Left Termination proof of /tmp/tmpv4B4LP/gopher2.pl

Left Termination of the query pattern gopher_in_2(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 TRUE

(0) Obligation:

Clauses:

gopher(nil, nil).
gopher(X, cons(nil, T)) :- ','(no(empty(X)), ','(head(X, nil), tail(X, T))).
gopher(X, Y) :- ','(no(empty(X)), ','(head(X, H), ','(no(empty(H)), ','(head(H, U), ','(tail(H, V), ','(tail(X, W), gopher(cons(U, cons(V, W)), Y))))))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, X), X).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Queries:

gopher(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: gopher(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: gopher(T1, T2), SCOPE: 1, CLAUSE: 0 | gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: gopher(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: gopher(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: gopher(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(no(empty(T3)), ','(head(T3, nil), tail(T3, T5)))\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\nX9 is free\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(no(empty(T3)), ','(head(T3, nil), tail(T3, T5))), SCOPE: 2, CLAUSE: 8 | ','(no(empty(T3)), ','(head(T3, nil), tail(T3, T5))), SCOPE: 2, CLAUSE: 9 | ?_2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(empty(T6), ','(!_2, ','(failure(a), ','(head(T6, nil), tail(T6, T5))))) | ','(no(empty(T6)), ','(head(T6, nil), tail(T6, T5))), SCOPE: 2, CLAUSE: 9 | ?_2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(empty(T6), ','(!_2, ','(failure(a), ','(head(T6, nil), tail(T6, T5))))), SCOPE: 3, CLAUSE: 7 | ','(no(empty(T6)), ','(head(T6, nil), tail(T6, T5))), SCOPE: 2, CLAUSE: 9 | ?_2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(!_2, ','(failure(a), ','(head([], nil), tail([], T5)))) | ','(no(empty([])), ','(head([], nil), tail([], T5))), SCOPE: 2, CLAUSE: 9 | ?_2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty(T6)), ','(head(T6, nil), tail(T6, T5))), SCOPE: 2, CLAUSE: 9\n\nT6 is ground\nempty(T6) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(failure(a), ','(head([], nil), tail([], T5)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(failure(a), ','(head([], nil), tail([], T5))), SCOPE: 4, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T7, nil), tail(T7, T5))\n\nT7 is ground\nempty(T7) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(head(T7, nil), tail(T7, T5)), SCOPE: 5, CLAUSE: 3 | ','(head(T7, nil), tail(T7, T5)), SCOPE: 5, CLAUSE: 4\n\nT7 is ground\nempty(T7) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T7, nil), tail(T7, T5)), SCOPE: 5, CLAUSE: 4\n\nT7 is ground\nX16 is free; \nempty(T7) !=? empty([])\nhead(T7, nil) !=? head([], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: tail(.(nil, T9), T5)\n\nT9 is ground\nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\nX16 is free\nX19 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: tail(.(nil, T9), T5), SCOPE: 6, CLAUSE: 5 | tail(.(nil, T9), T5), SCOPE: 6, CLAUSE: 6\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: tail(.(nil, T9), T5), SCOPE: 6, CLAUSE: 6\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX23 is free\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(empty(T11)), ','(head(T11, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T11, X32), gopher(cons(X30, cons(X31, X32)), T13)))))))\n\nT11 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(empty(T11)), ','(head(T11, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T11, X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 8 | ','(no(empty(T11)), ','(head(T11, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T11, X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT11 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(empty(T14), ','(!_7, ','(failure(a), ','(head(T14, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T14, X32), gopher(cons(X30, cons(X31, X32)), T13))))))))) | ','(no(empty(T14)), ','(head(T14, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T14, X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT14 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(empty(T14), ','(!_7, ','(failure(a), ','(head(T14, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T14, X32), gopher(cons(X30, cons(X31, X32)), T13))))))))), SCOPE: 8, CLAUSE: 7 | ','(no(empty(T14)), ','(head(T14, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T14, X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nT14 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(!_7, ','(failure(a), ','(head([], X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail([], X32), gopher(cons(X30, cons(X31, X32)), T13)))))))) | ','(no(empty([])), ','(head([], X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail([], X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 9 | ?_7\n\nX29 is free\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(no(empty(T14)), ','(head(T14, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T14, X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 7, CLAUSE: 9\n\nT14 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nempty(T14) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(failure(a), ','(head([], X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail([], X32), gopher(cons(X30, cons(X31, X32)), T13)))))))\n\nX29 is free\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(failure(a), ','(head([], X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail([], X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 9, CLAUSE: 10\n\nX29 is free\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(head(T15, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T15, X32), gopher(cons(X30, cons(X31, X32)), T13))))))\n\nT15 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nempty(T15) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(head(T15, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T15, X32), gopher(cons(X30, cons(X31, X32)), T13)))))), SCOPE: 10, CLAUSE: 3 | ','(head(T15, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T15, X32), gopher(cons(X30, cons(X31, X32)), T13)))))), SCOPE: 10, CLAUSE: 4\n\nT15 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nempty(T15) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(head(T15, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(T15, X32), gopher(cons(X30, cons(X31, X32)), T13)))))), SCOPE: 10, CLAUSE: 4\n\nT15 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX38 is free; \nempty(T15) !=? empty([])\nhead(T15, X29) !=? head([], X38)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(no(empty(T16)), ','(head(T16, X30), ','(tail(T16, X31), ','(tail(.(T16, T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))))\n\nT16 is ground\nT17 is ground\nX29 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\nX29 is free\nX38 is free; \nX41 is free; \nX42 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(no(empty(T16)), ','(head(T16, X30), ','(tail(T16, X31), ','(tail(.(T16, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 8 | ','(no(empty(T16)), ','(head(T16, X30), ','(tail(T16, X31), ','(tail(.(T16, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 9 | ?_11\n\nT16 is ground\nT17 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(empty(T18), ','(!_11, ','(failure(a), ','(head(T18, X30), ','(tail(T18, X31), ','(tail(.(T18, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))))) | ','(no(empty(T18)), ','(head(T18, X30), ','(tail(T18, X31), ','(tail(.(T18, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 9 | ?_11\n\nT17 is ground\nT18 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(empty(T18), ','(!_11, ','(failure(a), ','(head(T18, X30), ','(tail(T18, X31), ','(tail(.(T18, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))))), SCOPE: 12, CLAUSE: 7 | ','(no(empty(T18)), ','(head(T18, X30), ','(tail(T18, X31), ','(tail(.(T18, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 9 | ?