Left Termination proof of /tmp/tmpXiF7QF/times2.pl

Left Termination of the query pattern times_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇐)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇐)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔)

        ↳41 QDP

        ↳42 DependencyGraphProof (⇔)

        ↳43 QDP

        ↳44 QDPOrderProof (⇔)

        ↳45 QDP

        ↳46 DependencyGraphProof (⇔)

        ↳47 TRUE

    ↳48 PiDP

        ↳49 UsableRulesProof (⇔)

        ↳50 PiDP

        ↳51 PiDPToQDPProof (⇐)

        ↳52 QDP

        ↳53 QDPSizeChangeProof (⇔)

        ↳54 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)).

Queries:

times(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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\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), X14), if(X14, s(T9), T10, T12))\n\nT9 is ground\nT10 is ground\nX14 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), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 0 | ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1 | ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(even(s(T9), X14), if(X14, s(T9), T10, T12)), SCOPE: 2, CLAUSE: 2\n\nT9 is ground\nT10 is ground\nX14 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, T23, X45), plus(T23, X45, T25))\n\nT23 is ground\nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: times(0, T23, X45)\n\nT23 is ground\nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: plus(T23, T29, T25)\n\nT23 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: times(0, T23, X45), SCOPE: 4, CLAUSE: 7 | times(0, T23, X45), SCOPE: 4, CLAUSE: 8\n\nT23 is ground\nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: times(0, T23, X45), SCOPE: 4, CLAUSE: 7\n\nT23 is ground\nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: times(0, T23, X45), SCOPE: 4, CLAUSE: 8\n\nT23 is ground\nX45 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(T23, T29, T25), SCOPE: 5, CLAUSE: 5 | plus(T23, T29, T25), SCOPE: 5, CLAUSE: 6\n\nT23 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: plus(T23, T29, T25), SCOPE: 5, CLAUSE: 5\n\nT23 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: plus(T23, T29, T25), SCOPE: 5, CLAUSE: 6\n\nT23 is ground\nT29 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(T49, T50, T52)\n\nT49 is ground\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(even(T57, X90), if(X90, s(s(T57)), T10, T12))\n\nT10 is ground\nT57 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: even(T57, X90)\n\nT57 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: if(T58, s(s(T57)), T10, T12)\n\nT10 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(T57, X90), SCOPE: 6, CLAUSE: 0 | even(T57, X90), SCOPE: 6, CLAUSE: 1 | even(T57, X90), SCOPE: 6, CLAUSE: 2\n\nT57 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: even(T57, X90), SCOPE: 6, CLAUSE: 0\n\nT57 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: even(T57, X90), SCOPE: 6, CLAUSE: 1 | even(T57, X90), SCOPE: 6, CLAUSE: 2\n\nT57 is ground\nX90 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(T57, X90), SCOPE: 6, CLAUSE: 1\n\nT57 is ground\nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: even(T57, X90), SCOPE: 6, CLAUSE: 2\n\nT57 is ground\nX90 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(T61, X101)\n\nT61 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: if(T58, s(s(T57)), T10, T12), SCOPE: 7, CLAUSE: 9 | if(T58, s(s(T57)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: if(T58, s(s(T57)), T10, T12), SCOPE: 7, CLAUSE: 9\n\nT10 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: if(T58, s(s(T57)), T10, T12), SCOPE: 7, CLAUSE: 10\n\nT10 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(half(s(s(T77)), X128), ','(times(X128, T78, X129), plus(X129, X129, T80)))\n\nT77 is ground\nT78 is ground\nX128 is free; \nX129 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: half(s(s(T77)), X128)\n\nT77 is ground\nX128 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(times(T82, T78, X129), plus(X129, X129, T80))\n\nT78 is ground\nT82 is ground\nX129 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: half(s(s(T77)), X128), SCOPE: 8, CLAUSE: 3 | half(s(s(T77)), X128), SCOPE: 8, CLAUSE: 4\n\nT77 is ground\nX128 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: half(s(s(T77)), X128), SCOPE: 8, CLAUSE: 4\n\nT77 is ground\nX128 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: half(T86, X147)\n\nT86 is ground\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: half(T86, X147), SCOPE: 9, CLAUSE: 3 | half(T86, X147), SCOPE: 9, CLAUSE: 4\n\nT86 is ground\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: half(T86, X147), SCOPE: 9, CLAUSE: 3\n\nT86 is ground\nX147 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: half(T86, X147), SCOPE: 9, CLAUSE: 4\n\nT86 is ground\nX147 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(T89, X156)\n\nT89 is ground\nX156 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: times(T82, T78, X129)\n\nT78 is ground\nT82 is ground\nX129 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: plus(T92, T92, T80)\n\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: times(T82, T78, X129), SCOPE: 10, CLAUSE: 7 | times(T82, T78, X129), SCOPE: 10, CLAUSE: 8\n\nT78 is ground\nT82 is ground\nX129 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: times(T82, T78, X129), SCOPE: 10, CLAUSE: 7\n\nT78 is ground\nT82 is ground\nX129 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: times(T82, T78, X129), SCOPE: 10, CLAUSE: 8\n\nT78 is ground\nT82 is ground\nX129 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(T104), X181), if(X181, s(T104), T105, X182))\n\nT104 is ground\nT105 is ground\nX182 is free; \nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: even(s(T104), X181)\n\nT104 is ground\nX181 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: if(T106, s(T104), T105, X182)\n\nT104 is ground\nT105 is ground\nT106 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: if(T106, s(T104), T105, X182), SCOPE: 11, CLAUSE: 9 | if(T106, s(T104), T105, X182), SCOPE: 11, CLAUSE: 10\n\nT104 is ground\nT105 is ground\nT106 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: if(T106, s(T104), T105, X182), SCOPE: 11, CLAUSE: 9\n\nT104 is ground\nT105 is ground\nT106 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: if(T106, s(T104), T105, X182), SCOPE: 11, CLAUSE: 10\n\nT104 is ground\nT105 is ground\nT106 is ground\nX182 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(half(s(T117), X215), ','(times(X215, T118, X216), plus(X216, X216, X217)))\n\nT117 is ground\nT118 is ground\nX217 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: half(s(T117), X215)\n\nT117 is ground\nX215 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(times(T120, T118, X216), plus(X216, X216, X217))\n\nT118 is ground\nT120 is ground\nX217 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: times(T120, T118, X216)\n\nT118 is ground\nT120 is ground\nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: plus(T124, T124, X217)\n\nT124 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: plus(T124, T124, X217), SCOPE: 12, CLAUSE: 5 | plus(T124, T124, X217), SCOPE: 12, CLAUSE: 6\n\nT124 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: plus(T124, T124, X217), SCOPE: 12, CLAUSE: 5\n\nT124 is ground\nX217 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: plus(T124, T124, X217), SCOPE: 12, CLAUSE: 6\n\nT124 is ground\nX217 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(T129, s(T129), X250)\n\nT129 is ground\nX250 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: plus(T129, s(T129), X250), SCOPE: 13, CLAUSE: 5 | plus(T129, s(T129), X250), SCOPE: 13, CLAUSE: 6\n\nT129 is ground\nX250 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: plus(T129, s(T129), X250), SCOPE: 13, CLAUSE: 5\n\nT129 is ground\nX250 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: plus(T129, s(T129), X250), SCOPE: 13, CLAUSE: 6\n\nT129 is ground\nX250 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(T132, s(s(T132)), X273)\n\nT132 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: plus(T132, s(s(T132)), X273), SCOPE: 14, CLAUSE: 5 | plus(T132, s(s(T132)), X273), SCOPE: 14, CLAUSE: 6\n\nT132 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: plus(T132, s(s(T132)), X273), SCOPE: 14, CLAUSE: 5\n\nT132 is ground\nX273 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: plus(T132, s(s(T132)), X273), SCOPE: 14, CLAUSE: 6\n\nT132 is ground\nX273 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(T135, s(s(s(T135))), X296)\n\nT135 is ground\nX296 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: plus(T135, s(s(s(T135))), X296), SCOPE: 15, CLAUSE: 5 | plus(T135, s(s(s(T135))), X296), SCOPE: 15, CLAUSE: 6\n\nT135 is ground\nX296 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: plus(T135, s(s(s(T135))), X296), SCOPE: 15, CLAUSE: 5\n\nT135 is ground\nX296 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: plus(T135, s(s(s(T135))), X296), SCOPE: 15, CLAUSE: 6\n\nT135 is ground\nX296 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(T138, s(s(s(s(T138)))), X319)\n\nT138 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: plus(T138, s(s(s(s(T138)))), X319), SCOPE: 16, CLAUSE: 5 | plus(T138, s(s(s(s(T138)))), X319), SCOPE: 16, CLAUSE: 6\n\nT138 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: plus(T138, s(s(s(s(T138)))), X319), SCOPE: 16, CLAUSE: 5\n\nT138 is ground\nX319 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: plus(T138, s(s(s(s(T138)))), X319), SCOPE: 16, CLAUSE: 6\n\nT138 is ground\nX319 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(T141, s(s(s(s(s(T141))))), X342)\n\nT141 is ground\nX342 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: plus(T141, s(s(s(s(s(T141))))), X342), SCOPE: 17, CLAUSE: 5 | plus(T141, s(s(s(s(s(T141))))), X342), SCOPE: 17, CLAUSE: 6\n\nT141 is ground\nX342 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: plus(T141, s(s(s(s(s(T141))))), X342), SCOPE: 17, CLAUSE: 5\n\nT141 is ground\nX342 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: plus(T141, s(s(s(s(s(T141))))), X342), SCOPE: 17, CLAUSE: 6\n\nT141 is ground\nX342 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(T144, s(s(s(s(s(s(T144)))))), X365)\n\nT144 is ground\nX365 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: plus(T144, s(s(s(s(s(s(T144)))))), X365), SCOPE: 18, CLAUSE: 5 | plus(T144, s(s(s(s(s(s(T144)))))), X365), SCOPE: 18, CLAUSE: 6\n\nT144 is ground\nX365 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: plus(T144, s(s(s(s(s(s(T144)))))), X365), SCOPE: 18, CLAUSE: 5\n\nT144 is ground\nX365 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: plus(T144, s(s(s(s(s(s(T144)))))), X365), SCOPE: 18, CLAUSE: 6\n\nT144 is ground\nX365 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(T147, s(s(s(s(s(s(s(T147))))))), X388)\n\nT147 is ground\nX388 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: plus(T147, s(s(s(s(s(s(s(T147))))))), X388), SCOPE: 19, CLAUSE: 5 | plus(T147, s(s(s(s(s(s(s(T147))))))), X388), SCOPE: 19, CLAUSE: 6\n\nT147 is ground\nX388 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: plus(T147, s(s(s(s(s(s(s(T147))))))), X388), SCOPE: 19, CLAUSE: 5\n\nT147 is ground\nX388 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: plus(T147, s(s(s(s(s(s(s(T147))))))), X388), SCOPE: 19, CLAUSE: 6\n\nT147 is ground\nX388 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(T150, s(s(s(s(s(s(s(s(T150)))))))), X411)\n\nT150 is ground\nX411 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: plus(T150, s(T151), X411)\n\nT150 is ground\nT151 is ground\nX411 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: plus(T150, s(T151), X411), SCOPE: 20, CLAUSE: 5 | plus(T150, s(T151), X411), SCOPE: 20, CLAUSE: 6\n\nT150 is ground\nT151 is ground\nX411 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: plus(T150, s(T151), X411), SCOPE: 20, CLAUSE: 5\n\nT150 is ground\nT151 is ground\nX411 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: plus(T150, s(T151), X411), SCOPE: 20, CLAUSE: 6\n\nT150 is ground\nT151 is ground\nX411 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(T163, s(T164), X436)\n\nT163 is ground\nT164 is ground\nX436 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: ','(times(T171, T172, X454), plus(T172, X454, X455))\n\nT171 is ground\nT172 is ground\nX455 is free; \nX454 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: times(T171, T172, X454)\n\nT171 is ground\nT172 is ground\nX454 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: plus(T172, T175, X455)\n\nT172 is ground\nT175 is ground\nX455 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: plus(T172, T175, X455), SCOPE: 21, CLAUSE: 5 | plus(T172, T175, X455), SCOPE: 21, CLAUSE: 6\n\nT172 is ground\nT175 is ground\nX455 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: plus(T172, T175, X455), SCOPE: 21, CLAUSE: 5\n\nT172 is ground\nT175 is ground\nX455 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: plus(T172, T175, X455), SCOPE: 21, CLAUSE: 6\n\nT172 is ground\nT175 is ground\nX455 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(T189, T190, X482)\n\nT189 is ground\nT190 is ground\nX482 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: plus(T92, T92, T80), SCOPE: 22, CLAUSE: 5 | plus(T92, T92, T80), SCOPE: 22, CLAUSE: 6\n\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: plus(T92, T92, T80), SCOPE: 22, CLAUSE: 5\n\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: plus(T92, T92, T80), SCOPE: 22, CLAUSE: 6\n\nT92 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(T197, s(T197), T199)\n\nT197 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="182: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: plus(T197, s(T197), T199), SCOPE: 23, CLAUSE: 5 | plus(T197, s(T197), T199), SCOPE: 23, CLAUSE: 6\n\nT197 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: plus(T197, s(T197), T199), SCOPE: 23, CLAUSE: 5\n\nT197 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: plus(T197, s(T197), T199), SCOPE: 23, CLAUSE: 6\n\nT197 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(T204, s(s(T204)), T206)\n\nT204 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: plus(T204, s(s(T204)), T206), SCOPE: 24, CLAUSE: 5 | plus(T204, s(s(T204)), T206), SCOPE: 24, CLAUSE: 6\n\nT204 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: plus(T204, s(s(T204)), T206), SCOPE: 24, CLAUSE: 5\n\nT204 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: plus(T204, s(s(T204)), T206), SCOPE: 24, CLAUSE: 6\n\nT204 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(T211, s(s(s(T211))), T213)\n\nT211 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: plus(T211, s(s(s(T211))), T213), SCOPE: 25, CLAUSE: 5 | plus(T211, s(s(s(T211))), T213), SCOPE: 25, CLAUSE: 6\n\nT211 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: plus(T211, s(s(s(T211))), T213), SCOPE: 25, CLAUSE: 5\n\nT211 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: plus(T211, s(s(s(T211))), T213), SCOPE: 25, CLAUSE: 6\n\nT211 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(T218, s(s(s(s(T218)))), T220)\n\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: plus(T218, s(s(s(s(T218)))), T220), SCOPE: 26, CLAUSE: 5 | plus(T218, s(s(s(s(T218)))), T220), SCOPE: 26, CLAUSE: 6\n\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: plus(T218, s(s(s(s(T218)))), T220), SCOPE: 26, CLAUSE: 5\n\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: plus(T218, s(s(s(s(T218)))), T220), SCOPE: 26, CLAUSE: 6\n\nT218 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(T225, s(s(s(s(s(T225))))), T227)\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: plus(T225, s(s(s(s(s(T225))))), T227), SCOPE: 27, CLAUSE: 5 | plus(T225, s(s(s(s(s(T225))))), T227), SCOPE: 27, CLAUSE: 6\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: plus(T225, s(s(s(s(s(T225))))), T227), SCOPE: 27, CLAUSE: 5\n\nT225 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: plus(T225, s(s(s(s(s(T225))))), T227), SCOPE: 27, CLAUSE: 6\n\nT225 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(T232, s(s(s(s(s(s(T232)))))), T234)\n\nT232 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: plus(T232, s(s(s(s(s(s(T232)))))), T234), SCOPE: 28, CLAUSE: 5 | plus(T232, s(s(s(s(s(s(T232)))))), T234), SCOPE: 28, CLAUSE: 6\n\nT232 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: plus(T232, s(s(s(s(s(s(T232)))))), T234), SCOPE: 28, CLAUSE: 5\n\nT232 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: plus(T232, s(s(s(s(s(s(T232)))))), T234), SCOPE: 28, CLAUSE: 6\n\nT232 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(T239, s(s(s(s(s(s(s(T239))))))), T241)\n\nT239 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: plus(T239, s(s(s(s(s(s(s(T239))))))), T241), SCOPE: 29, CLAUSE: 5 | plus(T239, s(s(s(s(s(s(s(T239))))))), T241), SCOPE: 29, CLAUSE: 6\n\nT239 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: plus(T239, s(s(s(s(s(s(s(T239))))))), T241), SCOPE: 29, CLAUSE: 5\n\nT239 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: plus(T239, s(s(s(s(s(s(s(T239))))))), T241), SCOPE: 29, CLAUSE: 6\n\nT239 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(T246, s(s(s(s(s(s(s(s(T246)))))))), T248)\n\nT246 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: plus(T246, s(T249), T248)\n\nT246 is ground\nT249 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: ','(times(s(T258), T259, X662), plus(T259, X662, T261))\n\nT258 is ground\nT259 is ground\nX662 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: times(s(T258), T259, X662)\n\nT258 is ground\nT259 is ground\nX662 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: plus(T259, T264, T261)\n\nT259 is ground\nT264 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 substitution\nT1 -> 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 = lightblue];
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(X11), X12, X13) :- ','(even(s(X11), X14), if(X14, s(X11), X12, X13)).\nand substitution\nX11 -> T9\nT1 -> s(T9)\nT2 -> T10\nX12 -> T10\nT3 -> T12\nX13 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 248 [arrowhead = none ];
248 -> 7;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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 = white];
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 substitution\nT9 -> 0\nX14 -> false", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 255 [arrowhead = none ];
255 -> 13;

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

257 [label="EVAL with clause\neven(s(s(X88)), X89) :- even(X88, X89).\nand substitution\nX88 -> T57\nT9 -> s(T57)\nX14 -> X90\nX89 -> X90", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 257 [arrowhead = none ];
257 -> 34;

258 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
15 -> 260 [arrowhead = none ];
260 -> 16;

261 [label="ONLY EVAL with clause\nif(false, s(X42), X43, X44) :- ','(times(X42, X43, X45), plus(X43, X45, X44)).\nand substitution\nX42 -> 0\nT10 -> T23\nX43 -> T23\nT12 -> T25\nX44 -> T25\nT24 -> T25", fontsize=14, style = filled, fillcolor = white];
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:\nT29 is ground\nreplacements:\nX45 -> T29", 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, X56, 0).\nand substitution\nT23 -> T35\nX56 -> T35\nX45 -> 0", fontsize=14, style = filled, fillcolor = white];
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 = white];
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, X66, X66).\nand substitution\nT23 -> 0\nT29 -> T42\nX66 -> T42\nT25 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 273 [arrowhead = none ];
273 -> 29;

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

275 [label="EVAL with clause\nplus(s(X74), X75, s(X76)) :- plus(X74, X75, X76).\nand substitution\nX74 -> T49\nT23 -> s(T49)\nT29 -> T50\nX75 -> T50\nX76 -> T52\nT25 -> s(T52)\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 275 [arrowhead = none ];
275 -> 32;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT23 -> T49\nT29 -> T50\nT25 -> T52", 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:\nT58 is ground\nreplacements:\nX90 -> T58", 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 substitution\nT57 -> 0\nX90 -> true", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 285 [arrowhead = none ];
285 -> 41;

286 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 substitution\nT57 -> s(0)\nX90 -> false", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 290 [arrowhead = none ];
290 -> 46;

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

292 [label="EVAL with clause\neven(s(s(X99)), X100) :- even(X99, X100).\nand substitution\nX99 -> T61\nT57 -> s(s(T61))\nX90 -> X101\nX100 -> X101", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 292 [arrowhead = none ];
292 -> 49;

293 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT57 -> T61\nX90 -> X101", 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(X125), X126, X127) :- ','(half(s(X125), X128), ','(times(X128, X126, X129), plus(X129, X129, X127))).\nand substitution\nT58 -> true\nT57 -> T77\nX125 -> s(T77)\nT10 -> T78\nX126 -> T78\nT12 -> T80\nX127 -> T80\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 298 [arrowhead = none ];
298 -> 54;

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

300 [label="EVAL with clause\nif(false, s(X659), X660, X661) :- ','(times(X659, X660, X662), plus(X660, X662, X661)).\nand substitution\nT58 -> false\nT57 -> T258\nX659 -> s(T258)\nT10 -> T259\nX660 -> T259\nT12 -> T261\nX661 -> T261\nT260 -> T261", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 300 [arrowhead = none ];
300 -> 240;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT82 is ground\nreplacements:\nX128 -> T82", 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:\nT92 is ground\nreplacements:\nX129 -> T92", 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 = white];
58 -> 307 [arrowhead = none ];
307 -> 59;

308 [label="ONLY EVAL with clause\nhalf(s(s(X145)), s(X146)) :- half(X145, X146).\nand substitution\nT77 -> T86\nX145 -> T86\nX146 -> X147\nX128 -> s(X147)", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT86 -> 0\nX147 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 312 [arrowhead = none ];
312 -> 64;

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

314 [label="EVAL with clause\nhalf(s(s(X154)), s(X155)) :- half(X154, X155).\nand substitution\nX154 -> T89\nT86 -> s(s(T89))\nX155 -> X156\nX147 -> s(X156)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 314 [arrowhead = none ];
314 -> 67;

315 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT86 -> T89\nX147 -> X156", 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, X167, 0).\nand substitution\nT82 -> 0\nT78 -> T99\nX167 -> T99\nX129 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 322 [arrowhead = none ];
322 -> 74;

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

324 [label="EVAL with clause\ntimes(s(X178), X179, X180) :- ','(even(s(X178), X181), if(X181, s(X178), X179, X180)).\nand substitution\nX178 -> T104\nT82 -> s(T104)\nT78 -> T105\nX179 -> T105\nX129 -> X182\nX180 -> X182", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 324 [arrowhead = none ];
324 -> 77;

325 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT106 is ground\nreplacements:\nX181 -> T106", fontsize=14, style = filled, fillcolor = violet];
77 -> 328 [arrowhead = none ];
328 -> 80;

329 [label="INSTANCE with matching:\nT57 -> s(T104)\nX90 -> X181", 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(X212), X213, X214) :- ','(half(s(X212), X215), ','(times(X215, X213, X216), plus(X216, X216, X214))).\nand substitution\nT106 -> true\nT104 -> T117\nX212 -> T117\nT105 -> T118\nX213 -> T118\nX182 -> X217\nX214 -> X217", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 333 [arrowhead = none ];
333 -> 84;

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

335 [label="EVAL with clause\nif(false, s(X451), X452, X453) :- ','(times(X451, X452, X454), plus(X452, X454, X453)).\nand substitution\nT106 -> false\nT104 -> T171\nX451 -> T171\nT105 -> T172\nX452 -> T172\nX182 -> X455\nX453 -> X455", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 335 [arrowhead = none ];
335 -> 163;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT120 is ground\nreplacements:\nX215 -> T120", fontsize=14, style = filled, fillcolor = violet];
84 -> 338 [arrowhead = none ];
338 -> 87;

339 [label="INSTANCE with matching:\nT86 -> s(T117)\nX147 -> X215", 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:\nT124 is ground\nreplacements:\nX216 -> T124", fontsize=14, style = filled, fillcolor = violet];
87 -> 341 [arrowhead = none ];
341 -> 89;

342 [label="INSTANCE with matching:\nT82 -> T120\nT78 -> T118\nX129 -> X216", 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, X237, X237).\nand substitution\nT124 -> 0\nX237 -> 0\nX217 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 346 [arrowhead = none ];
346 -> 93;

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

348 [label="EVAL with clause\nplus(s(X247), X248, s(X249)) :- plus(X247, X248, X249).\nand substitution\nX247 -> T129\nT124 -> s(T129)\nX248 -> s(T129)\nX249 -> X250\nX217 -> s(X250)", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 348 [arrowhead = none ];
348 -> 96;

349 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X260, X260).\nand substitution\nT129 -> 0\nX260 -> s(0)\nX250 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 354 [arrowhead = none ];
354 -> 101;

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

356 [label="EVAL with clause\nplus(s(X270), X271, s(X272)) :- plus(X270, X271, X272).\nand substitution\nX270 -> T132\nT129 -> s(T132)\nX271 -> s(s(T132))\nX272 -> X273\nX250 -> s(X273)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 356 [arrowhead = none ];
356 -> 104;

357 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X283, X283).\nand substitution\nT132 -> 0\nX283 -> s(s(0))\nX273 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 362 [arrowhead = none ];
362 -> 109;

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

364 [label="EVAL with clause\nplus(s(X293), X294, s(X295)) :- plus(X293, X294, X295).\nand substitution\nX293 -> T135\nT132 -> s(T135)\nX294 -> s(s(s(T135)))\nX295 -> X296\nX273 -> s(X296)", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 364 [arrowhead = none ];
364 -> 112;

365 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X306, X306).\nand substitution\nT135 -> 0\nX306 -> s(s(s(0)))\nX296 -> 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 = lightblue];
115 -> 371 [arrowhead = none ];
371 -> 118;

372 [label="EVAL with clause\nplus(s(X316), X317, s(X318)) :- plus(X316, X317, X318).\nand substitution\nX316 -> T138\nT135 -> s(T138)\nX317 -> s(s(s(s(T138))))\nX318 -> X319\nX296 -> s(X319)", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 372 [arrowhead = none ];
372 -> 120;

373 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X329, X329).\nand substitution\nT138 -> 0\nX329 -> s(s(s(s(0))))\nX319 -> 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 = lightblue];
123 -> 379 [arrowhead = none ];
379 -> 126;

380 [label="EVAL with clause\nplus(s(X339), X340, s(X341)) :- plus(X339, X340, X341).\nand substitution\nX339 -> T141\nT138 -> s(T141)\nX340 -> s(s(s(s(s(T141)))))\nX341 -> X342\nX319 -> s(X342)", fontsize=14, style = filled, fillcolor = lightblue];
124 -> 380 [arrowhead = none ];
380 -> 128;

381 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X352, X352).\nand substitution\nT141 -> 0\nX352 -> s(s(s(s(s(0)))))\nX342 -> 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 = lightblue];
131 -> 387 [arrowhead = none ];
387 -> 134;

388 [label="EVAL with clause\nplus(s(X362), X363, s(X364)) :- plus(X362, X363, X364).\nand substitution\nX362 -> T144\nT141 -> s(T144)\nX363 -> s(s(s(s(s(s(T144))))))\nX364 -> X365\nX342 -> s(X365)", fontsize=14, style = filled, fillcolor = lightblue];
132 -> 388 [arrowhead = none ];
388 -> 136;

389 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X375, X375).\nand substitution\nT144 -> 0\nX375 -> s(s(s(s(s(s(0))))))\nX365 -> 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 = lightblue];
139 -> 395 [arrowhead = none ];
395 -> 142;

396 [label="EVAL with clause\nplus(s(X385), X386, s(X387)) :- plus(X385, X386, X387).\nand substitution\nX385 -> T147\nT144 -> s(T147)\nX386 -> s(s(s(s(s(s(s(T147)))))))\nX387 -> X388\nX365 -> s(X388)", fontsize=14, style = filled, fillcolor = lightblue];
140 -> 396 [arrowhead = none ];
396 -> 144;

397 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X398, X398).\nand substitution\nT147 -> 0\nX398 -> s(s(s(s(s(s(s(0)))))))\nX388 -> 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 = lightblue];
147 -> 403 [arrowhead = none ];
403 -> 150;

404 [label="EVAL with clause\nplus(s(X408), X409, s(X410)) :- plus(X408, X409, X410).\nand substitution\nX408 -> T150\nT147 -> s(T150)\nX409 -> s(s(s(s(s(s(s(s(T150))))))))\nX410 -> X411\nX388 -> s(X411)", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 404 [arrowhead = none ];
404 -> 152;

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

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

407 [label="GENERALIZATION\nT151 <-- s(s(s(s(s(s(s(T150)))))))\n\nNew Knowledge:\nT151 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, X423, X423).\nand substitution\nT150 -> 0\nT151 -> T158\nX423 -> s(T158)\nX411 -> s(T158)", fontsize=14, style = filled, fillcolor = lightblue];
156 -> 411 [arrowhead = none ];
411 -> 158;

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

413 [label="EVAL with clause\nplus(s(X433), X434, s(X435)) :- plus(X433, X434, X435).\nand substitution\nX433 -> T163\nT150 -> s(T163)\nT151 -> T164\nX434 -> s(T164)\nX435 -> X436\nX411 -> s(X436)", fontsize=14, style = filled, fillcolor = lightblue];
157 -> 413 [arrowhead = none ];
413 -> 161;

414 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT150 -> T163\nT151 -> T164\nX411 -> X436", 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:\nT175 is ground\nreplacements:\nX454 -> T175", fontsize=14, style = filled, fillcolor = violet];
163 -> 418 [arrowhead = none ];
418 -> 166;

419 [label="INSTANCE with matching:\nT82 -> T171\nT78 -> T172\nX129 -> X454", 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, X469, X469).\nand substitution\nT172 -> 0\nT175 -> T184\nX469 -> T184\nX455 -> T184", fontsize=14, style = filled, fillcolor = lightblue];
168 -> 423 [arrowhead = none ];
423 -> 170;

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

425 [label="EVAL with clause\nplus(s(X479), X480, s(X481)) :- plus(X479, X480, X481).\nand substitution\nX479 -> T189\nT172 -> s(T189)\nT175 -> T190\nX480 -> T190\nX481 -> X482\nX455 -> s(X482)", fontsize=14, style = filled, fillcolor = lightblue];
169 -> 425 [arrowhead = none ];
425 -> 173;

426 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT172 -> T189\nT175 -> T190\nX455 -> X482", 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, X494, X494).\nand substitution\nT92 -> 0\nX494 -> 0\nT80 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
176 -> 431 [arrowhead = none ];
431 -> 178;

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

433 [label="EVAL with clause\nplus(s(X502), X503, s(X504)) :- plus(X502, X503, X504).\nand substitution\nX502 -> T197\nT92 -> s(T197)\nX503 -> s(T197)\nX504 -> T199\nT80 -> s(T199)\nT198 -> T199", fontsize=14, style = filled, fillcolor = lightblue];
177 -> 433 [arrowhead = none ];
433 -> 181;

434 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X514, X514).\nand substitution\nT197 -> 0\nX514 -> s(0)\nT199 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
184 -> 439 [arrowhead = none ];
439 -> 186;

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

441 [label="EVAL with clause\nplus(s(X522), X523, s(X524)) :- plus(X522, X523, X524).\nand substitution\nX522 -> T204\nT197 -> s(T204)\nX523 -> s(s(T204))\nX524 -> T206\nT199 -> s(T206)\nT205 -> T206", fontsize=14, style = filled, fillcolor = lightblue];
185 -> 441 [arrowhead = none ];
441 -> 189;

442 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X534, X534).\nand substitution\nT204 -> 0\nX534 -> s(s(0))\nT206 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
192 -> 447 [arrowhead = none ];
447 -> 194;

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

449 [label="EVAL with clause\nplus(s(X542), X543, s(X544)) :- plus(X542, X543, X544).\nand substitution\nX542 -> T211\nT204 -> s(T211)\nX543 -> s(s(s(T211)))\nX544 -> T213\nT206 -> s(T213)\nT212 -> T213", fontsize=14, style = filled, fillcolor = lightblue];
193 -> 449 [arrowhead = none ];
449 -> 197;

450 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X554, X554).\nand substitution\nT211 -> 0\nX554 -> s(s(s(0)))\nT213 -> 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 = lightblue];
200 -> 456 [arrowhead = none ];
456 -> 203;

457 [label="EVAL with clause\nplus(s(X562), X563, s(X564)) :- plus(X562, X563, X564).\nand substitution\nX562 -> T218\nT211 -> s(T218)\nX563 -> s(s(s(s(T218))))\nX564 -> T220\nT213 -> s(T220)\nT219 -> T220", fontsize=14, style = filled, fillcolor = lightblue];
201 -> 457 [arrowhead = none ];
457 -> 205;

458 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X574, X574).\nand substitution\nT218 -> 0\nX574 -> s(s(s(s(0))))\nT220 -> 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 = lightblue];
208 -> 464 [arrowhead = none ];
464 -> 211;

465 [label="EVAL with clause\nplus(s(X582), X583, s(X584)) :- plus(X582, X583, X584).\nand substitution\nX582 -> T225\nT218 -> s(T225)\nX583 -> s(s(s(s(s(T225)))))\nX584 -> T227\nT220 -> s(T227)\nT226 -> T227", fontsize=14, style = filled, fillcolor = lightblue];
209 -> 465 [arrowhead = none ];
465 -> 213;

466 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X594, X594).\nand substitution\nT225 -> 0\nX594 -> s(s(s(s(s(0)))))\nT227 -> 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 = lightblue];
216 -> 472 [arrowhead = none ];
472 -> 219;

473 [label="EVAL with clause\nplus(s(X602), X603, s(X604)) :- plus(X602, X603, X604).\nand substitution\nX602 -> T232\nT225 -> s(T232)\nX603 -> s(s(s(s(s(s(T232))))))\nX604 -> T234\nT227 -> s(T234)\nT233 -> T234", fontsize=14, style = filled, fillcolor = lightblue];
217 -> 473 [arrowhead = none ];
473 -> 221;

474 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X614, X614).\nand substitution\nT232 -> 0\nX614 -> s(s(s(s(s(s(0))))))\nT234 -> 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 = lightblue];
224 -> 480 [arrowhead = none ];
480 -> 227;

481 [label="EVAL with clause\nplus(s(X622), X623, s(X624)) :- plus(X622, X623, X624).\nand substitution\nX622 -> T239\nT232 -> s(T239)\nX623 -> s(s(s(s(s(s(s(T239)))))))\nX624 -> T241\nT234 -> s(T241)\nT240 -> T241", fontsize=14, style = filled, fillcolor = lightblue];
225 -> 481 [arrowhead = none ];
481 -> 229;

482 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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, X634, X634).\nand substitution\nT239 -> 0\nX634 -> s(s(s(s(s(s(s(0)))))))\nT241 -> 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 = lightblue];
232 -> 488 [arrowhead = none ];
488 -> 235;

489 [label="EVAL with clause\nplus(s(X642), X643, s(X644)) :- plus(X642, X643, X644).\nand substitution\nX642 -> T246\nT239 -> s(T246)\nX643 -> s(s(s(s(s(s(s(s(T246))))))))\nX644 -> T248\nT241 -> s(T248)\nT247 -> T248", fontsize=14, style = filled, fillcolor = lightblue];
233 -> 489 [arrowhead = none ];
489 -> 237;

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

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

492 [label="GENERALIZATION\nT249 <-- s(s(s(s(s(s(s(T246)))))))\n\nNew Knowledge:\nT249 is ground", fontsize=14, style = filled, fillcolor = violet];
237 -> 492 [arrowhead = none ];
492 -> 239;

493 [label="INSTANCE with matching:\nT23 -> T246\nT29 -> s(T249)\nT25 -> T248", 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:\nT264 is ground\nreplacements:\nX662 -> T264", fontsize=14, style = filled, fillcolor = violet];
240 -> 495 [arrowhead = none ];
495 -> 243;

496 [label="INSTANCE with matching:\nT82 -> s(T258)\nT78 -> T259\nX129 -> X662", fontsize=14, style = filled, fillcolor = lightgrey];
242 -> 496 [arrowhead = none , style=dashed];
496 -> 69[style=dashed];

497 [label="INSTANCE with matching:\nT23 -> T259\nT29 -> T264\nT25 -> T261", fontsize=14, style = filled, fillcolor = lightgrey];
243 -> 497 [arrowhead = none , style=dashed];
497 -> 19[style=dashed];

}

(2) Obligation:

Triples:

plus19(s(T49), T50, s(T52)) :- plus19(T49, T50, T52).
even36(s(s(T61)), X101) :- even36(T61, X101).
half60(s(s(T89)), s(X156)) :- half60(T89, X156).
times69(s(T104), T105, X182) :- even36(s(T104), X181).
times69(s(T117), T118, X217) :- ','(evenc36(s(T117), true), half60(s(T117), X215)).
times69(s(T117), T118, X217) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), times69(T120, T118, X216))).
times69(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), ','(timesc69(T120, T118, s(s(s(s(s(s(s(s(T150))))))))), plus154(T150, s(s(s(s(s(s(s(T150))))))), X411)))).
times69(s(T171), T172, X455) :- ','(evenc36(s(T171), false), times69(T171, T172, X454)).
times69(s(T171), T172, X455) :- ','(evenc36(s(T171), false), ','(timesc69(T171, T172, T175), plus166(T172, T175, X455))).
plus154(s(T163), T164, s(X436)) :- plus154(T163, T164, X436).
plus166(s(T189), T190, s(X482)) :- plus166(T189, T190, X482).
times1(s(0), T23, T25) :- ','(timesc18(T23, T29), plus19(T23, T29, T25)).
times1(s(s(T57)), T10, T12) :- even36(T57, X90).
times1(s(s(T86)), T78, T80) :- ','(evenc36(T86, true), half60(T86, X147)).
times1(s(s(T77)), T78, T80) :- ','(evenc36(T77, true), ','(halfc56(T77, T82), times69(T82, T78, X129))).
times1(s(s(T77)), T78, s(s(s(s(s(s(s(s(T248))))))))) :- ','(evenc36(T77, true), ','(halfc56(T77, T82), ','(timesc69(T82, T78, s(s(s(s(s(s(s(s(T246))))))))), plus19(T246, s(s(s(s(s(s(s(s(T246)))))))), T248)))).
times1(s(s(T258)), T259, T261) :- ','(evenc36(T258, false), times69(s(T258), T259, X662)).
times1(s(s(T258)), T259, T261) :- ','(evenc36(T258, false), ','(timesc69(s(T258), T259, T264), plus19(T259, T264, T261))).

