Left Termination proof of /tmp/tmpG98Zxt/select2.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

            ↳11 PiDP

                ↳12 PiDPToQDPProof (⇔)

                    ↳13 QDP

                        ↳14 QDPSizeChangeProof (⇔)

                            ↳15 TRUE

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

            ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                    ↳20 QDP

                        ↳21 QDPSizeChangeProof (⇔)

                            ↳22 TRUE

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(no(empty(T5)), ','(head(T5, T4), tail(T5, T7)))\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(empty(T5)), ','(head(T5, T4), tail(T5, T7))), SCOPE: 2, CLAUSE: 7 | ','(no(empty(T5)), ','(head(T5, T4), tail(T5, T7))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(empty(T8), ','(!_2, ','(failure(a), ','(head(T8, T4), tail(T8, T7))))) | ','(no(empty(T8)), ','(head(T8, T4), tail(T8, T7))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT4 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(empty(T8), ','(!_2, ','(failure(a), ','(head(T8, T4), tail(T8, T7))))), SCOPE: 3, CLAUSE: 6 | ','(no(empty(T8)), ','(head(T8, T4), tail(T8, T7))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT4 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_2, ','(failure(a), ','(head([], T4), tail([], T7)))) | ','(no(empty([])), ','(head([], T4), tail([], T7))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(no(empty(T8)), ','(head(T8, T4), tail(T8, T7))), SCOPE: 2, CLAUSE: 8\n\nT4 is ground\nT8 is ground\nempty(T8) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(failure(a), ','(head([], T4), tail([], T7)))\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(failure(a), ','(head([], T4), tail([], T7))), SCOPE: 4, CLAUSE: 9\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(head(T9, T4), tail(T9, T7))\n\nT4 is ground\nT9 is ground\nempty(T9) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(head(T9, T4), tail(T9, T7)), SCOPE: 5, CLAUSE: 2 | ','(head(T9, T4), tail(T9, T7)), SCOPE: 5, CLAUSE: 3\n\nT4 is ground\nT9 is ground\nempty(T9) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(head(T9, T4), tail(T9, T7)), SCOPE: 5, CLAUSE: 3\n\nT4 is ground\nT9 is ground\nX19 is free; \nempty(T9) !=? empty([])\nhead(T9, T4) !=? head([], X19)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: tail(.(T10, T11), T7)\n\nT10 is ground\nT11 is ground\nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX19 is free\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: tail(.(T10, T11), T7), SCOPE: 6, CLAUSE: 4 | tail(.(T10, T11), T7), SCOPE: 6, CLAUSE: 5\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: tail(.(T10, T11), T7), SCOPE: 6, CLAUSE: 5\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX26 is free\nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(no(empty(T15)), ','(head(T15, X34), ','(head(T17, X34), ','(tail(T15, X35), ','(tail(T17, X36), select(T14, X35, X36))))))\n\nT14 is ground\nT15 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(empty(T15)), ','(head(T15, X34), ','(head(T17, X34), ','(tail(T15, X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 7 | ','(no(empty(T15)), ','(head(T15, X34), ','(head(T17, X34), ','(tail(T15, X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 8 | ?_7\n\nT14 is ground\nT15 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(empty(T18), ','(!_7, ','(failure(a), ','(head(T18, X34), ','(head(T17, X34), ','(tail(T18, X35), ','(tail(T17, X36), select(T14, X35, X36)))))))) | ','(no(empty(T18)), ','(head(T18, X34), ','(head(T17, X34), ','(tail(T18, X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 8 | ?_7\n\nT14 is ground\nT18 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(empty(T18), ','(!_7, ','(failure(a), ','(head(T18, X34), ','(head(T17, X34), ','(tail(T18, X35), ','(tail(T17, X36), select(T14, X35, X36)))))))), SCOPE: 8, CLAUSE: 6 | ','(no(empty(T18)), ','(head(T18, X34), ','(head(T17, X34), ','(tail(T18, X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 8 | ?_7\n\nT14 is ground\nT18 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(!_7, ','(failure(a), ','(head([], X34), ','(head(T17, X34), ','(tail([], X35), ','(tail(T17, X36), select(T14, X35, X36))))))) | ','(no(empty([])), ','(head([], X34), ','(head(T17, X34), ','(tail([], X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 8 | ?_7\n\nT14 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(empty(T18)), ','(head(T18, X34), ','(head(T17, X34), ','(tail(T18, X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 7, CLAUSE: 8\n\nT14 is ground\nT18 is ground\nX34 is free; \nX35 is free; \nX36 is free; \nempty(T18) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(failure(a), ','(head([], X34), ','(head(T17, X34), ','(tail([], X35), ','(tail(T17, X36), select(T14, X35, X36))))))\n\nT14 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(failure(a), ','(head([], X34), ','(head(T17, X34), ','(tail([], X35), ','(tail(T17, X36), select(T14, X35, X36)))))), SCOPE: 9, CLAUSE: 9\n\nT14 is ground\nX34 is free; \nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(head(T19, X34), ','(head(T17, X34), ','(tail(T19, X35), ','(tail(T17, X36), select(T14, X35, X36)))))\n\nT14 is ground\nT19 is ground\nX34 is free; \nX35 is free; \nX36 is free; \nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(T19, X34), ','(head(T17, X34), ','(tail(T19, X35), ','(tail(T17, X36), select(T14, X35, X36))))), SCOPE: 10, CLAUSE: 2 | ','(head(T19, X34), ','(head(T17, X34), ','(tail(T19, X35), ','(tail(T17, X36), select(T14, X35, X36))))), SCOPE: 10, CLAUSE: 3\n\nT14 is ground\nT19 is ground\nX34 is free; \nX35 is free; \nX36 is free; \nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(head(T19, X34), ','(head(T17, X34), ','(tail(T19, X35), ','(tail(T17, X36), select(T14, X35, X36))))), SCOPE: 10, CLAUSE: 3\n\nT14 is ground\nT19 is ground\nX34 is free; \nX35 is free; \nX36 is free; \nX42 is free; \nempty(T19) !=? empty([])\nhead(T19, X34) !=? head([], X42)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(head(T17, T20), ','(tail(.(T20, T21), X35), ','(tail(T17, X36), select(T14, X35, X36))))\n\nT14 is ground\nT20 is ground\nT21 is ground\nX34 is free; \nX35 is free; \nX36 is free; \nX42 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX34 is free\nX42 is free; \nX45 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(T17, T20), ','(tail(.(T20, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 2 | ','(head(T17, T20), ','(tail(.(T20, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT20 is ground\nT21 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(tail(.(T22, T21), X35), ','(tail([], X36), select(T14, X35, X36))) | ','(head(T17, T22), ','(tail(.(T22, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT21 is ground\nT22 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(head(T17, T20), ','(tail(.(T20, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT20 is ground\nT21 is ground\nX35 is free; \nX36 is free; \nX48 is free; \nhead(T17, T20) !=? head([], X48)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(.(T22, T21), X35), ','(tail([], X36), select(T14, X35, X36))), SCOPE: 12, CLAUSE: 4 | ','(tail(.(T22, T21), X35), ','(tail([], X36), select(T14, X35, X36))), SCOPE: 12, CLAUSE: 5 | ','(head(T17, T22), ','(tail(.(T22, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT21 is ground\nT22 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(.(T22, T21), X35), ','(tail([], X36), select(T14, X35, X36))), SCOPE: 12, CLAUSE: 5 | ','(head(T17, T22), ','(tail(.(T22, T21), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT21 is ground\nT22 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail([], X36), select(T14, T24, X36)) | ','(head(T17, T23), ','(tail(.(T23, T24), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT23 is ground\nT24 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail([], X36), select(T14, T24, X36)), SCOPE: 13, CLAUSE: 4 | ','(tail([], X36), select(T14, T24, X36)), SCOPE: 13, CLAUSE: 5 | ','(head(T17, T23), ','(tail(.(T23, T24), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT23 is ground\nT24 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: select(T14, T24, []) | ','(tail([], X36), select(T14, T24, X36)), SCOPE: 13, CLAUSE: 5 | ','(head(T17, T23), ','(tail(.(T23, T24), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT23 is ground\nT24 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: select(T14, T24, [])\n\nT14 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail([], X36), select(T14, T24, X36)), SCOPE: 13, CLAUSE: 5 | ','(head(T17, T23), ','(tail(.(T23, T24), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT23 is ground\nT24 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: select(T14, T24, []), SCOPE: 14, CLAUSE: 0 | select(T14, T24, []), SCOPE: 14, CLAUSE: 1\n\nT14 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: select(T14, T24, []), SCOPE: 14, CLAUSE: 0\n\nT14 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: select(T14, T24, []), SCOPE: 14, CLAUSE: 1\n\nT14 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(no(empty(T26)), ','(head(T26, T25), tail(T26, [])))\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(no(empty(T26)), ','(head(T26, T25), tail(T26, []))), SCOPE: 15, CLAUSE: 7 | ','(no(empty(T26)), ','(head(T26, T25), tail(T26, []))), SCOPE: 15, CLAUSE: 8 | ?_15\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(empty(T27), ','(!_15, ','(failure(a), ','(head(T27, T25), tail(T27, []))))) | ','(no(empty(T27)), ','(head(T27, T25), tail(T27, []))), SCOPE: 15, CLAUSE: 8 | ?_15\n\nT25 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(empty(T27), ','(!_15, ','(failure(a), ','(head(T27, T25), tail(T27, []))))), SCOPE: 16, CLAUSE: 6 | ','(no(empty(T27)), ','(head(T27, T25), tail(T27, []))), SCOPE: 15, CLAUSE: 8 | ?_15\n\nT25 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(!_15, ','(failure(a), ','(head([], T25), tail([], [])))) | ','(no(empty([])), ','(head([], T25), tail([], []))), SCOPE: 15, CLAUSE: 8 | ?