_11\n\nT17 is ground\nT18 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(!_11, ','(failure(a), ','(head([], X30), ','(tail([], X31), ','(tail(.([], T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))))) | ','(no(empty([])), ','(head([], X30), ','(tail([], X31), ','(tail(.([], T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 9 | ?_11\n\nT17 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(no(empty(T18)), ','(head(T18, X30), ','(tail(T18, X31), ','(tail(.(T18, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 11, CLAUSE: 9\n\nT17 is ground\nT18 is ground\nX30 is free; \nX31 is free; \nX32 is free; \nempty(T18) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(failure(a), ','(head([], X30), ','(tail([], X31), ','(tail(.([], T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))))\n\nT17 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(failure(a), ','(head([], X30), ','(tail([], X31), ','(tail(.([], T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))), SCOPE: 13, CLAUSE: 10\n\nT17 is ground\nX30 is free; \nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(T19, X30), ','(tail(T19, X31), ','(tail(.(T19, T17), X32), gopher(cons(X30, cons(X31, X32)), T13))))\n\nT17 is ground\nT19 is ground\nX30 is free; \nX31 is free; \nX32 is free; \nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(head(T19, X30), ','(tail(T19, X31), ','(tail(.(T19, T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))), SCOPE: 14, CLAUSE: 3 | ','(head(T19, X30), ','(tail(T19, X31), ','(tail(.(T19, T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))), SCOPE: 14, CLAUSE: 4\n\nT17 is ground\nT19 is ground\nX30 is free; \nX31 is free; \nX32 is free; \nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(T19, X30), ','(tail(T19, X31), ','(tail(.(T19, T17), X32), gopher(cons(X30, cons(X31, X32)), T13)))), SCOPE: 14, CLAUSE: 4\n\nT17 is ground\nT19 is ground\nX30 is free; \nX31 is free; \nX32 is free; \nX48 is free; \nempty(T19) !=? empty([])\nhead(T19, X30) !=? head([], X48)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(tail(.(T20, T21), X31), ','(tail(.(.(T20, T21), T17), X32), gopher(cons(T20, cons(X31, X32)), T13)))\n\nT17 is ground\nT20 is ground\nT21 is ground\nX30 is free; \nX31 is free; \nX32 is free; \nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\nX30 is free\nX48 is free; \nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(tail(.(T20, T21), X31), ','(tail(.(.(T20, T21), T17), X32), gopher(cons(T20, cons(X31, X32)), T13))), SCOPE: 15, CLAUSE: 5 | ','(tail(.(T20, T21), X31), ','(tail(.(.(T20, T21), T17), X32), gopher(cons(T20, cons(X31, X32)), T13))), SCOPE: 15, CLAUSE: 6\n\nT17 is ground\nT20 is ground\nT21 is ground\nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(tail(.(T20, T21), X31), ','(tail(.(.(T20, T21), T17), X32), gopher(cons(T20, cons(X31, X32)), T13))), SCOPE: 15, CLAUSE: 6\n\nT17 is ground\nT20 is ground\nT21 is ground\nX31 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(tail(.(.(T22, T23), T17), X32), gopher(cons(T22, cons(T23, X32)), T13))\n\nT17 is ground\nT22 is ground\nT23 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail(.(.(T22, T23), T17), X32), gopher(cons(T22, cons(T23, X32)), T13)), SCOPE: 16, CLAUSE: 5 | ','(tail(.(.(T22, T23), T17), X32), gopher(cons(T22, cons(T23, X32)), T13)), SCOPE: 16, CLAUSE: 6\n\nT17 is ground\nT22 is ground\nT23 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(tail(.(.(T22, T23), T17), X32), gopher(cons(T22, cons(T23, X32)), T13)), SCOPE: 16, CLAUSE: 6\n\nT17 is ground\nT22 is ground\nT23 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: gopher(cons(T24, cons(T25, T26)), T13)\n\nT24 is ground\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 63 [arrowhead = none ];
63 -> 2;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 64 [arrowhead = none ];
64 -> 3;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 65 [arrowhead = none ];
65 -> 4;

66 [label="EVAL with clause\ngopher(nil, nil).\nand substitution\nT1 -> nil\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 66 [arrowhead = none ];
66 -> 5;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 67 [arrowhead = none ];
67 -> 6;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 68 [arrowhead = none ];
68 -> 8;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 69 [arrowhead = none ];
69 -> 9;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 70 [arrowhead = none ];
70 -> 7;

71 [label="EVAL with clause\ngopher(X9, cons(nil, X10)) :- ','(no(empty(X9)), ','(head(X9, nil), tail(X9, X10))).\nand substitution\nT1 -> T3\nX9 -> T3\nX10 -> T5\nT2 -> cons(nil, T5)\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 71 [arrowhead = none ];
71 -> 10;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 72 [arrowhead = none ];
72 -> 11;

73 [label="EVAL with clause\ngopher(X27, X28) :- ','(no(empty(X27)), ','(head(X27, X29), ','(no(empty(X29)), ','(head(X29, X30), ','(tail(X29, X31), ','(tail(X27, X32), gopher(cons(X30, cons(X31, X32)), X28))))))).\nand substitution\nT1 -> T11\nX27 -> T11\nT2 -> T13\nX28 -> T13\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 74 [arrowhead = none ];
74 -> 12;

75 [label="EVAL with clause\nno(X12) :- ','(X12, ','(!, failure(a))).\nand substitution\nT3 -> T6\nX12 -> empty(T6)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 75 [arrowhead = none ];
75 -> 13;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 76 [arrowhead = none ];
76 -> 14;

77 [label="EVAL with clause\nempty([]).\nand substitution\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 77 [arrowhead = none ];
77 -> 15;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 78 [arrowhead = none ];
78 -> 16;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="EVAL with clause\nno(X14).\nand substitution\nT6 -> T7\nX14 -> empty(T7)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 80 [arrowhead = none ];
80 -> 20;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
18 -> 82 [arrowhead = none ];
82 -> 19;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 83 [arrowhead = none ];
83 -> 21;

84 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T7), empty([]))", fontsize=14, style = filled, fillcolor = white];
21 -> 84 [arrowhead = none ];
84 -> 22;

85 [label="EVAL with clause\nhead(.(X19, X20), X19).\nand substitution\nX19 -> nil\nX20 -> T9\nT7 -> .(nil, T9)\nT8 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 86 [arrowhead = none ];
86 -> 24;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
25 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="EVAL with clause\ntail(.(X23, X24), X24).\nand substitution\nX23 -> nil\nT9 -> T10\nX24 -> T10\nT5 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 89 [arrowhead = none ];
89 -> 27;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 90 [arrowhead = none ];
90 -> 28;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 91 [arrowhead = none ];
91 -> 29;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="EVAL with clause\nno(X34) :- ','(X34, ','(!, failure(a))).\nand substitution\nT11 -> T14\nX34 -> empty(T14)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 94 [arrowhead = none ];
94 -> 33;

95 [label="EVAL with clause\nempty([]).\nand substitution\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 95 [arrowhead = none ];
95 -> 34;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 96 [arrowhead = none ];
96 -> 35;

97 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 97 [arrowhead = none ];
97 -> 36;

98 [label="EVAL with clause\nno(X36).\nand substitution\nT14 -> T15\nX36 -> empty(T15)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 98 [arrowhead = none ];
98 -> 39;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 99 [arrowhead = none ];
99 -> 37;

100 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
37 -> 100 [arrowhead = none ];
100 -> 38;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 101 [arrowhead = none ];
101 -> 40;

102 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T15), empty([]))", fontsize=14, style = filled, fillcolor = white];
40 -> 102 [arrowhead = none ];
102 -> 41;

103 [label="EVAL with clause\nhead(.(X41, X42), X41).\nand substitution\nX41 -> T16\nX42 -> T17\nT15 -> .(T16, T17)\nX29 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 103 [arrowhead = none ];
103 -> 42;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 104 [arrowhead = none ];
104 -> 43;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 105 [arrowhead = none ];
105 -> 44;

106 [label="EVAL with clause\nno(X44) :- ','(X44, ','(!, failure(a))).\nand substitution\nT16 -> T18\nX44 -> empty(T18)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 106 [arrowhead = none ];
106 -> 45;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 107 [arrowhead = none ];
107 -> 46;

108 [label="EVAL with clause\nempty([]).\nand substitution\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 108 [arrowhead = none ];
108 -> 47;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 109 [arrowhead = none ];
109 -> 48;

110 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 110 [arrowhead = none ];
110 -> 49;

111 [label="EVAL with clause\nno(X46).\nand substitution\nT18 -> T19\nX46 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 111 [arrowhead = none ];
111 -> 52;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 112 [arrowhead = none ];
112 -> 50;

113 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
50 -> 113 [arrowhead = none ];
113 -> 51;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 114 [arrowhead = none ];
114 -> 53;

115 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T19), empty([]))", fontsize=14, style = filled, fillcolor = white];
53 -> 115 [arrowhead = none ];
115 -> 54;

116 [label="EVAL with clause\nhead(.(X51, X52), X51).\nand substitution\nX51 -> T20\nX52 -> T21\nT19 -> .(T20, T21)\nX30 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 116 [arrowhead = none ];
116 -> 55;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 117 [arrowhead = none ];
117 -> 56;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 118 [arrowhead = none ];
118 -> 57;

119 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 119 [arrowhead = none ];
119 -> 58;

120 [label="EVAL with clause\ntail(.(X55, X56), X56).\nand substitution\nT20 -> T22\nX55 -> T22\nT21 -> T23\nX56 -> T23\nX31 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 120 [arrowhead = none ];
120 -> 59;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 121 [arrowhead = none ];
121 -> 60;

122 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
60 -> 122 [arrowhead = none ];
122 -> 61;

123 [label="EVAL with clause\ntail(.(X59, X60), X60).\nand substitution\nT22 -> T24\nT23 -> T25\nX59 -> .(T24, T25)\nT17 -> T26\nX60 -> T26\nX32 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 123 [arrowhead = none ];
123 -> 62;

124 [label="INSTANCE with matching:\nT1 -> cons(T24, cons(T25, T26))\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 124 [arrowhead = none , style=dashed];
124 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

gopher1(nil, nil).
gopher1(.(nil, T10), cons(nil, T10)).
gopher1(.(.(T24, T25), T26), T13) :- gopher1(cons(T24, cons(T25, T26)), T13).

Queries:

gopher1(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:
gopher1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(.(nil, T10), cons(nil, T10)) → gopher1_out_ga(.(nil, T10), cons(nil, T10))
gopher1_in_ga(.(.(T24, T25), T26), T13) → U1_ga(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
U1_ga(T24, T25, T26, T13, gopher1_out_ga(cons(T24, cons(T25, T26)), T13)) → gopher1_out_ga(.(.(T24, T25), T26), T13)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(.(nil, T10), cons(nil, T10)) → gopher1_out_ga(.(nil, T10), cons(nil, T10))
gopher1_in_ga(.(.(T24, T25), T26), T13) → U1_ga(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
U1_ga(T24, T25, T26, T13, gopher1_out_ga(cons(T24, cons(T25, T26)), T13)) → gopher1_out_ga(.(.(T24, T25), T26), T13)

(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:

GOPHER1_IN_GA(.(.(T24, T25), T26), T13) → U1_GA(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
GOPHER1_IN_GA(.(.(T24, T25), T26), T13) → GOPHER1_IN_GA(cons(T24, cons(T25, T26)), T13)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(.(nil, T10), cons(nil, T10)) → gopher1_out_ga(.(nil, T10), cons(nil, T10))
gopher1_in_ga(.(.(T24, T25), T26), T13) → U1_ga(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
U1_ga(T24, T25, T26, T13, gopher1_out_ga(cons(T24, cons(T25, T26)), T13)) → gopher1_out_ga(.(.(T24, T25), T26), T13)

The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2) = gopher1_in_ga(x1)
nil = nil
gopher1_out_ga(x1, x2) = gopher1_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
cons(x1, x2) = cons(x1, x2)
GOPHER1_IN_GA(x1, x2) = GOPHER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)

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

(6) Obligation:

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

GOPHER1_IN_GA(.(.(T24, T25), T26), T13) → U1_GA(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
GOPHER1_IN_GA(.(.(T24, T25), T26), T13) → GOPHER1_IN_GA(cons(T24, cons(T25, T26)), T13)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga(.(nil, T10), cons(nil, T10)) → gopher1_out_ga(.(nil, T10), cons(nil, T10))
gopher1_in_ga(.(.(T24, T25), T26), T13) → U1_ga(T24, T25, T26, T13, gopher1_in_ga(cons(T24, cons(T25, T26)), T13))
U1_ga(T24, T25, T26, T13, gopher1_out_ga(cons(T24, cons(T25, T26)), T13)) → gopher1_out_ga(.(.(T24, T25), T26), T13)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 0 SCCs with 2 less nodes.

(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) TRUE