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

(0) Obligation:

Clauses:

shuffle(A, [], A) :- !.
shuffle([], B, B) :- !.
shuffle(.(A, RestA), B, .(A, Shuffled)) :- shuffle(RestA, B, Shuffled).
shuffle(A, .(B, RestB), .(B, Shuffled)) :- shuffle(A, RestB, Shuffled).

Query: shuffle(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: shuffle(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | shuffle(T5, [], T3), SCOPE: 1, CLAUSE: 1 | shuffle(T5, [], T3), SCOPE: 1, CLAUSE: 2 | shuffle(T5, [], T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX2 is free\nshuffle(T1, T2, T3) !=? shuffle(X2, [], X2)", 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: !_1 | shuffle([], T7, T3), SCOPE: 1, CLAUSE: 2 | shuffle([], T7, T3), SCOPE: 1, CLAUSE: 3\n\nT7 is ground\nX2 is free\nshuffle([], T7, T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX2 is free\nX4 is free\nshuffle(T1, T2, T3) !=? shuffle(X2, [], X2)\nshuffle(T1, T2, T3) !=? shuffle([], X4, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: shuffle(T13, T14, T16) | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX2 is free\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nshuffle(T1, T2, T3) !=? shuffle(X2, [], X2)\nshuffle(T1, T2, T3) !=? shuffle([], X4, X4)\nshuffle(T1, T2, T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 0 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 1 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 2 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: !_2 | shuffle(T19, [], T16), SCOPE: 2, CLAUSE: 1 | shuffle(T19, [], T16), SCOPE: 2, CLAUSE: 2 | shuffle(T19, [], T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, T19), [], T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT19 is ground\nX2 is free\nshuffle(.(T12, T19), [], T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 1 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 2 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | shuffle(.(T12, T19), [], T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT19 is ground\nX2 is free\nshuffle(.(T12, T19), [], T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: shuffle(.(T12, T19), [], T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT19 is ground\nX2 is free\nshuffle(.(T12, T19), [], T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: !_2 | shuffle([], T24, T16), SCOPE: 2, CLAUSE: 2 | shuffle([], T24, T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, []), T24, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT24 is ground\nX2 is free\nX15 is free\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X15, [], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 2 | shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true | shuffle(.(T12, []), T24, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT24 is ground\nX2 is free\nX15 is free\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X15, [], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: shuffle(.(T12, []), T24, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT24 is ground\nX2 is free\nX15 is free\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X15, [], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: shuffle(.(T33, []), T35, T37)\n\nT33 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 2\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 3 | ?_2 | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: shuffle(T56, T57, T59)\n\nT12 is ground\nT55 is ground\nT56 is ground\nT57 is ground\nX2 is free\nX15 is free\nshuffle(.(T12, .(T55, T56)), T57, T3) !=? shuffle(X2, [], X2)\nshuffle(.(T55, T56), T57, T16) !=? shuffle(X15, [], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: shuffle(T13, T14, T16), SCOPE: 2, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_2 | shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: shuffle(T77, T79, T81)\n\nT77 is ground\nT78 is ground\nT79 is ground\nX22 is free\nshuffle(T77, .(T78, T79), T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: shuffle(.(T12, T13), T14, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT13 is ground\nT14 is ground\nX2 is free\nX15 is free\nX22 is free\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X15, [], X15)\nshuffle(T13, T14, T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: shuffle(.(T93, T94), T96, T98)\n\nT93 is ground\nT94 is ground\nT95 is ground\nT96 is ground\nX22 is free\nshuffle(T94, .(T95, T96), T16) !=? shuffle([], X22, X22)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: shuffle(T105, T107, T109)\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 0 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 1 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 2 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 3\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: !_3 | shuffle(T112, [], T109), SCOPE: 3, CLAUSE: 1 | shuffle(T112, [], T109), SCOPE: 3, CLAUSE: 2 | shuffle(T112, [], T109), SCOPE: 3, CLAUSE: 3\n\nT106 is ground\nT112 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nshuffle(T112, .(T106, []), T3) !=? shuffle([], X4, X4)\nshuffle(T112, .(T106, []), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 1 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 2 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 3\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX101 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(T105, T107, T109) !=? shuffle(X101, [], X101)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: !_3 | shuffle([], T115, T109), SCOPE: 3, CLAUSE: 2 | shuffle([], T115, T109), SCOPE: 3, CLAUSE: 3\n\nT106 is ground\nT115 is ground\nX4 is free\nX101 is free\nshuffle([], .(T106, T115), T3) !=? shuffle([], X4, X4)\nshuffle([], T115, T109) !=? shuffle(X101, [], X101)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 2 | shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 3\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX101 is free\nX104 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(T105, T107, T109) !=? shuffle(X101, [], X101)\nshuffle(T105, T107, T109) !=? shuffle([], X104, X104)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 2\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX101 is free\nX104 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(T105, T107, T109) !=? shuffle(X101, [], X101)\nshuffle(T105, T107, T109) !=? shuffle([], X104, X104)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 3\n\nT105 is ground\nT106 is ground\nT107 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX101 is free\nX104 is free\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X4, X4)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(T105, T107, T109) !=? shuffle(X101, [], X101)\nshuffle(T105, T107, T109) !=? shuffle([], X104, X104)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: shuffle(T133, T134, T136)\n\nT106 is ground\nT132 is ground\nT133 is ground\nT134 is ground\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX101 is free\nshuffle(.(T132, T133), .(T106, T134), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(.(T132, T133), T134, T109) !=? shuffle(X101, [], X101)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: shuffle(T146, T148, T150)\n\nT106 is ground\nT146 is ground\nT147 is ground\nT148 is ground\nX4 is free\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX104 is free\nshuffle(T146, .(T106, .(T147, T148)), T3) !=? shuffle([], X4, X4)\nshuffle(T146, .(T106, .(T147, T148)), T3) !=? shuffle(.(X9, X10), X11, .(X9, X12))\nshuffle(T146, .(T147, T148), T109) !=? shuffle([], X104, X104)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 53 [arrowhead = none ];
53 -> 2;