_15\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(empty(T27)), ','(head(T27, T25), tail(T27, []))), SCOPE: 15, CLAUSE: 8\n\nT25 is ground\nT27 is ground\nempty(T27) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(failure(a), ','(head([], T25), tail([], [])))\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(failure(a), ','(head([], T25), tail([], []))), SCOPE: 17, CLAUSE: 9\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(head(T28, T25), tail(T28, []))\n\nT25 is ground\nT28 is ground\nempty(T28) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(T28, T25), tail(T28, [])), SCOPE: 18, CLAUSE: 2 | ','(head(T28, T25), tail(T28, [])), SCOPE: 18, CLAUSE: 3\n\nT25 is ground\nT28 is ground\nempty(T28) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(head(T28, T25), tail(T28, [])), SCOPE: 18, CLAUSE: 3\n\nT25 is ground\nT28 is ground\nX67 is free; \nempty(T28) !=? empty([])\nhead(T28, T25) !=? head([], X67)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: tail(.(T29, T30), [])\n\nT29 is ground\nT30 is ground\nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\nX67 is free\nX70 is free; \nX71 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: tail(.(T29, T30), []), SCOPE: 19, CLAUSE: 4 | tail(.(T29, T30), []), SCOPE: 19, CLAUSE: 5\n\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: tail(.(T29, T30), []), SCOPE: 19, CLAUSE: 5\n\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\nX74 is free\nX75 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(no(empty(T34)), ','(head(T34, X82), ','(head([], X82), ','(tail(T34, X83), ','(tail([], X84), select(T33, X83, X84))))))\n\nT33 is ground\nT34 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(no(empty(T34)), ','(head(T34, X82), ','(head([], X82), ','(tail(T34, X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 7 | ','(no(empty(T34)), ','(head(T34, X82), ','(head([], X82), ','(tail(T34, X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 8 | ?_20\n\nT33 is ground\nT34 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(empty(T35), ','(!_20, ','(failure(a), ','(head(T35, X82), ','(head([], X82), ','(tail(T35, X83), ','(tail([], X84), select(T33, X83, X84)))))))) | ','(no(empty(T35)), ','(head(T35, X82), ','(head([], X82), ','(tail(T35, X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 8 | ?_20\n\nT33 is ground\nT35 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(empty(T35), ','(!_20, ','(failure(a), ','(head(T35, X82), ','(head([], X82), ','(tail(T35, X83), ','(tail([], X84), select(T33, X83, X84)))))))), SCOPE: 21, CLAUSE: 6 | ','(no(empty(T35)), ','(head(T35, X82), ','(head([], X82), ','(tail(T35, X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 8 | ?_20\n\nT33 is ground\nT35 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(!_20, ','(failure(a), ','(head([], X82), ','(head([], X82), ','(tail([], X83), ','(tail([], X84), select(T33, X83, X84))))))) | ','(no(empty([])), ','(head([], X82), ','(head([], X82), ','(tail([], X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 8 | ?_20\n\nT33 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(no(empty(T35)), ','(head(T35, X82), ','(head([], X82), ','(tail(T35, X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 20, CLAUSE: 8\n\nT33 is ground\nT35 is ground\nX82 is free; \nX83 is free; \nX84 is free; \nempty(T35) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(failure(a), ','(head([], X82), ','(head([], X82), ','(tail([], X83), ','(tail([], X84), select(T33, X83, X84))))))\n\nT33 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(failure(a), ','(head([], X82), ','(head([], X82), ','(tail([], X83), ','(tail([], X84), select(T33, X83, X84)))))), SCOPE: 22, CLAUSE: 9\n\nT33 is ground\nX82 is free; \nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(head(T36, X82), ','(head([], X82), ','(tail(T36, X83), ','(tail([], X84), select(T33, X83, X84)))))\n\nT33 is ground\nT36 is ground\nX82 is free; \nX83 is free; \nX84 is free; \nempty(T36) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(head(T36, X82), ','(head([], X82), ','(tail(T36, X83), ','(tail([], X84), select(T33, X83, X84))))), SCOPE: 23, CLAUSE: 2 | ','(head(T36, X82), ','(head([], X82), ','(tail(T36, X83), ','(tail([], X84), select(T33, X83, X84))))), SCOPE: 23, CLAUSE: 3\n\nT33 is ground\nT36 is ground\nX82 is free; \nX83 is free; \nX84 is free; \nempty(T36) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(head(T36, X82), ','(head([], X82), ','(tail(T36, X83), ','(tail([], X84), select(T33, X83, X84))))), SCOPE: 23, CLAUSE: 3\n\nT33 is ground\nT36 is ground\nX82 is free; \nX83 is free; \nX84 is free; \nX90 is free; \nempty(T36) !=? empty([])\nhead(T36, X82) !=? head([], X90)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(head([], T37), ','(tail(.(T37, T38), X83), ','(tail([], X84), select(T33, X83, X84))))\n\nT33 is ground\nT37 is ground\nT38 is ground\nX82 is free; \nX83 is free; \nX84 is free; \nX90 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\nX82 is free\nX90 is free; \nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(head([], T37), ','(tail(.(T37, T38), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 2 | ','(head([], T37), ','(tail(.(T37, T38), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT37 is ground\nT38 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84))) | ','(head([], T39), ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT38 is ground\nT39 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84))), SCOPE: 25, CLAUSE: 4 | ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84))), SCOPE: 25, CLAUSE: 5 | ','(head([], T39), ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT38 is ground\nT39 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84))), SCOPE: 25, CLAUSE: 5 | ','(head([], T39), ','(tail(.(T39, T38), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT38 is ground\nT39 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(tail([], X84), select(T33, T41, X84)) | ','(head([], T40), ','(tail(.(T40, T41), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nT41 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(tail([], X84), select(T33, T41, X84)), SCOPE: 26, CLAUSE: 4 | ','(tail([], X84), select(T33, T41, X84)), SCOPE: 26, CLAUSE: 5 | ','(head([], T40), ','(tail(.(T40, T41), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nT41 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: select(T33, T41, []) | ','(tail([], X84), select(T33, T41, X84)), SCOPE: 26, CLAUSE: 5 | ','(head([], T40), ','(tail(.(T40, T41), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nT41 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: select(T33, T41, [])\n\nT33 is ground\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(tail([], X84), select(T33, T41, X84)), SCOPE: 26, CLAUSE: 5 | ','(head([], T40), ','(tail(.(T40, T41), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nT41 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(head([], T40), ','(tail(.(T40, T41), X83), ','(tail([], X84), select(T33, X83, X84)))), SCOPE: 24, CLAUSE: 3\n\nT33 is ground\nT40 is ground\nT41 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(head(T17, T23), ','(tail(.(T23, T24), X35), ','(tail(T17, X36), select(T14, X35, X36)))), SCOPE: 11, CLAUSE: 3\n\nT14 is ground\nT23 is ground\nT24 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(tail(.(T42, T24), X35), ','(tail(.(T42, T44), X36), select(T14, X35, X36)))\n\nT14 is ground\nT24 is ground\nT42 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\nX109 is free\nX110 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(tail(.(T42, T24), X35), ','(tail(.(T42, T44), X36), select(T14, X35, X36))), SCOPE: 27, CLAUSE: 4 | ','(tail(.(T42, T24), X35), ','(tail(.(T42, T44), X36), select(T14, X35, X36))), SCOPE: 27, CLAUSE: 5\n\nT14 is ground\nT24 is ground\nT42 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(tail(.(T42, T24), X35), ','(tail(.(T42, T44), X36), select(T14, X35, X36))), SCOPE: 27, CLAUSE: 5\n\nT14 is ground\nT24 is ground\nT42 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(tail(.(T45, T44), X36), select(T14, T46, X36))\n\nT14 is ground\nT45 is ground\nT46 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(tail(.(T45, T44), X36), select(T14, T46, X36)), SCOPE: 28, CLAUSE: 4 | ','(tail(.(T45, T44), X36), select(T14, T46, X36)), SCOPE: 28, CLAUSE: 5\n\nT14 is ground\nT45 is ground\nT46 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(tail(.(T45, T44), X36), select(T14, T46, X36)), SCOPE: 28, CLAUSE: 5\n\nT14 is ground\nT45 is ground\nT46 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: select(T14, T46, T49)\n\nT14 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(tail(.(T50, T21), X35), ','(tail(.(T50, T52), X36), select(T14, X35, X36)))\n\nT14 is ground\nT21 is ground\nT50 is ground\nX35 is free; \nX36 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\nX48 is free\nX121 is free; \nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(tail(.(T50, T21), X35), ','(tail(.(T50, T52), X36), select(T14, X35, X36))), SCOPE: 29, CLAUSE: 4 | ','(tail(.(T50, T21), X35), ','(tail(.(T50, T52), X36), select(T14, X35, X36))), SCOPE: 29, CLAUSE: 5\n\nT14 is ground\nT21 is ground\nT50 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(tail(.(T50, T21), X35), ','(tail(.(T50, T52), X36), select(T14, X35, X36))), SCOPE: 29, CLAUSE: 5\n\nT14 is ground\nT21 is ground\nT50 is ground\nX35 is free; \nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(tail(.(T53, T52), X36), select(T14, T54, X36))\n\nT14 is ground\nT53 is ground\nT54 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: ','(tail(.(T53, T52), X36), select(T14, T54, X36)), SCOPE: 30, CLAUSE: 4 | ','(tail(.(T53, T52), X36), select(T14, T54, X36)), SCOPE: 30, CLAUSE: 5\n\nT14 is ground\nT53 is ground\nT54 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(tail(.(T53, T52), X36), select(T14, T54, X36)), SCOPE: 30, CLAUSE: 5\n\nT14 is ground\nT53 is ground\nT54 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: select(T14, T54, T57)\n\nT14 is ground\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 112 [arrowhead = none ];
112 -> 2;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 113 [arrowhead = none ];
113 -> 3;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 114 [arrowhead = none ];
114 -> 4;

115 [label="EVAL with clause\nselect(X11, X12, X13) :- ','(no(empty(X12)), ','(head(X12, X11), tail(X12, X13))).\nand substitution\nT1 -> T4\nX11 -> T4\nT2 -> T5\nX12 -> T5\nT3 -> T7\nX13 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 115 [arrowhead = none ];
115 -> 5;

116 [label="EVAL with clause\nselect(X31, X32, X33) :- ','(no(empty(X32)), ','(head(X32, X34), ','(head(X33, X34), ','(tail(X32, X35), ','(tail(X33, X36), select(X31, X35, X36)))))).\nand substitution\nT1 -> T14\nX31 -> T14\nT2 -> T15\nX32 -> T15\nT3 -> T17\nX33 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 116 [arrowhead = none ];
116 -> 24;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 117 [arrowhead = none ];
117 -> 6;

118 [label="EVAL with clause\nno(X15) :- ','(X15, ','(!, failure(a))).\nand substitution\nT5 -> T8\nX15 -> empty(T8)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 118 [arrowhead = none ];
118 -> 7;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 119 [arrowhead = none ];
119 -> 8;

120 [label="EVAL with clause\nempty([]).\nand substitution\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 120 [arrowhead = none ];
120 -> 9;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 121 [arrowhead = none ];
121 -> 10;

122 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 122 [arrowhead = none ];
122 -> 11;

123 [label="EVAL with clause\nno(X17).\nand substitution\nT8 -> T9\nX17 -> empty(T9)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 123 [arrowhead = none ];
123 -> 14;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 124 [arrowhead = none ];
124 -> 12;

125 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
12 -> 125 [arrowhead = none ];
125 -> 13;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 126 [arrowhead = none ];
126 -> 15;

127 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T9), empty([]))", fontsize=14, style = filled, fillcolor = white];
15 -> 127 [arrowhead = none ];
127 -> 16;

128 [label="EVAL with clause\nhead(.(X22, X23), X22).\nand substitution\nX22 -> T10\nX23 -> T11\nT9 -> .(T10, T11)\nT4 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 128 [arrowhead = none ];
128 -> 17;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 129 [arrowhead = none ];
129 -> 18;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 130 [arrowhead = none ];
130 -> 19;

131 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 131 [arrowhead = none ];
131 -> 20;

132 [label="EVAL with clause\ntail(.(X26, X27), X27).\nand substitution\nT10 -> T12\nX26 -> T12\nT11 -> T13\nX27 -> T13\nT7 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 132 [arrowhead = none ];
132 -> 21;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 133 [arrowhead = none ];
133 -> 22;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 134 [arrowhead = none ];
134 -> 23;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 135 [arrowhead = none ];
135 -> 25;

136 [label="EVAL with clause\nno(X38) :- ','(X38, ','(!, failure(a))).\nand substitution\nT15 -> T18\nX38 -> empty(T18)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 136 [arrowhead = none ];
136 -> 26;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 137 [arrowhead = none ];
137 -> 27;

138 [label="EVAL with clause\nempty([]).\nand substitution\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 138 [arrowhead = none ];
138 -> 28;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 139 [arrowhead = none ];
139 -> 29;

140 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 140 [arrowhead = none ];
140 -> 30;

141 [label="EVAL with clause\nno(X40).\nand substitution\nT18 -> T19\nX40 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 141 [arrowhead = none ];
141 -> 33;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 142 [arrowhead = none ];
142 -> 31;

143 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
31 -> 143 [arrowhead = none ];
143 -> 32;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 144 [arrowhead = none ];
144 -> 34;

145 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T19), empty([]))", fontsize=14, style = filled, fillcolor = white];
34 -> 145 [arrowhead = none ];
145 -> 35;

146 [label="EVAL with clause\nhead(.(X45, X46), X45).\nand substitution\nX45 -> T20\nX46 -> T21\nT19 -> .(T20, T21)\nX34 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 146 [arrowhead = none ];
146 -> 36;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 147 [arrowhead = none ];
147 -> 37;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 148 [arrowhead = none ];
148 -> 38;

149 [label="EVAL with clause\nhead([], X48).\nand substitution\nT17 -> []\nT20 -> T22\nX48 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 149 [arrowhead = none ];
149 -> 39;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 150 [arrowhead = none ];
150 -> 40;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 151 [arrowhead = none ];
151 -> 41;

152 [label="EVAL with clause\nhead(.(X121, X122), X121).\nand substitution\nX121 -> T50\nX122 -> T52\nT17 -> .(T50, T52)\nT20 -> T50\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 152 [arrowhead = none ];
152 -> 104;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 153 [arrowhead = none ];
153 -> 105;

154 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
41 -> 154 [arrowhead = none ];
154 -> 42;

155 [label="EVAL with clause\ntail(.(X51, X52), X52).\nand substitution\nT22 -> T23\nX51 -> T23\nT21 -> T24\nX52 -> T24\nX35 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 155 [arrowhead = none ];
155 -> 43;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 156 [arrowhead = none ];
156 -> 44;

157 [label="EVAL with clause\ntail([], []).\nand substitution\nX36 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 157 [arrowhead = none ];
157 -> 45;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 158 [arrowhead = none ];
158 -> 46;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 159 [arrowhead = none ];
159 -> 47;

160 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 160 [arrowhead = none ];
160 -> 48;

161 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 161 [arrowhead = none ];
161 -> 95;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 162 [arrowhead = none ];
162 -> 49;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 163 [arrowhead = none ];
163 -> 50;

164 [label="EVAL with clause\nselect(X59, X60, X61) :- ','(no(empty(X60)), ','(head(X60, X59), tail(X60, X61))).\nand substitution\nT14 -> T25\nX59 -> T25\nT24 -> T26\nX60 -> T26\nX61 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 164 [arrowhead = none ];
164 -> 51;

165 [label="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 substitution\nT14 -> T33\nX79 -> T33\nT24 -> T34\nX80 -> T34\nX81 -> []", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 165 [arrowhead = none ];
165 -> 70;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 166 [arrowhead = none ];
166 -> 52;

167 [label="EVAL with clause\nno(X63) :- ','(X63, ','(!, failure(a))).\nand substitution\nT26 -> T27\nX63 -> empty(T27)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 167 [arrowhead = none ];
167 -> 53;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 168 [arrowhead = none ];
168 -> 54;

169 [label="EVAL with clause\nempty([]).\nand substitution\nT27 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 169 [arrowhead = none ];
169 -> 55;

170 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 170 [arrowhead = none ];
170 -> 56;

171 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 171 [arrowhead = none ];
171 -> 57;

172 [label="EVAL with clause\nno(X65).\nand substitution\nT27 -> T28\nX65 -> empty(T28)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 172 [arrowhead = none ];
172 -> 60;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 173 [arrowhead = none ];
173 -> 58;

174 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
58 -> 174 [arrowhead = none ];
174 -> 59;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 175 [arrowhead = none ];
175 -> 61;

176 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T28), empty([]))", fontsize=14, style = filled, fillcolor = white];
61 -> 176 [arrowhead = none ];
176 -> 62;

177 [label="EVAL with clause\nhead(.(X70, X71), X70).\nand substitution\nX70 -> T29\nX71 -> T30\nT28 -> .(T29, T30)\nT25 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 177 [arrowhead = none ];
177 -> 63;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 178 [arrowhead = none ];
178 -> 64;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 179 [arrowhead = none ];
179 -> 65;

180 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 180 [arrowhead = none ];
180 -> 66;

181 [label="EVAL with clause\ntail(.(X74, X75), X75).\nand substitution\nT29 -> T31\nX74 -> T31\nT30 -> []\nX75 -> []\nT32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 181 [arrowhead = none ];
181 -> 67;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 182 [arrowhead = none ];
182 -> 68;

183 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 183 [arrowhead = none ];
183 -> 69;

184 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 184 [arrowhead = none ];
184 -> 71;

185 [label="EVAL with clause\nno(X86) :- ','(X86, ','(!, failure(a))).\nand substitution\nT34 -> T35\nX86 -> empty(T35)", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 185 [arrowhead = none ];
185 -> 72;

186 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 186 [arrowhead = none ];
186 -> 73;

