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

(0) Obligation:

Clauses:

duplicate([], []).
duplicate(.(X, Y), .(X, .(X, Z))) :- duplicate(Y, Z).

Query: duplicate(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: duplicate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0 | duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nduplicate(T1, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: duplicate(T7, T9)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: duplicate(T7, T9), SCOPE: 2, CLAUSE: 0 | duplicate(T7, T9), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: duplicate(T7, T9), SCOPE: 2, CLAUSE: 0\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: duplicate(T7, T9), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: duplicate(T17, T19)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 43 [arrowhead = none ];
43 -> 10;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 44 [arrowhead = none ];
44 -> 11;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="EVAL with clause\nduplicate(.(X7, X8), .(X7, .(X7, X9))) :- duplicate(X8, X9).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> .(T6, T7),\nX9 -> T9,\nT2 -> .(T6, .(T6, T9)),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="BACKTRACK\nfor clause: duplicate(.(X, Y), .(X, .(X, Z))) :- duplicate(Y, Z)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 48 [arrowhead = none ];
48 -> 16;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 49 [arrowhead = none ];
49 -> 26;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 50 [arrowhead = none ];
50 -> 30;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 51 [arrowhead = none ];
51 -> 31;

52 [label="EVAL with clause\nduplicate([], []).\nand substitutionT7 -> [],\nT9 -> []", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 52 [arrowhead = none ];
52 -> 33;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 53 [arrowhead = none ];
53 -> 36;

54 [label="EVAL with clause\nduplicate(.(X16, X17), .(X16, .(X16, X18))) :- duplicate(X17, X18).\nand substitutionX16 -> T16,\nX17 -> T17,\nT7 -> .(T16, T17),\nX18 -> T19,\nT9 -> .(T16, .(T16, T19)),\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 54 [arrowhead = none ];
54 -> 39;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 55 [arrowhead = none ];
55 -> 41;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 56 [arrowhead = none ];
56 -> 37;

57 [label="INSTANCE with matching:\nT1 -> T17\nT2 -> T19", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 57 [arrowhead = none , style=dashed];
57 -> 2[style=dashed];

}

(2) Obligation:

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

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T6, []), .(T6, .(T6, []))) → duplicateA_out_ga(.(T6, []), .(T6, .(T6, [])))
duplicateA_in_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_ga(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
U1_ga(T6, T16, T17, T19, duplicateA_out_ga(T17, T19)) → duplicateA_out_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19)))))

The argument filtering Pi contains the following mapping:
duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
[]  =  []
duplicateA_out_ga(x1, x2)  =  duplicateA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)

(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:

DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_GA(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → DUPLICATEA_IN_GA(T17, T19)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T6, []), .(T6, .(T6, []))) → duplicateA_out_ga(.(T6, []), .(T6, .(T6, [])))
duplicateA_in_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_ga(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
U1_ga(T6, T16, T17, T19, duplicateA_out_ga(T17, T19)) → duplicateA_out_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19)))))

The argument filtering Pi contains the following mapping:
duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
[]  =  []
duplicateA_out_ga(x1, x2)  =  duplicateA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)

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

(4) Obligation:

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

DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_GA(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → DUPLICATEA_IN_GA(T17, T19)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T6, []), .(T6, .(T6, []))) → duplicateA_out_ga(.(T6, []), .(T6, .(T6, [])))
duplicateA_in_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_ga(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
U1_ga(T6, T16, T17, T19, duplicateA_out_ga(T17, T19)) → duplicateA_out_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19)))))

The argument filtering Pi contains the following mapping:
duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
[]  =  []
duplicateA_out_ga(x1, x2)  =  duplicateA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → DUPLICATEA_IN_GA(T17, T19)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T6, []), .(T6, .(T6, []))) → duplicateA_out_ga(.(T6, []), .(T6, .(T6, [])))
duplicateA_in_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → U1_ga(T6, T16, T17, T19, duplicateA_in_ga(T17, T19))
U1_ga(T6, T16, T17, T19, duplicateA_out_ga(T17, T19)) → duplicateA_out_ga(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19)))))

The argument filtering Pi contains the following mapping:
duplicateA_in_ga(x1, x2)  =  duplicateA_in_ga(x1)
[]  =  []
duplicateA_out_ga(x1, x2)  =  duplicateA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
DUPLICATEA_IN_GA(x1, x2)  =  DUPLICATEA_IN_GA(x1)

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:

DUPLICATEA_IN_GA(.(T6, .(T16, T17)), .(T6, .(T6, .(T16, .(T16, T19))))) → DUPLICATEA_IN_GA(T17, T19)

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

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:

DUPLICATEA_IN_GA(.(T6, .(T16, T17))) → DUPLICATEA_IN_GA(T17)

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:

• DUPLICATEA_IN_GA(.(T6, .(T16, T17))) → DUPLICATEA_IN_GA(T17)
  The graph contains the following edges 1 > 1

(12) YES