Left Termination of the query pattern suffix_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

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

Queries:

suffix(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

app3(.(X29, X30), T17, .(X29, T19)) :- app3(X30, T17, T19).
suffix1(T5, T7) :- app3(X6, T5, T7).

Clauses:

appc3([], T12, T12).
appc3(.(X29, X30), T17, .(X29, T19)) :- appc3(X30, T17, T19).

Afs:

suffix1(x1, x2)  =  suffix1(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:
suffix1_in: (b,f)
app3_in: (f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

R is empty.
The argument filtering Pi contains the following mapping:
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
.(x1, x2)  =  .(x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, 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:

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

R is empty.
The argument filtering Pi contains the following mapping:
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
.(x1, x2)  =  .(x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(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 3 less nodes.

(6) Obligation:

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

APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(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:

APP3_IN_AGA(T17) → APP3_IN_AGA(T17)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(9) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP3_IN_AGA(T17) evaluates to t =APP3_IN_AGA(T17)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP3_IN_AGA(T17) to APP3_IN_AGA(T17).

(10) NO