97)\n\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: double(T34, X97), SCOPE: 8, CLAUSE: 3 | double(T34, X97), SCOPE: 8, CLAUSE: 4\n\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: double(T34, X97), SCOPE: 8, CLAUSE: 3\n\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: double(T34, X97), SCOPE: 8, CLAUSE: 4\n\nT34 is ground\nX97 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(pred(s(T37), X107), double(X107, X108))\n\nT37 is ground\nX108 is free; \nX107 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 68 [arrowhead = none ];
68 -> 2;

69 [label="EVAL with clause\nf(s(X2)) :- ','(half(s(X2), X3), ','(double(X3, X4), f(X4))).\nand substitution\nX2 -> T3\nT1 -> s(T3)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 3;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 70 [arrowhead = none ];
70 -> 4;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 71 [arrowhead = none ];
71 -> 5;

72 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 72 [arrowhead = none ];
72 -> 6;

73 [label="EVAL with clause\nhalf(s(s(X14)), s(X15)) :- ','(pred(s(s(X14)), X16), ','(pred(X16, X17), half(X17, X15))).\nand substitution\nX14 -> T6\nT3 -> s(T6)\nX15 -> X18\nX3 -> s(X18)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 73 [arrowhead = none ];
73 -> 7;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 74 [arrowhead = none ];
74 -> 8;

75 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 75 [arrowhead = none ];
75 -> 9;

76 [label="SPLIT 2\nnew knowledge:\nT8 is ground\nreplacements:\nX16 -> T8", fontsize=14, style = filled, fillcolor = violet];
7 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
10 -> 78 [arrowhead = none ];
78 -> 24;

79 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nreplacements:\nX17 -> T16", fontsize=14, style = filled, fillcolor = violet];
10 -> 79 [arrowhead = none ];
79 -> 25;

80 [label="BACKTRACK\nfor clause: pred(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="BACKTRACK\nfor clause: pred(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="ONLY EVAL with clause\npred(s(s(X33)), s(X34)) :- pred(s(X33), X34).\nand substitution\nT6 -> T12\nX33 -> T12\nX34 -> X35\nX16 -> s(X35)", fontsize=14, style = filled, fillcolor = white];
13 -> 82 [arrowhead = none ];
82 -> 14;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 83 [arrowhead = none ];
83 -> 15;

84 [label="BACKTRACK\nfor clause: pred(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 84 [arrowhead = none ];
84 -> 16;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 85 [arrowhead = none ];
85 -> 17;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="EVAL with clause\npred(s(0), 0).\nand substitution\nT12 -> 0\nX35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 87 [arrowhead = none ];
87 -> 19;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 88 [arrowhead = none ];
88 -> 20;

89 [label="EVAL with clause\npred(s(s(X42)), s(X43)) :- pred(s(X42), X43).\nand substitution\nX42 -> T15\nT12 -> s(T15)\nX43 -> X44\nX35 -> s(X44)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 89 [arrowhead = none ];
89 -> 22;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 90 [arrowhead = none ];
90 -> 23;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 91 [arrowhead = none ];
91 -> 21;

92 [label="INSTANCE with matching:\nT12 -> T15\nX35 -> X44", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 92 [arrowhead = none , style=dashed];
92 -> 14[style=dashed];

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 93 [arrowhead = none ];
93 -> 26;

94 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="SPLIT 2\nnew knowledge:\nT20 is ground\nreplacements:\nX18 -> T20", fontsize=14, style = filled, fillcolor = violet];
25 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 96 [arrowhead = none ];
96 -> 27;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="EVAL with clause\npred(0, 0).\nand substitution\nT8 -> 0\nX17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 99 [arrowhead = none ];
99 -> 30;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 100 [arrowhead = none ];
100 -> 32;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 101 [arrowhead = none ];
101 -> 33;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="EVAL with clause\npred(s(0), 0).\nand substitution\nT8 -> s(0)\nX17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 103 [arrowhead = none ];
103 -> 34;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 104 [arrowhead = none ];
104 -> 35;

105 [label="EVAL with clause\npred(s(s(X53)), s(X54)) :- pred(s(X53), X54).\nand substitution\nX53 -> T19\nT8 -> s(s(T19))\nX54 -> X55\nX17 -> s(X55)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 105 [arrowhead = none ];
105 -> 37;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 106 [arrowhead = none ];
106 -> 38;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 107 [arrowhead = none ];
107 -> 36;

108 [label="INSTANCE with matching:\nT8 -> s(T19)\nX17 -> X55", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 108 [arrowhead = none , style=dashed];
108 -> 24[style=dashed];

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 109 [arrowhead = none ];
109 -> 41;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 110 [arrowhead = none ];
110 -> 53;

111 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:\nX4 -> T29", fontsize=14, style = filled, fillcolor = violet];
40 -> 111 [arrowhead = none ];
111 -> 54;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 112 [arrowhead = none ];
112 -> 42;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 113 [arrowhead = none ];
113 -> 43;

114 [label="EVAL with clause\nhalf(0, 0).\nand substitution\nT16 -> 0\nX18 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 115 [arrowhead = none ];
115 -> 45;

116 [label="EVAL with clause\nhalf(s(s(X66)), s(X67)) :- ','(pred(s(s(X66)), X68), ','(pred(X68, X69), half(X69, X67))).\nand substitution\nX66 -> T23\nT16 -> s(s(T23))\nX67 -> X70\nX18 -> s(X70)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 116 [arrowhead = none ];
116 -> 47;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 117 [arrowhead = none ];
117 -> 48;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 118 [arrowhead = none ];
118 -> 46;

119 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 119 [arrowhead = none ];
119 -> 49;

120 [label="SPLIT 2\nnew knowledge:\nT25 is ground\nreplacements:\nX68 -> T25", fontsize=14, style = filled, fillcolor = violet];
47 -> 120 [arrowhead = none ];
120 -> 50;

121 [label="INSTANCE with matching:\nT6 -> T23\nX16 -> X68", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 121 [arrowhead = none , style=dashed];
121 -> 9[style=dashed];

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 122 [arrowhead = none ];
122 -> 51;

123 [label="SPLIT 2\nnew knowledge:\nT27 is ground\nreplacements:\nX69 -> T27", fontsize=14, style = filled, fillcolor = violet];
50 -> 123 [arrowhead = none ];
123 -> 52;

124 [label="INSTANCE with matching:\nT8 -> T25\nX17 -> X69", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 124 [arrowhead = none , style=dashed];
124 -> 24[style=dashed];

125 [label="INSTANCE with matching:\nT16 -> T27\nX18 -> X70", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 125 [arrowhead = none , style=dashed];
125 -> 39[style=dashed];

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 126 [arrowhead = none ];
126 -> 55;

127 [label="INSTANCE with matching:\nT1 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 127 [arrowhead = none , style=dashed];
127 -> 1[style=dashed];

128 [label="BACKTRACK\nfor clause: double(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 128 [arrowhead = none ];
128 -> 56;

129 [label="ONLY EVAL with clause\ndouble(s(X94), s(s(X95))) :- ','(pred(s(X94), X96), double(X96, X95)).\nand substitution\nT20 -> T33\nX94 -> T33\nX95 -> X97\nX4 -> s(s(X97))", fontsize=14, style = filled, fillcolor = white];
56 -> 129 [arrowhead = none ];
129 -> 57;

130 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 130 [arrowhead = none ];
130 -> 58;

131 [label="SPLIT 2\nnew knowledge:\nT34 is ground\nreplacements:\nX96 -> T34", fontsize=14, style = filled, fillcolor = violet];
57 -> 131 [arrowhead = none ];
131 -> 59;

132 [label="INSTANCE with matching:\nT8 -> s(T33)\nX17 -> X96", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 132 [arrowhead = none , style=dashed];
132 -> 24[style=dashed];

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 133 [arrowhead = none ];
133 -> 60;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 134 [arrowhead = none ];
134 -> 61;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 135 [arrowhead = none ];
135 -> 62;

136 [label="EVAL with clause\ndouble(0, 0).\nand substitution\nT34 -> 0\nX97 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 136 [arrowhead = none ];
136 -> 63;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 137 [arrowhead = none ];
137 -> 64;

138 [label="EVAL with clause\ndouble(s(X105), s(s(X106))) :- ','(pred(s(X105), X107), double(X107, X106)).\nand substitution\nX105 -> T37\nT34 -> s(T37)\nX106 -> X108\nX97 -> s(s(X108))", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 138 [arrowhead = none ];
138 -> 66;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 139 [arrowhead = none ];
139 -> 67;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 140 [arrowhead = none ];
140 -> 65;

141 [label="INSTANCE with matching:\nT33 -> T37\nX96 -> X107\nX97 -> X108", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 141 [arrowhead = none , style=dashed];
141 -> 57[style=dashed];

}

(2) Obligation:

Clauses:

pred14(0, 0).
pred14(s(T15), s(X44)) :- pred14(T15, X44).
pred24(0, 0).
pred24(s(0), 0).
pred24(s(s(T19)), s(X55)) :- pred24(s(T19), X55).
pred9(T12, s(X35)) :- pred14(T12, X35).
half39(0, 0).
half39(s(s(T23)), s(X70)) :- pred9(T23, X68).
half39(s(s(T23)), s(X70)) :- ','(pred9(T23, T25), pred24(T25, X69)).
half39(s(s(T23)), s(X70)) :- ','(pred9(T23, T25), ','(pred24(T25, T27), half39(T27, X70))).
p57(T33, X96, X97) :- pred24(s(T33), X96).
p57(T33, 0, 0) :- pred24(s(T33), 0).
p57(T33, s(T37), s(s(X108))) :- ','(pred24(s(T33), s(T37)), p57(T37, X107, X108)).
double53(T33, s(s(X97))) :- p57(T33, X96, X97).
f1(s(s(T6))) :- pred9(T6, X16).
f1(s(s(T6))) :- ','(pred9(T6, T8), pred24(T8, X17)).
f1(s(s(T6))) :- ','(pred9(T6, T8), ','(pred24(T8, T16), half39(T16, X18))).
f1(s(s(T6))) :- ','(pred9(T6, T8), ','(pred24(T8, T16), ','(half39(T16, T20), double53(T20, X4)))).
f1(s(s(T6))) :- ','(pred9(T6, T8), ','(pred24(T8, T16), ','(half39(T16, T20), ','(double53(T20, T29), f1(T29))))).

Queries:

f1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
f1_in: (b) (f)
pred9_in: (b,f) (f,f)
pred14_in: (b,f) (f,f)
pred24_in: (b,f) (f,f) (f,b) (b,b)
half39_in: (b,f)
double53_in: (f,f)
p57_in: (f,f,f) (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

F1_IN_G(s(s(T6))) → U14_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U15_G(T6, pred9_in_ga(T6, T8))
U15_G(T6, pred9_out_ga(T6, T8)) → U16_G(T6, pred24_in_ga(T8, X17))
U15_G(T6, pred9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U15_G(T6, pred9_out_ga(T6, T8)) → U17_G(T6, pred24_in_ga(T8, T16))
U17_G(T6, pred24_out_ga(T8, T16)) → U18_G(T6, half39_in_ga(T16, X18))
U17_G(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U17_G(T6, pred24_out_ga(T8, T16)) → U19_G(T6, half39_in_ga(T16, T20))
U19_G(T6, half39_out_ga(T16, T20)) → U20_G(T6, double53_in_aa(T20, X4))
U19_G(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
DOUBLE53_IN_AA(T33, s(s(X97))) → U13_AA(T33, X97, p57_in_aaa(T33, X96, X97))
DOUBLE53_IN_AA(T33, s(s(X97))) → P57_IN_AAA(T33, X96, X97)
P57_IN_AAA(T33, X96, X97) → U9_AAA(T33, X96, X97, pred24_in_aa(s(T33), X96))
P57_IN_AAA(T33, X96, X97) → PRED24_IN_AA(s(T33), X96)
PRED24_IN_AA(s(s(T19)), s(X55)) → U2_AA(T19, X55, pred24_in_aa(s(T19), X55))
PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)
P57_IN_AAA(T33, 0, 0) → U10_AAA(T33, pred24_in_ag(s(T33), 0))
P57_IN_AAA(T33, 0, 0) → PRED24_IN_AG(s(T33), 0)
PRED24_IN_AG(s(s(T19)), s(X55)) → U2_AG(T19, X55, pred24_in_ag(s(T19), X55))
PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)
P57_IN_AAA(T33, s(T37), s(s(X108))) → U11_AAA(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
P57_IN_AAA(T33, s(T37), s(s(X108))) → PRED24_IN_AA(s(T33), s(T37))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_AAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, 0, 0) → U10_GAA(T33, pred24_in_gg(s(T33), 0))
P57_IN_GAA(T33, 0, 0) → PRED24_IN_GG(s(T33), 0)
PRED24_IN_GG(s(s(T19)), s(X55)) → U2_GG(T19, X55, pred24_in_gg(s(T19), X55))
PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
P57_IN_GAA(T33, s(T37), s(s(X108))) → PRED24_IN_GA(s(T33), s(T37))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U19_G(T6, half39_out_ga(T16, T20)) → U21_G(T6, double53_in_aa(T20, T29))
U21_G(T6, double53_out_aa(T20, T29)) → U22_G(T6, f1_in_a(T29))
U21_G(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U14_A(T6, pred9_in_aa(T6, X16))
F1_IN_A(s(s(T6))) → PRED9_IN_AA(T6, X16)
PRED9_IN_AA(T12, s(X35)) → U3_AA(T12, X35, pred14_in_aa(T12, X35))
PRED9_IN_AA(T12, s(X35)) → PRED14_IN_AA(T12, X35)
PRED14_IN_AA(s(T15), s(X44)) → U1_AA(T15, X44, pred14_in_aa(T15, X44))
PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U16_A(T6, pred24_in_ga(T8, X17))
U15_A(T6, pred9_out_aa(T6, T8)) → PRED24_IN_GA(T8, X17)
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))
U17_A(T6, pred24_out_ga(T8, T16)) → U18_A(T6, half39_in_ga(T16, X18))
U17_A(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U20_A(T6, double53_in_aa(T20, X4))
U19_A(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → U22_A(T6, f1_in_a(T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
DOUBLE53_IN_AA(x1, x2)  =  DOUBLE53_IN_AA
U13_AA(x1, x2, x3)  =  U13_AA(x3)
P57_IN_AAA(x1, x2, x3)  =  P57_IN_AAA
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U10_AAA(x1, x2)  =  U10_AAA(x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U11_AAA(x1, x2, x3, x4)  =  U11_AAA(x4)
U12_AAA(x1, x2, x3, x4)  =  U12_AAA(x1, x2, x4)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2)  =  U10_GAA(x1, x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x2, x4)
U21_G(x1, x2)  =  U21_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U14_A(x1, x2)  =  U14_A(x2)
PRED9_IN_AA(x1, x2)  =  PRED9_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U15_A(x1, x2)  =  U15_A(x2)
U16_A(x1, x2)  =  U16_A(x1, x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U18_A(x1, x2)  =  U18_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U20_A(x1, x2)  =  U20_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)
U22_A(x1, x2)  =  U22_A(x1, x2)

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

(6) Obligation:

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

F1_IN_G(s(s(T6))) → U14_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U15_G(T6, pred9_in_ga(T6, T8))
U15_G(T6, pred9_out_ga(T6, T8)) → U16_G(T6, pred24_in_ga(T8, X17))
U15_G(T6, pred9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U15_G(T6, pred9_out_ga(T6, T8)) → U17_G(T6, pred24_in_ga(T8, T16))
U17_G(T6, pred24_out_ga(T8, T16)) → U18_G(T6, half39_in_ga(T16, X18))
U17_G(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U17_G(T6, pred24_out_ga(T8, T16)) → U19_G(T6, half39_in_ga(T16, T20))
U19_G(T6, half39_out_ga(T16, T20)) → U20_G(T6, double53_in_aa(T20, X4))
U19_G(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
DOUBLE53_IN_AA(T33, s(s(X97))) → U13_AA(T33, X97, p57_in_aaa(T33, X96, X97))
DOUBLE53_IN_AA(T33, s(s(X97))) → P57_IN_AAA(T33, X96, X97)
P57_IN_AAA(T33, X96, X97) → U9_AAA(T33, X96, X97, pred24_in_aa(s(T33), X96))
P57_IN_AAA(T33, X96, X97) → PRED24_IN_AA(s(T33), X96)
PRED24_IN_AA(s(s(T19)), s(X55)) → U2_AA(T19, X55, pred24_in_aa(s(T19), X55))
PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)
P57_IN_AAA(T33, 0, 0) → U10_AAA(T33, pred24_in_ag(s(T33), 0))
P57_IN_AAA(T33, 0, 0) → PRED24_IN_AG(s(T33), 0)
PRED24_IN_AG(s(s(T19)), s(X55)) → U2_AG(T19, X55, pred24_in_ag(s(T19), X55))
PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)
P57_IN_AAA(T33, s(T37), s(s(X108))) → U11_AAA(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
P57_IN_AAA(T33, s(T37), s(s(X108))) → PRED24_IN_AA(s(T33), s(T37))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_AAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, 0, 0) → U10_GAA(T33, pred24_in_gg(s(T33), 0))
P57_IN_GAA(T33, 0, 0) → PRED24_IN_GG(s(T33), 0)
PRED24_IN_GG(s(s(T19)), s(X55)) → U2_GG(T19, X55, pred24_in_gg(s(T19), X55))
PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
P57_IN_GAA(T33, s(T37), s(s(X108))) → PRED24_IN_GA(s(T33), s(T37))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U19_G(T6, half39_out_ga(T16, T20)) → U21_G(T6, double53_in_aa(T20, T29))
U21_G(T6, double53_out_aa(T20, T29)) → U22_G(T6, f1_in_a(T29))
U21_G(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U14_A(T6, pred9_in_aa(T6, X16))
F1_IN_A(s(s(T6))) → PRED9_IN_AA(T6, X16)
PRED9_IN_AA(T12, s(X35)) → U3_AA(T12, X35, pred14_in_aa(T12, X35))
PRED9_IN_AA(T12, s(X35)) → PRED14_IN_AA(T12, X35)
PRED14_IN_AA(s(T15), s(X44)) → U1_AA(T15, X44, pred14_in_aa(T15, X44))
PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U16_A(T6, pred24_in_ga(T8, X17))
U15_A(T6, pred9_out_aa(T6, T8)) → PRED24_IN_GA(T8, X17)
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))
U17_A(T6, pred24_out_ga(T8, T16)) → U18_A(T6, half39_in_ga(T16, X18))
U17_A(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U20_A(T6, double53_in_aa(T20, X4))
U19_A(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → U22_A(T6, f1_in_a(T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
DOUBLE53_IN_AA(x1, x2)  =  DOUBLE53_IN_AA
U13_AA(x1, x2, x3)  =  U13_AA(x3)
P57_IN_AAA(x1, x2, x3)  =  P57_IN_AAA
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U10_AAA(x1, x2)  =  U10_AAA(x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U11_AAA(x1, x2, x3, x4)  =  U11_AAA(x4)
U12_AAA(x1, x2, x3, x4)  =  U12_AAA(x1, x2, x4)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2)  =  U10_GAA(x1, x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x1, x2, x4)
U21_G(x1, x2)  =  U21_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U14_A(x1, x2)  =  U14_A(x2)
PRED9_IN_AA(x1, x2)  =  PRED9_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U15_A(x1, x2)  =  U15_A(x2)
U16_A(x1, x2)  =  U16_A(x1, x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U18_A(x1, x2)  =  U18_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U20_A(x1, x2)  =  U20_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)
U22_A(x1, x2)  =  U22_A(x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 9 SCCs with 54 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA

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:

PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA

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:

PRED14_IN_AA → PRED14_IN_AA

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

(14) 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 = PRED14_IN_AA evaluates to t =PRED14_IN_AA

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 PRED14_IN_AA to PRED14_IN_AA.

(15) NO

(16) Obligation:

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

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

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:

• PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
  The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(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:

PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(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:

PRED24_IN_AG(s(X55)) → PRED24_IN_AG(X55)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• PRED24_IN_AG(s(X55)) → PRED24_IN_AG(X55)
  The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA

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:

PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA

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:

PRED24_IN_AA → PRED24_IN_AA

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

(35) 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 = PRED24_IN_AA evaluates to t =PRED24_IN_AA

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 PRED24_IN_AA to PRED24_IN_AA.

(36) NO

(37) Obligation:

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

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))

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

(42) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))
  The graph contains the following edges 1 > 1

(43) YES

(44) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

P57_IN_GAA(T33) → U11_GAA(T33, pred24_in_ga(s(T33)))
U11_GAA(T33, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37)

The TRS R consists of the following rules:

pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)

The set Q consists of the following terms:

pred24_in_ga(x0)
U2_ga(x0, x1)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11_GAA(T33, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37)

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

POL(0) = 1   
POL(P57_IN_GAA(x₁)) = 1 + x₁   
POL(U11_GAA(x₁, x₂)) = x₂   
POL(U2_ga(x₁, x₂)) = 1 + x₂   
POL(pred24_in_ga(x₁)) = x₁   
POL(pred24_out_ga(x₁, x₂)) = 1 + x₂   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))

(50) Obligation:

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

P57_IN_GAA(T33) → U11_GAA(T33, pred24_in_ga(s(T33)))

The TRS R consists of the following rules:

pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)

The set Q consists of the following terms:

pred24_in_ga(x0)
U2_ga(x0, x1)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(52) TRUE

(53) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)

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

(58) 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:

• PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)
  The graph contains the following edges 1 > 1

(59) YES

(60) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)

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

(61) UsableRulesProof (EQUIVALENT transformation)

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

(62) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)

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

(63) PiDPToQDPProof (SOUND transformation)

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

(64) Obligation:

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

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, pred9_in_ga(T23))
U5_GA(T23, pred9_out_ga(T23, T25)) → U7_GA(T23, pred24_in_ga(T25))
U7_GA(T23, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

pred9_in_ga(x0)
pred24_in_ga(x0)
U3_ga(x0, x1)
U2_ga(x0, x1)
pred14_in_ga(x0)
U1_ga(x0, x1)

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

(65) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule HALF39_IN_GA(s(s(T23))) → U5_GA(T23, pred9_in_ga(T23)) at position [1] we obtained the following new rules [LPAR04]:

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U3_ga(T23, pred14_in_ga(T23)))

(66) Obligation:

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

U5_GA(T23, pred9_out_ga(T23, T25)) → U7_GA(T23, pred24_in_ga(T25))
U7_GA(T23, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U3_ga(T23, pred14_in_ga(T23)))

The TRS R consists of the following rules:

pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

pred9_in_ga(x0)
pred24_in_ga(x0)
U3_ga(x0, x1)
U2_ga(x0, x1)
pred14_in_ga(x0)
U1_ga(x0, x1)

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

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) Obligation:

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

U5_GA(T23, pred9_out_ga(T23, T25)) → U7_GA(T23, pred24_in_ga(T25))
U7_GA(T23, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U3_ga(T23, pred14_in_ga(T23)))

The TRS R consists of the following rules:

pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

pred9_in_ga(x0)
pred24_in_ga(x0)
U3_ga(x0, x1)
U2_ga(x0, x1)
pred14_in_ga(x0)
U1_ga(x0, x1)

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

(69) QReductionProof (EQUIVALENT transformation)

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

pred9_in_ga(x0)

(70) Obligation:

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

U5_GA(T23, pred9_out_ga(T23, T25)) → U7_GA(T23, pred24_in_ga(T25))
U7_GA(T23, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U3_ga(T23, pred14_in_ga(T23)))

The TRS R consists of the following rules:

pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

pred24_in_ga(x0)
U3_ga(x0, x1)
U2_ga(x0, x1)
pred14_in_ga(x0)
U1_ga(x0, x1)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U3_ga(T23, pred14_in_ga(T23)))

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

POL(0) = 0   
POL(HALF39_IN_GA(x₁)) = x₁   
POL(U1_ga(x₁, x₂)) = 1 + x₂   
POL(U2_ga(x₁, x₂)) = 1 + x₂   
POL(U3_ga(x₁, x₂)) = 1 + x₂   
POL(U5_GA(x₁, x₂)) = x₂   
POL(U7_GA(x₁, x₂)) = x₂   
POL(pred14_in_ga(x₁)) = x₁   
POL(pred14_out_ga(x₁, x₂)) = x₂   
POL(pred24_in_ga(x₁)) = x₁   
POL(pred24_out_ga(x₁, x₂)) = x₂   
POL(pred9_out_ga(x₁, x₂)) = x₂   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))

(72) Obligation:

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

U5_GA(T23, pred9_out_ga(T23, T25)) → U7_GA(T23, pred24_in_ga(T25))
U7_GA(T23, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

pred24_in_ga(x0)
U3_ga(x0, x1)
U2_ga(x0, x1)
pred14_in_ga(x0)
U1_ga(x0, x1)

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

(73) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(74) TRUE

(75) Obligation:

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

U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
f1_out_g(x1)  =  f1_out_g(x1)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
U19_g(x1, x2)  =  U19_g(x1, x2)
U20_g(x1, x2)  =  U20_g(x1, x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U15_A(x1, x2)  =  U15_A(x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)

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

(76) UsableRulesProof (EQUIVALENT transformation)

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

(77) Obligation:

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

U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))

The TRS R consists of the following rules:

half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1, x2)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x1, x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x1, x2)
U10_gaa(x1, x2)  =  U10_gaa(x1, x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg(x1, x2)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x1, x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x1, x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U15_A(x1, x2)  =  U15_A(x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)

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

(78) PiDPToQDPProof (SOUND transformation)

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

(79) Obligation:

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

U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16))
U19_A(T6, half39_out_ga(T16)) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(80) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16)) at position [1] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, pred9_in_ga(x0)))

(81) Obligation:

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

U19_A(T6, half39_out_ga(T16)) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, pred9_in_ga(x0)))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(82) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, pred9_in_ga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))

(83) Obligation:

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

U19_A(T6, half39_out_ga(T16)) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(84) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, pred9_in_ga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))

(85) Obligation:

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

U19_A(T6, half39_out_ga(T16)) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(86) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_A(T6, half39_out_ga(T16)) → U21_A(T6, double53_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))

(87) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(88) UsableRulesProof (EQUIVALENT transformation)

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

(89) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(90) QReductionProof (EQUIVALENT transformation)

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

double53_in_aa

(91) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(92) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_A → U15_A(pred9_in_aa) at position [0] we obtained the following new rules [LPAR04]:

F1_IN_A → U15_A(U3_aa(pred14_in_aa))

(93) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(94) UsableRulesProof (EQUIVALENT transformation)

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

(95) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(96) QReductionProof (EQUIVALENT transformation)

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

pred9_in_aa

(97) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(98) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8)) at position [1] we obtained the following new rules [LPAR04]:

U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(s(0), 0))
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(x0, pred24_in_ga(s(x0))))
U15_A(pred9_out_aa(y0, 0)) → U17_A(y0, pred24_out_ga(0, 0))

(99) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))
U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(s(0), 0))
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(x0, pred24_in_ga(s(x0))))
U15_A(pred9_out_aa(y0, 0)) → U17_A(y0, pred24_out_ga(0, 0))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(100) DependencyGraphProof (EQUIVALENT transformation)

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

(101) Obligation:

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

F1_IN_A → U15_A(U3_aa(pred14_in_aa))
U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(s(0), 0))
U17_A(y0, pred24_out_ga(y1, 0)) → U19_A(y0, half39_out_ga(0))
U19_A(y0, half39_out_ga(y1)) → U21_A(y0, U13_aa(p57_in_aaa))
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(x0, pred24_in_ga(s(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U4_ga(x0, U3_ga(x0, pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(y1, s(s(x0)))) → U19_A(y0, U5_ga(x0, U3_ga(x0, pred14_in_ga(x0))))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(T33, pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(T33, pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19))) → U2_ga(T19, pred24_in_ga(s(T19)))
U11_gaa(T33, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, p57_in_gaa(T37))
U12_gaa(T33, T37, p57_out_gaa(T37, X107)) → p57_out_gaa(T33, s(T37))
U2_ga(T19, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0)
U9_gaa(T33, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0), 0)
U10_aaa(pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15)) → U1_ga(T15, pred14_in_ga(T15))
U3_ga(T12, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U5_ga(T23, pred9_out_ga(T23, T25)) → U6_ga(T23, pred24_in_ga(T25))
U5_ga(T23, pred9_out_ga(T23, T25)) → U7_ga(T23, pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0, 0)
U7_ga(T23, pred24_out_ga(T25, T27)) → U8_ga(T23, half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga(0)
half39_in_ga(s(s(T23))) → U4_ga(T23, pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(T23, pred9_in_ga(T23))
U8_ga(T23, half39_out_ga(T27)) → half39_out_ga(s(s(T23)))
pred9_in_ga(T12) → U3_ga(T12, pred14_in_ga(T12))
U4_ga(T23, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)))
U6_ga(T23, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)))
U1_ga(T15, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0, x1)
pred9_in_ga(x0)
U6_ga(x0, x1)
U7_ga(x0, x1)
p57_in_aaa
pred14_in_aa
U3_ga(x0, x1)
U8_ga(x0, x1)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0, x1)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0, x1)
U10_gaa(x0, x1)
U11_gaa(x0, x1)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1, x2)

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

(102) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
f1_in: (b) (f)
pred9_in: (b,f) (f,f)
pred14_in: (b,f) (f,f)
pred24_in: (b,f) (f,f) (f,b) (b,b)
half39_in: (b,f)
double53_in: (f,f)
p57_in: (f,f,f) (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(103) Obligation:

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

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)

(104) 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:

F1_IN_G(s(s(T6))) → U14_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U15_G(T6, pred9_in_ga(T6, T8))
U15_G(T6, pred9_out_ga(T6, T8)) → U16_G(T6, pred24_in_ga(T8, X17))
U15_G(T6, pred9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U15_G(T6, pred9_out_ga(T6, T8)) → U17_G(T6, pred24_in_ga(T8, T16))
U17_G(T6, pred24_out_ga(T8, T16)) → U18_G(T6, half39_in_ga(T16, X18))
U17_G(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U17_G(T6, pred24_out_ga(T8, T16)) → U19_G(T6, half39_in_ga(T16, T20))
U19_G(T6, half39_out_ga(T16, T20)) → U20_G(T6, double53_in_aa(T20, X4))
U19_G(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
DOUBLE53_IN_AA(T33, s(s(X97))) → U13_AA(T33, X97, p57_in_aaa(T33, X96, X97))
DOUBLE53_IN_AA(T33, s(s(X97))) → P57_IN_AAA(T33, X96, X97)
P57_IN_AAA(T33, X96, X97) → U9_AAA(T33, X96, X97, pred24_in_aa(s(T33), X96))
P57_IN_AAA(T33, X96, X97) → PRED24_IN_AA(s(T33), X96)
PRED24_IN_AA(s(s(T19)), s(X55)) → U2_AA(T19, X55, pred24_in_aa(s(T19), X55))
PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)
P57_IN_AAA(T33, 0, 0) → U10_AAA(T33, pred24_in_ag(s(T33), 0))
P57_IN_AAA(T33, 0, 0) → PRED24_IN_AG(s(T33), 0)
PRED24_IN_AG(s(s(T19)), s(X55)) → U2_AG(T19, X55, pred24_in_ag(s(T19), X55))
PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)
P57_IN_AAA(T33, s(T37), s(s(X108))) → U11_AAA(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
P57_IN_AAA(T33, s(T37), s(s(X108))) → PRED24_IN_AA(s(T33), s(T37))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_AAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, 0, 0) → U10_GAA(T33, pred24_in_gg(s(T33), 0))
P57_IN_GAA(T33, 0, 0) → PRED24_IN_GG(s(T33), 0)
PRED24_IN_GG(s(s(T19)), s(X55)) → U2_GG(T19, X55, pred24_in_gg(s(T19), X55))
PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
P57_IN_GAA(T33, s(T37), s(s(X108))) → PRED24_IN_GA(s(T33), s(T37))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U19_G(T6, half39_out_ga(T16, T20)) → U21_G(T6, double53_in_aa(T20, T29))
U21_G(T6, double53_out_aa(T20, T29)) → U22_G(T6, f1_in_a(T29))
U21_G(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U14_A(T6, pred9_in_aa(T6, X16))
F1_IN_A(s(s(T6))) → PRED9_IN_AA(T6, X16)
PRED9_IN_AA(T12, s(X35)) → U3_AA(T12, X35, pred14_in_aa(T12, X35))
PRED9_IN_AA(T12, s(X35)) → PRED14_IN_AA(T12, X35)
PRED14_IN_AA(s(T15), s(X44)) → U1_AA(T15, X44, pred14_in_aa(T15, X44))
PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U16_A(T6, pred24_in_ga(T8, X17))
U15_A(T6, pred9_out_aa(T6, T8)) → PRED24_IN_GA(T8, X17)
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))
U17_A(T6, pred24_out_ga(T8, T16)) → U18_A(T6, half39_in_ga(T16, X18))
U17_A(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U20_A(T6, double53_in_aa(T20, X4))
U19_A(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → U22_A(T6, f1_in_a(T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1, x2)  =  U18_G(x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U19_G(x1, x2)  =  U19_G(x2)
U20_G(x1, x2)  =  U20_G(x2)
DOUBLE53_IN_AA(x1, x2)  =  DOUBLE53_IN_AA
U13_AA(x1, x2, x3)  =  U13_AA(x3)
P57_IN_AAA(x1, x2, x3)  =  P57_IN_AAA
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U10_AAA(x1, x2)  =  U10_AAA(x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U11_AAA(x1, x2, x3, x4)  =  U11_AAA(x4)
U12_AAA(x1, x2, x3, x4)  =  U12_AAA(x1, x2, x4)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U10_GAA(x1, x2)  =  U10_GAA(x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x2, x4)
U21_G(x1, x2)  =  U21_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
F1_IN_A(x1)  =  F1_IN_A
U14_A(x1, x2)  =  U14_A(x2)
PRED9_IN_AA(x1, x2)  =  PRED9_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U15_A(x1, x2)  =  U15_A(x2)
U16_A(x1, x2)  =  U16_A(x1, x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U18_A(x1, x2)  =  U18_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U20_A(x1, x2)  =  U20_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)
U22_A(x1, x2)  =  U22_A(x1, x2)

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

(105) Obligation:

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

F1_IN_G(s(s(T6))) → U14_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U15_G(T6, pred9_in_ga(T6, T8))
U15_G(T6, pred9_out_ga(T6, T8)) → U16_G(T6, pred24_in_ga(T8, X17))
U15_G(T6, pred9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U15_G(T6, pred9_out_ga(T6, T8)) → U17_G(T6, pred24_in_ga(T8, T16))
U17_G(T6, pred24_out_ga(T8, T16)) → U18_G(T6, half39_in_ga(T16, X18))
U17_G(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U17_G(T6, pred24_out_ga(T8, T16)) → U19_G(T6, half39_in_ga(T16, T20))
U19_G(T6, half39_out_ga(T16, T20)) → U20_G(T6, double53_in_aa(T20, X4))
U19_G(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
DOUBLE53_IN_AA(T33, s(s(X97))) → U13_AA(T33, X97, p57_in_aaa(T33, X96, X97))
DOUBLE53_IN_AA(T33, s(s(X97))) → P57_IN_AAA(T33, X96, X97)
P57_IN_AAA(T33, X96, X97) → U9_AAA(T33, X96, X97, pred24_in_aa(s(T33), X96))
P57_IN_AAA(T33, X96, X97) → PRED24_IN_AA(s(T33), X96)
PRED24_IN_AA(s(s(T19)), s(X55)) → U2_AA(T19, X55, pred24_in_aa(s(T19), X55))
PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)
P57_IN_AAA(T33, 0, 0) → U10_AAA(T33, pred24_in_ag(s(T33), 0))
P57_IN_AAA(T33, 0, 0) → PRED24_IN_AG(s(T33), 0)
PRED24_IN_AG(s(s(T19)), s(X55)) → U2_AG(T19, X55, pred24_in_ag(s(T19), X55))
PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)
P57_IN_AAA(T33, s(T37), s(s(X108))) → U11_AAA(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
P57_IN_AAA(T33, s(T37), s(s(X108))) → PRED24_IN_AA(s(T33), s(T37))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_AAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_AAA(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, 0, 0) → U10_GAA(T33, pred24_in_gg(s(T33), 0))
P57_IN_GAA(T33, 0, 0) → PRED24_IN_GG(s(T33), 0)
PRED24_IN_GG(s(s(T19)), s(X55)) → U2_GG(T19, X55, pred24_in_gg(s(T19), X55))
PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
P57_IN_GAA(T33, s(T37), s(s(X108))) → PRED24_IN_GA(s(T33), s(T37))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U19_G(T6, half39_out_ga(T16, T20)) → U21_G(T6, double53_in_aa(T20, T29))
U21_G(T6, double53_out_aa(T20, T29)) → U22_G(T6, f1_in_a(T29))
U21_G(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U14_A(T6, pred9_in_aa(T6, X16))
F1_IN_A(s(s(T6))) → PRED9_IN_AA(T6, X16)
PRED9_IN_AA(T12, s(X35)) → U3_AA(T12, X35, pred14_in_aa(T12, X35))
PRED9_IN_AA(T12, s(X35)) → PRED14_IN_AA(T12, X35)
PRED14_IN_AA(s(T15), s(X44)) → U1_AA(T15, X44, pred14_in_aa(T15, X44))
PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U16_A(T6, pred24_in_ga(T8, X17))
U15_A(T6, pred9_out_aa(T6, T8)) → PRED24_IN_GA(T8, X17)
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))
U17_A(T6, pred24_out_ga(T8, T16)) → U18_A(T6, half39_in_ga(T16, X18))
U17_A(T6, pred24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U20_A(T6, double53_in_aa(T20, X4))
U19_A(T6, half39_out_ga(T16, T20)) → DOUBLE53_IN_AA(T20, X4)
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → U22_A(T6, f1_in_a(T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U14_G(x1, x2)  =  U14_G(x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1, x2)  =  U18_G(x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U8_GA(x1, x2, x3)  =  U8_GA(x3)
U19_G(x1, x2)  =  U19_G(x2)
U20_G(x1, x2)  =  U20_G(x2)
DOUBLE53_IN_AA(x1, x2)  =  DOUBLE53_IN_AA
U13_AA(x1, x2, x3)  =  U13_AA(x3)
P57_IN_AAA(x1, x2, x3)  =  P57_IN_AAA
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U10_AAA(x1, x2)  =  U10_AAA(x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U11_AAA(x1, x2, x3, x4)  =  U11_AAA(x4)
U12_AAA(x1, x2, x3, x4)  =  U12_AAA(x1, x2, x4)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x4)
U10_GAA(x1, x2)  =  U10_GAA(x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)
U12_GAA(x1, x2, x3, x4)  =  U12_GAA(x2, x4)
U21_G(x1, x2)  =  U21_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
F1_IN_A(x1)  =  F1_IN_A
U14_A(x1, x2)  =  U14_A(x2)
PRED9_IN_AA(x1, x2)  =  PRED9_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA
U1_AA(x1, x2, x3)  =  U1_AA(x3)
U15_A(x1, x2)  =  U15_A(x2)
U16_A(x1, x2)  =  U16_A(x1, x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U18_A(x1, x2)  =  U18_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U20_A(x1, x2)  =  U20_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)
U22_A(x1, x2)  =  U22_A(x1, x2)

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

(106) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 9 SCCs with 54 less nodes.

(107) Complex Obligation (AND)

(108) Obligation:

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

PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA

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

(109) UsableRulesProof (EQUIVALENT transformation)

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

(110) Obligation:

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

PRED14_IN_AA(s(T15), s(X44)) → PRED14_IN_AA(T15, X44)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED14_IN_AA(x1, x2)  =  PRED14_IN_AA

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

(111) PiDPToQDPProof (SOUND transformation)

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

(112) Obligation:

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

PRED14_IN_AA → PRED14_IN_AA

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

(113) 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 = PRED14_IN_AA evaluates to t =PRED14_IN_AA

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 PRED14_IN_AA to PRED14_IN_AA.

(114) NO

(115) Obligation:

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

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_GG(x1, x2)  =  PRED24_IN_GG(x1, x2)

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

(116) UsableRulesProof (EQUIVALENT transformation)

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

(117) Obligation:

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

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

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

(118) PiDPToQDPProof (EQUIVALENT transformation)

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

(119) Obligation:

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

PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)

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

(120) 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:

• PRED24_IN_GG(s(s(T19)), s(X55)) → PRED24_IN_GG(s(T19), X55)
  The graph contains the following edges 1 > 1, 2 > 2

(121) YES

(122) Obligation:

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

PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)

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

(123) UsableRulesProof (EQUIVALENT transformation)

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

(124) Obligation:

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

PRED24_IN_AG(s(s(T19)), s(X55)) → PRED24_IN_AG(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_AG(x1, x2)  =  PRED24_IN_AG(x2)

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

(125) PiDPToQDPProof (SOUND transformation)

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

(126) Obligation:

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

PRED24_IN_AG(s(X55)) → PRED24_IN_AG(X55)

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

(127) 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:

• PRED24_IN_AG(s(X55)) → PRED24_IN_AG(X55)
  The graph contains the following edges 1 > 1

(128) YES

(129) Obligation:

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

PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA

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

(130) UsableRulesProof (EQUIVALENT transformation)

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

(131) Obligation:

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

PRED24_IN_AA(s(s(T19)), s(X55)) → PRED24_IN_AA(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_AA(x1, x2)  =  PRED24_IN_AA

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

(132) PiDPToQDPProof (SOUND transformation)

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

(133) Obligation:

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

PRED24_IN_AA → PRED24_IN_AA

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

(134) 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 = PRED24_IN_AA evaluates to t =PRED24_IN_AA

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 PRED24_IN_AA to PRED24_IN_AA.

(135) NO

(136) Obligation:

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

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(137) UsableRulesProof (EQUIVALENT transformation)

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

(138) Obligation:

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

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(139) PiDPToQDPProof (SOUND transformation)

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

(140) Obligation:

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

PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))

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

(141) 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:

• PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))
  The graph contains the following edges 1 > 1

(142) YES

(143) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)

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

(144) UsableRulesProof (EQUIVALENT transformation)

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

(145) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U11_GAA(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_GAA(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x4)

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

(146) PiDPToQDPProof (SOUND transformation)

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

(147) Obligation:

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

P57_IN_GAA(T33) → U11_GAA(pred24_in_ga(s(T33)))
U11_GAA(pred24_out_ga(s(T37))) → P57_IN_GAA(T37)

The TRS R consists of the following rules:

pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_ga(s(0)) → pred24_out_ga(0)

The set Q consists of the following terms:

pred24_in_ga(x0)
U2_ga(x0)

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

(148) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

P57_IN_GAA(T33) → U11_GAA(pred24_in_ga(s(T33)))
U11_GAA(pred24_out_ga(s(T37))) → P57_IN_GAA(T37)

Strictly oriented rules of the TRS R:

pred24_in_ga(s(0)) → pred24_out_ga(0)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P57_IN_GAA(x₁)) = 4 + x₁   
POL(U11_GAA(x₁)) = x₁   
POL(U2_ga(x₁)) = 3 + x₁   
POL(pred24_in_ga(x₁)) = x₁   
POL(pred24_out_ga(x₁)) = 2 + x₁   
POL(s(x₁)) = 3 + x₁   

(149) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))

The set Q consists of the following terms:

pred24_in_ga(x0)
U2_ga(x0)

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

(150) PisEmptyProof (EQUIVALENT transformation)

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

(151) YES

(152) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(153) UsableRulesProof (EQUIVALENT transformation)

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

(154) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(155) PiDPToQDPProof (SOUND transformation)

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

(156) Obligation:

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

PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)

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

(157) 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:

• PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)
  The graph contains the following edges 1 > 1

(158) YES

(159) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)

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

(160) UsableRulesProof (EQUIVALENT transformation)

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

(161) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, pred9_in_ga(T23, T25))
U5_GA(T23, X70, pred9_out_ga(T23, T25)) → U7_GA(T23, X70, pred24_in_ga(T25, T27))
U7_GA(T23, X70, pred24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)

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

(162) PiDPToQDPProof (SOUND transformation)

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

(163) Obligation:

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

HALF39_IN_GA(s(s(T23))) → U5_GA(pred9_in_ga(T23))
U5_GA(pred9_out_ga(T25)) → U7_GA(pred24_in_ga(T25))
U7_GA(pred24_out_ga(T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

pred9_in_ga(x0)
pred24_in_ga(x0)
U3_ga(x0)
U2_ga(x0)
pred14_in_ga(x0)
U1_ga(x0)

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

(164) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

HALF39_IN_GA(s(s(T23))) → U5_GA(pred9_in_ga(T23))
U5_GA(pred9_out_ga(T25)) → U7_GA(pred24_in_ga(T25))
U7_GA(pred24_out_ga(T27)) → HALF39_IN_GA(T27)

Strictly oriented rules of the TRS R:

pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
pred14_in_ga(0) → pred14_out_ga(0)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(HALF39_IN_GA(x₁)) = x₁   
POL(U1_ga(x₁)) = 7 + x₁   
POL(U2_ga(x₁)) = 7 + x₁   
POL(U3_ga(x₁)) = x₁   
POL(U5_GA(x₁)) = x₁   
POL(U7_GA(x₁)) = 1 + x₁   
POL(pred14_in_ga(x₁)) = 12 + x₁   
POL(pred14_out_ga(x₁)) = 11 + x₁   
POL(pred24_in_ga(x₁)) = 1 + x₁   
POL(pred24_out_ga(x₁)) = x₁   
POL(pred9_in_ga(x₁)) = 13 + x₁   
POL(pred9_out_ga(x₁)) = 3 + x₁   
POL(s(x₁)) = 7 + x₁   

(165) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

pred9_in_ga(x0)
pred24_in_ga(x0)
U3_ga(x0)
U2_ga(x0)
pred14_in_ga(x0)
U1_ga(x0)

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

(166) PisEmptyProof (EQUIVALENT transformation)

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

(167) YES

(168) Obligation:

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

U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))

The TRS R consists of the following rules:

f1_in_g(s(s(T6))) → U14_g(T6, pred9_in_ga(T6, X16))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U14_g(T6, pred9_out_ga(T6, X16)) → f1_out_g(s(s(T6)))
f1_in_g(s(s(T6))) → U15_g(T6, pred9_in_ga(T6, T8))
U15_g(T6, pred9_out_ga(T6, T8)) → U16_g(T6, pred24_in_ga(T8, X17))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
U16_g(T6, pred24_out_ga(T8, X17)) → f1_out_g(s(s(T6)))
U15_g(T6, pred9_out_ga(T6, T8)) → U17_g(T6, pred24_in_ga(T8, T16))
U17_g(T6, pred24_out_ga(T8, T16)) → U18_g(T6, half39_in_ga(T16, X18))
half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U18_g(T6, half39_out_ga(T16, X18)) → f1_out_g(s(s(T6)))
U17_g(T6, pred24_out_ga(T8, T16)) → U19_g(T6, half39_in_ga(T16, T20))
U19_g(T6, half39_out_ga(T16, T20)) → U20_g(T6, double53_in_aa(T20, X4))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
pred24_in_aa(0, 0) → pred24_out_aa(0, 0)
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
pred24_in_ag(0, 0) → pred24_out_ag(0, 0)
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
pred24_in_ag(s(s(T19)), s(X55)) → U2_ag(T19, X55, pred24_in_ag(s(T19), X55))
U2_ag(T19, X55, pred24_out_ag(s(T19), X55)) → pred24_out_ag(s(s(T19)), s(X55))
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
pred24_in_gg(0, 0) → pred24_out_gg(0, 0)
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
pred24_in_gg(s(s(T19)), s(X55)) → U2_gg(T19, X55, pred24_in_gg(s(T19), X55))
U2_gg(T19, X55, pred24_out_gg(s(T19), X55)) → pred24_out_gg(s(s(T19)), s(X55))
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U20_g(T6, double53_out_aa(T20, X4)) → f1_out_g(s(s(T6)))
U19_g(T6, half39_out_ga(T16, T20)) → U21_g(T6, double53_in_aa(T20, T29))
U21_g(T6, double53_out_aa(T20, T29)) → U22_g(T6, f1_in_a(T29))
f1_in_a(s(s(T6))) → U14_a(T6, pred9_in_aa(T6, X16))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U14_a(T6, pred9_out_aa(T6, X16)) → f1_out_a(s(s(T6)))
f1_in_a(s(s(T6))) → U15_a(T6, pred9_in_aa(T6, T8))
U15_a(T6, pred9_out_aa(T6, T8)) → U16_a(T6, pred24_in_ga(T8, X17))
U16_a(T6, pred24_out_ga(T8, X17)) → f1_out_a(s(s(T6)))
U15_a(T6, pred9_out_aa(T6, T8)) → U17_a(T6, pred24_in_ga(T8, T16))
U17_a(T6, pred24_out_ga(T8, T16)) → U18_a(T6, half39_in_ga(T16, X18))
U18_a(T6, half39_out_ga(T16, X18)) → f1_out_a(s(s(T6)))
U17_a(T6, pred24_out_ga(T8, T16)) → U19_a(T6, half39_in_ga(T16, T20))
U19_a(T6, half39_out_ga(T16, T20)) → U20_a(T6, double53_in_aa(T20, X4))
U20_a(T6, double53_out_aa(T20, X4)) → f1_out_a(s(s(T6)))
U19_a(T6, half39_out_ga(T16, T20)) → U21_a(T6, double53_in_aa(T20, T29))
U21_a(T6, double53_out_aa(T20, T29)) → U22_a(T6, f1_in_a(T29))
U22_a(T6, f1_out_a(T29)) → f1_out_a(s(s(T6)))
U22_g(T6, f1_out_a(T29)) → f1_out_g(s(s(T6)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U14_g(x1, x2)  =  U14_g(x2)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
f1_out_g(x1)  =  f1_out_g
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
U19_g(x1, x2)  =  U19_g(x2)
U20_g(x1, x2)  =  U20_g(x2)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
f1_in_a(x1)  =  f1_in_a
U14_a(x1, x2)  =  U14_a(x2)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
f1_out_a(x1)  =  f1_out_a(x1)
U15_a(x1, x2)  =  U15_a(x2)
U16_a(x1, x2)  =  U16_a(x1, x2)
U17_a(x1, x2)  =  U17_a(x1, x2)
U18_a(x1, x2)  =  U18_a(x1, x2)
U19_a(x1, x2)  =  U19_a(x1, x2)
U20_a(x1, x2)  =  U20_a(x1, x2)
U21_a(x1, x2)  =  U21_a(x1, x2)
U22_a(x1, x2)  =  U22_a(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U15_A(x1, x2)  =  U15_A(x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)

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

(169) UsableRulesProof (EQUIVALENT transformation)

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

(170) Obligation:

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

U17_A(T6, pred24_out_ga(T8, T16)) → U19_A(T6, half39_in_ga(T16, T20))
U19_A(T6, half39_out_ga(T16, T20)) → U21_A(T6, double53_in_aa(T20, T29))
U21_A(T6, double53_out_aa(T20, T29)) → F1_IN_A(T29)
F1_IN_A(s(s(T6))) → U15_A(T6, pred9_in_aa(T6, T8))
U15_A(T6, pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8, T16))

The TRS R consists of the following rules:

half39_in_ga(0, 0) → half39_out_ga(0, 0)
half39_in_ga(s(s(T23)), s(X70)) → U4_ga(T23, X70, pred9_in_ga(T23, X68))
half39_in_ga(s(s(T23)), s(X70)) → U5_ga(T23, X70, pred9_in_ga(T23, T25))
double53_in_aa(T33, s(s(X97))) → U13_aa(T33, X97, p57_in_aaa(T33, X96, X97))
pred9_in_aa(T12, s(X35)) → U3_aa(T12, X35, pred14_in_aa(T12, X35))
pred24_in_ga(0, 0) → pred24_out_ga(0, 0)
pred24_in_ga(s(0), 0) → pred24_out_ga(s(0), 0)
pred24_in_ga(s(s(T19)), s(X55)) → U2_ga(T19, X55, pred24_in_ga(s(T19), X55))
U4_ga(T23, X70, pred9_out_ga(T23, X68)) → half39_out_ga(s(s(T23)), s(X70))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U6_ga(T23, X70, pred24_in_ga(T25, X69))
U5_ga(T23, X70, pred9_out_ga(T23, T25)) → U7_ga(T23, X70, pred24_in_ga(T25, T27))
U13_aa(T33, X97, p57_out_aaa(T33, X96, X97)) → double53_out_aa(T33, s(s(X97)))
U3_aa(T12, X35, pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(T19, X55, pred24_out_ga(s(T19), X55)) → pred24_out_ga(s(s(T19)), s(X55))
pred9_in_ga(T12, s(X35)) → U3_ga(T12, X35, pred14_in_ga(T12, X35))
U6_ga(T23, X70, pred24_out_ga(T25, X69)) → half39_out_ga(s(s(T23)), s(X70))
U7_ga(T23, X70, pred24_out_ga(T25, T27)) → U8_ga(T23, X70, half39_in_ga(T27, X70))
p57_in_aaa(T33, X96, X97) → U9_aaa(T33, X96, X97, pred24_in_aa(s(T33), X96))
p57_in_aaa(T33, 0, 0) → U10_aaa(T33, pred24_in_ag(s(T33), 0))
p57_in_aaa(T33, s(T37), s(s(X108))) → U11_aaa(T33, T37, X108, pred24_in_aa(s(T33), s(T37)))
pred14_in_aa(0, 0) → pred14_out_aa(0, 0)
pred14_in_aa(s(T15), s(X44)) → U1_aa(T15, X44, pred14_in_aa(T15, X44))
U3_ga(T12, X35, pred14_out_ga(T12, X35)) → pred9_out_ga(T12, s(X35))
U8_ga(T23, X70, half39_out_ga(T27, X70)) → half39_out_ga(s(s(T23)), s(X70))
U9_aaa(T33, X96, X97, pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96, X97)
U10_aaa(T33, pred24_out_ag(s(T33), 0)) → p57_out_aaa(T33, 0, 0)
U11_aaa(T33, T37, X108, pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U1_aa(T15, X44, pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0, 0) → pred14_out_ga(0, 0)
pred14_in_ga(s(T15), s(X44)) → U1_ga(T15, X44, pred14_in_ga(T15, X44))
pred24_in_aa(s(0), 0) → pred24_out_aa(s(0), 0)
pred24_in_aa(s(s(T19)), s(X55)) → U2_aa(T19, X55, pred24_in_aa(s(T19), X55))
pred24_in_ag(s(0), 0) → pred24_out_ag(s(0), 0)
U12_aaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_aaa(T33, s(T37), s(s(X108)))
U1_ga(T15, X44, pred14_out_ga(T15, X44)) → pred14_out_ga(s(T15), s(X44))
U2_aa(T19, X55, pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33, X96, X97) → U9_gaa(T33, X96, X97, pred24_in_ga(s(T33), X96))
p57_in_gaa(T33, 0, 0) → U10_gaa(T33, pred24_in_gg(s(T33), 0))
p57_in_gaa(T33, s(T37), s(s(X108))) → U11_gaa(T33, T37, X108, pred24_in_ga(s(T33), s(T37)))
U9_gaa(T33, X96, X97, pred24_out_ga(s(T33), X96)) → p57_out_gaa(T33, X96, X97)
U10_gaa(T33, pred24_out_gg(s(T33), 0)) → p57_out_gaa(T33, 0, 0)
U11_gaa(T33, T37, X108, pred24_out_ga(s(T33), s(T37))) → U12_gaa(T33, T37, X108, p57_in_gaa(T37, X107, X108))
pred24_in_gg(s(0), 0) → pred24_out_gg(s(0), 0)
U12_gaa(T33, T37, X108, p57_out_gaa(T37, X107, X108)) → p57_out_gaa(T33, s(T37), s(s(X108)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
0  =  0
pred14_out_ga(x1, x2)  =  pred14_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
pred9_out_ga(x1, x2)  =  pred9_out_ga(x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
pred24_out_ga(x1, x2)  =  pred24_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
half39_out_ga(x1, x2)  =  half39_out_ga
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U8_ga(x1, x2, x3)  =  U8_ga(x3)
double53_in_aa(x1, x2)  =  double53_in_aa
U13_aa(x1, x2, x3)  =  U13_aa(x3)
p57_in_aaa(x1, x2, x3)  =  p57_in_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
pred24_in_aa(x1, x2)  =  pred24_in_aa
pred24_out_aa(x1, x2)  =  pred24_out_aa(x1, x2)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
p57_out_aaa(x1, x2, x3)  =  p57_out_aaa(x1, x2)
U10_aaa(x1, x2)  =  U10_aaa(x2)
pred24_in_ag(x1, x2)  =  pred24_in_ag(x2)
pred24_out_ag(x1, x2)  =  pred24_out_ag(x1)
U11_aaa(x1, x2, x3, x4)  =  U11_aaa(x4)
U12_aaa(x1, x2, x3, x4)  =  U12_aaa(x1, x2, x4)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
U9_gaa(x1, x2, x3, x4)  =  U9_gaa(x4)
p57_out_gaa(x1, x2, x3)  =  p57_out_gaa(x2)
U10_gaa(x1, x2)  =  U10_gaa(x2)
pred24_in_gg(x1, x2)  =  pred24_in_gg(x1, x2)
pred24_out_gg(x1, x2)  =  pred24_out_gg
U11_gaa(x1, x2, x3, x4)  =  U11_gaa(x4)
U12_gaa(x1, x2, x3, x4)  =  U12_gaa(x2, x4)
double53_out_aa(x1, x2)  =  double53_out_aa(x1)
pred9_in_aa(x1, x2)  =  pred9_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
pred14_in_aa(x1, x2)  =  pred14_in_aa
pred14_out_aa(x1, x2)  =  pred14_out_aa(x1, x2)
U1_aa(x1, x2, x3)  =  U1_aa(x3)
pred9_out_aa(x1, x2)  =  pred9_out_aa(x1, x2)
F1_IN_A(x1)  =  F1_IN_A
U15_A(x1, x2)  =  U15_A(x2)
U17_A(x1, x2)  =  U17_A(x1, x2)
U19_A(x1, x2)  =  U19_A(x1, x2)
U21_A(x1, x2)  =  U21_A(x1, x2)

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

(171) PiDPToQDPProof (SOUND transformation)

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

(172) Obligation:

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

U17_A(T6, pred24_out_ga(T16)) → U19_A(T6, half39_in_ga(T16))
U19_A(T6, half39_out_ga) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U8_ga(half39_out_ga) → half39_out_ga
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(173) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U17_A(T6, pred24_out_ga(T16)) → U19_A(T6, half39_in_ga(T16)) at position [1] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(pred9_in_ga(x0)))

(174) Obligation:

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

U19_A(T6, half39_out_ga) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(pred9_in_ga(x0)))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U8_ga(half39_out_ga) → half39_out_ga
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(175) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(pred9_in_ga(x0))) at position [1,0] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))

(176) Obligation:

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

U19_A(T6, half39_out_ga) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(pred9_in_ga(x0)))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U8_ga(half39_out_ga) → half39_out_ga
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(177) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(pred9_in_ga(x0))) at position [1,0] we obtained the following new rules [LPAR04]:

U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))

(178) Obligation:

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

U19_A(T6, half39_out_ga) → U21_A(T6, double53_in_aa)
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U8_ga(half39_out_ga) → half39_out_ga
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(179) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_A(T6, half39_out_ga) → U21_A(T6, double53_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))

(180) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
double53_in_aa → U13_aa(p57_in_aaa)
pred9_in_aa → U3_aa(pred14_in_aa)
pred24_in_ga(0) → pred24_out_ga(0)
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U8_ga(half39_out_ga) → half39_out_ga
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
pred24_in_ag(0) → pred24_out_ag(s(0))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
pred24_in_gg(s(0), 0) → pred24_out_gg
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(181) UsableRulesProof (EQUIVALENT transformation)

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

(182) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
double53_in_aa
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(183) QReductionProof (EQUIVALENT transformation)

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

double53_in_aa

(184) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
F1_IN_A → U15_A(pred9_in_aa)
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(185) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_A → U15_A(pred9_in_aa) at position [0] we obtained the following new rules [LPAR04]:

F1_IN_A → U15_A(U3_aa(pred14_in_aa))

(186) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))
pred9_in_aa → U3_aa(pred14_in_aa)
pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(187) UsableRulesProof (EQUIVALENT transformation)

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

(188) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred9_in_aa
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(189) QReductionProof (EQUIVALENT transformation)

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

pred9_in_aa

(190) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(191) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_A(pred9_out_aa(T6, T8)) → U17_A(T6, pred24_in_ga(T8)) at position [1] we obtained the following new rules [LPAR04]:

U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(0))
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(pred24_in_ga(s(x0))))
U15_A(pred9_out_aa(y0, 0)) → U17_A(y0, pred24_out_ga(0))

(192) Obligation:

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

U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))
F1_IN_A → U15_A(U3_aa(pred14_in_aa))
U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(0))
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(pred24_in_ga(s(x0))))
U15_A(pred9_out_aa(y0, 0)) → U17_A(y0, pred24_out_ga(0))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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

(193) DependencyGraphProof (EQUIVALENT transformation)

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

(194) Obligation:

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

F1_IN_A → U15_A(U3_aa(pred14_in_aa))
U15_A(pred9_out_aa(y0, s(0))) → U17_A(y0, pred24_out_ga(0))
U17_A(y0, pred24_out_ga(0)) → U19_A(y0, half39_out_ga)
U19_A(y0, half39_out_ga) → U21_A(y0, U13_aa(p57_in_aaa))
U21_A(T6, double53_out_aa(T20)) → F1_IN_A
U15_A(pred9_out_aa(y0, s(s(x0)))) → U17_A(y0, U2_ga(pred24_in_ga(s(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U4_ga(U3_ga(pred14_in_ga(x0))))
U17_A(y0, pred24_out_ga(s(s(x0)))) → U19_A(y0, U5_ga(U3_ga(pred14_in_ga(x0))))

The TRS R consists of the following rules:

pred14_in_aa → pred14_out_aa(0, 0)
pred14_in_aa → U1_aa(pred14_in_aa)
U3_aa(pred14_out_aa(T12, X35)) → pred9_out_aa(T12, s(X35))
U1_aa(pred14_out_aa(T15, X44)) → pred14_out_aa(s(T15), s(X44))
p57_in_aaa → U9_aaa(pred24_in_aa)
p57_in_aaa → U10_aaa(pred24_in_ag(0))
p57_in_aaa → U11_aaa(pred24_in_aa)
U13_aa(p57_out_aaa(T33, X96)) → double53_out_aa(T33)
pred24_in_aa → pred24_out_aa(s(0), 0)
pred24_in_aa → U2_aa(pred24_in_aa)
U11_aaa(pred24_out_aa(s(T33), s(T37))) → U12_aaa(T33, T37, p57_in_gaa(T37))
p57_in_gaa(T33) → U9_gaa(pred24_in_ga(s(T33)))
p57_in_gaa(T33) → U10_gaa(pred24_in_gg(s(T33), 0))
p57_in_gaa(T33) → U11_gaa(pred24_in_ga(s(T33)))
U12_aaa(T33, T37, p57_out_gaa(X107)) → p57_out_aaa(T33, s(T37))
pred24_in_ga(s(0)) → pred24_out_ga(0)
pred24_in_ga(s(s(T19))) → U2_ga(pred24_in_ga(s(T19)))
U11_gaa(pred24_out_ga(s(T37))) → U12_gaa(T37, p57_in_gaa(T37))
U12_gaa(T37, p57_out_gaa(X107)) → p57_out_gaa(s(T37))
U2_ga(pred24_out_ga(X55)) → pred24_out_ga(s(X55))
pred24_in_gg(s(0), 0) → pred24_out_gg
U10_gaa(pred24_out_gg) → p57_out_gaa(0)
U9_gaa(pred24_out_ga(X96)) → p57_out_gaa(X96)
U2_aa(pred24_out_aa(s(T19), X55)) → pred24_out_aa(s(s(T19)), s(X55))
pred24_in_ag(0) → pred24_out_ag(s(0))
U10_aaa(pred24_out_ag(s(T33))) → p57_out_aaa(T33, 0)
U9_aaa(pred24_out_aa(s(T33), X96)) → p57_out_aaa(T33, X96)
pred14_in_ga(0) → pred14_out_ga(0)
pred14_in_ga(s(T15)) → U1_ga(pred14_in_ga(T15))
U3_ga(pred14_out_ga(X35)) → pred9_out_ga(s(X35))
U5_ga(pred9_out_ga(T25)) → U6_ga(pred24_in_ga(T25))
U5_ga(pred9_out_ga(T25)) → U7_ga(pred24_in_ga(T25))
pred24_in_ga(0) → pred24_out_ga(0)
U7_ga(pred24_out_ga(T27)) → U8_ga(half39_in_ga(T27))
half39_in_ga(0) → half39_out_ga
half39_in_ga(s(s(T23))) → U4_ga(pred9_in_ga(T23))
half39_in_ga(s(s(T23))) → U5_ga(pred9_in_ga(T23))
U8_ga(half39_out_ga) → half39_out_ga
pred9_in_ga(T12) → U3_ga(pred14_in_ga(T12))
U4_ga(pred9_out_ga(X68)) → half39_out_ga
U6_ga(pred24_out_ga(X69)) → half39_out_ga
U1_ga(pred14_out_ga(X44)) → pred14_out_ga(s(X44))

The set Q consists of the following terms:

half39_in_ga(x0)
pred24_in_ga(x0)
U4_ga(x0)
U5_ga(x0)
U13_aa(x0)
U3_aa(x0)
U2_ga(x0)
pred9_in_ga(x0)
U6_ga(x0)
U7_ga(x0)
p57_in_aaa
pred14_in_aa
U3_ga(x0)
U8_ga(x0)
U9_aaa(x0)
U10_aaa(x0)
U11_aaa(x0)
U1_aa(x0)
pred14_in_ga(x0)
pred24_in_aa
pred24_in_ag(x0)
U12_aaa(x0, x1, x2)
U1_ga(x0)
U2_aa(x0)
p57_in_gaa(x0)
U9_gaa(x0)
U10_gaa(x0)
U11_gaa(x0)
pred24_in_gg(x0, x1)
U12_gaa(x0, x1)

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