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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
  ↳9 RisEmptyProof (⇔)
    ↳10 TRUE
  ↳11 RisEmptyProof (⇔)
    ↳12 TRUE
  ↳13 RisEmptyProof (⇔)
    ↳14 TRUE

(0) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2 + x1   
POL(U12(x1)) = 1 + x1   
POL(__(x1, x2)) = x1 + x2   
POL(active(x1)) = x1   
POL(isNePal(x1)) = 3 + x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))


(2) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = x1   
POL(U12(x1)) = 2·x1   
POL(__(x1, x2)) = x1 + x2   
POL(active(x1)) = x1   
POL(isNePal(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(nil) = 2   
POL(tt) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)


(4) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2·x1   
POL(U12(x1)) = 2·x1   
POL(__(x1, x2)) = 2 + 2·x1 + x2   
POL(active(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))


(6) Obligation:

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

mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > _2
mark1 > active1
mark1 > U111
mark1 > U121
mark1 > isNePal1

Status:
trivial

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)


(8) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE

(11) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(14) TRUE