(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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

__1(__(X, Y), Z) → __1(X, __(Y, Z))
__1(__(X, Y), Z) → __1(Y, Z)
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
ISLISTU111(isNeList)
ISLISTISNELIST
ISLISTU211(isList)
ISLISTISLIST
ISNELISTU311(isQid)
ISNELISTISQID
ISNELISTU411(isList)
ISNELISTISLIST
ISNELISTU511(isNeList)
ISNELISTISNELIST
ISNEPALU611(isQid)
ISNEPALISQID
ISNEPALU711(isQid)
ISPALU811(isNePal)
ISPALISNEPAL

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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 10 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

U711(tt) → ISPAL
ISPALISNEPAL
ISNEPALU711(isQid)

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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

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

(6) Obligation:

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

U211(tt) → ISLIST
ISLISTISNELIST
ISNELISTU411(isList)
U411(tt) → ISNELIST
ISNELISTISLIST
ISLISTU211(isList)
ISLISTISLIST
ISNELISTU511(isNeList)
U511(tt) → ISLIST
ISNELISTISNELIST

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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

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

(7) Obligation:

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

__1(__(X, Y), Z) → __1(Y, Z)
__1(__(X, Y), Z) → __1(X, __(Y, Z))

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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


__1(__(X, Y), Z) → __1(Y, Z)
__1(__(X, Y), Z) → __1(X, __(Y, Z))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__1(x1, x2)  =  __1(x1, x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21
U22(x1)  =  U22
isList  =  isList
U31(x1)  =  x1
U41(x1)  =  x1
U42(x1)  =  x1
isNeList  =  isNeList
U51(x1)  =  U51
U52(x1)  =  U52
U61(x1)  =  x1
U71(x1)  =  x1
U72(x1)  =  U72
isPal  =  isPal
U81(x1)  =  x1
isQid  =  isQid
isNePal  =  isNePal

Lexicographic path order with status [LPO].
Quasi-Precedence:
_^12 > _2
[tt, U21, U22, isList, isNeList, U51, U52, U72, isPal, isQid, isNePal]

Status:
_^12: [1,2]
_2: [1,2]
nil: []
tt: []
U21: []
U22: []
isList: []
isNeList: []
U51: []
U52: []
U72: []
isPal: []
isQid: []
isNePal: []


The following usable rules [FROCOS05] were oriented:

__(__(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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

(9) Obligation:

Q DP problem:
P is empty.
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
isListU11(isNeList)
isListtt
isListU21(isList)
isNeListU31(isQid)
isNeListU41(isList)
isNeListU51(isNeList)
isNePalU61(isQid)
isNePalU71(isQid)
isPalU81(isNePal)
isPaltt
isQidtt

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE