Left Termination proof of /tmp/tmpCnxbHu/times2.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 486 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 348 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 67 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 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇒, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇒, 0 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 6 ms)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔, 78 ms)

        ↳41 QDP

        ↳42 DependencyGraphProof (⇔, 0 ms)

        ↳43 QDP

        ↳44 QDPQMonotonicMRRProof (⇔, 77 ms)

        ↳45 QDP

        ↳46 QDPQMonotonicMRRProof (⇔, 101 ms)

        ↳47 QDP

        ↳48 DependencyGraphProof (⇔, 0 ms)

        ↳49 TRUE

    ↳50 PiDP

        ↳51 UsableRulesProof (⇔, 0 ms)

        ↳52 PiDP

        ↳53 PiDPToQDPProof (⇒, 0 ms)

        ↳54 QDP

        ↳55 QDPSizeChangeProof (⇔, 0 ms)

        ↳56 YES

(0) Obligation:

Clauses:

even(0, true).
even(s(0), false).
even(s(s(X)), B) :- even(X, B).
half(0, 0).
half(s(s(X)), s(Y)) :- half(X, Y).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z)).
if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z))).
if(false, s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).

Query: times(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="I: times(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: times(T1, T2, T3), SCOPE: 1, CLAUSE: 7 | times(T1, T2, T3), SCOPE: 1, CLAUSE: 8\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | times(0, T5, T3), SCOPE: 1, CLAUSE: 8\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: times(T1, T2, T3), SCOPE: 1, CLAUSE: 8\n\nT1 is ground\nT2 is ground\nX3 is free\ntimes(T1, T2, T3) !=? times(0, X3, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: times(0, T5, T3), SCOPE: 1, CLAUSE: 8\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(even(s(T9), X13), if(X13, s(T9), T10, T12))\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(even(s(T9), X13), if(X13, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: if(false, s(0), T10, T12)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 9 | if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 10\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: if(false, s(0), T10, T12), SCOPE: 3, CLAUSE: 10\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(times(0, T21, X37), plus(T21, X37, T23))\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="G: times(0, T21, X37)\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="A: plus(T21, T27, T23)\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: times(0, T21, X37), SCOPE: 4, CLAUSE: 7 | times(0, T21, X37), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: times(0, T21, X37), SCOPE: 4, CLAUSE: 7\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: times(0, T21, X37), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 5 | plus(T21, T27, T23), SCOPE: 5, CLAUSE: 6\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 5\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: plus(T21, T27, T23), SCOPE: 5, CLAUSE: 6\n\nT21 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: plus(T47, T48, T50)\n\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(even(T55, X78), if(X78, s(s(T55)), T10, T12))\n\nT10 is ground\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="B: even(T55, X78)\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: if(T56, s(s(T55)), T10, T12)\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(T55, X78), SCOPE: 6, CLAUSE: 0 | even(T55, X78), SCOPE: 6, CLAUSE: 1 | even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: even(T55, X78), SCOPE: 6, CLAUSE: 0\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: even(T55, X78), SCOPE: 6, CLAUSE: 1 | even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: even(T55, X78), SCOPE: 6, CLAUSE: 1\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: even(T55, X78), SCOPE: 6, CLAUSE: 2\n\nT55 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: even(T59, X87)\n\nT59 is ground\nX87 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 9 | if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 9\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: if(T56, s(s(T55)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT55 is ground\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(half(s(s(T75)), X112), ','(times(X112, T76, X113), plus(X113, X113, T78)))\n\nT75 is ground\nT76 is ground\nX112 is free\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="H: half(s(s(T75)), X112)\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(times(T79, T76, X113), plus(X113, X113, T78))\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 3 | half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 4\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: half(s(s(T75)), X112), SCOPE: 8, CLAUSE: 4\n\nT75 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="C: half(T82, X122)\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: half(T82, X122), SCOPE: 9, CLAUSE: 3 | half(T82, X122), SCOPE: 9, CLAUSE: 4\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: half(T82, X122), SCOPE: 9, CLAUSE: 3\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: half(T82, X122), SCOPE: 9, CLAUSE: 4\n\nT82 is ground\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: half(T85, X131)\n\nT85 is ground\nX131 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="D: times(T79, T76, X113)\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: plus(T88, T88, T78)\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: times(T79, T76, X113), SCOPE: 10, CLAUSE: 7 | times(T79, T76, X113), SCOPE: 10, CLAUSE: 8\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: times(T79, T76, X113), SCOPE: 10, CLAUSE: 7\n\nT76 is ground\nT79 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: times(T79, T76, X113), SCOPE: 10, CLAUSE: 8\n\nT76 is ground\nT79 is ground\nX113 is free", 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: ','(even(s(T100), X153), if(X153, s(T100), T101, X154))\n\nT100 is ground\nT101 is ground\nX154 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: even(s(T100), X153)\n\nT100 is ground\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: if(T102, s(T100), T101, X154)\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 9 | if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 10\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 9\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: if(T102, s(T100), T101, X154), SCOPE: 11, CLAUSE: 10\n\nT100 is ground\nT101 is ground\nT102 is ground\nX154 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(half(s(T113), X184), ','(times(X184, T114, X185), plus(X185, X185, X186)))\n\nT113 is ground\nT114 is ground\nX186 is free\nX184 is free\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: half(s(T113), X184)\n\nT113 is ground\nX184 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(times(T115, T114, X185), plus(X185, X185, X186))\n\nT114 is ground\nT115 is ground\nX186 is free\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: times(T115, T114, X185)\n\nT114 is ground\nT115 is ground\nX185 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: plus(T118, T118, X186)\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 5 | plus(T118, T118, X186), SCOPE: 12, CLAUSE: 6\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 5\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: plus(T118, T118, X186), SCOPE: 12, CLAUSE: 6\n\nT118 is ground\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: plus(T123, s(T123), X209)\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 5 | plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 6\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 5\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: plus(T123, s(T123), X209), SCOPE: 13, CLAUSE: 6\n\nT123 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: plus(T126, s(s(T126)), X228)\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 5 | plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 6\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 5\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: plus(T126, s(s(T126)), X228), SCOPE: 14, CLAUSE: 6\n\nT126 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: plus(T129, s(s(s(T129))), X247)\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 5 | plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 6\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 5\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: plus(T129, s(s(s(T129))), X247), SCOPE: 15, CLAUSE: 6\n\nT129 is ground\nX247 is free", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: plus(T132, s(s(s(s(T132)))), X266)\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 5 | plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 6\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 5\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: plus(T132, s(s(s(s(T132)))), X266), SCOPE: 16, CLAUSE: 6\n\nT132 is ground\nX266 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: plus(T135, s(s(s(s(s(T135))))), X285)\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 5 | plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 6\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 5\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: plus(T135, s(s(s(s(s(T135))))), X285), SCOPE: 17, CLAUSE: 6\n\nT135 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: plus(T138, s(s(s(s(s(s(T138)))))), X304)\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 5 | plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 6\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 5\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: plus(T138, s(s(s(s(s(s(T138)))))), X304), SCOPE: 18, CLAUSE: 6\n\nT138 is ground\nX304 is free", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: plus(T141, s(s(s(s(s(s(s(T141))))))), X323)\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 5 | plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 6\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 5\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: plus(T141, s(s(s(s(s(s(s(T141))))))), X323), SCOPE: 19, CLAUSE: 6\n\nT141 is ground\nX323 is free", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
152 [label="152: plus(T144, s(s(s(s(s(s(s(s(T144)))))))), X342)\n\nT144 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="E: plus(T144, s(T145), X342)\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 5 | plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 6\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 5\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: plus(T144, s(T145), X342), SCOPE: 20, CLAUSE: 6\n\nT144 is ground\nT145 is ground\nX342 is free", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: plus(T157, s(T158), X363)\n\nT157 is ground\nT158 is ground\nX363 is free", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: ','(times(T165, T166, X378), plus(T166, X378, X379))\n\nT165 is ground\nT166 is ground\nX379 is free\nX378 is free", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: times(T165, T166, X378)\n\nT165 is ground\nT166 is ground\nX378 is free", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="F: plus(T166, T169, X379)\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 5 | plus(T166, T169, X379), SCOPE: 21, CLAUSE: 6\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 5\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: plus(T166, T169, X379), SCOPE: 21, CLAUSE: 6\n\nT166 is ground\nT169 is ground\nX379 is free", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: plus(T183, T184, X402)\n\nT183 is ground\nT184 is ground\nX402 is free", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 5 | plus(T88, T88, T78), SCOPE: 22, CLAUSE: 6\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 5\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: plus(T88, T88, T78), SCOPE: 22, CLAUSE: 6\n\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="179: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
181 [label="181: plus(T191, s(T191), T193)\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="182: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 5 | plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 6\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 5\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: plus(T191, s(T191), T193), SCOPE: 23, CLAUSE: 6\n\nT191 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: plus(T198, s(s(T198)), T200)\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 5 | plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 6\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 5\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: plus(T198, s(s(T198)), T200), SCOPE: 24, CLAUSE: 6\n\nT198 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: plus(T205, s(s(s(T205))), T207)\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 5 | plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 6\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 5\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: plus(T205, s(s(s(T205))), T207), SCOPE: 25, CLAUSE: 6\n\nT205 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: plus(T212, s(s(s(s(T212)))), T214)\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 5 | plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 6\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 5\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: plus(T212, s(s(s(s(T212)))), T214), SCOPE: 26, CLAUSE: 6\n\nT212 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: plus(T219, s(s(s(s(s(T219))))), T221)\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 5 | plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 6\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 5\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: plus(T219, s(s(s(s(s(T219))))), T221), SCOPE: 27, CLAUSE: 6\n\nT219 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: plus(T226, s(s(s(s(s(s(T226)))))), T228)\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 5 | plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 6\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 5\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: plus(T226, s(s(s(s(s(s(T226)))))), T228), SCOPE: 28, CLAUSE: 6\n\nT226 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: plus(T233, s(s(s(s(s(s(s(T233))))))), T235)\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 5 | plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 6\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 5\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: plus(T233, s(s(s(s(s(s(s(T233))))))), T235), SCOPE: 29, CLAUSE: 6\n\nT233 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="236: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: plus(T240, s(s(s(s(s(s(s(s(T240)))))))), T242)\n\nT240 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: plus(T240, s(T243), T242)\n\nT240 is ground\nT243 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: ','(times(s(T252), T253, X547), plus(T253, X547, T255))\n\nT252 is ground\nT253 is ground\nX547 is free", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: times(s(T252), T253, X547)\n\nT252 is ground\nT253 is ground\nX547 is free", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: plus(T253, T256, T255)\n\nT253 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 244 [arrowhead = none ];
244 -> 2;

245 [label="EVAL with clause\ntimes(0, X3, 0).\nand substitutionT1 -> 0,\nT2 -> T5,\nX3 -> T5,\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 245 [arrowhead = none ];
245 -> 3;

246 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 246 [arrowhead = none ];
246 -> 4;

247 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 247 [arrowhead = none ];
247 -> 5;

248 [label="EVAL with clause\ntimes(s(X10), X11, X12) :- ','(even(s(X10), X13), if(X13, s(X10), X11, X12)).\nand substitutionX10 -> T9,\nT1 -> s(T9),\nT2 -> T10,\nX11 -> T10,\nT3 -> T12,\nX12 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 248 [arrowhead = none ];
248 -> 7;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 249 [arrowhead = none ];
249 -> 8;

250 [label="BACKTRACK\nfor clause: times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 250 [arrowhead = none ];
250 -> 6;

251 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 251 [arrowhead = none ];
251 -> 9;

252 [label="BACKTRACK\nfor clause: even(0, true)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 252 [arrowhead = none ];
252 -> 10;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 253 [arrowhead = none ];
253 -> 11;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 254 [arrowhead = none ];
254 -> 12;

255 [label="EVAL with clause\neven(s(0), false).\nand substitutionT9 -> 0,\nX13 -> false", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 255 [arrowhead = none ];
255 -> 13;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 256 [arrowhead = none ];
256 -> 14;

257 [label="EVAL with clause\neven(s(s(X76)), X77) :- even(X76, X77).\nand substitutionX76 -> T55,\nT9 -> s(T55),\nX13 -> X78,\nX77 -> X78", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 257 [arrowhead = none ];
257 -> 34;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 258 [arrowhead = none ];
258 -> 35;

259 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 259 [arrowhead = none ];
259 -> 15;

260 [label="BACKTRACK\nfor clause: if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 260 [arrowhead = none ];
260 -> 16;

261 [label="ONLY EVAL with clause\nif(false, s(X34), X35, X36) :- ','(times(X34, X35, X37), plus(X35, X37, X36)).\nand substitutionX34 -> 0,\nT10 -> T21,\nX35 -> T21,\nT12 -> T23,\nX36 -> T23,\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 261 [arrowhead = none ];
261 -> 17;

262 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 262 [arrowhead = none ];
262 -> 18;

263 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nT27 is ground\nreplacements:X37 -> T27", fontsize=14, style = filled, fillcolor = violet];
17 -> 263 [arrowhead = none ];
263 -> 19;

264 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 264 [arrowhead = none ];
264 -> 20;

265 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 265 [arrowhead = none ];
265 -> 26;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 266 [arrowhead = none ];
266 -> 21;

267 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 267 [arrowhead = none ];
267 -> 22;

268 [label="ONLY EVAL with clause\ntimes(0, X48, 0).\nand substitutionT21 -> T33,\nX48 -> T33,\nX37 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 268 [arrowhead = none ];
268 -> 23;

269 [label="BACKTRACK\nfor clause: times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 269 [arrowhead = none ];
269 -> 25;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 270 [arrowhead = none ];
270 -> 24;

271 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 271 [arrowhead = none ];
271 -> 27;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 272 [arrowhead = none ];
272 -> 28;

273 [label="EVAL with clause\nplus(0, X58, X58).\nand substitutionT21 -> 0,\nT27 -> T40,\nX58 -> T40,\nT23 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 273 [arrowhead = none ];
273 -> 29;

274 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 274 [arrowhead = none ];
274 -> 30;

275 [label="EVAL with clause\nplus(s(X65), X66, s(X67)) :- plus(X65, X66, X67).\nand substitutionX65 -> T47,\nT21 -> s(T47),\nT27 -> T48,\nX66 -> T48,\nX67 -> T50,\nT23 -> s(T50),\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 275 [arrowhead = none ];
275 -> 32;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 276 [arrowhead = none ];
276 -> 33;

277 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 277 [arrowhead = none ];
277 -> 31;

278 [label="INSTANCE with matching:\nT21 -> T47\nT27 -> T48\nT23 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 278 [arrowhead = none , style=dashed];
278 -> 19[style=dashed];

279 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 279 [arrowhead = none ];
279 -> 36;

280 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT56 is ground\nreplacements:X78 -> T56", fontsize=14, style = filled, fillcolor = violet];
34 -> 280 [arrowhead = none ];
280 -> 37;

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 281 [arrowhead = none ];
281 -> 38;

282 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 282 [arrowhead = none ];
282 -> 51;

283 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 283 [arrowhead = none ];
283 -> 39;

284 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 284 [arrowhead = none ];
284 -> 40;

285 [label="EVAL with clause\neven(0, true).\nand substitutionT55 -> 0,\nX78 -> true", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 285 [arrowhead = none ];
285 -> 41;

286 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 286 [arrowhead = none ];
286 -> 42;

287 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 287 [arrowhead = none ];
287 -> 44;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 288 [arrowhead = none ];
288 -> 45;

289 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 289 [arrowhead = none ];
289 -> 43;

290 [label="EVAL with clause\neven(s(0), false).\nand substitutionT55 -> s(0),\nX78 -> false", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 290 [arrowhead = none ];
290 -> 46;

291 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 291 [arrowhead = none ];
291 -> 47;

292 [label="EVAL with clause\neven(s(s(X85)), X86) :- even(X85, X86).\nand substitutionX85 -> T59,\nT55 -> s(s(T59)),\nX78 -> X87,\nX86 -> X87", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 292 [arrowhead = none ];
292 -> 49;

293 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 293 [arrowhead = none ];
293 -> 50;

294 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 294 [arrowhead = none ];
294 -> 48;

295 [label="INSTANCE with matching:\nT55 -> T59\nX78 -> X87", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 295 [arrowhead = none , style=dashed];
295 -> 36[style=dashed];

296 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 296 [arrowhead = none ];
296 -> 52;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 297 [arrowhead = none ];
297 -> 53;

298 [label="EVAL with clause\nif(true, s(X109), X110, X111) :- ','(half(s(X109), X112), ','(times(X112, X110, X113), plus(X113, X113, X111))).\nand substitutionT56 -> true,\nT55 -> T75,\nX109 -> s(T75),\nT10 -> T76,\nX110 -> T76,\nT12 -> T78,\nX111 -> T78,\nT77 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 298 [arrowhead = none ];
298 -> 54;

299 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 299 [arrowhead = none ];
299 -> 55;

300 [label="EVAL with clause\nif(false, s(X544), X545, X546) :- ','(times(X544, X545, X547), plus(X545, X547, X546)).\nand substitutionT56 -> false,\nT55 -> T252,\nX544 -> s(T252),\nT10 -> T253,\nX545 -> T253,\nT12 -> T255,\nX546 -> T255,\nT254 -> T255", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 300 [arrowhead = none ];
300 -> 240;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 301 [arrowhead = none ];
301 -> 241;

302 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 302 [arrowhead = none ];
302 -> 56;

303 [label="SPLIT 2\nnew knowledge:\nT75 is ground\nT79 is ground\nreplacements:X112 -> T79", fontsize=14, style = filled, fillcolor = violet];
54 -> 303 [arrowhead = none ];
303 -> 57;

304 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 304 [arrowhead = none ];
304 -> 58;

305 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 305 [arrowhead = none ];
305 -> 69;

306 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nT76 is ground\nT88 is ground\nreplacements:X113 -> T88", fontsize=14, style = filled, fillcolor = violet];
57 -> 306 [arrowhead = none ];
306 -> 70;

307 [label="BACKTRACK\nfor clause: half(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 307 [arrowhead = none ];
307 -> 59;

308 [label="ONLY EVAL with clause\nhalf(s(s(X120)), s(X121)) :- half(X120, X121).\nand substitutionT75 -> T82,\nX120 -> T82,\nX121 -> X122,\nX112 -> s(X122)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 308 [arrowhead = none ];
308 -> 60;

309 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 309 [arrowhead = none ];
309 -> 61;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 310 [arrowhead = none ];
310 -> 62;

311 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 311 [arrowhead = none ];
311 -> 63;

312 [label="EVAL with clause\nhalf(0, 0).\nand substitutionT82 -> 0,\nX122 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 312 [arrowhead = none ];
312 -> 64;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 313 [arrowhead = none ];
313 -> 65;

314 [label="EVAL with clause\nhalf(s(s(X129)), s(X130)) :- half(X129, X130).\nand substitutionX129 -> T85,\nT82 -> s(s(T85)),\nX130 -> X131,\nX122 -> s(X131)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 314 [arrowhead = none ];
314 -> 67;

315 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 315 [arrowhead = none ];
315 -> 68;

316 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 316 [arrowhead = none ];
316 -> 66;

317 [label="INSTANCE with matching:\nT82 -> T85\nX122 -> X131", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 317 [arrowhead = none , style=dashed];
317 -> 60[style=dashed];

318 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 318 [arrowhead = none ];
318 -> 71;

319 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 319 [arrowhead = none ];
319 -> 175;

320 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 320 [arrowhead = none ];
320 -> 72;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 321 [arrowhead = none ];
321 -> 73;

322 [label="EVAL with clause\ntimes(0, X140, 0).\nand substitutionT79 -> 0,\nT76 -> T95,\nX140 -> T95,\nX113 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 322 [arrowhead = none ];
322 -> 74;

323 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 323 [arrowhead = none ];
323 -> 75;

324 [label="EVAL with clause\ntimes(s(X150), X151, X152) :- ','(even(s(X150), X153), if(X153, s(X150), X151, X152)).\nand substitutionX150 -> T100,\nT79 -> s(T100),\nT76 -> T101,\nX151 -> T101,\nX113 -> X154,\nX152 -> X154", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 324 [arrowhead = none ];
324 -> 77;

325 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 325 [arrowhead = none ];
325 -> 78;

326 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 326 [arrowhead = none ];
326 -> 76;

327 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
77 -> 327 [arrowhead = none ];
327 -> 79;

328 [label="SPLIT 2\nnew knowledge:\nT100 is ground\nT102 is ground\nreplacements:X153 -> T102", fontsize=14, style = filled, fillcolor = violet];
77 -> 328 [arrowhead = none ];
328 -> 80;

329 [label="INSTANCE with matching:\nT55 -> s(T100)\nX78 -> X153", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 329 [arrowhead = none , style=dashed];
329 -> 36[style=dashed];

330 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 330 [arrowhead = none ];
330 -> 81;

331 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 331 [arrowhead = none ];
331 -> 82;

332 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 332 [arrowhead = none ];
332 -> 83;

