Left Termination proof of /tmp/tmpjVcBCL/select2.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 181 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 6 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

(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).

Query: select(g,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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, X82), ','(head(T54, X82), ','(tail(T52, X83), ','(tail(T54, X84), select(T51, X83, X84))))))\n\nT51 is ground\nT52 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T52) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(no(empty(T52)), ','(head(T52, X82), ','(head(T54, X82), ','(tail(T52, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 7 | ','(no(empty(T52)), ','(head(T52, X82), ','(head(T54, X82), ','(tail(T52, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT52 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T52) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(','(call(empty(T58)), ','(!_12, failure(a))), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))) | ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(empty(T58), ','(','(!_12, failure(a)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84))))))) | ?_13 | ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(empty(T58), ','(','(!_12, failure(a)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84))))))), SCOPE: 14, CLAUSE: 6 | ?_14 | ?_13 | ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ?_14 | ?_13 | ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_13 | ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(no(empty(T58)), ','(head(T58, X82), ','(head(T54, X82), ','(tail(T58, X83), ','(tail(T54, X84), select(T51, X83, X84)))))), SCOPE: 12, CLAUSE: 8\n\nT51 is ground\nT58 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T58) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(head(T62, X82), ','(head(T54, X82), ','(tail(T62, X83), ','(tail(T54, X84), select(T51, X83, X84)))))\n\nT51 is ground\nT62 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(head(T62, X82), ','(head(T54, X82), ','(tail(T62, X83), ','(tail(T54, X84), select(T51, X83, X84))))), SCOPE: 15, CLAUSE: 2 | ','(head(T62, X82), ','(head(T54, X82), ','(tail(T62, X83), ','(tail(T54, X84), select(T51, X83, X84))))), SCOPE: 15, CLAUSE: 3\n\nT51 is ground\nT62 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(head(T62, X82), ','(head(T54, X82), ','(tail(T62, X83), ','(tail(T54, X84), select(T51, X83, X84))))), SCOPE: 15, CLAUSE: 3\n\nT51 is ground\nT62 is ground\nX82 is free\nX83 is free\nX84 is free\nempty(T62) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(T54, T68), ','(tail(.(T68, T69), X83), ','(tail(T54, X84), select(T51, X83, X84))))\n\nT51 is ground\nT68 is ground\nT69 is ground\nX83 is free\nX84 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), X83), ','(tail(T54, X84), select(T51, X83, X84)))), SCOPE: 16, CLAUSE: 2 | ','(head(T54, T68), ','(tail(.(T68, T69), X83), ','(tail(T54, X84), select(T51, X83, X84)))), SCOPE: 16, CLAUSE: 3\n\nT51 is ground\nT68 is ground\nT69 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(head(T54, T68), ','(tail(.(T68, T69), X83), ','(tail(T54, X84), select(T51, X83, X84)))), SCOPE: 16, CLAUSE: 2\n\nT51 is ground\nT68 is ground\nT69 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T54, T68), ','(tail(.(T68, T69), X83), ','(tail(T54, X84), select(T51, X83, X84)))), SCOPE: 16, CLAUSE: 3\n\nT51 is ground\nT68 is ground\nT69 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail(.(T75, T69), X83), ','(tail([], X84), select(T51, X83, X84)))\n\nT51 is ground\nT69 is ground\nT75 is ground\nX83 is free\nX84 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), X83), ','(tail([], X84), select(T51, X83, X84))), SCOPE: 17, CLAUSE: 4 | ','(tail(.(T75, T69), X83), ','(tail([], X84), select(T51, X83, X84))), SCOPE: 17, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT75 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(.(T75, T69), X83), ','(tail([], X84), select(T51, X83, X84))), SCOPE: 17, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT75 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail([], X84), select(T51, T84, X84))\n\nT51 is ground\nT84 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail([], X84), select(T51, T84, X84)), SCOPE: 18, CLAUSE: 4 | ','(tail([], X84), select(T51, T84, X84)), SCOPE: 18, CLAUSE: 5\n\nT51 is ground\nT84 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail([], X84), select(T51, T84, X84)), SCOPE: 18, CLAUSE: 4\n\nT51 is ground\nT84 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail([], X84), select(T51, T84, X84)), SCOPE: 18, CLAUSE: 5\n\nT51 is ground\nT84 is ground\nX84 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, X203), ','(head([], X203), ','(tail(T128, X204), ','(tail([], X205), select(T127, X204, X205))))))\n\nT127 is ground\nT128 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(no(empty(T128)), ','(head(T128, X203), ','(head([], X203), ','(tail(T128, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 7 | ','(no(empty(T128)), ','(head(T128, X203), ','(head([], X203), ','(tail(T128, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT128 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(','(call(empty(T132)), ','(!_26, failure(a))), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))) | ','(no(empty(T132)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(empty(T132), ','(','(!_26, failure(a)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205))))))) | ?_27 | ','(no(empty(T132)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(empty(T132), ','(','(!_26, failure(a)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205))))))), SCOPE: 28, CLAUSE: 6 | ?_28 | ?_27 | ','(no(empty(T132)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(','(!_26, failure(a)), ','(head([], X203), ','(head([], X203), ','(tail([], X204), ','(tail([], X205), select(T127, X204, X205)))))) | ?_28 | ?_27 | ','(no(empty([])), ','(head([], X203), ','(head([], X203), ','(tail([], X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ?_28 | ?_27 | ','(no(empty(T132)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(failure(a), ','(head([], X203), ','(head([], X203), ','(tail([], X204), ','(tail([], X205), select(T127, X204, X205))))))\n\nT127 is ground\nX203 is free\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(failure(a), ','(head([], X203), ','(head([], X203), ','(tail([], X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 29, CLAUSE: 9\n\nT127 is ground\nX203 is free\nX204 is free\nX205 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, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(no(empty(T132)), ','(head(T132, X203), ','(head([], X203), ','(tail(T132, X204), ','(tail([], X205), select(T127, X204, X205)))))), SCOPE: 26, CLAUSE: 8\n\nT127 is ground\nT132 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T132) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(head(T137, X203), ','(head([], X203), ','(tail(T137, X204), ','(tail([], X205), select(T127, X204, X205)))))\n\nT127 is ground\nT137 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(head(T137, X203), ','(head([], X203), ','(tail(T137, X204), ','(tail([], X205), select(T127, X204, X205))))), SCOPE: 30, CLAUSE: 2 | ','(head(T137, X203), ','(head([], X203), ','(tail(T137, X204), ','(tail([], X205), select(T127, X204, X205))))), SCOPE: 30, CLAUSE: 3\n\nT127 is ground\nT137 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(head(T137, X203), ','(head([], X203), ','(tail(T137, X204), ','(tail([], X205), select(T127, X204, X205))))), SCOPE: 30, CLAUSE: 3\n\nT127 is ground\nT137 is ground\nX203 is free\nX204 is free\nX205 is free\nempty(T137) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(head([], T143), ','(tail(.(T143, T144), X204), ','(tail([], X205), select(T127, X204, X205))))\n\nT127 is ground\nT143 is ground\nT144 is ground\nX204 is free\nX205 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), X204), ','(tail([], X205), select(T127, X204, X205)))), SCOPE: 31, CLAUSE: 2 | ','(head([], T143), ','(tail(.(T143, T144), X204), ','(tail([], X205), select(T127, X204, X205)))), SCOPE: 31, CLAUSE: 3\n\nT127 is ground\nT143 is ground\nT144 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(head([], T143), ','(tail(.(T143, T144), X204), ','(tail([], X205), select(T127, X204, X205)))), SCOPE: 31, CLAUSE: 2\n\nT127 is ground\nT143 is ground\nT144 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(head([], T143), ','(tail(.(T143, T144), X204), ','(tail([], X205), select(T127, X204, X205)))), SCOPE: 31, CLAUSE: 3\n\nT127 is ground\nT143 is ground\nT144 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: ','(tail(.(T152, T144), X204), ','(tail([], X205), select(T127, X204, X205)))\n\nT127 is ground\nT144 is ground\nT152 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(tail(.(T152, T144), X204), ','(tail([], X205), select(T127, X204, X205))), SCOPE: 32, CLAUSE: 4 | ','(tail(.(T152, T144), X204), ','(tail([], X205), select(T127, X204, X205))), SCOPE: 32, CLAUSE: 5\n\nT127 is ground\nT144 is ground\nT152 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(tail(.(T152, T144), X204), ','(tail([], X205), select(T127, X204, X205))), SCOPE: 32, CLAUSE: 5\n\nT127 is ground\nT144 is ground\nT152 is ground\nX204 is free\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(tail([], X205), select(T127, T161, X205))\n\nT127 is ground\nT161 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(tail([], X205), select(T127, T161, X205)), SCOPE: 33, CLAUSE: 4 | ','(tail([], X205), select(T127, T161, X205)), SCOPE: 33, CLAUSE: 5\n\nT127 is ground\nT161 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(tail([], X205), select(T127, T161, X205)), SCOPE: 33, CLAUSE: 4\n\nT127 is ground\nT161 is ground\nX205 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: ','(tail([], X205), select(T127, T161, X205)), SCOPE: 33, CLAUSE: 5\n\nT127 is ground\nT161 is ground\nX205 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), X83), ','(tail(.(T174, T176), X84), select(T51, X83, X84)))\n\nT51 is ground\nT69 is ground\nT174 is ground\nX83 is free\nX84 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), X83), ','(tail(.(T174, T176), X84), select(T51, X83, X84))), SCOPE: 34, CLAUSE: 4 | ','(tail(.(T174, T69), X83), ','(tail(.(T174, T176), X84), select(T51, X83, X84))), SCOPE: 34, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT174 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(tail(.(T174, T69), X83), ','(tail(.(T174, T176), X84), select(T51, X83, X84))), SCOPE: 34, CLAUSE: 5\n\nT51 is ground\nT69 is ground\nT174 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(tail(.(T184, T176), X84), select(T51, T185, X84))\n\nT51 is ground\nT184 is ground\nT185 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(tail(.(T184, T176), X84), select(T51, T185, X84)), SCOPE: 35, CLAUSE: 4 | ','(tail(.(T184, T176), X84), select(T51, T185, X84)), SCOPE: 35, CLAUSE: 5\n\nT51 is ground\nT184 is ground\nT185 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: ','(tail(.(T184, T176), X84), select(T51, T185, X84)), SCOPE: 35, CLAUSE: 5\n\nT51 is ground\nT184 is ground\nT185 is ground\nX84 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 substitutionT1 -> T7,\nX8 -> T7,\nT2 -> T8,\nX9 -> T8,\nT3 -> T10,\nX10 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT8 -> T13,\nX13 -> empty(T13)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 131 [arrowhead = none ];
131 -> 5;

