Left Termination proof of /tmp/tmpVIbIj7/gopher1.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, L) :- ','(!, eq(L, nil)).
gopher(X, Y) :- ','(head(X, nil), ','(!, ','(tail(X, T), eq(Y, cons(nil, T))))).
gopher(X, Y) :- ','(head(X, 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).
eq(X, X).

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 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T4, nil)) | gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: gopher(T1, T2), SCOPE: 1, CLAUSE: 1 | gopher(T1, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground\nX5 is free; \ngopher(T1, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T4, nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T4, nil), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T6, nil), ','(!_1, ','(tail(T6, X12), eq(T8, cons(nil, X12))))) | gopher(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX5 is free; \nX12 is free; \ngopher(T6, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T6, nil), ','(!_1, ','(tail(T6, X12), eq(T8, cons(nil, X12))))), SCOPE: 3, CLAUSE: 3 | ','(head(T6, nil), ','(!_1, ','(tail(T6, X12), eq(T8, cons(nil, X12))))), SCOPE: 3, CLAUSE: 4 | gopher(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX5 is free; \nX12 is free; \ngopher(T6, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_1, ','(tail([], X12), eq(T8, cons(nil, X12)))) | ','(head([], nil), ','(!_1, ','(tail([], X12), eq(T8, cons(nil, X12))))), SCOPE: 3, CLAUSE: 4 | gopher([], T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nX5 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T6, nil), ','(!_1, ','(tail(T6, X12), eq(T8, cons(nil, X12))))), SCOPE: 3, CLAUSE: 4 | gopher(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX5 is free; \nX12 is free; \nX14 is free; \ngopher(T6, T2) !=? gopher(nil, X5)\nhead(T6, nil) !=? head([], X14)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail([], X12), eq(T8, cons(nil, X12)))\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 5 | ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: eq(T8, cons(nil, [])) | ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: eq(T8, cons(nil, [])), SCOPE: 5, CLAUSE: 7 | ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true | ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free\nX16 is free; \neq(T8, cons(nil, [])) !=? eq(X16, X16)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(tail([], X12), eq(T8, cons(nil, X12))), SCOPE: 4, CLAUSE: 6\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(!_1, ','(tail(.(nil, T11), X12), eq(T8, cons(nil, X12)))) | gopher(.(nil, T11), T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT11 is ground\nX5 is free; \nX12 is free; \nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: gopher(T6, T2), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX5 is free; \nX14 is free; \nX23 is free; \nX24 is free; \ngopher(T6, T2) !=? gopher(nil, X5)\nhead(T6, nil) !=? head([], X14)\nhead(T6, nil) !=? head(.(X23, X24), X23)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(tail(.(nil, T11), X12), eq(T8, cons(nil, X12)))\n\nT11 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(.(nil, T11), X12), eq(T8, cons(nil, X12))), SCOPE: 6, CLAUSE: 5 | ','(tail(.(nil, T11), X12), eq(T8, cons(nil, X12))), SCOPE: 6, CLAUSE: 6\n\nT11 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(nil, T11), X12), eq(T8, cons(nil, X12))), SCOPE: 6, CLAUSE: 6\n\nT11 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: eq(T8, cons(nil, T12))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: eq(T8, cons(nil, T12)), SCOPE: 7, CLAUSE: 7\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(head(T15, X35), ','(head(X35, X36), ','(tail(X35, X37), ','(tail(T15, X38), gopher(cons(X36, cons(X37, X38)), T17)))))\n\nT15 is ground\nX5 is free; \nX14 is free; \nX23 is free; \nX24 is free; \nX35 is free; \nX36 is free; \nX37 is free; \nX38 is free; \ngopher(T15, T2) !=? gopher(nil, X5)\nhead(T15, nil) !=? head([], X14)\nhead(T15, nil) !=? head(.(X23, X24), X23)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(head(T15, X35), ','(head(X35, X36), ','(tail(X35, X37), ','(tail(T15, X38), gopher(cons(X36, cons(X37, X38)), T17))))), SCOPE: 8, CLAUSE: 3 | ','(head(T15, X35), ','(head(X35, X36), ','(tail(X35, X37), ','(tail(T15, X38), gopher(cons(X36, cons(X37, X38)), T17))))), SCOPE: 8, CLAUSE: 4\n\nT15 is ground\nX5 is free; \nX14 is free; \nX23 is free; \nX24 is free; \nX35 is free; \nX36 is free; \nX37 is free; \nX38 is free; \ngopher(T15, T2) !=? gopher(nil, X5)\nhead(T15, nil) !=? head([], X14)\nhead(T15, nil) !=? head(.(X23, X24), X23)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(head(T15, X35), ','(head(X35, X36), ','(tail(X35, X37), ','(tail(T15, X38), gopher(cons(X36, cons(X37, X38)), T17))))), SCOPE: 8, CLAUSE: 4\n\nT15 is ground\nX5 is free; \nX14 is free; \nX23 is free; \nX24 is free; \nX35 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nX40 is free; \ngopher(T15, T2) !=? gopher(nil, X5)\nhead(T15, nil) !=? head([], X14)\nhead(T15, nil) !=? head(.(X23, X24), X23)\nhead(T15, X35) !=? head([], X40)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(head(T18, X36), ','(tail(T18, X37), ','(tail(.(T18, T19), X38), gopher(cons(X36, cons(X37, X38)), T17))))\n\nT18 is ground\nT19 is ground\nX5 is free; \nX14 is free; \nX23 is free; \nX24 is free; \nX35 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nX40 is free; \nhead(.(T18, T19), nil) !=? head(.(X23, X24), X23)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX5 is free\nX14 is free; \nX23 is free; \nX24 is free; \nX35 is free; \nX40 is free; \nX43 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(T18, X36), ','(tail(T18, X37), ','(tail(.(T18, T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 3 | ','(head(T18, X36), ','(tail(T18, X37), ','(tail(.(T18, T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT18 is ground\nT19 is ground\nX23 is free; \nX24 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nhead(.(T18, T19), nil) !=? head(.(X23, X24), X23)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))) | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT19 is ground\nX23 is free; \nX24 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(head(T18, X36), ','(tail(T18, X37), ','(tail(.(T18, T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT18 is ground\nT19 is ground\nX23 is free; \nX24 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nX46 is free; \nhead(.(T18, T19), nil) !=? head(.(X23, X24), X23)\nhead(T18, X36) !=? head([], X46)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 5 | ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(tail(.([], T19), X38), gopher(cons(X47, cons([], X38)), T17)) | ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail(.([], T19), X38), gopher(cons(X47, cons([], X38)), T17)), SCOPE: 11, CLAUSE: 5 | ','(tail(.([], T19), X38), gopher(cons(X47, cons([], X38)), T17)), SCOPE: 11, CLAUSE: 6 | ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(.([], T19), X38), gopher(cons(X47, cons([], X38)), T17)), SCOPE: 11, CLAUSE: 6 | ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T19), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: gopher(cons(X47, cons([], T20)), T17) | ','(tail([], X37), ','(tail(.([], T20), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T20), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT20 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: gopher(cons(X47, cons([], T20)), T17)\n\nT20 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail([], X37), ','(tail(.([], T20), X38), gopher(cons(X47, cons(X37, X38)), T17))), SCOPE: 10, CLAUSE: 6 | ','(head([], X36), ','(tail([], X37), ','(tail(.([], T20), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT20 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: gopher(cons(X47, cons([], T20)), T17), SCOPE: 12, CLAUSE: 0 | gopher(cons(X47, cons([], T20)), T17), SCOPE: 12, CLAUSE: 1 | gopher(cons(X47, cons([], T20)), T17), SCOPE: 12, CLAUSE: 2 | ?_12\n\nT20 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: gopher(cons(X47, cons([], T20)), T17), SCOPE: 12, CLAUSE: 1 | gopher(cons(X47, cons([], T20)), T17), SCOPE: 12, CLAUSE: 2 | ?_12\n\nT20 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(head(cons(X58, cons([], T21)), nil), ','(!_12, ','(tail(cons(X58, cons([], T21)), X57), eq(T23, cons(nil, X57))))) | gopher(cons(X47, cons([], T21)), T17), SCOPE: 12, CLAUSE: 2 | ?_12\n\nT21 is ground\nX47 is free; \nX58 is free; \nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(head(cons(X58, cons([], T21)), nil), ','(!_12, ','(tail(cons(X58, cons([], T21)), X57), eq(T23, cons(nil, X57))))), SCOPE: 13, CLAUSE: 3 | ','(head(cons(X58, cons([], T21)), nil), ','(!_12, ','(tail(cons(X58, cons([], T21)), X57), eq(T23, cons(nil, X57))))), SCOPE: 13, CLAUSE: 4 | gopher(cons(X47, cons([], T21)), T17), SCOPE: 12, CLAUSE: 2 | ?_12\n\nT21 is ground\nX47 is free; \nX58 is free; \nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(head(cons(X58, cons([], T21)), nil), ','(!_12, ','(tail(cons(X58, cons([], T21)), X57), eq(T23, cons(nil, X57))))), SCOPE: 13, CLAUSE: 4 | gopher(cons(X47, cons([], T21)), T17), SCOPE: 12, CLAUSE: 2 | ?_12\n\nT21 is ground\nX47 is free; \nX58 is free; \nX57 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: gopher(cons(X47, cons([], T21)), T17), SCOPE: 12, CLAUSE: 2\n\nT21 is ground\nX47 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(cons(X70, cons([], T24)), X66), ','(head(X66, X67), ','(tail(X66, X68), ','(tail(cons(X70, cons([], T24)), X69), gopher(cons(X67, cons(X68, X69)), T26)))))\n\nT24 is ground\nX70 is free; \nX66 is free; \nX67 is free; \nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(head(cons(X70, cons([], T24)), X66), ','(head(X66, X67), ','(tail(X66, X68), ','(tail(cons(X70, cons([], T24)), X69), gopher(cons(X67, cons(X68, X69)), T26))))), SCOPE: 14, CLAUSE: 3 | ','(head(cons(X70, cons([], T24)), X66), ','(head(X66, X67), ','(tail(X66, X68), ','(tail(cons(X70, cons([], T24)), X69), gopher(cons(X67, cons(X68, X69)), T26))))), SCOPE: 14, CLAUSE: 4\n\nT24 is ground\nX70 is free; \nX66 is free; \nX67 is free; \nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(head(cons(X70, cons([], T24)), X66), ','(head(X66, X67), ','(tail(X66, X68), ','(tail(cons(X70, cons([], T24)), X69), gopher(cons(X67, cons(X68, X69)), T26))))), SCOPE: 14, CLAUSE: 4\n\nT24 is ground\nX70 is free; \nX66 is free; \nX67 is free; \nX68 is free; \nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(head([], X36), ','(tail([], X37), ','(tail(.([], T20), X38), gopher(cons(X36, cons(X37, X38)), T17)))), SCOPE: 9, CLAUSE: 4\n\nT20 is ground\nX36 is free; \nX37 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(tail(.(T27, T28), X37), ','(tail(.(.(T27, T28), T19), X38), gopher(cons(T27, cons(X37, X38)), T17)))\n\nT19 is ground\nT27 is ground\nT28 is ground\nX23 is free; \nX24 is free; \nX36 is free; \nX37 is free; \nX38 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\nX23 is free\nX24 is free; \nX36 is free; \nX46 is free; \nX80 is free; \nX81 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(tail(.(T27, T28), X37), ','(tail(.(.(T27, T28), T19), X38), gopher(cons(T27, cons(X37, X38)), T17))), SCOPE: 15, CLAUSE: 5 | ','(tail(.(T27, T28), X37), ','(tail(.(.(T27, T28), T19), X38), gopher(cons(T27, cons(X37, X38)), T17))), SCOPE: 15, CLAUSE: 6\n\nT19 is ground\nT27 is ground\nT28 is ground\nX37 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(tail(.(T27, T28), X37), ','(tail(.(.(T27, T28), T19), X38), gopher(cons(T27, cons(X37, X38)), T17))), SCOPE: 15, CLAUSE: 6\n\nT19 is ground\nT27 is ground\nT28 is ground\nX37 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(.(.(T29, T30), T19), X38), gopher(cons(T29, cons(T30, X38)), T17))\n\nT19 is ground\nT29 is ground\nT30 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(tail(.(.(T29, T30), T19), X38), gopher(cons(T29, cons(T30, X38)), T17)), SCOPE: 16, CLAUSE: 5 | ','(tail(.(.(T29, T30), T19), X38), gopher(cons(T29, cons(T30, X38)), T17)), SCOPE: 16, CLAUSE: 6\n\nT19 is ground\nT29 is ground\nT30 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(tail(.(.(T29, T30), T19), X38), gopher(cons(T29, cons(T30, X38)), T17)), SCOPE: 16, CLAUSE: 6\n\nT19 is ground\nT29 is ground\nT30 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gopher(cons(T31, cons(T32, T33)), T17)\n\nT31 is ground\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 68 [arrowhead = none ];
68 -> 2;

69 [label="EVAL with clause\ngopher(nil, X5) :- ','(!, eq(X5, nil)).\nand substitution\nT1 -> nil\nT2 -> T4\nX5 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 3;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 70 [arrowhead = none ];
70 -> 4;

71 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 71 [arrowhead = none ];
71 -> 5;

72 [label="EVAL with clause\ngopher(X10, X11) :- ','(head(X10, nil), ','(!, ','(tail(X10, X12), eq(X11, cons(nil, X12))))).\nand substitution\nT1 -> T6\nX10 -> T6\nT2 -> T8\nX11 -> T8\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 72 [arrowhead = none ];
72 -> 10;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="EVAL with clause\neq(X7, X7).\nand substitution\nT4 -> nil\nX7 -> nil\nT5 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 76 [arrowhead = none ];
76 -> 9;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="EVAL with clause\nhead([], X14).\nand substitution\nT6 -> []\nX14 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 80 [arrowhead = none ];
80 -> 14;

81 [label="EVAL with clause\nhead(.(X23, X24), X23).\nand substitution\nX23 -> nil\nX24 -> T11\nT6 -> .(nil, T11)\nT10 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 81 [arrowhead = none ];
81 -> 23;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 82 [arrowhead = none ];
82 -> 24;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 83 [arrowhead = none ];
83 -> 15;

84 [label="EVAL with clause\ntail([], []).\nand substitution\nX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 84 [arrowhead = none ];
84 -> 16;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 85 [arrowhead = none ];
85 -> 17;

86 [label="EVAL with clause\neq(X16, X16).\nand substitution\nT8 -> cons(nil, [])\nX16 -> cons(nil, [])\nT9 -> cons(nil, [])", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 87 [arrowhead = none ];
87 -> 19;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 88 [arrowhead = none ];
88 -> 20;

89 [label="BACKTRACK\nfor clause: tail(.(X3, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 89 [arrowhead = none ];
89 -> 22;

90 [label="BACKTRACK\nfor clause: tail(.(X3, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 90 [arrowhead = none ];
90 -> 21;

91 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 91 [arrowhead = none ];
91 -> 25;

92 [label="EVAL with clause\ngopher(X33, X34) :- ','(head(X33, X35), ','(head(X35, X36), ','(tail(X35, X37), ','(tail(X33, X38), gopher(cons(X36, cons(X37, X38)), X34))))).\nand substitution\nT6 -> T15\nX33 -> T15\nT2 -> T17\nX34 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 92 [arrowhead = none ];
92 -> 33;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 93 [arrowhead = none ];
93 -> 26;

94 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
26 -> 94 [arrowhead = none ];
94 -> 27;

95 [label="EVAL with clause\ntail(.(X27, X28), X28).\nand substitution\nX27 -> nil\nT11 -> T12\nX28 -> T12\nX12 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 95 [arrowhead = none ];
95 -> 28;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 96 [arrowhead = none ];
96 -> 29;

97 [label="EVAL with clause\neq(X30, X30).\nand substitution\nT8 -> cons(nil, T14)\nX30 -> cons(nil, T14)\nT12 -> T14\nT13 -> cons(nil, T14)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 97 [arrowhead = none ];
97 -> 30;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 98 [arrowhead = none ];
98 -> 31;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 99 [arrowhead = none ];
99 -> 32;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 100 [arrowhead = none ];
100 -> 34;

101 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (head(T15, nil), head([], X14))", fontsize=14, style = filled, fillcolor = white];
34 -> 101 [arrowhead = none ];
101 -> 35;

102 [label="EVAL with clause\nhead(.(X43, X44), X43).\nand substitution\nX43 -> T18\nX44 -> T19\nT15 -> .(T18, T19)\nX35 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 102 [arrowhead = none ];
102 -> 36;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 103 [arrowhead = none ];
103 -> 37;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 104 [arrowhead = none ];
104 -> 38;

105 [label="EVAL with clause\nhead([], X46).\nand substitution\nT18 -> []\nX36 -> X47\nX46 -> X47", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 105 [arrowhead = none ];
105 -> 39;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 106 [arrowhead = none ];
106 -> 40;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 107 [arrowhead = none ];
107 -> 41;

108 [label="EVAL with clause\nhead(.(X80, X81), X80).\nand substitution\nX80 -> T27\nX81 -> T28\nT18 -> .(T27, T28)\nX36 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 108 [arrowhead = none ];
108 -> 60;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 109 [arrowhead = none ];
109 -> 61;

110 [label="EVAL with clause\ntail([], []).\nand substitution\nX37 -> []", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 110 [arrowhead = none ];
110 -> 42;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 111 [arrowhead = none ];
111 -> 43;

112 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
43 -> 112 [arrowhead = none ];
112 -> 44;

113 [label="EVAL with clause\ntail(.(X50, X51), X51).\nand substitution\nX50 -> []\nT19 -> T20\nX51 -> T20\nX38 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 113 [arrowhead = none ];
113 -> 45;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 114 [arrowhead = none ];
114 -> 46;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 115 [arrowhead = none ];
115 -> 47;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 116 [arrowhead = none ];
116 -> 48;

117 [label="BACKTRACK\nfor clause: tail(.(X3, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 117 [arrowhead = none ];
117 -> 58;

118 [label="BACKTRACK\nfor clause: gopher(nil, L) :- ','(!, eq(L, nil))because of non-unification", fontsize=14, style = filled, fillcolor = white];
48 -> 118 [arrowhead = none ];
118 -> 49;

119 [label="EVAL with clause\ngopher(X55, X56) :- ','(head(X55, nil), ','(!, ','(tail(X55, X57), eq(X56, cons(nil, X57))))).\nand substitution\nX47 -> X58\nT20 -> T21\nX55 -> cons(X58, cons([], T21))\nT17 -> T23\nX56 -> T23\nT22 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 119 [arrowhead = none ];
119 -> 50;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 120 [arrowhead = none ];
120 -> 51;

121 [label="BACKTRACK\nfor clause: head([], X1)because of non-unification", fontsize=14, style = filled, fillcolor = white];
51 -> 121 [arrowhead = none ];
121 -> 52;

122 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 122 [arrowhead = none ];
122 -> 53;

123 [label="EVAL with clause\ngopher(X64, X65) :- ','(head(X64, X66), ','(head(X66, X67), ','(tail(X66, X68), ','(tail(X64, X69), gopher(cons(X67, cons(X68, X69)), X65))))).\nand substitution\nX47 -> X70\nT21 -> T24\nX64 -> cons(X70, cons([], T24))\nT17 -> T26\nX65 -> T26\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 123 [arrowhead = none ];
123 -> 54;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 124 [arrowhead = none ];
124 -> 55;

125 [label="BACKTRACK\nfor clause: head([], X1)because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 125 [arrowhead = none ];
125 -> 56;

126 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
56 -> 126 [arrowhead = none ];
126 -> 57;

127 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
58 -> 127 [arrowhead = none ];
127 -> 59;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 128 [arrowhead = none ];
128 -> 62;

129 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 129 [arrowhead = none ];
129 -> 63;

130 [label="EVAL with clause\ntail(.(X84, X85), X85).\nand substitution\nT27 -> T29\nX84 -> T29\nT28 -> T30\nX85 -> T30\nX37 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 130 [arrowhead = none ];
130 -> 64;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 131 [arrowhead = none ];
131 -> 65;

132 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 132 [arrowhead = none ];
132 -> 66;

133 [label="EVAL with clause\ntail(.(X88, X89), X89).\nand substitution\nT29 -> T31\nT30 -> T32\nX88 -> .(T31, T32)\nT19 -> T33\nX89 -> T33\nX38 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 133 [arrowhead = none ];
133 -> 67;

134 [label="INSTANCE with matching:\nT1 -> cons(T31, cons(T32, T33))\nT2 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 134 [arrowhead = none , style=dashed];
134 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

gopher1(nil, nil).
gopher1([], cons(nil, [])).
gopher1(.(nil, T14), cons(nil, T14)).
gopher1(.(.(T31, T32), T33), T17) :- gopher1(cons(T31, cons(T32, T33)), T17).

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([], cons(nil, [])) → gopher1_out_ga([], cons(nil, []))
gopher1_in_ga(.(nil, T14), cons(nil, T14)) → gopher1_out_ga(.(nil, T14), cons(nil, T14))
gopher1_in_ga(.(.(T31, T32), T33), T17) → U1_ga(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
U1_ga(T31, T32, T33, T17, gopher1_out_ga(cons(T31, cons(T32, T33)), T17)) → gopher1_out_ga(.(.(T31, T32), T33), T17)

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([], cons(nil, [])) → gopher1_out_ga([], cons(nil, []))
gopher1_in_ga(.(nil, T14), cons(nil, T14)) → gopher1_out_ga(.(nil, T14), cons(nil, T14))
gopher1_in_ga(.(.(T31, T32), T33), T17) → U1_ga(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
U1_ga(T31, T32, T33, T17, gopher1_out_ga(cons(T31, cons(T32, T33)), T17)) → gopher1_out_ga(.(.(T31, T32), T33), T17)

(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(.(.(T31, T32), T33), T17) → U1_GA(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
GOPHER1_IN_GA(.(.(T31, T32), T33), T17) → GOPHER1_IN_GA(cons(T31, cons(T32, T33)), T17)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga([], cons(nil, [])) → gopher1_out_ga([], cons(nil, []))
gopher1_in_ga(.(nil, T14), cons(nil, T14)) → gopher1_out_ga(.(nil, T14), cons(nil, T14))
gopher1_in_ga(.(.(T31, T32), T33), T17) → U1_ga(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
U1_ga(T31, T32, T33, T17, gopher1_out_ga(cons(T31, cons(T32, T33)), T17)) → gopher1_out_ga(.(.(T31, T32), T33), T17)

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(x1, x2)
[] = []
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x3, 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(x1, x2, x3, 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(.(.(T31, T32), T33), T17) → U1_GA(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
GOPHER1_IN_GA(.(.(T31, T32), T33), T17) → GOPHER1_IN_GA(cons(T31, cons(T32, T33)), T17)

The TRS R consists of the following rules:

gopher1_in_ga(nil, nil) → gopher1_out_ga(nil, nil)
gopher1_in_ga([], cons(nil, [])) → gopher1_out_ga([], cons(nil, []))
gopher1_in_ga(.(nil, T14), cons(nil, T14)) → gopher1_out_ga(.(nil, T14), cons(nil, T14))
gopher1_in_ga(.(.(T31, T32), T33), T17) → U1_ga(T31, T32, T33, T17, gopher1_in_ga(cons(T31, cons(T32, T33)), T17))
U1_ga(T31, T32, T33, T17, gopher1_out_ga(cons(T31, cons(T32, T33)), T17)) → gopher1_out_ga(.(.(T31, T32), T33), T17)

(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