333 [label="EVAL with clause\nif(true, s(X181), X182, X183) :- ','(half(s(X181), X184), ','(times(X184, X182, X185), plus(X185, X185, X183))).\nand substitutionT102 -> true,\nT100 -> T113,\nX181 -> T113,\nT101 -> T114,\nX182 -> T114,\nX154 -> X186,\nX183 -> X186", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 333 [arrowhead = none ];
333 -> 84;

334 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 334 [arrowhead = none ];
334 -> 85;

335 [label="EVAL with clause\nif(false, s(X375), X376, X377) :- ','(times(X375, X376, X378), plus(X376, X378, X377)).\nand substitutionT102 -> false,\nT100 -> T165,\nX375 -> T165,\nT101 -> T166,\nX376 -> T166,\nX154 -> X379,\nX377 -> X379", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 335 [arrowhead = none ];
335 -> 163;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 336 [arrowhead = none ];
336 -> 164;

337 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
84 -> 337 [arrowhead = none ];
337 -> 86;

338 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nT115 is ground\nreplacements:X184 -> T115", fontsize=14, style = filled, fillcolor = violet];
84 -> 338 [arrowhead = none ];
338 -> 87;

339 [label="INSTANCE with matching:\nT82 -> s(T113)\nX122 -> X184", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 339 [arrowhead = none , style=dashed];
339 -> 60[style=dashed];

340 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 340 [arrowhead = none ];
340 -> 88;

341 [label="SPLIT 2\nnew knowledge:\nT115 is ground\nT114 is ground\nT118 is ground\nreplacements:X185 -> T118", fontsize=14, style = filled, fillcolor = violet];
87 -> 341 [arrowhead = none ];
341 -> 89;

342 [label="INSTANCE with matching:\nT79 -> T115\nT76 -> T114\nX113 -> X185", fontsize=14, style = filled, fillcolor = lightgrey];
88 -> 342 [arrowhead = none , style=dashed];
342 -> 69[style=dashed];

343 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
89 -> 343 [arrowhead = none ];
343 -> 90;

344 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 344 [arrowhead = none ];
344 -> 91;

345 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 345 [arrowhead = none ];
345 -> 92;

346 [label="EVAL with clause\nplus(0, X197, X197).\nand substitutionT118 -> 0,\nX197 -> 0,\nX186 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 346 [arrowhead = none ];
346 -> 93;

347 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 347 [arrowhead = none ];
347 -> 94;

348 [label="EVAL with clause\nplus(s(X206), X207, s(X208)) :- plus(X206, X207, X208).\nand substitutionX206 -> T123,\nT118 -> s(T123),\nX207 -> s(T123),\nX208 -> X209,\nX186 -> s(X209)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 348 [arrowhead = none ];
348 -> 96;

349 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 349 [arrowhead = none ];
349 -> 97;

350 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 350 [arrowhead = none ];
350 -> 95;

351 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 351 [arrowhead = none ];
351 -> 98;

352 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 352 [arrowhead = none ];
352 -> 99;

353 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
98 -> 353 [arrowhead = none ];
353 -> 100;

354 [label="EVAL with clause\nplus(0, X216, X216).\nand substitutionT123 -> 0,\nX216 -> s(0),\nX209 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 354 [arrowhead = none ];
354 -> 101;

355 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 355 [arrowhead = none ];
355 -> 102;

356 [label="EVAL with clause\nplus(s(X225), X226, s(X227)) :- plus(X225, X226, X227).\nand substitutionX225 -> T126,\nT123 -> s(T126),\nX226 -> s(s(T126)),\nX227 -> X228,\nX209 -> s(X228)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 356 [arrowhead = none ];
356 -> 104;

357 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 357 [arrowhead = none ];
357 -> 105;

358 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 358 [arrowhead = none ];
358 -> 103;

359 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
104 -> 359 [arrowhead = none ];
359 -> 106;

360 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 360 [arrowhead = none ];
360 -> 107;

361 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 361 [arrowhead = none ];
361 -> 108;

362 [label="EVAL with clause\nplus(0, X235, X235).\nand substitutionT126 -> 0,\nX235 -> s(s(0)),\nX228 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 362 [arrowhead = none ];
362 -> 109;

363 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
107 -> 363 [arrowhead = none ];
363 -> 110;

364 [label="EVAL with clause\nplus(s(X244), X245, s(X246)) :- plus(X244, X245, X246).\nand substitutionX244 -> T129,\nT126 -> s(T129),\nX245 -> s(s(s(T129))),\nX246 -> X247,\nX228 -> s(X247)", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 364 [arrowhead = none ];
364 -> 112;

365 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 365 [arrowhead = none ];
365 -> 113;

366 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 366 [arrowhead = none ];
366 -> 111;

367 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
112 -> 367 [arrowhead = none ];
367 -> 114;

368 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 368 [arrowhead = none ];
368 -> 115;

369 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 369 [arrowhead = none ];
369 -> 116;

370 [label="EVAL with clause\nplus(0, X254, X254).\nand substitutionT129 -> 0,\nX254 -> s(s(s(0))),\nX247 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 370 [arrowhead = none ];
370 -> 117;

371 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 371 [arrowhead = none ];
371 -> 118;

372 [label="EVAL with clause\nplus(s(X263), X264, s(X265)) :- plus(X263, X264, X265).\nand substitutionX263 -> T132,\nT129 -> s(T132),\nX264 -> s(s(s(s(T132)))),\nX265 -> X266,\nX247 -> s(X266)", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 372 [arrowhead = none ];
372 -> 120;

373 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 373 [arrowhead = none ];
373 -> 121;

374 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
117 -> 374 [arrowhead = none ];
374 -> 119;

375 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
120 -> 375 [arrowhead = none ];
375 -> 122;

376 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 376 [arrowhead = none ];
376 -> 123;

377 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 377 [arrowhead = none ];
377 -> 124;

378 [label="EVAL with clause\nplus(0, X273, X273).\nand substitutionT132 -> 0,\nX273 -> s(s(s(s(0)))),\nX266 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 378 [arrowhead = none ];
378 -> 125;

379 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 379 [arrowhead = none ];
379 -> 126;

380 [label="EVAL with clause\nplus(s(X282), X283, s(X284)) :- plus(X282, X283, X284).\nand substitutionX282 -> T135,\nT132 -> s(T135),\nX283 -> s(s(s(s(s(T135))))),\nX284 -> X285,\nX266 -> s(X285)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 380 [arrowhead = none ];
380 -> 128;

381 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 381 [arrowhead = none ];
381 -> 129;

382 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
125 -> 382 [arrowhead = none ];
382 -> 127;

383 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
128 -> 383 [arrowhead = none ];
383 -> 130;

384 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
130 -> 384 [arrowhead = none ];
384 -> 131;

385 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
130 -> 385 [arrowhead = none ];
385 -> 132;

386 [label="EVAL with clause\nplus(0, X292, X292).\nand substitutionT135 -> 0,\nX292 -> s(s(s(s(s(0))))),\nX285 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
131 -> 386 [arrowhead = none ];
386 -> 133;

387 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
131 -> 387 [arrowhead = none ];
387 -> 134;

388 [label="EVAL with clause\nplus(s(X301), X302, s(X303)) :- plus(X301, X302, X303).\nand substitutionX301 -> T138,\nT135 -> s(T138),\nX302 -> s(s(s(s(s(s(T138)))))),\nX303 -> X304,\nX285 -> s(X304)", fontsize=14, style = filled, fillcolor = lightblue];
132 -> 388 [arrowhead = none ];
388 -> 136;

389 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
132 -> 389 [arrowhead = none ];
389 -> 137;

390 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
133 -> 390 [arrowhead = none ];
390 -> 135;

391 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
136 -> 391 [arrowhead = none ];
391 -> 138;

392 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
138 -> 392 [arrowhead = none ];
392 -> 139;

393 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
138 -> 393 [arrowhead = none ];
393 -> 140;

394 [label="EVAL with clause\nplus(0, X311, X311).\nand substitutionT138 -> 0,\nX311 -> s(s(s(s(s(s(0)))))),\nX304 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 394 [arrowhead = none ];
394 -> 141;

395 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
139 -> 395 [arrowhead = none ];
395 -> 142;

396 [label="EVAL with clause\nplus(s(X320), X321, s(X322)) :- plus(X320, X321, X322).\nand substitutionX320 -> T141,\nT138 -> s(T141),\nX321 -> s(s(s(s(s(s(s(T141))))))),\nX322 -> X323,\nX304 -> s(X323)", fontsize=14, style = filled, fillcolor = lightblue];
140 -> 396 [arrowhead = none ];
396 -> 144;

397 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 397 [arrowhead = none ];
397 -> 145;

398 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
141 -> 398 [arrowhead = none ];
398 -> 143;

399 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
144 -> 399 [arrowhead = none ];
399 -> 146;

400 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 400 [arrowhead = none ];
400 -> 147;

401 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 401 [arrowhead = none ];
401 -> 148;

402 [label="EVAL with clause\nplus(0, X330, X330).\nand substitutionT141 -> 0,\nX330 -> s(s(s(s(s(s(s(0))))))),\nX323 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
147 -> 402 [arrowhead = none ];
402 -> 149;

403 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
147 -> 403 [arrowhead = none ];
403 -> 150;

404 [label="EVAL with clause\nplus(s(X339), X340, s(X341)) :- plus(X339, X340, X341).\nand substitutionX339 -> T144,\nT141 -> s(T144),\nX340 -> s(s(s(s(s(s(s(s(T144)))))))),\nX341 -> X342,\nX323 -> s(X342)", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 404 [arrowhead = none ];
404 -> 152;

405 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
148 -> 405 [arrowhead = none ];
405 -> 153;

406 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
149 -> 406 [arrowhead = none ];
406 -> 151;

407 [label="GENERALIZATION\nT145 <-- s(s(s(s(s(s(s(T144)))))))\n\nNew Knowledge:\nT145 is ground", fontsize=14, style = filled, fillcolor = violet];
152 -> 407 [arrowhead = none ];
407 -> 154;

408 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
154 -> 408 [arrowhead = none ];
408 -> 155;

409 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
155 -> 409 [arrowhead = none ];
409 -> 156;

410 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
155 -> 410 [arrowhead = none ];
410 -> 157;

411 [label="EVAL with clause\nplus(0, X351, X351).\nand substitutionT144 -> 0,\nT145 -> T152,\nX351 -> s(T152),\nX342 -> s(T152)", fontsize=14, style = filled, fillcolor = lightblue];
156 -> 411 [arrowhead = none ];
411 -> 158;

412 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
156 -> 412 [arrowhead = none ];
412 -> 159;

413 [label="EVAL with clause\nplus(s(X360), X361, s(X362)) :- plus(X360, X361, X362).\nand substitutionX360 -> T157,\nT144 -> s(T157),\nT145 -> T158,\nX361 -> s(T158),\nX362 -> X363,\nX342 -> s(X363)", fontsize=14, style = filled, fillcolor = lightblue];
157 -> 413 [arrowhead = none ];
413 -> 161;

414 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
157 -> 414 [arrowhead = none ];
414 -> 162;

415 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
158 -> 415 [arrowhead = none ];
415 -> 160;

416 [label="INSTANCE with matching:\nT144 -> T157\nT145 -> T158\nX342 -> X363", fontsize=14, style = filled, fillcolor = lightgrey];
161 -> 416 [arrowhead = none , style=dashed];
416 -> 154[style=dashed];

417 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
163 -> 417 [arrowhead = none ];
417 -> 165;

418 [label="SPLIT 2\nnew knowledge:\nT165 is ground\nT166 is ground\nT169 is ground\nreplacements:X378 -> T169", fontsize=14, style = filled, fillcolor = violet];
163 -> 418 [arrowhead = none ];
418 -> 166;

419 [label="INSTANCE with matching:\nT79 -> T165\nT76 -> T166\nX113 -> X378", fontsize=14, style = filled, fillcolor = lightgrey];
165 -> 419 [arrowhead = none , style=dashed];
419 -> 69[style=dashed];

420 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
166 -> 420 [arrowhead = none ];
420 -> 167;

421 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
167 -> 421 [arrowhead = none ];
421 -> 168;

422 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
167 -> 422 [arrowhead = none ];
422 -> 169;

423 [label="EVAL with clause\nplus(0, X390, X390).\nand substitutionT166 -> 0,\nT169 -> T178,\nX390 -> T178,\nX379 -> T178", fontsize=14, style = filled, fillcolor = lightblue];
168 -> 423 [arrowhead = none ];
423 -> 170;

424 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
168 -> 424 [arrowhead = none ];
424 -> 171;

425 [label="EVAL with clause\nplus(s(X399), X400, s(X401)) :- plus(X399, X400, X401).\nand substitutionX399 -> T183,\nT166 -> s(T183),\nT169 -> T184,\nX400 -> T184,\nX401 -> X402,\nX379 -> s(X402)", fontsize=14, style = filled, fillcolor = lightblue];
169 -> 425 [arrowhead = none ];
425 -> 173;

426 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
169 -> 426 [arrowhead = none ];
426 -> 174;

427 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
170 -> 427 [arrowhead = none ];
427 -> 172;

428 [label="INSTANCE with matching:\nT166 -> T183\nT169 -> T184\nX379 -> X402", fontsize=14, style = filled, fillcolor = lightgrey];
173 -> 428 [arrowhead = none , style=dashed];
428 -> 166[style=dashed];

429 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
175 -> 429 [arrowhead = none ];
429 -> 176;

430 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
175 -> 430 [arrowhead = none ];
430 -> 177;

431 [label="EVAL with clause\nplus(0, X411, X411).\nand substitutionT88 -> 0,\nX411 -> 0,\nT78 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
176 -> 431 [arrowhead = none ];
431 -> 178;

432 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
176 -> 432 [arrowhead = none ];
432 -> 179;

433 [label="EVAL with clause\nplus(s(X418), X419, s(X420)) :- plus(X418, X419, X420).\nand substitutionX418 -> T191,\nT88 -> s(T191),\nX419 -> s(T191),\nX420 -> T193,\nT78 -> s(T193),\nT192 -> T193", fontsize=14, style = filled, fillcolor = lightblue];
177 -> 433 [arrowhead = none ];
433 -> 181;

434 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
177 -> 434 [arrowhead = none ];
434 -> 182;

435 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
178 -> 435 [arrowhead = none ];
435 -> 180;

436 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
181 -> 436 [arrowhead = none ];
436 -> 183;

437 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
183 -> 437 [arrowhead = none ];
437 -> 184;

438 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
183 -> 438 [arrowhead = none ];
438 -> 185;

439 [label="EVAL with clause\nplus(0, X427, X427).\nand substitutionT191 -> 0,\nX427 -> s(0),\nT193 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
184 -> 439 [arrowhead = none ];
439 -> 186;

440 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
184 -> 440 [arrowhead = none ];
440 -> 187;

441 [label="EVAL with clause\nplus(s(X434), X435, s(X436)) :- plus(X434, X435, X436).\nand substitutionX434 -> T198,\nT191 -> s(T198),\nX435 -> s(s(T198)),\nX436 -> T200,\nT193 -> s(T200),\nT199 -> T200", fontsize=14, style = filled, fillcolor = lightblue];
185 -> 441 [arrowhead = none ];
441 -> 189;

442 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
185 -> 442 [arrowhead = none ];
442 -> 190;

443 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
186 -> 443 [arrowhead = none ];
443 -> 188;

444 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
189 -> 444 [arrowhead = none ];
444 -> 191;

445 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
191 -> 445 [arrowhead = none ];
445 -> 192;

446 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
191 -> 446 [arrowhead = none ];
446 -> 193;

447 [label="EVAL with clause\nplus(0, X443, X443).\nand substitutionT198 -> 0,\nX443 -> s(s(0)),\nT200 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
192 -> 447 [arrowhead = none ];
447 -> 194;

448 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
192 -> 448 [arrowhead = none ];
448 -> 195;

449 [label="EVAL with clause\nplus(s(X450), X451, s(X452)) :- plus(X450, X451, X452).\nand substitutionX450 -> T205,\nT198 -> s(T205),\nX451 -> s(s(s(T205))),\nX452 -> T207,\nT200 -> s(T207),\nT206 -> T207", fontsize=14, style = filled, fillcolor = lightblue];
193 -> 449 [arrowhead = none ];
449 -> 197;

450 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
193 -> 450 [arrowhead = none ];
450 -> 198;

451 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
194 -> 451 [arrowhead = none ];
451 -> 196;

452 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
197 -> 452 [arrowhead = none ];
452 -> 199;

453 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
199 -> 453 [arrowhead = none ];
453 -> 200;

454 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
199 -> 454 [arrowhead = none ];
454 -> 201;

455 [label="EVAL with clause\nplus(0, X459, X459).\nand substitutionT205 -> 0,\nX459 -> s(s(s(0))),\nT207 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
200 -> 455 [arrowhead = none ];
455 -> 202;

456 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
200 -> 456 [arrowhead = none ];
456 -> 203;

457 [label="EVAL with clause\nplus(s(X466), X467, s(X468)) :- plus(X466, X467, X468).\nand substitutionX466 -> T212,\nT205 -> s(T212),\nX467 -> s(s(s(s(T212)))),\nX468 -> T214,\nT207 -> s(T214),\nT213 -> T214", fontsize=14, style = filled, fillcolor = lightblue];
201 -> 457 [arrowhead = none ];
457 -> 205;

458 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
201 -> 458 [arrowhead = none ];
458 -> 206;

459 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
202 -> 459 [arrowhead = none ];
459 -> 204;

460 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
205 -> 460 [arrowhead = none ];
460 -> 207;

461 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
207 -> 461 [arrowhead = none ];
461 -> 208;

462 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
207 -> 462 [arrowhead = none ];
462 -> 209;

463 [label="EVAL with clause\nplus(0, X475, X475).\nand substitutionT212 -> 0,\nX475 -> s(s(s(s(0)))),\nT214 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
208 -> 463 [arrowhead = none ];
463 -> 210;

464 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
208 -> 464 [arrowhead = none ];
464 -> 211;

465 [label="EVAL with clause\nplus(s(X482), X483, s(X484)) :- plus(X482, X483, X484).\nand substitutionX482 -> T219,\nT212 -> s(T219),\nX483 -> s(s(s(s(s(T219))))),\nX484 -> T221,\nT214 -> s(T221),\nT220 -> T221", fontsize=14, style = filled, fillcolor = lightblue];
209 -> 465 [arrowhead = none ];
465 -> 213;

466 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
209 -> 466 [arrowhead = none ];
466 -> 214;

467 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
210 -> 467 [arrowhead = none ];
467 -> 212;

468 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
213 -> 468 [arrowhead = none ];
468 -> 215;

469 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
215 -> 469 [arrowhead = none ];
469 -> 216;

470 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
215 -> 470 [arrowhead = none ];
470 -> 217;

471 [label="EVAL with clause\nplus(0, X491, X491).\nand substitutionT219 -> 0,\nX491 -> s(s(s(s(s(0))))),\nT221 -> s(s(s(s(s(0)))))", fontsize=14, style = filled, fillcolor = lightblue];
216 -> 471 [arrowhead = none ];
471 -> 218;

472 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
216 -> 472 [arrowhead = none ];
472 -> 219;

473 [label="EVAL with clause\nplus(s(X498), X499, s(X500)) :- plus(X498, X499, X500).\nand substitutionX498 -> T226,\nT219 -> s(T226),\nX499 -> s(s(s(s(s(s(T226)))))),\nX500 -> T228,\nT221 -> s(T228),\nT227 -> T228", fontsize=14, style = filled, fillcolor = lightblue];
217 -> 473 [arrowhead = none ];
473 -> 221;

474 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
217 -> 474 [arrowhead = none ];
474 -> 222;

475 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
218 -> 475 [arrowhead = none ];
475 -> 220;

476 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
221 -> 476 [arrowhead = none ];
476 -> 223;

477 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
223 -> 477 [arrowhead = none ];
477 -> 224;

478 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
223 -> 478 [arrowhead = none ];
478 -> 225;

479 [label="EVAL with clause\nplus(0, X507, X507).\nand substitutionT226 -> 0,\nX507 -> s(s(s(s(s(s(0)))))),\nT228 -> s(s(s(s(s(s(0))))))", fontsize=14, style = filled, fillcolor = lightblue];
224 -> 479 [arrowhead = none ];
479 -> 226;

480 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 480 [arrowhead = none ];
480 -> 227;

481 [label="EVAL with clause\nplus(s(X514), X515, s(X516)) :- plus(X514, X515, X516).\nand substitutionX514 -> T233,\nT226 -> s(T233),\nX515 -> s(s(s(s(s(s(s(T233))))))),\nX516 -> T235,\nT228 -> s(T235),\nT234 -> T235", fontsize=14, style = filled, fillcolor = lightblue];
225 -> 481 [arrowhead = none ];
481 -> 229;

482 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
225 -> 482 [arrowhead = none ];
482 -> 230;

483 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
226 -> 483 [arrowhead = none ];
483 -> 228;

484 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
229 -> 484 [arrowhead = none ];
484 -> 231;

485 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
231 -> 485 [arrowhead = none ];
485 -> 232;

486 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
231 -> 486 [arrowhead = none ];
486 -> 233;

487 [label="EVAL with clause\nplus(0, X523, X523).\nand substitutionT233 -> 0,\nX523 -> s(s(s(s(s(s(s(0))))))),\nT235 -> s(s(s(s(s(s(s(0)))))))", fontsize=14, style = filled, fillcolor = lightblue];
232 -> 487 [arrowhead = none ];
487 -> 234;

488 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
232 -> 488 [arrowhead = none ];
488 -> 235;

489 [label="EVAL with clause\nplus(s(X530), X531, s(X532)) :- plus(X530, X531, X532).\nand substitutionX530 -> T240,\nT233 -> s(T240),\nX531 -> s(s(s(s(s(s(s(s(T240)))))))),\nX532 -> T242,\nT235 -> s(T242),\nT241 -> T242", fontsize=14, style = filled, fillcolor = lightblue];
233 -> 489 [arrowhead = none ];
489 -> 237;

490 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
233 -> 490 [arrowhead = none ];
490 -> 238;

491 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
234 -> 491 [arrowhead = none ];
491 -> 236;

492 [label="GENERALIZATION\nT243 <-- s(s(s(s(s(s(s(T240)))))))\n\nNew Knowledge:\nT243 is ground", fontsize=14, style = filled, fillcolor = violet];
237 -> 492 [arrowhead = none ];
492 -> 239;

493 [label="INSTANCE with matching:\nT21 -> T240\nT27 -> s(T243)\nT23 -> T242", fontsize=14, style = filled, fillcolor = lightgrey];
239 -> 493 [arrowhead = none , style=dashed];
493 -> 19[style=dashed];

494 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
240 -> 494 [arrowhead = none ];
494 -> 242;

495 [label="SPLIT 2\nnew knowledge:\nT252 is ground\nT253 is ground\nT256 is ground\nreplacements:X547 -> T256", fontsize=14, style = filled, fillcolor = violet];
240 -> 495 [arrowhead = none ];
495 -> 243;

496 [label="INSTANCE with matching:\nT79 -> s(T252)\nT76 -> T253\nX113 -> X547", fontsize=14, style = filled, fillcolor = lightgrey];
242 -> 496 [arrowhead = none , style=dashed];
496 -> 69[style=dashed];

497 [label="INSTANCE with matching:\nT21 -> T253\nT27 -> T256\nT23 -> T255", fontsize=14, style = filled, fillcolor = lightgrey];
243 -> 497 [arrowhead = none , style=dashed];
497 -> 19[style=dashed];

}

(2) Obligation:

Triples:

plusA(s(X1), X2, s(X3)) :- plusA(X1, X2, X3).
evenB(s(s(X1)), X2) :- evenB(X1, X2).
halfC(s(s(X1)), s(X2)) :- halfC(X1, X2).
timesD(s(X1), X2, X3) :- evenB(s(X1), X4).
timesD(s(X1), X2, X3) :- ','(evencB(s(X1), true), halfC(s(X1), X4)).
timesD(s(X1), X2, X3) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X4), timesD(X4, X2, X5))).
timesD(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X4), ','(timescD(X4, X2, s(s(s(s(s(s(s(s(X5))))))))), plusE(X5, s(s(s(s(s(s(s(X5))))))), X3)))).
timesD(s(X1), X2, X3) :- ','(evencB(s(X1), false), timesD(X1, X2, X4)).
timesD(s(X1), X2, X3) :- ','(evencB(s(X1), false), ','(timescD(X1, X2, X4), plusF(X2, X4, X3))).
plusE(s(X1), X2, s(X3)) :- plusE(X1, X2, X3).
plusF(s(X1), X2, s(X3)) :- plusF(X1, X2, X3).
timesI(s(0), X1, X2) :- ','(timescG(X1, X3), plusA(X1, X3, X2)).
timesI(s(s(X1)), X2, X3) :- evenB(X1, X4).
timesI(s(s(X1)), X2, X3) :- ','(evencB(X1, true), halfC(X1, X4)).
timesI(s(s(X1)), X2, X3) :- ','(evencB(X1, true), ','(halfcH(X1, X4), timesD(X4, X2, X5))).
timesI(s(s(X1)), X2, s(s(s(s(s(s(s(s(X3))))))))) :- ','(evencB(X1, true), ','(halfcH(X1, X4), ','(timescD(X4, X2, s(s(s(s(s(s(s(s(X5))))))))), plusA(X5, s(s(s(s(s(s(s(s(X5)))))))), X3)))).
timesI(s(s(X1)), X2, X3) :- ','(evencB(X1, false), timesD(s(X1), X2, X4)).
timesI(s(s(X1)), X2, X3) :- ','(evencB(X1, false), ','(timescD(s(X1), X2, X4), plusA(X2, X4, X3))).

Clauses:

pluscA(0, X1, X1).
pluscA(s(X1), X2, s(X3)) :- pluscA(X1, X2, X3).
evencB(0, true).
evencB(s(0), false).
evencB(s(s(X1)), X2) :- evencB(X1, X2).
halfcC(0, 0).
halfcC(s(s(X1)), s(X2)) :- halfcC(X1, X2).
timescD(0, X1, 0).
timescD(s(X1), X2, 0) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, 0))).
timescD(s(X1), X2, s(s(0))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(0)))).
timescD(s(X1), X2, s(s(s(s(0))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(0))))).
timescD(s(X1), X2, s(s(s(s(s(s(0))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(s(0)))))).
timescD(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(s(s(0))))))).
timescD(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(s(s(s(0)))))))).
timescD(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(s(s(s(s(0))))))))).
timescD(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X3), timescD(X3, X2, s(s(s(s(s(s(s(0)))))))))).
timescD(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) :- ','(evencB(s(X1), true), ','(halfcC(s(X1), X4), ','(timescD(X4, X2, s(s(s(s(s(s(s(s(X5))))))))), pluscE(X5, s(s(s(s(s(s(s(X5))))))), X3)))).
timescD(s(X1), X2, X3) :- ','(evencB(s(X1), false), ','(timescD(X1, X2, X4), pluscF(X2, X4, X3))).
pluscE(0, X1, s(X1)).
pluscE(s(X1), X2, s(X3)) :- pluscE(X1, X2, X3).
pluscF(0, X1, X1).
pluscF(s(X1), X2, s(X3)) :- pluscF(X1, X2, X3).
timescG(X1, 0).
halfcH(X1, s(X2)) :- halfcC(X1, X2).

Afs:

timesI(x1, x2, x3) = timesI(x1, x2)

(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:
timesI_in: (b,b,f)
plusA_in: (b,b,f)
evenB_in: (b,f)
evencB_in: (b,b)
halfC_in: (b,f)
halfcH_in: (b,f)
halfcC_in: (b,f)
timesD_in: (b,b,f)
timescD_in: (b,b,f) (b,b,b)
pluscF_in: (b,b,f) (b,b,b)
pluscE_in: (b,b,f) (b,b,b)
plusE_in: (b,b,f)
plusF_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TIMESI_IN_GGA(s(0), X1, X2) → U19_GGA(X1, X2, timescG_in_ga(X1, X3))
U19_GGA(X1, X2, timescG_out_ga(X1, X3)) → U20_GGA(X1, X2, plusA_in_gga(X1, X3, X2))
U19_GGA(X1, X2, timescG_out_ga(X1, X3)) → PLUSA_IN_GGA(X1, X3, X2)
PLUSA_IN_GGA(s(X1), X2, s(X3)) → U1_GGA(X1, X2, X3, plusA_in_gga(X1, X2, X3))
PLUSA_IN_GGA(s(X1), X2, s(X3)) → PLUSA_IN_GGA(X1, X2, X3)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U21_GGA(X1, X2, X3, evenB_in_ga(X1, X4))
TIMESI_IN_GGA(s(s(X1)), X2, X3) → EVENB_IN_GA(X1, X4)
EVENB_IN_GA(s(s(X1)), X2) → U2_GA(X1, X2, evenB_in_ga(X1, X2))
EVENB_IN_GA(s(s(X1)), X2) → EVENB_IN_GA(X1, X2)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U22_GGA(X1, X2, X3, evencB_in_gg(X1, true))
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U23_GGA(X1, X2, X3, halfC_in_ga(X1, X4))
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → HALFC_IN_GA(X1, X4)
HALFC_IN_GA(s(s(X1)), s(X2)) → U3_GA(X1, X2, halfC_in_ga(X1, X2))
HALFC_IN_GA(s(s(X1)), s(X2)) → HALFC_IN_GA(X1, X2)
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U24_GGA(X1, X2, X3, halfcH_in_ga(X1, X4))
U24_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → U25_GGA(X1, X2, X3, timesD_in_gga(X4, X2, X5))
U24_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, X3) → U4_GGA(X1, X2, X3, evenB_in_ga(s(X1), X4))
TIMESD_IN_GGA(s(X1), X2, X3) → EVENB_IN_GA(s(X1), X4)
TIMESD_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U6_GGA(X1, X2, X3, halfC_in_ga(s(X1), X4))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → HALFC_IN_GA(s(X1), X4)
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U8_GGA(X1, X2, X3, timesD_in_gga(X4, X2, X5))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U9_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U9_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U10_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U10_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U11_GGA(X1, X2, X3, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
U11_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U12_GGA(X1, X2, X3, plusE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
U11_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → PLUSE_IN_GGA(X5, s(s(s(s(s(s(s(X5))))))), X3)
PLUSE_IN_GGA(s(X1), X2, s(X3)) → U17_GGA(X1, X2, X3, plusE_in_gga(X1, X2, X3))
PLUSE_IN_GGA(s(X1), X2, s(X3)) → PLUSE_IN_GGA(X1, X2, X3)
TIMESD_IN_GGA(s(X1), X2, X3) → U13_GGA(X1, X2, X3, evencB_in_gg(s(X1), false))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → U14_GGA(X1, X2, X3, timesD_in_gga(X1, X2, X4))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2, X4)
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → U15_GGA(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U15_GGA(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U16_GGA(X1, X2, X3, plusF_in_gga(X2, X4, X3))
U15_GGA(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → PLUSF_IN_GGA(X2, X4, X3)
PLUSF_IN_GGA(s(X1), X2, s(X3)) → U18_GGA(X1, X2, X3, plusF_in_gga(X1, X2, X3))
PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)
TIMESI_IN_GGA(s(s(X1)), X2, s(s(s(s(s(s(s(s(X3))))))))) → U26_GGA(X1, X2, X3, evencB_in_gg(X1, true))
U26_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U27_GGA(X1, X2, X3, halfcH_in_ga(X1, X4))
U27_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → U28_GGA(X1, X2, X3, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
U28_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U29_GGA(X1, X2, X3, plusA_in_gga(X5, s(s(s(s(s(s(s(s(X5)))))))), X3))
U28_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → PLUSA_IN_GGA(X5, s(s(s(s(s(s(s(s(X5)))))))), X3)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U30_GGA(X1, X2, X3, evencB_in_gg(X1, false))
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → U31_GGA(X1, X2, X3, timesD_in_gga(s(X1), X2, X4))
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → TIMESD_IN_GGA(s(X1), X2, X4)
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → U32_GGA(X1, X2, X3, timescD_in_gga(s(X1), X2, X4))
U32_GGA(X1, X2, X3, timescD_out_gga(s(X1), X2, X4)) → U33_GGA(X1, X2, X3, plusA_in_gga(X2, X4, X3))
U32_GGA(X1, X2, X3, timescD_out_gga(s(X1), X2, X4)) → PLUSA_IN_GGA(X2, X4, X3)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
plusA_in_gga(x1, x2, x3) = plusA_in_gga(x1, x2)
evenB_in_ga(x1, x2) = evenB_in_ga(x1)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfC_in_ga(x1, x2) = halfC_in_ga(x1)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timesD_in_gga(x1, x2, x3) = timesD_in_gga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusE_in_gga(x1, x2, x3) = plusE_in_gga(x1, x2)
plusF_in_gga(x1, x2, x3) = plusF_in_gga(x1, x2)
TIMESI_IN_GGA(x1, x2, x3) = TIMESI_IN_GGA(x1, x2)
U19_GGA(x1, x2, x3) = U19_GGA(x1, x3)
U20_GGA(x1, x2, x3) = U20_GGA(x1, x3)
PLUSA_IN_GGA(x1, x2, x3) = PLUSA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
U21_GGA(x1, x2, x3, x4) = U21_GGA(x1, x2, x4)
EVENB_IN_GA(x1, x2) = EVENB_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U22_GGA(x1, x2, x3, x4) = U22_GGA(x1, x2, x4)
U23_GGA(x1, x2, x3, x4) = U23_GGA(x1, x2, x4)
HALFC_IN_GA(x1, x2) = HALFC_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
U24_GGA(x1, x2, x3, x4) = U24_GGA(x1, x2, x4)
U25_GGA(x1, x2, x3, x4) = U25_GGA(x1, x2, x4)
TIMESD_IN_GGA(x1, x2, x3) = TIMESD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4) = U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4) = U12_GGA(x1, x2, x4)
PLUSE_IN_GGA(x1, x2, x3) = PLUSE_IN_GGA(x1, x2)
U17_GGA(x1, x2, x3, x4) = U17_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4) = U13_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4) = U14_GGA(x1, x2, x4)
U15_GGA(x1, x2, x3, x4) = U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4) = U16_GGA(x1, x2, x4)
PLUSF_IN_GGA(x1, x2, x3) = PLUSF_IN_GGA(x1, x2)
U18_GGA(x1, x2, x3, x4) = U18_GGA(x1, x2, x4)
U26_GGA(x1, x2, x3, x4) = U26_GGA(x1, x2, x4)
U27_GGA(x1, x2, x3, x4) = U27_GGA(x1, x2, x4)
U28_GGA(x1, x2, x3, x4) = U28_GGA(x1, x2, x4)
U29_GGA(x1, x2, x3, x4) = U29_GGA(x1, x2, x4)
U30_GGA(x1, x2, x3, x4) = U30_GGA(x1, x2, x4)
U31_GGA(x1, x2, x3, x4) = U31_GGA(x1, x2, x4)
U32_GGA(x1, x2, x3, x4) = U32_GGA(x1, x2, x4)
U33_GGA(x1, x2, x3, x4) = U33_GGA(x1, x2, x4)