132 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
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 substitutionT13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 134 [arrowhead = none ];
134 -> 8;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT7 -> T20,\nX28 -> T20,\nX29 -> [],\nT3 -> T22,\nX30 -> T22,\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 141 [arrowhead = none ];
141 -> 15;

142 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX41 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 144 [arrowhead = none ];
144 -> 17;

145 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
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 = lightblue];
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 = lightgreen];
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 substitutionT13 -> T29,\nX51 -> empty(T29)", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
29 -> 157 [arrowhead = none ];
157 -> 30;

158 [label="EVAL with clause\nhead(.(X57, X58), X57).\nand substitutionX57 -> 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 = lightgreen];
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 = lightgreen];
33 -> 161 [arrowhead = none ];
161 -> 34;

162 [label="EVAL with clause\ntail(.(X63, X64), X64).\nand substitutionT35 -> T41,\nX63 -> T41,\nT36 -> T42,\nX64 -> T42,\nT10 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 162 [arrowhead = none ];
162 -> 35;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X79, X80, X81) :- ','(no(empty(X80)), ','(head(X80, X82), ','(head(X81, X82), ','(tail(X80, X83), ','(tail(X81, X84), select(X79, X83, X84)))))).\nand substitutionT7 -> T51,\nX79 -> T51,\nT13 -> T52,\nX80 -> T52,\nT3 -> T54,\nX81 -> T54,\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
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(X92) :- ','(call(X92), ','(!_12, failure(a))).\nand substitutionT52 -> T58,\nX92 -> empty(T58)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 167 [arrowhead = none ];
167 -> 41;

168 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
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 = lightgreen];
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(X100).\nand substitutionT58 -> T62,\nX100 -> empty(T62)", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
48 -> 175 [arrowhead = none ];
175 -> 49;

176 [label="EVAL with clause\nhead(.(X112, X113), X112).\nand substitutionX112 -> T68,\nX113 -> T69,\nT62 -> .(T68, T69),\nX82 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 176 [arrowhead = none ];
176 -> 50;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], X123).\nand substitutionT54 -> [],\nT68 -> T75,\nX123 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 181 [arrowhead = none ];
181 -> 55;

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

183 [label="EVAL with clause\nhead(.(X298, X299), X298).\nand substitutionX298 -> T174,\nX299 -> 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 = lightgreen];
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 = lightgreen];
57 -> 186 [arrowhead = none ];
186 -> 58;

187 [label="ONLY EVAL with clause\ntail(.(X137, X138), X138).\nand substitutionT75 -> T83,\nX137 -> T83,\nT69 -> T84,\nX138 -> T84,\nX83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionX84 -> []", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 191 [arrowhead = none ];
191 -> 63;

192 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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(X164, X165, X166) :- ','(no(empty(X165)), ','(head(X165, X164), tail(X165, X166))).\nand substitutionT51 -> T101,\nX164 -> T101,\nT84 -> T102,\nX165 -> T102,\nX166 -> []", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 196 [arrowhead = none ];
196 -> 67;

197 [label="ONLY EVAL with clause\nselect(X200, X201, X202) :- ','(no(empty(X201)), ','(head(X201, X203), ','(head(X202, X203), ','(tail(X201, X204), ','(tail(X202, X205), select(X200, X204, X205)))))).\nand substitutionT51 -> T127,\nX200 -> T127,\nT84 -> T128,\nX201 -> T128,\nX202 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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(X169) :- ','(call(X169), ','(!_20, failure(a))).\nand substitutionT102 -> T105,\nX169 -> empty(T105)", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 199 [arrowhead = none ];
199 -> 69;

200 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
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 substitutionT105 -> []", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 202 [arrowhead = none ];
202 -> 72;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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(X172).\nand substitutionT105 -> T108,\nX172 -> empty(T108)", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
80 -> 211 [arrowhead = none ];
211 -> 81;

212 [label="EVAL with clause\nhead(.(X178, X179), X178).\nand substitutionX178 -> T114,\nX179 -> 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 = lightgreen];
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 = lightgreen];
84 -> 215 [arrowhead = none ];
215 -> 85;

216 [label="EVAL with clause\ntail(.(X184, X185), X185).\nand substitutionT114 -> T120,\nX184 -> T120,\nT115 -> [],\nX185 -> [],\nT121 -> []", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 216 [arrowhead = none ];
216 -> 86;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X213) :- ','(call(X213), ','(!_26, failure(a))).\nand substitutionT128 -> T132,\nX213 -> empty(T132)", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 220 [arrowhead = none ];
220 -> 91;

221 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
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 substitutionT132 -> []", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 223 [arrowhead = none ];
223 -> 94;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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(X226).\nand substitutionT132 -> T137,\nX226 -> empty(T137)", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
102 -> 232 [arrowhead = none ];
232 -> 103;

233 [label="EVAL with clause\nhead(.(X238, X239), X238).\nand substitutionX238 -> T143,\nX239 -> T144,\nT137 -> .(T143, T144),\nX203 -> T143", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 233 [arrowhead = none ];
233 -> 104;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], X259).\nand substitutionT143 -> T152,\nX259 -> T152", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 238 [arrowhead = none ];
238 -> 109;

239 [label="BACKTRACK\nfor clause: head(.(H, X2), H)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
110 -> 241 [arrowhead = none ];
241 -> 111;

242 [label="ONLY EVAL with clause\ntail(.(X273, X274), X274).\nand substitutionT152 -> T160,\nX273 -> T160,\nT144 -> T161,\nX274 -> T161,\nX204 -> T161", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionX205 -> []", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 246 [arrowhead = none ];
246 -> 116;

247 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
122 -> 250 [arrowhead = none ];
250 -> 123;

251 [label="ONLY EVAL with clause\ntail(.(X313, X314), X314).\nand substitutionT174 -> T184,\nX313 -> T184,\nT69 -> T185,\nX314 -> T185,\nX83 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
125 -> 253 [arrowhead = none ];
253 -> 126;

254 [label="ONLY EVAL with clause\ntail(.(X326, X327), X327).\nand substitutionT184 -> T196,\nX326 -> T196,\nT176 -> T198,\nX327 -> T198,\nX84 -> T198,\nT197 -> T198", fontsize=14, style = filled, fillcolor = lightblue];
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:

Clauses:

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

Query: selectB(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
selectB_in: (b,b,f)
selectA_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SELECTB_IN_GGA(T51, .(T83, T84), []) → U2_GGA(T51, T83, T84, selectA_in_gg(T51, T84))
SELECTB_IN_GGA(T51, .(T83, T84), []) → SELECTA_IN_GG(T51, T84)
SELECTA_IN_GG(T127, .(T160, T161)) → U1_GG(T127, T160, T161, selectA_in_gg(T127, T161))
SELECTA_IN_GG(T127, .(T160, T161)) → SELECTA_IN_GG(T127, T161)
SELECTB_IN_GGA(T51, .(T196, T185), .(T196, T198)) → U3_GGA(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
SELECTB_IN_GGA(T51, .(T196, T185), .(T196, T198)) → SELECTB_IN_GGA(T51, T185, T198)

The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

The argument filtering Pi contains the following mapping:
selectB_in_gga(x1, x2, x3) = selectB_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
selectB_out_gga(x1, x2, x3) = selectB_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
selectA_in_gg(x1, x2) = selectA_in_gg(x1, x2)
[] = []
selectA_out_gg(x1, x2) = selectA_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x2, x5)
SELECTB_IN_GGA(x1, x2, x3) = SELECTB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
SELECTA_IN_GG(x1, x2) = SELECTA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4) = U1_GG(x4)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x2, x5)

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

(6) Obligation:

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

SELECTB_IN_GGA(T51, .(T83, T84), []) → U2_GGA(T51, T83, T84, selectA_in_gg(T51, T84))
SELECTB_IN_GGA(T51, .(T83, T84), []) → SELECTA_IN_GG(T51, T84)
SELECTA_IN_GG(T127, .(T160, T161)) → U1_GG(T127, T160, T161, selectA_in_gg(T127, T161))
SELECTA_IN_GG(T127, .(T160, T161)) → SELECTA_IN_GG(T127, T161)
SELECTB_IN_GGA(T51, .(T196, T185), .(T196, T198)) → U3_GGA(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
SELECTB_IN_GGA(T51, .(T196, T185), .(T196, T198)) → SELECTB_IN_GGA(T51, T185, T198)

The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

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

The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

The argument filtering Pi contains the following mapping:
selectB_in_gga(x1, x2, x3) = selectB_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
selectB_out_gga(x1, x2, x3) = selectB_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
selectA_in_gg(x1, x2) = selectA_in_gg(x1, x2)
[] = []
selectA_out_gg(x1, x2) = selectA_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x2, x5)
SELECTA_IN_GG(x1, x2) = SELECTA_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

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

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

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

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

(14) 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:

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

(15) YES

(16) Obligation:

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

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

The TRS R consists of the following rules:

selectB_in_gga(T41, .(T41, T42), T42) → selectB_out_gga(T41, .(T41, T42), T42)
selectB_in_gga(T51, .(T83, T84), []) → U2_gga(T51, T83, T84, selectA_in_gg(T51, T84))
selectA_in_gg(T120, .(T120, [])) → selectA_out_gg(T120, .(T120, []))
selectA_in_gg(T127, .(T160, T161)) → U1_gg(T127, T160, T161, selectA_in_gg(T127, T161))
U1_gg(T127, T160, T161, selectA_out_gg(T127, T161)) → selectA_out_gg(T127, .(T160, T161))
U2_gga(T51, T83, T84, selectA_out_gg(T51, T84)) → selectB_out_gga(T51, .(T83, T84), [])
selectB_in_gga(T51, .(T196, T185), .(T196, T198)) → U3_gga(T51, T196, T185, T198, selectB_in_gga(T51, T185, T198))
U3_gga(T51, T196, T185, T198, selectB_out_gga(T51, T185, T198)) → selectB_out_gga(T51, .(T196, T185), .(T196, T198))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

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

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

(21) 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:

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES