Left Termination of the query pattern append_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
append([], Y, Y).

Queries:

append(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append1(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) :- append1(T30, T31, T33).

Clauses:

appendc1(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) :- appendc1(T30, T31, T33).
appendc1(.(T8, []), T42, .([], T42)).
appendc1([], T44, T44).

Afs:

append1(x1, x2, x3)  =  append1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
append1_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

R is empty.
The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)

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:

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

R is empty.
The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)

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:

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_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:

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31) → APPEND1_IN_GGA(T30, T31)

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:

• APPEND1_IN_GGA(.(T8, .(T29, T30)), T31) → APPEND1_IN_GGA(T30, T31)
  The graph contains the following edges 1 > 1, 2 >= 2

(10) YES