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([], []).
duplicate(X, .(H, .(H, Z))) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), duplicate(T, Z)))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

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\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(empty(T3)), ','(head(T3, T6), ','(tail(T3, X11), duplicate(X11, T7))))\n\nT3 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\nX8 is free\nX9 is free; \nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(no(empty(T3)), ','(head(T3, T6), ','(tail(T3, X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty(T3)), ','(head(T3, T6), ','(tail(T3, X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT3 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(empty(T8), ','(!_2, ','(failure(a), ','(head(T8, T6), ','(tail(T8, X11), duplicate(X11, T7)))))) | ','(no(empty(T8)), ','(head(T8, T6), ','(tail(T8, X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT8 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(empty(T8), ','(!_2, ','(failure(a), ','(head(T8, T6), ','(tail(T8, X11), duplicate(X11, T7)))))), SCOPE: 3, CLAUSE: 6 | ','(no(empty(T8)), ','(head(T8, T6), ','(tail(T8, X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nT8 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_2, ','(failure(a), ','(head([], T6), ','(tail([], X11), duplicate(X11, T7))))) | ','(no(empty([])), ','(head([], T6), ','(tail([], X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 8 | ?_2\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(empty(T8)), ','(head(T8, T6), ','(tail(T8, X11), duplicate(X11, T7)))), SCOPE: 2, CLAUSE: 8\n\nT8 is ground\nX11 is free; \nempty(T8) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(failure(a), ','(head([], T6), ','(tail([], X11), duplicate(X11, T7))))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(failure(a), ','(head([], T6), ','(tail([], X11), duplicate(X11, T7)))), SCOPE: 4, CLAUSE: 9\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T9, T6), ','(tail(T9, X11), duplicate(X11, T7)))\n\nT9 is ground\nX11 is free; \nempty(T9) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T9, T6), ','(tail(T9, X11), duplicate(X11, T7))), SCOPE: 5, CLAUSE: 2 | ','(head(T9, T6), ','(tail(T9, X11), duplicate(X11, T7))), SCOPE: 5, CLAUSE: 3\n\nT9 is ground\nX11 is free; \nempty(T9) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T9, T6), ','(tail(T9, X11), duplicate(X11, T7))), SCOPE: 5, CLAUSE: 3\n\nT9 is ground\nX11 is free; \nX17 is free; \nempty(T9) !=? empty([])\nhead(T9, T6) !=? head([], X17)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(.(T10, T11), X11), duplicate(X11, T12))\n\nT10 is ground\nT11 is ground\nX11 is free; \nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX17 is free\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(.(T10, T11), X11), duplicate(X11, T12)), SCOPE: 6, CLAUSE: 4 | ','(tail(.(T10, T11), X11), duplicate(X11, T12)), SCOPE: 6, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T10, T11), X11), duplicate(X11, T12)), SCOPE: 6, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: duplicate(T14, T12)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="EVAL with clause\nduplicate([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL with clause\nduplicate(X8, .(X9, .(X9, X10))) :- ','(no(empty(X8)), ','(head(X8, X9), ','(tail(X8, X11), duplicate(X11, X10)))).\nand substitution\nT1 -> T3\nX8 -> T3\nX9 -> T6\nX10 -> T7\nT2 -> .(T6, .(T6, T7))\nT4 -> T6\nT5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 9;

33 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="EVAL with clause\nno(X13) :- ','(X13, ','(!, failure(a))).\nand substitution\nT3 -> T8\nX13 -> empty(T8)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="EVAL with clause\nempty([]).\nand substitution\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="EVAL with clause\nno(X15).\nand substitution\nT8 -> T9\nX15 -> empty(T9)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 43 [arrowhead = none ];
43 -> 19;

44 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T9), empty([]))", fontsize=14, style = filled, fillcolor = white];
19 -> 44 [arrowhead = none ];
44 -> 20;

45 [label="EVAL with clause\nhead(.(X20, X21), X20).\nand substitution\nX20 -> T10\nX21 -> T11\nT9 -> .(T10, T11)\nT6 -> T10\nT7 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 45 [arrowhead = none ];
45 -> 21;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 46 [arrowhead = none ];
46 -> 22;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="EVAL with clause\ntail(.(X24, X25), X25).\nand substitution\nT10 -> T13\nX24 -> T13\nT11 -> T14\nX25 -> T14\nX11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 49 [arrowhead = none ];
49 -> 25;

50 [label="INSTANCE with matching:\nT1 -> T14\nT2 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 50 [arrowhead = none , style=dashed];
50 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

duplicate1([], []).
duplicate1(.(T13, T14), .(T13, .(T13, T12))) :- duplicate1(T14, T12).

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(.(T13, T14), .(T13, .(T13, T12))) → U1_ga(T13, T14, T12, duplicate1_in_ga(T14, T12))
U1_ga(T13, T14, T12, duplicate1_out_ga(T14, T12)) → duplicate1_out_ga(.(T13, T14), .(T13, .(T13, T12)))

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(.(T13, T14), .(T13, .(T13, T12))) → U1_ga(T13, T14, T12, duplicate1_in_ga(T14, T12))
U1_ga(T13, T14, T12, duplicate1_out_ga(T14, T12)) → duplicate1_out_ga(.(T13, T14), .(T13, .(T13, T12)))

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(.(T13, T14), .(T13, .(T13, T12))) → U1_GA(T13, T14, T12, duplicate1_in_ga(T14, T12))
DUPLICATE1_IN_GA(.(T13, T14), .(T13, .(T13, T12))) → DUPLICATE1_IN_GA(T14, T12)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T13, T14), .(T13, .(T13, T12))) → U1_ga(T13, T14, T12, duplicate1_in_ga(T14, T12))
U1_ga(T13, T14, T12, duplicate1_out_ga(T14, T12)) → duplicate1_out_ga(.(T13, T14), .(T13, .(T13, T12)))

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(.(T13, T14), .(T13, .(T13, T12))) → U1_GA(T13, T14, T12, duplicate1_in_ga(T14, T12))
DUPLICATE1_IN_GA(.(T13, T14), .(T13, .(T13, T12))) → DUPLICATE1_IN_GA(T14, T12)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T13, T14), .(T13, .(T13, T12))) → U1_ga(T13, T14, T12, duplicate1_in_ga(T14, T12))
U1_ga(T13, T14, T12, duplicate1_out_ga(T14, T12)) → duplicate1_out_ga(.(T13, T14), .(T13, .(T13, T12)))

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(.(T13, T14), .(T13, .(T13, T12))) → DUPLICATE1_IN_GA(T14, T12)

The TRS R consists of the following rules:

duplicate1_in_ga([], []) → duplicate1_out_ga([], [])
duplicate1_in_ga(.(T13, T14), .(T13, .(T13, T12))) → U1_ga(T13, T14, T12, duplicate1_in_ga(T14, T12))
U1_ga(T13, T14, T12, duplicate1_out_ga(T14, T12)) → duplicate1_out_ga(.(T13, T14), .(T13, .(T13, T12)))

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(.(T13, T14), .(T13, .(T13, T12))) → DUPLICATE1_IN_GA(T14, T12)

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(.(T13, T14)) → DUPLICATE1_IN_GA(T14)

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(.(T13, T14)) → DUPLICATE1_IN_GA(T14)
  The graph contains the following edges 1 > 1

(14) TRUE