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 RisEmptyProof (⇔)
    ↳12 TRUE
  ↳13 RisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 RisEmptyProof (⇔)
    ↳16 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__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(a__and(x1, x2)) = 1 + x1 + x2   
POL(a__isNePal(x1)) = 1 + x1   
POL(and(x1, x2)) = 1 + x1 + x2   
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__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt


(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____(X, nil) → mark(X)
a____(nil, X) → mark(X)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNePal(x1)) = x1   
POL(and(x1, x2)) = x1 + x2   
POL(isNePal(x1)) = 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)


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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 2 + 2·x1 + x2   
POL(a____(x1, x2)) = 2 + 2·x1 + x2   
POL(a__and(x1, x2)) = x1 + 2·x2   
POL(a__isNePal(x1)) = 2·x1   
POL(and(x1, x2)) = x1 + 2·x2   
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(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + x1 + x2   
POL(a____(x1, x2)) = 2 + x1 + x2   
POL(a__and(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(a__isNePal(x1)) = 2 + x1   
POL(and(x1, x2)) = 1 + 2·x1 + x2   
POL(isNePal(x1)) = 1 + 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____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)


(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 2 + x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNePal(x1)) = x1   
POL(and(x1, x2)) = 1 + x1 + x2   
POL(isNePal(x1)) = 1 + x1   
POL(mark(x1)) = 1 + 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(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil


(10) Obligation:

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

(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

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE