Left Termination of the query pattern shuffle_in_3(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).

Queries:

shuffle(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | shuffle(T4, [], T3), SCOPE: 1, CLAUSE: 1 | shuffle(T4, [], T3), SCOPE: 1, CLAUSE: 2 | shuffle(T4, [], T3), SCOPE: 1, CLAUSE: 3 | ?_1\n\nT4 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 | ?_1\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([], T5, T3), SCOPE: 1, CLAUSE: 2 | shuffle([], T5, T3), SCOPE: 1, CLAUSE: 3 | ?_1\n\nT5 is ground\nX2 is free; \nshuffle([], T5, 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\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: shuffle(T1, T2, T3), SCOPE: 1, CLAUSE: 2\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];
12 [label="12: 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];
13 [label="13: shuffle(T7, T8, T10)\n\nT7 is ground\nT8 is ground\nX2 is free; \nX4 is free; \nshuffle(.(T6, T7), T8, T3) !=? shuffle(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX2 is free\nX4 is free; \nX13 is free; \nX14 is free; \nX15 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: shuffle(T11, T13, T15)\n\nT11 is ground\nT13 is ground\nX2 is free; \nX4 is free; \nshuffle(T11, .(T12, T13), T3) !=? shuffle([], X4, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\nX2 is free\nX4 is free; \nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 17 [arrowhead = none ];
17 -> 2;

18 [label="EVAL with clause\nshuffle(X2, [], X2) :- !.\nand substitution\nT1 -> T4\nX2 -> T4\nT2 -> []\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 18 [arrowhead = none ];
18 -> 3;

19 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 19 [arrowhead = none ];
19 -> 4;

20 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 20 [arrowhead = none ];
20 -> 5;

21 [label="EVAL with clause\nshuffle([], X4, X4) :- !.\nand substitution\nT1 -> []\nT2 -> T5\nX4 -> T5\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 21 [arrowhead = none ];
21 -> 7;

22 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 22 [arrowhead = none ];
22 -> 8;

23 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 23 [arrowhead = none ];
23 -> 6;

24 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 25 [arrowhead = none ];
25 -> 11;

26 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 26 [arrowhead = none ];
26 -> 12;

27 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 27 [arrowhead = none ];
27 -> 10;

28 [label="EVAL with clause\nshuffle(.(X13, X14), X15, .(X13, X16)) :- shuffle(X14, X15, X16).\nand substitution\nX13 -> T6\nX14 -> T7\nT1 -> .(T6, T7)\nT2 -> T8\nX15 -> T8\nX16 -> T10\nT3 -> .(T6, T10)\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 28 [arrowhead = none ];
28 -> 13;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 29 [arrowhead = none ];
29 -> 14;

30 [label="EVAL with clause\nshuffle(X21, .(X22, X23), .(X22, X24)) :- shuffle(X21, X23, X24).\nand substitution\nT1 -> T11\nX21 -> T11\nX22 -> T12\nX23 -> T13\nT2 -> .(T12, T13)\nX24 -> T15\nT3 -> .(T12, T15)\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 30 [arrowhead = none ];
30 -> 15;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 31 [arrowhead = none ];
31 -> 16;

32 [label="INSTANCE with matching:\nT1 -> T7\nT2 -> T8\nT3 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
13 -> 32 [arrowhead = none , style=dashed];
32 -> 1[style=dashed];

33 [label="INSTANCE with matching:\nT1 -> T11\nT2 -> T13\nT3 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
15 -> 33 [arrowhead = none , style=dashed];
33 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

shuffle1(T4, [], T4).
shuffle1([], T5, T5).
shuffle1(.(T6, T7), T8, .(T6, T10)) :- shuffle1(T7, T8, T10).
shuffle1(T11, .(T12, T13), .(T12, T15)) :- shuffle1(T11, T13, T15).

Queries:

shuffle1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
shuffle1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → U1_GGA(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → U2_GGA(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x5)

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

(6) Obligation:

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

SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → U1_GGA(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → U2_GGA(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

SHUFFLE1_IN_GGA(T11, .(T12, T13)) → SHUFFLE1_IN_GGA(T11, T13)
SHUFFLE1_IN_GGA(.(T6, T7), T8) → SHUFFLE1_IN_GGA(T7, T8)

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

(13) 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(T11, .(T12, T13)) → SHUFFLE1_IN_GGA(T11, T13)
  The graph contains the following edges 1 >= 1, 2 > 2

• SHUFFLE1_IN_GGA(.(T6, T7), T8) → SHUFFLE1_IN_GGA(T7, T8)
  The graph contains the following edges 1 > 1, 2 >= 2

(14) TRUE