Left Termination proof of /tmp/tmp

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 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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: ','(!_1, eq(T5, nil)) | gopher(nil, T2), SCOPE: 1, CLAUSE: 1 | gopher(nil, T2), SCOPE: 1, CLAUSE: 2\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\n\nT1 is ground\ngopher(T1, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T5, nil)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T5, 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\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T11, nil), ','(!_1, ','(tail(T11, X15), eq(T13, cons(nil, X15))))) | gopher(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX15 is free; \ngopher(T11, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T11, nil), ','(!_1, ','(tail(T11, X15), eq(T13, cons(nil, X15))))), SCOPE: 3, CLAUSE: 3 | ','(head(T11, nil), ','(!_1, ','(tail(T11, X15), eq(T13, cons(nil, X15))))), SCOPE: 3, CLAUSE: 4 | ?_3 | gopher(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX15 is free; \ngopher(T11, T2) !=? gopher(nil, X5)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_1, ','(tail([], X15), eq(T13, cons(nil, X15)))) | ','(head([], nil), ','(!_1, ','(tail([], X15), eq(T13, cons(nil, X15))))), SCOPE: 3, CLAUSE: 4 | ?_3 | gopher([], T2), SCOPE: 1, CLAUSE: 2\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T11, nil), ','(!_1, ','(tail(T11, X15), eq(T13, cons(nil, X15))))), SCOPE: 3, CLAUSE: 4 | ?_3 | gopher(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\nX15 is free; \ngopher(T11, T2) !=? gopher(nil, X5)\nhead(T11, nil) !=? head([], X18)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail([], X15), eq(T13, cons(nil, X15)))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail([], X15), eq(T13, cons(nil, X15))), SCOPE: 4, CLAUSE: 5 | ','(tail([], X15), eq(T13, cons(nil, X15))), SCOPE: 4, CLAUSE: 6\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail([], X15), eq(T13, cons(nil, X15))), SCOPE: 4, CLAUSE: 5\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail([], X15), eq(T13, cons(nil, X15))), SCOPE: 4, CLAUSE: 6\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: eq(T13, cons(nil, []))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: eq(T13, cons(nil, [])), SCOPE: 5, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_1, ','(tail(.(nil, T22), X15), eq(T13, cons(nil, X15)))) | ?_3 | gopher(.(nil, T22), T2), SCOPE: 1, CLAUSE: 2\n\nT22 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_3 | gopher(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\ngopher(T11, T2) !=? gopher(nil, X5)\nhead(T11, nil) !=? head([], X18)\nhead(T11, nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(.(nil, T22), X15), eq(T13, cons(nil, X15)))\n\nT22 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(nil, T22), X15), eq(T13, cons(nil, X15))), SCOPE: 6, CLAUSE: 5 | ','(tail(.(nil, T22), X15), eq(T13, cons(nil, X15))), SCOPE: 6, CLAUSE: 6\n\nT22 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(.(nil, T22), X15), eq(T13, cons(nil, X15))), SCOPE: 6, CLAUSE: 6\n\nT22 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: eq(T13, cons(nil, T25))\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: eq(T13, cons(nil, T25)), SCOPE: 7, CLAUSE: 7\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: gopher(T11, T2), SCOPE: 1, CLAUSE: 2\n\nT11 is ground\ngopher(T11, T2) !=? gopher(nil, X5)\nhead(T11, nil) !=? head([], X18)\nhead(T11, nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(head(T38, X61), ','(head(X61, X62), ','(tail(X61, X63), ','(tail(T38, X64), gopher(cons(X62, cons(X63, X64)), T40)))))\n\nT38 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \ngopher(T38, T2) !=? gopher(nil, X5)\nhead(T38, nil) !=? head([], X18)\nhead(T38, nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(head(T38, X61), ','(head(X61, X62), ','(tail(X61, X63), ','(tail(T38, X64), gopher(cons(X62, cons(X63, X64)), T40))))), SCOPE: 8, CLAUSE: 3 | ','(head(T38, X61), ','(head(X61, X62), ','(tail(X61, X63), ','(tail(T38, X64), gopher(cons(X62, cons(X63, X64)), T40))))), SCOPE: 8, CLAUSE: 4\n\nT38 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \ngopher(T38, T2) !=? gopher(nil, X5)\nhead(T38, nil) !=? head([], X18)\nhead(T38, nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(head(T38, X61), ','(head(X61, X62), ','(tail(X61, X63), ','(tail(T38, X64), gopher(cons(X62, cons(X63, X64)), T40))))), SCOPE: 8, CLAUSE: 4\n\nT38 is ground\nX61 is free; \nX62 is free; \nX63 is free; \nX64 is free; \ngopher(T38, T2) !=? gopher(nil, X5)\nhead(T38, nil) !=? head([], X18)\nhead(T38, nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(T46, X62), ','(tail(T46, X63), ','(tail(.(T46, T47), X64), gopher(cons(X62, cons(X63, X64)), T40))))\n\nT46 is ground\nT47 is ground\nX62 is free; \nX63 is free; \nX64 is free; \nhead(.(T46, T47), nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(head(T46, X62), ','(tail(T46, X63), ','(tail(.(T46, T47), X64), gopher(cons(X62, cons(X63, X64)), T40)))), SCOPE: 9, CLAUSE: 3 | ','(head(T46, X62), ','(tail(T46, X63), ','(tail(.(T46, T47), X64), gopher(cons(X62, cons(X63, X64)), T40)))), SCOPE: 9, CLAUSE: 4\n\nT46 is ground\nT47 is ground\nX62 is free; \nX63 is free; \nX64 is free; \nhead(.(T46, T47), nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(head(T46, X62), ','(tail(T46, X63), ','(tail(.(T46, T47), X64), gopher(cons(X62, cons(X63, X64)), T40)))), SCOPE: 9, CLAUSE: 3\n\nT46 is ground\nT47 is ground\nX62 is free; \nX63 is free; \nX64 is free; \nhead(.(T46, T47), nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(head(T46, X62), ','(tail(T46, X63), ','(tail(.(T46, T47), X64), gopher(cons(X62, cons(X63, X64)), T40)))), SCOPE: 9, CLAUSE: 4\n\nT46 is ground\nT47 is ground\nX62 is free; \nX63 is free; \nX64 is free; \nhead(.(T46, T47), nil) !=? head(.(X30, X31), X30)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail([], X63), ','(tail(.([], T47), X64), gopher(cons(X98, cons(X63, X64)), T40)))\n\nT47 is ground\nX63 is free; \nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(tail([], X63), ','(tail(.([], T47), X64), gopher(cons(X98, cons(X63, X64)), T40))), SCOPE: 10, CLAUSE: 5 | ','(tail([], X63), ','(tail(.([], T47), X64), gopher(cons(X98, cons(X63, X64)), T40))), SCOPE: 10, CLAUSE: 6\n\nT47 is ground\nX63 is free; \nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(tail([], X63), ','(tail(.([], T47), X64), gopher(cons(X98, cons(X63, X64)), T40))), SCOPE: 10, CLAUSE: 5\n\nT47 is ground\nX63 is free; \nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(tail([], X63), ','(tail(.([], T47), X64), gopher(cons(X98, cons(X63, X64)), T40))), SCOPE: 10, CLAUSE: 6\n\nT47 is ground\nX63 is free; \nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(tail(.([], T47), X64), gopher(cons(X98, cons([], X64)), T40))\n\nT47 is ground\nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(tail(.([], T47), X64), gopher(cons(X98, cons([], X64)), T40)), SCOPE: 11, CLAUSE: 5 | ','(tail(.([], T47), X64), gopher(cons(X98, cons([], X64)), T40)), SCOPE: 11, CLAUSE: 6\n\nT47 is ground\nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(tail(.([], T47), X64), gopher(cons(X98, cons([], X64)), T40)), SCOPE: 11, CLAUSE: 6\n\nT47 is ground\nX64 is free; \nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: gopher(cons(X98, cons([], T57)), T40)\n\nT57 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: gopher(cons(X98, cons([], T57)), T40), SCOPE: 12, CLAUSE: 0 | gopher(cons(X98, cons([], T57)), T40), SCOPE: 12, CLAUSE: 1 | gopher(cons(X98, cons([], T57)), T40), SCOPE: 12, CLAUSE: 2\n\nT57 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: gopher(cons(X98, cons([], T57)), T40), SCOPE: 12, CLAUSE: 1 | gopher(cons(X98, cons([], T57)), T40), SCOPE: 12, CLAUSE: 2\n\nT57 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(head(cons(X152, cons([], T65)), nil), ','(!_12, ','(tail(cons(X152, cons([], T65)), X151), eq(T67, cons(nil, X151))))) | gopher(cons(X98, cons([], T65)), T40), SCOPE: 12, CLAUSE: 2\n\nT65 is ground\nX98 is free; \nX152 is free; \nX151 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(head(cons(X152, cons([], T65)), nil), ','(!_12, ','(tail(cons(X152, cons([], T65)), X151), eq(T67, cons(nil, X151))))), SCOPE: 13, CLAUSE: 3 | ','(head(cons(X152, cons([], T65)), nil), ','(!_12, ','(tail(cons(X152, cons([], T65)), X151), eq(T67, cons(nil, X151))))), SCOPE: 13, CLAUSE: 4 | ?_13 | gopher(cons(X98, cons([], T65)), T40), SCOPE: 12, CLAUSE: 2\n\nT65 is ground\nX98 is free; \nX152 is free; \nX151 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(head(cons(X152, cons([], T65)), nil), ','(!_12, ','(tail(cons(X152, cons([], T65)), X151), eq(T67, cons(nil, X151))))), SCOPE: 13, CLAUSE: 4 | ?_13 | gopher(cons(X98, cons([], T65)), T40), SCOPE: 12, CLAUSE: 2\n\nT65 is ground\nX98 is free; \nX152 is free; \nX151 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ?_13 | gopher(cons(X98, cons([], T65)), T40), SCOPE: 12, CLAUSE: 2\n\nT65 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: gopher(cons(X98, cons([], T65)), T40), SCOPE: 12, CLAUSE: 2\n\nT65 is ground\nX98 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(head(cons(X179, cons([], T74)), X175), ','(head(X175, X176), ','(tail(X175, X177), ','(tail(cons(X179, cons([], T74)), X178), gopher(cons(X176, cons(X177, X178)), T76)))))\n\nT74 is ground\nX179 is free; \nX175 is free; \nX176 is free; \nX177 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(head(cons(X179, cons([], T74)), X175), ','(head(X175, X176), ','(tail(X175, X177), ','(tail(cons(X179, cons([], T74)), X178), gopher(cons(X176, cons(X177, X178)), T76))))), SCOPE: 14, CLAUSE: 3 | ','(head(cons(X179, cons([], T74)), X175), ','(head(X175, X176), ','(tail(X175, X177), ','(tail(cons(X179, cons([], T74)), X178), gopher(cons(X176, cons(X177, X178)), T76))))), SCOPE: 14, CLAUSE: 4\n\nT74 is ground\nX179 is free; \nX175 is free; \nX176 is free; \nX177 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(head(cons(X179, cons([], T74)), X175), ','(head(X175, X176), ','(tail(X175, X177), ','(tail(cons(X179, cons([], T74)), X178), gopher(cons(X176, cons(X177, X178)), T76))))), SCOPE: 14, CLAUSE: 4\n\nT74 is ground\nX179 is free; \nX175 is free; \nX176 is free; \nX177 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(tail(.(T82, T83), X63), ','(tail(.(.(T82, T83), T47), X64), gopher(cons(T82, cons(X63, X64)), T40)))\n\nT47 is ground\nT82 is ground\nT83 is ground\nX63 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(tail(.(T82, T83), X63), ','(tail(.(.(T82, T83), T47), X64), gopher(cons(T82, cons(X63, X64)), T40))), SCOPE: 15, CLAUSE: 5 | ','(tail(.(T82, T83), X63), ','(tail(.(.(T82, T83), T47), X64), gopher(cons(T82, cons(X63, X64)), T40))), SCOPE: 15, CLAUSE: 6\n\nT47 is ground\nT82 is ground\nT83 is ground\nX63 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(tail(.(T82, T83), X63), ','(tail(.(.(T82, T83), T47), X64), gopher(cons(T82, cons(X63, X64)), T40))), SCOPE: 15, CLAUSE: 6\n\nT47 is ground\nT82 is ground\nT83 is ground\nX63 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(tail(.(.(T93, T94), T47), X64), gopher(cons(T93, cons(T94, X64)), T40))\n\nT47 is ground\nT93 is ground\nT94 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(tail(.(.(T93, T94), T47), X64), gopher(cons(T93, cons(T94, X64)), T40)), SCOPE: 16, CLAUSE: 5 | ','(tail(.(.(T93, T94), T47), X64), gopher(cons(T93, cons(T94, X64)), T40)), SCOPE: 16, CLAUSE: 6\n\nT47 is ground\nT93 is ground\nT94 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(tail(.(.(T93, T94), T47), X64), gopher(cons(T93, cons(T94, X64)), T40)), SCOPE: 16, CLAUSE: 6\n\nT47 is ground\nT93 is ground\nT94 is ground\nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: gopher(cons(T108, cons(T109, T110)), T40)\n\nT108 is ground\nT109 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 72 [arrowhead = none ];
72 -> 2;

73 [label="EVAL with clause\ngopher(nil, X5) :- ','(!_1, eq(X5, nil)).\nand substitution\nT1 -> nil\nT2 -> T5\nX5 -> T5\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 73 [arrowhead = none ];
73 -> 3;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 4;

75 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 75 [arrowhead = none ];
75 -> 5;

76 [label="ONLY EVAL with clause\ngopher(X13, X14) :- ','(head(X13, nil), ','(!_1, ','(tail(X13, X15), eq(X14, cons(nil, X15))))).\nand substitution\nT1 -> T11\nX13 -> T11\nT2 -> T13\nX14 -> T13\nT12 -> T13", fontsize=14, style = filled, fillcolor = white];
4 -> 76 [arrowhead = none ];
76 -> 10;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 77 [arrowhead = none ];
77 -> 6;

78 [label="EVAL with clause\neq(X9, X9).\nand substitution\nT5 -> nil\nX9 -> nil\nT8 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 78 [arrowhead = none ];
78 -> 7;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 79 [arrowhead = none ];
79 -> 8;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 80 [arrowhead = none ];
80 -> 9;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 81 [arrowhead = none ];
81 -> 11;

82 [label="EVAL with clause\nhead([], X18).\nand substitution\nT11 -> []\nX18 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 82 [arrowhead = none ];
82 -> 12;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 83 [arrowhead = none ];
83 -> 13;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="EVAL with clause\nhead(.(X30, X31), X30).\nand substitution\nX30 -> nil\nX31 -> T22\nT11 -> .(nil, T22)\nT21 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 87 [arrowhead = none ];
87 -> 15;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 88 [arrowhead = none ];
88 -> 16;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 89 [arrowhead = none ];
89 -> 17;

90 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX15 -> []", fontsize=14, style = filled, fillcolor = white];
16 -> 90 [arrowhead = none ];
90 -> 18;

91 [label="BACKTRACK\nfor clause: tail(.(X3, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
17 -> 91 [arrowhead = none ];
91 -> 23;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 92 [arrowhead = none ];
92 -> 19;

93 [label="EVAL with clause\neq(X22, X22).\nand substitution\nT13 -> cons(nil, [])\nX22 -> cons(nil, [])\nT16 -> cons(nil, [])", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 93 [arrowhead = none ];
93 -> 20;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 94 [arrowhead = none ];
94 -> 21;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 95 [arrowhead = none ];
95 -> 22;

96 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 96 [arrowhead = none ];
96 -> 26;

97 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 98 [arrowhead = none ];
98 -> 27;

99 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 99 [arrowhead = none ];
99 -> 28;

100 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitution\nX38 -> nil\nT22 -> T25\nX39 -> T25\nX15 -> T25", fontsize=14, style = filled, fillcolor = white];
28 -> 100 [arrowhead = none ];
100 -> 29;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 101 [arrowhead = none ];
101 -> 30;

102 [label="EVAL with clause\neq(X42, X42).\nand substitution\nT13 -> cons(nil, T31)\nX42 -> cons(nil, T31)\nT25 -> T31\nT30 -> cons(nil, T31)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 102 [arrowhead = none ];
102 -> 31;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 103 [arrowhead = none ];
103 -> 32;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 104 [arrowhead = none ];
104 -> 33;

105 [label="ONLY EVAL with clause\ngopher(X59, X60) :- ','(head(X59, X61), ','(head(X61, X62), ','(tail(X61, X63), ','(tail(X59, X64), gopher(cons(X62, cons(X63, X64)), X60))))).\nand substitution\nT11 -> T38\nX59 -> T38\nT2 -> T40\nX60 -> T40\nT39 -> T40", fontsize=14, style = filled, fillcolor = white];
34 -> 105 [arrowhead = none ];
105 -> 35;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 106 [arrowhead = none ];
106 -> 36;

107 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (head(T38, nil), head([], X18))", fontsize=14, style = filled, fillcolor = white];
36 -> 107 [arrowhead = none ];
107 -> 37;

108 [label="EVAL with clause\nhead(.(X78, X79), X78).\nand substitution\nX78 -> T46\nX79 -> T47\nT38 -> .(T46, T47)\nX61 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 108 [arrowhead = none ];
108 -> 38;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 109 [arrowhead = none ];
109 -> 39;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 110 [arrowhead = none ];
110 -> 40;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 111 [arrowhead = none ];
111 -> 41;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 112 [arrowhead = none ];
112 -> 42;

113 [label="EVAL with clause\nhead([], X97).\nand substitution\nT46 -> []\nX62 -> X98\nX97 -> X98", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 113 [arrowhead = none ];
113 -> 43;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL with clause\nhead(.(X196, X197), X196).\nand substitution\nX196 -> T82\nX197 -> T83\nT46 -> .(T82, T83)\nX62 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 115 [arrowhead = none ];
115 -> 64;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 116 [arrowhead = none ];
116 -> 65;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 117 [arrowhead = none ];
117 -> 45;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 118 [arrowhead = none ];
118 -> 46;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 119 [arrowhead = none ];
119 -> 47;

120 [label="ONLY EVAL with clause\ntail([], []).\nand substitution\nX63 -> []", fontsize=14, style = filled, fillcolor = white];
46 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="BACKTRACK\nfor clause: tail(.(X3, X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
47 -> 121 [arrowhead = none ];
121 -> 63;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 122 [arrowhead = none ];
122 -> 49;

123 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 123 [arrowhead = none ];
123 -> 50;

124 [label="ONLY EVAL with clause\ntail(.(X134, X135), X135).\nand substitution\nX134 -> []\nT47 -> T57\nX135 -> T57\nX64 -> T57", fontsize=14, style = filled, fillcolor = white];
50 -> 124 [arrowhead = none ];
124 -> 51;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="BACKTRACK\nfor clause: gopher(nil, L) :- ','(!, eq(L, nil))because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 126 [arrowhead = none ];
126 -> 53;

127 [label="ONLY EVAL with clause\ngopher(X149, X150) :- ','(head(X149, nil), ','(!_12, ','(tail(X149, X151), eq(X150, cons(nil, X151))))).\nand substitution\nX98 -> X152\nT57 -> T65\nX149 -> cons(X152, cons([], T65))\nT40 -> T67\nX150 -> T67\nT66 -> T67", fontsize=14, style = filled, fillcolor = white];
53 -> 127 [arrowhead = none ];
127 -> 54;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 128 [arrowhead = none ];
128 -> 55;

129 [label="BACKTRACK\nfor clause: head([], X1)because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 129 [arrowhead = none ];
129 -> 56;

130 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
56 -> 130 [arrowhead = none ];
130 -> 57;

131 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 131 [arrowhead = none ];
131 -> 58;

132 [label="ONLY EVAL with clause\ngopher(X173, X174) :- ','(head(X173, X175), ','(head(X175, X176), ','(tail(X175, X177), ','(tail(X173, X178), gopher(cons(X176, cons(X177, X178)), X174))))).\nand substitution\nX98 -> X179\nT65 -> T74\nX173 -> cons(X179, cons([], T74))\nT40 -> T76\nX174 -> T76\nT75 -> T76", fontsize=14, style = filled, fillcolor = white];
58 -> 132 [arrowhead = none ];
132 -> 59;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 133 [arrowhead = none ];
133 -> 60;

134 [label="BACKTRACK\nfor clause: head([], X1)because of non-unification", fontsize=14, style = filled, fillcolor = white];
60 -> 134 [arrowhead = none ];
134 -> 61;

135 [label="BACKTRACK\nfor clause: head(.(X, X2), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
61 -> 135 [arrowhead = none ];
135 -> 62;

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 136 [arrowhead = none ];
136 -> 66;

137 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
66 -> 137 [arrowhead = none ];
137 -> 67;

138 [label="ONLY EVAL with clause\ntail(.(X215, X216), X216).\nand substitution\nT82 -> T93\nX215 -> T93\nT83 -> T94\nX216 -> T94\nX63 -> T94", fontsize=14, style = filled, fillcolor = white];
67 -> 138 [arrowhead = none ];
138 -> 68;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 139 [arrowhead = none ];
139 -> 69;

140 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
69 -> 140 [arrowhead = none ];
140 -> 70;

141 [label="ONLY EVAL with clause\ntail(.(X230, X231), X231).\nand substitution\nT93 -> T108\nT94 -> T109\nX230 -> .(T108, T109)\nT47 -> T110\nX231 -> T110\nX64 -> T110", fontsize=14, style = filled, fillcolor = white];
70 -> 141 [arrowhead = none ];
141 -> 71;

142 [label="INSTANCE with matching:\nT1 -> cons(T108, cons(T109, T110))\nT2 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 142 [arrowhead = none , style=dashed];
142 -> 1[style=dashed];

}

(2) Obligation:

Triples:

gopher1(.(.(T108, T109), T110), T40) :- gopher1(cons(T108, cons(T109, T110)), T40).

Clauses:

gopherc1(nil, nil).
gopherc1([], cons(nil, [])).
gopherc1(.(nil, T31), cons(nil, T31)).
gopherc1(.(.(T108, T109), T110), T40) :- gopherc1(cons(T108, cons(T109, T110)), T40).

Afs:

gopher1(x1, x2) = gopher1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
gopher1_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOPHER1_IN_GA(.(.(T108, T109), T110), T40) → U1_GA(T108, T109, T110, T40, gopher1_in_ga(cons(T108, cons(T109, T110)), T40))
GOPHER1_IN_GA(.(.(T108, T109), T110), T40) → GOPHER1_IN_GA(cons(T108, cons(T109, T110)), T40)

R is empty.
The argument filtering Pi contains the following mapping:
gopher1_in_ga(x1, x2) = gopher1_in_ga(x1)
.(x1, x2) = .(x1, x2)
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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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