Left Termination of the query pattern p2_in_1(a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 PiDP
↳7 PiDPToQDPProof (⇐)
↳8 QDP
↳9 NonTerminationProof (⇔)
↳10 NO

(0) Obligation:

Clauses:

p1(f(X)) :- p1(X).
p2(f(X)) :- p2(X).

Queries:

p2(a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

p21(f(f(T8))) :- p21(T8).

Clauses:

p2c1(f(f(T8))) :- p2c1(T8).

Afs:

p21(x1)  =  p21

(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:
p21_in: (f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

P21_IN_A(f(f(T8))) → U1_A(T8, p21_in_a(T8))
P21_IN_A(f(f(T8))) → P21_IN_A(T8)

R is empty.
The argument filtering Pi contains the following mapping:
p21_in_a(x1)  =  p21_in_a
f(x1)  =  f(x1)
P21_IN_A(x1)  =  P21_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:

P21_IN_A(f(f(T8))) → U1_A(T8, p21_in_a(T8))
P21_IN_A(f(f(T8))) → P21_IN_A(T8)

R is empty.
The argument filtering Pi contains the following mapping:
p21_in_a(x1)  =  p21_in_a
f(x1)  =  f(x1)
P21_IN_A(x1)  =  P21_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 1 less node.

(6) Obligation:

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

P21_IN_A(f(f(T8))) → P21_IN_A(T8)

R is empty.
The argument filtering Pi contains the following mapping:
f(x1)  =  f(x1)
P21_IN_A(x1)  =  P21_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:

P21_IN_A → P21_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 = P21_IN_A evaluates to t =P21_IN_A

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 P21_IN_A to P21_IN_A.



(10) NO