Clauses:

plusc19(0, T42, T42).
plusc19(s(T49), T50, s(T52)) :- plusc19(T49, T50, T52).
evenc36(0, true).
evenc36(s(0), false).
evenc36(s(s(T61)), X101) :- evenc36(T61, X101).
halfc60(0, 0).
halfc60(s(s(T89)), s(X156)) :- halfc60(T89, X156).
timesc69(0, T99, 0).
timesc69(s(T117), T118, 0) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, 0))).
timesc69(s(T117), T118, s(s(0))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(0)))).
timesc69(s(T117), T118, s(s(s(s(0))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(0))))).
timesc69(s(T117), T118, s(s(s(s(s(s(0))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(s(0)))))).
timesc69(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(s(s(0))))))).
timesc69(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(s(s(s(0)))))))).
timesc69(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(s(s(s(s(0))))))))).
timesc69(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), timesc69(T120, T118, s(s(s(s(s(s(s(0)))))))))).
timesc69(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) :- ','(evenc36(s(T117), true), ','(halfc60(s(T117), T120), ','(timesc69(T120, T118, s(s(s(s(s(s(s(s(T150))))))))), plusc154(T150, s(s(s(s(s(s(s(T150))))))), X411)))).
timesc69(s(T171), T172, X455) :- ','(evenc36(s(T171), false), ','(timesc69(T171, T172, T175), plusc166(T172, T175, X455))).
plusc154(0, T158, s(T158)).
plusc154(s(T163), T164, s(X436)) :- plusc154(T163, T164, X436).
plusc166(0, T184, T184).
plusc166(s(T189), T190, s(X482)) :- plusc166(T189, T190, X482).
timesc18(T35, 0).
halfc56(T86, s(X147)) :- halfc60(T86, X147).

Afs:

times1(x1, x2, x3) = times1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
times1_in: (b,b,f)
plus19_in: (b,b,f)
even36_in: (b,f)
evenc36_in: (b,b)
half60_in: (b,f)
halfc56_in: (b,f)
halfc60_in: (b,f)
times69_in: (b,b,f)
timesc69_in: (b,b,f) (b,b,b)
plusc166_in: (b,b,f) (b,b,b)
plusc154_in: (b,b,f) (b,b,b)
plus154_in: (b,b,f)
plus166_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TIMES1_IN_GGA(s(0), T23, T25) → U19_GGA(T23, T25, timesc18_in_ga(T23, T29))
U19_GGA(T23, T25, timesc18_out_ga(T23, T29)) → U20_GGA(T23, T25, plus19_in_gga(T23, T29, T25))
U19_GGA(T23, T25, timesc18_out_ga(T23, T29)) → PLUS19_IN_GGA(T23, T29, T25)
PLUS19_IN_GGA(s(T49), T50, s(T52)) → U1_GGA(T49, T50, T52, plus19_in_gga(T49, T50, T52))
PLUS19_IN_GGA(s(T49), T50, s(T52)) → PLUS19_IN_GGA(T49, T50, T52)
TIMES1_IN_GGA(s(s(T57)), T10, T12) → U21_GGA(T57, T10, T12, even36_in_ga(T57, X90))
TIMES1_IN_GGA(s(s(T57)), T10, T12) → EVEN36_IN_GA(T57, X90)
EVEN36_IN_GA(s(s(T61)), X101) → U2_GA(T61, X101, even36_in_ga(T61, X101))
EVEN36_IN_GA(s(s(T61)), X101) → EVEN36_IN_GA(T61, X101)
TIMES1_IN_GGA(s(s(T86)), T78, T80) → U22_GGA(T86, T78, T80, evenc36_in_gg(T86, true))
U22_GGA(T86, T78, T80, evenc36_out_gg(T86, true)) → U23_GGA(T86, T78, T80, half60_in_ga(T86, X147))
U22_GGA(T86, T78, T80, evenc36_out_gg(T86, true)) → HALF60_IN_GA(T86, X147)
HALF60_IN_GA(s(s(T89)), s(X156)) → U3_GA(T89, X156, half60_in_ga(T89, X156))
HALF60_IN_GA(s(s(T89)), s(X156)) → HALF60_IN_GA(T89, X156)
TIMES1_IN_GGA(s(s(T77)), T78, T80) → U24_GGA(T77, T78, T80, evenc36_in_gg(T77, true))
U24_GGA(T77, T78, T80, evenc36_out_gg(T77, true)) → U25_GGA(T77, T78, T80, halfc56_in_ga(T77, T82))
U25_GGA(T77, T78, T80, halfc56_out_ga(T77, T82)) → U26_GGA(T77, T78, T80, times69_in_gga(T82, T78, X129))
U25_GGA(T77, T78, T80, halfc56_out_ga(T77, T82)) → TIMES69_IN_GGA(T82, T78, X129)
TIMES69_IN_GGA(s(T104), T105, X182) → U4_GGA(T104, T105, X182, even36_in_ga(s(T104), X181))
TIMES69_IN_GGA(s(T104), T105, X182) → EVEN36_IN_GA(s(T104), X181)
TIMES69_IN_GGA(s(T117), T118, X217) → U5_GGA(T117, T118, X217, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U6_GGA(T117, T118, X217, half60_in_ga(s(T117), X215))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → HALF60_IN_GA(s(T117), X215)
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, X217, halfc60_in_ga(s(T117), T120))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → U8_GGA(T117, T118, X217, times69_in_gga(T120, T118, X216))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118, X216)
TIMES69_IN_GGA(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U9_GGA(T117, T118, X411, evenc36_in_gg(s(T117), true))
U9_GGA(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U10_GGA(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U10_GGA(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U11_GGA(T117, T118, X411, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
U11_GGA(T117, T118, X411, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U12_GGA(T117, T118, X411, plus154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
U11_GGA(T117, T118, X411, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → PLUS154_IN_GGA(T150, s(s(s(s(s(s(s(T150))))))), X411)
PLUS154_IN_GGA(s(T163), T164, s(X436)) → U17_GGA(T163, T164, X436, plus154_in_gga(T163, T164, X436))
PLUS154_IN_GGA(s(T163), T164, s(X436)) → PLUS154_IN_GGA(T163, T164, X436)
TIMES69_IN_GGA(s(T171), T172, X455) → U13_GGA(T171, T172, X455, evenc36_in_gg(s(T171), false))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U14_GGA(T171, T172, X455, times69_in_gga(T171, T172, X454))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172, X454)
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U15_GGA(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U15_GGA(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U16_GGA(T171, T172, X455, plus166_in_gga(T172, T175, X455))
U15_GGA(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → PLUS166_IN_GGA(T172, T175, X455)
PLUS166_IN_GGA(s(T189), T190, s(X482)) → U18_GGA(T189, T190, X482, plus166_in_gga(T189, T190, X482))
PLUS166_IN_GGA(s(T189), T190, s(X482)) → PLUS166_IN_GGA(T189, T190, X482)
TIMES1_IN_GGA(s(s(T77)), T78, s(s(s(s(s(s(s(s(T248))))))))) → U27_GGA(T77, T78, T248, evenc36_in_gg(T77, true))
U27_GGA(T77, T78, T248, evenc36_out_gg(T77, true)) → U28_GGA(T77, T78, T248, halfc56_in_ga(T77, T82))
U28_GGA(T77, T78, T248, halfc56_out_ga(T77, T82)) → U29_GGA(T77, T78, T248, timesc69_in_gga(T82, T78, s(s(s(s(s(s(s(s(T246))))))))))
U29_GGA(T77, T78, T248, timesc69_out_gga(T82, T78, s(s(s(s(s(s(s(s(T246)))))))))) → U30_GGA(T77, T78, T248, plus19_in_gga(T246, s(s(s(s(s(s(s(s(T246)))))))), T248))
U29_GGA(T77, T78, T248, timesc69_out_gga(T82, T78, s(s(s(s(s(s(s(s(T246)))))))))) → PLUS19_IN_GGA(T246, s(s(s(s(s(s(s(s(T246)))))))), T248)
TIMES1_IN_GGA(s(s(T258)), T259, T261) → U31_GGA(T258, T259, T261, evenc36_in_gg(T258, false))
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → U32_GGA(T258, T259, T261, times69_in_gga(s(T258), T259, X662))
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → TIMES69_IN_GGA(s(T258), T259, X662)
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → U33_GGA(T258, T259, T261, timesc69_in_gga(s(T258), T259, T264))
U33_GGA(T258, T259, T261, timesc69_out_gga(s(T258), T259, T264)) → U34_GGA(T258, T259, T261, plus19_in_gga(T259, T264, T261))
U33_GGA(T258, T259, T261, timesc69_out_gga(s(T258), T259, T264)) → PLUS19_IN_GGA(T259, T264, T261)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
plus19_in_gga(x1, x2, x3) = plus19_in_gga(x1, x2)
even36_in_ga(x1, x2) = even36_in_ga(x1)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
half60_in_ga(x1, x2) = half60_in_ga(x1)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
times69_in_gga(x1, x2, x3) = times69_in_gga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
plus154_in_gga(x1, x2, x3) = plus154_in_gga(x1, x2)
plus166_in_gga(x1, x2, x3) = plus166_in_gga(x1, x2)
TIMES1_IN_GGA(x1, x2, x3) = TIMES1_IN_GGA(x1, x2)
U19_GGA(x1, x2, x3) = U19_GGA(x1, x3)
U20_GGA(x1, x2, x3) = U20_GGA(x1, x3)
PLUS19_IN_GGA(x1, x2, x3) = PLUS19_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)
EVEN36_IN_GA(x1, x2) = EVEN36_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)
HALF60_IN_GA(x1, x2) = HALF60_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)
U26_GGA(x1, x2, x3, x4) = U26_GGA(x1, x2, x4)
TIMES69_IN_GGA(x1, x2, x3) = TIMES69_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)
PLUS154_IN_GGA(x1, x2, x3) = PLUS154_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)
PLUS166_IN_GGA(x1, x2, x3) = PLUS166_IN_GGA(x1, x2)
U18_GGA(x1, x2, x3, x4) = U18_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)
U34_GGA(x1, x2, x3, x4) = U34_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:

TIMES1_IN_GGA(s(0), T23, T25) → U19_GGA(T23, T25, timesc18_in_ga(T23, T29))
U19_GGA(T23, T25, timesc18_out_ga(T23, T29)) → U20_GGA(T23, T25, plus19_in_gga(T23, T29, T25))
U19_GGA(T23, T25, timesc18_out_ga(T23, T29)) → PLUS19_IN_GGA(T23, T29, T25)
PLUS19_IN_GGA(s(T49), T50, s(T52)) → U1_GGA(T49, T50, T52, plus19_in_gga(T49, T50, T52))
PLUS19_IN_GGA(s(T49), T50, s(T52)) → PLUS19_IN_GGA(T49, T50, T52)
TIMES1_IN_GGA(s(s(T57)), T10, T12) → U21_GGA(T57, T10, T12, even36_in_ga(T57, X90))
TIMES1_IN_GGA(s(s(T57)), T10, T12) → EVEN36_IN_GA(T57, X90)
EVEN36_IN_GA(s(s(T61)), X101) → U2_GA(T61, X101, even36_in_ga(T61, X101))
EVEN36_IN_GA(s(s(T61)), X101) → EVEN36_IN_GA(T61, X101)
TIMES1_IN_GGA(s(s(T86)), T78, T80) → U22_GGA(T86, T78, T80, evenc36_in_gg(T86, true))
U22_GGA(T86, T78, T80, evenc36_out_gg(T86, true)) → U23_GGA(T86, T78, T80, half60_in_ga(T86, X147))
U22_GGA(T86, T78, T80, evenc36_out_gg(T86, true)) → HALF60_IN_GA(T86, X147)
HALF60_IN_GA(s(s(T89)), s(X156)) → U3_GA(T89, X156, half60_in_ga(T89, X156))
HALF60_IN_GA(s(s(T89)), s(X156)) → HALF60_IN_GA(T89, X156)
TIMES1_IN_GGA(s(s(T77)), T78, T80) → U24_GGA(T77, T78, T80, evenc36_in_gg(T77, true))
U24_GGA(T77, T78, T80, evenc36_out_gg(T77, true)) → U25_GGA(T77, T78, T80, halfc56_in_ga(T77, T82))
U25_GGA(T77, T78, T80, halfc56_out_ga(T77, T82)) → U26_GGA(T77, T78, T80, times69_in_gga(T82, T78, X129))
U25_GGA(T77, T78, T80, halfc56_out_ga(T77, T82)) → TIMES69_IN_GGA(T82, T78, X129)
TIMES69_IN_GGA(s(T104), T105, X182) → U4_GGA(T104, T105, X182, even36_in_ga(s(T104), X181))
TIMES69_IN_GGA(s(T104), T105, X182) → EVEN36_IN_GA(s(T104), X181)
TIMES69_IN_GGA(s(T117), T118, X217) → U5_GGA(T117, T118, X217, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U6_GGA(T117, T118, X217, half60_in_ga(s(T117), X215))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → HALF60_IN_GA(s(T117), X215)
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, X217, halfc60_in_ga(s(T117), T120))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → U8_GGA(T117, T118, X217, times69_in_gga(T120, T118, X216))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118, X216)
TIMES69_IN_GGA(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U9_GGA(T117, T118, X411, evenc36_in_gg(s(T117), true))
U9_GGA(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U10_GGA(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U10_GGA(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U11_GGA(T117, T118, X411, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
U11_GGA(T117, T118, X411, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U12_GGA(T117, T118, X411, plus154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
U11_GGA(T117, T118, X411, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → PLUS154_IN_GGA(T150, s(s(s(s(s(s(s(T150))))))), X411)
PLUS154_IN_GGA(s(T163), T164, s(X436)) → U17_GGA(T163, T164, X436, plus154_in_gga(T163, T164, X436))
PLUS154_IN_GGA(s(T163), T164, s(X436)) → PLUS154_IN_GGA(T163, T164, X436)
TIMES69_IN_GGA(s(T171), T172, X455) → U13_GGA(T171, T172, X455, evenc36_in_gg(s(T171), false))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U14_GGA(T171, T172, X455, times69_in_gga(T171, T172, X454))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172, X454)
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U15_GGA(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U15_GGA(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U16_GGA(T171, T172, X455, plus166_in_gga(T172, T175, X455))
U15_GGA(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → PLUS166_IN_GGA(T172, T175, X455)
PLUS166_IN_GGA(s(T189), T190, s(X482)) → U18_GGA(T189, T190, X482, plus166_in_gga(T189, T190, X482))
PLUS166_IN_GGA(s(T189), T190, s(X482)) → PLUS166_IN_GGA(T189, T190, X482)
TIMES1_IN_GGA(s(s(T77)), T78, s(s(s(s(s(s(s(s(T248))))))))) → U27_GGA(T77, T78, T248, evenc36_in_gg(T77, true))
U27_GGA(T77, T78, T248, evenc36_out_gg(T77, true)) → U28_GGA(T77, T78, T248, halfc56_in_ga(T77, T82))
U28_GGA(T77, T78, T248, halfc56_out_ga(T77, T82)) → U29_GGA(T77, T78, T248, timesc69_in_gga(T82, T78, s(s(s(s(s(s(s(s(T246))))))))))
U29_GGA(T77, T78, T248, timesc69_out_gga(T82, T78, s(s(s(s(s(s(s(s(T246)))))))))) → U30_GGA(T77, T78, T248, plus19_in_gga(T246, s(s(s(s(s(s(s(s(T246)))))))), T248))
U29_GGA(T77, T78, T248, timesc69_out_gga(T82, T78, s(s(s(s(s(s(s(s(T246)))))))))) → PLUS19_IN_GGA(T246, s(s(s(s(s(s(s(s(T246)))))))), T248)
TIMES1_IN_GGA(s(s(T258)), T259, T261) → U31_GGA(T258, T259, T261, evenc36_in_gg(T258, false))
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → U32_GGA(T258, T259, T261, times69_in_gga(s(T258), T259, X662))
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → TIMES69_IN_GGA(s(T258), T259, X662)
U31_GGA(T258, T259, T261, evenc36_out_gg(T258, false)) → U33_GGA(T258, T259, T261, timesc69_in_gga(s(T258), T259, T264))
U33_GGA(T258, T259, T261, timesc69_out_gga(s(T258), T259, T264)) → U34_GGA(T258, T259, T261, plus19_in_gga(T259, T264, T261))
U33_GGA(T258, T259, T261, timesc69_out_gga(s(T258), T259, T264)) → PLUS19_IN_GGA(T259, T264, T261)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 42 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PLUS166_IN_GGA(s(T189), T190, s(X482)) → PLUS166_IN_GGA(T189, T190, X482)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
PLUS166_IN_GGA(x1, x2, x3) = PLUS166_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:

PLUS166_IN_GGA(s(T189), T190, s(X482)) → PLUS166_IN_GGA(T189, T190, X482)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS166_IN_GGA(x1, x2, x3) = PLUS166_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:

PLUS166_IN_GGA(s(T189), T190) → PLUS166_IN_GGA(T189, T190)

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:

PLUS166_IN_GGA(s(T189), T190) → PLUS166_IN_GGA(T189, T190)
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:

PLUS154_IN_GGA(s(T163), T164, s(X436)) → PLUS154_IN_GGA(T163, T164, X436)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
PLUS154_IN_GGA(x1, x2, x3) = PLUS154_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:

PLUS154_IN_GGA(s(T163), T164, s(X436)) → PLUS154_IN_GGA(T163, T164, X436)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS154_IN_GGA(x1, x2, x3) = PLUS154_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:

PLUS154_IN_GGA(s(T163), T164) → PLUS154_IN_GGA(T163, T164)

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:

PLUS154_IN_GGA(s(T163), T164) → PLUS154_IN_GGA(T163, T164)
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:

HALF60_IN_GA(s(s(T89)), s(X156)) → HALF60_IN_GA(T89, X156)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
HALF60_IN_GA(x1, x2) = HALF60_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:

HALF60_IN_GA(s(s(T89)), s(X156)) → HALF60_IN_GA(T89, X156)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
HALF60_IN_GA(x1, x2) = HALF60_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:

HALF60_IN_GA(s(s(T89))) → HALF60_IN_GA(T89)

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:

HALF60_IN_GA(s(s(T89))) → HALF60_IN_GA(T89)
The graph contains the following edges 1 > 1

(27) YES

(28) Obligation:

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

EVEN36_IN_GA(s(s(T61)), X101) → EVEN36_IN_GA(T61, X101)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
EVEN36_IN_GA(x1, x2) = EVEN36_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:

EVEN36_IN_GA(s(s(T61)), X101) → EVEN36_IN_GA(T61, X101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
EVEN36_IN_GA(x1, x2) = EVEN36_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:

EVEN36_IN_GA(s(s(T61))) → EVEN36_IN_GA(T61)

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:

