Left Termination of the query pattern duplicate_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

duplicate([], L) :- ','(!, eq(L, [])).
duplicate(X, .(H, .(H, Z))) :- ','(head(X, H), ','(tail(X, T), duplicate(T, Z))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Queries:

duplicate(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: duplicate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0 | duplicate(T1, T2), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T4, [])) | duplicate([], T2), SCOPE: 1, CLAUSE: 1 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX5 is free; \nduplicate(T1, T2) !=? duplicate([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T4, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T4, []), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T6, T9), ','(tail(T6, X14), duplicate(X14, T10)))\n\nT6 is ground\nX5 is free; \nX14 is free; \nduplicate(T6, T2) !=? duplicate([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\nX5 is free\nX11 is free; \nX12 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T6, T9), ','(tail(T6, X14), duplicate(X14, T10))), SCOPE: 3, CLAUSE: 2 | ','(head(T6, T9), ','(tail(T6, X14), duplicate(X14, T10))), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nX5 is free; \nX14 is free; \nduplicate(T6, T2) !=? duplicate([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T6, T9), ','(tail(T6, X14), duplicate(X14, T10))), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nX5 is free; \nX14 is free; \nX16 is free; \nduplicate(T6, T2) !=? duplicate([], X5)\nhead(T6, T9) !=? head([], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail(.(T11, T12), X14), duplicate(X14, T13))\n\nT11 is ground\nT12 is ground\nX5 is free; \nX14 is free; \nX16 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\nX5 is free\nX16 is free; \nX19 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T11, T12), X14), duplicate(X14, T13)), SCOPE: 4, CLAUSE: 4 | ','(tail(.(T11, T12), X14), duplicate(X14, T13)), SCOPE: 4, CLAUSE: 5\n\nT11 is ground\nT12 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(.(T11, T12), X14), duplicate(X14, T13)), SCOPE: 4, CLAUSE: 5\n\nT11 is ground\nT12 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: duplicate(T15, T13)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 19 [arrowhead = none ];
19 -> 2;

20 [label="EVAL with clause\nduplicate([], X5) :- ','(!, eq(X5, [])).\nand substitution\nT1 -> []\nT2 -> T4\nX5 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 20 [arrowhead = none ];
20 -> 3;

21 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 21 [arrowhead = none ];
21 -> 4;

22 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 22 [arrowhead = none ];
22 -> 5;

23 [label="EVAL with clause\nduplicate(X11, .(X12, .(X12, X13))) :- ','(head(X11, X12), ','(tail(X11, X14), duplicate(X14, X13))).\nand substitution\nT1 -> T6\nX11 -> T6\nX12 -> T9\nX13 -> T10\nT2 -> .(T9, .(T9, T10))\nT7 -> T9\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 23 [arrowhead = none ];
23 -> 10;

24 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 24 [arrowhead = none ];
24 -> 11;

25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 25 [arrowhead = none ];
25 -> 6;

26 [label="EVAL with clause\neq(X7, X7).\nand substitution\nT4 -> []\nX7 -> []\nT5 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 26 [arrowhead = none ];
26 -> 7;

27 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 27 [arrowhead = none ];
27 -> 8;

28 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 28 [arrowhead = none ];
28 -> 9;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 29 [arrowhead = none ];
29 -> 12;

30 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (duplicate(T6, T2), duplicate([], X5))", fontsize=14, style = filled, fillcolor = white];
12 -> 30 [arrowhead = none ];
30 -> 13;

31 [label="EVAL with clause\nhead(.(X19, X20), X19).\nand substitution\nX19 -> T11\nX20 -> T12\nT6 -> .(T11, T12)\nT9 -> T11\nT10 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 31 [arrowhead = none ];
31 -> 14;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 32 [arrowhead = none ];
32 -> 15;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 33 [arrowhead = none ];
33 -> 16;

34 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 34 [arrowhead = none ];
34 -> 17;

35 [label="EVAL with clause\ntail(.(X23, X24), X24).\nand substitution\nT11 -> T14\nX23 -> T14\nT12 -> T15\nX24 -> T15\nX14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 35 [arrowhead = none ];
35 -> 18;

36 [label="INSTANCE with matching:\nT1 -> T15\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 36 [arrowhead = none , style=dashed];
36 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

duplicate1([], []).
duplicate1(.(T14, T15), .(T14, .(T14, T13))) :- duplicate1(T15, T13).

Queries:

duplicate1(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:
duplicate1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T14, T15), .(T14, .(T14, T13))) → U1_ga(T14, T15, T13, duplicate1_in_ga(T15, T13))
U1_ga(T14, T15, T13, duplicate1_out_ga(T15, T13)) → duplicate1_out_ga(.(T14, T15), .(T14, .(T14, T13)))

The argument filtering Pi contains the following mapping:
duplicate1_in_ga(x1, x2)  =  duplicate1_in_ga(x1)
[]  =  []
duplicate1_out_ga(x1, x2)  =  duplicate1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T14, T15), .(T14, .(T14, T13))) → U1_ga(T14, T15, T13, duplicate1_in_ga(T15, T13))
U1_ga(T14, T15, T13, duplicate1_out_ga(T15, T13)) → duplicate1_out_ga(.(T14, T15), .(T14, .(T14, T13)))

The argument filtering Pi contains the following mapping:
duplicate1_in_ga(x1, x2)  =  duplicate1_in_ga(x1)
[]  =  []
duplicate1_out_ga(x1, x2)  =  duplicate1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

(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:

DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → U1_GA(T14, T15, T13, duplicate1_in_ga(T15, T13))
DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → DUPLICATE1_IN_GA(T15, T13)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T14, T15), .(T14, .(T14, T13))) → U1_ga(T14, T15, T13, duplicate1_in_ga(T15, T13))
U1_ga(T14, T15, T13, duplicate1_out_ga(T15, T13)) → duplicate1_out_ga(.(T14, T15), .(T14, .(T14, T13)))

The argument filtering Pi contains the following mapping:
duplicate1_in_ga(x1, x2)  =  duplicate1_in_ga(x1)
[]  =  []
duplicate1_out_ga(x1, x2)  =  duplicate1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
DUPLICATE1_IN_GA(x1, x2)  =  DUPLICATE1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

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

(6) Obligation:

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

DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → U1_GA(T14, T15, T13, duplicate1_in_ga(T15, T13))
DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → DUPLICATE1_IN_GA(T15, T13)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T14, T15), .(T14, .(T14, T13))) → U1_ga(T14, T15, T13, duplicate1_in_ga(T15, T13))
U1_ga(T14, T15, T13, duplicate1_out_ga(T15, T13)) → duplicate1_out_ga(.(T14, T15), .(T14, .(T14, T13)))

The argument filtering Pi contains the following mapping:
duplicate1_in_ga(x1, x2)  =  duplicate1_in_ga(x1)
[]  =  []
duplicate1_out_ga(x1, x2)  =  duplicate1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
DUPLICATE1_IN_GA(x1, x2)  =  DUPLICATE1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → DUPLICATE1_IN_GA(T15, T13)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T14, T15), .(T14, .(T14, T13))) → U1_ga(T14, T15, T13, duplicate1_in_ga(T15, T13))
U1_ga(T14, T15, T13, duplicate1_out_ga(T15, T13)) → duplicate1_out_ga(.(T14, T15), .(T14, .(T14, T13)))

The argument filtering Pi contains the following mapping:
duplicate1_in_ga(x1, x2)  =  duplicate1_in_ga(x1)
[]  =  []
duplicate1_out_ga(x1, x2)  =  duplicate1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
DUPLICATE1_IN_GA(x1, x2)  =  DUPLICATE1_IN_GA(x1)

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:

DUPLICATE1_IN_GA(.(T14, T15), .(T14, .(T14, T13))) → DUPLICATE1_IN_GA(T15, T13)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DUPLICATE1_IN_GA(x1, x2)  =  DUPLICATE1_IN_GA(x1)

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:

DUPLICATE1_IN_GA(.(T14, T15)) → DUPLICATE1_IN_GA(T15)

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:

• DUPLICATE1_IN_GA(.(T14, T15)) → DUPLICATE1_IN_GA(T15)
  The graph contains the following edges 1 > 1

(14) TRUE