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

(0) Obligation:

Clauses:

f([], RES, RES).
f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES).
g(A, B, C, RES) :- f(A, .(B, C), RES).

Query: f(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: f(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | f([], T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: f(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX2 is free\nf(T1, T2, T3) !=? f([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: f([], T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: g(T11, T12, .(T10, T11), T14)\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: g(T11, T12, .(T10, T11), T14), SCOPE: 2, CLAUSE: 2\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: f(T28, .(T29, .(T30, T28)), T32)\n\nT28 is ground\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 11 [arrowhead = none ];
11 -> 2;

12 [label="EVAL with clause\nf([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 12 [arrowhead = none ];
12 -> 3;

13 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 13 [arrowhead = none ];
13 -> 4;

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

15 [label="EVAL with clause\nf(.(X11, X12), X13, X14) :- g(X12, X13, .(X11, X12), X14).\nand substitutionX11 -> T10,\nX12 -> T11,\nT1 -> .(T10, T11),\nT2 -> T12,\nX13 -> T12,\nT3 -> T14,\nX14 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 15 [arrowhead = none ];
15 -> 7;

16 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 16 [arrowhead = none ];
16 -> 8;

17 [label="BACKTRACK\nfor clause: f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 17 [arrowhead = none ];
17 -> 6;

18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 18 [arrowhead = none ];
18 -> 9;

19 [label="ONLY EVAL with clause\ng(X25, X26, X27, X28) :- f(X25, .(X26, X27), X28).\nand substitutionT11 -> T28,\nX25 -> T28,\nT12 -> T29,\nX26 -> T29,\nT10 -> T30,\nX27 -> .(T30, T28),\nT14 -> T32,\nX28 -> T32,\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 19 [arrowhead = none ];
19 -> 10;

20 [label="INSTANCE with matching:\nT1 -> T28\nT2 -> .(T29, .(T30, T28))\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
10 -> 20 [arrowhead = none , style=dashed];
20 -> 1[style=dashed];

}

(2) Obligation:

Triples:

fA(.(X1, X2), X3, X4) :- fA(X2, .(X3, .(X1, X2)), X4).

Clauses:

fcA([], X1, X1).
fcA(.(X1, X2), X3, X4) :- fcA(X2, .(X3, .(X1, X2)), X4).

Afs:

fA(x1, x2, x3)  =  fA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
fA_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FA_IN_GGA(.(X1, X2), X3, X4) → U1_GGA(X1, X2, X3, X4, fA_in_gga(X2, .(X3, .(X1, X2)), X4))
FA_IN_GGA(.(X1, X2), X3, X4) → FA_IN_GGA(X2, .(X3, .(X1, X2)), X4)

R is empty.
The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

FA_IN_GGA(.(X1, X2), X3, X4) → U1_GGA(X1, X2, X3, X4, fA_in_gga(X2, .(X3, .(X1, X2)), X4))
FA_IN_GGA(.(X1, X2), X3, X4) → FA_IN_GGA(X2, .(X3, .(X1, X2)), X4)

R is empty.
The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(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:

FA_IN_GGA(.(X1, X2), X3, X4) → FA_IN_GGA(X2, .(X3, .(X1, X2)), X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

FA_IN_GGA(.(X1, X2), X3) → FA_IN_GGA(X2, .(X3, .(X1, X2)))

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

(9) 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:

• FA_IN_GGA(.(X1, X2), X3) → FA_IN_GGA(X2, .(X3, .(X1, X2)))
  The graph contains the following edges 1 > 1

(10) YES