NO Termination w.r.t. Q proof of Transformed_CSR_04_Ex1_Zan97_L.ari

Termination w.r.t. Q of the given QTRS could be disproven:

0 QTRS
↳1 QTRSRRRProof (⇔, 19 ms)
↳2 QTRS
↳3 AAECC Innermost (⇔, 0 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 0 ms)
↳6 QDP
↳7 UsableRulesProof (⇔, 0 ms)
↳8 QDP
↳9 QReductionProof (⇔, 0 ms)
↳10 QDP
↳11 NonTerminationLoopProof (⇒, 0 ms)
↳12 NO

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

g → h
c → d
h → g

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(c) = 1   
POL(d) = 0   
POL(g) = 0   
POL(h) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

c → d


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

g → h
h → g

Q is empty.

(3) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

g → h
h → g

The signature Sigma is {g, h}

(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

g → h
h → g

The set Q consists of the following terms:

g
h

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(6) Obligation:

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

G → H
H → G

The TRS R consists of the following rules:

g → h
h → g

The set Q consists of the following terms:

g
h

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

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(8) Obligation:

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

G → H
H → G

R is empty.
The set Q consists of the following terms:

g
h

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

(9) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

g
h

(10) Obligation:

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

G → H
H → G

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

(11) NonTerminationLoopProof (COMPLETE 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 = H evaluates to t =H

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



Rewriting sequence

H → G
with rule H → G at position [] and matcher [ ]

G → H
with rule G → H

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.



(12) NO