Left Termination proof of /tmp/tmpkpeUDk/shuffle.pl

Left Termination of the query pattern shuffle_in_3(g, 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 PiDP

↳7 PiDPToQDPProof (⇐)

↳8 QDP

↳9 QDPSizeChangeProof (⇔)

↳10 YES

(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).

Queries:

shuffle(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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\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\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\nshuffle(T1, T2, T3) !=? shuffle(X2, [], X2)\nshuffle(T1, T2, T3) !=? shuffle([], X5, X5)", 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\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\nshuffle(T1, T2, T3) !=? shuffle(X2, [], X2)\nshuffle(T1, T2, T3) !=? shuffle([], X5, X5)\nshuffle(T1, T2, T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))", 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\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\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\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)", 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\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\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\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X21, [], X21)", 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\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", 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\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X21, [], X21)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: shuffle(.(T12, []), T24, T3), SCOPE: 1, CLAUSE: 3\n\nT12 is ground\nT24 is ground\nshuffle(.(T12, []), T24, T3) !=? shuffle(X2, [], X2)\nshuffle([], T24, T16) !=? shuffle(X21, [], X21)", 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\nT13 is ground\nT14 is ground\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", 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\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: shuffle(T56, T57, T59)\n\nT56 is ground\nT57 is ground\nshuffle(.(T12, .(T55, T56)), T57, T3) !=? shuffle(X2, [], X2)\nshuffle(.(T55, T56), T57, T16) !=? shuffle(X21, [], X21)", 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\nT13 is ground\nT14 is ground\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", 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\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: shuffle(T77, T79, T81)\n\nT77 is ground\nT79 is ground\nshuffle(T77, .(T78, T79), T16) !=? shuffle([], X29, X29)", 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\nshuffle(.(T12, T13), T14, T3) !=? shuffle(X2, [], X2)\nshuffle(T13, T14, T16) !=? shuffle(X21, [], X21)\nshuffle(T13, T14, T16) !=? shuffle([], X29, X29)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: shuffle(.(T93, T94), T96, T98)\n\nT93 is ground\nT94 is ground\nT96 is ground\nshuffle(T94, .(T95, T96), T16) !=? shuffle([], X29, X29)", 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\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))", 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\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))", 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\nT112 is ground\nshuffle(T112, .(T106, []), T3) !=? shuffle([], X5, X5)\nshuffle(T112, .(T106, []), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))", 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\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))\nshuffle(T105, T107, T109) !=? shuffle(X129, [], X129)", 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\nT115 is ground\nshuffle([], T115, T109) !=? shuffle(X129, [], X129)", 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\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))\nshuffle(T105, T107, T109) !=? shuffle(X129, [], X129)\nshuffle(T105, T107, T109) !=? shuffle([], X133, X133)", 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\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))\nshuffle(T105, T107, T109) !=? shuffle(X129, [], X129)\nshuffle(T105, T107, T109) !=? shuffle([], X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: shuffle(T105, T107, T109), SCOPE: 3, CLAUSE: 3\n\nT105 is ground\nT107 is ground\nshuffle(T105, .(T106, T107), T3) !=? shuffle([], X5, X5)\nshuffle(T105, .(T106, T107), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))\nshuffle(T105, T107, T109) !=? shuffle(X129, [], X129)\nshuffle(T105, T107, T109) !=? shuffle([], X133, X133)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: shuffle(T133, T134, T136)\n\nT133 is ground\nT134 is ground\nshuffle(.(T132, T133), T134, T109) !=? shuffle(X129, [], X129)", 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\nT146 is ground\nT148 is ground\nshuffle(T146, .(T106, .(T147, T148)), T3) !=? shuffle([], X5, X5)\nshuffle(T146, .(T106, .(T147, T148)), T3) !=? shuffle(.(X11, X12), X13, .(X11, X14))\nshuffle(T146, .(T147, T148), T109) !=? shuffle([], X133, X133)", 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 substitution\nT1 -> 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 = lightblue];
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([], X5, X5) :- !_1.\nand substitution\nT1 -> []\nT2 -> T7\nX5 -> T7\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X11, X12), X13, .(X11, X14)) :- shuffle(X12, X13, X14).\nand substitution\nX11 -> T12\nX12 -> T13\nT1 -> .(T12, T13)\nT2 -> T14\nX13 -> T14\nX14 -> 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 = lightblue];
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(X119, .(X120, X121), .(X120, X122)) :- shuffle(X119, X121, X122).\nand substitution\nT1 -> T105\nX119 -> T105\nX120 -> T106\nX121 -> T107\nT2 -> .(T106, T107)\nX122 -> 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 = lightblue];
12 -> 66 [arrowhead = none ];
66 -> 37;

67 [label="EVAL with clause\nshuffle(X21, [], X21) :- !_2.\nand substitution\nT13 -> T19\nX21 -> 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 = lightblue];
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([], X29, X29) :- !_2.\nand substitution\nT13 -> []\nT14 -> T24\nX29 -> T24\nT16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 19;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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 = white];
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(X39, .(X40, X41), .(X40, X42)) :- shuffle(X39, X41, X42).\nand substitution\nT12 -> T33\nX39 -> .(T33, [])\nX40 -> T34\nX41 -> T35\nT24 -> .(T34, T35)\nX42 -> 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 = lightblue];
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(.(X64, X65), X66, .(X64, X67)) :- shuffle(X65, X66, X67).\nand substitution\nX64 -> T55\nX65 -> T56\nT13 -> .(T55, T56)\nT14 -> T57\nX66 -> T57\nX67 -> 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 = lightblue];
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(X89, .(X90, X91), .(X90, X92)) :- shuffle(X89, X91, X92).\nand substitution\nT13 -> T77\nX89 -> T77\nX90 -> T78\nX91 -> T79\nT14 -> .(T78, T79)\nX92 -> 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 = lightblue];
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(X106, .(X107, X108), .(X107, X109)) :- shuffle(X106, X108, X109).\nand substitution\nT12 -> T93\nT13 -> T94\nX106 -> .(T93, T94)\nX107 -> T95\nX108 -> T96\nT14 -> .(T95, T96)\nX109 -> 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 = lightblue];
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(X129, [], X129) :- !_3.\nand substitution\nT105 -> T112\nX129 -> 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 = lightblue];
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([], X133, X133) :- !_3.\nand substitution\nT105 -> []\nT107 -> T115\nX133 -> T115\nT109 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 97 [arrowhead = none ];
97 -> 43;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X151, X152), X153, .(X151, X154)) :- shuffle(X152, X153, X154).\nand substitution\nX151 -> T132\nX152 -> T133\nT105 -> .(T132, T133)\nT107 -> T134\nX153 -> T134\nX154 -> 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 = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL with clause\nshuffle(X168, .(X169, X170), .(X169, X171)) :- shuffle(X168, X170, X171).\nand substitution\nT105 -> T146\nX168 -> T146\nX169 -> T147\nX170 -> T148\nT107 -> .(T147, T148)\nX171 -> 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 = lightblue];
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:

shuffle1(.(T33, []), .(T34, T35), .(T34, T37)) :- shuffle1(.(T33, []), T35, T37).
shuffle1(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) :- shuffle1(T56, T57, T59).
shuffle1(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) :- shuffle1(T77, T79, T81).
shuffle1(.(T93, T94), .(T95, T96), .(T95, T98)) :- shuffle1(.(T93, T94), T96, T98).
shuffle1(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) :- shuffle1(T133, T134, T136).
shuffle1(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) :- shuffle1(T146, T148, T150).

Clauses:

shufflec1(T5, [], T5).
shufflec1([], T7, T7).
shufflec1(.(T12, T19), [], .(T12, T19)).
shufflec1(.(T12, []), T24, .(T12, T24)).
shufflec1(.(T33, []), .(T34, T35), .(T34, T37)) :- shufflec1(.(T33, []), T35, T37).
shufflec1(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) :- shufflec1(T56, T57, T59).
shufflec1(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) :- shufflec1(T77, T79, T81).
shufflec1(.(T93, T94), .(T95, T96), .(T95, T98)) :- shufflec1(.(T93, T94), T96, T98).
shufflec1(T112, .(T106, []), .(T106, T112)).
shufflec1([], .(T106, T115), .(T106, T115)).
shufflec1(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) :- shufflec1(T133, T134, T136).
shufflec1(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) :- shufflec1(T146, T148, T150).

Afs:

shuffle1(x1, x2, x3) = shuffle1(x1, x2)

(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:
shuffle1_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → U1_GGA(T33, T34, T35, T37, shuffle1_in_gga(.(T33, []), T35, T37))
SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → SHUFFLE1_IN_GGA(.(T33, []), T35, T37)
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_GGA(T12, T55, T56, T57, T59, shuffle1_in_gga(T56, T57, T59))
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → SHUFFLE1_IN_GGA(T56, T57, T59)
SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_GGA(T12, T77, T78, T79, T81, shuffle1_in_gga(T77, T79, T81))
SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → SHUFFLE1_IN_GGA(T77, T79, T81)
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_GGA(T93, T94, T95, T96, T98, shuffle1_in_gga(.(T93, T94), T96, T98))
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → SHUFFLE1_IN_GGA(.(T93, T94), T96, T98)
SHUFFLE1_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_GGA(T132, T133, T106, T134, T136, shuffle1_in_gga(T133, T134, T136))
SHUFFLE1_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → SHUFFLE1_IN_GGA(T133, T134, T136)
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_GGA(T146, T106, T147, T148, T150, shuffle1_in_gga(T146, T148, T150))
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → SHUFFLE1_IN_GGA(T146, T148, T150)

R is empty.
The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3) = shuffle1_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
[] = []
SHUFFLE1_IN_GGA(x1, x2, x3) = SHUFFLE1_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:

SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → U1_GGA(T33, T34, T35, T37, shuffle1_in_gga(.(T33, []), T35, T37))
SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → SHUFFLE1_IN_GGA(.(T33, []), T35, T37)
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_GGA(T12, T55, T56, T57, T59, shuffle1_in_gga(T56, T57, T59))
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → SHUFFLE1_IN_GGA(T56, T57, T59)
SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_GGA(T12, T77, T78, T79, T81, shuffle1_in_gga(T77, T79, T81))
SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → SHUFFLE1_IN_GGA(T77, T79, T81)
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_GGA(T93, T94, T95, T96, T98, shuffle1_in_gga(.(T93, T94), T96, T98))
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → SHUFFLE1_IN_GGA(.(T93, T94), T96, T98)
SHUFFLE1_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_GGA(T132, T133, T106, T134, T136, shuffle1_in_gga(T133, T134, T136))
SHUFFLE1_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → SHUFFLE1_IN_GGA(T133, T134, T136)
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_GGA(T146, T106, T147, T148, T150, shuffle1_in_gga(T146, T148, T150))
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → SHUFFLE1_IN_GGA(T146, T148, T150)

(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:

SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → SHUFFLE1_IN_GGA(T77, T79, T81)
SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → SHUFFLE1_IN_GGA(.(T33, []), T35, T37)
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → SHUFFLE1_IN_GGA(.(T93, T94), T96, T98)
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → SHUFFLE1_IN_GGA(T56, T57, T59)
SHUFFLE1_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → SHUFFLE1_IN_GGA(T133, T134, T136)
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → SHUFFLE1_IN_GGA(T146, T148, T150)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
SHUFFLE1_IN_GGA(x1, x2, x3) = SHUFFLE1_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:

SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79)) → SHUFFLE1_IN_GGA(T77, T79)
SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35)) → SHUFFLE1_IN_GGA(.(T33, []), T35)
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96)) → SHUFFLE1_IN_GGA(.(T93, T94), T96)
SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57) → SHUFFLE1_IN_GGA(T56, T57)
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148))) → SHUFFLE1_IN_GGA(T146, T148)

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:

SHUFFLE1_IN_GGA(.(T12, .(T55, T56)), T57) → SHUFFLE1_IN_GGA(T56, T57)
The graph contains the following edges 1 > 1, 2 >= 2
SHUFFLE1_IN_GGA(.(T33, []), .(T34, T35)) → SHUFFLE1_IN_GGA(.(T33, []), T35)
The graph contains the following edges 1 >= 1, 2 > 2
SHUFFLE1_IN_GGA(.(T12, T77), .(T78, T79)) → SHUFFLE1_IN_GGA(T77, T79)
The graph contains the following edges 1 > 1, 2 > 2
SHUFFLE1_IN_GGA(.(T93, T94), .(T95, T96)) → SHUFFLE1_IN_GGA(.(T93, T94), T96)
The graph contains the following edges 1 >= 1, 2 > 2
SHUFFLE1_IN_GGA(T146, .(T106, .(T147, T148))) → SHUFFLE1_IN_GGA(T146, T148)
The graph contains the following edges 1 >= 1, 2 > 2