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

0 Prolog
↳1 CutEliminatorProof (⇐)
↳2 Prolog
  ↳3 PrologToPiTRSProof (⇐)
    ↳4 PiTRS
    ↳5 DependencyPairsProof (⇔)
    ↳6 PiDP
    ↳7 DependencyGraphProof (⇔)
    ↳8 PiDP
    ↳9 UsableRulesProof (⇔)
    ↳10 PiDP
    ↳11 PiDPToQDPProof (⇔)
    ↳12 QDP
    ↳13 NonTerminationProof (⇔)
    ↳14 FALSE
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 PiDP
    ↳21 UsableRulesProof (⇔)
    ↳22 PiDP
    ↳23 PiDPToQDPProof (⇔)
    ↳24 QDP
    ↳25 NonTerminationProof (⇔)
    ↳26 FALSE

(0) Obligation:

Clauses:

a :- b.
b :- c.
c :- d.
d :- ','(e, !).
e.
e :- a.

Queries:

a().

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

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

Queries:

a().

(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:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_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:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_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:

A_IN_ → U1_1(b_in_)
A_IN_ → B_IN_
B_IN_ → U2_1(c_in_)
B_IN_ → C_IN_
C_IN_ → U3_1(d_in_)
C_IN_ → D_IN_
D_IN_ → U4_1(e_in_)
D_IN_ → E_IN_
E_IN_ → U5_1(a_in_)
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_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:

A_IN_ → U1_1(b_in_)
A_IN_ → B_IN_
B_IN_ → U2_1(c_in_)
B_IN_ → C_IN_
C_IN_ → U3_1(d_in_)
C_IN_ → D_IN_
D_IN_ → U4_1(e_in_)
D_IN_ → E_IN_
E_IN_ → U5_1(a_in_)
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_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 1 SCC with 5 less nodes.

(8) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

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

(11) PiDPToQDPProof (EQUIVALENT transformation)

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

(12) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

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

(13) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = B_IN_ evaluates to t =B_IN_

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



Rewriting sequence

B_IN_ → C_IN_
with rule B_IN_ → C_IN_ at position [] and matcher [ ]

C_IN_ → D_IN_
with rule C_IN_ → D_IN_ at position [] and matcher [ ]

D_IN_ → E_IN_
with rule D_IN_ → E_IN_ at position [] and matcher [ ]

E_IN_ → A_IN_
with rule E_IN_ → A_IN_ at position [] and matcher [ ]

A_IN_ → B_IN_
with rule A_IN_ → B_IN_

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(14) FALSE

(15) 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:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

Pi is empty.

(17) 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:

A_IN_ → U1_1(b_in_)
A_IN_ → B_IN_
B_IN_ → U2_1(c_in_)
B_IN_ → C_IN_
C_IN_ → U3_1(d_in_)
C_IN_ → D_IN_
D_IN_ → U4_1(e_in_)
D_IN_ → E_IN_
E_IN_ → U5_1(a_in_)
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

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

(18) Obligation:

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

A_IN_ → U1_1(b_in_)
A_IN_ → B_IN_
B_IN_ → U2_1(c_in_)
B_IN_ → C_IN_
C_IN_ → U3_1(d_in_)
C_IN_ → D_IN_
D_IN_ → U4_1(e_in_)
D_IN_ → E_IN_
E_IN_ → U5_1(a_in_)
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(20) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

The TRS R consists of the following rules:

a_in_ → U1_(b_in_)
b_in_ → U2_(c_in_)
c_in_ → U3_(d_in_)
d_in_ → U4_(e_in_)
e_in_ → e_out_
e_in_ → U5_(a_in_)
U5_(a_out_) → e_out_
U4_(e_out_) → d_out_
U3_(d_out_) → c_out_
U2_(c_out_) → b_out_
U1_(b_out_) → a_out_

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

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

(23) PiDPToQDPProof (EQUIVALENT transformation)

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

(24) Obligation:

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

A_IN_ → B_IN_
B_IN_ → C_IN_
C_IN_ → D_IN_
D_IN_ → E_IN_
E_IN_ → A_IN_

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

(25) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = B_IN_ evaluates to t =B_IN_

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



Rewriting sequence

B_IN_ → C_IN_
with rule B_IN_ → C_IN_ at position [] and matcher [ ]

C_IN_ → D_IN_
with rule C_IN_ → D_IN_ at position [] and matcher [ ]

D_IN_ → E_IN_
with rule D_IN_ → E_IN_ at position [] and matcher [ ]

E_IN_ → A_IN_
with rule E_IN_ → A_IN_ at position [] and matcher [ ]

A_IN_ → B_IN_
with rule A_IN_ → B_IN_

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(26) FALSE