Left Termination of the query pattern log2_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

log2(X, Y) :- log2(X, 0, Y).
log2(0, I, I).
log2(s(0), I, I).
log2(s(s(X)), I, Y) :- ','(half(s(s(X)), X1), log2(X1, s(I), Y)).
half(0, 0).
half(s(0), 0).
half(s(s(X)), s(Y)) :- half(X, Y).

Queries:

log2(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: log2(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: log2(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: log2(T5, 0, T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: log2(T5, 0, T7), SCOPE: 2, CLAUSE: 1 | log2(T5, 0, T7), SCOPE: 2, CLAUSE: 2 | log2(T5, 0, T7), SCOPE: 2, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: log2(T5, 0, T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: log2(T5, 0, T7), SCOPE: 2, CLAUSE: 2 | log2(T5, 0, T7), SCOPE: 2, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: log2(T5, 0, T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: log2(T5, 0, T7), SCOPE: 2, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(half(s(s(T12)), X28), log2(X28, s(0), T14))\n\nT12 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: half(s(s(T12)), X28)\n\nT12 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: log2(T16, s(0), T14)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 4 | half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 5 | half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 5 | half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: half(s(s(T12)), X28), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX28 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: half(T20, X46)\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: half(T20, X46), SCOPE: 4, CLAUSE: 4 | half(T20, X46), SCOPE: 4, CLAUSE: 5 | half(T20, X46), SCOPE: 4, CLAUSE: 6\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: half(T20, X46), SCOPE: 4, CLAUSE: 4\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: half(T20, X46), SCOPE: 4, CLAUSE: 5 | half(T20, X46), SCOPE: 4, CLAUSE: 6\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: half(T20, X46), SCOPE: 4, CLAUSE: 5\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: half(T20, X46), SCOPE: 4, CLAUSE: 6\n\nT20 is ground\nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: half(T23, X55)\n\nT23 is ground\nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 1 | log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 2 | log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 1\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 2 | log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: log2(T16, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT16 is ground", 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: ','(half(s(s(T28)), X82), log2(X82, s(s(0)), T30))\n\nT28 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: half(s(s(T28)), X82)\n\nT28 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: log2(T32, s(s(0)), T30)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 1 | log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 2 | log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 3\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 1\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 2 | log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 3\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 2\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: log2(T32, s(s(0)), T30), SCOPE: 6, CLAUSE: 3\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(half(s(s(T38)), X116), log2(X116, s(s(s(0))), T40))\n\nT38 is ground\nX116 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: half(s(s(T38)), X116)\n\nT38 is ground\nX116 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: log2(T42, s(s(s(0))), T40)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 1 | log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 2 | log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 3\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 1\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 2 | log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 3\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 2\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: log2(T42, s(s(s(0))), T40), SCOPE: 7, CLAUSE: 3\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(half(s(s(T48)), X150), log2(X150, s(s(s(s(0)))), T50))\n\nT48 is ground\nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: half(s(s(T48)), X150)\n\nT48 is ground\nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: log2(T52, s(s(s(s(0)))), T50)\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 1 | log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 2 | log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 3\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 1\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 2 | log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 3\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 2\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: log2(T52, s(s(s(s(0)))), T50), SCOPE: 8, CLAUSE: 3\n\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(half(s(s(T58)), X184), log2(X184, s(s(s(s(s(0))))), T60))\n\nT58 is ground\nX184 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: half(s(s(T58)), X184)\n\nT58 is ground\nX184 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: log2(T62, s(s(s(s(s(0))))), T60)\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 1 | log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 2 | log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 3\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 1\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 2 | log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 3\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 2\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: log2(T62, s(s(s(s(s(0))))), T60), SCOPE: 9, CLAUSE: 3\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(half(s(s(T68)), X218), log2(X218, s(s(s(s(s(s(0)))))), T70))\n\nT68 is ground\nX218 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: half(s(s(T68)), X218)\n\nT68 is ground\nX218 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: log2(T72, s(s(s(s(s(s(0)))))), T70)\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 1 | log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 2 | log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 3\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 1\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 2 | log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 3\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 2\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: log2(T72, s(s(s(s(s(s(0)))))), T70), SCOPE: 10, CLAUSE: 3\n\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(half(s(s(T78)), X252), log2(X252, s(s(s(s(s(s(s(0))))))), T80))\n\nT78 is ground\nX252 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: half(s(s(T78)), X252)\n\nT78 is ground\nX252 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: log2(T82, s(s(s(s(s(s(s(0))))))), T80)\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 1 | log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 2 | log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 3\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 1\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 2 | log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 3\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 2\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: log2(T82, s(s(s(s(s(s(s(0))))))), T80), SCOPE: 11, CLAUSE: 3\n\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: ','(half(s(s(T88)), X286), log2(X286, s(s(s(s(s(s(s(s(0)))))))), T90))\n\nT88 is ground\nX286 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: ','(half(s(s(T88)), X286), log2(X286, s(T92), T90))\n\nT88 is ground\nT92 is ground\nX286 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: half(s(s(T88)), X286)\n\nT88 is ground\nX286 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: log2(T94, s(T92), T90)\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 1 | log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 2 | log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 3\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 1\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 2 | log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 3\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 2\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: log2(T94, s(T92), T90), SCOPE: 12, CLAUSE: 3\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: ','(half(s(s(T114)), X323), log2(X323, s(s(T115)), T117))\n\nT114 is ground\nT115 is ground\nX323 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 155 [arrowhead = none ];
155 -> 2;

156 [label="ONLY EVAL with clause\nlog2(X4, X5) :- log2(X4, 0, X5).\nand substitution\nT1 -> T5\nX4 -> T5\nT2 -> T7\nX5 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 156 [arrowhead = none ];
156 -> 3;

157 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 157 [arrowhead = none ];
157 -> 4;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 158 [arrowhead = none ];
158 -> 5;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 159 [arrowhead = none ];
159 -> 6;

160 [label="EVAL with clause\nlog2(0, X10, X10).\nand substitution\nT5 -> 0\nX10 -> 0\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 160 [arrowhead = none ];
160 -> 7;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 161 [arrowhead = none ];
161 -> 8;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 162 [arrowhead = none ];
162 -> 10;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 163 [arrowhead = none ];
163 -> 11;

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 164 [arrowhead = none ];
164 -> 9;

165 [label="EVAL with clause\nlog2(s(0), X16, X16).\nand substitution\nT5 -> s(0)\nX16 -> 0\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 165 [arrowhead = none ];
165 -> 12;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 166 [arrowhead = none ];
166 -> 13;

167 [label="EVAL with clause\nlog2(s(s(X25)), X26, X27) :- ','(half(s(s(X25)), X28), log2(X28, s(X26), X27)).\nand substitution\nX25 -> T12\nT5 -> s(s(T12))\nX26 -> 0\nT7 -> T14\nX27 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 167 [arrowhead = none ];
167 -> 15;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 168 [arrowhead = none ];
168 -> 16;

169 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 169 [arrowhead = none ];
169 -> 14;

170 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 170 [arrowhead = none ];
170 -> 17;

171 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nreplacements:\nX28 -> T16", fontsize=14, style = filled, fillcolor = violet];
15 -> 171 [arrowhead = none ];
171 -> 18;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 172 [arrowhead = none ];
172 -> 19;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 173 [arrowhead = none ];
173 -> 36;

174 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 174 [arrowhead = none ];
174 -> 20;

175 [label="BACKTRACK\nfor clause: half(s(0), 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 175 [arrowhead = none ];
175 -> 21;

176 [label="ONLY EVAL with clause\nhalf(s(s(X44)), s(X45)) :- half(X44, X45).\nand substitution\nT12 -> T20\nX44 -> T20\nX45 -> X46\nX28 -> s(X46)", fontsize=14, style = filled, fillcolor = white];
21 -> 176 [arrowhead = none ];
176 -> 22;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 177 [arrowhead = none ];
177 -> 23;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 178 [arrowhead = none ];
178 -> 24;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 179 [arrowhead = none ];
179 -> 25;

180 [label="EVAL with clause\nhalf(0, 0).\nand substitution\nT20 -> 0\nX46 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 180 [arrowhead = none ];
180 -> 26;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 181 [arrowhead = none ];
181 -> 27;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 182 [arrowhead = none ];
182 -> 29;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 183 [arrowhead = none ];
183 -> 30;

184 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 184 [arrowhead = none ];
184 -> 28;

185 [label="EVAL with clause\nhalf(s(0), 0).\nand substitution\nT20 -> s(0)\nX46 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 185 [arrowhead = none ];
185 -> 31;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 186 [arrowhead = none ];
186 -> 32;

187 [label="EVAL with clause\nhalf(s(s(X53)), s(X54)) :- half(X53, X54).\nand substitution\nX53 -> T23\nT20 -> s(s(T23))\nX54 -> X55\nX46 -> s(X55)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 187 [arrowhead = none ];
187 -> 34;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 188 [arrowhead = none ];
188 -> 35;

189 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 189 [arrowhead = none ];
189 -> 33;

190 [label="INSTANCE with matching:\nT20 -> T23\nX46 -> X55", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 190 [arrowhead = none , style=dashed];
190 -> 22[style=dashed];

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 191 [arrowhead = none ];
191 -> 37;

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 192 [arrowhead = none ];
192 -> 38;

193 [label="EVAL with clause\nlog2(0, X64, X64).\nand substitution\nT16 -> 0\nX64 -> s(0)\nT14 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 193 [arrowhead = none ];
193 -> 39;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 194 [arrowhead = none ];
194 -> 40;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 195 [arrowhead = none ];
195 -> 42;

196 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 196 [arrowhead = none ];
196 -> 43;

197 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 197 [arrowhead = none ];
197 -> 41;

198 [label="EVAL with clause\nlog2(s(0), X70, X70).\nand substitution\nT16 -> s(0)\nX70 -> s(0)\nT14 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 198 [arrowhead = none ];
198 -> 44;

199 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 199 [arrowhead = none ];
199 -> 45;

200 [label="EVAL with clause\nlog2(s(s(X79)), X80, X81) :- ','(half(s(s(X79)), X82), log2(X82, s(X80), X81)).\nand substitution\nX79 -> T28\nT16 -> s(s(T28))\nX80 -> s(0)\nT14 -> T30\nX81 -> T30\nT29 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 200 [arrowhead = none ];
200 -> 47;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 201 [arrowhead = none ];
201 -> 48;

202 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 202 [arrowhead = none ];
202 -> 46;

203 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 203 [arrowhead = none ];
203 -> 49;

204 [label="SPLIT 2\nnew knowledge:\nT32 is ground\nreplacements:\nX82 -> T32", fontsize=14, style = filled, fillcolor = violet];
47 -> 204 [arrowhead = none ];
204 -> 50;

205 [label="INSTANCE with matching:\nT12 -> T28\nX28 -> X82", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 205 [arrowhead = none , style=dashed];
205 -> 17[style=dashed];

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 206 [arrowhead = none ];
206 -> 51;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 207 [arrowhead = none ];
207 -> 52;

208 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 208 [arrowhead = none ];
208 -> 53;

209 [label="EVAL with clause\nlog2(0, X98, X98).\nand substitution\nT32 -> 0\nX98 -> s(s(0))\nT30 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 209 [arrowhead = none ];
209 -> 54;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 210 [arrowhead = none ];
210 -> 55;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 211 [arrowhead = none ];
211 -> 57;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 212 [arrowhead = none ];
212 -> 58;

213 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 213 [arrowhead = none ];
213 -> 56;

214 [label="EVAL with clause\nlog2(s(0), X104, X104).\nand substitution\nT32 -> s(0)\nX104 -> s(s(0))\nT30 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 214 [arrowhead = none ];
214 -> 59;

215 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 215 [arrowhead = none ];
215 -> 60;

216 [label="EVAL with clause\nlog2(s(s(X113)), X114, X115) :- ','(half(s(s(X113)), X116), log2(X116, s(X114), X115)).\nand substitution\nX113 -> T38\nT32 -> s(s(T38))\nX114 -> s(s(0))\nT30 -> T40\nX115 -> T40\nT39 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 216 [arrowhead = none ];
216 -> 62;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 217 [arrowhead = none ];
217 -> 63;

218 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 218 [arrowhead = none ];
218 -> 61;

219 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
62 -> 219 [arrowhead = none ];
219 -> 64;

220 [label="SPLIT 2\nnew knowledge:\nT42 is ground\nreplacements:\nX116 -> T42", fontsize=14, style = filled, fillcolor = violet];
62 -> 220 [arrowhead = none ];
220 -> 65;

221 [label="INSTANCE with matching:\nT12 -> T38\nX28 -> X116", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 221 [arrowhead = none , style=dashed];
221 -> 17[style=dashed];

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 222 [arrowhead = none ];
222 -> 66;

223 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 223 [arrowhead = none ];
223 -> 67;

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 224 [arrowhead = none ];
224 -> 68;

225 [label="EVAL with clause\nlog2(0, X132, X132).\nand substitution\nT42 -> 0\nX132 -> s(s(s(0)))\nT40 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 225 [arrowhead = none ];
225 -> 69;

226 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 226 [arrowhead = none ];
226 -> 70;

227 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 227 [arrowhead = none ];
227 -> 72;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 228 [arrowhead = none ];
228 -> 73;

229 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 229 [arrowhead = none ];
229 -> 71;

230 [label="EVAL with clause\nlog2(s(0), X138, X138).\nand substitution\nT42 -> s(0)\nX138 -> s(s(s(0)))\nT40 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 230 [arrowhead = none ];
230 -> 74;

231 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 231 [arrowhead = none ];
231 -> 75;

232 [label="EVAL with clause\nlog2(s(s(X147)), X148, X149) :- ','(half(s(s(X147)), X150), log2(X150, s(X148), X149)).\nand substitution\nX147 -> T48\nT42 -> s(s(T48))\nX148 -> s(s(s(0)))\nT40 -> T50\nX149 -> T50\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 232 [arrowhead = none ];
232 -> 77;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 233 [arrowhead = none ];
233 -> 78;

234 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 234 [arrowhead = none ];
234 -> 76;

235 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
77 -> 235 [arrowhead = none ];
235 -> 79;

236 [label="SPLIT 2\nnew knowledge:\nT52 is ground\nreplacements:\nX150 -> T52", fontsize=14, style = filled, fillcolor = violet];
77 -> 236 [arrowhead = none ];
236 -> 80;

237 [label="INSTANCE with matching:\nT12 -> T48\nX28 -> X150", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 237 [arrowhead = none , style=dashed];
237 -> 17[style=dashed];

238 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 238 [arrowhead = none ];
238 -> 81;

239 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 239 [arrowhead = none ];
239 -> 82;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 240 [arrowhead = none ];
240 -> 83;

241 [label="EVAL with clause\nlog2(0, X166, X166).\nand substitution\nT52 -> 0\nX166 -> s(s(s(s(0))))\nT50 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 241 [arrowhead = none ];
241 -> 84;

242 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 242 [arrowhead = none ];
242 -> 85;

243 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 243 [arrowhead = none ];
243 -> 87;

244 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 244 [arrowhead = none ];
244 -> 88;

245 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 245 [arrowhead = none ];
245 -> 86;

246 [label="EVAL with clause\nlog2(s(0), X172, X172).\nand substitution\nT52 -> s(0)\nX172 -> s(s(s(s(0))))\nT50 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 246 [arrowhead = none ];
246 -> 89;

247 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 247 [arrowhead = none ];
247 -> 90;

248 [label="EVAL with clause\nlog2(s(s(X181)), X182, X183) :- ','(half(s(s(X181)), X184), log2(X184, s(X182), X183)).\nand substitution\nX181 -> T58\nT52 -> s(s(T58))\nX182 -> s(s(s(s(0))))\nT50 -> T60\nX183 -> T60\nT59 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 248 [arrowhead = none ];
248 -> 92;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 249 [arrowhead = none ];
249 -> 93;

250 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 250 [arrowhead = none ];
250 -> 91;

251 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 251 [arrowhead = none ];
251 -> 94;

252 [label="SPLIT 2\nnew knowledge:\nT62 is ground\nreplacements:\nX184 -> T62", fontsize=14, style = filled, fillcolor = violet];
92 -> 252 [arrowhead = none ];
252 -> 95;

253 [label="INSTANCE with matching:\nT12 -> T58\nX28 -> X184", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 253 [arrowhead = none , style=dashed];
253 -> 17[style=dashed];

254 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
95 -> 254 [arrowhead = none ];
254 -> 96;

255 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 255 [arrowhead = none ];
255 -> 97;

256 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 256 [arrowhead = none ];
256 -> 98;

257 [label="EVAL with clause\nlog2(0, X200, X200).\nand substitution\nT62 -> 0\nX200 -> s(s(s(s(s(0)))))\nT60 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 257 [arrowhead = none ];
257 -> 99;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 258 [arrowhead = none ];
258 -> 100;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 259 [arrowhead = none ];
259 -> 102;

260 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 260 [arrowhead = none ];
260 -> 103;

261 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 261 [arrowhead = none ];
261 -> 101;

262 [label="EVAL with clause\nlog2(s(0), X206, X206).\nand substitution\nT62 -> s(0)\nX206 -> s(s(s(s(s(0)))))\nT60 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 262 [arrowhead = none ];
262 -> 104;

263 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 263 [arrowhead = none ];
263 -> 105;

264 [label="EVAL with clause\nlog2(s(s(X215)), X216, X217) :- ','(half(s(s(X215)), X218), log2(X218, s(X216), X217)).\nand substitution\nX215 -> T68\nT62 -> s(s(T68))\nX216 -> s(s(s(s(s(0)))))\nT60 -> T70\nX217 -> T70\nT69 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 264 [arrowhead = none ];
264 -> 107;

265 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 265 [arrowhead = none ];
265 -> 108;

266 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 266 [arrowhead = none ];
266 -> 106;

267 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
107 -> 267 [arrowhead = none ];
267 -> 109;

268 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nreplacements:\nX218 -> T72", fontsize=14, style = filled, fillcolor = violet];
107 -> 268 [arrowhead = none ];
268 -> 110;

269 [label="INSTANCE with matching:\nT12 -> T68\nX28 -> X218", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 269 [arrowhead = none , style=dashed];
269 -> 17[style=dashed];

270 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
110 -> 270 [arrowhead = none ];
270 -> 111;

271 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
111 -> 271 [arrowhead = none ];
271 -> 112;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
111 -> 272 [arrowhead = none ];
272 -> 113;

273 [label="EVAL with clause\nlog2(0, X234, X234).\nand substitution\nT72 -> 0\nX234 -> s(s(s(s(s(s(0))))))\nT70 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
112 -> 273 [arrowhead = none ];
273 -> 114;

274 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
112 -> 274 [arrowhead = none ];
274 -> 115;

275 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
113 -> 275 [arrowhead = none ];
275 -> 117;

276 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
113 -> 276 [arrowhead = none ];
276 -> 118;

277 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 277 [arrowhead = none ];
277 -> 116;

278 [label="EVAL with clause\nlog2(s(0), X240, X240).\nand substitution\nT72 -> s(0)\nX240 -> s(s(s(s(s(s(0))))))\nT70 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 278 [arrowhead = none ];
278 -> 119;

279 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 279 [arrowhead = none ];
279 -> 120;

280 [label="EVAL with clause\nlog2(s(s(X249)), X250, X251) :- ','(half(s(s(X249)), X252), log2(X252, s(X250), X251)).\nand substitution\nX249 -> T78\nT72 -> s(s(T78))\nX250 -> s(s(s(s(s(s(0))))))\nT70 -> T80\nX251 -> T80\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 280 [arrowhead = none ];
280 -> 122;

281 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 281 [arrowhead = none ];
281 -> 123;

282 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 282 [arrowhead = none ];
282 -> 121;

283 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
122 -> 283 [arrowhead = none ];
283 -> 124;

284 [label="SPLIT 2\nnew knowledge:\nT82 is ground\nreplacements:\nX252 -> T82", fontsize=14, style = filled, fillcolor = violet];
122 -> 284 [arrowhead = none ];
284 -> 125;

285 [label="INSTANCE with matching:\nT12 -> T78\nX28 -> X252", fontsize=14, style = filled, fillcolor = lightgrey];
124 -> 285 [arrowhead = none , style=dashed];
285 -> 17[style=dashed];

286 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 286 [arrowhead = none ];
286 -> 126;

287 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
126 -> 287 [arrowhead = none ];
287 -> 127;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
126 -> 288 [arrowhead = none ];
288 -> 128;

289 [label="EVAL with clause\nlog2(0, X268, X268).\nand substitution\nT82 -> 0\nX268 -> s(s(s(s(s(s(s(0)))))))\nT80 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 289 [arrowhead = none ];
289 -> 129;

290 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 290 [arrowhead = none ];
290 -> 130;

291 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 291 [arrowhead = none ];
291 -> 132;

292 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 292 [arrowhead = none ];
292 -> 133;

293 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 293 [arrowhead = none ];
293 -> 131;

294 [label="EVAL with clause\nlog2(s(0), X274, X274).\nand substitution\nT82 -> s(0)\nX274 -> s(s(s(s(s(s(s(0)))))))\nT80 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
132 -> 294 [arrowhead = none ];
294 -> 134;

295 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
132 -> 295 [arrowhead = none ];
295 -> 135;

296 [label="EVAL with clause\nlog2(s(s(X283)), X284, X285) :- ','(half(s(s(X283)), X286), log2(X286, s(X284), X285)).\nand substitution\nX283 -> T88\nT82 -> s(s(T88))\nX284 -> s(s(s(s(s(s(s(0)))))))\nT80 -> T90\nX285 -> T90\nT89 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 296 [arrowhead = none ];
296 -> 137;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 297 [arrowhead = none ];
297 -> 138;

298 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
134 -> 298 [arrowhead = none ];
298 -> 136;

299 [label="GENERALIZATION\nT92 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT92 is ground", fontsize=14, style = filled, fillcolor = violet];
137 -> 299 [arrowhead = none ];
299 -> 139;

300 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
139 -> 300 [arrowhead = none ];
300 -> 140;

301 [label="SPLIT 2\nnew knowledge:\nT94 is ground\nreplacements:\nX286 -> T94", fontsize=14, style = filled, fillcolor = violet];
139 -> 301 [arrowhead = none ];
301 -> 141;

302 [label="INSTANCE with matching:\nT12 -> T88\nX28 -> X286", fontsize=14, style = filled, fillcolor = lightgrey];
140 -> 302 [arrowhead = none , style=dashed];
302 -> 17[style=dashed];

303 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
141 -> 303 [arrowhead = none ];
303 -> 142;

304 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 304 [arrowhead = none ];
304 -> 143;

305 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 305 [arrowhead = none ];
305 -> 144;

306 [label="EVAL with clause\nlog2(0, X305, X305).\nand substitution\nT94 -> 0\nT92 -> T102\nX305 -> s(T102)\nT90 -> s(T102)", fontsize=14, style = filled, fillcolor = lightblue];
143 -> 306 [arrowhead = none ];
306 -> 145;

307 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
143 -> 307 [arrowhead = none ];
307 -> 146;

308 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 308 [arrowhead = none ];
308 -> 148;

309 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 309 [arrowhead = none ];
309 -> 149;

310 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
145 -> 310 [arrowhead = none ];
310 -> 147;

311 [label="EVAL with clause\nlog2(s(0), X311, X311).\nand substitution\nT94 -> s(0)\nT92 -> T107\nX311 -> s(T107)\nT90 -> s(T107)", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 311 [arrowhead = none ];
311 -> 150;

312 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 312 [arrowhead = none ];
312 -> 151;

313 [label="EVAL with clause\nlog2(s(s(X320)), X321, X322) :- ','(half(s(s(X320)), X323), log2(X323, s(X321), X322)).\nand substitution\nX320 -> T114\nT94 -> s(s(T114))\nT92 -> T115\nX321 -> s(T115)\nT90 -> T117\nX322 -> T117\nT116 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 313 [arrowhead = none ];
313 -> 153;

314 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 314 [arrowhead = none ];
314 -> 154;

315 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
150 -> 315 [arrowhead = none ];
315 -> 152;

316 [label="INSTANCE with matching:\nT88 -> T114\nX286 -> X323\nT92 -> s(T115)\nT90 -> T117", fontsize=14, style = filled, fillcolor = lightgrey];
153 -> 316 [arrowhead = none , style=dashed];
316 -> 139[style=dashed];

}

(2) Obligation:

Triples:

half22(s(s(T23)), s(X55)) :- half22(T23, X55).
half17(T20, s(X46)) :- half22(T20, X46).
p139(T88, X286, T92, T90) :- half17(T88, X286).
p139(T88, s(s(T114)), T115, T117) :- ','(halfc17(T88, s(s(T114))), p139(T114, X323, s(T115), T117)).
log21(s(s(T12)), T14) :- half17(T12, X28).
log21(s(s(T12)), T30) :- ','(halfc17(T12, s(s(T28))), half17(T28, X82)).
log21(s(s(T12)), T40) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), half17(T38, X116))).
log21(s(s(T12)), T50) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), ','(halfc17(T38, s(s(T48))), half17(T48, X150)))).
log21(s(s(T12)), T60) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), ','(halfc17(T38, s(s(T48))), ','(halfc17(T48, s(s(T58))), half17(T58, X184))))).
log21(s(s(T12)), T70) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), ','(halfc17(T38, s(s(T48))), ','(halfc17(T48, s(s(T58))), ','(halfc17(T58, s(s(T68))), half17(T68, X218)))))).
log21(s(s(T12)), T80) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), ','(halfc17(T38, s(s(T48))), ','(halfc17(T48, s(s(T58))), ','(halfc17(T58, s(s(T68))), ','(halfc17(T68, s(s(T78))), half17(T78, X252))))))).
log21(s(s(T12)), T90) :- ','(halfc17(T12, s(s(T28))), ','(halfc17(T28, s(s(T38))), ','(halfc17(T38, s(s(T48))), ','(halfc17(T48, s(s(T58))), ','(halfc17(T58, s(s(T68))), ','(halfc17(T68, s(s(T78))), ','(halfc17(T78, s(s(T88))), p139(T88, X286, s(s(s(s(s(s(s(0))))))), T90)))))))).

Clauses:

halfc22(0, 0).
halfc22(s(0), 0).
halfc22(s(s(T23)), s(X55)) :- halfc22(T23, X55).
halfc17(T20, s(X46)) :- halfc22(T20, X46).
qc139(T88, 0, T102, s(T102)) :- halfc17(T88, 0).
qc139(T88, s(0), T107, s(T107)) :- halfc17(T88, s(0)).
qc139(T88, s(s(T114)), T115, T117) :- ','(halfc17(T88, s(s(T114))), qc139(T114, X323, s(T115), T117)).

Afs:

log21(x1, x2)  =  log21(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:
log21_in: (b,f)
half17_in: (b,f)
half22_in: (b,f)
halfc17_in: (b,f)
halfc22_in: (b,f)
p139_in: (b,f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

LOG21_IN_GA(s(s(T12)), T14) → U6_GA(T12, T14, half17_in_ga(T12, X28))
LOG21_IN_GA(s(s(T12)), T14) → HALF17_IN_GA(T12, X28)
HALF17_IN_GA(T20, s(X46)) → U2_GA(T20, X46, half22_in_ga(T20, X46))
HALF17_IN_GA(T20, s(X46)) → HALF22_IN_GA(T20, X46)
HALF22_IN_GA(s(s(T23)), s(X55)) → U1_GA(T23, X55, half22_in_ga(T23, X55))
HALF22_IN_GA(s(s(T23)), s(X55)) → HALF22_IN_GA(T23, X55)
LOG21_IN_GA(s(s(T12)), T30) → U7_GA(T12, T30, halfc17_in_ga(T12, s(s(T28))))
U7_GA(T12, T30, halfc17_out_ga(T12, s(s(T28)))) → U8_GA(T12, T30, half17_in_ga(T28, X82))
U7_GA(T12, T30, halfc17_out_ga(T12, s(s(T28)))) → HALF17_IN_GA(T28, X82)
LOG21_IN_GA(s(s(T12)), T40) → U9_GA(T12, T40, halfc17_in_ga(T12, s(s(T28))))
U9_GA(T12, T40, halfc17_out_ga(T12, s(s(T28)))) → U10_GA(T12, T40, halfc17_in_ga(T28, s(s(T38))))
U10_GA(T12, T40, halfc17_out_ga(T28, s(s(T38)))) → U11_GA(T12, T40, half17_in_ga(T38, X116))
U10_GA(T12, T40, halfc17_out_ga(T28, s(s(T38)))) → HALF17_IN_GA(T38, X116)
LOG21_IN_GA(s(s(T12)), T50) → U12_GA(T12, T50, halfc17_in_ga(T12, s(s(T28))))
U12_GA(T12, T50, halfc17_out_ga(T12, s(s(T28)))) → U13_GA(T12, T50, halfc17_in_ga(T28, s(s(T38))))
U13_GA(T12, T50, halfc17_out_ga(T28, s(s(T38)))) → U14_GA(T12, T50, halfc17_in_ga(T38, s(s(T48))))
U14_GA(T12, T50, halfc17_out_ga(T38, s(s(T48)))) → U15_GA(T12, T50, half17_in_ga(T48, X150))
U14_GA(T12, T50, halfc17_out_ga(T38, s(s(T48)))) → HALF17_IN_GA(T48, X150)
LOG21_IN_GA(s(s(T12)), T60) → U16_GA(T12, T60, halfc17_in_ga(T12, s(s(T28))))
U16_GA(T12, T60, halfc17_out_ga(T12, s(s(T28)))) → U17_GA(T12, T60, halfc17_in_ga(T28, s(s(T38))))
U17_GA(T12, T60, halfc17_out_ga(T28, s(s(T38)))) → U18_GA(T12, T60, halfc17_in_ga(T38, s(s(T48))))
U18_GA(T12, T60, halfc17_out_ga(T38, s(s(T48)))) → U19_GA(T12, T60, halfc17_in_ga(T48, s(s(T58))))
U19_GA(T12, T60, halfc17_out_ga(T48, s(s(T58)))) → U20_GA(T12, T60, half17_in_ga(T58, X184))
U19_GA(T12, T60, halfc17_out_ga(T48, s(s(T58)))) → HALF17_IN_GA(T58, X184)
LOG21_IN_GA(s(s(T12)), T70) → U21_GA(T12, T70, halfc17_in_ga(T12, s(s(T28))))
U21_GA(T12, T70, halfc17_out_ga(T12, s(s(T28)))) → U22_GA(T12, T70, halfc17_in_ga(T28, s(s(T38))))
U22_GA(T12, T70, halfc17_out_ga(T28, s(s(T38)))) → U23_GA(T12, T70, halfc17_in_ga(T38, s(s(T48))))
U23_GA(T12, T70, halfc17_out_ga(T38, s(s(T48)))) → U24_GA(T12, T70, halfc17_in_ga(T48, s(s(T58))))
U24_GA(T12, T70, halfc17_out_ga(T48, s(s(T58)))) → U25_GA(T12, T70, halfc17_in_ga(T58, s(s(T68))))
U25_GA(T12, T70, halfc17_out_ga(T58, s(s(T68)))) → U26_GA(T12, T70, half17_in_ga(T68, X218))
U25_GA(T12, T70, halfc17_out_ga(T58, s(s(T68)))) → HALF17_IN_GA(T68, X218)
LOG21_IN_GA(s(s(T12)), T80) → U27_GA(T12, T80, halfc17_in_ga(T12, s(s(T28))))
U27_GA(T12, T80, halfc17_out_ga(T12, s(s(T28)))) → U28_GA(T12, T80, halfc17_in_ga(T28, s(s(T38))))
U28_GA(T12, T80, halfc17_out_ga(T28, s(s(T38)))) → U29_GA(T12, T80, halfc17_in_ga(T38, s(s(T48))))
U29_GA(T12, T80, halfc17_out_ga(T38, s(s(T48)))) → U30_GA(T12, T80, halfc17_in_ga(T48, s(s(T58))))
U30_GA(T12, T80, halfc17_out_ga(T48, s(s(T58)))) → U31_GA(T12, T80, halfc17_in_ga(T58, s(s(T68))))
U31_GA(T12, T80, halfc17_out_ga(T58, s(s(T68)))) → U32_GA(T12, T80, halfc17_in_ga(T68, s(s(T78))))
U32_GA(T12, T80, halfc17_out_ga(T68, s(s(T78)))) → U33_GA(T12, T80, half17_in_ga(T78, X252))
U32_GA(T12, T80, halfc17_out_ga(T68, s(s(T78)))) → HALF17_IN_GA(T78, X252)
LOG21_IN_GA(s(s(T12)), T90) → U34_GA(T12, T90, halfc17_in_ga(T12, s(s(T28))))
U34_GA(T12, T90, halfc17_out_ga(T12, s(s(T28)))) → U35_GA(T12, T90, halfc17_in_ga(T28, s(s(T38))))
U35_GA(T12, T90, halfc17_out_ga(T28, s(s(T38)))) → U36_GA(T12, T90, halfc17_in_ga(T38, s(s(T48))))
U36_GA(T12, T90, halfc17_out_ga(T38, s(s(T48)))) → U37_GA(T12, T90, halfc17_in_ga(T48, s(s(T58))))
U37_GA(T12, T90, halfc17_out_ga(T48, s(s(T58)))) → U38_GA(T12, T90, halfc17_in_ga(T58, s(s(T68))))
U38_GA(T12, T90, halfc17_out_ga(T58, s(s(T68)))) → U39_GA(T12, T90, halfc17_in_ga(T68, s(s(T78))))
U39_GA(T12, T90, halfc17_out_ga(T68, s(s(T78)))) → U40_GA(T12, T90, halfc17_in_ga(T78, s(s(T88))))
U40_GA(T12, T90, halfc17_out_ga(T78, s(s(T88)))) → U41_GA(T12, T90, p139_in_gaga(T88, X286, s(s(s(s(s(s(s(0))))))), T90))
U40_GA(T12, T90, halfc17_out_ga(T78, s(s(T88)))) → P139_IN_GAGA(T88, X286, s(s(s(s(s(s(s(0))))))), T90)
P139_IN_GAGA(T88, X286, T92, T90) → U3_GAGA(T88, X286, T92, T90, half17_in_ga(T88, X286))
P139_IN_GAGA(T88, X286, T92, T90) → HALF17_IN_GA(T88, X286)
P139_IN_GAGA(T88, s(s(T114)), T115, T117) → U4_GAGA(T88, T114, T115, T117, halfc17_in_ga(T88, s(s(T114))))
U4_GAGA(T88, T114, T115, T117, halfc17_out_ga(T88, s(s(T114)))) → U5_GAGA(T88, T114, T115, T117, p139_in_gaga(T114, X323, s(T115), T117))
U4_GAGA(T88, T114, T115, T117, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, X323, s(T115), T117)

The TRS R consists of the following rules:

halfc17_in_ga(T20, s(X46)) → U44_ga(T20, X46, halfc22_in_ga(T20, X46))
halfc22_in_ga(0, 0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0), 0) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23)), s(X55)) → U43_ga(T23, X55, halfc22_in_ga(T23, X55))
U43_ga(T23, X55, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, X46, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
half17_in_ga(x1, x2)  =  half17_in_ga(x1)
half22_in_ga(x1, x2)  =  half22_in_ga(x1)
halfc17_in_ga(x1, x2)  =  halfc17_in_ga(x1)
U44_ga(x1, x2, x3)  =  U44_ga(x1, x3)
halfc22_in_ga(x1, x2)  =  halfc22_in_ga(x1)
0  =  0
halfc22_out_ga(x1, x2)  =  halfc22_out_ga(x1, x2)
U43_ga(x1, x2, x3)  =  U43_ga(x1, x3)
halfc17_out_ga(x1, x2)  =  halfc17_out_ga(x1, x2)
p139_in_gaga(x1, x2, x3, x4)  =  p139_in_gaga(x1, x3)
LOG21_IN_GA(x1, x2)  =  LOG21_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
HALF17_IN_GA(x1, x2)  =  HALF17_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
HALF22_IN_GA(x1, x2)  =  HALF22_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U18_GA(x1, x2, x3)  =  U18_GA(x1, x3)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
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)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)
U40_GA(x1, x2, x3)  =  U40_GA(x1, x3)
U41_GA(x1, x2, x3)  =  U41_GA(x1, x3)
P139_IN_GAGA(x1, x2, x3, x4)  =  P139_IN_GAGA(x1, x3)
U3_GAGA(x1, x2, x3, x4, x5)  =  U3_GAGA(x1, x3, x5)
U4_GAGA(x1, x2, x3, x4, x5)  =  U4_GAGA(x1, x3, x5)
U5_GAGA(x1, x2, x3, x4, x5)  =  U5_GAGA(x1, x2, x3, x5)

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:

LOG21_IN_GA(s(s(T12)), T14) → U6_GA(T12, T14, half17_in_ga(T12, X28))
LOG21_IN_GA(s(s(T12)), T14) → HALF17_IN_GA(T12, X28)
HALF17_IN_GA(T20, s(X46)) → U2_GA(T20, X46, half22_in_ga(T20, X46))
HALF17_IN_GA(T20, s(X46)) → HALF22_IN_GA(T20, X46)
HALF22_IN_GA(s(s(T23)), s(X55)) → U1_GA(T23, X55, half22_in_ga(T23, X55))
HALF22_IN_GA(s(s(T23)), s(X55)) → HALF22_IN_GA(T23, X55)
LOG21_IN_GA(s(s(T12)), T30) → U7_GA(T12, T30, halfc17_in_ga(T12, s(s(T28))))
U7_GA(T12, T30, halfc17_out_ga(T12, s(s(T28)))) → U8_GA(T12, T30, half17_in_ga(T28, X82))
U7_GA(T12, T30, halfc17_out_ga(T12, s(s(T28)))) → HALF17_IN_GA(T28, X82)
LOG21_IN_GA(s(s(T12)), T40) → U9_GA(T12, T40, halfc17_in_ga(T12, s(s(T28))))
U9_GA(T12, T40, halfc17_out_ga(T12, s(s(T28)))) → U10_GA(T12, T40, halfc17_in_ga(T28, s(s(T38))))
U10_GA(T12, T40, halfc17_out_ga(T28, s(s(T38)))) → U11_GA(T12, T40, half17_in_ga(T38, X116))
U10_GA(T12, T40, halfc17_out_ga(T28, s(s(T38)))) → HALF17_IN_GA(T38, X116)
LOG21_IN_GA(s(s(T12)), T50) → U12_GA(T12, T50, halfc17_in_ga(T12, s(s(T28))))
U12_GA(T12, T50, halfc17_out_ga(T12, s(s(T28)))) → U13_GA(T12, T50, halfc17_in_ga(T28, s(s(T38))))
U13_GA(T12, T50, halfc17_out_ga(T28, s(s(T38)))) → U14_GA(T12, T50, halfc17_in_ga(T38, s(s(T48))))
U14_GA(T12, T50, halfc17_out_ga(T38, s(s(T48)))) → U15_GA(T12, T50, half17_in_ga(T48, X150))
U14_GA(T12, T50, halfc17_out_ga(T38, s(s(T48)))) → HALF17_IN_GA(T48, X150)
LOG21_IN_GA(s(s(T12)), T60) → U16_GA(T12, T60, halfc17_in_ga(T12, s(s(T28))))
U16_GA(T12, T60, halfc17_out_ga(T12, s(s(T28)))) → U17_GA(T12, T60, halfc17_in_ga(T28, s(s(T38))))
U17_GA(T12, T60, halfc17_out_ga(T28, s(s(T38)))) → U18_GA(T12, T60, halfc17_in_ga(T38, s(s(T48))))
U18_GA(T12, T60, halfc17_out_ga(T38, s(s(T48)))) → U19_GA(T12, T60, halfc17_in_ga(T48, s(s(T58))))
U19_GA(T12, T60, halfc17_out_ga(T48, s(s(T58)))) → U20_GA(T12, T60, half17_in_ga(T58, X184))
U19_GA(T12, T60, halfc17_out_ga(T48, s(s(T58)))) → HALF17_IN_GA(T58, X184)
LOG21_IN_GA(s(s(T12)), T70) → U21_GA(T12, T70, halfc17_in_ga(T12, s(s(T28))))
U21_GA(T12, T70, halfc17_out_ga(T12, s(s(T28)))) → U22_GA(T12, T70, halfc17_in_ga(T28, s(s(T38))))
U22_GA(T12, T70, halfc17_out_ga(T28, s(s(T38)))) → U23_GA(T12, T70, halfc17_in_ga(T38, s(s(T48))))
U23_GA(T12, T70, halfc17_out_ga(T38, s(s(T48)))) → U24_GA(T12, T70, halfc17_in_ga(T48, s(s(T58))))
U24_GA(T12, T70, halfc17_out_ga(T48, s(s(T58)))) → U25_GA(T12, T70, halfc17_in_ga(T58, s(s(T68))))
U25_GA(T12, T70, halfc17_out_ga(T58, s(s(T68)))) → U26_GA(T12, T70, half17_in_ga(T68, X218))
U25_GA(T12, T70, halfc17_out_ga(T58, s(s(T68)))) → HALF17_IN_GA(T68, X218)
LOG21_IN_GA(s(s(T12)), T80) → U27_GA(T12, T80, halfc17_in_ga(T12, s(s(T28))))
U27_GA(T12, T80, halfc17_out_ga(T12, s(s(T28)))) → U28_GA(T12, T80, halfc17_in_ga(T28, s(s(T38))))
U28_GA(T12, T80, halfc17_out_ga(T28, s(s(T38)))) → U29_GA(T12, T80, halfc17_in_ga(T38, s(s(T48))))
U29_GA(T12, T80, halfc17_out_ga(T38, s(s(T48)))) → U30_GA(T12, T80, halfc17_in_ga(T48, s(s(T58))))
U30_GA(T12, T80, halfc17_out_ga(T48, s(s(T58)))) → U31_GA(T12, T80, halfc17_in_ga(T58, s(s(T68))))
U31_GA(T12, T80, halfc17_out_ga(T58, s(s(T68)))) → U32_GA(T12, T80, halfc17_in_ga(T68, s(s(T78))))
U32_GA(T12, T80, halfc17_out_ga(T68, s(s(T78)))) → U33_GA(T12, T80, half17_in_ga(T78, X252))
U32_GA(T12, T80, halfc17_out_ga(T68, s(s(T78)))) → HALF17_IN_GA(T78, X252)
LOG21_IN_GA(s(s(T12)), T90) → U34_GA(T12, T90, halfc17_in_ga(T12, s(s(T28))))
U34_GA(T12, T90, halfc17_out_ga(T12, s(s(T28)))) → U35_GA(T12, T90, halfc17_in_ga(T28, s(s(T38))))
U35_GA(T12, T90, halfc17_out_ga(T28, s(s(T38)))) → U36_GA(T12, T90, halfc17_in_ga(T38, s(s(T48))))
U36_GA(T12, T90, halfc17_out_ga(T38, s(s(T48)))) → U37_GA(T12, T90, halfc17_in_ga(T48, s(s(T58))))
U37_GA(T12, T90, halfc17_out_ga(T48, s(s(T58)))) → U38_GA(T12, T90, halfc17_in_ga(T58, s(s(T68))))
U38_GA(T12, T90, halfc17_out_ga(T58, s(s(T68)))) → U39_GA(T12, T90, halfc17_in_ga(T68, s(s(T78))))
U39_GA(T12, T90, halfc17_out_ga(T68, s(s(T78)))) → U40_GA(T12, T90, halfc17_in_ga(T78, s(s(T88))))
U40_GA(T12, T90, halfc17_out_ga(T78, s(s(T88)))) → U41_GA(T12, T90, p139_in_gaga(T88, X286, s(s(s(s(s(s(s(0))))))), T90))
U40_GA(T12, T90, halfc17_out_ga(T78, s(s(T88)))) → P139_IN_GAGA(T88, X286, s(s(s(s(s(s(s(0))))))), T90)
P139_IN_GAGA(T88, X286, T92, T90) → U3_GAGA(T88, X286, T92, T90, half17_in_ga(T88, X286))
P139_IN_GAGA(T88, X286, T92, T90) → HALF17_IN_GA(T88, X286)
P139_IN_GAGA(T88, s(s(T114)), T115, T117) → U4_GAGA(T88, T114, T115, T117, halfc17_in_ga(T88, s(s(T114))))
U4_GAGA(T88, T114, T115, T117, halfc17_out_ga(T88, s(s(T114)))) → U5_GAGA(T88, T114, T115, T117, p139_in_gaga(T114, X323, s(T115), T117))
U4_GAGA(T88, T114, T115, T117, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, X323, s(T115), T117)

The TRS R consists of the following rules:

halfc17_in_ga(T20, s(X46)) → U44_ga(T20, X46, halfc22_in_ga(T20, X46))
halfc22_in_ga(0, 0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0), 0) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23)), s(X55)) → U43_ga(T23, X55, halfc22_in_ga(T23, X55))
U43_ga(T23, X55, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, X46, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
half17_in_ga(x1, x2)  =  half17_in_ga(x1)
half22_in_ga(x1, x2)  =  half22_in_ga(x1)
halfc17_in_ga(x1, x2)  =  halfc17_in_ga(x1)
U44_ga(x1, x2, x3)  =  U44_ga(x1, x3)
halfc22_in_ga(x1, x2)  =  halfc22_in_ga(x1)
0  =  0
halfc22_out_ga(x1, x2)  =  halfc22_out_ga(x1, x2)
U43_ga(x1, x2, x3)  =  U43_ga(x1, x3)
halfc17_out_ga(x1, x2)  =  halfc17_out_ga(x1, x2)
p139_in_gaga(x1, x2, x3, x4)  =  p139_in_gaga(x1, x3)
LOG21_IN_GA(x1, x2)  =  LOG21_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
HALF17_IN_GA(x1, x2)  =  HALF17_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
HALF22_IN_GA(x1, x2)  =  HALF22_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U9_GA(x1, x2, x3)  =  U9_GA(x1, x3)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
U15_GA(x1, x2, x3)  =  U15_GA(x1, x3)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
U17_GA(x1, x2, x3)  =  U17_GA(x1, x3)
U18_GA(x1, x2, x3)  =  U18_GA(x1, x3)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3)  =  U22_GA(x1, x3)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
U24_GA(x1, x2, x3)  =  U24_GA(x1, x3)
U25_GA(x1, x2, x3)  =  U25_GA(x1, x3)
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)
U29_GA(x1, x2, x3)  =  U29_GA(x1, x3)
U30_GA(x1, x2, x3)  =  U30_GA(x1, x3)
U31_GA(x1, x2, x3)  =  U31_GA(x1, x3)
U32_GA(x1, x2, x3)  =  U32_GA(x1, x3)
U33_GA(x1, x2, x3)  =  U33_GA(x1, x3)
U34_GA(x1, x2, x3)  =  U34_GA(x1, x3)
U35_GA(x1, x2, x3)  =  U35_GA(x1, x3)
U36_GA(x1, x2, x3)  =  U36_GA(x1, x3)
U37_GA(x1, x2, x3)  =  U37_GA(x1, x3)
U38_GA(x1, x2, x3)  =  U38_GA(x1, x3)
U39_GA(x1, x2, x3)  =  U39_GA(x1, x3)
U40_GA(x1, x2, x3)  =  U40_GA(x1, x3)
U41_GA(x1, x2, x3)  =  U41_GA(x1, x3)
P139_IN_GAGA(x1, x2, x3, x4)  =  P139_IN_GAGA(x1, x3)
U3_GAGA(x1, x2, x3, x4, x5)  =  U3_GAGA(x1, x3, x5)
U4_GAGA(x1, x2, x3, x4, x5)  =  U4_GAGA(x1, x3, x5)
U5_GAGA(x1, x2, x3, x4, x5)  =  U5_GAGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 50 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

HALF22_IN_GA(s(s(T23)), s(X55)) → HALF22_IN_GA(T23, X55)

The TRS R consists of the following rules:

halfc17_in_ga(T20, s(X46)) → U44_ga(T20, X46, halfc22_in_ga(T20, X46))
halfc22_in_ga(0, 0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0), 0) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23)), s(X55)) → U43_ga(T23, X55, halfc22_in_ga(T23, X55))
U43_ga(T23, X55, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, X46, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
halfc17_in_ga(x1, x2)  =  halfc17_in_ga(x1)
U44_ga(x1, x2, x3)  =  U44_ga(x1, x3)
halfc22_in_ga(x1, x2)  =  halfc22_in_ga(x1)
0  =  0
halfc22_out_ga(x1, x2)  =  halfc22_out_ga(x1, x2)
U43_ga(x1, x2, x3)  =  U43_ga(x1, x3)
halfc17_out_ga(x1, x2)  =  halfc17_out_ga(x1, x2)
HALF22_IN_GA(x1, x2)  =  HALF22_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:

HALF22_IN_GA(s(s(T23)), s(X55)) → HALF22_IN_GA(T23, X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
HALF22_IN_GA(x1, x2)  =  HALF22_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:

HALF22_IN_GA(s(s(T23))) → HALF22_IN_GA(T23)

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:

• HALF22_IN_GA(s(s(T23))) → HALF22_IN_GA(T23)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

P139_IN_GAGA(T88, s(s(T114)), T115, T117) → U4_GAGA(T88, T114, T115, T117, halfc17_in_ga(T88, s(s(T114))))
U4_GAGA(T88, T114, T115, T117, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, X323, s(T115), T117)

The TRS R consists of the following rules:

halfc17_in_ga(T20, s(X46)) → U44_ga(T20, X46, halfc22_in_ga(T20, X46))
halfc22_in_ga(0, 0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0), 0) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23)), s(X55)) → U43_ga(T23, X55, halfc22_in_ga(T23, X55))
U43_ga(T23, X55, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, X46, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
halfc17_in_ga(x1, x2)  =  halfc17_in_ga(x1)
U44_ga(x1, x2, x3)  =  U44_ga(x1, x3)
halfc22_in_ga(x1, x2)  =  halfc22_in_ga(x1)
0  =  0
halfc22_out_ga(x1, x2)  =  halfc22_out_ga(x1, x2)
U43_ga(x1, x2, x3)  =  U43_ga(x1, x3)
halfc17_out_ga(x1, x2)  =  halfc17_out_ga(x1, x2)
P139_IN_GAGA(x1, x2, x3, x4)  =  P139_IN_GAGA(x1, x3)
U4_GAGA(x1, x2, x3, x4, x5)  =  U4_GAGA(x1, x3, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, halfc17_in_ga(T88))
U4_GAGA(T88, T115, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, s(T115))

The TRS R consists of the following rules:

halfc17_in_ga(T20) → U44_ga(T20, halfc22_in_ga(T20))
halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The set Q consists of the following terms:

halfc17_in_ga(x0)
halfc22_in_ga(x0)
U43_ga(x0, x1)
U44_ga(x0, x1)

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

(17) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, halfc17_in_ga(T88)) at position [2] we obtained the following new rules [LPAR04]:

P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, U44_ga(T88, halfc22_in_ga(T88)))

(18) Obligation:

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

U4_GAGA(T88, T115, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, s(T115))
P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, U44_ga(T88, halfc22_in_ga(T88)))

The TRS R consists of the following rules:

halfc17_in_ga(T20) → U44_ga(T20, halfc22_in_ga(T20))
halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))

The set Q consists of the following terms:

halfc17_in_ga(x0)
halfc22_in_ga(x0)
U43_ga(x0, x1)
U44_ga(x0, x1)

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

(19) 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.

(20) Obligation:

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

U4_GAGA(T88, T115, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, s(T115))
P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, U44_ga(T88, halfc22_in_ga(T88)))

The TRS R consists of the following rules:

halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))

The set Q consists of the following terms:

halfc17_in_ga(x0)
halfc22_in_ga(x0)
U43_ga(x0, x1)
U44_ga(x0, x1)

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

(21) 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].

halfc17_in_ga(x0)

(22) Obligation:

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

U4_GAGA(T88, T115, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, s(T115))
P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, U44_ga(T88, halfc22_in_ga(T88)))

The TRS R consists of the following rules:

halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))

The set Q consists of the following terms:

halfc22_in_ga(x0)
U43_ga(x0, x1)
U44_ga(x0, x1)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U4_GAGA(T88, T115, halfc17_out_ga(T88, s(s(T114)))) → P139_IN_GAGA(T114, s(T115))

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

POL(0) = 0   
POL(P139_IN_GAGA(x₁, x₂)) = 1 + x₁   
POL(U43_ga(x₁, x₂)) = 1 + x₂   
POL(U44_ga(x₁, x₂)) = x₂   
POL(U4_GAGA(x₁, x₂, x₃)) = x₃   
POL(halfc17_out_ga(x₁, x₂)) = x₂   
POL(halfc22_in_ga(x₁)) = 1 + x₁   
POL(halfc22_out_ga(x₁, x₂)) = 1 + x₂   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))

(24) Obligation:

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

P139_IN_GAGA(T88, T115) → U4_GAGA(T88, T115, U44_ga(T88, halfc22_in_ga(T88)))

The TRS R consists of the following rules:

halfc22_in_ga(0) → halfc22_out_ga(0, 0)
halfc22_in_ga(s(0)) → halfc22_out_ga(s(0), 0)
halfc22_in_ga(s(s(T23))) → U43_ga(T23, halfc22_in_ga(T23))
U44_ga(T20, halfc22_out_ga(T20, X46)) → halfc17_out_ga(T20, s(X46))
U43_ga(T23, halfc22_out_ga(T23, X55)) → halfc22_out_ga(s(s(T23)), s(X55))

The set Q consists of the following terms:

halfc22_in_ga(x0)
U43_ga(x0, x1)
U44_ga(x0, x1)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE