Left Termination proof of /tmp/tmpie3C8v/log2a.pl

Left Termination of the query pattern
log2(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 285 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 262 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 PiDPToQDPProof (⇒, 0 ms)

        ↳16 QDP

        ↳17 Rewriting (⇔, 0 ms)

        ↳18 QDP

        ↳19 UsableRulesProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QReductionProof (⇔, 0 ms)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔, 26 ms)

        ↳24 QDP

        ↳25 DependencyGraphProof (⇔, 0 ms)

        ↳26 TRUE

(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).

Query: log2(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="D: 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)), X26), log2(X26, s(0), T14))\n\nT12 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="B: half(s(s(T12)), X26)\n\nT12 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: log2(T15, s(0), T14)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 4 | half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 5 | half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 5 | half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: half(s(s(T12)), X26), SCOPE: 3, CLAUSE: 6\n\nT12 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="A: half(T18, X35)\n\nT18 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: half(T18, X35), SCOPE: 4, CLAUSE: 4 | half(T18, X35), SCOPE: 4, CLAUSE: 5 | half(T18, X35), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: half(T18, X35), SCOPE: 4, CLAUSE: 4\n\nT18 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: half(T18, X35), SCOPE: 4, CLAUSE: 5 | half(T18, X35), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nX35 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(T18, X35), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: half(T18, X35), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nX35 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(T21, X44)\n\nT21 is ground\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 1 | log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 2 | log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 1\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 2 | log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT15 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(T15, s(0), T14), SCOPE: 5, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: log2(T15, s(0), T14), SCOPE: 5, CLAUSE: 3\n\nT15 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(T26)), X67), log2(X67, s(s(0)), T28))\n\nT26 is ground\nX67 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(T26)), X67)\n\nT26 is ground\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: log2(T29, s(s(0)), T28)\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 1 | log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 2 | log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 3\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 1\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 2 | log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 3\n\nT29 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(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 2\n\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: log2(T29, s(s(0)), T28), SCOPE: 6, CLAUSE: 3\n\nT29 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(T34)), X90), log2(X90, s(s(s(0))), T36))\n\nT34 is ground\nX90 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(T34)), X90)\n\nT34 is ground\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: log2(T37, s(s(s(0))), T36)\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 1 | log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 2 | log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 3\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 1\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 2 | log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 3\n\nT37 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(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 2\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: log2(T37, s(s(s(0))), T36), SCOPE: 7, CLAUSE: 3\n\nT37 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(T42)), X113), log2(X113, s(s(s(s(0)))), T44))\n\nT42 is ground\nX113 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(T42)), X113)\n\nT42 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: log2(T45, s(s(s(s(0)))), T44)\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 1 | log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 2 | log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 3\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 1\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 2 | log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 3\n\nT45 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(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 2\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: log2(T45, s(s(s(s(0)))), T44), SCOPE: 8, CLAUSE: 3\n\nT45 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(T50)), X136), log2(X136, s(s(s(s(s(0))))), T52))\n\nT50 is ground\nX136 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(T50)), X136)\n\nT50 is ground\nX136 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: log2(T53, s(s(s(s(s(0))))), T52)\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 1 | log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 2 | log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 3\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 1\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 2 | log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 3\n\nT53 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(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 2\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: log2(T53, s(s(s(s(s(0))))), T52), SCOPE: 9, CLAUSE: 3\n\nT53 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(T58)), X159), log2(X159, s(s(s(s(s(s(0)))))), T60))\n\nT58 is ground\nX159 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(T58)), X159)\n\nT58 is ground\nX159 is free", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: log2(T61, s(s(s(s(s(s(0)))))), T60)\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 1 | log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 2 | log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 3\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 1\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 2 | log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 3\n\nT61 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(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 2\n\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: log2(T61, s(s(s(s(s(s(0)))))), T60), SCOPE: 10, CLAUSE: 3\n\nT61 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(T66)), X182), log2(X182, s(s(s(s(s(s(s(0))))))), T68))\n\nT66 is ground\nX182 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(T66)), X182)\n\nT66 is ground\nX182 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: log2(T69, s(s(s(s(s(s(s(0))))))), T68)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 1 | log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 2 | log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 3\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 1\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 2 | log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 3\n\nT69 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(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: log2(T69, s(s(s(s(s(s(s(0))))))), T68), SCOPE: 11, CLAUSE: 3\n\nT69 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(T74)), X205), log2(X205, s(s(s(s(s(s(s(s(0)))))))), T76))\n\nT74 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="C: ','(half(s(s(T74)), X205), log2(X205, s(T77), T76))\n\nT74 is ground\nT77 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: half(s(s(T74)), X205)\n\nT74 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: log2(T78, s(T77), T76)\n\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 1 | log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 2 | log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 3\n\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 1\n\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 2 | log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 3\n\nT77 is ground\nT78 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(T78, s(T77), T76), SCOPE: 12, CLAUSE: 2\n\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: log2(T78, s(T77), T76), SCOPE: 12, CLAUSE: 3\n\nT77 is ground\nT78 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(T97)), X228), log2(X228, s(s(T98)), T100))\n\nT97 is ground\nT98 is ground\nX228 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 substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT5 -> 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 = lightgreen];
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), X15, X15).\nand substitutionT5 -> s(0),\nX15 -> 0,\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 165 [arrowhead = none ];
165 -> 12;

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

167 [label="EVAL with clause\nlog2(s(s(X23)), X24, X25) :- ','(half(s(s(X23)), X26), log2(X26, s(X24), X25)).\nand substitutionX23 -> T12,\nT5 -> s(s(T12)),\nX24 -> 0,\nT7 -> T14,\nX25 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 167 [arrowhead = none ];
167 -> 15;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT12 is ground\nT15 is ground\nreplacements:X26 -> T15", 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 = lightgreen];
19 -> 174 [arrowhead = none ];
174 -> 20;

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

176 [label="ONLY EVAL with clause\nhalf(s(s(X33)), s(X34)) :- half(X33, X34).\nand substitutionT12 -> T18,\nX33 -> T18,\nX34 -> X35,\nX26 -> s(X35)", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT18 -> 0,\nX35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 180 [arrowhead = none ];
180 -> 26;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT18 -> s(0),\nX35 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 185 [arrowhead = none ];
185 -> 31;

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

187 [label="EVAL with clause\nhalf(s(s(X42)), s(X43)) :- half(X42, X43).\nand substitutionX42 -> T21,\nT18 -> s(s(T21)),\nX43 -> X44,\nX35 -> s(X44)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 187 [arrowhead = none ];
187 -> 34;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT18 -> T21\nX35 -> X44", 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, X51, X51).\nand substitutionT15 -> 0,\nX51 -> 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 = lightgreen];
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), X56, X56).\nand substitutionT15 -> s(0),\nX56 -> 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 = lightgreen];
42 -> 199 [arrowhead = none ];
199 -> 45;

200 [label="EVAL with clause\nlog2(s(s(X64)), X65, X66) :- ','(half(s(s(X64)), X67), log2(X67, s(X65), X66)).\nand substitutionX64 -> T26,\nT15 -> s(s(T26)),\nX65 -> s(0),\nT14 -> T28,\nX66 -> T28,\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 200 [arrowhead = none ];
200 -> 47;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT26 is ground\nT29 is ground\nreplacements:X67 -> T29", fontsize=14, style = filled, fillcolor = violet];
47 -> 204 [arrowhead = none ];
204 -> 50;

205 [label="INSTANCE with matching:\nT12 -> T26\nX26 -> X67", 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, X74, X74).\nand substitutionT29 -> 0,\nX74 -> s(s(0)),\nT28 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 209 [arrowhead = none ];
209 -> 54;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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), X79, X79).\nand substitutionT29 -> s(0),\nX79 -> s(s(0)),\nT28 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 214 [arrowhead = none ];
214 -> 59;

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

216 [label="EVAL with clause\nlog2(s(s(X87)), X88, X89) :- ','(half(s(s(X87)), X90), log2(X90, s(X88), X89)).\nand substitutionX87 -> T34,\nT29 -> s(s(T34)),\nX88 -> s(s(0)),\nT28 -> T36,\nX89 -> T36,\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 216 [arrowhead = none ];
216 -> 62;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT34 is ground\nT37 is ground\nreplacements:X90 -> T37", fontsize=14, style = filled, fillcolor = violet];
62 -> 220 [arrowhead = none ];
220 -> 65;

221 [label="INSTANCE with matching:\nT12 -> T34\nX26 -> X90", 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, X97, X97).\nand substitutionT37 -> 0,\nX97 -> s(s(s(0))),\nT36 -> 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 = lightgreen];
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), X102, X102).\nand substitutionT37 -> s(0),\nX102 -> s(s(s(0))),\nT36 -> 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 = lightgreen];
72 -> 231 [arrowhead = none ];
231 -> 75;

232 [label="EVAL with clause\nlog2(s(s(X110)), X111, X112) :- ','(half(s(s(X110)), X113), log2(X113, s(X111), X112)).\nand substitutionX110 -> T42,\nT37 -> s(s(T42)),\nX111 -> s(s(s(0))),\nT36 -> T44,\nX112 -> T44,\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 232 [arrowhead = none ];
232 -> 77;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT42 is ground\nT45 is ground\nreplacements:X113 -> T45", fontsize=14, style = filled, fillcolor = violet];
77 -> 236 [arrowhead = none ];
236 -> 80;

237 [label="INSTANCE with matching:\nT12 -> T42\nX26 -> X113", 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, X120, X120).\nand substitutionT45 -> 0,\nX120 -> s(s(s(s(0)))),\nT44 -> 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 = lightgreen];
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), X125, X125).\nand substitutionT45 -> s(0),\nX125 -> s(s(s(s(0)))),\nT44 -> 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 = lightgreen];
87 -> 247 [arrowhead = none ];
247 -> 90;

248 [label="EVAL with clause\nlog2(s(s(X133)), X134, X135) :- ','(half(s(s(X133)), X136), log2(X136, s(X134), X135)).\nand substitutionX133 -> T50,\nT45 -> s(s(T50)),\nX134 -> s(s(s(s(0)))),\nT44 -> T52,\nX135 -> T52,\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 248 [arrowhead = none ];
248 -> 92;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT50 is ground\nT53 is ground\nreplacements:X136 -> T53", fontsize=14, style = filled, fillcolor = violet];
92 -> 252 [arrowhead = none ];
252 -> 95;

253 [label="INSTANCE with matching:\nT12 -> T50\nX26 -> X136", 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, X143, X143).\nand substitutionT53 -> 0,\nX143 -> s(s(s(s(s(0))))),\nT52 -> 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 = lightgreen];
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), X148, X148).\nand substitutionT53 -> s(0),\nX148 -> s(s(s(s(s(0))))),\nT52 -> 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 = lightgreen];
102 -> 263 [arrowhead = none ];
263 -> 105;

264 [label="EVAL with clause\nlog2(s(s(X156)), X157, X158) :- ','(half(s(s(X156)), X159), log2(X159, s(X157), X158)).\nand substitutionX156 -> T58,\nT53 -> s(s(T58)),\nX157 -> s(s(s(s(s(0))))),\nT52 -> T60,\nX158 -> T60,\nT59 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 264 [arrowhead = none ];
264 -> 107;

265 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT58 is ground\nT61 is ground\nreplacements:X159 -> T61", fontsize=14, style = filled, fillcolor = violet];
107 -> 268 [arrowhead = none ];
268 -> 110;

269 [label="INSTANCE with matching:\nT12 -> T58\nX26 -> X159", 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, X166, X166).\nand substitutionT61 -> 0,\nX166 -> s(s(s(s(s(s(0)))))),\nT60 -> 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 = lightgreen];
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), X171, X171).\nand substitutionT61 -> s(0),\nX171 -> s(s(s(s(s(s(0)))))),\nT60 -> 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 = lightgreen];
117 -> 279 [arrowhead = none ];
279 -> 120;

280 [label="EVAL with clause\nlog2(s(s(X179)), X180, X181) :- ','(half(s(s(X179)), X182), log2(X182, s(X180), X181)).\nand substitutionX179 -> T66,\nT61 -> s(s(T66)),\nX180 -> s(s(s(s(s(s(0)))))),\nT60 -> T68,\nX181 -> T68,\nT67 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 280 [arrowhead = none ];
280 -> 122;

281 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT66 is ground\nT69 is ground\nreplacements:X182 -> T69", fontsize=14, style = filled, fillcolor = violet];
122 -> 284 [arrowhead = none ];
284 -> 125;

285 [label="INSTANCE with matching:\nT12 -> T66\nX26 -> X182", 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, X189, X189).\nand substitutionT69 -> 0,\nX189 -> s(s(s(s(s(s(s(0))))))),\nT68 -> 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 = lightgreen];
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), X194, X194).\nand substitutionT69 -> s(0),\nX194 -> s(s(s(s(s(s(s(0))))))),\nT68 -> 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 = lightgreen];
132 -> 295 [arrowhead = none ];
295 -> 135;

296 [label="EVAL with clause\nlog2(s(s(X202)), X203, X204) :- ','(half(s(s(X202)), X205), log2(X205, s(X203), X204)).\nand substitutionX202 -> T74,\nT69 -> s(s(T74)),\nX203 -> s(s(s(s(s(s(s(0))))))),\nT68 -> T76,\nX204 -> T76,\nT75 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
133 -> 296 [arrowhead = none ];
296 -> 137;

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

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

299 [label="GENERALIZATION\nT77 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT77 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:\nT74 is ground\nT78 is ground\nreplacements:X205 -> T78", fontsize=14, style = filled, fillcolor = violet];
139 -> 301 [arrowhead = none ];
301 -> 141;

302 [label="INSTANCE with matching:\nT12 -> T74\nX26 -> X205", 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, X212, X212).\nand substitutionT78 -> 0,\nT77 -> T85,\nX212 -> s(T85),\nT76 -> s(T85)", fontsize=14, style = filled, fillcolor = lightblue];
143 -> 306 [arrowhead = none ];
306 -> 145;

307 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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), X217, X217).\nand substitutionT78 -> s(0),\nT77 -> T90,\nX217 -> s(T90),\nT76 -> s(T90)", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 311 [arrowhead = none ];
311 -> 150;

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

313 [label="EVAL with clause\nlog2(s(s(X225)), X226, X227) :- ','(half(s(s(X225)), X228), log2(X228, s(X226), X227)).\nand substitutionX225 -> T97,\nT78 -> s(s(T97)),\nT77 -> T98,\nX226 -> s(T98),\nT76 -> T100,\nX227 -> T100,\nT99 -> T100", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 313 [arrowhead = none ];
313 -> 153;

314 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT74 -> T97\nX205 -> X228\nT77 -> s(T98)\nT76 -> T100", fontsize=14, style = filled, fillcolor = lightgrey];
153 -> 316 [arrowhead = none , style=dashed];
316 -> 139[style=dashed];

}

(2) Obligation:

Triples:

halfA(s(s(X1)), s(X2)) :- halfA(X1, X2).
halfB(X1, s(X2)) :- halfA(X1, X2).
pC(X1, X2, X3, X4) :- halfB(X1, X2).
pC(X1, s(s(X2)), X3, X4) :- ','(halfcB(X1, s(s(X2))), pC(X2, X5, s(X3), X4)).
log2D(s(s(X1)), X2) :- halfB(X1, X3).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), halfB(X3, X4)).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), halfB(X4, X5))).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), ','(halfcB(X4, s(s(X5))), halfB(X5, X6)))).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), ','(halfcB(X4, s(s(X5))), ','(halfcB(X5, s(s(X6))), halfB(X6, X7))))).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), ','(halfcB(X4, s(s(X5))), ','(halfcB(X5, s(s(X6))), ','(halfcB(X6, s(s(X7))), halfB(X7, X8)))))).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), ','(halfcB(X4, s(s(X5))), ','(halfcB(X5, s(s(X6))), ','(halfcB(X6, s(s(X7))), ','(halfcB(X7, s(s(X8))), halfB(X8, X9))))))).
log2D(s(s(X1)), X2) :- ','(halfcB(X1, s(s(X3))), ','(halfcB(X3, s(s(X4))), ','(halfcB(X4, s(s(X5))), ','(halfcB(X5, s(s(X6))), ','(halfcB(X6, s(s(X7))), ','(halfcB(X7, s(s(X8))), ','(halfcB(X8, s(s(X9))), pC(X9, X10, s(s(s(s(s(s(s(0))))))), X2)))))))).

Clauses:

halfcA(0, 0).
halfcA(s(0), 0).
halfcA(s(s(X1)), s(X2)) :- halfcA(X1, X2).
halfcB(X1, s(X2)) :- halfcA(X1, X2).
qcC(X1, 0, X2, s(X2)) :- halfcB(X1, 0).
qcC(X1, s(0), X2, s(X2)) :- halfcB(X1, s(0)).
qcC(X1, s(s(X2)), X3, X4) :- ','(halfcB(X1, s(s(X2))), qcC(X2, X5, s(X3), X4)).

Afs:

log2D(x1, x2) = log2D(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
log2D_in: (b,f)
halfB_in: (b,f)
halfA_in: (b,f)
halfcB_in: (b,f)
halfcA_in: (b,f)
pC_in: (b,f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

LOG2D_IN_GA(s(s(X1)), X2) → U6_GA(X1, X2, halfB_in_ga(X1, X3))
LOG2D_IN_GA(s(s(X1)), X2) → HALFB_IN_GA(X1, X3)
HALFB_IN_GA(X1, s(X2)) → U2_GA(X1, X2, halfA_in_ga(X1, X2))
HALFB_IN_GA(X1, s(X2)) → HALFA_IN_GA(X1, X2)
HALFA_IN_GA(s(s(X1)), s(X2)) → U1_GA(X1, X2, halfA_in_ga(X1, X2))
HALFA_IN_GA(s(s(X1)), s(X2)) → HALFA_IN_GA(X1, X2)
LOG2D_IN_GA(s(s(X1)), X2) → U7_GA(X1, X2, halfcB_in_ga(X1, s(s(X3))))
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → U8_GA(X1, X2, halfB_in_ga(X3, X4))
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → HALFB_IN_GA(X3, X4)
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → U9_GA(X1, X2, halfcB_in_ga(X3, s(s(X4))))
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → U10_GA(X1, X2, halfB_in_ga(X4, X5))
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → HALFB_IN_GA(X4, X5)
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → U11_GA(X1, X2, halfcB_in_ga(X4, s(s(X5))))
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → U12_GA(X1, X2, halfB_in_ga(X5, X6))
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → HALFB_IN_GA(X5, X6)
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → U13_GA(X1, X2, halfcB_in_ga(X5, s(s(X6))))
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → U14_GA(X1, X2, halfB_in_ga(X6, X7))
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → HALFB_IN_GA(X6, X7)
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → U15_GA(X1, X2, halfcB_in_ga(X6, s(s(X7))))
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → U16_GA(X1, X2, halfB_in_ga(X7, X8))
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → HALFB_IN_GA(X7, X8)
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → U17_GA(X1, X2, halfcB_in_ga(X7, s(s(X8))))
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → U18_GA(X1, X2, halfB_in_ga(X8, X9))
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → HALFB_IN_GA(X8, X9)
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → U19_GA(X1, X2, halfcB_in_ga(X8, s(s(X9))))
U19_GA(X1, X2, halfcB_out_ga(X8, s(s(X9)))) → U20_GA(X1, X2, pC_in_gaga(X9, X10, s(s(s(s(s(s(s(0))))))), X2))
U19_GA(X1, X2, halfcB_out_ga(X8, s(s(X9)))) → PC_IN_GAGA(X9, X10, s(s(s(s(s(s(s(0))))))), X2)
PC_IN_GAGA(X1, X2, X3, X4) → U3_GAGA(X1, X2, X3, X4, halfB_in_ga(X1, X2))
PC_IN_GAGA(X1, X2, X3, X4) → HALFB_IN_GA(X1, X2)
PC_IN_GAGA(X1, s(s(X2)), X3, X4) → U4_GAGA(X1, X2, X3, X4, halfcB_in_ga(X1, s(s(X2))))
U4_GAGA(X1, X2, X3, X4, halfcB_out_ga(X1, s(s(X2)))) → U5_GAGA(X1, X2, X3, X4, pC_in_gaga(X2, X5, s(X3), X4))
U4_GAGA(X1, X2, X3, X4, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, X5, s(X3), X4)

The TRS R consists of the following rules:

halfcB_in_ga(X1, s(X2)) → U23_ga(X1, X2, halfcA_in_ga(X1, X2))
halfcA_in_ga(0, 0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0), 0) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1)), s(X2)) → U22_ga(X1, X2, halfcA_in_ga(X1, X2))
U22_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
halfB_in_ga(x1, x2) = halfB_in_ga(x1)
halfA_in_ga(x1, x2) = halfA_in_ga(x1)
halfcB_in_ga(x1, x2) = halfcB_in_ga(x1)
U23_ga(x1, x2, x3) = U23_ga(x1, x3)
halfcA_in_ga(x1, x2) = halfcA_in_ga(x1)
0 = 0
halfcA_out_ga(x1, x2) = halfcA_out_ga(x1, x2)
U22_ga(x1, x2, x3) = U22_ga(x1, x3)
halfcB_out_ga(x1, x2) = halfcB_out_ga(x1, x2)
pC_in_gaga(x1, x2, x3, x4) = pC_in_gaga(x1, x3)
LOG2D_IN_GA(x1, x2) = LOG2D_IN_GA(x1)
U6_GA(x1, x2, x3) = U6_GA(x1, x3)
HALFB_IN_GA(x1, x2) = HALFB_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
HALFA_IN_GA(x1, x2) = HALFA_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)
PC_IN_GAGA(x1, x2, x3, x4) = PC_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:

LOG2D_IN_GA(s(s(X1)), X2) → U6_GA(X1, X2, halfB_in_ga(X1, X3))
LOG2D_IN_GA(s(s(X1)), X2) → HALFB_IN_GA(X1, X3)
HALFB_IN_GA(X1, s(X2)) → U2_GA(X1, X2, halfA_in_ga(X1, X2))
HALFB_IN_GA(X1, s(X2)) → HALFA_IN_GA(X1, X2)
HALFA_IN_GA(s(s(X1)), s(X2)) → U1_GA(X1, X2, halfA_in_ga(X1, X2))
HALFA_IN_GA(s(s(X1)), s(X2)) → HALFA_IN_GA(X1, X2)
LOG2D_IN_GA(s(s(X1)), X2) → U7_GA(X1, X2, halfcB_in_ga(X1, s(s(X3))))
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → U8_GA(X1, X2, halfB_in_ga(X3, X4))
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → HALFB_IN_GA(X3, X4)
U7_GA(X1, X2, halfcB_out_ga(X1, s(s(X3)))) → U9_GA(X1, X2, halfcB_in_ga(X3, s(s(X4))))
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → U10_GA(X1, X2, halfB_in_ga(X4, X5))
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → HALFB_IN_GA(X4, X5)
U9_GA(X1, X2, halfcB_out_ga(X3, s(s(X4)))) → U11_GA(X1, X2, halfcB_in_ga(X4, s(s(X5))))
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → U12_GA(X1, X2, halfB_in_ga(X5, X6))
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → HALFB_IN_GA(X5, X6)
U11_GA(X1, X2, halfcB_out_ga(X4, s(s(X5)))) → U13_GA(X1, X2, halfcB_in_ga(X5, s(s(X6))))
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → U14_GA(X1, X2, halfB_in_ga(X6, X7))
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → HALFB_IN_GA(X6, X7)
U13_GA(X1, X2, halfcB_out_ga(X5, s(s(X6)))) → U15_GA(X1, X2, halfcB_in_ga(X6, s(s(X7))))
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → U16_GA(X1, X2, halfB_in_ga(X7, X8))
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → HALFB_IN_GA(X7, X8)
U15_GA(X1, X2, halfcB_out_ga(X6, s(s(X7)))) → U17_GA(X1, X2, halfcB_in_ga(X7, s(s(X8))))
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → U18_GA(X1, X2, halfB_in_ga(X8, X9))
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → HALFB_IN_GA(X8, X9)
U17_GA(X1, X2, halfcB_out_ga(X7, s(s(X8)))) → U19_GA(X1, X2, halfcB_in_ga(X8, s(s(X9))))
U19_GA(X1, X2, halfcB_out_ga(X8, s(s(X9)))) → U20_GA(X1, X2, pC_in_gaga(X9, X10, s(s(s(s(s(s(s(0))))))), X2))
U19_GA(X1, X2, halfcB_out_ga(X8, s(s(X9)))) → PC_IN_GAGA(X9, X10, s(s(s(s(s(s(s(0))))))), X2)
PC_IN_GAGA(X1, X2, X3, X4) → U3_GAGA(X1, X2, X3, X4, halfB_in_ga(X1, X2))
PC_IN_GAGA(X1, X2, X3, X4) → HALFB_IN_GA(X1, X2)
PC_IN_GAGA(X1, s(s(X2)), X3, X4) → U4_GAGA(X1, X2, X3, X4, halfcB_in_ga(X1, s(s(X2))))
U4_GAGA(X1, X2, X3, X4, halfcB_out_ga(X1, s(s(X2)))) → U5_GAGA(X1, X2, X3, X4, pC_in_gaga(X2, X5, s(X3), X4))
U4_GAGA(X1, X2, X3, X4, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, X5, s(X3), X4)

The TRS R consists of the following rules:

halfcB_in_ga(X1, s(X2)) → U23_ga(X1, X2, halfcA_in_ga(X1, X2))
halfcA_in_ga(0, 0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0), 0) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1)), s(X2)) → U22_ga(X1, X2, halfcA_in_ga(X1, X2))
U22_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 29 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

HALFA_IN_GA(s(s(X1)), s(X2)) → HALFA_IN_GA(X1, X2)

The TRS R consists of the following rules:

halfcB_in_ga(X1, s(X2)) → U23_ga(X1, X2, halfcA_in_ga(X1, X2))
halfcA_in_ga(0, 0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0), 0) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1)), s(X2)) → U22_ga(X1, X2, halfcA_in_ga(X1, X2))
U22_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
halfcB_in_ga(x1, x2) = halfcB_in_ga(x1)
U23_ga(x1, x2, x3) = U23_ga(x1, x3)
halfcA_in_ga(x1, x2) = halfcA_in_ga(x1)
0 = 0
halfcA_out_ga(x1, x2) = halfcA_out_ga(x1, x2)
U22_ga(x1, x2, x3) = U22_ga(x1, x3)
halfcB_out_ga(x1, x2) = halfcB_out_ga(x1, x2)
HALFA_IN_GA(x1, x2) = HALFA_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:

HALFA_IN_GA(s(s(X1)), s(X2)) → HALFA_IN_GA(X1, X2)

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

HALFA_IN_GA(s(s(X1))) → HALFA_IN_GA(X1)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(13) YES

(14) Obligation:

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

PC_IN_GAGA(X1, s(s(X2)), X3, X4) → U4_GAGA(X1, X2, X3, X4, halfcB_in_ga(X1, s(s(X2))))
U4_GAGA(X1, X2, X3, X4, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, X5, s(X3), X4)

The TRS R consists of the following rules:

halfcB_in_ga(X1, s(X2)) → U23_ga(X1, X2, halfcA_in_ga(X1, X2))
halfcA_in_ga(0, 0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0), 0) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1)), s(X2)) → U22_ga(X1, X2, halfcA_in_ga(X1, X2))
U22_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, X2, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
halfcB_in_ga(x1, x2) = halfcB_in_ga(x1)
U23_ga(x1, x2, x3) = U23_ga(x1, x3)
halfcA_in_ga(x1, x2) = halfcA_in_ga(x1)
0 = 0
halfcA_out_ga(x1, x2) = halfcA_out_ga(x1, x2)
U22_ga(x1, x2, x3) = U22_ga(x1, x3)
halfcB_out_ga(x1, x2) = halfcB_out_ga(x1, x2)
PC_IN_GAGA(x1, x2, x3, x4) = PC_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:

PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, halfcB_in_ga(X1))
U4_GAGA(X1, X3, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, s(X3))

The TRS R consists of the following rules:

halfcB_in_ga(X1) → U23_ga(X1, halfcA_in_ga(X1))
halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

The set Q consists of the following terms:

halfcB_in_ga(x0)
halfcA_in_ga(x0)
U22_ga(x0, x1)
U23_ga(x0, x1)

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

(17) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, halfcB_in_ga(X1)) at position [2] we obtained the following new rules [LPAR04]:

PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, U23_ga(X1, halfcA_in_ga(X1)))

(18) Obligation:

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

U4_GAGA(X1, X3, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, s(X3))
PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, U23_ga(X1, halfcA_in_ga(X1)))

The TRS R consists of the following rules:

halfcB_in_ga(X1) → U23_ga(X1, halfcA_in_ga(X1))
halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))

The set Q consists of the following terms:

halfcB_in_ga(x0)
halfcA_in_ga(x0)
U22_ga(x0, x1)
U23_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(X1, X3, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, s(X3))
PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, U23_ga(X1, halfcA_in_ga(X1)))

The TRS R consists of the following rules:

halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))

The set Q consists of the following terms:

halfcB_in_ga(x0)
halfcA_in_ga(x0)
U22_ga(x0, x1)
U23_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].

halfcB_in_ga(x0)

(22) Obligation:

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

U4_GAGA(X1, X3, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, s(X3))
PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, U23_ga(X1, halfcA_in_ga(X1)))

The TRS R consists of the following rules:

halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))

The set Q consists of the following terms:

halfcA_in_ga(x0)
U22_ga(x0, x1)
U23_ga(x0, x1)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U4_GAGA(X1, X3, halfcB_out_ga(X1, s(s(X2)))) → PC_IN_GAGA(X2, s(X3))

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

POL(0) = 0
POL(PC_IN_GAGA(x₁, x₂)) = 1 + x₁
POL(U22_ga(x₁, x₂)) = 1 + x₂
POL(U23_ga(x₁, x₂)) = x₂
POL(U4_GAGA(x₁, x₂, x₃)) = x₃
POL(halfcA_in_ga(x₁)) = 1 + x₁
POL(halfcA_out_ga(x₁, x₂)) = 1 + x₂
POL(halfcB_out_ga(x₁, x₂)) = x₂
POL(s(x₁)) = 1 + x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))

(24) Obligation:

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

PC_IN_GAGA(X1, X3) → U4_GAGA(X1, X3, U23_ga(X1, halfcA_in_ga(X1)))

The TRS R consists of the following rules:

halfcA_in_ga(0) → halfcA_out_ga(0, 0)
halfcA_in_ga(s(0)) → halfcA_out_ga(s(0), 0)
halfcA_in_ga(s(s(X1))) → U22_ga(X1, halfcA_in_ga(X1))
U23_ga(X1, halfcA_out_ga(X1, X2)) → halfcB_out_ga(X1, s(X2))
U22_ga(X1, halfcA_out_ga(X1, X2)) → halfcA_out_ga(s(s(X1)), s(X2))

The set Q consists of the following terms:

halfcA_in_ga(x0)
U22_ga(x0, x1)
U23_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.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) PiDPToQDPProof (SOUND transformation)

(16) Obligation:

(17) Rewriting (EQUIVALENT transformation)

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) QReductionProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) TRUE