187 [label="EVAL with clause\nempty([]).\nand substitution\nT35 -> []", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 187 [arrowhead = none ];
187 -> 74;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 188 [arrowhead = none ];
188 -> 75;

189 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 189 [arrowhead = none ];
189 -> 76;

190 [label="EVAL with clause\nno(X88).\nand substitution\nT35 -> T36\nX88 -> empty(T36)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 190 [arrowhead = none ];
190 -> 79;

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 191 [arrowhead = none ];
191 -> 77;

192 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
77 -> 192 [arrowhead = none ];
192 -> 78;

193 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 193 [arrowhead = none ];
193 -> 80;

194 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T36), empty([]))", fontsize=14, style = filled, fillcolor = white];
80 -> 194 [arrowhead = none ];
194 -> 81;

195 [label="EVAL with clause\nhead(.(X93, X94), X93).\nand substitution\nX93 -> T37\nX94 -> T38\nT36 -> .(T37, T38)\nX82 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 195 [arrowhead = none ];
195 -> 82;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 196 [arrowhead = none ];
196 -> 83;

197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
82 -> 197 [arrowhead = none ];
197 -> 84;

198 [label="EVAL with clause\nhead([], X96).\nand substitution\nT37 -> T39\nX96 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 198 [arrowhead = none ];
198 -> 85;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 199 [arrowhead = none ];
199 -> 86;

200 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
86 -> 200 [arrowhead = none ];
200 -> 87;

201 [label="EVAL with clause\ntail(.(X99, X100), X100).\nand substitution\nT39 -> T40\nX99 -> T40\nT38 -> T41\nX100 -> T41\nX83 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 201 [arrowhead = none ];
201 -> 88;

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 202 [arrowhead = none ];
202 -> 89;

203 [label="EVAL with clause\ntail([], []).\nand substitution\nX84 -> []", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 203 [arrowhead = none ];
203 -> 90;

204 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 204 [arrowhead = none ];
204 -> 91;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 205 [arrowhead = none ];
205 -> 92;

206 [label="INSTANCE with matching:\nT14 -> T33\nT24 -> T41", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 206 [arrowhead = none , style=dashed];
206 -> 46[style=dashed];

207 [label="BACKTRACK\nfor clause: tail(.(X3, T), T)because of non-unification", fontsize=14, style = filled, fillcolor = white];
92 -> 207 [arrowhead = none ];
207 -> 93;

208 [label="BACKTRACK\nfor clause: head(.(H, X2), H)because of non-unification", fontsize=14, style = filled, fillcolor = white];
93 -> 208 [arrowhead = none ];
208 -> 94;

209 [label="EVAL with clause\nhead(.(X109, X110), X109).\nand substitution\nX109 -> T42\nX110 -> T44\nT17 -> .(T42, T44)\nT23 -> T42\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 209 [arrowhead = none ];
209 -> 96;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 210 [arrowhead = none ];
210 -> 97;

211 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
96 -> 211 [arrowhead = none ];
211 -> 98;

212 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
98 -> 212 [arrowhead = none ];
212 -> 99;

213 [label="EVAL with clause\ntail(.(X113, X114), X114).\nand substitution\nT42 -> T45\nX113 -> T45\nT24 -> T46\nX114 -> T46\nX35 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 213 [arrowhead = none ];
213 -> 100;

214 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
100 -> 214 [arrowhead = none ];
214 -> 101;

215 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
101 -> 215 [arrowhead = none ];
215 -> 102;

216 [label="EVAL with clause\ntail(.(X117, X118), X118).\nand substitution\nT45 -> T47\nX117 -> T47\nT44 -> T49\nX118 -> T49\nX36 -> T49\nT48 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 216 [arrowhead = none ];
216 -> 103;

217 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T46\nT3 -> T49", fontsize=14, style = filled, fillcolor = lightgrey];
103 -> 217 [arrowhead = none , style=dashed];
217 -> 1[style=dashed];

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
104 -> 218 [arrowhead = none ];
218 -> 106;

219 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
106 -> 219 [arrowhead = none ];
219 -> 107;

220 [label="EVAL with clause\ntail(.(X125, X126), X126).\nand substitution\nT50 -> T53\nX125 -> T53\nT21 -> T54\nX126 -> T54\nX35 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 220 [arrowhead = none ];
220 -> 108;

221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
108 -> 221 [arrowhead = none ];
221 -> 109;

222 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
109 -> 222 [arrowhead = none ];
222 -> 110;

223 [label="EVAL with clause\ntail(.(X129, X130), X130).\nand substitution\nT53 -> T55\nX129 -> T55\nT52 -> T57\nX130 -> T57\nX36 -> T57\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 223 [arrowhead = none ];
223 -> 111;

224 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T54\nT3 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
111 -> 224 [arrowhead = none , style=dashed];
224 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

select46(T31, .(T31, [])).
select46(T33, .(T40, T41)) :- select46(T33, T41).
select1(T12, .(T12, T13), T13).
select1(T31, .(T23, .(T31, [])), []).
select1(T33, .(T23, .(T40, T41)), []) :- select46(T33, T41).
select1(T14, .(T47, T46), .(T47, T49)) :- select1(T14, T46, T49).
select1(T14, .(T55, T54), .(T55, T57)) :- select1(T14, T54, T57).

Queries:

select1(g,g,a).

(3) PrologToPiTRSProof (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)
select46_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

(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:

SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → U2_GGA(T33, T23, T40, T41, select46_in_gg(T33, T41))
SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → SELECT46_IN_GG(T33, T41)
SELECT46_IN_GG(T33, .(T40, T41)) → U1_GG(T33, T40, T41, select46_in_gg(T33, T41))
SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → U3_GGA(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)
SELECT1_IN_GGA(T14, .(T55, T54), .(T55, T57)) → U4_GGA(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3) = select1_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
select1_out_gga(x1, x2, x3) = select1_out_gga(x3)
[] = []
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5)
select46_in_gg(x1, x2) = select46_in_gg(x1, x2)
select46_out_gg(x1, x2) = select46_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x5)
SELECT1_IN_GGA(x1, x2, x3) = SELECT1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x5)
SELECT46_IN_GG(x1, x2) = SELECT46_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4) = U1_GG(x4)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x2, x5)
U4_GGA(x1, x2, x3, x4, x5) = U4_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:

SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → U2_GGA(T33, T23, T40, T41, select46_in_gg(T33, T41))
SELECT1_IN_GGA(T33, .(T23, .(T40, T41)), []) → SELECT46_IN_GG(T33, T41)
SELECT46_IN_GG(T33, .(T40, T41)) → U1_GG(T33, T40, T41, select46_in_gg(T33, T41))
SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → U3_GGA(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)
SELECT1_IN_GGA(T14, .(T55, T54), .(T55, T57)) → U4_GGA(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3) = select1_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
select1_out_gga(x1, x2, x3) = select1_out_gga(x3)
[] = []
U2_gga(x1, x2, x3, x4, x5) = U2_gga(x5)
select46_in_gg(x1, x2) = select46_in_gg(x1, x2)
select46_out_gg(x1, x2) = select46_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x2, x5)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x5)
SELECT46_IN_GG(x1, x2) = SELECT46_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:

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

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:

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)

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:

SELECT46_IN_GG(T33, .(T40, T41)) → SELECT46_IN_GG(T33, T41)
The graph contains the following edges 1 >= 1, 2 > 2

(15) TRUE

(16) Obligation:

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

SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)

The TRS R consists of the following rules:

select1_in_gga(T12, .(T12, T13), T13) → select1_out_gga(T12, .(T12, T13), T13)
select1_in_gga(T31, .(T23, .(T31, [])), []) → select1_out_gga(T31, .(T23, .(T31, [])), [])
select1_in_gga(T33, .(T23, .(T40, T41)), []) → U2_gga(T33, T23, T40, T41, select46_in_gg(T33, T41))
select46_in_gg(T31, .(T31, [])) → select46_out_gg(T31, .(T31, []))
select46_in_gg(T33, .(T40, T41)) → U1_gg(T33, T40, T41, select46_in_gg(T33, T41))
U1_gg(T33, T40, T41, select46_out_gg(T33, T41)) → select46_out_gg(T33, .(T40, T41))
U2_gga(T33, T23, T40, T41, select46_out_gg(T33, T41)) → select1_out_gga(T33, .(T23, .(T40, T41)), [])
select1_in_gga(T14, .(T47, T46), .(T47, T49)) → U3_gga(T14, T47, T46, T49, select1_in_gga(T14, T46, T49))
select1_in_gga(T14, .(T55, T54), .(T55, T57)) → U4_gga(T14, T55, T54, T57, select1_in_gga(T14, T54, T57))
U4_gga(T14, T55, T54, T57, select1_out_gga(T14, T54, T57)) → select1_out_gga(T14, .(T55, T54), .(T55, T57))
U3_gga(T14, T47, T46, T49, select1_out_gga(T14, T46, T49)) → select1_out_gga(T14, .(T47, T46), .(T47, T49))

(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:

SELECT1_IN_GGA(T14, .(T47, T46), .(T47, T49)) → SELECT1_IN_GGA(T14, T46, T49)

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

(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:

SELECT1_IN_GGA(T14, .(T47, T46)) → SELECT1_IN_GGA(T14, T46)

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:

SELECT1_IN_GGA(T14, .(T47, T46)) → SELECT1_IN_GGA(T14, T46)
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) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE