Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

p(a).
p(X) :- p(Y).
q(b).
goal(X) :- ','(p(X), q(X)).

Queries:

goal(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

p12(X15) :- p12(X14).
goal1(T6) :- p12(X7).

Clauses:

pc12(a).
pc12(X15) :- pc12(X14).

Afs:

goal1(x1)  =  goal1(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:
goal1_in: (b)
p12_in: (f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)

R is empty.
The argument filtering Pi contains the following mapping:
p12_in_a(x1)  =  p12_in_a
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)

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:

GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)

R is empty.
The argument filtering Pi contains the following mapping:
p12_in_a(x1)  =  p12_in_a
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x1, x2)
P12_IN_A(x1)  =  P12_IN_A
U1_A(x1, x2)  =  U1_A(x2)

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:

P12_IN_A(X15) → P12_IN_A(X14)

R is empty.
The argument filtering Pi contains the following mapping:
P12_IN_A(x1)  =  P12_IN_A

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:

P12_IN_A → P12_IN_A

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 = P12_IN_A evaluates to t =P12_IN_A

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

• Semiunifier: [ ]
• Matcher: [ ]

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Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P12_IN_A to P12_IN_A.

(10) NO