Left Termination of the query pattern prefix_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

prefix(Xs, Ys) :- app(Xs, X1, Ys).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

prefix(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

app3(.(T19, T22), X32, .(T19, T21)) :- app3(T22, X32, T21).
prefix1(T7, T6) :- app3(T7, X6, T6).

Clauses:

appc3([], T12, T12).
appc3(.(T19, T22), X32, .(T19, T21)) :- appc3(T22, X32, T21).

Afs:

prefix1(x1, x2)  =  prefix1(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:
prefix1_in: (f,b)
app3_in: (f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PREFIX1_IN_AG(T7, T6) → U2_AG(T7, T6, app3_in_aag(T7, X6, T6))
PREFIX1_IN_AG(T7, T6) → APP3_IN_AAG(T7, X6, T6)
APP3_IN_AAG(.(T19, T22), X32, .(T19, T21)) → U1_AAG(T19, T22, X32, T21, app3_in_aag(T22, X32, T21))
APP3_IN_AAG(.(T19, T22), X32, .(T19, T21)) → APP3_IN_AAG(T22, X32, T21)

R is empty.
The argument filtering Pi contains the following mapping:
app3_in_aag(x1, x2, x3)  =  app3_in_aag(x3)
.(x1, x2)  =  .(x1, x2)
PREFIX1_IN_AG(x1, x2)  =  PREFIX1_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
APP3_IN_AAG(x1, x2, x3)  =  APP3_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x4, 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:

PREFIX1_IN_AG(T7, T6) → U2_AG(T7, T6, app3_in_aag(T7, X6, T6))
PREFIX1_IN_AG(T7, T6) → APP3_IN_AAG(T7, X6, T6)
APP3_IN_AAG(.(T19, T22), X32, .(T19, T21)) → U1_AAG(T19, T22, X32, T21, app3_in_aag(T22, X32, T21))
APP3_IN_AAG(.(T19, T22), X32, .(T19, T21)) → APP3_IN_AAG(T22, X32, T21)

R is empty.
The argument filtering Pi contains the following mapping:
app3_in_aag(x1, x2, x3)  =  app3_in_aag(x3)
.(x1, x2)  =  .(x1, x2)
PREFIX1_IN_AG(x1, x2)  =  PREFIX1_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
APP3_IN_AAG(x1, x2, x3)  =  APP3_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x4, 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 3 less nodes.

(6) Obligation:

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

APP3_IN_AAG(.(T19, T22), X32, .(T19, T21)) → APP3_IN_AAG(T22, X32, T21)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP3_IN_AAG(x1, x2, x3)  =  APP3_IN_AAG(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:

APP3_IN_AAG(.(T19, T21)) → APP3_IN_AAG(T21)

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:

• APP3_IN_AAG(.(T19, T21)) → APP3_IN_AAG(T21)
  The graph contains the following edges 1 > 1

(10) YES