EVEN36_IN_GA(s(s(T61))) → EVEN36_IN_GA(T61)
The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

TIMES69_IN_GGA(s(T117), T118, X217) → U5_GGA(T117, T118, X217, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, X217, halfc60_in_ga(s(T117), T120))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118, X216)
TIMES69_IN_GGA(s(T171), T172, X455) → U13_GGA(T171, T172, X455, evenc36_in_gg(s(T171), false))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172, X454)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
TIMES69_IN_GGA(x1, x2, x3) = TIMES69_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:

TIMES69_IN_GGA(s(T117), T118, X217) → U5_GGA(T117, T118, X217, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, X217, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, X217, halfc60_in_ga(s(T117), T120))
U7_GGA(T117, T118, X217, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118, X216)
TIMES69_IN_GGA(s(T171), T172, X455) → U13_GGA(T171, T172, X455, evenc36_in_gg(s(T171), false))
U13_GGA(T171, T172, X455, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172, X454)

The TRS R consists of the following rules:

evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
TIMES69_IN_GGA(x1, x2, x3) = TIMES69_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:

TIMES69_IN_GGA(s(T117), T118) → U5_GGA(T117, T118, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, halfc60_in_ga(s(T117)))
U7_GGA(T117, T118, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118)
TIMES69_IN_GGA(s(T171), T172) → U13_GGA(T171, T172, evenc36_in_gg(s(T171), false))
U13_GGA(T171, T172, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172)

The TRS R consists of the following rules:

evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
halfc60_in_ga(0) → halfc60_out_ga(0, 0)

The set Q consists of the following terms:

evenc36_in_gg(x0, x1)
halfc60_in_ga(x0)
U37_gg(x0, x1, x2)
U38_ga(x0, x1)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TIMES69_IN_GGA(s(T171), T172) → U13_GGA(T171, T172, evenc36_in_gg(s(T171), false))

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

POL(0) = 0
POL(TIMES69_IN_GGA(x₁, x₂)) = x₁
POL(U13_GGA(x₁, x₂, x₃)) = x₁
POL(U37_gg(x₁, x₂, x₃)) = 0
POL(U38_ga(x₁, x₂)) = 1 + x₂
POL(U5_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(U7_GGA(x₁, x₂, x₃)) = x₃
POL(evenc36_in_gg(x₁, x₂)) = 0
POL(evenc36_out_gg(x₁, x₂)) = 0
POL(false) = 0
POL(halfc60_in_ga(x₁)) = x₁
POL(halfc60_out_ga(x₁, x₂)) = x₂
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
halfc60_in_ga(0) → halfc60_out_ga(0, 0)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))

(41) Obligation:

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

TIMES69_IN_GGA(s(T117), T118) → U5_GGA(T117, T118, evenc36_in_gg(s(T117), true))
U5_GGA(T117, T118, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, halfc60_in_ga(s(T117)))
U7_GGA(T117, T118, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118)
U13_GGA(T171, T172, evenc36_out_gg(s(T171), false)) → TIMES69_IN_GGA(T171, T172)

The TRS R consists of the following rules:

evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
halfc60_in_ga(0) → halfc60_out_ga(0, 0)

The set Q consists of the following terms:

evenc36_in_gg(x0, x1)
halfc60_in_ga(x0)
U37_gg(x0, x1, x2)
U38_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(T117, T118, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, halfc60_in_ga(s(T117)))
U7_GGA(T117, T118, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118)
TIMES69_IN_GGA(s(T117), T118) → U5_GGA(T117, T118, evenc36_in_gg(s(T117), true))

The TRS R consists of the following rules:

evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
halfc60_in_ga(0) → halfc60_out_ga(0, 0)

The set Q consists of the following terms:

evenc36_in_gg(x0, x1)
halfc60_in_ga(x0)
U37_gg(x0, x1, x2)
U38_ga(x0, x1)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TIMES69_IN_GGA(s(T117), T118) → U5_GGA(T117, T118, evenc36_in_gg(s(T117), true))

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

POL(U5_GGA(x₁, x₂, x₃)) = 0 +
[ 0, 1 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃

POL(evenc36_out_gg(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(true) =
/ 1 \

\ 1 /

POL(U7_GGA(x₁, x₂, x₃)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 1 ]

· x₃

POL(halfc60_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(halfc60_out_ga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(TIMES69_IN_GGA(x₁, x₂)) = 0 +
[ 0, 1 ]

· x₁ +
[ 0, 0 ]

· x₂

POL(evenc36_in_gg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(U38_ga(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(U37_gg(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂ +
/ 1 0 \

\ 1 0 /

· x₃

POL(0) =
/ 0 \

\ 0 /

POL(false) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
halfc60_in_ga(0) → halfc60_out_ga(0, 0)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))

(45) Obligation:

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

U5_GGA(T117, T118, evenc36_out_gg(s(T117), true)) → U7_GGA(T117, T118, halfc60_in_ga(s(T117)))
U7_GGA(T117, T118, halfc60_out_ga(s(T117), T120)) → TIMES69_IN_GGA(T120, T118)

The TRS R consists of the following rules:

evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
halfc60_in_ga(s(s(T89))) → U38_ga(T89, halfc60_in_ga(T89))
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
U38_ga(T89, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
halfc60_in_ga(0) → halfc60_out_ga(0, 0)

The set Q consists of the following terms:

evenc36_in_gg(x0, x1)
halfc60_in_ga(x0)
U37_gg(x0, x1, x2)
U38_ga(x0, x1)

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

(46) DependencyGraphProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

PLUS19_IN_GGA(s(T49), T50, s(T52)) → PLUS19_IN_GGA(T49, T50, T52)

The TRS R consists of the following rules:

