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 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
↳15 QTRSRRRProof (⇔)
↳16 QTRS
↳17 QTRSRRRProof (⇔)
↳18 QTRS
↳19 QTRSRRRProof (⇔)
↳20 QTRS
↳21 QTRSRRRProof (⇔)
↳22 QTRS
  ↳23 RisEmptyProof (⇔)
    ↳24 TRUE
  ↳25 RisEmptyProof (⇔)
    ↳26 TRUE
  ↳27 RisEmptyProof (⇔)
    ↳28 TRUE

(0) Obligation:

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt) → a__U12(tt)
a__U12(tt) → tt
a__isNePal(__(I, __(P, I))) → a__U11(tt)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U12(X)) → a__U12(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U12(X) → U12(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2·x1   
POL(U12(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a__U11(x1)) = 2·x1   
POL(a__U12(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isNePal(x1)) = 2·x1   
POL(isNePal(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(tt) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a____(X, nil) → mark(X)
a____(nil, X) → mark(X)


(2) Obligation:

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__U11(tt) → a__U12(tt)
a__U12(tt) → tt
a__isNePal(__(I, __(P, I))) → a__U11(tt)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U12(X)) → a__U12(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U12(X) → U12(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2 + x1   
POL(U12(x1)) = 1 + x1   
POL(__(x1, x2)) = 1 + x1 + x2   
POL(a__U11(x1)) = 2 + x1   
POL(a__U12(x1)) = 1 + x1   
POL(a____(x1, x2)) = 1 + x1 + x2   
POL(a__isNePal(x1)) = 1 + x1   
POL(isNePal(x1)) = 1 + 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:

a__U11(tt) → a__U12(tt)
a__U12(tt) → tt
a__isNePal(__(I, __(P, I))) → a__U11(tt)


(4) Obligation:

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U12(X)) → a__U12(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U12(X) → U12(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = x1   
POL(U12(x1)) = x1   
POL(__(x1, x2)) = 1 + 2·x1 + x2   
POL(a__U11(x1)) = x1   
POL(a__U12(x1)) = x1   
POL(a____(x1, x2)) = 1 + 2·x1 + x2   
POL(a__isNePal(x1)) = 2·x1   
POL(isNePal(x1)) = 2·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:

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


(6) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U12(X)) → a__U12(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U12(X) → U12(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 2·x1   
POL(U12(x1)) = 1 + 2·x1   
POL(__(x1, x2)) = 2·x1 + 2·x2   
POL(a__U11(x1)) = 2·x1   
POL(a__U12(x1)) = 2 + 2·x1   
POL(a____(x1, x2)) = 2·x1 + 2·x2   
POL(a__isNePal(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(mark(x1)) = 2·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:

a__U12(X) → U12(X)


(8) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U12(X)) → a__U12(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = x1   
POL(U12(x1)) = 1 + x1   
POL(__(x1, x2)) = x1 + x2   
POL(a__U11(x1)) = x1   
POL(a__U12(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isNePal(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:

mark(U12(X)) → a__U12(mark(X))


(10) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__isNePal(X) → isNePal(X)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 1 + 2·x1   
POL(__(x1, x2)) = 2·x1 + x2   
POL(a__U11(x1)) = 2 + 2·x1   
POL(a____(x1, x2)) = 2·x1 + x2   
POL(a__isNePal(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(mark(x1)) = 2·x1   
POL(nil) = 0   
POL(tt) = 1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(tt) → tt
a__U11(X) → U11(X)


(12) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = 1 + x1   
POL(__(x1, x2)) = x1 + x2   
POL(a__U11(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isNePal(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(U11(X)) → a__U11(mark(X))


(14) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

mark(nil) → nil
a__isNePal(X) → isNePal(X)


(16) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNePal(X)) → a__isNePal(mark(X))
a____(X1, X2) → __(X1, X2)

Q is empty.

(17) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isNePal(x1)) = x1   
POL(isNePal(x1)) = 1 + x1   
POL(mark(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(isNePal(X)) → a__isNePal(mark(X))


(18) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(X1, X2) → __(X1, X2)

Q is empty.

(19) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + x1 + 2·x2   
POL(a____(x1, x2)) = 2 + x1 + 2·x2   
POL(mark(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a____(X1, X2) → __(X1, X2)


(20) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))

Q is empty.

(21) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(mark(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))


(22) Obligation:

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

(23) RisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) RisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) RisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE