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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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:

Triples:

pred14(s(T15), s(X44)) :- pred14(T15, X44).
pred24(s(s(T19)), s(X55)) :- pred24(s(T19), X55).
pred9(T12, s(X35)) :- pred14(T12, X35).
half39(s(s(T23)), s(X70)) :- pred9(T23, X68).
half39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), pred24(T25, X69)).
half39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), ','(predc24(T25, T27), half39(T27, X70))).
p57(T33, X96, X97) :- pred24(s(T33), X96).
p57(T33, s(T37), s(s(X108))) :- ','(predc24(s(T33), s(T37)), p57(T37, X107, X108)).
f1(s(s(T6))) :- pred9(T6, X16).
f1(s(s(T6))) :- ','(predc9(T6, T8), pred24(T8, X17)).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), half39(T16, X18))).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T33), p57(T33, X96, X97)))).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T20), ','(doublec53(T20, T29), f1(T29))))).

Clauses:

predc14(0, 0).
predc14(s(T15), s(X44)) :- predc14(T15, X44).
predc24(0, 0).
predc24(s(0), 0).
predc24(s(s(T19)), s(X55)) :- predc24(s(T19), X55).
predc9(T12, s(X35)) :- predc14(T12, X35).
halfc39(0, 0).
halfc39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), ','(predc24(T25, T27), halfc39(T27, X70))).
fc1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T20), ','(doublec53(T20, T29), fc1(T29))))).
qc57(T33, 0, 0) :- predc24(s(T33), 0).
qc57(T33, s(T37), s(s(X108))) :- ','(predc24(s(T33), s(T37)), qc57(T37, X107, X108)).
doublec53(T33, s(s(X97))) :- qc57(T33, X96, X97).

Afs:

f1(x1)  =  f1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
f1_in: (b)
pred9_in: (b,f)
pred14_in: (b,f)
predc9_in: (b,f)
predc14_in: (b,f)
pred24_in: (b,f)
predc24_in: (b,f) (b,b)
half39_in: (b,f)
halfc39_in: (b,f)
p57_in: (b,f,f)
doublec53_in: (b,f)
qc57_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

F1_IN_G(s(s(T6))) → U12_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))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U14_G(T6, pred24_in_ga(T8, X17))
U13_G(T6, predc9_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)
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U16_G(T6, half39_in_ga(T16, X18))
U15_G(T6, predc24_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, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U15_G(T6, predc24_out_ga(T8, T16)) → U17_G(T6, halfc39_in_ga(T16, T33))
U17_G(T6, halfc39_out_ga(T16, T33)) → U18_G(T6, p57_in_gaa(T33, X96, X97))
U17_G(T6, halfc39_out_ga(T16, T33)) → P57_IN_GAA(T33, X96, X97)
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, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U11_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → U21_G(T6, f1_in_g(T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U12_G(x1, x2)  =  U12_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)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_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)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x2, x4)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

F1_IN_G(s(s(T6))) → U12_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))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U14_G(T6, pred24_in_ga(T8, X17))
U13_G(T6, predc9_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)
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U16_G(T6, half39_in_ga(T16, X18))
U15_G(T6, predc24_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, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U15_G(T6, predc24_out_ga(T8, T16)) → U17_G(T6, halfc39_in_ga(T16, T33))
U17_G(T6, halfc39_out_ga(T16, T33)) → U18_G(T6, p57_in_gaa(T33, X96, X97))
U17_G(T6, halfc39_out_ga(T16, T33)) → P57_IN_GAA(T33, X96, X97)
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, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U11_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → U21_G(T6, f1_in_g(T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U12_G(x1, x2)  =  U12_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)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_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)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x2, x4)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) 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:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(13) YES

(14) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

P57_IN_GAA(T33) → U10_GAA(T33, predc24_in_ga(s(T33)))
U10_GAA(T33, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37)

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)

The set Q consists of the following terms:

predc24_in_ga(x0)
U24_ga(x0, x1)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U10_GAA(T33, predc24_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(U10_GAA(x₁, x₂)) = x₂   
POL(U24_ga(x₁, x₂)) = 1 + x₂   
POL(predc24_in_ga(x₁)) = x₁   
POL(predc24_out_ga(x₁, x₂)) = 1 + x₂   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

(20) Obligation:

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

P57_IN_GAA(T33) → U10_GAA(T33, predc24_in_ga(s(T33)))

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)

The set Q consists of the following terms:

predc24_in_ga(x0)
U24_ga(x0, x1)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE

(23) 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:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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:

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

(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:

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

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:

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

(29) YES

(30) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(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

(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:

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_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

(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:

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, predc9_in_ga(T23))
U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule HALF39_IN_GA(s(s(T23))) → U5_GA(T23, predc9_in_ga(T23)) at position [1] we obtained the following new rules [LPAR04]:

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

(36) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(37) 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.

(38) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(39) 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].

predc9_in_ga(x0)

(40) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(41) 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, U25_ga(T23, predc14_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(U23_ga(x₁, x₂)) = 1 + x₂   
POL(U24_ga(x₁, x₂)) = 1 + x₂   
POL(U25_ga(x₁, x₂)) = 1 + x₂   
POL(U5_GA(x₁, x₂)) = x₂   
POL(U7_GA(x₁, x₂)) = x₂   
POL(predc14_in_ga(x₁)) = x₁   
POL(predc14_out_ga(x₁, x₂)) = x₂   
POL(predc24_in_ga(x₁)) = x₁   
POL(predc24_out_ga(x₁, x₂)) = x₂   
POL(predc9_out_ga(x₁, x₂)) = x₂   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

(42) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

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

(44) TRUE

(45) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(50) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6)) at position [1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))

(51) Obligation:

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

U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(52) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8)) at position [1] we obtained the following new rules [LPAR04]:

U13_G(y0, predc9_out_ga(y0, 0)) → U15_G(y0, predc24_out_ga(0, 0))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

(53) Obligation:

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

U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, 0)) → U15_G(y0, predc24_out_ga(0, 0))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20)) at position [1] we obtained the following new rules [LPAR04]:

U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

(57) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(58) 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.

(59) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(60) 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].

doublec53_in_ga(x0)

(61) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(62) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6))) at position [1,1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))

(63) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(64) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0))) at position [1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))

(65) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(66) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16)) at position [1] we obtained the following new rules [LPAR04]:

U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0)))

(67) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(68) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))

(69) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(70) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0)) we obtained the following new rules [LPAR04]:

U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))

(71) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0)))) we obtained the following new rules [LPAR04]:

U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))

(73) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(74) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0)) we obtained the following new rules [LPAR04]:

U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))

(75) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(76) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0)))) we obtained the following new rules [LPAR04]:

U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))

(77) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(78) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0))) we obtained the following new rules [LPAR04]:

U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))

(79) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(80) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29) we obtained the following new rules [LPAR04]:

U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)

(81) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(82) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2) we obtained the following new rules [LPAR04]:

U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))

(83) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(84) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2) we obtained the following new rules [LPAR04]:

U20_G(0, doublec53_out_ga(x0, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(0, doublec53_out_ga(x0, s(s(0)))) → F1_IN_G(s(s(0)))

(85) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))
U20_G(0, doublec53_out_ga(x0, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(0, doublec53_out_ga(x0, s(s(0)))) → F1_IN_G(s(s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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