timesc18_in_ga(T35, 0) → timesc18_out_ga(T35, 0)
evenc36_in_gg(0, true) → evenc36_out_gg(0, true)
evenc36_in_gg(s(0), false) → evenc36_out_gg(s(0), false)
evenc36_in_gg(s(s(T61)), X101) → U37_gg(T61, X101, evenc36_in_gg(T61, X101))
U37_gg(T61, X101, evenc36_out_gg(T61, X101)) → evenc36_out_gg(s(s(T61)), X101)
halfc56_in_ga(T86, s(X147)) → U72_ga(T86, X147, halfc60_in_ga(T86, X147))
halfc60_in_ga(0, 0) → halfc60_out_ga(0, 0)
halfc60_in_ga(s(s(T89)), s(X156)) → U38_ga(T89, X156, halfc60_in_ga(T89, X156))
U38_ga(T89, X156, halfc60_out_ga(T89, X156)) → halfc60_out_ga(s(s(T89)), s(X156))
U72_ga(T86, X147, halfc60_out_ga(T86, X147)) → halfc56_out_ga(T86, s(X147))
timesc69_in_gga(0, T99, 0) → timesc69_out_gga(0, T99, 0)
timesc69_in_gga(s(T117), T118, 0) → U39_gga(T117, T118, evenc36_in_gg(s(T117), true))
U39_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U40_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U40_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_gga(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(0, T99, 0) → timesc69_out_ggg(0, T99, 0)
timesc69_in_ggg(s(T117), T118, 0) → U39_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U39_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U40_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U40_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U41_ggg(T117, T118, timesc69_in_ggg(T120, T118, 0))
timesc69_in_ggg(s(T117), T118, s(s(0))) → U42_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U42_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U43_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U43_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T117), T118, s(s(s(s(0))))) → U45_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U45_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U46_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U46_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U48_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U49_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U49_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U51_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U52_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U52_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U54_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U55_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U55_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U57_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U58_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U58_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_ggg(T117, T118, evenc36_in_gg(s(T117), true))
U60_ggg(T117, T118, evenc36_out_gg(s(T117), true)) → U61_ggg(T117, T118, halfc60_in_ga(s(T117), T120))
U61_ggg(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_ggg(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
timesc69_in_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_ggg(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_ggg(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_ggg(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_ggg(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_ggg(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T117), T118, s(s(0))) → U42_gga(T117, T118, evenc36_in_gg(s(T117), true))
U42_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U43_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U43_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U44_gga(T117, T118, timesc69_in_ggg(T120, T118, s(0)))
timesc69_in_ggg(s(T171), T172, X455) → U67_ggg(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_ggg(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_ggg(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
timesc69_in_gga(s(T117), T118, s(s(s(s(0))))) → U45_gga(T117, T118, evenc36_in_gg(s(T117), true))
U45_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U46_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U46_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U47_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(0))))
U47_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_gga(s(T117), T118, s(s(s(s(0)))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(0))))))) → U48_gga(T117, T118, evenc36_in_gg(s(T117), true))
U48_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U49_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U49_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U50_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(0)))))
U50_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(0)))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(0))))))))) → U51_gga(T117, T118, evenc36_in_gg(s(T117), true))
U51_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U52_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U52_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U53_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(0))))))
U53_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0))))))))))) → U54_gga(T117, T118, evenc36_in_gg(s(T117), true))
U54_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U55_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U55_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U56_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(0)))))))
U56_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → U57_gga(T117, T118, evenc36_in_gg(s(T117), true))
U57_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U58_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U58_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U59_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(0))))))))
U59_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → U60_gga(T117, T118, evenc36_in_gg(s(T117), true))
U60_gga(T117, T118, evenc36_out_gg(s(T117), true)) → U61_gga(T117, T118, halfc60_in_ga(s(T117), T120))
U61_gga(T117, T118, halfc60_out_ga(s(T117), T120)) → U62_gga(T117, T118, timesc69_in_ggg(T120, T118, s(s(s(s(s(s(s(0)))))))))
U62_gga(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
timesc69_in_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411))))))))) → U63_gga(T117, T118, X411, evenc36_in_gg(s(T117), true))
U63_gga(T117, T118, X411, evenc36_out_gg(s(T117), true)) → U64_gga(T117, T118, X411, halfc60_in_ga(s(T117), T120))
U64_gga(T117, T118, X411, halfc60_out_ga(s(T117), T120)) → U65_gga(T117, T118, X411, T120, timesc69_in_gga(T120, T118, s(s(s(s(s(s(s(s(T150))))))))))
timesc69_in_gga(s(T171), T172, X455) → U67_gga(T171, T172, X455, evenc36_in_gg(s(T171), false))
U67_gga(T171, T172, X455, evenc36_out_gg(s(T171), false)) → U68_gga(T171, T172, X455, timesc69_in_gga(T171, T172, T175))
U68_gga(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_gga(T171, T172, X455, plusc166_in_gga(T172, T175, X455))
plusc166_in_gga(0, T184, T184) → plusc166_out_gga(0, T184, T184)
plusc166_in_gga(s(T189), T190, s(X482)) → U71_gga(T189, T190, X482, plusc166_in_gga(T189, T190, X482))
U71_gga(T189, T190, X482, plusc166_out_gga(T189, T190, X482)) → plusc166_out_gga(s(T189), T190, s(X482))
U69_gga(T171, T172, X455, plusc166_out_gga(T172, T175, X455)) → timesc69_out_gga(s(T171), T172, X455)
U65_gga(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_gga(T117, T118, X411, plusc154_in_gga(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_gga(0, T158, s(T158)) → plusc154_out_gga(0, T158, s(T158))
plusc154_in_gga(s(T163), T164, s(X436)) → U70_gga(T163, T164, X436, plusc154_in_gga(T163, T164, X436))
U70_gga(T163, T164, X436, plusc154_out_gga(T163, T164, X436)) → plusc154_out_gga(s(T163), T164, s(X436))
U66_gga(T117, T118, X411, plusc154_out_gga(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_gga(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U68_ggg(T171, T172, X455, timesc69_out_gga(T171, T172, T175)) → U69_ggg(T171, T172, X455, plusc166_in_ggg(T172, T175, X455))
plusc166_in_ggg(0, T184, T184) → plusc166_out_ggg(0, T184, T184)
plusc166_in_ggg(s(T189), T190, s(X482)) → U71_ggg(T189, T190, X482, plusc166_in_ggg(T189, T190, X482))
U71_ggg(T189, T190, X482, plusc166_out_ggg(T189, T190, X482)) → plusc166_out_ggg(s(T189), T190, s(X482))
U69_ggg(T171, T172, X455, plusc166_out_ggg(T172, T175, X455)) → timesc69_out_ggg(s(T171), T172, X455)
U44_gga(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_gga(s(T117), T118, s(s(0)))
U65_ggg(T117, T118, X411, T120, timesc69_out_gga(T120, T118, s(s(s(s(s(s(s(s(T150)))))))))) → U66_ggg(T117, T118, X411, plusc154_in_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411))
plusc154_in_ggg(0, T158, s(T158)) → plusc154_out_ggg(0, T158, s(T158))
plusc154_in_ggg(s(T163), T164, s(X436)) → U70_ggg(T163, T164, X436, plusc154_in_ggg(T163, T164, X436))
U70_ggg(T163, T164, X436, plusc154_out_ggg(T163, T164, X436)) → plusc154_out_ggg(s(T163), T164, s(X436))
U66_ggg(T117, T118, X411, plusc154_out_ggg(T150, s(s(s(s(s(s(s(T150))))))), X411)) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(X411)))))))))
U62_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(s(0))))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
U59_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(s(0)))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
U56_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(s(0))))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(s(s(0)))))))))))
U53_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(s(0)))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(s(s(0)))))))))
U50_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(s(0))))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(s(s(0)))))))
U47_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(s(0)))) → timesc69_out_ggg(s(T117), T118, s(s(s(s(0)))))
U44_ggg(T117, T118, timesc69_out_ggg(T120, T118, s(0))) → timesc69_out_ggg(s(T117), T118, s(s(0)))
U41_ggg(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_ggg(s(T117), T118, 0)
U41_gga(T117, T118, timesc69_out_ggg(T120, T118, 0)) → timesc69_out_gga(s(T117), T118, 0)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
timesc18_in_ga(x1, x2) = timesc18_in_ga(x1)
timesc18_out_ga(x1, x2) = timesc18_out_ga(x1, x2)
evenc36_in_gg(x1, x2) = evenc36_in_gg(x1, x2)
true = true
evenc36_out_gg(x1, x2) = evenc36_out_gg(x1, x2)
false = false
U37_gg(x1, x2, x3) = U37_gg(x1, x2, x3)
halfc56_in_ga(x1, x2) = halfc56_in_ga(x1)
U72_ga(x1, x2, x3) = U72_ga(x1, x3)
halfc60_in_ga(x1, x2) = halfc60_in_ga(x1)
halfc60_out_ga(x1, x2) = halfc60_out_ga(x1, x2)
U38_ga(x1, x2, x3) = U38_ga(x1, x3)
halfc56_out_ga(x1, x2) = halfc56_out_ga(x1, x2)
timesc69_in_gga(x1, x2, x3) = timesc69_in_gga(x1, x2)
timesc69_out_gga(x1, x2, x3) = timesc69_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3) = U39_gga(x1, x2, x3)
U40_gga(x1, x2, x3) = U40_gga(x1, x2, x3)
U41_gga(x1, x2, x3) = U41_gga(x1, x2, x3)
timesc69_in_ggg(x1, x2, x3) = timesc69_in_ggg(x1, x2, x3)
timesc69_out_ggg(x1, x2, x3) = timesc69_out_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) = U62_ggg(x1, x2, x3)
U63_ggg(x1, x2, x3, x4) = U63_ggg(x1, x2, x3, x4)
U64_ggg(x1, x2, x3, x4) = U64_ggg(x1, x2, x3, x4)
U65_ggg(x1, x2, x3, x4, x5) = U65_ggg(x1, x2, x3, x5)
U42_gga(x1, x2, x3) = U42_gga(x1, x2, x3)
U43_gga(x1, x2, x3) = U43_gga(x1, x2, x3)
U44_gga(x1, x2, x3) = U44_gga(x1, x2, x3)
U67_ggg(x1, x2, x3, x4) = U67_ggg(x1, x2, x3, x4)
U68_ggg(x1, x2, x3, x4) = U68_ggg(x1, x2, x3, x4)
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) = U62_gga(x1, x2, x3)
U63_gga(x1, x2, x3, x4) = U63_gga(x1, x2, x4)
U64_gga(x1, x2, x3, x4) = U64_gga(x1, x2, x4)
U65_gga(x1, x2, x3, x4, x5) = U65_gga(x1, x2, x5)
U67_gga(x1, x2, x3, x4) = U67_gga(x1, x2, x4)
U68_gga(x1, x2, x3, x4) = U68_gga(x1, x2, x4)
U69_gga(x1, x2, x3, x4) = U69_gga(x1, x2, x4)
plusc166_in_gga(x1, x2, x3) = plusc166_in_gga(x1, x2)
plusc166_out_gga(x1, x2, x3) = plusc166_out_gga(x1, x2, x3)
U71_gga(x1, x2, x3, x4) = U71_gga(x1, x2, x4)
U66_gga(x1, x2, x3, x4) = U66_gga(x1, x2, x4)
plusc154_in_gga(x1, x2, x3) = plusc154_in_gga(x1, x2)
plusc154_out_gga(x1, x2, x3) = plusc154_out_gga(x1, x2, x3)
U70_gga(x1, x2, x3, x4) = U70_gga(x1, x2, x4)
U69_ggg(x1, x2, x3, x4) = U69_ggg(x1, x2, x3, x4)
plusc166_in_ggg(x1, x2, x3) = plusc166_in_ggg(x1, x2, x3)
plusc166_out_ggg(x1, x2, x3) = plusc166_out_ggg(x1, x2, x3)
U71_ggg(x1, x2, x3, x4) = U71_ggg(x1, x2, x3, x4)
U66_ggg(x1, x2, x3, x4) = U66_ggg(x1, x2, x3, x4)
plusc154_in_ggg(x1, x2, x3) = plusc154_in_ggg(x1, x2, x3)
plusc154_out_ggg(x1, x2, x3) = plusc154_out_ggg(x1, x2, x3)
U70_ggg(x1, x2, x3, x4) = U70_ggg(x1, x2, x3, x4)
PLUS19_IN_GGA(x1, x2, x3) = PLUS19_IN_GGA(x1, x2)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) Obligation:

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

PLUS19_IN_GGA(s(T49), T50, s(T52)) → PLUS19_IN_GGA(T49, T50, T52)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS19_IN_GGA(x1, x2, x3) = PLUS19_IN_GGA(x1, x2)

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

(51) PiDPToQDPProof (SOUND transformation)

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

(52) Obligation:

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

PLUS19_IN_GGA(s(T49), T50) → PLUS19_IN_GGA(T49, T50)

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

(53) 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:

PLUS19_IN_GGA(s(T49), T50) → PLUS19_IN_GGA(T49, T50)
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) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) DependencyGraphProof (EQUIVALENT transformation)

(47) TRUE

(48) Obligation:

(49) UsableRulesProof (EQUIVALENT transformation)

(50) Obligation:

(51) PiDPToQDPProof (SOUND transformation)

(52) Obligation:

(53) QDPSizeChangeProof (EQUIVALENT transformation)

(54) YES