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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 PiDPToQDPProof (⇔)
        ↳9 QDP
        ↳10 NonTerminationProof (⇔)
        ↳11 NO
    ↳12 PiDP
        ↳13 PiDPToQDPProof (⇔)
        ↳14 QDP
        ↳15 NonTerminationProof (⇔)
        ↳16 NO

(0) Obligation:

Clauses:

a :- b.
a :- e.
b :- c.
c :- d.
d :- b.
e :- f.
f :- g.
g :- e.

Queries:

a().

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

c7 :- c7.
e15 :- e15.
a1 :- c7.
a1 :- e15.

Clauses:

cc7 :- cc7.
ec15 :- ec15.

Afs:

a1  =  a1

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

A1_IN_ → U3_1(c7_in_)
A1_IN_ → C7_IN_
C7_IN_ → U1_1(c7_in_)
C7_IN_ → C7_IN_
A1_IN_ → U4_1(e15_in_)
A1_IN_ → E15_IN_
E15_IN_ → U2_1(e15_in_)
E15_IN_ → E15_IN_

R is empty.
Pi is empty.
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:

A1_IN_ → U3_1(c7_in_)
A1_IN_ → C7_IN_
C7_IN_ → U1_1(c7_in_)
C7_IN_ → C7_IN_
A1_IN_ → U4_1(e15_in_)
A1_IN_ → E15_IN_
E15_IN_ → U2_1(e15_in_)
E15_IN_ → E15_IN_

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

E15_IN_ → E15_IN_

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

(8) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(9) Obligation:

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

E15_IN_ → E15_IN_

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

(10) 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 = E15_IN_ evaluates to t =E15_IN_

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 E15_IN_ to E15_IN_.



(11) NO

(12) Obligation:

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

C7_IN_ → C7_IN_

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

(13) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(14) Obligation:

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

C7_IN_ → C7_IN_

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

(15) 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 = C7_IN_ evaluates to t =C7_IN_

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from C7_IN_ to C7_IN_.



(16) NO