Left Termination proof of /tmp/tmpqQhBUg/average1.pl

Left Termination of the query pattern average_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 PiDP

↳7 PiDPToQDPProof (⇐)

↳8 QDP

↳9 Instantiation (⇔)

↳10 QDP

↳11 Instantiation (⇔)

↳12 QDP

↳13 ForwardInstantiation (⇔)

↳14 QDP

↳15 ForwardInstantiation (⇔)

↳16 QDP

↳17 NonTerminationProof (⇔)

↳18 NO

(0) Obligation:

Clauses:

average(0, 0, Z) :- ','(!, eq(Z, 0)).
average(0, s(0), Z) :- ','(!, eq(Z, 0)).
average(0, s(s(0)), Z) :- ','(!, eq(Z, s(0))).
average(X, Y, Z) :- ','(p(X, P), average(P, s(Y), Z)).
average(X, Y, Z) :- ','(p(Y, P1), ','(p(P1, P2), ','(p(P2, P3), ','(average(s(X), P3, U), p(Z, U))))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

average(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: average(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: average(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T6, 0)) | average(0, 0, T3), SCOPE: 1, CLAUSE: 1 | average(0, 0, T3), SCOPE: 1, CLAUSE: 2 | average(0, 0, T3), SCOPE: 1, CLAUSE: 3 | average(0, 0, T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: average(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T6, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T6, 0), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_1, eq(T12, 0)) | average(0, s(0), T3), SCOPE: 1, CLAUSE: 2 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(0), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: average(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T12, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: eq(T12, 0), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(!_1, eq(T18, s(0))) | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 3 | average(0, s(s(0)), T3), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: average(T1, T2, T3), SCOPE: 1, CLAUSE: 3 | average(T1, T2, T3), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\naverage(T1, T2, T3) !=? average(0, 0, X2)\naverage(T1, T2, T3) !=? average(0, s(0), X9)\naverage(T1, T2, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: eq(T18, s(0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: eq(T18, s(0)), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T25, X28), average(X28, s(T26), T28)) | average(T25, T26, T3), SCOPE: 1, CLAUSE: 4\n\nT25 is ground\nT26 is ground\nX28 is free; \naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T25, X28), average(X28, s(T26), T28)), SCOPE: 5, CLAUSE: 5 | ','(p(T25, X28), average(X28, s(T26), T28)), SCOPE: 5, CLAUSE: 6 | ?_5 | average(T25, T26, T3), SCOPE: 1, CLAUSE: 4\n\nT25 is ground\nT26 is ground\nX28 is free; \naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T25, X28), average(X28, s(T26), T28)), SCOPE: 5, CLAUSE: 5\n\nT25 is ground\nT26 is ground\nX28 is free; \naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(p(T25, X28), average(X28, s(T26), T28)), SCOPE: 5, CLAUSE: 6 | ?_5 | average(T25, T26, T3), SCOPE: 1, CLAUSE: 4\n\nT25 is ground\nT26 is ground\nX28 is free; \naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: average(0, s(T26), T28)\n\nT26 is ground\naverage(0, T26, T3) !=? average(0, 0, X2)\naverage(0, T26, T3) !=? average(0, s(0), X9)\naverage(0, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T25, X28), average(X28, s(T26), T28)), SCOPE: 5, CLAUSE: 6\n\nT25 is ground\nT26 is ground\nX28 is free; \naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ?_5 | average(T25, T26, T3), SCOPE: 1, CLAUSE: 4\n\nT25 is ground\nT26 is ground\naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: average(T34, s(T26), T28)\n\nT26 is ground\nT34 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: average(T25, T26, T3), SCOPE: 1, CLAUSE: 4\n\nT25 is ground\nT26 is ground\naverage(T25, T26, T3) !=? average(0, 0, X2)\naverage(T25, T26, T3) !=? average(0, s(0), X9)\naverage(T25, T26, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T45, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))))\n\nT44 is ground\nT45 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, T45, T3) !=? average(0, 0, X2)\naverage(T44, T45, T3) !=? average(0, s(0), X9)\naverage(T44, T45, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T45, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))), SCOPE: 6, CLAUSE: 5 | ','(p(T45, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))), SCOPE: 6, CLAUSE: 6\n\nT44 is ground\nT45 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, T45, T3) !=? average(0, 0, X2)\naverage(T44, T45, T3) !=? average(0, s(0), X9)\naverage(T44, T45, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T45, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))), SCOPE: 6, CLAUSE: 5\n\nT44 is ground\nT45 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, T45, T3) !=? average(0, 0, X2)\naverage(T44, T45, T3) !=? average(0, s(0), X9)\naverage(T44, T45, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T45, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))), SCOPE: 6, CLAUSE: 6\n\nT44 is ground\nT45 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, T45, T3) !=? average(0, 0, X2)\naverage(T44, T45, T3) !=? average(0, s(0), X9)\naverage(T44, T45, T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(0, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))\n\nT44 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(0, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 7, CLAUSE: 5 | ','(p(0, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 7, CLAUSE: 6\n\nT44 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(p(0, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 7, CLAUSE: 5\n\nT44 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(0, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 7, CLAUSE: 6\n\nT44 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64)))\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 8, CLAUSE: 5 | ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 8, CLAUSE: 6\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 8, CLAUSE: 5\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 8, CLAUSE: 6\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(average(s(T44), 0, X64), p(T47, X64))\n\nT44 is ground\nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: average(s(T44), 0, X64)\n\nT44 is ground\nX64 is free; \naverage(T44, 0, T3) !=? average(0, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: p(T47, T56)\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: p(T47, T56), SCOPE: 9, CLAUSE: 5 | p(T47, T56), SCOPE: 9, CLAUSE: 6\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: p(T47, T56), SCOPE: 9, CLAUSE: 5\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: p(T47, T56), SCOPE: 9, CLAUSE: 6\n\nT56 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(T63, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64))))\n\nT44 is ground\nT63 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, s(T63), T3) !=? average(0, s(0), X9)\naverage(T44, s(T63), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(T63, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 10, CLAUSE: 5 | ','(p(T63, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 10, CLAUSE: 6\n\nT44 is ground\nT63 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, s(T63), T3) !=? average(0, s(0), X9)\naverage(T44, s(T63), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(p(T63, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 10, CLAUSE: 5\n\nT44 is ground\nT63 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, s(T63), T3) !=? average(0, s(0), X9)\naverage(T44, s(T63), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(p(T63, X62), ','(p(X62, X63), ','(average(s(T44), X63, X64), p(T47, X64)))), SCOPE: 10, CLAUSE: 6\n\nT44 is ground\nT63 is ground\nX62 is free; \nX63 is free; \nX64 is free; \naverage(T44, s(T63), T3) !=? average(0, s(0), X9)\naverage(T44, s(T63), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64)))\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 11, CLAUSE: 5 | ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 11, CLAUSE: 6\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 11, CLAUSE: 5\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(p(0, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 11, CLAUSE: 6\n\nT44 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(average(s(T44), 0, X64), p(T47, X64))\n\nT44 is ground\nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: average(s(T44), 0, X64)\n\nT44 is ground\nX64 is free; \naverage(T44, s(0), T3) !=? average(0, s(0), X9)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: p(T47, T69)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: p(T47, T69), SCOPE: 12, CLAUSE: 5 | p(T47, T69), SCOPE: 12, CLAUSE: 6\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: p(T47, T69), SCOPE: 12, CLAUSE: 5\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: p(T47, T69), SCOPE: 12, CLAUSE: 6\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(p(T76, X63), ','(average(s(T44), X63, X64), p(T47, X64)))\n\nT44 is ground\nT76 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(s(T76)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(p(T76, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 13, CLAUSE: 5 | ','(p(T76, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 13, CLAUSE: 6\n\nT44 is ground\nT76 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(s(T76)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(p(T76, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 13, CLAUSE: 5\n\nT44 is ground\nT76 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(s(T76)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(p(T76, X63), ','(average(s(T44), X63, X64), p(T47, X64))), SCOPE: 13, CLAUSE: 6\n\nT44 is ground\nT76 is ground\nX63 is free; \nX64 is free; \naverage(T44, s(s(T76)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(average(s(T44), 0, X64), p(T47, X64))\n\nT44 is ground\nX64 is free; \naverage(T44, s(s(0)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: average(s(T44), 0, X64)\n\nT44 is ground\nX64 is free; \naverage(T44, s(s(0)), T3) !=? average(0, s(s(0)), X16)", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: p(T47, T79)\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: p(T47, T79), SCOPE: 14, CLAUSE: 5 | p(T47, T79), SCOPE: 14, CLAUSE: 6\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: p(T47, T79), SCOPE: 14, CLAUSE: 5\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: p(T47, T79), SCOPE: 14, CLAUSE: 6\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(average(s(T44), T86, X64), p(T47, X64))\n\nT44 is ground\nT86 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: average(s(T44), T86, X64)\n\nT44 is ground\nT86 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: p(T47, T89)\n\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: p(T47, T89), SCOPE: 15, CLAUSE: 5 | p(T47, T89), SCOPE: 15, CLAUSE: 6\n\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: p(T47, T89), SCOPE: 15, CLAUSE: 5\n\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: p(T47, T89), SCOPE: 15, CLAUSE: 6\n\nT89 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 116 [arrowhead = none ];
116 -> 2;

117 [label="EVAL with clause\naverage(0, 0, X2) :- ','(!_1, eq(X2, 0)).\nand substitution\nT1 -> 0\nT2 -> 0\nT3 -> T6\nX2 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 117 [arrowhead = none ];
117 -> 3;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 118 [arrowhead = none ];
118 -> 4;

119 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 119 [arrowhead = none ];
119 -> 5;

120 [label="EVAL with clause\naverage(0, s(0), X9) :- ','(!_1, eq(X9, 0)).\nand substitution\nT1 -> 0\nT2 -> s(0)\nT3 -> T12\nX9 -> T12\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 120 [arrowhead = none ];
120 -> 10;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 121 [arrowhead = none ];
121 -> 11;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 122 [arrowhead = none ];
122 -> 6;

123 [label="EVAL with clause\neq(X6, X6).\nand substitution\nT6 -> 0\nX6 -> 0\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 123 [arrowhead = none ];
123 -> 7;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 124 [arrowhead = none ];
124 -> 8;

125 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 125 [arrowhead = none ];
125 -> 9;

126 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 126 [arrowhead = none ];
126 -> 12;

127 [label="EVAL with clause\naverage(0, s(s(0)), X16) :- ','(!_1, eq(X16, s(0))).\nand substitution\nT1 -> 0\nT2 -> s(s(0))\nT3 -> T18\nX16 -> T18\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 127 [arrowhead = none ];
127 -> 17;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 128 [arrowhead = none ];
128 -> 18;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 129 [arrowhead = none ];
129 -> 13;

130 [label="EVAL with clause\neq(X13, X13).\nand substitution\nT12 -> 0\nX13 -> 0\nT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 130 [arrowhead = none ];
130 -> 14;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 131 [arrowhead = none ];
131 -> 15;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 132 [arrowhead = none ];
132 -> 16;

133 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 133 [arrowhead = none ];
133 -> 19;

134 [label="ONLY EVAL with clause\naverage(X25, X26, X27) :- ','(p(X25, X28), average(X28, s(X26), X27)).\nand substitution\nT1 -> T25\nX25 -> T25\nT2 -> T26\nX26 -> T26\nT3 -> T28\nX27 -> T28\nT27 -> T28", fontsize=14, style = filled, fillcolor = white];
18 -> 134 [arrowhead = none ];
134 -> 24;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 135 [arrowhead = none ];
135 -> 20;

136 [label="EVAL with clause\neq(X20, X20).\nand substitution\nT18 -> s(0)\nX20 -> s(0)\nT21 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 136 [arrowhead = none ];
136 -> 21;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 137 [arrowhead = none ];
137 -> 22;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 138 [arrowhead = none ];
138 -> 23;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 139 [arrowhead = none ];
139 -> 25;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 140 [arrowhead = none ];
140 -> 26;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 141 [arrowhead = none ];
141 -> 27;

142 [label="EVAL with clause\np(0, 0).\nand substitution\nT25 -> 0\nX28 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 142 [arrowhead = none ];
142 -> 28;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 143 [arrowhead = none ];
143 -> 29;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 144 [arrowhead = none ];
144 -> 30;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 145 [arrowhead = none ];
145 -> 31;

146 [label="INSTANCE with matching:\nT1 -> 0\nT2 -> s(T26)\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 146 [arrowhead = none , style=dashed];
146 -> 1[style=dashed];

147 [label="EVAL with clause\np(s(X35), X35).\nand substitution\nX35 -> T34\nT25 -> s(T34)\nX28 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 147 [arrowhead = none ];
147 -> 32;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 148 [arrowhead = none ];
148 -> 33;

149 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 149 [arrowhead = none ];
149 -> 34;

150 [label="INSTANCE with matching:\nT1 -> T34\nT2 -> s(T26)\nT3 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

151 [label="ONLY EVAL with clause\naverage(X58, X59, X60) :- ','(p(X59, X61), ','(p(X61, X62), ','(p(X62, X63), ','(average(s(X58), X63, X64), p(X60, X64))))).\nand substitution\nT25 -> T44\nX58 -> T44\nT26 -> T45\nX59 -> T45\nT3 -> T47\nX60 -> T47\nT46 -> T47", fontsize=14, style = filled, fillcolor = white];
34 -> 151 [arrowhead = none ];
151 -> 35;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 152 [arrowhead = none ];
152 -> 36;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 153 [arrowhead = none ];
153 -> 37;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 154 [arrowhead = none ];
154 -> 38;

155 [label="EVAL with clause\np(0, 0).\nand substitution\nT45 -> 0\nX61 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 155 [arrowhead = none ];
155 -> 39;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 156 [arrowhead = none ];
156 -> 40;

157 [label="EVAL with clause\np(s(X150), X150).\nand substitution\nX150 -> T63\nT45 -> s(T63)\nX61 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 157 [arrowhead = none ];
157 -> 62;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 158 [arrowhead = none ];
158 -> 63;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 159 [arrowhead = none ];
159 -> 41;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 160 [arrowhead = none ];
160 -> 42;

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 161 [arrowhead = none ];
161 -> 43;

162 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX62 -> 0", fontsize=14, style = filled, fillcolor = white];
42 -> 162 [arrowhead = none ];
162 -> 44;

163 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 163 [arrowhead = none ];
163 -> 61;

164 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 164 [arrowhead = none ];
164 -> 45;

165 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 165 [arrowhead = none ];
165 -> 46;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 166 [arrowhead = none ];
166 -> 47;

167 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX63 -> 0", fontsize=14, style = filled, fillcolor = white];
46 -> 167 [arrowhead = none ];
167 -> 48;

168 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 168 [arrowhead = none ];
168 -> 60;

169 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 169 [arrowhead = none ];
169 -> 49;

170 [label="SPLIT 2\nnew knowledge:\nT56 is ground\nreplacements:\nX64 -> T56", fontsize=14, style = filled, fillcolor = violet];
48 -> 170 [arrowhead = none ];
170 -> 50;

171 [label="INSTANCE with matching:\nT1 -> s(T44)\nT2 -> 0\nT3 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 171 [arrowhead = none , style=dashed];
171 -> 1[style=dashed];

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 172 [arrowhead = none ];
172 -> 51;

173 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 173 [arrowhead = none ];
173 -> 52;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 174 [arrowhead = none ];
174 -> 53;

175 [label="EVAL with clause\np(0, 0).\nand substitution\nT47 -> 0\nT56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 175 [arrowhead = none ];
175 -> 54;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 176 [arrowhead = none ];
176 -> 55;

177 [label="EVAL with clause\np(s(X144), X144).\nand substitution\nX144 -> T60\nT47 -> s(T60)\nT56 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 177 [arrowhead = none ];
177 -> 57;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 178 [arrowhead = none ];
178 -> 58;

179 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 179 [arrowhead = none ];
179 -> 56;

180 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 180 [arrowhead = none ];
180 -> 59;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 181 [arrowhead = none ];
181 -> 64;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 182 [arrowhead = none ];
182 -> 65;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 183 [arrowhead = none ];
183 -> 66;

184 [label="EVAL with clause\np(0, 0).\nand substitution\nT63 -> 0\nX62 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 184 [arrowhead = none ];
184 -> 67;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 185 [arrowhead = none ];
185 -> 68;

186 [label="EVAL with clause\np(s(X209), X209).\nand substitution\nX209 -> T76\nT63 -> s(T76)\nX62 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 186 [arrowhead = none ];
186 -> 85;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 187 [arrowhead = none ];
187 -> 86;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 188 [arrowhead = none ];
188 -> 69;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 189 [arrowhead = none ];
189 -> 70;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 190 [arrowhead = none ];
190 -> 71;

191 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX63 -> 0", fontsize=14, style = filled, fillcolor = white];
70 -> 191 [arrowhead = none ];
191 -> 72;

192 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
71 -> 192 [arrowhead = none ];
192 -> 84;

193 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 193 [arrowhead = none ];
193 -> 73;

194 [label="SPLIT 2\nnew knowledge:\nT69 is ground\nreplacements:\nX64 -> T69", fontsize=14, style = filled, fillcolor = violet];
72 -> 194 [arrowhead = none ];
194 -> 74;

195 [label="INSTANCE with matching:\nT1 -> s(T44)\nT2 -> 0\nT3 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 195 [arrowhead = none , style=dashed];
195 -> 1[style=dashed];

196 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 196 [arrowhead = none ];
196 -> 75;

197 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 197 [arrowhead = none ];
197 -> 76;

198 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 198 [arrowhead = none ];
198 -> 77;

199 [label="EVAL with clause\np(0, 0).\nand substitution\nT47 -> 0\nT69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 199 [arrowhead = none ];
199 -> 78;

200 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 200 [arrowhead = none ];
200 -> 79;

201 [label="EVAL with clause\np(s(X204), X204).\nand substitution\nX204 -> T73\nT47 -> s(T73)\nT69 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 201 [arrowhead = none ];
201 -> 81;

202 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 202 [arrowhead = none ];
202 -> 82;

203 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 203 [arrowhead = none ];
203 -> 80;

204 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 204 [arrowhead = none ];
204 -> 83;

205 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 205 [arrowhead = none ];
205 -> 87;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 206 [arrowhead = none ];
206 -> 88;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 207 [arrowhead = none ];
207 -> 89;

208 [label="EVAL with clause\np(0, 0).\nand substitution\nT76 -> 0\nX63 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 208 [arrowhead = none ];
208 -> 90;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 209 [arrowhead = none ];
209 -> 91;

210 [label="EVAL with clause\np(s(X242), X242).\nand substitution\nX242 -> T86\nT76 -> s(T86)\nX63 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 210 [arrowhead = none ];
210 -> 103;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 211 [arrowhead = none ];
211 -> 104;

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 212 [arrowhead = none ];
212 -> 92;

213 [label="SPLIT 2\nnew knowledge:\nT79 is ground\nreplacements:\nX64 -> T79", fontsize=14, style = filled, fillcolor = violet];
90 -> 213 [arrowhead = none ];
213 -> 93;

214 [label="INSTANCE with matching:\nT1 -> s(T44)\nT2 -> 0\nT3 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
92 -> 214 [arrowhead = none , style=dashed];
214 -> 1[style=dashed];

215 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
93 -> 215 [arrowhead = none ];
215 -> 94;

216 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 216 [arrowhead = none ];
216 -> 95;

217 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
94 -> 217 [arrowhead = none ];
217 -> 96;

218 [label="EVAL with clause\np(0, 0).\nand substitution\nT47 -> 0\nT79 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 218 [arrowhead = none ];
218 -> 97;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 219 [arrowhead = none ];
219 -> 98;

220 [label="EVAL with clause\np(s(X238), X238).\nand substitution\nX238 -> T83\nT47 -> s(T83)\nT79 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 220 [arrowhead = none ];
220 -> 100;

221 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 221 [arrowhead = none ];
221 -> 101;

222 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 222 [arrowhead = none ];
222 -> 99;

223 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 223 [arrowhead = none ];
223 -> 102;

224 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
103 -> 224 [arrowhead = none ];
224 -> 105;

225 [label="SPLIT 2\nnew knowledge:\nT89 is ground\nreplacements:\nX64 -> T89", fontsize=14, style = filled, fillcolor = violet];
103 -> 225 [arrowhead = none ];
225 -> 106;

226 [label="INSTANCE with matching:\nT1 -> s(T44)\nT2 -> T86\nT3 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 226 [arrowhead = none , style=dashed];
226 -> 1[style=dashed];

227 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
106 -> 227 [arrowhead = none ];
227 -> 107;

228 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 228 [arrowhead = none ];
228 -> 108;

229 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 229 [arrowhead = none ];
229 -> 109;

230 [label="EVAL with clause\np(0, 0).\nand substitution\nT47 -> 0\nT89 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 230 [arrowhead = none ];
230 -> 110;

231 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 231 [arrowhead = none ];
231 -> 111;

232 [label="EVAL with clause\np(s(X262), X262).\nand substitution\nX262 -> T94\nT47 -> s(T94)\nT89 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 232 [arrowhead = none ];
232 -> 113;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 233 [arrowhead = none ];
233 -> 114;

234 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 234 [arrowhead = none ];
234 -> 112;

235 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 235 [arrowhead = none ];
235 -> 115;

}

(2) Obligation:

Triples:

average1(0, T26, T28) :- average1(0, s(T26), T28).
average1(s(T34), T26, T28) :- average1(T34, s(T26), T28).
average1(T44, 0, T47) :- average1(s(T44), 0, X64).
average1(T44, s(0), T47) :- average1(s(T44), 0, X64).
average1(T44, s(s(0)), T47) :- average1(s(T44), 0, X64).
average1(T44, s(s(s(T86))), T47) :- average1(s(T44), T86, X64).

Clauses:

averagec1(0, 0, 0).
averagec1(0, s(0), 0).
averagec1(0, s(s(0)), s(0)).
averagec1(0, T26, T28) :- averagec1(0, s(T26), T28).
averagec1(s(T34), T26, T28) :- averagec1(T34, s(T26), T28).
averagec1(T44, 0, 0) :- averagec1(s(T44), 0, 0).
averagec1(T44, 0, s(T60)) :- averagec1(s(T44), 0, T60).
averagec1(T44, s(0), 0) :- averagec1(s(T44), 0, 0).
averagec1(T44, s(0), s(T73)) :- averagec1(s(T44), 0, T73).
averagec1(T44, s(s(0)), 0) :- averagec1(s(T44), 0, 0).
averagec1(T44, s(s(0)), s(T83)) :- averagec1(s(T44), 0, T83).
averagec1(T44, s(s(s(T86))), 0) :- averagec1(s(T44), T86, 0).
averagec1(T44, s(s(s(T86))), s(T94)) :- averagec1(s(T44), T86, T94).

Afs:

average1(x1, x2, x3) = average1(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:
average1_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

AVERAGE1_IN_GGA(0, T26, T28) → U1_GGA(T26, T28, average1_in_gga(0, s(T26), T28))
AVERAGE1_IN_GGA(0, T26, T28) → AVERAGE1_IN_GGA(0, s(T26), T28)
AVERAGE1_IN_GGA(s(T34), T26, T28) → U2_GGA(T34, T26, T28, average1_in_gga(T34, s(T26), T28))
AVERAGE1_IN_GGA(s(T34), T26, T28) → AVERAGE1_IN_GGA(T34, s(T26), T28)
AVERAGE1_IN_GGA(T44, 0, T47) → U3_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, 0, T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(0), T47) → U4_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, s(0), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(s(0)), T47) → U5_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, s(s(0)), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(s(s(T86))), T47) → U6_GGA(T44, T86, T47, average1_in_gga(s(T44), T86, X64))
AVERAGE1_IN_GGA(T44, s(s(s(T86))), T47) → AVERAGE1_IN_GGA(s(T44), T86, X64)

R is empty.
The argument filtering Pi contains the following mapping:
average1_in_gga(x1, x2, x3) = average1_in_gga(x1, x2)
0 = 0
s(x1) = s(x1)
AVERAGE1_IN_GGA(x1, x2, x3) = AVERAGE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3) = U1_GGA(x1, x3)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3) = U3_GGA(x1, x3)
U4_GGA(x1, x2, x3) = U4_GGA(x1, x3)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x3)
U6_GGA(x1, x2, x3, x4) = U6_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:

AVERAGE1_IN_GGA(0, T26, T28) → U1_GGA(T26, T28, average1_in_gga(0, s(T26), T28))
AVERAGE1_IN_GGA(0, T26, T28) → AVERAGE1_IN_GGA(0, s(T26), T28)
AVERAGE1_IN_GGA(s(T34), T26, T28) → U2_GGA(T34, T26, T28, average1_in_gga(T34, s(T26), T28))
AVERAGE1_IN_GGA(s(T34), T26, T28) → AVERAGE1_IN_GGA(T34, s(T26), T28)
AVERAGE1_IN_GGA(T44, 0, T47) → U3_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, 0, T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(0), T47) → U4_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, s(0), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(s(0)), T47) → U5_GGA(T44, T47, average1_in_gga(s(T44), 0, X64))
AVERAGE1_IN_GGA(T44, s(s(0)), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(s(s(T86))), T47) → U6_GGA(T44, T86, T47, average1_in_gga(s(T44), T86, X64))
AVERAGE1_IN_GGA(T44, s(s(s(T86))), T47) → AVERAGE1_IN_GGA(s(T44), T86, X64)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(6) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(s(T34), T26, T28) → AVERAGE1_IN_GGA(T34, s(T26), T28)
AVERAGE1_IN_GGA(0, T26, T28) → AVERAGE1_IN_GGA(0, s(T26), T28)
AVERAGE1_IN_GGA(T44, s(s(0)), T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, 0, T47) → AVERAGE1_IN_GGA(s(T44), 0, X64)
AVERAGE1_IN_GGA(T44, s(s(s(T86))), T47) → AVERAGE1_IN_GGA(s(T44), T86, X64)

R is empty.
The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
AVERAGE1_IN_GGA(x1, x2, x3) = AVERAGE1_IN_GGA(x1, x2)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0)) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(s(T34), T26) → AVERAGE1_IN_GGA(T34, s(T26))
AVERAGE1_IN_GGA(0, T26) → AVERAGE1_IN_GGA(0, s(T26))
AVERAGE1_IN_GGA(T44, s(s(0))) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, 0) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(s(T86)))) → AVERAGE1_IN_GGA(s(T44), T86)

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

(9) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(0, T26) → AVERAGE1_IN_GGA(0, s(T26)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

(10) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0)) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(s(T34), T26) → AVERAGE1_IN_GGA(T34, s(T26))
AVERAGE1_IN_GGA(T44, s(s(0))) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, 0) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(s(T86)))) → AVERAGE1_IN_GGA(s(T44), T86)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))

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

(11) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AVERAGE1_IN_GGA(T44, 0) → AVERAGE1_IN_GGA(s(T44), 0) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

(12) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0)) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(s(T34), T26) → AVERAGE1_IN_GGA(T34, s(T26))
AVERAGE1_IN_GGA(T44, s(s(0))) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(s(T86)))) → AVERAGE1_IN_GGA(s(T44), T86)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)

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

(13) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(s(T34), T26) → AVERAGE1_IN_GGA(T34, s(T26)) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

(14) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0)) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(0))) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(s(T86)))) → AVERAGE1_IN_GGA(s(T44), T86)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))

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

(15) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule AVERAGE1_IN_GGA(T44, s(s(s(T86)))) → AVERAGE1_IN_GGA(s(T44), T86) we obtained the following new rules [LPAR04]:

AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

(16) Obligation:

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

AVERAGE1_IN_GGA(T44, s(0)) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(T44, s(s(0))) → AVERAGE1_IN_GGA(s(T44), 0)
AVERAGE1_IN_GGA(0, s(z1)) → AVERAGE1_IN_GGA(0, s(s(z1)))
AVERAGE1_IN_GGA(s(z0), 0) → AVERAGE1_IN_GGA(s(s(z0)), 0)
AVERAGE1_IN_GGA(s(x0), 0) → AVERAGE1_IN_GGA(x0, s(0))
AVERAGE1_IN_GGA(s(x0), s(0)) → AVERAGE1_IN_GGA(x0, s(s(0)))
AVERAGE1_IN_GGA(s(x0), s(s(y_1))) → AVERAGE1_IN_GGA(x0, s(s(s(y_1))))
AVERAGE1_IN_GGA(s(s(y_0)), x1) → AVERAGE1_IN_GGA(s(y_0), s(x1))
AVERAGE1_IN_GGA(s(0), x1) → AVERAGE1_IN_GGA(0, s(x1))
AVERAGE1_IN_GGA(x0, s(s(s(s(0))))) → AVERAGE1_IN_GGA(s(x0), s(0))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(0)))))) → AVERAGE1_IN_GGA(s(x0), s(s(0)))
AVERAGE1_IN_GGA(x0, s(s(s(s(s(s(y_1))))))) → AVERAGE1_IN_GGA(s(x0), s(s(s(y_1))))
AVERAGE1_IN_GGA(x0, s(s(s(0)))) → AVERAGE1_IN_GGA(s(x0), 0)
AVERAGE1_IN_GGA(x0, s(s(s(s(s(y_1)))))) → AVERAGE1_IN_GGA(s(x0), s(s(y_1)))
AVERAGE1_IN_GGA(s(y_0), s(s(s(x1)))) → AVERAGE1_IN_GGA(s(s(y_0)), x1)
AVERAGE1_IN_GGA(0, s(s(s(x1)))) → AVERAGE1_IN_GGA(s(0), x1)

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

(17) NonTerminationProof (EQUIVALENT transformation)

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

s = AVERAGE1_IN_GGA(0, s(z1)) evaluates to t =AVERAGE1_IN_GGA(0, s(s(z1)))

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

Semiunifier: [ ]
Matcher: [z1 / s(z1)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from AVERAGE1_IN_GGA(0, s(z1)) to AVERAGE1_IN_GGA(0, s(s(z1))).

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) PiDPToQDPProof (SOUND transformation)

(8) Obligation:

(9) Instantiation (EQUIVALENT transformation)

(10) Obligation:

(11) Instantiation (EQUIVALENT transformation)

(12) Obligation:

(13) ForwardInstantiation (EQUIVALENT transformation)

(14) Obligation:

(15) ForwardInstantiation (EQUIVALENT transformation)

(16) Obligation:

(17) NonTerminationProof (EQUIVALENT transformation)

(18) NO