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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: shuffle(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: 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];
8 [label="8: !_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];
10 [label="10: 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];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: !_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];
22 [label="22: 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];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: 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];
35 [label="35: 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];
37 [label="37: 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];
39 [label="39: !_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];
41 [label="41: 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];
43 [label="43: 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];
44 [label="44: 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];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: !_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];
51 [label="51: 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];
54 [label="54: 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];
57 [label="57: 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];
59 [label="59: shuffle(.(T33, []), T35, T37)\n\nT33 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: 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];
65 [label="65: 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];
69 [label="69: 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];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: 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];
75 [label="75: ?_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];
77 [label="77: 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];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: 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];
84 [label="84: 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];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: 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];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: 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];
90 [label="90: !_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];
93 [label="93: 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];
94 [label="94: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: !_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];
97 [label="97: 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];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: 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];
101 [label="101: 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];
102 [label="102: 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];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: 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];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 112 [arrowhead = none ];
112 -> 5;

113 [label="EVAL with clause\nshuffle(X2, [], X2) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> [],\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 113 [arrowhead = none ];
113 -> 8;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 114 [arrowhead = none ];
114 -> 10;

115 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 115 [arrowhead = none ];
115 -> 15;

116 [label="EVAL with clause\nshuffle([], X4, X4) :- !_1.\nand substitutionT1 -> [],\nT2 -> T7,\nX4 -> T7,\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 116 [arrowhead = none ];
116 -> 20;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 118 [arrowhead = none ];
118 -> 16;

119 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 119 [arrowhead = none ];
119 -> 25;

120 [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];
22 -> 120 [arrowhead = none ];
120 -> 31;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 121 [arrowhead = none ];
121 -> 35;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 122 [arrowhead = none ];
122 -> 26;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 123 [arrowhead = none ];
123 -> 37;

124 [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];
35 -> 124 [arrowhead = none ];
124 -> 87;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 125 [arrowhead = none ];
125 -> 88;

126 [label="EVAL with clause\nshuffle(X15, [], X15) :- !_2.\nand substitutionT13 -> T19,\nX15 -> T19,\nT14 -> [],\nT16 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 126 [arrowhead = none ];
126 -> 39;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 127 [arrowhead = none ];
127 -> 41;

128 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 128 [arrowhead = none ];
128 -> 43;

129 [label="EVAL with clause\nshuffle([], X22, X22) :- !_2.\nand substitutionT13 -> [],\nT14 -> T24,\nX22 -> T24,\nT16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 130 [arrowhead = none ];
130 -> 51;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 131 [arrowhead = none ];
131 -> 44;

132 [label="BACKTRACK\nfor clause: shuffle(A, .(B, RestB), .(B, Shuffled)) :- shuffle(A, RestB, Shuffled)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 132 [arrowhead = none ];
132 -> 47;

133 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 133 [arrowhead = none ];
133 -> 54;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 134 [arrowhead = none ];
134 -> 64;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 135 [arrowhead = none ];
135 -> 65;

136 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 136 [arrowhead = none ];
136 -> 57;

137 [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];
57 -> 137 [arrowhead = none ];
137 -> 59;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 138 [arrowhead = none ];
138 -> 61;

139 [label="INSTANCE with matching:\nT1 -> .(T33, [])\nT2 -> T35\nT3 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 139 [arrowhead = none , style=dashed];
139 -> 2[style=dashed];

140 [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];
64 -> 140 [arrowhead = none ];
140 -> 69;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 141 [arrowhead = none ];
141 -> 71;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 142 [arrowhead = none ];
142 -> 73;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 143 [arrowhead = none ];
143 -> 75;

144 [label="INSTANCE with matching:\nT1 -> T56\nT2 -> T57\nT3 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 144 [arrowhead = none , style=dashed];
144 -> 2[style=dashed];

145 [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];
73 -> 145 [arrowhead = none ];
145 -> 77;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 146 [arrowhead = none ];
146 -> 79;

147 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 147 [arrowhead = none ];
147 -> 81;

148 [label="INSTANCE with matching:\nT1 -> T77\nT2 -> T79\nT3 -> T81", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 148 [arrowhead = none , style=dashed];
148 -> 2[style=dashed];

149 [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];
81 -> 149 [arrowhead = none ];
149 -> 84;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 150 [arrowhead = none ];
150 -> 86;

151 [label="INSTANCE with matching:\nT1 -> .(T93, T94)\nT2 -> T96\nT3 -> T98", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 151 [arrowhead = none , style=dashed];
151 -> 2[style=dashed];

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 152 [arrowhead = none ];
152 -> 89;

153 [label="EVAL with clause\nshuffle(X101, [], X101) :- !_3.\nand substitutionT105 -> T112,\nX101 -> T112,\nT107 -> [],\nT109 -> T112", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 153 [arrowhead = none ];
153 -> 90;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 154 [arrowhead = none ];
154 -> 93;

155 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 155 [arrowhead = none ];
155 -> 94;

156 [label="EVAL with clause\nshuffle([], X104, X104) :- !_3.\nand substitutionT105 -> [],\nT107 -> T115,\nX104 -> T115,\nT109 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 156 [arrowhead = none ];
156 -> 96;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 157 [arrowhead = none ];
157 -> 97;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 158 [arrowhead = none ];
158 -> 95;

159 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 159 [arrowhead = none ];
159 -> 98;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
97 -> 160 [arrowhead = none ];
160 -> 100;

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
97 -> 161 [arrowhead = none ];
161 -> 101;

162 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 162 [arrowhead = none ];
162 -> 99;

163 [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];
100 -> 163 [arrowhead = none ];
163 -> 102;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 164 [arrowhead = none ];
164 -> 103;

165 [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];
101 -> 165 [arrowhead = none ];
165 -> 109;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 166 [arrowhead = none ];
166 -> 111;

167 [label="INSTANCE with matching:\nT1 -> T133\nT2 -> T134\nT3 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
102 -> 167 [arrowhead = none , style=dashed];
167 -> 2[style=dashed];

168 [label="INSTANCE with matching:\nT1 -> T146\nT2 -> T148\nT3 -> T150", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 168 [arrowhead = none , style=dashed];
168 -> 2[style=dashed];

}

(2) Obligation:

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

shuffleA_in_gga(T5, [], T5) → shuffleA_out_gga(T5, [], T5)
shuffleA_in_gga([], T7, T7) → shuffleA_out_gga([], T7, T7)
shuffleA_in_gga(.(T12, T19), [], .(T12, T19)) → shuffleA_out_gga(.(T12, T19), [], .(T12, T19))
shuffleA_in_gga(.(T12, []), T24, .(T12, T24)) → shuffleA_out_gga(.(T12, []), T24, .(T12, T24))
shuffleA_in_gga(.(T33, []), .(T34, T35), .(T34, T37)) → U1_gga(T33, T34, T35, T37, shuffleA_in_gga(.(T33, []), T35, T37))
shuffleA_in_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_gga(T12, T55, T56, T57, T59, shuffleA_in_gga(T56, T57, T59))
shuffleA_in_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_gga(T12, T77, T78, T79, T81, shuffleA_in_gga(T77, T79, T81))
shuffleA_in_gga(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_gga(T93, T94, T95, T96, T98, shuffleA_in_gga(.(T93, T94), T96, T98))
shuffleA_in_gga(T112, .(T106, []), .(T106, T112)) → shuffleA_out_gga(T112, .(T106, []), .(T106, T112))
shuffleA_in_gga([], .(T106, T115), .(T106, T115)) → shuffleA_out_gga([], .(T106, T115), .(T106, T115))
shuffleA_in_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_gga(T132, T133, T106, T134, T136, shuffleA_in_gga(T133, T134, T136))
shuffleA_in_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_gga(T146, T106, T147, T148, T150, shuffleA_in_gga(T146, T148, T150))
U6_gga(T146, T106, T147, T148, T150, shuffleA_out_gga(T146, T148, T150)) → shuffleA_out_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150)))
U5_gga(T132, T133, T106, T134, T136, shuffleA_out_gga(T133, T134, T136)) → shuffleA_out_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136)))
U4_gga(T93, T94, T95, T96, T98, shuffleA_out_gga(.(T93, T94), T96, T98)) → shuffleA_out_gga(.(T93, T94), .(T95, T96), .(T95, T98))
U3_gga(T12, T77, T78, T79, T81, shuffleA_out_gga(T77, T79, T81)) → shuffleA_out_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81)))
U2_gga(T12, T55, T56, T57, T59, shuffleA_out_gga(T56, T57, T59)) → shuffleA_out_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59)))
U1_gga(T33, T34, T35, T37, shuffleA_out_gga(.(T33, []), T35, T37)) → shuffleA_out_gga(.(T33, []), .(T34, T35), .(T34, T37))

The argument filtering Pi contains the following mapping:
shuffleA_in_gga(x1, x2, x3)  =  shuffleA_in_gga(x1, x2)
[]  =  []
shuffleA_out_gga(x1, x2, x3)  =  shuffleA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(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)

(3) 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:

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

The TRS R consists of the following rules:

shuffleA_in_gga(T5, [], T5) → shuffleA_out_gga(T5, [], T5)
shuffleA_in_gga([], T7, T7) → shuffleA_out_gga([], T7, T7)
shuffleA_in_gga(.(T12, T19), [], .(T12, T19)) → shuffleA_out_gga(.(T12, T19), [], .(T12, T19))
shuffleA_in_gga(.(T12, []), T24, .(T12, T24)) → shuffleA_out_gga(.(T12, []), T24, .(T12, T24))
shuffleA_in_gga(.(T33, []), .(T34, T35), .(T34, T37)) → U1_gga(T33, T34, T35, T37, shuffleA_in_gga(.(T33, []), T35, T37))
shuffleA_in_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_gga(T12, T55, T56, T57, T59, shuffleA_in_gga(T56, T57, T59))
shuffleA_in_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_gga(T12, T77, T78, T79, T81, shuffleA_in_gga(T77, T79, T81))
shuffleA_in_gga(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_gga(T93, T94, T95, T96, T98, shuffleA_in_gga(.(T93, T94), T96, T98))
shuffleA_in_gga(T112, .(T106, []), .(T106, T112)) → shuffleA_out_gga(T112, .(T106, []), .(T106, T112))
shuffleA_in_gga([], .(T106, T115), .(T106, T115)) → shuffleA_out_gga([], .(T106, T115), .(T106, T115))
shuffleA_in_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_gga(T132, T133, T106, T134, T136, shuffleA_in_gga(T133, T134, T136))
shuffleA_in_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_gga(T146, T106, T147, T148, T150, shuffleA_in_gga(T146, T148, T150))
U6_gga(T146, T106, T147, T148, T150, shuffleA_out_gga(T146, T148, T150)) → shuffleA_out_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150)))
U5_gga(T132, T133, T106, T134, T136, shuffleA_out_gga(T133, T134, T136)) → shuffleA_out_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136)))
U4_gga(T93, T94, T95, T96, T98, shuffleA_out_gga(.(T93, T94), T96, T98)) → shuffleA_out_gga(.(T93, T94), .(T95, T96), .(T95, T98))
U3_gga(T12, T77, T78, T79, T81, shuffleA_out_gga(T77, T79, T81)) → shuffleA_out_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81)))
U2_gga(T12, T55, T56, T57, T59, shuffleA_out_gga(T56, T57, T59)) → shuffleA_out_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59)))
U1_gga(T33, T34, T35, T37, shuffleA_out_gga(.(T33, []), T35, T37)) → shuffleA_out_gga(.(T33, []), .(T34, T35), .(T34, T37))

The argument filtering Pi contains the following mapping:
shuffleA_in_gga(x1, x2, x3)  =  shuffleA_in_gga(x1, x2)
[]  =  []
shuffleA_out_gga(x1, x2, x3)  =  shuffleA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(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)
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

(4) Obligation:

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

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

The TRS R consists of the following rules:

shuffleA_in_gga(T5, [], T5) → shuffleA_out_gga(T5, [], T5)
shuffleA_in_gga([], T7, T7) → shuffleA_out_gga([], T7, T7)
shuffleA_in_gga(.(T12, T19), [], .(T12, T19)) → shuffleA_out_gga(.(T12, T19), [], .(T12, T19))
shuffleA_in_gga(.(T12, []), T24, .(T12, T24)) → shuffleA_out_gga(.(T12, []), T24, .(T12, T24))
shuffleA_in_gga(.(T33, []), .(T34, T35), .(T34, T37)) → U1_gga(T33, T34, T35, T37, shuffleA_in_gga(.(T33, []), T35, T37))
shuffleA_in_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_gga(T12, T55, T56, T57, T59, shuffleA_in_gga(T56, T57, T59))
shuffleA_in_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_gga(T12, T77, T78, T79, T81, shuffleA_in_gga(T77, T79, T81))
shuffleA_in_gga(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_gga(T93, T94, T95, T96, T98, shuffleA_in_gga(.(T93, T94), T96, T98))
shuffleA_in_gga(T112, .(T106, []), .(T106, T112)) → shuffleA_out_gga(T112, .(T106, []), .(T106, T112))
shuffleA_in_gga([], .(T106, T115), .(T106, T115)) → shuffleA_out_gga([], .(T106, T115), .(T106, T115))
shuffleA_in_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_gga(T132, T133, T106, T134, T136, shuffleA_in_gga(T133, T134, T136))
shuffleA_in_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_gga(T146, T106, T147, T148, T150, shuffleA_in_gga(T146, T148, T150))
U6_gga(T146, T106, T147, T148, T150, shuffleA_out_gga(T146, T148, T150)) → shuffleA_out_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150)))
U5_gga(T132, T133, T106, T134, T136, shuffleA_out_gga(T133, T134, T136)) → shuffleA_out_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136)))
U4_gga(T93, T94, T95, T96, T98, shuffleA_out_gga(.(T93, T94), T96, T98)) → shuffleA_out_gga(.(T93, T94), .(T95, T96), .(T95, T98))
U3_gga(T12, T77, T78, T79, T81, shuffleA_out_gga(T77, T79, T81)) → shuffleA_out_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81)))
U2_gga(T12, T55, T56, T57, T59, shuffleA_out_gga(T56, T57, T59)) → shuffleA_out_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59)))
U1_gga(T33, T34, T35, T37, shuffleA_out_gga(.(T33, []), T35, T37)) → shuffleA_out_gga(.(T33, []), .(T34, T35), .(T34, T37))

The argument filtering Pi contains the following mapping:
shuffleA_in_gga(x1, x2, x3)  =  shuffleA_in_gga(x1, x2)
[]  =  []
shuffleA_out_gga(x1, x2, x3)  =  shuffleA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(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)
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(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → SHUFFLEA_IN_GGA(T77, T79, T81)
SHUFFLEA_IN_GGA(.(T33, []), .(T34, T35), .(T34, T37)) → SHUFFLEA_IN_GGA(.(T33, []), T35, T37)
SHUFFLEA_IN_GGA(.(T93, T94), .(T95, T96), .(T95, T98)) → SHUFFLEA_IN_GGA(.(T93, T94), T96, T98)
SHUFFLEA_IN_GGA(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → SHUFFLEA_IN_GGA(T56, T57, T59)
SHUFFLEA_IN_GGA(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → SHUFFLEA_IN_GGA(T133, T134, T136)
SHUFFLEA_IN_GGA(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → SHUFFLEA_IN_GGA(T146, T148, T150)

The TRS R consists of the following rules:

shuffleA_in_gga(T5, [], T5) → shuffleA_out_gga(T5, [], T5)
shuffleA_in_gga([], T7, T7) → shuffleA_out_gga([], T7, T7)
shuffleA_in_gga(.(T12, T19), [], .(T12, T19)) → shuffleA_out_gga(.(T12, T19), [], .(T12, T19))
shuffleA_in_gga(.(T12, []), T24, .(T12, T24)) → shuffleA_out_gga(.(T12, []), T24, .(T12, T24))
shuffleA_in_gga(.(T33, []), .(T34, T35), .(T34, T37)) → U1_gga(T33, T34, T35, T37, shuffleA_in_gga(.(T33, []), T35, T37))
shuffleA_in_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59))) → U2_gga(T12, T55, T56, T57, T59, shuffleA_in_gga(T56, T57, T59))
shuffleA_in_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81))) → U3_gga(T12, T77, T78, T79, T81, shuffleA_in_gga(T77, T79, T81))
shuffleA_in_gga(.(T93, T94), .(T95, T96), .(T95, T98)) → U4_gga(T93, T94, T95, T96, T98, shuffleA_in_gga(.(T93, T94), T96, T98))
shuffleA_in_gga(T112, .(T106, []), .(T106, T112)) → shuffleA_out_gga(T112, .(T106, []), .(T106, T112))
shuffleA_in_gga([], .(T106, T115), .(T106, T115)) → shuffleA_out_gga([], .(T106, T115), .(T106, T115))
shuffleA_in_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136))) → U5_gga(T132, T133, T106, T134, T136, shuffleA_in_gga(T133, T134, T136))
shuffleA_in_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150))) → U6_gga(T146, T106, T147, T148, T150, shuffleA_in_gga(T146, T148, T150))
U6_gga(T146, T106, T147, T148, T150, shuffleA_out_gga(T146, T148, T150)) → shuffleA_out_gga(T146, .(T106, .(T147, T148)), .(T106, .(T147, T150)))
U5_gga(T132, T133, T106, T134, T136, shuffleA_out_gga(T133, T134, T136)) → shuffleA_out_gga(.(T132, T133), .(T106, T134), .(T106, .(T132, T136)))
U4_gga(T93, T94, T95, T96, T98, shuffleA_out_gga(.(T93, T94), T96, T98)) → shuffleA_out_gga(.(T93, T94), .(T95, T96), .(T95, T98))
U3_gga(T12, T77, T78, T79, T81, shuffleA_out_gga(T77, T79, T81)) → shuffleA_out_gga(.(T12, T77), .(T78, T79), .(T12, .(T78, T81)))
U2_gga(T12, T55, T56, T57, T59, shuffleA_out_gga(T56, T57, T59)) → shuffleA_out_gga(.(T12, .(T55, T56)), T57, .(T12, .(T55, T59)))
U1_gga(T33, T34, T35, T37, shuffleA_out_gga(.(T33, []), T35, T37)) → shuffleA_out_gga(.(T33, []), .(T34, T35), .(T34, T37))

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

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

(7) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(8) Obligation:

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

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

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

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

(11) 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(.(T12, .(T55, T56)), T57) → SHUFFLEA_IN_GGA(T56, T57)
  The graph contains the following edges 1 > 1, 2 >= 2

• SHUFFLEA_IN_GGA(.(T33, []), .(T34, T35)) → SHUFFLEA_IN_GGA(.(T33, []), T35)
  The graph contains the following edges 1 >= 1, 2 > 2

• SHUFFLEA_IN_GGA(.(T12, T77), .(T78, T79)) → SHUFFLEA_IN_GGA(T77, T79)
  The graph contains the following edges 1 > 1, 2 > 2

• SHUFFLEA_IN_GGA(.(T93, T94), .(T95, T96)) → SHUFFLEA_IN_GGA(.(T93, T94), T96)
  The graph contains the following edges 1 >= 1, 2 > 2

• SHUFFLEA_IN_GGA(T146, .(T106, .(T147, T148))) → SHUFFLEA_IN_GGA(T146, T148)
  The graph contains the following edges 1 >= 1, 2 > 2

(12) YES