Left Termination of the query pattern select_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

select(X, Y, Zs) :- ','(no(empty(Y)), ','(head(Y, X), tail(Y, Zs))).
select(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(head(Z, H), ','(tail(Y, T), ','(tail(Z, Zs), select(X, T, Zs)))))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

select(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: select(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(no(empty(T8)), ','(head(T8, T7), tail(T8, T10))) | select(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(no(empty(T8)), ','(head(T8, T7), tail(T8, T10))), SCOPE: 2, CLAUSE: 7 | ','(no(empty(T8)), ','(head(T8, T7), tail(T8, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T8, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(call(empty(T13)), ','(!_2, ','(failure(a), ','(head(T13, T7), tail(T13, T10))))) | ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(empty(T13), ','(!_2, ','(failure(a), ','(head(T13, T7), tail(T13, T10))))) | ?_3 | ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(empty(T13), ','(!_2, ','(failure(a), ','(head(T13, T7), tail(T13, T10))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_2, ','(failure(a), ','(head([], T7), tail([], T10)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T7), tail([], T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, [], T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ?_4 | ?_3 | ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(failure(a), ','(head([], T7), tail([], T10))) | select(T7, [], T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(failure(a), ','(head([], T7), tail([], T10)))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: select(T7, [], T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(head([], T7), tail([], T10))), SCOPE: 5, CLAUSE: 9\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33))))))\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 7 | ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 8\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(call(empty([])), ','(!_6, ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))))) | ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 8\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(empty([]), ','(!_6, ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))))) | ?_7 | ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 8\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(empty([]), ','(!_6, ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 8\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_6, ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33))))))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 6, CLAUSE: 8\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33))))))\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), ','(head([], X31), ','(head(T22, X31), ','(tail([], X32), ','(tail(T22, X33), select(T20, X32, X33)))))), SCOPE: 9, CLAUSE: 9\n\nT20 is ground\nX31 is free; \nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ?_3 | ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8 | ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(no(empty(T13)), ','(head(T13, T7), tail(T13, T10))), SCOPE: 2, CLAUSE: 8\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_2 | select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(head(T29, T7), tail(T29, T10))\n\nT7 is ground\nT29 is ground\nempty(T29) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T29, T7), tail(T29, T10)), SCOPE: 10, CLAUSE: 2 | ','(head(T29, T7), tail(T29, T10)), SCOPE: 10, CLAUSE: 3\n\nT7 is ground\nT29 is ground\nempty(T29) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T29, T7), tail(T29, T10)), SCOPE: 10, CLAUSE: 3\n\nT7 is ground\nT29 is ground\nempty(T29) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: tail(.(T35, T36), T10)\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: tail(.(T35, T36), T10), SCOPE: 11, CLAUSE: 4 | tail(.(T35, T36), T10), SCOPE: 11, CLAUSE: 5\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: tail(.(T35, T36), T10), SCOPE: 11, CLAUSE: 5\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: select(T7, T13, T3), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT13 is ground\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(no(empty(T52)), ','(head(T52, X86), ','(head(T54, X86), ','(tail(T52, X87), ','(tail(T54, X88), select(T51, X87, X88))))))\n\nT51 is ground\nT52 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T52) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(no(empty(T52)), ','(head(T52, X86), ','(head(T54, X86), ','(tail(T52, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 7 | ','(no(empty(T52)), ','(head(T52, X86), ','(head(T54, X86), ','(tail(T52, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT52 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T52) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(call(empty(T58)), ','(!_12, ','(failure(a), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))))) | ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(empty(T58), ','(!_12, ','(failure(a), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))))) | ?_13 | ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(empty(T58), ','(!_12, ','(failure(a), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))))), SCOPE: 14, CLAUSE: 6 | ?_14 | ?_13 | ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ?_14 | ?_13 | ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_13 | ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(no(empty(T58)), ','(head(T58, X86), ','(head(T54, X86), ','(tail(T58, X87), ','(tail(T54, X88), select(T51, X87, X88)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(head(T62, X86), ','(head(T54, X86), ','(tail(T62, X87), ','(tail(T54, X88), select(T51, X87, X88)))))\n\nT51 is ground\nT62 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(head(T62, X86), ','(head(T54, X86), ','(tail(T62, X87), ','(tail(T54, X88), select(T51, X87, X88))))), SCOPE: 15, CLAUSE: 2 | ','(head(T62, X86), ','(head(T54, X86), ','(tail(T62, X87), ','(tail(T54, X88), select(T51, X87, X88))))), SCOPE: 15, CLAUSE: 3\n\nT51 is ground\nT62 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(head(T62, X86), ','(head(T54, X86), ','(tail(T62, X87), ','(tail(T54, X88), select(T51, X87, X88))))), SCOPE: 15, CLAUSE: 3\n\nT51 is ground\nT62 is ground\nX86 is free; \nX87 is free; \nX88 is free; \nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(T54, T68), ','(tail(.(T68, T69), X87), ','(tail(T54, X88), select(T51, X87, X88))))\n\nT51 is ground\nT68 is ground\nT69 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T54, T68), ','(tail(.(T68, T69), X87), ','(tail(T54, X88), select(T51, X87, X88)))), SCOPE: 16, CLAUSE: 2 | ','(head(T54, T68), ','(tail(.(T68, T69), X87), ','(tail(T54, X88), select(T51, X87, X88)))), SCOPE: 16, CLAUSE: 3\n\nT51 is ground\nT68 is ground\nT69 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(head(T54, T68), ','(tail(.(T68, T69), X87), ','(tail(T54, X88), select(T51, X87, X88)))), SCOPE: 16, CLAUSE: 2\n\nT51 is ground\nT68 is ground\nT69 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T54, T68), ','(tail(.(T68, T69), X87), ','(tail(T54, X88), select(T51, X87, X88)))), SCOPE: 16, CLAUSE: 3\n\nT51 is ground\nT68 is ground\nT69 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail(.(T75, T69), X87), ','(tail([], X88), select(T51, X87, X88)))\n\nT51 is ground\nT69 is ground\nT75 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail(.(T75, T69), X87), ','(tail([], X88), select(T51, X87, X88))), SCOPE: 17, CLAUSE: 4 | ','(tail(.(T75, T69), X87), ','(tail([], X88), select(T51, X87, X88))), SCOPE: 17, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT75 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(.(T75, T69), X87), ','(tail([], X88), select(T51, X87, X88))), SCOPE: 17, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT75 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail([], X88), select(T51, T84, X88))\n\nT51 is ground\nT84 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail([], X88), select(T51, T84, X88)), SCOPE: 18, CLAUSE: 4 | ','(tail([], X88), select(T51, T84, X88)), SCOPE: 18, CLAUSE: 5\n\nT51 is ground\nT84 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail([], X88), select(T51, T84, X88)), SCOPE: 18, CLAUSE: 4\n\nT51 is ground\nT84 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail([], X88), select(T51, T84, X88)), SCOPE: 18, CLAUSE: 5\n\nT51 is ground\nT84 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: select(T51, T84, [])\n\nT51 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: select(T51, T84, []), SCOPE: 19, CLAUSE: 0 | select(T51, T84, []), SCOPE: 19, CLAUSE: 1\n\nT51 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: select(T51, T84, []), SCOPE: 19, CLAUSE: 0\n\nT51 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: select(T51, T84, []), SCOPE: 19, CLAUSE: 1\n\nT51 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(no(empty(T102)), ','(head(T102, T101), tail(T102, [])))\n\nT101 is ground\nT102 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(no(empty(T102)), ','(head(T102, T101), tail(T102, []))), SCOPE: 20, CLAUSE: 7 | ','(no(empty(T102)), ','(head(T102, T101), tail(T102, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT102 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(call(empty(T105)), ','(!_20, ','(failure(a), ','(head(T105, T101), tail(T105, []))))) | ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(empty(T105), ','(!_20, ','(failure(a), ','(head(T105, T101), tail(T105, []))))) | ?_21 | ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(empty(T105), ','(!_20, ','(failure(a), ','(head(T105, T101), tail(T105, []))))), SCOPE: 22, CLAUSE: 6 | ?_22 | ?_21 | ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(!_20, ','(failure(a), ','(head([], T101), tail([], [])))) | ?_22 | ?_21 | ','(no(empty([])), ','(head([], T101), tail([], []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ?_22 | ?_21 | ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground\nempty(T105) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(failure(a), ','(head([], T101), tail([], [])))\n\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(failure(a), ','(head([], T101), tail([], []))), SCOPE: 23, CLAUSE: 9\n\nT101 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ?_21 | ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground\nempty(T105) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(no(empty(T105)), ','(head(T105, T101), tail(T105, []))), SCOPE: 20, CLAUSE: 8\n\nT101 is ground\nT105 is ground\nempty(T105) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(head(T108, T101), tail(T108, []))\n\nT101 is ground\nT108 is ground\nempty(T108) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head(T108, T101), tail(T108, [])), SCOPE: 24, CLAUSE: 2 | ','(head(T108, T101), tail(T108, [])), SCOPE: 24, CLAUSE: 3\n\nT101 is ground\nT108 is ground\nempty(T108) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(head(T108, T101), tail(T108, [])), SCOPE: 24, CLAUSE: 3\n\nT101 is ground\nT108 is ground\nempty(T108) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: tail(.(T114, T115), [])\n\nT114 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: tail(.(T114, T115), []), SCOPE: 25, CLAUSE: 4 | tail(.(T114, T115), []), SCOPE: 25, CLAUSE: 5\n\nT114 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: tail(.(T114, T115), []), SCOPE: 25, CLAUSE: 5\n\nT114 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(no(empty(T128)), ','(head(T128, X214), ','(head([], X214), ','(tail(T128, X215), ','(tail([], X216), select(T127, X215, X216))))))\n\nT127 is ground\nT128 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(no(empty(T128)), ','(head(T128, X214), ','(head([], X214), ','(tail(T128, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 7 | ','(no(empty(T128)), ','(head(T128, X214), ','(head([], X214), ','(tail(T128, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT128 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(call(empty(T132)), ','(!_26, ','(failure(a), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))))) | ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(empty(T132), ','(!_26, ','(failure(a), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))))) | ?_27 | ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(empty(T132), ','(!_26, ','(failure(a), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))))), SCOPE: 28, CLAUSE: 6 | ?_28 | ?_27 | ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(!_26, ','(failure(a), ','(head([], X214), ','(head([], X214), ','(tail([], X215), ','(tail([], X216), select(T127, X215, X216))))))) | ?_28 | ?_27 | ','(no(empty([])), ','(head([], X214), ','(head([], X214), ','(tail([], X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ?_28 | ?_27 | ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(failure(a), ','(head([], X214), ','(head([], X214), ','(tail([], X215), ','(tail([], X216), select(T127, X215, X216))))))\n\nT127 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(failure(a), ','(head([], X214), ','(head([], X214), ','(tail([], X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 29, CLAUSE: 9\n\nT127 is ground\nX214 is free; \nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ?_27 | ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(no(empty(T132)), ','(head(T132, X214), ','(head([], X214), ','(tail(T132, X215), ','(tail([], X216), select(T127, X215, X216)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(head(T137, X214), ','(head([], X214), ','(tail(T137, X215), ','(tail([], X216), select(T127, X215, X216)))))\n\nT127 is ground\nT137 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(head(T137, X214), ','(head([], X214), ','(tail(T137, X215), ','(tail([], X216), select(T127, X215, X216))))), SCOPE: 30, CLAUSE: 2 | ','(head(T137, X214), ','(head([], X214), ','(tail(T137, X215), ','(tail([], X216), select(T127, X215, X216))))), SCOPE: 30, CLAUSE: 3\n\nT127 is ground\nT137 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(head(T137, X214), ','(head([], X214), ','(tail(T137, X215), ','(tail([], X216), select(T127, X215, X216))))), SCOPE: 30, CLAUSE: 3\n\nT127 is ground\nT137 is ground\nX214 is free; \nX215 is free; \nX216 is free; \nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(head([], T143), ','(tail(.(T143, T144), X215), ','(tail([], X216), select(T127, X215, X216))))\n\nT127 is ground\nT143 is ground\nT144 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(head([], T143), ','(tail(.(T143, T144), X215), ','(tail([], X216), select(T127, X215, X216)))), SCOPE: 31, CLAUSE: 2 | ','(head([], T143), ','(tail(.(T143, T144), X215), ','(tail([], X216), select(T127, X215, X216)))), SCOPE: 31, CLAUSE: 3\n\nT127 is ground\nT143 is ground\nT144 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(head([], T143), ','(tail(.(T143, T144), X215), ','(tail([], X216), select(T127, X215, X216)))), SCOPE: 31, CLAUSE: 2\n\nT127 is ground\nT143 is ground\nT144 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(head([], T143), ','(tail(.(T143, T144), X215), ','(tail([], X216), select(T127, X215, X216)))), SCOPE: 31, CLAUSE: 3\n\nT127 is ground\nT143 is ground\nT144 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: ','(tail(.(T152, T144), X215), ','(tail([], X216), select(T127, X215, X216)))\n\nT127 is ground\nT144 is ground\nT152 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(tail(.(T152, T144), X215), ','(tail([], X216), select(T127, X215, X216))), SCOPE: 32, CLAUSE: 4 | ','(tail(.(T152, T144), X215), ','(tail([], X216), select(T127, X215, X216))), SCOPE: 32, CLAUSE: 5\n\nT127 is ground\nT144 is ground\nT152 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(tail(.(T152, T144), X215), ','(tail([], X216), select(T127, X215, X216))), SCOPE: 32, CLAUSE: 5\n\nT127 is ground\nT144 is ground\nT152 is ground\nX215 is free; \nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(tail([], X216), select(T127, T161, X216))\n\nT127 is ground\nT161 is ground\nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(tail([], X216), select(T127, T161, X216)), SCOPE: 33, CLAUSE: 4 | ','(tail([], X216), select(T127, T161, X216)), SCOPE: 33, CLAUSE: 5\n\nT127 is ground\nT161 is ground\nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(tail([], X216), select(T127, T161, X216)), SCOPE: 33, CLAUSE: 4\n\nT127 is ground\nT161 is ground\nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: ','(tail([], X216), select(T127, T161, X216)), SCOPE: 33, CLAUSE: 5\n\nT127 is ground\nT161 is ground\nX216 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: select(T127, T161, [])\n\nT127 is ground\nT161 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \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: ','(tail(.(T174, T69), X87), ','(tail(.(T174, T176), X88), select(T51, X87, X88)))\n\nT51 is ground\nT69 is ground\nT174 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(tail(.(T174, T69), X87), ','(tail(.(T174, T176), X88), select(T51, X87, X88))), SCOPE: 34, CLAUSE: 4 | ','(tail(.(T174, T69), X87), ','(tail(.(T174, T176), X88), select(T51, X87, X88))), SCOPE: 34, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT174 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(tail(.(T174, T69), X87), ','(tail(.(T174, T176), X88), select(T51, X87, X88))), SCOPE: 34, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT174 is ground\nX87 is free; \nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(tail(.(T184, T176), X88), select(T51, T185, X88))\n\nT51 is ground\nT184 is ground\nT185 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(tail(.(T184, T176), X88), select(T51, T185, X88)), SCOPE: 35, CLAUSE: 4 | ','(tail(.(T184, T176), X88), select(T51, T185, X88)), SCOPE: 35, CLAUSE: 5\n\nT51 is ground\nT184 is ground\nT185 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: ','(tail(.(T184, T176), X88), select(T51, T185, X88)), SCOPE: 35, CLAUSE: 5\n\nT51 is ground\nT184 is ground\nT185 is ground\nX88 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: select(T51, T185, T198)\n\nT51 is ground\nT185 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 128 [arrowhead = none ];
128 -> 2;

129 [label="ONLY EVAL with clause\nselect(X8, X9, X10) :- ','(no(empty(X9)), ','(head(X9, X8), tail(X9, X10))).\nand substitution\nT1 -> T7\nX8 -> T7\nT2 -> T8\nX9 -> T8\nT3 -> T10\nX10 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = white];
2 -> 129 [arrowhead = none ];
129 -> 3;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 130 [arrowhead = none ];
130 -> 4;

131 [label="ONLY EVAL with clause\nno(X13) :- ','(call(X13), ','(!_2, failure(a))).\nand substitution\nT8 -> T13\nX13 -> empty(T13)", fontsize=14, style = filled, fillcolor = white];
4 -> 131 [arrowhead = none ];
131 -> 5;

132 [label="CALL", fontsize=14, style = filled, fillcolor = white];
5 -> 132 [arrowhead = none ];
132 -> 6;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 133 [arrowhead = none ];
133 -> 7;

134 [label="EVAL with clause\nempty([]).\nand substitution\nT13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 134 [arrowhead = none ];
134 -> 8;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 135 [arrowhead = none ];
135 -> 9;

136 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 136 [arrowhead = none ];
136 -> 10;

137 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 137 [arrowhead = none ];
137 -> 24;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 138 [arrowhead = none ];
138 -> 11;

139 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 139 [arrowhead = none ];
139 -> 12;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 140 [arrowhead = none ];
140 -> 13;

141 [label="ONLY EVAL with clause\nselect(X28, X29, X30) :- ','(no(empty(X29)), ','(head(X29, X31), ','(head(X30, X31), ','(tail(X29, X32), ','(tail(X30, X33), select(X28, X32, X33)))))).\nand substitution\nT7 -> T20\nX28 -> T20\nX29 -> []\nT3 -> T22\nX30 -> T22\nT21 -> T22", fontsize=14, style = filled, fillcolor = white];
12 -> 141 [arrowhead = none ];
141 -> 15;

142 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 142 [arrowhead = none ];
142 -> 14;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 143 [arrowhead = none ];
143 -> 16;

144 [label="ONLY EVAL with clause\nno(X41) :- ','(call(X41), ','(!_6, failure(a))).\nand substitution\nX41 -> empty([])", fontsize=14, style = filled, fillcolor = white];
16 -> 144 [arrowhead = none ];
144 -> 17;

145 [label="CALL", fontsize=14, style = filled, fillcolor = white];
17 -> 145 [arrowhead = none ];
145 -> 18;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 146 [arrowhead = none ];
146 -> 19;

147 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = white];
19 -> 147 [arrowhead = none ];
147 -> 20;

148 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 148 [arrowhead = none ];
148 -> 21;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 149 [arrowhead = none ];
149 -> 22;

150 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 150 [arrowhead = none ];
150 -> 23;

151 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 151 [arrowhead = none ];
151 -> 25;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 152 [arrowhead = none ];
152 -> 26;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 153 [arrowhead = none ];
153 -> 27;

154 [label="ONLY EVAL with clause\nno(X51).\nand substitution\nT13 -> T29\nX51 -> empty(T29)", fontsize=14, style = filled, fillcolor = white];
26 -> 154 [arrowhead = none ];
154 -> 28;

155 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 155 [arrowhead = none ];
155 -> 38;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 156 [arrowhead = none ];
156 -> 29;

157 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T29), empty([]))", fontsize=14, style = filled, fillcolor = white];
29 -> 157 [arrowhead = none ];
157 -> 30;

158 [label="EVAL with clause\nhead(.(X57, X58), X57).\nand substitution\nX57 -> T35\nX58 -> T36\nT29 -> .(T35, T36)\nT7 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 158 [arrowhead = none ];
158 -> 31;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 159 [arrowhead = none ];
159 -> 32;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 160 [arrowhead = none ];
160 -> 33;

161 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
33 -> 161 [arrowhead = none ];
161 -> 34;

162 [label="EVAL with clause\ntail(.(X65, X66), X66).\nand substitution\nT35 -> T41\nX65 -> T41\nT36 -> T42\nX66 -> T42\nT10 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 162 [arrowhead = none ];
162 -> 35;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 163 [arrowhead = none ];
163 -> 36;

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 164 [arrowhead = none ];
164 -> 37;

165 [label="ONLY EVAL with clause\nselect(X83, X84, X85) :- ','(no(empty(X84)), ','(head(X84, X86), ','(head(X85, X86), ','(tail(X84, X87), ','(tail(X85, X88), select(X83, X87, X88)))))).\nand substitution\nT7 -> T51\nX83 -> T51\nT13 -> T52\nX84 -> T52\nT3 -> T54\nX85 -> T54\nT53 -> T54", fontsize=14, style = filled, fillcolor = white];
38 -> 165 [arrowhead = none ];
165 -> 39;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 166 [arrowhead = none ];
166 -> 40;

167 [label="ONLY EVAL with clause\nno(X96) :- ','(call(X96), ','(!_12, failure(a))).\nand substitution\nT52 -> T58\nX96 -> empty(T58)", fontsize=14, style = filled, fillcolor = white];
40 -> 167 [arrowhead = none ];
167 -> 41;

168 [label="CALL", fontsize=14, style = filled, fillcolor = white];
41 -> 168 [arrowhead = none ];
168 -> 42;

169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 169 [arrowhead = none ];
169 -> 43;

170 [label="BACKTRACK\nfor clause: empty([])\nwith clash: (empty(T58), empty([]))", fontsize=14, style = filled, fillcolor = white];
43 -> 170 [arrowhead = none ];
170 -> 44;

171 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 171 [arrowhead = none ];
171 -> 45;

172 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 172 [arrowhead = none ];
172 -> 46;

173 [label="ONLY EVAL with clause\nno(X104).\nand substitution\nT58 -> T62\nX104 -> empty(T62)", fontsize=14, style = filled, fillcolor = white];
46 -> 173 [arrowhead = none ];
173 -> 47;

174 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 174 [arrowhead = none ];
174 -> 48;

175 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T62), empty([]))", fontsize=14, style = filled, fillcolor = white];
48 -> 175 [arrowhead = none ];
175 -> 49;

176 [label="EVAL with clause\nhead(.(X116, X117), X116).\nand substitution\nX116 -> T68\nX117 -> T69\nT62 -> .(T68, T69)\nX86 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 176 [arrowhead = none ];
176 -> 50;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 177 [arrowhead = none ];
177 -> 51;

178 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 178 [arrowhead = none ];
178 -> 52;

179 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 179 [arrowhead = none ];
179 -> 53;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 180 [arrowhead = none ];
180 -> 54;

181 [label="EVAL with clause\nhead([], X129).\nand substitution\nT54 -> []\nT68 -> T75\nX129 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 181 [arrowhead = none ];
181 -> 55;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 182 [arrowhead = none ];
182 -> 56;

183 [label="EVAL with clause\nhead(.(X311, X312), X311).\nand substitution\nX311 -> T174\nX312 -> T176\nT54 -> .(T174, T176)\nT68 -> T174\nT175 -> T176", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 183 [arrowhead = none ];
183 -> 120;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 184 [arrowhead = none ];
184 -> 121;

185 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 185 [arrowhead = none ];
185 -> 57;

186 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 186 [arrowhead = none ];
186 -> 58;

187 [label="ONLY EVAL with clause\ntail(.(X144, X145), X145).\nand substitution\nT75 -> T83\nX144 -> T83\nT69 -> T84\nX145 -> T84\nX87 -> T84", fontsize=14, style = filled, fillcolor = white];
58 -> 187 [arrowhead = none ];
187 -> 59;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 188 [arrowhead = none ];
188 -> 60;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 189 [arrowhead = none ];
189 -> 61;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 190 [arrowhead = none ];
190 -> 62;

191 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX88 -> []", fontsize=14, style = filled, fillcolor = white];
61 -> 191 [arrowhead = none ];
191 -> 63;

192 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 192 [arrowhead = none ];
192 -> 119;

193 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 193 [arrowhead = none ];
193 -> 64;

194 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 194 [arrowhead = none ];
194 -> 65;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 195 [arrowhead = none ];
195 -> 66;

196 [label="ONLY EVAL with clause\nselect(X171, X172, X173) :- ','(no(empty(X172)), ','(head(X172, X171), tail(X172, X173))).\nand substitution\nT51 -> T101\nX171 -> T101\nT84 -> T102\nX172 -> T102\nX173 -> []", fontsize=14, style = filled, fillcolor = white];
65 -> 196 [arrowhead = none ];
196 -> 67;

197 [label="ONLY EVAL with clause\nselect(X211, X212, X213) :- ','(no(empty(X212)), ','(head(X212, X214), ','(head(X213, X214), ','(tail(X212, X215), ','(tail(X213, X216), select(X211, X215, X216)))))).\nand substitution\nT51 -> T127\nX211 -> T127\nT84 -> T128\nX212 -> T128\nX213 -> []", fontsize=14, style = filled, fillcolor = white];
66 -> 197 [arrowhead = none ];
197 -> 89;

198 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 198 [arrowhead = none ];
198 -> 68;

199 [label="ONLY EVAL with clause\nno(X176) :- ','(call(X176), ','(!_20, failure(a))).\nand substitution\nT102 -> T105\nX176 -> empty(T105)", fontsize=14, style = filled, fillcolor = white];
68 -> 199 [arrowhead = none ];
199 -> 69;

200 [label="CALL", fontsize=14, style = filled, fillcolor = white];
69 -> 200 [arrowhead = none ];
200 -> 70;

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 201 [arrowhead = none ];
201 -> 71;

202 [label="EVAL with clause\nempty([]).\nand substitution\nT105 -> []", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 202 [arrowhead = none ];
202 -> 72;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 203 [arrowhead = none ];
203 -> 73;

204 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 204 [arrowhead = none ];
204 -> 74;

205 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 205 [arrowhead = none ];
205 -> 77;

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 206 [arrowhead = none ];
206 -> 75;

207 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
75 -> 207 [arrowhead = none ];
207 -> 76;

208 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 208 [arrowhead = none ];
208 -> 78;

209 [label="ONLY EVAL with clause\nno(X179).\nand substitution\nT105 -> T108\nX179 -> empty(T108)", fontsize=14, style = filled, fillcolor = white];
78 -> 209 [arrowhead = none ];
209 -> 79;

210 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 210 [arrowhead = none ];
210 -> 80;

211 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T108), empty([]))", fontsize=14, style = filled, fillcolor = white];
80 -> 211 [arrowhead = none ];
211 -> 81;

212 [label="EVAL with clause\nhead(.(X185, X186), X185).\nand substitution\nX185 -> T114\nX186 -> T115\nT108 -> .(T114, T115)\nT101 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 212 [arrowhead = none ];
212 -> 82;

213 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 213 [arrowhead = none ];
213 -> 83;

214 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 214 [arrowhead = none ];
214 -> 84;

215 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
84 -> 215 [arrowhead = none ];
215 -> 85;

216 [label="EVAL with clause\ntail(.(X193, X194), X194).\nand substitution\nT114 -> T120\nX193 -> T120\nT115 -> []\nX194 -> []\nT121 -> []", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 216 [arrowhead = none ];
216 -> 86;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 217 [arrowhead = none ];
217 -> 87;

218 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 218 [arrowhead = none ];
218 -> 88;

219 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
89 -> 219 [arrowhead = none ];
219 -> 90;

220 [label="ONLY EVAL with clause\nno(X224) :- ','(call(X224), ','(!_26, failure(a))).\nand substitution\nT128 -> T132\nX224 -> empty(T132)", fontsize=14, style = filled, fillcolor = white];
90 -> 220 [arrowhead = none ];
220 -> 91;

221 [label="CALL", fontsize=14, style = filled, fillcolor = white];
91 -> 221 [arrowhead = none ];
221 -> 92;

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
92 -> 222 [arrowhead = none ];
222 -> 93;

223 [label="EVAL with clause\nempty([]).\nand substitution\nT132 -> []", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 223 [arrowhead = none ];
223 -> 94;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 224 [arrowhead = none ];
224 -> 95;

225 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 225 [arrowhead = none ];
225 -> 96;

226 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 226 [arrowhead = none ];
226 -> 99;

227 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 227 [arrowhead = none ];
227 -> 97;

228 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
97 -> 228 [arrowhead = none ];
228 -> 98;

229 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 229 [arrowhead = none ];
229 -> 100;

230 [label="ONLY EVAL with clause\nno(X237).\nand substitution\nT132 -> T137\nX237 -> empty(T137)", fontsize=14, style = filled, fillcolor = white];
100 -> 230 [arrowhead = none ];
230 -> 101;

231 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
101 -> 231 [arrowhead = none ];
231 -> 102;

232 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T137), empty([]))", fontsize=14, style = filled, fillcolor = white];
102 -> 232 [arrowhead = none ];
232 -> 103;

233 [label="EVAL with clause\nhead(.(X249, X250), X249).\nand substitution\nX249 -> T143\nX250 -> T144\nT137 -> .(T143, T144)\nX214 -> T143", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 233 [arrowhead = none ];
233 -> 104;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 234 [arrowhead = none ];
234 -> 105;

235 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
104 -> 235 [arrowhead = none ];
235 -> 106;

236 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 236 [arrowhead = none ];
236 -> 107;

237 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 237 [arrowhead = none ];
237 -> 108;

238 [label="ONLY EVAL with clause\nhead([], X272).\nand substitution\nT143 -> T152\nX272 -> T152", fontsize=14, style = filled, fillcolor = white];
107 -> 238 [arrowhead = none ];
238 -> 109;

239 [label="BACKTRACK\nfor clause: head(.(H, X2), H)because of non-unification", fontsize=14, style = filled, fillcolor = white];
108 -> 239 [arrowhead = none ];
239 -> 118;

240 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
109 -> 240 [arrowhead = none ];
240 -> 110;

241 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
110 -> 241 [arrowhead = none ];
241 -> 111;

242 [label="ONLY EVAL with clause\ntail(.(X286, X287), X287).\nand substitution\nT152 -> T160\nX286 -> T160\nT144 -> T161\nX287 -> T161\nX215 -> T161", fontsize=14, style = filled, fillcolor = white];
111 -> 242 [arrowhead = none ];
242 -> 112;

243 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
112 -> 243 [arrowhead = none ];
243 -> 113;

244 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
113 -> 244 [arrowhead = none ];
244 -> 114;

245 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
113 -> 245 [arrowhead = none ];
245 -> 115;

246 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX216 -> []", fontsize=14, style = filled, fillcolor = white];
114 -> 246 [arrowhead = none ];
246 -> 116;

247 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
115 -> 247 [arrowhead = none ];
247 -> 117;

248 [label="INSTANCE with matching:\nT51 -> T127\nT84 -> T161", fontsize=14, style = filled, fillcolor = lightgrey];
116 -> 248 [arrowhead = none , style=dashed];
248 -> 63[style=dashed];

249 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
120 -> 249 [arrowhead = none ];
249 -> 122;

250 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
122 -> 250 [arrowhead = none ];
250 -> 123;

251 [label="ONLY EVAL with clause\ntail(.(X328, X329), X329).\nand substitution\nT174 -> T184\nX328 -> T184\nT69 -> T185\nX329 -> T185\nX87 -> T185", fontsize=14, style = filled, fillcolor = white];
123 -> 251 [arrowhead = none ];
251 -> 124;

252 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
124 -> 252 [arrowhead = none ];
252 -> 125;

253 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
125 -> 253 [arrowhead = none ];
253 -> 126;

254 [label="ONLY EVAL with clause\ntail(.(X341, X342), X342).\nand substitution\nT184 -> T196\nX341 -> T196\nT176 -> T198\nX342 -> T198\nX88 -> T198\nT197 -> T198", fontsize=14, style = filled, fillcolor = white];
126 -> 254 [arrowhead = none ];
254 -> 127;

255 [label="INSTANCE with matching:\nT1 -> T51\nT2 -> T185\nT3 -> T198", fontsize=14, style = filled, fillcolor = lightgrey];
127 -> 255 [arrowhead = none , style=dashed];
255 -> 1[style=dashed];

}

(2) Obligation:

Triples:

select63(T127, .(T160, T161)) :- select63(T127, T161).
select1(T51, .(T83, T84), []) :- select63(T51, T84).
select1(T51, .(T196, T185), .(T196, T198)) :- select1(T51, T185, T198).

Clauses:

selectc63(T120, .(T120, [])).
selectc63(T127, .(T160, T161)) :- selectc63(T127, T161).
selectc1(T41, .(T41, T42), T42).
selectc1(T51, .(T83, T84), []) :- selectc63(T51, T84).
selectc1(T51, .(T196, T185), .(T196, T198)) :- selectc1(T51, T185, T198).

Afs:

select1(x1, x2, x3)  =  select1(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:
select1_in: (b,b,f)
select63_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SELECT1_IN_GGA(T51, .(T83, T84), []) → U2_GGA(T51, T83, T84, select63_in_gg(T51, T84))
SELECT1_IN_GGA(T51, .(T83, T84), []) → SELECT63_IN_GG(T51, T84)
SELECT63_IN_GG(T127, .(T160, T161)) → U1_GG(T127, T160, T161, select63_in_gg(T127, T161))
SELECT63_IN_GG(T127, .(T160, T161)) → SELECT63_IN_GG(T127, T161)
SELECT1_IN_GGA(T51, .(T196, T185), .(T196, T198)) → U3_GGA(T51, T196, T185, T198, select1_in_gga(T51, T185, T198))
SELECT1_IN_GGA(T51, .(T196, T185), .(T196, T198)) → SELECT1_IN_GGA(T51, T185, T198)

R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select63_in_gg(x1, x2)  =  select63_in_gg(x1, x2)
[]  =  []
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x3, x4)
SELECT63_IN_GG(x1, x2)  =  SELECT63_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SELECT1_IN_GGA(T51, .(T83, T84), []) → U2_GGA(T51, T83, T84, select63_in_gg(T51, T84))
SELECT1_IN_GGA(T51, .(T83, T84), []) → SELECT63_IN_GG(T51, T84)
SELECT63_IN_GG(T127, .(T160, T161)) → U1_GG(T127, T160, T161, select63_in_gg(T127, T161))
SELECT63_IN_GG(T127, .(T160, T161)) → SELECT63_IN_GG(T127, T161)
SELECT1_IN_GGA(T51, .(T196, T185), .(T196, T198)) → U3_GGA(T51, T196, T185, T198, select1_in_gga(T51, T185, T198))
SELECT1_IN_GGA(T51, .(T196, T185), .(T196, T198)) → SELECT1_IN_GGA(T51, T185, T198)

R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
select63_in_gg(x1, x2)  =  select63_in_gg(x1, x2)
[]  =  []
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x3, x4)
SELECT63_IN_GG(x1, x2)  =  SELECT63_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

SELECT63_IN_GG(T127, .(T160, T161)) → SELECT63_IN_GG(T127, T161)

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

(8) PiDPToQDPProof (EQUIVALENT transformation)

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

(9) Obligation:

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

SELECT63_IN_GG(T127, .(T160, T161)) → SELECT63_IN_GG(T127, T161)

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

(10) 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:

• SELECT63_IN_GG(T127, .(T160, T161)) → SELECT63_IN_GG(T127, T161)
  The graph contains the following edges 1 >= 1, 2 > 2

(11) YES

(12) Obligation:

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

SELECT1_IN_GGA(T51, .(T196, T185), .(T196, T198)) → SELECT1_IN_GGA(T51, T185, T198)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

SELECT1_IN_GGA(T51, .(T196, T185)) → SELECT1_IN_GGA(T51, T185)

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

(15) 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:

• SELECT1_IN_GGA(T51, .(T196, T185)) → SELECT1_IN_GGA(T51, T185)
  The graph contains the following edges 1 >= 1, 2 > 2

(16) YES