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:

TIMESI_IN_GGA(s(0), X1, X2) → U19_GGA(X1, X2, timescG_in_ga(X1, X3))
U19_GGA(X1, X2, timescG_out_ga(X1, X3)) → U20_GGA(X1, X2, plusA_in_gga(X1, X3, X2))
U19_GGA(X1, X2, timescG_out_ga(X1, X3)) → PLUSA_IN_GGA(X1, X3, X2)
PLUSA_IN_GGA(s(X1), X2, s(X3)) → U1_GGA(X1, X2, X3, plusA_in_gga(X1, X2, X3))
PLUSA_IN_GGA(s(X1), X2, s(X3)) → PLUSA_IN_GGA(X1, X2, X3)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U21_GGA(X1, X2, X3, evenB_in_ga(X1, X4))
TIMESI_IN_GGA(s(s(X1)), X2, X3) → EVENB_IN_GA(X1, X4)
EVENB_IN_GA(s(s(X1)), X2) → U2_GA(X1, X2, evenB_in_ga(X1, X2))
EVENB_IN_GA(s(s(X1)), X2) → EVENB_IN_GA(X1, X2)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U22_GGA(X1, X2, X3, evencB_in_gg(X1, true))
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U23_GGA(X1, X2, X3, halfC_in_ga(X1, X4))
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → HALFC_IN_GA(X1, X4)
HALFC_IN_GA(s(s(X1)), s(X2)) → U3_GA(X1, X2, halfC_in_ga(X1, X2))
HALFC_IN_GA(s(s(X1)), s(X2)) → HALFC_IN_GA(X1, X2)
U22_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U24_GGA(X1, X2, X3, halfcH_in_ga(X1, X4))
U24_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → U25_GGA(X1, X2, X3, timesD_in_gga(X4, X2, X5))
U24_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, X3) → U4_GGA(X1, X2, X3, evenB_in_ga(s(X1), X4))
TIMESD_IN_GGA(s(X1), X2, X3) → EVENB_IN_GA(s(X1), X4)
TIMESD_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U6_GGA(X1, X2, X3, halfC_in_ga(s(X1), X4))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → HALFC_IN_GA(s(X1), X4)
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U8_GGA(X1, X2, X3, timesD_in_gga(X4, X2, X5))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U9_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U9_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U10_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U10_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U11_GGA(X1, X2, X3, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
U11_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U12_GGA(X1, X2, X3, plusE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
U11_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → PLUSE_IN_GGA(X5, s(s(s(s(s(s(s(X5))))))), X3)
PLUSE_IN_GGA(s(X1), X2, s(X3)) → U17_GGA(X1, X2, X3, plusE_in_gga(X1, X2, X3))
PLUSE_IN_GGA(s(X1), X2, s(X3)) → PLUSE_IN_GGA(X1, X2, X3)
TIMESD_IN_GGA(s(X1), X2, X3) → U13_GGA(X1, X2, X3, evencB_in_gg(s(X1), false))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → U14_GGA(X1, X2, X3, timesD_in_gga(X1, X2, X4))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2, X4)
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → U15_GGA(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U15_GGA(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U16_GGA(X1, X2, X3, plusF_in_gga(X2, X4, X3))
U15_GGA(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → PLUSF_IN_GGA(X2, X4, X3)
PLUSF_IN_GGA(s(X1), X2, s(X3)) → U18_GGA(X1, X2, X3, plusF_in_gga(X1, X2, X3))
PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)
TIMESI_IN_GGA(s(s(X1)), X2, s(s(s(s(s(s(s(s(X3))))))))) → U26_GGA(X1, X2, X3, evencB_in_gg(X1, true))
U26_GGA(X1, X2, X3, evencB_out_gg(X1, true)) → U27_GGA(X1, X2, X3, halfcH_in_ga(X1, X4))
U27_GGA(X1, X2, X3, halfcH_out_ga(X1, X4)) → U28_GGA(X1, X2, X3, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
U28_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U29_GGA(X1, X2, X3, plusA_in_gga(X5, s(s(s(s(s(s(s(s(X5)))))))), X3))
U28_GGA(X1, X2, X3, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → PLUSA_IN_GGA(X5, s(s(s(s(s(s(s(s(X5)))))))), X3)
TIMESI_IN_GGA(s(s(X1)), X2, X3) → U30_GGA(X1, X2, X3, evencB_in_gg(X1, false))
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → U31_GGA(X1, X2, X3, timesD_in_gga(s(X1), X2, X4))
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → TIMESD_IN_GGA(s(X1), X2, X4)
U30_GGA(X1, X2, X3, evencB_out_gg(X1, false)) → U32_GGA(X1, X2, X3, timescD_in_gga(s(X1), X2, X4))
U32_GGA(X1, X2, X3, timescD_out_gga(s(X1), X2, X4)) → U33_GGA(X1, X2, X3, plusA_in_gga(X2, X4, X3))
U32_GGA(X1, X2, X3, timescD_out_gga(s(X1), X2, X4)) → PLUSA_IN_GGA(X2, X4, X3)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 41 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
PLUSF_IN_GGA(x1, x2, x3) = PLUSF_IN_GGA(x1, x2)

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:

PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUSF_IN_GGA(x1, x2, x3) = PLUSF_IN_GGA(x1, x2)

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:

PLUSF_IN_GGA(s(X1), X2) → PLUSF_IN_GGA(X1, X2)

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:

PLUSF_IN_GGA(s(X1), X2) → PLUSF_IN_GGA(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

PLUSE_IN_GGA(s(X1), X2, s(X3)) → PLUSE_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
PLUSE_IN_GGA(x1, x2, x3) = PLUSE_IN_GGA(x1, x2)

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:

PLUSE_IN_GGA(s(X1), X2, s(X3)) → PLUSE_IN_GGA(X1, X2, X3)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUSE_IN_GGA(x1, x2, x3) = PLUSE_IN_GGA(x1, x2)

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:

PLUSE_IN_GGA(s(X1), X2) → PLUSE_IN_GGA(X1, X2)

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

(19) 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:

PLUSE_IN_GGA(s(X1), X2) → PLUSE_IN_GGA(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

HALFC_IN_GA(s(s(X1)), s(X2)) → HALFC_IN_GA(X1, X2)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
HALFC_IN_GA(x1, x2) = HALFC_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

HALFC_IN_GA(s(s(X1)), s(X2)) → HALFC_IN_GA(X1, X2)

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

HALFC_IN_GA(s(s(X1))) → HALFC_IN_GA(X1)

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

(26) 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:

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

(27) YES

(28) Obligation:

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

EVENB_IN_GA(s(s(X1)), X2) → EVENB_IN_GA(X1, X2)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
EVENB_IN_GA(x1, x2) = EVENB_IN_GA(x1)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

EVENB_IN_GA(s(s(X1)), X2) → EVENB_IN_GA(X1, X2)

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

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

EVENB_IN_GA(s(s(X1))) → EVENB_IN_GA(X1)

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

(33) 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:

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

(34) YES

(35) Obligation:

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

TIMESD_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, X3) → U13_GGA(X1, X2, X3, evencB_in_gg(s(X1), false))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2, X4)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
TIMESD_IN_GGA(x1, x2, x3) = TIMESD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4) = U13_GGA(x1, x2, x4)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

TIMESD_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, X3, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U7_GGA(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2, X5)
TIMESD_IN_GGA(s(X1), X2, X3) → U13_GGA(X1, X2, X3, evencB_in_gg(s(X1), false))
U13_GGA(X1, X2, X3, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2, X4)

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
TIMESD_IN_GGA(x1, x2, x3) = TIMESD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4) = U13_GGA(x1, x2, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

TIMESD_IN_GGA(s(X1), X2) → U5_GGA(X1, X2, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, halfcC_in_ga(s(X1)))
U7_GGA(X1, X2, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2)
TIMESD_IN_GGA(s(X1), X2) → U13_GGA(X1, X2, evencB_in_gg(s(X1), false))
U13_GGA(X1, X2, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2)

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0) → halfcC_out_ga(0, 0)

The set Q consists of the following terms:

evencB_in_gg(x0, x1)
halfcC_in_ga(x0)
U36_gg(x0, x1, x2)
U37_ga(x0, x1)

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

(40) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

TIMESD_IN_GGA(s(X1), X2) → U13_GGA(X1, X2, evencB_in_gg(s(X1), false))

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

POL(0) = 0
POL(TIMESD_IN_GGA(x₁, x₂)) = x₁
POL(U13_GGA(x₁, x₂, x₃)) = x₁
POL(U36_gg(x₁, x₂, x₃)) = 0
POL(U37_ga(x₁, x₂)) = 1 + x₂
POL(U5_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(U7_GGA(x₁, x₂, x₃)) = x₃
POL(evencB_in_gg(x₁, x₂)) = 0
POL(evencB_out_gg(x₁, x₂)) = 0
POL(false) = 0
POL(halfcC_in_ga(x₁)) = x₁
POL(halfcC_out_ga(x₁, x₂)) = x₂
POL(s(x₁)) = 1 + x₁
POL(true) = 0

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

halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))
halfcC_in_ga(0) → halfcC_out_ga(0, 0)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))

(41) Obligation:

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

TIMESD_IN_GGA(s(X1), X2) → U5_GGA(X1, X2, evencB_in_gg(s(X1), true))
U5_GGA(X1, X2, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, halfcC_in_ga(s(X1)))
U7_GGA(X1, X2, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2)
U13_GGA(X1, X2, evencB_out_gg(s(X1), false)) → TIMESD_IN_GGA(X1, X2)

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0) → halfcC_out_ga(0, 0)

The set Q consists of the following terms:

evencB_in_gg(x0, x1)
halfcC_in_ga(x0)
U36_gg(x0, x1, x2)
U37_ga(x0, x1)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) Obligation:

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

U5_GGA(X1, X2, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, halfcC_in_ga(s(X1)))
U7_GGA(X1, X2, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2)
TIMESD_IN_GGA(s(X1), X2) → U5_GGA(X1, X2, evencB_in_gg(s(X1), true))

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0) → halfcC_out_ga(0, 0)

The set Q consists of the following terms:

evencB_in_gg(x0, x1)
halfcC_in_ga(x0)
U36_gg(x0, x1, x2)
U37_ga(x0, x1)

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

(44) QDPQMonotonicMRRProof (EQUIVALENT transformation)

By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented rules of the TRS R:

evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(TIMESD_IN_GGA(x₁, x₂)) = 0
POL(U36_gg(x₁, x₂, x₃)) = x₃
POL(U37_ga(x₁, x₂)) = 0
POL(U5_GGA(x₁, x₂, x₃)) = 2·x₃
POL(U7_GGA(x₁, x₂, x₃)) = 0
POL(evencB_in_gg(x₁, x₂)) = x₂
POL(evencB_out_gg(x₁, x₂)) = 0
POL(false) = 2
POL(halfcC_in_ga(x₁)) = 0
POL(halfcC_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(true) = 0

(45) Obligation:

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

U5_GGA(X1, X2, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, halfcC_in_ga(s(X1)))
U7_GGA(X1, X2, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2)
TIMESD_IN_GGA(s(X1), X2) → U5_GGA(X1, X2, evencB_in_gg(s(X1), true))

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0) → halfcC_out_ga(0, 0)

The set Q consists of the following terms:

evencB_in_gg(x0, x1)
halfcC_in_ga(x0)
U36_gg(x0, x1, x2)
U37_ga(x0, x1)

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

(46) QDPQMonotonicMRRProof (EQUIVALENT transformation)

halfcC_in_ga(s(s(X1))) → U37_ga(X1, halfcC_in_ga(X1))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(TIMESD_IN_GGA(x₁, x₂)) = 2·x₁
POL(U36_gg(x₁, x₂, x₃)) = x₂
POL(U37_ga(x₁, x₂)) = 2 + x₂
POL(U5_GGA(x₁, x₂, x₃)) = 2·x₁ + 2·x₃
POL(U7_GGA(x₁, x₂, x₃)) = 2·x₃
POL(evencB_in_gg(x₁, x₂)) = x₂
POL(evencB_out_gg(x₁, x₂)) = x₂
POL(halfcC_in_ga(x₁)) = x₁
POL(halfcC_out_ga(x₁, x₂)) = x₂
POL(s(x₁)) = 2 + x₁
POL(true) = 2

(47) Obligation:

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

U5_GGA(X1, X2, evencB_out_gg(s(X1), true)) → U7_GGA(X1, X2, halfcC_in_ga(s(X1)))
U7_GGA(X1, X2, halfcC_out_ga(s(X1), X4)) → TIMESD_IN_GGA(X4, X2)
TIMESD_IN_GGA(s(X1), X2) → U5_GGA(X1, X2, evencB_in_gg(s(X1), true))

The TRS R consists of the following rules:

evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
U37_ga(X1, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
evencB_in_gg(0, true) → evencB_out_gg(0, true)
halfcC_in_ga(0) → halfcC_out_ga(0, 0)

The set Q consists of the following terms:

evencB_in_gg(x0, x1)
halfcC_in_ga(x0)
U36_gg(x0, x1, x2)
U37_ga(x0, x1)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(49) TRUE

(50) Obligation:

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

PLUSA_IN_GGA(s(X1), X2, s(X3)) → PLUSA_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