54 [label="EVAL with clause\nshuffle(X2, [], X2) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> [],\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 54 [arrowhead = none ];
54 -> 3;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 55 [arrowhead = none ];
55 -> 4;

56 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 56 [arrowhead = none ];
56 -> 5;

57 [label="EVAL with clause\nshuffle([], X4, X4) :- !_1.\nand substitutionT1 -> [],\nT2 -> T7,\nX4 -> T7,\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="EVAL with clause\nshuffle(.(X9, X10), X11, .(X9, X12)) :- shuffle(X10, X11, X12).\nand substitutionX9 -> T12,\nX10 -> T13,\nT1 -> .(T12, T13),\nT2 -> T14,\nX11 -> T14,\nX12 -> T16,\nT3 -> .(T12, T16),\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 62 [arrowhead = none ];
62 -> 12;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 63 [arrowhead = none ];
63 -> 10;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="EVAL with clause\nshuffle(X95, .(X96, X97), .(X96, X98)) :- shuffle(X95, X97, X98).\nand substitutionT1 -> T105,\nX95 -> T105,\nX96 -> T106,\nX97 -> T107,\nT2 -> .(T106, T107),\nX98 -> T109,\nT3 -> .(T106, T109),\nT108 -> T109", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 65 [arrowhead = none ];
65 -> 36;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 66 [arrowhead = none ];
66 -> 37;

67 [label="EVAL with clause\nshuffle(X15, [], X15) :- !_2.\nand substitutionT13 -> T19,\nX15 -> T19,\nT14 -> [],\nT16 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\nshuffle([], X22, X22) :- !_2.\nand substitutionT13 -> [],\nT14 -> T24,\nX22 -> T24,\nT16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 19;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 71 [arrowhead = none ];
71 -> 20;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="BACKTRACK\nfor clause: shuffle(A, .(B, RestB), .(B, Shuffled)) :- shuffle(A, RestB, Shuffled)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 73 [arrowhead = none ];
73 -> 18;

74 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 75 [arrowhead = none ];
75 -> 25;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 76 [arrowhead = none ];
76 -> 26;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 77 [arrowhead = none ];
77 -> 22;

78 [label="EVAL with clause\nshuffle(X31, .(X32, X33), .(X32, X34)) :- shuffle(X31, X33, X34).\nand substitutionT12 -> T33,\nX31 -> .(T33, []),\nX32 -> T34,\nX33 -> T35,\nT24 -> .(T34, T35),\nX34 -> T37,\nT3 -> .(T34, T37),\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 78 [arrowhead = none ];
78 -> 23;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 79 [arrowhead = none ];
79 -> 24;

80 [label="INSTANCE with matching:\nT1 -> .(T33, [])\nT2 -> T35\nT3 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 80 [arrowhead = none , style=dashed];
80 -> 1[style=dashed];

81 [label="EVAL with clause\nshuffle(.(X52, X53), X54, .(X52, X55)) :- shuffle(X53, X54, X55).\nand substitutionX52 -> T55,\nX53 -> T56,\nT13 -> .(T55, T56),\nT14 -> T57,\nX54 -> T57,\nX55 -> T59,\nT16 -> .(T55, T59),\nT58 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 81 [arrowhead = none ];
81 -> 27;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="INSTANCE with matching:\nT1 -> T56\nT2 -> T57\nT3 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 85 [arrowhead = none , style=dashed];
85 -> 1[style=dashed];

86 [label="EVAL with clause\nshuffle(X73, .(X74, X75), .(X74, X76)) :- shuffle(X73, X75, X76).\nand substitutionT13 -> T77,\nX73 -> T77,\nX74 -> T78,\nX75 -> T79,\nT14 -> .(T78, T79),\nX76 -> T81,\nT16 -> .(T78, T81),\nT80 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 86 [arrowhead = none ];
86 -> 31;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 87 [arrowhead = none ];
87 -> 32;

88 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 88 [arrowhead = none ];
88 -> 33;

89 [label="INSTANCE with matching:\nT1 -> T77\nT2 -> T79\nT3 -> T81", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 89 [arrowhead = none , style=dashed];
89 -> 1[style=dashed];

90 [label="EVAL with clause\nshuffle(X86, .(X87, X88), .(X87, X89)) :- shuffle(X86, X88, X89).\nand substitutionT12 -> T93,\nT13 -> T94,\nX86 -> .(T93, T94),\nX87 -> T95,\nX88 -> T96,\nT14 -> .(T95, T96),\nX89 -> T98,\nT3 -> .(T95, T98),\nT97 -> T98", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 90 [arrowhead = none ];
90 -> 34;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 91 [arrowhead = none ];
91 -> 35;

92 [label="INSTANCE with matching:\nT1 -> .(T93, T94)\nT2 -> T96\nT3 -> T98", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 92 [arrowhead = none , style=dashed];
92 -> 1[style=dashed];

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 93 [arrowhead = none ];
93 -> 38;

94 [label="EVAL with clause\nshuffle(X101, [], X101) :- !_3.\nand substitutionT105 -> T112,\nX101 -> T112,\nT107 -> [],\nT109 -> T112", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 95 [arrowhead = none ];
95 -> 40;

96 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="EVAL with clause\nshuffle([], X104, X104) :- !_3.\nand substitutionT105 -> [],\nT107 -> T115,\nX104 -> T115,\nT109 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 97 [arrowhead = none ];
97 -> 43;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 98 [arrowhead = none ];
98 -> 44;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 99 [arrowhead = none ];
99 -> 42;

100 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 100 [arrowhead = none ];
100 -> 45;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 101 [arrowhead = none ];
101 -> 47;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 103 [arrowhead = none ];
103 -> 46;

104 [label="EVAL with clause\nshuffle(.(X121, X122), X123, .(X121, X124)) :- shuffle(X122, X123, X124).\nand substitutionX121 -> T132,\nX122 -> T133,\nT105 -> .(T132, T133),\nT107 -> T134,\nX123 -> T134,\nX124 -> T136,\nT109 -> .(T132, T136),\nT135 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 104 [arrowhead = none ];
104 -> 49;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL with clause\nshuffle(X134, .(X135, X136), .(X135, X137)) :- shuffle(X134, X136, X137).\nand substitutionT105 -> T146,\nX134 -> T146,\nX135 -> T147,\nX136 -> T148,\nT107 -> .(T147, T148),\nX137 -> T150,\nT109 -> .(T147, T150),\nT149 -> T150", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 106 [arrowhead = none ];
106 -> 51;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 107 [arrowhead = none ];
107 -> 52;

108 [label="INSTANCE with matching:\nT1 -> T133\nT2 -> T134\nT3 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 108 [arrowhead = none , style=dashed];
108 -> 1[style=dashed];

109 [label="INSTANCE with matching:\nT1 -> T146\nT2 -> T148\nT3 -> T150", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 109 [arrowhead = none , style=dashed];
109 -> 1[style=dashed];

}

(2) Obligation:

Triples:

shuffleA(.(X1, []), .(X2, X3), .(X2, X4)) :- shuffleA(.(X1, []), X3, X4).
shuffleA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- shuffleA(X3, X4, X5).
shuffleA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) :- shuffleA(X2, X4, X5).
shuffleA(.(X1, X2), .(X3, X4), .(X3, X5)) :- shuffleA(.(X1, X2), X4, X5).
shuffleA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) :- shuffleA(X2, X4, X5).
shuffleA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- shuffleA(X1, X4, X5).

Clauses:

shufflecA(X1, [], X1).
shufflecA([], X1, X1).
shufflecA(.(X1, X2), [], .(X1, X2)).
shufflecA(.(X1, []), X2, .(X1, X2)).
shufflecA(.(X1, []), .(X2, X3), .(X2, X4)) :- shufflecA(.(X1, []), X3, X4).
shufflecA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) :- shufflecA(X3, X4, X5).
shufflecA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) :- shufflecA(X2, X4, X5).
shufflecA(.(X1, X2), .(X3, X4), .(X3, X5)) :- shufflecA(.(X1, X2), X4, X5).
shufflecA(X1, .(X2, []), .(X2, X1)).
shufflecA([], .(X1, X2), .(X1, X2)).
shufflecA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) :- shufflecA(X2, X4, X5).
shufflecA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) :- shufflecA(X1, X4, X5).

Afs:

shuffleA(x1, x2, x3)  =  shuffleA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
shuffleA_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3), .(X2, X4)) → U1_GGA(X1, X2, X3, X4, shuffleA_in_gga(.(X1, []), X3, X4))
SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3), .(X2, X4)) → SHUFFLEA_IN_GGA(.(X1, []), X3, X4)
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → U2_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X3, X4, X5))
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → SHUFFLEA_IN_GGA(X3, X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) → U3_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X2, X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U4_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(.(X1, X2), X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → SHUFFLEA_IN_GGA(.(X1, X2), X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) → U5_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X2, X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → U6_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X1, X4, X5))
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SHUFFLEA_IN_GGA(X1, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
shuffleA_in_gga(x1, x2, x3)  =  shuffleA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
[]  =  []
SHUFFLEA_IN_GGA(x1, x2, x3)  =  SHUFFLEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)

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:

SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3), .(X2, X4)) → U1_GGA(X1, X2, X3, X4, shuffleA_in_gga(.(X1, []), X3, X4))
SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3), .(X2, X4)) → SHUFFLEA_IN_GGA(.(X1, []), X3, X4)
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → U2_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X3, X4, X5))
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → SHUFFLEA_IN_GGA(X3, X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) → U3_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X2, X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U4_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(.(X1, X2), X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → SHUFFLEA_IN_GGA(.(X1, X2), X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) → U5_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X2, X4, X5))
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → U6_GGA(X1, X2, X3, X4, X5, shuffleA_in_gga(X1, X4, X5))
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SHUFFLEA_IN_GGA(X1, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
shuffleA_in_gga(x1, x2, x3)  =  shuffleA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
[]  =  []
SHUFFLEA_IN_GGA(x1, x2, x3)  =  SHUFFLEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(6) Obligation:

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

SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X1, .(X3, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3), .(X2, X4)) → SHUFFLEA_IN_GGA(.(X1, []), X3, X4)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → SHUFFLEA_IN_GGA(.(X1, X2), X4, X5)
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4, .(X1, .(X2, X5))) → SHUFFLEA_IN_GGA(X3, X4, X5)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, .(X1, X5))) → SHUFFLEA_IN_GGA(X2, X4, X5)
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4)), .(X2, .(X3, X5))) → SHUFFLEA_IN_GGA(X1, X4, X5)

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

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4)) → SHUFFLEA_IN_GGA(X2, X4)
SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3)) → SHUFFLEA_IN_GGA(.(X1, []), X3)
SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4)) → SHUFFLEA_IN_GGA(.(X1, X2), X4)
SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4) → SHUFFLEA_IN_GGA(X3, X4)
SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4))) → SHUFFLEA_IN_GGA(X1, X4)

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

(9) 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:

• SHUFFLEA_IN_GGA(.(X1, .(X2, X3)), X4) → SHUFFLEA_IN_GGA(X3, X4)
  The graph contains the following edges 1 > 1, 2 >= 2

• SHUFFLEA_IN_GGA(.(X1, []), .(X2, X3)) → SHUFFLEA_IN_GGA(.(X1, []), X3)
  The graph contains the following edges 1 >= 1, 2 > 2

• SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4)) → SHUFFLEA_IN_GGA(X2, X4)
  The graph contains the following edges 1 > 1, 2 > 2

• SHUFFLEA_IN_GGA(.(X1, X2), .(X3, X4)) → SHUFFLEA_IN_GGA(.(X1, X2), X4)
  The graph contains the following edges 1 >= 1, 2 > 2

• SHUFFLEA_IN_GGA(X1, .(X2, .(X3, X4))) → SHUFFLEA_IN_GGA(X1, X4)
  The graph contains the following edges 1 >= 1, 2 > 2

(10) YES