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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: f(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: 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];
13 [label="13: true | f([], T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: 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];
15 [label="15: f([], T5, T3), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: g(T11, T12, .(T10, T11), T14)\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: 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];
29 [label="29: f(T28, .(T29, .(T30, T28)), T32)\n\nT28 is ground\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 30 [arrowhead = none ];
30 -> 5;

31 [label="EVAL with clause\nf([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 31 [arrowhead = none ];
31 -> 13;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 32 [arrowhead = none ];
32 -> 14;

33 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 33 [arrowhead = none ];
33 -> 15;

34 [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];
14 -> 34 [arrowhead = none ];
34 -> 21;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 35 [arrowhead = none ];
35 -> 24;

36 [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];
15 -> 36 [arrowhead = none ];
36 -> 16;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 37 [arrowhead = none ];
37 -> 25;

38 [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];
25 -> 38 [arrowhead = none ];
38 -> 29;

39 [label="INSTANCE with matching:\nT1 -> T28\nT2 -> .(T29, .(T30, T28))\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 39 [arrowhead = none , style=dashed];
39 -> 3[style=dashed];

}

(2) Obligation:

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

fA_in_gga([], T5, T5) → fA_out_gga([], T5, T5)
fA_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, fA_out_gga(T28, .(T29, .(T30, T28)), T32)) → fA_out_gga(.(T30, T28), T29, T32)

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

FA_IN_GGA(.(T30, T28), T29, T32) → U1_GGA(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
FA_IN_GGA(.(T30, T28), T29, T32) → FA_IN_GGA(T28, .(T29, .(T30, T28)), T32)

The TRS R consists of the following rules:

fA_in_gga([], T5, T5) → fA_out_gga([], T5, T5)
fA_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, fA_out_gga(T28, .(T29, .(T30, T28)), T32)) → fA_out_gga(.(T30, T28), T29, T32)

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

(4) Obligation:

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

FA_IN_GGA(.(T30, T28), T29, T32) → U1_GGA(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
FA_IN_GGA(.(T30, T28), T29, T32) → FA_IN_GGA(T28, .(T29, .(T30, T28)), T32)

The TRS R consists of the following rules:

fA_in_gga([], T5, T5) → fA_out_gga([], T5, T5)
fA_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, fA_out_gga(T28, .(T29, .(T30, T28)), T32)) → fA_out_gga(.(T30, T28), T29, T32)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
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(.(T30, T28), T29, T32) → FA_IN_GGA(T28, .(T29, .(T30, T28)), T32)

The TRS R consists of the following rules:

fA_in_gga([], T5, T5) → fA_out_gga([], T5, T5)
fA_in_gga(.(T30, T28), T29, T32) → U1_gga(T30, T28, T29, T32, fA_in_gga(T28, .(T29, .(T30, T28)), T32))
U1_gga(T30, T28, T29, T32, fA_out_gga(T28, .(T29, .(T30, T28)), T32)) → fA_out_gga(.(T30, T28), T29, T32)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)

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:

FA_IN_GGA(.(T30, T28), T29, T32) → FA_IN_GGA(T28, .(T29, .(T30, T28)), T32)

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

(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:

FA_IN_GGA(.(T30, T28), T29) → FA_IN_GGA(T28, .(T29, .(T30, T28)))

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:

• FA_IN_GGA(.(T30, T28), T29) → FA_IN_GGA(T28, .(T29, .(T30, T28)))
  The graph contains the following edges 1 > 1

(12) YES