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

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 AND
            ↳9 PiDP
                ↳10 UsableRulesProof (⇔)
                ↳11 PiDP
                ↳12 PiDPToQDPProof (⇔)
                ↳13 QDP
                ↳14 NonTerminationProof (⇔)
                ↳15 NO
            ↳16 PiDP
                ↳17 UsableRulesProof (⇔)
                ↳18 PiDP
                ↳19 PiDPToQDPProof (⇔)
                ↳20 QDP
                ↳21 NonTerminationProof (⇔)
                ↳22 NO
    ↳23 PrologToPiTRSProof (⇐)
        ↳24 PiTRS
        ↳25 DependencyPairsProof (⇔)
        ↳26 PiDP
        ↳27 DependencyGraphProof (⇔)
        ↳28 AND
            ↳29 PiDP
                ↳30 UsableRulesProof (⇔)
                ↳31 PiDP
                ↳32 PiDPToQDPProof (⇔)
                ↳33 QDP
                ↳34 NonTerminationProof (⇔)
                ↳35 NO
            ↳36 PiDP
                ↳37 UsableRulesProof (⇔)
                ↳38 PiDP
                ↳39 PiDPToQDPProof (⇔)
                ↳40 QDP
                ↳41 NonTerminationProof (⇔)
                ↳42 NO

(0) Obligation:

Clauses:

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

Queries:

a().

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

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

Queries:

a1().

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

Pi is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(6) 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_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

E15_IN_ → E15_IN_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) 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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) 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.

(14) 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:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from E15_IN_ to E15_IN_.



(15) NO

(16) Obligation:

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

C7_IN_ → C7_IN_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) 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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) 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.

(21) 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:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

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



(22) NO

(23) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

Pi is empty.

(25) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(26) 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_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Complex Obligation (AND)

(29) Obligation:

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

E15_IN_ → E15_IN_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(30) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(31) 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

(32) PiDPToQDPProof (EQUIVALENT transformation)

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

(33) 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.

(34) 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:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from E15_IN_ to E15_IN_.



(35) NO

(36) Obligation:

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

C7_IN_ → C7_IN_

The TRS R consists of the following rules:

a1_in_ → U3_(c7_in_)
c7_in_ → U1_(c7_in_)
U1_(c7_out_) → c7_out_
U3_(c7_out_) → a1_out_
a1_in_ → U4_(e15_in_)
e15_in_ → U2_(e15_in_)
U2_(e15_out_) → e15_out_
U4_(e15_out_) → a1_out_

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

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) 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

(39) PiDPToQDPProof (EQUIVALENT transformation)

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

(40) 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.

(41) 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_.



(42) NO