timescG_in_ga(X1, 0) → timescG_out_ga(X1, 0)
evencB_in_gg(0, true) → evencB_out_gg(0, true)
evencB_in_gg(s(0), false) → evencB_out_gg(s(0), false)
evencB_in_gg(s(s(X1)), X2) → U36_gg(X1, X2, evencB_in_gg(X1, X2))
U36_gg(X1, X2, evencB_out_gg(X1, X2)) → evencB_out_gg(s(s(X1)), X2)
halfcH_in_ga(X1, s(X2)) → U71_ga(X1, X2, halfcC_in_ga(X1, X2))
halfcC_in_ga(0, 0) → halfcC_out_ga(0, 0)
halfcC_in_ga(s(s(X1)), s(X2)) → U37_ga(X1, X2, halfcC_in_ga(X1, X2))
U37_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcC_out_ga(s(s(X1)), s(X2))
U71_ga(X1, X2, halfcC_out_ga(X1, X2)) → halfcH_out_ga(X1, s(X2))
timescD_in_gga(0, X1, 0) → timescD_out_gga(0, X1, 0)
timescD_in_gga(s(X1), X2, 0) → U38_gga(X1, X2, evencB_in_gg(s(X1), true))
U38_gga(X1, X2, evencB_out_gg(s(X1), true)) → U39_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U39_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_gga(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(0, X1, 0) → timescD_out_ggg(0, X1, 0)
timescD_in_ggg(s(X1), X2, 0) → U38_ggg(X1, X2, evencB_in_gg(s(X1), true))
U38_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U39_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U39_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U40_ggg(X1, X2, timescD_in_ggg(X3, X2, 0))
timescD_in_ggg(s(X1), X2, s(s(0))) → U41_ggg(X1, X2, evencB_in_gg(s(X1), true))
U41_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U42_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U42_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_ggg(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, s(s(s(s(0))))) → U44_ggg(X1, X2, evencB_in_gg(s(X1), true))
U44_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U45_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U45_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_ggg(X1, X2, evencB_in_gg(s(X1), true))
U47_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U48_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U48_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_ggg(X1, X2, evencB_in_gg(s(X1), true))
U50_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U51_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U51_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_ggg(X1, X2, evencB_in_gg(s(X1), true))
U53_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U54_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U54_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_ggg(X1, X2, evencB_in_gg(s(X1), true))
U56_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U57_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U57_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_ggg(X1, X2, evencB_in_gg(s(X1), true))
U59_ggg(X1, X2, evencB_out_gg(s(X1), true)) → U60_ggg(X1, X2, halfcC_in_ga(s(X1), X3))
U60_ggg(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_ggg(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
timescD_in_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_ggg(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_ggg(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_ggg(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_ggg(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_ggg(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, s(s(0))) → U41_gga(X1, X2, evencB_in_gg(s(X1), true))
U41_gga(X1, X2, evencB_out_gg(s(X1), true)) → U42_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U42_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U43_gga(X1, X2, timescD_in_ggg(X3, X2, s(0)))
timescD_in_ggg(s(X1), X2, X3) → U66_ggg(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_ggg(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_ggg(X1, X2, X3, timescD_in_gga(X1, X2, X4))
timescD_in_gga(s(X1), X2, s(s(s(s(0))))) → U44_gga(X1, X2, evencB_in_gg(s(X1), true))
U44_gga(X1, X2, evencB_out_gg(s(X1), true)) → U45_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U45_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U46_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(0))))
U46_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_gga(s(X1), X2, s(s(s(s(0)))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(0))))))) → U47_gga(X1, X2, evencB_in_gg(s(X1), true))
U47_gga(X1, X2, evencB_out_gg(s(X1), true)) → U48_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U48_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U49_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(0)))))
U49_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(0)))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(0))))))))) → U50_gga(X1, X2, evencB_in_gg(s(X1), true))
U50_gga(X1, X2, evencB_out_gg(s(X1), true)) → U51_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U51_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U52_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(0))))))
U52_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U53_gga(X1, X2, evencB_in_gg(s(X1), true))
U53_gga(X1, X2, evencB_out_gg(s(X1), true)) → U54_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U54_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U55_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(0)))))))
U55_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U56_gga(X1, X2, evencB_in_gg(s(X1), true))
U56_gga(X1, X2, evencB_out_gg(s(X1), true)) → U57_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U57_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U58_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(0))))))))
U58_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U59_gga(X1, X2, evencB_in_gg(s(X1), true))
U59_gga(X1, X2, evencB_out_gg(s(X1), true)) → U60_gga(X1, X2, halfcC_in_ga(s(X1), X3))
U60_gga(X1, X2, halfcC_out_ga(s(X1), X3)) → U61_gga(X1, X2, timescD_in_ggg(X3, X2, s(s(s(s(s(s(s(0)))))))))
U61_gga(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timescD_in_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3))))))))) → U62_gga(X1, X2, X3, evencB_in_gg(s(X1), true))
U62_gga(X1, X2, X3, evencB_out_gg(s(X1), true)) → U63_gga(X1, X2, X3, halfcC_in_ga(s(X1), X4))
U63_gga(X1, X2, X3, halfcC_out_ga(s(X1), X4)) → U64_gga(X1, X2, X3, X4, timescD_in_gga(X4, X2, s(s(s(s(s(s(s(s(X5))))))))))
timescD_in_gga(s(X1), X2, X3) → U66_gga(X1, X2, X3, evencB_in_gg(s(X1), false))
U66_gga(X1, X2, X3, evencB_out_gg(s(X1), false)) → U67_gga(X1, X2, X3, timescD_in_gga(X1, X2, X4))
U67_gga(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U70_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U70_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U68_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescD_out_gga(s(X1), X2, X3)
U64_gga(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_gga(X1, X2, X3, pluscE_in_gga(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_gga(0, X1, s(X1)) → pluscE_out_gga(0, X1, s(X1))
pluscE_in_gga(s(X1), X2, s(X3)) → U69_gga(X1, X2, X3, pluscE_in_gga(X1, X2, X3))
U69_gga(X1, X2, X3, pluscE_out_gga(X1, X2, X3)) → pluscE_out_gga(s(X1), X2, s(X3))
U65_gga(X1, X2, X3, pluscE_out_gga(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_gga(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U67_ggg(X1, X2, X3, timescD_out_gga(X1, X2, X4)) → U68_ggg(X1, X2, X3, pluscF_in_ggg(X2, X4, X3))
pluscF_in_ggg(0, X1, X1) → pluscF_out_ggg(0, X1, X1)
pluscF_in_ggg(s(X1), X2, s(X3)) → U70_ggg(X1, X2, X3, pluscF_in_ggg(X1, X2, X3))
U70_ggg(X1, X2, X3, pluscF_out_ggg(X1, X2, X3)) → pluscF_out_ggg(s(X1), X2, s(X3))
U68_ggg(X1, X2, X3, pluscF_out_ggg(X2, X4, X3)) → timescD_out_ggg(s(X1), X2, X3)
U43_gga(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_gga(s(X1), X2, s(s(0)))
U64_ggg(X1, X2, X3, X4, timescD_out_gga(X4, X2, s(s(s(s(s(s(s(s(X5)))))))))) → U65_ggg(X1, X2, X3, pluscE_in_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3))
pluscE_in_ggg(0, X1, s(X1)) → pluscE_out_ggg(0, X1, s(X1))
pluscE_in_ggg(s(X1), X2, s(X3)) → U69_ggg(X1, X2, X3, pluscE_in_ggg(X1, X2, X3))
U69_ggg(X1, X2, X3, pluscE_out_ggg(X1, X2, X3)) → pluscE_out_ggg(s(X1), X2, s(X3))
U65_ggg(X1, X2, X3, pluscE_out_ggg(X5, s(s(s(s(s(s(s(X5))))))), X3)) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(X3)))))))))
U61_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(s(0))))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U58_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(s(0)))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U55_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(s(0))))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U52_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(s(0)))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(s(s(0)))))))))
U49_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(s(0))))) → timescD_out_ggg(s(X1), X2, s(s(s(s(s(s(0)))))))
U46_ggg(X1, X2, timescD_out_ggg(X3, X2, s(s(0)))) → timescD_out_ggg(s(X1), X2, s(s(s(s(0)))))
U43_ggg(X1, X2, timescD_out_ggg(X3, X2, s(0))) → timescD_out_ggg(s(X1), X2, s(s(0)))
U40_ggg(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_ggg(s(X1), X2, 0)
U40_gga(X1, X2, timescD_out_ggg(X3, X2, 0)) → timescD_out_gga(s(X1), X2, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timescG_in_ga(x1, x2) = timescG_in_ga(x1)
timescG_out_ga(x1, x2) = timescG_out_ga(x1, x2)
evencB_in_gg(x1, x2) = evencB_in_gg(x1, x2)
true = true
evencB_out_gg(x1, x2) = evencB_out_gg(x1, x2)
false = false
U36_gg(x1, x2, x3) = U36_gg(x1, x2, x3)
halfcH_in_ga(x1, x2) = halfcH_in_ga(x1)
U71_ga(x1, x2, x3) = U71_ga(x1, x3)
halfcC_in_ga(x1, x2) = halfcC_in_ga(x1)
halfcC_out_ga(x1, x2) = halfcC_out_ga(x1, x2)
U37_ga(x1, x2, x3) = U37_ga(x1, x3)
halfcH_out_ga(x1, x2) = halfcH_out_ga(x1, x2)
timescD_in_gga(x1, x2, x3) = timescD_in_gga(x1, x2)
timescD_out_gga(x1, x2, x3) = timescD_out_gga(x1, x2, x3)
U38_gga(x1, x2, x3) = U38_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
timescD_in_ggg(x1, x2, x3) = timescD_in_ggg(x1, x2, x3)
timescD_out_ggg(x1, x2, x3) = timescD_out_ggg(x1, x2, x3)
U38_ggg(x1, x2, x3) = U38_ggg(x1, x2, x3)
U39_ggg(x1, x2, x3) = U39_ggg(x1, x2, x3)
U40_ggg(x1, x2, x3) = U40_ggg(x1, x2, x3)
U41_ggg(x1, x2, x3) = U41_ggg(x1, x2, x3)
U42_ggg(x1, x2, x3) = U42_ggg(x1, x2, x3)
U43_ggg(x1, x2, x3) = U43_ggg(x1, x2, x3)
U44_ggg(x1, x2, x3) = U44_ggg(x1, x2, x3)
U45_ggg(x1, x2, x3) = U45_ggg(x1, x2, x3)
U46_ggg(x1, x2, x3) = U46_ggg(x1, x2, x3)
U47_ggg(x1, x2, x3) = U47_ggg(x1, x2, x3)
U48_ggg(x1, x2, x3) = U48_ggg(x1, x2, x3)
U49_ggg(x1, x2, x3) = U49_ggg(x1, x2, x3)
U50_ggg(x1, x2, x3) = U50_ggg(x1, x2, x3)
U51_ggg(x1, x2, x3) = U51_ggg(x1, x2, x3)
U52_ggg(x1, x2, x3) = U52_ggg(x1, x2, x3)
U53_ggg(x1, x2, x3) = U53_ggg(x1, x2, x3)
U54_ggg(x1, x2, x3) = U54_ggg(x1, x2, x3)
U55_ggg(x1, x2, x3) = U55_ggg(x1, x2, x3)
U56_ggg(x1, x2, x3) = U56_ggg(x1, x2, x3)
U57_ggg(x1, x2, x3) = U57_ggg(x1, x2, x3)
U58_ggg(x1, x2, x3) = U58_ggg(x1, x2, x3)
U59_ggg(x1, x2, x3) = U59_ggg(x1, x2, x3)
U60_ggg(x1, x2, x3) = U60_ggg(x1, x2, x3)
U61_ggg(x1, x2, x3) = U61_ggg(x1, x2, x3)
U62_ggg(x1, x2, x3, x4) = U62_ggg(x1, x2, x3, x4)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4, x5) = U64_ggg(x1, x2, x3, x5)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U45_gga(x1, x2, x3) = U45_gga(x1, x2, x3)
U46_gga(x1, x2, x3) = U46_gga(x1, x2, x3)
U47_gga(x1, x2, x3) = U47_gga(x1, x2, x3)
U48_gga(x1, x2, x3) = U48_gga(x1, x2, x3)
U49_gga(x1, x2, x3) = U49_gga(x1, x2, x3)
U50_gga(x1, x2, x3) = U50_gga(x1, x2, x3)
U51_gga(x1, x2, x3) = U51_gga(x1, x2, x3)
U52_gga(x1, x2, x3) = U52_gga(x1, x2, x3)
U53_gga(x1, x2, x3) = U53_gga(x1, x2, x3)
U54_gga(x1, x2, x3) = U54_gga(x1, x2, x3)
U55_gga(x1, x2, x3) = U55_gga(x1, x2, x3)
U56_gga(x1, x2, x3) = U56_gga(x1, x2, x3)
U57_gga(x1, x2, x3) = U57_gga(x1, x2, x3)
U58_gga(x1, x2, x3) = U58_gga(x1, x2, x3)
U59_gga(x1, x2, x3) = U59_gga(x1, x2, x3)
U60_gga(x1, x2, x3) = U60_gga(x1, x2, x3)
U61_gga(x1, x2, x3) = U61_gga(x1, x2, x3)
U62_gga(x1, x2, x3, x4) = U62_gga(x1, x2, x4)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4, x5) = U64_gga(x1, x2, x5)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3) = pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3) = pluscF_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4) = U65_gga(x1, x2, x4)
pluscE_in_gga(x1, x2, x3) = pluscE_in_gga(x1, x2)
pluscE_out_gga(x1, x2, x3) = pluscE_out_gga(x1, x2, x3)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
pluscF_in_ggg(x1, x2, x3) = pluscF_in_ggg(x1, x2, x3)
pluscF_out_ggg(x1, x2, x3) = pluscF_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4) = U65_ggg(x1, x2, x3, x4)
pluscE_in_ggg(x1, x2, x3) = pluscE_in_ggg(x1, x2, x3)
pluscE_out_ggg(x1, x2, x3) = pluscE_out_ggg(x1, x2, x3)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
PLUSA_IN_GGA(x1, x2, x3) = PLUSA_IN_GGA(x1, x2)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

PLUSA_IN_GGA(s(X1), X2, s(X3)) → PLUSA_IN_GGA(X1, X2, X3)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUSA_IN_GGA(x1, x2, x3) = PLUSA_IN_GGA(x1, x2)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

PLUSA_IN_GGA(s(X1), X2) → PLUSA_IN_GGA(X1, X2)

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

(55) 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:

PLUSA_IN_GGA(s(X1), X2) → PLUSA_IN_GGA(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

(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) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

(38) PiDPToQDPProof (SOUND transformation)

(39) Obligation:

(40) QDPOrderProof (EQUIVALENT transformation)

(41) Obligation:

(42) DependencyGraphProof (EQUIVALENT transformation)

(43) Obligation:

(44) QDPQMonotonicMRRProof (EQUIVALENT transformation)

(45) Obligation:

(46) QDPQMonotonicMRRProof (EQUIVALENT transformation)

(47) Obligation:

(48) DependencyGraphProof (EQUIVALENT transformation)

(49) TRUE

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) PiDPToQDPProof (SOUND transformation)

(54) Obligation:

(55) QDPSizeChangeProof (EQUIVALENT transformation)

(56) YES