Left Termination of the query pattern app_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

app([], Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
eq(X, X).

Queries:

app(g,a,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

app1(.(T41, T42), T31, .(T41, T32)) :- app1(T42, T31, T32).

Clauses:

appc1([], T12, T12).
appc1(.(T41, T42), T31, .(T41, T32)) :- appc1(T42, T31, T32).

Afs:

app1(x1, x2, x3)  =  app1(x1)

(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:
app1_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

APP1_IN_GAA(.(T41, T42), T31, .(T41, T32)) → U1_GAA(T41, T42, T31, T32, app1_in_gaa(T42, T31, T32))
APP1_IN_GAA(.(T41, T42), T31, .(T41, T32)) → APP1_IN_GAA(T42, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3)  =  app1_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
APP1_IN_GAA(x1, x2, x3)  =  APP1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, 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:

APP1_IN_GAA(.(T41, T42), T31, .(T41, T32)) → U1_GAA(T41, T42, T31, T32, app1_in_gaa(T42, T31, T32))
APP1_IN_GAA(.(T41, T42), T31, .(T41, T32)) → APP1_IN_GAA(T42, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
app1_in_gaa(x1, x2, x3)  =  app1_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
APP1_IN_GAA(x1, x2, x3)  =  APP1_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, 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:

APP1_IN_GAA(.(T41, T42), T31, .(T41, T32)) → APP1_IN_GAA(T42, T31, T32)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP1_IN_GAA(x1, x2, x3)  =  APP1_IN_GAA(x1)

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:

APP1_IN_GAA(.(T41, T42)) → APP1_IN_GAA(T42)

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:

• APP1_IN_GAA(.(T41, T42)) → APP1_IN_GAA(T42)
  The graph contains the following edges 1 > 1

(10) YES