Left Termination proof of /tmp/tmplyc7Bw/log2a-oi.pl

Left Termination of the query pattern log2_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 Rewriting (⇔)

        ↳20 QDP

        ↳21 UsableRulesProof (⇔)

        ↳22 QDP

        ↳23 QReductionProof (⇔)

        ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 DependencyGraphProof (⇔)

        ↳28 TRUE

    ↳29 PiDP

        ↳30 UsableRulesProof (⇔)

        ↳31 PiDP

        ↳32 PiDPToQDPProof (⇐)

        ↳33 QDP

        ↳34 NonTerminationProof (⇔)

        ↳35 NO

(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(a,g).

(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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: log2(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: log2(T7, 0, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: log2(T7, 0, T6), SCOPE: 2, CLAUSE: 1 | log2(T7, 0, T6), SCOPE: 2, CLAUSE: 2 | log2(T7, 0, T6), SCOPE: 2, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: log2(T7, 0, T6), SCOPE: 2, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: log2(T7, 0, T6), SCOPE: 2, CLAUSE: 2 | log2(T7, 0, T6), SCOPE: 2, CLAUSE: 3\n\nT6 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(T7, 0, T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: log2(T7, 0, T6), SCOPE: 2, CLAUSE: 3\n\nT6 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(T14)), X28), log2(X28, s(0), T13))\n\nT13 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(T14)), X28)\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: log2(T16, s(0), T13)\n\nT13 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 4 | half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 5 | half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 6\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 5 | half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 6\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: half(s(s(T14)), X28), SCOPE: 3, CLAUSE: 6\n\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: half(T22, X46)\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: half(T22, X46), SCOPE: 4, CLAUSE: 4 | half(T22, X46), SCOPE: 4, CLAUSE: 5 | half(T22, X46), SCOPE: 4, CLAUSE: 6\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: half(T22, X46), SCOPE: 4, CLAUSE: 4\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: half(T22, X46), SCOPE: 4, CLAUSE: 5 | half(T22, X46), SCOPE: 4, CLAUSE: 6\n\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(T22, X46), SCOPE: 4, CLAUSE: 5\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: half(T22, X46), SCOPE: 4, CLAUSE: 6\n\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(T26, X55)\n\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), T13), SCOPE: 5, CLAUSE: 1 | log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 2 | log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 3\n\nT13 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 1\n\nT13 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 2 | log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 3\n\nT13 is ground\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), T13), SCOPE: 5, CLAUSE: 2\n\nT13 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: log2(T16, s(0), T13), SCOPE: 5, CLAUSE: 3\n\nT13 is ground\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(T31)), X82), log2(X82, s(s(0)), T32))\n\nT31 is ground\nT32 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(T31)), X82)\n\nT31 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: log2(T34, s(s(0)), T32)\n\nT32 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 1 | log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 2 | log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 3\n\nT32 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 1\n\nT32 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 2 | log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 3\n\nT32 is ground\nT34 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(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 2\n\nT32 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: log2(T34, s(s(0)), T32), SCOPE: 6, CLAUSE: 3\n\nT32 is ground\nT34 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(T40)), X116), log2(X116, s(s(s(0))), T41))\n\nT40 is ground\nT41 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(T40)), X116)\n\nT40 is ground\nX116 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: log2(T43, s(s(s(0))), T41)\n\nT41 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 1 | log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 2 | log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 3\n\nT41 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 1\n\nT41 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 2 | log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 3\n\nT41 is ground\nT43 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(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 2\n\nT41 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: log2(T43, s(s(s(0))), T41), SCOPE: 7, CLAUSE: 3\n\nT41 is ground\nT43 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(T49)), X150), log2(X150, s(s(s(s(0)))), T50))\n\nT49 is ground\nT50 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(T49)), X150)\n\nT49 is ground\nX150 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: log2(T52, s(s(s(s(0)))), T50)\n\nT50 is ground\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\nT50 is ground\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\nT50 is ground\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\nT50 is ground\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\nT50 is ground\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\nT50 is ground\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))))), T59))\n\nT58 is ground\nT59 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(T61, s(s(s(s(s(0))))), T59)\n\nT59 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 1 | log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 2 | log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 3\n\nT59 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 1\n\nT59 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 2 | log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 3\n\nT59 is ground\nT61 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(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 2\n\nT59 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: log2(T61, s(s(s(s(s(0))))), T59), SCOPE: 9, CLAUSE: 3\n\nT59 is ground\nT61 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(T67)), X218), log2(X218, s(s(s(s(s(s(0)))))), T68))\n\nT67 is ground\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(T67)), X218)\n\nT67 is ground\nX218 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: log2(T70, s(s(s(s(s(s(0)))))), T68)\n\nT68 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 1 | log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 2 | log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 3\n\nT68 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 1\n\nT68 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 2 | log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 3\n\nT68 is ground\nT70 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(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 2\n\nT68 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: log2(T70, s(s(s(s(s(s(0)))))), T68), SCOPE: 10, CLAUSE: 3\n\nT68 is ground\nT70 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(T76)), X252), log2(X252, s(s(s(s(s(s(s(0))))))), T77))\n\nT76 is ground\nT77 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(T76)), X252)\n\nT76 is ground\nX252 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: log2(T79, s(s(s(s(s(s(s(0))))))), T77)\n\nT77 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 1 | log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 2 | log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 3\n\nT77 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 1\n\nT77 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 2 | log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 3\n\nT77 is ground\nT79 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(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 2\n\nT77 is ground\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: log2(T79, s(s(s(s(s(s(s(0))))))), T77), SCOPE: 11, CLAUSE: 3\n\nT77 is ground\nT79 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(T85)), X286), log2(X286, s(s(s(s(s(s(s(s(0)))))))), T86))\n\nT85 is ground\nT86 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(T85)), X286), log2(X286, s(T88), T86))\n\nT85 is ground\nT86 is ground\nT88 is ground\nX286 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: half(s(s(T85)), X286)\n\nT85 is ground\nX286 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: log2(T90, s(T88), T86)\n\nT86 is ground\nT88 is ground\nT90 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 1 | log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 2 | log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 3\n\nT86 is ground\nT88 is ground\nT90 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 1\n\nT86 is ground\nT88 is ground\nT90 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 2 | log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 3\n\nT86 is ground\nT88 is ground\nT90 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(T90, s(T88), T86), SCOPE: 12, CLAUSE: 2\n\nT86 is ground\nT88 is ground\nT90 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: log2(T90, s(T88), T86), SCOPE: 12, CLAUSE: 3\n\nT86 is ground\nT88 is ground\nT90 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(T110)), X323), log2(X323, s(s(T111)), T112))\n\nT110 is ground\nT111 is ground\nT112 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 -> T7\nX4 -> T7\nT2 -> T6\nX5 -> T6\nT5 -> 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\nT7 -> 0\nX10 -> 0\nT6 -> 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\nT7 -> s(0)\nX16 -> 0\nT6 -> 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 -> T14\nT7 -> s(s(T14))\nX26 -> 0\nT6 -> T13\nX27 -> T13\nT12 -> 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:\nT14 is ground\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\nT14 -> T22\nX44 -> T22\nX45 -> X46\nX28 -> s(X46)\nT21 -> T22", 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\nT22 -> 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\nT22 -> 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 -> T26\nT22 -> s(s(T26))\nX54 -> X55\nX46 -> s(X55)\nT25 -> T26", 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:\nT22 -> T26\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)\nT13 -> 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)\nT13 -> 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 -> T31\nT16 -> s(s(T31))\nX80 -> s(0)\nT13 -> T32\nX81 -> T32", 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:\nT34 is ground\nreplacements:\nX82 -> T34", fontsize=14, style = filled, fillcolor = violet];
47 -> 204 [arrowhead = none ];
204 -> 50;

205 [label="INSTANCE with matching:\nT14 -> T31\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\nT34 -> 0\nX98 -> s(s(0))\nT32 -> 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\nT34 -> s(0)\nX104 -> s(s(0))\nT32 -> 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 -> T40\nT34 -> s(s(T40))\nX114 -> s(s(0))\nT32 -> T41\nX115 -> T41", 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:\nT43 is ground\nreplacements:\nX116 -> T43", fontsize=14, style = filled, fillcolor = violet];
62 -> 220 [arrowhead = none ];
220 -> 65;

221 [label="INSTANCE with matching:\nT14 -> T40\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\nT43 -> 0\nX132 -> s(s(s(0)))\nT41 -> 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\nT43 -> s(0)\nX138 -> s(s(s(0)))\nT41 -> 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 -> T49\nT43 -> s(s(T49))\nX148 -> s(s(s(0)))\nT41 -> T50\nX149 -> 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:\nT14 -> T49\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 -> T59\nX183 -> T59", 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:\nT61 is ground\nreplacements:\nX184 -> T61", fontsize=14, style = filled, fillcolor = violet];
92 -> 252 [arrowhead = none ];
252 -> 95;

253 [label="INSTANCE with matching:\nT14 -> 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\nT61 -> 0\nX200 -> s(s(s(s(s(0)))))\nT59 -> 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\nT61 -> s(0)\nX206 -> s(s(s(s(s(0)))))\nT59 -> 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 -> T67\nT61 -> s(s(T67))\nX216 -> s(s(s(s(s(0)))))\nT59 -> T68\nX217 -> T68", 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:\nT70 is ground\nreplacements:\nX218 -> T70", fontsize=14, style = filled, fillcolor = violet];
107 -> 268 [arrowhead = none ];
268 -> 110;

269 [label="INSTANCE with matching:\nT14 -> T67\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\nT70 -> 0\nX234 -> s(s(s(s(s(s(0))))))\nT68 -> 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\nT70 -> s(0)\nX240 -> s(s(s(s(s(s(0))))))\nT68 -> 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 -> T76\nT70 -> s(s(T76))\nX250 -> s(s(s(s(s(s(0))))))\nT68 -> T77\nX251 -> T77", 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:\nT79 is ground\nreplacements:\nX252 -> T79", fontsize=14, style = filled, fillcolor = violet];
122 -> 284 [arrowhead = none ];
284 -> 125;

285 [label="INSTANCE with matching:\nT14 -> T76\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\nT79 -> 0\nX268 -> s(s(s(s(s(s(s(0)))))))\nT77 -> 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\nT79 -> s(0)\nX274 -> s(s(s(s(s(s(s(0)))))))\nT77 -> 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 -> T85\nT79 -> s(s(T85))\nX284 -> s(s(s(s(s(s(s(0)))))))\nT77 -> T86\nX285 -> T86", 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\nT88 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT88 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:\nT90 is ground\nreplacements:\nX286 -> T90", fontsize=14, style = filled, fillcolor = violet];
139 -> 301 [arrowhead = none ];
301 -> 141;

302 [label="INSTANCE with matching:\nT14 -> T85\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\nT90 -> 0\nT88 -> T98\nX305 -> s(T98)\nT86 -> s(T98)", 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\nT90 -> s(0)\nT88 -> T103\nX311 -> s(T103)\nT86 -> s(T103)", 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 -> T110\nT90 -> s(s(T110))\nT88 -> T111\nX321 -> s(T111)\nT86 -> T112\nX322 -> T112", 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:\nT85 -> T110\nX286 -> X323\nT88 -> s(T111)\nT86 -> T112", fontsize=14, style = filled, fillcolor = lightgrey];
153 -> 316 [arrowhead = none , style=dashed];
316 -> 139[style=dashed];

}

(2) Obligation:

Triples:

half22(s(s(T26)), s(X55)) :- half22(T26, X55).
half17(T22, s(X46)) :- half22(T22, X46).
p139(T85, X286, T88, T86) :- half17(T85, X286).
p139(T85, s(s(T110)), T111, T112) :- ','(halfc17(T85, s(s(T110))), p139(T110, X323, s(T111), T112)).
log21(s(s(T14)), T13) :- half17(T14, X28).
log21(s(s(T14)), T32) :- ','(halfc17(T14, s(s(T31))), half17(T31, X82)).
log21(s(s(T14)), T41) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), half17(T40, X116))).
log21(s(s(T14)), T50) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), ','(halfc17(T40, s(s(T49))), half17(T49, X150)))).
log21(s(s(T14)), T59) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), ','(halfc17(T40, s(s(T49))), ','(halfc17(T49, s(s(T58))), half17(T58, X184))))).
log21(s(s(T14)), T68) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), ','(halfc17(T40, s(s(T49))), ','(halfc17(T49, s(s(T58))), ','(halfc17(T58, s(s(T67))), half17(T67, X218)))))).
log21(s(s(T14)), T77) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), ','(halfc17(T40, s(s(T49))), ','(halfc17(T49, s(s(T58))), ','(halfc17(T58, s(s(T67))), ','(halfc17(T67, s(s(T76))), half17(T76, X252))))))).
log21(s(s(T14)), T86) :- ','(halfc17(T14, s(s(T31))), ','(halfc17(T31, s(s(T40))), ','(halfc17(T40, s(s(T49))), ','(halfc17(T49, s(s(T58))), ','(halfc17(T58, s(s(T67))), ','(halfc17(T67, s(s(T76))), ','(halfc17(T76, s(s(T85))), p139(T85, X286, s(s(s(s(s(s(s(0))))))), T86)))))))).

Clauses:

halfc22(0, 0).
halfc22(s(0), 0).
halfc22(s(s(T26)), s(X55)) :- halfc22(T26, X55).
halfc17(T22, s(X46)) :- halfc22(T22, X46).
qc139(T85, 0, T98, s(T98)) :- halfc17(T85, 0).
qc139(T85, s(0), T103, s(T103)) :- halfc17(T85, s(0)).
qc139(T85, s(s(T110)), T111, T112) :- ','(halfc17(T85, s(s(T110))), qc139(T110, X323, s(T111), T112)).

Afs:

log21(x1, x2) = log21(x2)

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

LOG21_IN_AG(s(s(T14)), T13) → U6_AG(T14, T13, half17_in_aa(T14, X28))
LOG21_IN_AG(s(s(T14)), T13) → HALF17_IN_AA(T14, X28)
HALF17_IN_AA(T22, s(X46)) → U2_AA(T22, X46, half22_in_aa(T22, X46))
HALF17_IN_AA(T22, s(X46)) → HALF22_IN_AA(T22, X46)
HALF22_IN_AA(s(s(T26)), s(X55)) → U1_AA(T26, X55, half22_in_aa(T26, X55))
HALF22_IN_AA(s(s(T26)), s(X55)) → HALF22_IN_AA(T26, X55)
LOG21_IN_AG(s(s(T14)), T32) → U7_AG(T14, T32, halfc17_in_aa(T14, s(s(T31))))
U7_AG(T14, T32, halfc17_out_aa(T14, s(s(T31)))) → U8_AG(T14, T32, half17_in_ga(T31, X82))
U7_AG(T14, T32, halfc17_out_aa(T14, s(s(T31)))) → HALF17_IN_GA(T31, X82)
HALF17_IN_GA(T22, s(X46)) → U2_GA(T22, X46, half22_in_ga(T22, X46))
HALF17_IN_GA(T22, s(X46)) → HALF22_IN_GA(T22, X46)
HALF22_IN_GA(s(s(T26)), s(X55)) → U1_GA(T26, X55, half22_in_ga(T26, X55))
HALF22_IN_GA(s(s(T26)), s(X55)) → HALF22_IN_GA(T26, X55)
LOG21_IN_AG(s(s(T14)), T41) → U9_AG(T14, T41, halfc17_in_aa(T14, s(s(T31))))
U9_AG(T14, T41, halfc17_out_aa(T14, s(s(T31)))) → U10_AG(T14, T41, halfc17_in_ga(T31, s(s(T40))))
U10_AG(T14, T41, halfc17_out_ga(T31, s(s(T40)))) → U11_AG(T14, T41, half17_in_ga(T40, X116))
U10_AG(T14, T41, halfc17_out_ga(T31, s(s(T40)))) → HALF17_IN_GA(T40, X116)
LOG21_IN_AG(s(s(T14)), T50) → U12_AG(T14, T50, halfc17_in_aa(T14, s(s(T31))))
U12_AG(T14, T50, halfc17_out_aa(T14, s(s(T31)))) → U13_AG(T14, T50, halfc17_in_ga(T31, s(s(T40))))
U13_AG(T14, T50, halfc17_out_ga(T31, s(s(T40)))) → U14_AG(T14, T50, halfc17_in_ga(T40, s(s(T49))))
U14_AG(T14, T50, halfc17_out_ga(T40, s(s(T49)))) → U15_AG(T14, T50, half17_in_ga(T49, X150))
U14_AG(T14, T50, halfc17_out_ga(T40, s(s(T49)))) → HALF17_IN_GA(T49, X150)
LOG21_IN_AG(s(s(T14)), T59) → U16_AG(T14, T59, halfc17_in_aa(T14, s(s(T31))))
U16_AG(T14, T59, halfc17_out_aa(T14, s(s(T31)))) → U17_AG(T14, T59, halfc17_in_ga(T31, s(s(T40))))
U17_AG(T14, T59, halfc17_out_ga(T31, s(s(T40)))) → U18_AG(T14, T59, halfc17_in_ga(T40, s(s(T49))))
U18_AG(T14, T59, halfc17_out_ga(T40, s(s(T49)))) → U19_AG(T14, T59, halfc17_in_ga(T49, s(s(T58))))
U19_AG(T14, T59, halfc17_out_ga(T49, s(s(T58)))) → U20_AG(T14, T59, half17_in_ga(T58, X184))
U19_AG(T14, T59, halfc17_out_ga(T49, s(s(T58)))) → HALF17_IN_GA(T58, X184)
LOG21_IN_AG(s(s(T14)), T68) → U21_AG(T14, T68, halfc17_in_aa(T14, s(s(T31))))
U21_AG(T14, T68, halfc17_out_aa(T14, s(s(T31)))) → U22_AG(T14, T68, halfc17_in_ga(T31, s(s(T40))))
U22_AG(T14, T68, halfc17_out_ga(T31, s(s(T40)))) → U23_AG(T14, T68, halfc17_in_ga(T40, s(s(T49))))
U23_AG(T14, T68, halfc17_out_ga(T40, s(s(T49)))) → U24_AG(T14, T68, halfc17_in_ga(T49, s(s(T58))))
U24_AG(T14, T68, halfc17_out_ga(T49, s(s(T58)))) → U25_AG(T14, T68, halfc17_in_ga(T58, s(s(T67))))
U25_AG(T14, T68, halfc17_out_ga(T58, s(s(T67)))) → U26_AG(T14, T68, half17_in_ga(T67, X218))
U25_AG(T14, T68, halfc17_out_ga(T58, s(s(T67)))) → HALF17_IN_GA(T67, X218)
LOG21_IN_AG(s(s(T14)), T77) → U27_AG(T14, T77, halfc17_in_aa(T14, s(s(T31))))
U27_AG(T14, T77, halfc17_out_aa(T14, s(s(T31)))) → U28_AG(T14, T77, halfc17_in_ga(T31, s(s(T40))))
U28_AG(T14, T77, halfc17_out_ga(T31, s(s(T40)))) → U29_AG(T14, T77, halfc17_in_ga(T40, s(s(T49))))
U29_AG(T14, T77, halfc17_out_ga(T40, s(s(T49)))) → U30_AG(T14, T77, halfc17_in_ga(T49, s(s(T58))))
U30_AG(T14, T77, halfc17_out_ga(T49, s(s(T58)))) → U31_AG(T14, T77, halfc17_in_ga(T58, s(s(T67))))
U31_AG(T14, T77, halfc17_out_ga(T58, s(s(T67)))) → U32_AG(T14, T77, halfc17_in_ga(T67, s(s(T76))))
U32_AG(T14, T77, halfc17_out_ga(T67, s(s(T76)))) → U33_AG(T14, T77, half17_in_ga(T76, X252))
U32_AG(T14, T77, halfc17_out_ga(T67, s(s(T76)))) → HALF17_IN_GA(T76, X252)
LOG21_IN_AG(s(s(T14)), T86) → U34_AG(T14, T86, halfc17_in_aa(T14, s(s(T31))))
U34_AG(T14, T86, halfc17_out_aa(T14, s(s(T31)))) → U35_AG(T14, T86, halfc17_in_ga(T31, s(s(T40))))
U35_AG(T14, T86, halfc17_out_ga(T31, s(s(T40)))) → U36_AG(T14, T86, halfc17_in_ga(T40, s(s(T49))))
U36_AG(T14, T86, halfc17_out_ga(T40, s(s(T49)))) → U37_AG(T14, T86, halfc17_in_ga(T49, s(s(T58))))
U37_AG(T14, T86, halfc17_out_ga(T49, s(s(T58)))) → U38_AG(T14, T86, halfc17_in_ga(T58, s(s(T67))))
U38_AG(T14, T86, halfc17_out_ga(T58, s(s(T67)))) → U39_AG(T14, T86, halfc17_in_ga(T67, s(s(T76))))
U39_AG(T14, T86, halfc17_out_ga(T67, s(s(T76)))) → U40_AG(T14, T86, halfc17_in_ga(T76, s(s(T85))))
U40_AG(T14, T86, halfc17_out_ga(T76, s(s(T85)))) → U41_AG(T14, T86, p139_in_gagg(T85, X286, s(s(s(s(s(s(s(0))))))), T86))
U40_AG(T14, T86, halfc17_out_ga(T76, s(s(T85)))) → P139_IN_GAGG(T85, X286, s(s(s(s(s(s(s(0))))))), T86)
P139_IN_GAGG(T85, X286, T88, T86) → U3_GAGG(T85, X286, T88, T86, half17_in_ga(T85, X286))
P139_IN_GAGG(T85, X286, T88, T86) → HALF17_IN_GA(T85, X286)
P139_IN_GAGG(T85, s(s(T110)), T111, T112) → U4_GAGG(T85, T110, T111, T112, halfc17_in_ga(T85, s(s(T110))))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → U5_GAGG(T85, T110, T111, T112, p139_in_gagg(T110, X323, s(T111), T112))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, X323, s(T111), T112)

The TRS R consists of the following rules:

halfc17_in_aa(T22, s(X46)) → U44_aa(T22, X46, halfc22_in_aa(T22, X46))
halfc22_in_aa(0, 0) → halfc22_out_aa(0, 0)
halfc22_in_aa(s(0), 0) → halfc22_out_aa(s(0), 0)
halfc22_in_aa(s(s(T26)), s(X55)) → U43_aa(T26, X55, halfc22_in_aa(T26, X55))
U43_aa(T26, X55, halfc22_out_aa(T26, X55)) → halfc22_out_aa(s(s(T26)), s(X55))
U44_aa(T22, X46, halfc22_out_aa(T22, X46)) → halfc17_out_aa(T22, s(X46))
halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, 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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))

The argument filtering Pi contains the following mapping:
half17_in_aa(x1, x2) = half17_in_aa
half22_in_aa(x1, x2) = half22_in_aa
halfc17_in_aa(x1, x2) = halfc17_in_aa
U44_aa(x1, x2, x3) = U44_aa(x3)
halfc22_in_aa(x1, x2) = halfc22_in_aa
halfc22_out_aa(x1, x2) = halfc22_out_aa(x1, x2)
U43_aa(x1, x2, x3) = U43_aa(x3)
halfc17_out_aa(x1, x2) = halfc17_out_aa(x1, x2)
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_gagg(x1, x2, x3, x4) = p139_in_gagg(x1, x3, x4)
LOG21_IN_AG(x1, x2) = LOG21_IN_AG(x2)
U6_AG(x1, x2, x3) = U6_AG(x2, x3)
HALF17_IN_AA(x1, x2) = HALF17_IN_AA
U2_AA(x1, x2, x3) = U2_AA(x3)
HALF22_IN_AA(x1, x2) = HALF22_IN_AA
U1_AA(x1, x2, x3) = U1_AA(x3)
U7_AG(x1, x2, x3) = U7_AG(x2, x3)
U8_AG(x1, x2, x3) = U8_AG(x1, x2, 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)
U9_AG(x1, x2, x3) = U9_AG(x2, x3)
U10_AG(x1, x2, x3) = U10_AG(x1, x2, x3)
U11_AG(x1, x2, x3) = U11_AG(x1, x2, x3)
U12_AG(x1, x2, x3) = U12_AG(x2, x3)
U13_AG(x1, x2, x3) = U13_AG(x1, x2, x3)
U14_AG(x1, x2, x3) = U14_AG(x1, x2, x3)
U15_AG(x1, x2, x3) = U15_AG(x1, x2, x3)
U16_AG(x1, x2, x3) = U16_AG(x2, x3)
U17_AG(x1, x2, x3) = U17_AG(x1, x2, x3)
U18_AG(x1, x2, x3) = U18_AG(x1, x2, x3)
U19_AG(x1, x2, x3) = U19_AG(x1, x2, x3)
U20_AG(x1, x2, x3) = U20_AG(x1, x2, x3)
U21_AG(x1, x2, x3) = U21_AG(x2, x3)
U22_AG(x1, x2, x3) = U22_AG(x1, x2, x3)
U23_AG(x1, x2, x3) = U23_AG(x1, x2, x3)
U24_AG(x1, x2, x3) = U24_AG(x1, x2, x3)
U25_AG(x1, x2, x3) = U25_AG(x1, x2, x3)
U26_AG(x1, x2, x3) = U26_AG(x1, x2, x3)
U27_AG(x1, x2, x3) = U27_AG(x2, x3)
U28_AG(x1, x2, x3) = U28_AG(x1, x2, x3)
U29_AG(x1, x2, x3) = U29_AG(x1, x2, x3)
U30_AG(x1, x2, x3) = U30_AG(x1, x2, x3)
U31_AG(x1, x2, x3) = U31_AG(x1, x2, x3)
U32_AG(x1, x2, x3) = U32_AG(x1, x2, x3)
U33_AG(x1, x2, x3) = U33_AG(x1, x2, x3)
U34_AG(x1, x2, x3) = U34_AG(x2, x3)
U35_AG(x1, x2, x3) = U35_AG(x1, x2, x3)
U36_AG(x1, x2, x3) = U36_AG(x1, x2, x3)
U37_AG(x1, x2, x3) = U37_AG(x1, x2, x3)
U38_AG(x1, x2, x3) = U38_AG(x1, x2, x3)
U39_AG(x1, x2, x3) = U39_AG(x1, x2, x3)
U40_AG(x1, x2, x3) = U40_AG(x1, x2, x3)
U41_AG(x1, x2, x3) = U41_AG(x1, x2, x3)
P139_IN_GAGG(x1, x2, x3, x4) = P139_IN_GAGG(x1, x3, x4)
U3_GAGG(x1, x2, x3, x4, x5) = U3_GAGG(x1, x3, x4, x5)
U4_GAGG(x1, x2, x3, x4, x5) = U4_GAGG(x1, x3, x4, x5)
U5_GAGG(x1, x2, x3, x4, x5) = U5_GAGG(x1, x2, x3, x4, 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_AG(s(s(T14)), T13) → U6_AG(T14, T13, half17_in_aa(T14, X28))
LOG21_IN_AG(s(s(T14)), T13) → HALF17_IN_AA(T14, X28)
HALF17_IN_AA(T22, s(X46)) → U2_AA(T22, X46, half22_in_aa(T22, X46))
HALF17_IN_AA(T22, s(X46)) → HALF22_IN_AA(T22, X46)
HALF22_IN_AA(s(s(T26)), s(X55)) → U1_AA(T26, X55, half22_in_aa(T26, X55))
HALF22_IN_AA(s(s(T26)), s(X55)) → HALF22_IN_AA(T26, X55)
LOG21_IN_AG(s(s(T14)), T32) → U7_AG(T14, T32, halfc17_in_aa(T14, s(s(T31))))
U7_AG(T14, T32, halfc17_out_aa(T14, s(s(T31)))) → U8_AG(T14, T32, half17_in_ga(T31, X82))
U7_AG(T14, T32, halfc17_out_aa(T14, s(s(T31)))) → HALF17_IN_GA(T31, X82)
HALF17_IN_GA(T22, s(X46)) → U2_GA(T22, X46, half22_in_ga(T22, X46))
HALF17_IN_GA(T22, s(X46)) → HALF22_IN_GA(T22, X46)
HALF22_IN_GA(s(s(T26)), s(X55)) → U1_GA(T26, X55, half22_in_ga(T26, X55))
HALF22_IN_GA(s(s(T26)), s(X55)) → HALF22_IN_GA(T26, X55)
LOG21_IN_AG(s(s(T14)), T41) → U9_AG(T14, T41, halfc17_in_aa(T14, s(s(T31))))
U9_AG(T14, T41, halfc17_out_aa(T14, s(s(T31)))) → U10_AG(T14, T41, halfc17_in_ga(T31, s(s(T40))))
U10_AG(T14, T41, halfc17_out_ga(T31, s(s(T40)))) → U11_AG(T14, T41, half17_in_ga(T40, X116))
U10_AG(T14, T41, halfc17_out_ga(T31, s(s(T40)))) → HALF17_IN_GA(T40, X116)
LOG21_IN_AG(s(s(T14)), T50) → U12_AG(T14, T50, halfc17_in_aa(T14, s(s(T31))))
U12_AG(T14, T50, halfc17_out_aa(T14, s(s(T31)))) → U13_AG(T14, T50, halfc17_in_ga(T31, s(s(T40))))
U13_AG(T14, T50, halfc17_out_ga(T31, s(s(T40)))) → U14_AG(T14, T50, halfc17_in_ga(T40, s(s(T49))))
U14_AG(T14, T50, halfc17_out_ga(T40, s(s(T49)))) → U15_AG(T14, T50, half17_in_ga(T49, X150))
U14_AG(T14, T50, halfc17_out_ga(T40, s(s(T49)))) → HALF17_IN_GA(T49, X150)
LOG21_IN_AG(s(s(T14)), T59) → U16_AG(T14, T59, halfc17_in_aa(T14, s(s(T31))))
U16_AG(T14, T59, halfc17_out_aa(T14, s(s(T31)))) → U17_AG(T14, T59, halfc17_in_ga(T31, s(s(T40))))
U17_AG(T14, T59, halfc17_out_ga(T31, s(s(T40)))) → U18_AG(T14, T59, halfc17_in_ga(T40, s(s(T49))))
U18_AG(T14, T59, halfc17_out_ga(T40, s(s(T49)))) → U19_AG(T14, T59, halfc17_in_ga(T49, s(s(T58))))
U19_AG(T14, T59, halfc17_out_ga(T49, s(s(T58)))) → U20_AG(T14, T59, half17_in_ga(T58, X184))
U19_AG(T14, T59, halfc17_out_ga(T49, s(s(T58)))) → HALF17_IN_GA(T58, X184)
LOG21_IN_AG(s(s(T14)), T68) → U21_AG(T14, T68, halfc17_in_aa(T14, s(s(T31))))
U21_AG(T14, T68, halfc17_out_aa(T14, s(s(T31)))) → U22_AG(T14, T68, halfc17_in_ga(T31, s(s(T40))))
U22_AG(T14, T68, halfc17_out_ga(T31, s(s(T40)))) → U23_AG(T14, T68, halfc17_in_ga(T40, s(s(T49))))
U23_AG(T14, T68, halfc17_out_ga(T40, s(s(T49)))) → U24_AG(T14, T68, halfc17_in_ga(T49, s(s(T58))))
U24_AG(T14, T68, halfc17_out_ga(T49, s(s(T58)))) → U25_AG(T14, T68, halfc17_in_ga(T58, s(s(T67))))
U25_AG(T14, T68, halfc17_out_ga(T58, s(s(T67)))) → U26_AG(T14, T68, half17_in_ga(T67, X218))
U25_AG(T14, T68, halfc17_out_ga(T58, s(s(T67)))) → HALF17_IN_GA(T67, X218)
LOG21_IN_AG(s(s(T14)), T77) → U27_AG(T14, T77, halfc17_in_aa(T14, s(s(T31))))
U27_AG(T14, T77, halfc17_out_aa(T14, s(s(T31)))) → U28_AG(T14, T77, halfc17_in_ga(T31, s(s(T40))))
U28_AG(T14, T77, halfc17_out_ga(T31, s(s(T40)))) → U29_AG(T14, T77, halfc17_in_ga(T40, s(s(T49))))
U29_AG(T14, T77, halfc17_out_ga(T40, s(s(T49)))) → U30_AG(T14, T77, halfc17_in_ga(T49, s(s(T58))))
U30_AG(T14, T77, halfc17_out_ga(T49, s(s(T58)))) → U31_AG(T14, T77, halfc17_in_ga(T58, s(s(T67))))
U31_AG(T14, T77, halfc17_out_ga(T58, s(s(T67)))) → U32_AG(T14, T77, halfc17_in_ga(T67, s(s(T76))))
U32_AG(T14, T77, halfc17_out_ga(T67, s(s(T76)))) → U33_AG(T14, T77, half17_in_ga(T76, X252))
U32_AG(T14, T77, halfc17_out_ga(T67, s(s(T76)))) → HALF17_IN_GA(T76, X252)
LOG21_IN_AG(s(s(T14)), T86) → U34_AG(T14, T86, halfc17_in_aa(T14, s(s(T31))))
U34_AG(T14, T86, halfc17_out_aa(T14, s(s(T31)))) → U35_AG(T14, T86, halfc17_in_ga(T31, s(s(T40))))
U35_AG(T14, T86, halfc17_out_ga(T31, s(s(T40)))) → U36_AG(T14, T86, halfc17_in_ga(T40, s(s(T49))))
U36_AG(T14, T86, halfc17_out_ga(T40, s(s(T49)))) → U37_AG(T14, T86, halfc17_in_ga(T49, s(s(T58))))
U37_AG(T14, T86, halfc17_out_ga(T49, s(s(T58)))) → U38_AG(T14, T86, halfc17_in_ga(T58, s(s(T67))))
U38_AG(T14, T86, halfc17_out_ga(T58, s(s(T67)))) → U39_AG(T14, T86, halfc17_in_ga(T67, s(s(T76))))
U39_AG(T14, T86, halfc17_out_ga(T67, s(s(T76)))) → U40_AG(T14, T86, halfc17_in_ga(T76, s(s(T85))))
U40_AG(T14, T86, halfc17_out_ga(T76, s(s(T85)))) → U41_AG(T14, T86, p139_in_gagg(T85, X286, s(s(s(s(s(s(s(0))))))), T86))
U40_AG(T14, T86, halfc17_out_ga(T76, s(s(T85)))) → P139_IN_GAGG(T85, X286, s(s(s(s(s(s(s(0))))))), T86)
P139_IN_GAGG(T85, X286, T88, T86) → U3_GAGG(T85, X286, T88, T86, half17_in_ga(T85, X286))
P139_IN_GAGG(T85, X286, T88, T86) → HALF17_IN_GA(T85, X286)
P139_IN_GAGG(T85, s(s(T110)), T111, T112) → U4_GAGG(T85, T110, T111, T112, halfc17_in_ga(T85, s(s(T110))))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → U5_GAGG(T85, T110, T111, T112, p139_in_gagg(T110, X323, s(T111), T112))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, X323, s(T111), T112)

The TRS R consists of the following rules:

halfc17_in_aa(T22, s(X46)) → U44_aa(T22, X46, halfc22_in_aa(T22, X46))
halfc22_in_aa(0, 0) → halfc22_out_aa(0, 0)
halfc22_in_aa(s(0), 0) → halfc22_out_aa(s(0), 0)
halfc22_in_aa(s(s(T26)), s(X55)) → U43_aa(T26, X55, halfc22_in_aa(T26, X55))
U43_aa(T26, X55, halfc22_out_aa(T26, X55)) → halfc22_out_aa(s(s(T26)), s(X55))
U44_aa(T22, X46, halfc22_out_aa(T22, X46)) → halfc17_out_aa(T22, s(X46))
halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, 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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 53 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

HALF22_IN_GA(s(s(T26)), s(X55)) → HALF22_IN_GA(T26, X55)

The TRS R consists of the following rules:

halfc17_in_aa(T22, s(X46)) → U44_aa(T22, X46, halfc22_in_aa(T22, X46))
halfc22_in_aa(0, 0) → halfc22_out_aa(0, 0)
halfc22_in_aa(s(0), 0) → halfc22_out_aa(s(0), 0)
halfc22_in_aa(s(s(T26)), s(X55)) → U43_aa(T26, X55, halfc22_in_aa(T26, X55))
U43_aa(T26, X55, halfc22_out_aa(T26, X55)) → halfc22_out_aa(s(s(T26)), s(X55))
U44_aa(T22, X46, halfc22_out_aa(T22, X46)) → halfc17_out_aa(T22, s(X46))
halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, 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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))

The argument filtering Pi contains the following mapping:
halfc17_in_aa(x1, x2) = halfc17_in_aa
U44_aa(x1, x2, x3) = U44_aa(x3)
halfc22_in_aa(x1, x2) = halfc22_in_aa
halfc22_out_aa(x1, x2) = halfc22_out_aa(x1, x2)
U43_aa(x1, x2, x3) = U43_aa(x3)
halfc17_out_aa(x1, x2) = halfc17_out_aa(x1, x2)
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(T26)), s(X55)) → HALF22_IN_GA(T26, 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(T26))) → HALF22_IN_GA(T26)

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(T26))) → HALF22_IN_GA(T26)
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_GAGG(T85, s(s(T110)), T111, T112) → U4_GAGG(T85, T110, T111, T112, halfc17_in_ga(T85, s(s(T110))))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, X323, s(T111), T112)

The TRS R consists of the following rules:

halfc17_in_aa(T22, s(X46)) → U44_aa(T22, X46, halfc22_in_aa(T22, X46))
halfc22_in_aa(0, 0) → halfc22_out_aa(0, 0)
halfc22_in_aa(s(0), 0) → halfc22_out_aa(s(0), 0)
halfc22_in_aa(s(s(T26)), s(X55)) → U43_aa(T26, X55, halfc22_in_aa(T26, X55))
U43_aa(T26, X55, halfc22_out_aa(T26, X55)) → halfc22_out_aa(s(s(T26)), s(X55))
U44_aa(T22, X46, halfc22_out_aa(T22, X46)) → halfc17_out_aa(T22, s(X46))
halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, 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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))

The argument filtering Pi contains the following mapping:
halfc17_in_aa(x1, x2) = halfc17_in_aa
U44_aa(x1, x2, x3) = U44_aa(x3)
halfc22_in_aa(x1, x2) = halfc22_in_aa
halfc22_out_aa(x1, x2) = halfc22_out_aa(x1, x2)
U43_aa(x1, x2, x3) = U43_aa(x3)
halfc17_out_aa(x1, x2) = halfc17_out_aa(x1, x2)
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_GAGG(x1, x2, x3, x4) = P139_IN_GAGG(x1, x3, x4)
U4_GAGG(x1, x2, x3, x4, x5) = U4_GAGG(x1, x3, x4, x5)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P139_IN_GAGG(T85, s(s(T110)), T111, T112) → U4_GAGG(T85, T110, T111, T112, halfc17_in_ga(T85, s(s(T110))))
U4_GAGG(T85, T110, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, X323, s(T111), T112)

The TRS R consists of the following rules:

halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, X46))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

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_GAGG(x1, x2, x3, x4) = P139_IN_GAGG(x1, x3, x4)
U4_GAGG(x1, x2, x3, x4, x5) = U4_GAGG(x1, x3, x4, x5)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, halfc17_in_ga(T85))
U4_GAGG(T85, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, s(T111), T112)

The TRS R consists of the following rules:

halfc17_in_ga(T22) → U44_ga(T22, halfc22_in_ga(T22))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

The set Q consists of the following terms:

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

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

(19) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, halfc17_in_ga(T85)) at position [3] we obtained the following new rules [LPAR04]:

P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, U44_ga(T85, halfc22_in_ga(T85)))

(20) Obligation:

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

U4_GAGG(T85, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, s(T111), T112)
P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, U44_ga(T85, halfc22_in_ga(T85)))

The TRS R consists of the following rules:

halfc17_in_ga(T22) → U44_ga(T22, halfc22_in_ga(T22))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

The set Q consists of the following terms:

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

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

(21) 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.

(22) Obligation:

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

U4_GAGG(T85, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, s(T111), T112)
P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, U44_ga(T85, halfc22_in_ga(T85)))

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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

The set Q consists of the following terms:

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

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

(23) 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)

(24) Obligation:

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

U4_GAGG(T85, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, s(T111), T112)
P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, U44_ga(T85, halfc22_in_ga(T85)))

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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

The set Q consists of the following terms:

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

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U4_GAGG(T85, T111, T112, halfc17_out_ga(T85, s(s(T110)))) → P139_IN_GAGG(T110, s(T111), T112)

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

POL(0) = 0
POL(P139_IN_GAGG(x₁, x₂, x₃)) = 1 + x₁
POL(U43_ga(x₁, x₂)) = 1 + x₂
POL(U44_ga(x₁, x₂)) = x₂
POL(U4_GAGG(x₁, x₂, x₃, x₄)) = x₄
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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

(26) Obligation:

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

P139_IN_GAGG(T85, T111, T112) → U4_GAGG(T85, T111, T112, U44_ga(T85, halfc22_in_ga(T85)))

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(T26))) → U43_ga(T26, halfc22_in_ga(T26))
U44_ga(T22, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))
U43_ga(T26, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))

The set Q consists of the following terms:

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

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

HALF22_IN_AA(s(s(T26)), s(X55)) → HALF22_IN_AA(T26, X55)

The TRS R consists of the following rules:

halfc17_in_aa(T22, s(X46)) → U44_aa(T22, X46, halfc22_in_aa(T22, X46))
halfc22_in_aa(0, 0) → halfc22_out_aa(0, 0)
halfc22_in_aa(s(0), 0) → halfc22_out_aa(s(0), 0)
halfc22_in_aa(s(s(T26)), s(X55)) → U43_aa(T26, X55, halfc22_in_aa(T26, X55))
U43_aa(T26, X55, halfc22_out_aa(T26, X55)) → halfc22_out_aa(s(s(T26)), s(X55))
U44_aa(T22, X46, halfc22_out_aa(T22, X46)) → halfc17_out_aa(T22, s(X46))
halfc17_in_ga(T22, s(X46)) → U44_ga(T22, X46, halfc22_in_ga(T22, 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(T26)), s(X55)) → U43_ga(T26, X55, halfc22_in_ga(T26, X55))
U43_ga(T26, X55, halfc22_out_ga(T26, X55)) → halfc22_out_ga(s(s(T26)), s(X55))
U44_ga(T22, X46, halfc22_out_ga(T22, X46)) → halfc17_out_ga(T22, s(X46))

The argument filtering Pi contains the following mapping:
halfc17_in_aa(x1, x2) = halfc17_in_aa
U44_aa(x1, x2, x3) = U44_aa(x3)
halfc22_in_aa(x1, x2) = halfc22_in_aa
halfc22_out_aa(x1, x2) = halfc22_out_aa(x1, x2)
U43_aa(x1, x2, x3) = U43_aa(x3)
halfc17_out_aa(x1, x2) = halfc17_out_aa(x1, x2)
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_AA(x1, x2) = HALF22_IN_AA

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

(30) UsableRulesProof (EQUIVALENT transformation)

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

(31) Obligation:

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

HALF22_IN_AA(s(s(T26)), s(X55)) → HALF22_IN_AA(T26, X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
HALF22_IN_AA(x1, x2) = HALF22_IN_AA

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

(32) PiDPToQDPProof (SOUND transformation)

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

(33) Obligation:

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

HALF22_IN_AA → HALF22_IN_AA

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

(34) NonTerminationProof (EQUIVALENT transformation)

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

s = HALF22_IN_AA evaluates to t =HALF22_IN_AA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from HALF22_IN_AA to HALF22_IN_AA.

(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) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) Rewriting (EQUIVALENT transformation)

(20) Obligation:

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) QReductionProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) TRUE

(29) Obligation:

(30) UsableRulesProof (EQUIVALENT transformation)

(31) Obligation:

(32) PiDPToQDPProof (SOUND transformation)

(33) Obligation:

(34) NonTerminationProof (EQUIVALENT transformation)

(35) NO