NO Termination w.r.t. Q proof of Transformed_CSR_04_PALINDROME_nokinds-noand_L.ari

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

0 QTRS
↳1 QTRSRRRProof (⇔, 0 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 4 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 0 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 MNOCProof (⇔, 0 ms)
        ↳13 QDP
        ↳14 TransformationProof (⇔, 0 ms)
        ↳15 QDP
        ↳16 UsableRulesProof (⇔, 0 ms)
        ↳17 QDP
        ↳18 QReductionProof (⇔, 0 ms)
        ↳19 QDP
        ↳20 NonTerminationLoopProof (⇒, 0 ms)
        ↳21 NO
    ↳22 QDP
        ↳23 UsableRulesProof (⇔, 0 ms)
        ↳24 QDP
        ↳25 TransformationProof (⇔, 0 ms)
        ↳26 QDP
        ↳27 TransformationProof (⇔, 0 ms)
        ↳28 QDP
        ↳29 TransformationProof (⇔, 0 ms)
        ↳30 QDP
        ↳31 TransformationProof (⇔, 0 ms)
        ↳32 QDP
        ↳33 TransformationProof (⇔, 0 ms)
        ↳34 QDP
        ↳35 NonTerminationLoopProof (⇒, 0 ms)
        ↳36 NO

(0) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = x1   
POL(U21(x1)) = 2·x1   
POL(U22(x1)) = x1   
POL(U31(x1)) = 2·x1   
POL(U41(x1)) = x1   
POL(U42(x1)) = x1   
POL(U51(x1)) = 2·x1   
POL(U52(x1)) = 2·x1   
POL(U61(x1)) = x1   
POL(U71(x1)) = 2 + x1   
POL(U72(x1)) = x1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = 2 + 2·x1 + x2   
POL(isList) = 0   
POL(isNeList) = 0   
POL(isNePal) = 2   
POL(isPal) = 2   
POL(isQid) = 0   
POL(nil) = 2   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
isNePal → U61(isQid)
isPal → tt


(2) Obligation:

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

U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U71(isQid)
isPal → U81(isNePal)
isQid → tt

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2·x1   
POL(U21(x1)) = 2·x1   
POL(U22(x1)) = x1   
POL(U31(x1)) = 2·x1   
POL(U41(x1)) = x1   
POL(U42(x1)) = x1   
POL(U51(x1)) = x1   
POL(U52(x1)) = 2·x1   
POL(U61(x1)) = 1 + x1   
POL(U71(x1)) = x1   
POL(U72(x1)) = x1   
POL(U81(x1)) = 2·x1   
POL(isList) = 0   
POL(isNeList) = 0   
POL(isNePal) = 0   
POL(isPal) = 0   
POL(isQid) = 0   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U61(tt) → tt


(4) Obligation:

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

U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U71(isQid)
isPal → U81(isNePal)
isQid → tt

Q is empty.

(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:

U211(tt) → U221(isList)
U211(tt) → ISLIST
U411(tt) → U421(isNeList)
U411(tt) → ISNELIST
U511(tt) → U521(isList)
U511(tt) → ISLIST
U711(tt) → U721(isPal)
U711(tt) → ISPAL
ISLIST → U111(isNeList)
ISLIST → ISNELIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U311(isQid)
ISNELIST → ISQID
ISNELIST → U411(isList)
ISNELIST → ISLIST
ISNELIST → U511(isNeList)
ISNELIST → ISNELIST
ISNEPAL → U711(isQid)
ISNEPAL → ISQID
ISPAL → U811(isNePal)
ISPAL → ISNEPAL

The TRS R consists of the following rules:

U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U71(isQid)
isPal → U81(isNePal)
isQid → tt

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(isQid)

The TRS R consists of the following rules:

U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U71(isQid)
isPal → U81(isNePal)
isQid → tt

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

(10) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(11) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(isQid)

The TRS R consists of the following rules:

isQid → tt

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

(12) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(13) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(isQid)

The TRS R consists of the following rules:

isQid → tt

The set Q consists of the following terms:

isQid

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

(14) TransformationProof (EQUIVALENT transformation)

By rewriting [LPAR04] the rule ISNEPAL → U711(isQid) at position [0] we obtained the following new rules [LPAR04]:

ISNEPAL → U711(tt) → ISNEPAL → U711(tt)

(15) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(tt)

The TRS R consists of the following rules:

isQid → tt

The set Q consists of the following terms:

isQid

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

(16) 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.

(17) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(tt)

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

isQid

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

(18) 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].

isQid

(19) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(tt)

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

(20) 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 = ISPAL evaluates to t =ISPAL

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



Rewriting sequence

ISPAL → ISNEPAL
with rule ISPAL → ISNEPAL at position [] and matcher [ ]

ISNEPAL → U711(tt)
with rule ISNEPAL → U711(tt) at position [] and matcher [ ]

U711(tt) → ISPAL
with rule U711(tt) → ISPAL

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.



(21) NO

(22) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
ISNELIST → U411(isList)
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
ISNELIST → ISNELIST

The TRS R consists of the following rules:

U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U71(isQid)
isPal → U81(isNePal)
isQid → tt

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

(23) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(24) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
ISNELIST → U411(isList)
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
ISNELIST → ISNELIST

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(25) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNELIST → U411(isList) at position [0] we obtained the following new rules [LPAR04]:

ISNELIST → U411(U11(isNeList)) → ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt) → ISNELIST → U411(tt)
ISNELIST → U411(U21(isList)) → ISNELIST → U411(U21(isList))

(26) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
ISNELIST → ISNELIST
ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt)
ISNELIST → U411(U21(isList))

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(27) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISLIST → U211(isList) at position [0] we obtained the following new rules [LPAR04]:

ISLIST → U211(U11(isNeList)) → ISLIST → U211(U11(isNeList))
ISLIST → U211(tt) → ISLIST → U211(tt)
ISLIST → U211(U21(isList)) → ISLIST → U211(U21(isList))

(28) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
ISNELIST → ISNELIST
ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt)
ISNELIST → U411(U21(isList))
ISLIST → U211(U11(isNeList))
ISLIST → U211(tt)
ISLIST → U211(U21(isList))

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(29) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNELIST → U511(isNeList) at position [0] we obtained the following new rules [LPAR04]:

ISNELIST → U511(U31(isQid)) → ISNELIST → U511(U31(isQid))
ISNELIST → U511(U41(isList)) → ISNELIST → U511(U41(isList))
ISNELIST → U511(U51(isNeList)) → ISNELIST → U511(U51(isNeList))

(30) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → ISLIST
U511(tt) → ISLIST
ISNELIST → ISNELIST
ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt)
ISNELIST → U411(U21(isList))
ISLIST → U211(U11(isNeList))
ISLIST → U211(tt)
ISLIST → U211(U21(isList))
ISNELIST → U511(U31(isQid))
ISNELIST → U511(U41(isList))
ISNELIST → U511(U51(isNeList))

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(31) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNELIST → U511(U31(isQid)) at position [0] we obtained the following new rules [LPAR04]:

ISNELIST → U511(U31(tt)) → ISNELIST → U511(U31(tt))

(32) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → ISLIST
U511(tt) → ISLIST
ISNELIST → ISNELIST
ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt)
ISNELIST → U411(U21(isList))
ISLIST → U211(U11(isNeList))
ISLIST → U211(tt)
ISLIST → U211(U21(isList))
ISNELIST → U511(U41(isList))
ISNELIST → U511(U51(isNeList))
ISNELIST → U511(U31(tt))

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(33) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNELIST → U511(U31(tt)) at position [0] we obtained the following new rules [LPAR04]:

ISNELIST → U511(tt) → ISNELIST → U511(tt)

(34) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → ISLIST
U511(tt) → ISLIST
ISNELIST → ISNELIST
ISNELIST → U411(U11(isNeList))
ISNELIST → U411(tt)
ISNELIST → U411(U21(isList))
ISLIST → U211(U11(isNeList))
ISLIST → U211(tt)
ISLIST → U211(U21(isList))
ISNELIST → U511(U41(isList))
ISNELIST → U511(U51(isNeList))
ISNELIST → U511(tt)

The TRS R consists of the following rules:

isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
U51(tt) → U52(isList)
isList → U11(isNeList)
isList → tt
isList → U21(isList)
U52(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U11(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
isQid → tt
U31(tt) → tt

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

(35) NonTerminationLoopProof (COMPLETE 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 = ISLIST evaluates to t =ISLIST

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 ISLIST to ISLIST.



(36) NO