Left Termination of the query pattern app2_in_3(a, g, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).

Queries:

app2(a,g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

app21(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) :- app21(T33, T31, T32).

Clauses:

app2c1(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) :- app2c1(T33, T31, T32).
app2c1(.(T8, []), T42, .(T8, T42)).
app2c1([], .(T50, T51), .(T50, T51)).
app2c1([], T53, T53).

Afs:

app21(x1, x2, x3)  =  app21(x2, x3)

(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:
app21_in: (f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x4, x5, 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:

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → U1_AGG(T8, T29, T33, T31, T32, app21_in_agg(T33, T31, T32))
APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
app21_in_agg(x1, x2, x3)  =  app21_in_agg(x2, x3)
.(x1, x2)  =  .(x1, x2)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)
U1_AGG(x1, x2, x3, x4, x5, x6)  =  U1_AGG(x1, x2, x4, x5, 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:

APP21_IN_AGG(.(T8, .(T29, T33)), T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T33, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP21_IN_AGG(x1, x2, x3)  =  APP21_IN_AGG(x2, x3)

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:

APP21_IN_AGG(T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T31, T32)

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:

• APP21_IN_AGG(T31, .(T8, .(T29, T32))) → APP21_IN_AGG(T31, T32)
  The graph contains the following edges 1 >= 1, 2 > 2

(10) YES