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 RisEmptyProof (⇔)
    ↳20 TRUE
  ↳21 RisEmptyProof (⇔)
    ↳22 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__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 3 + x1 + x2   
POL(a) = 1   
POL(a____(x1, x2)) = 3 + x1 + x2   
POL(a__and(x1, x2)) = 1 + x1 + x2   
POL(a__isList(x1)) = 2 + x1   
POL(a__isNeList(x1)) = 1 + x1   
POL(a__isNePal(x1)) = 1 + x1   
POL(a__isPal(x1)) = 2 + x1   
POL(a__isQid(x1)) = x1   
POL(and(x1, x2)) = 1 + x1 + x2   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = 2 + x1   
POL(isNeList(x1)) = 1 + x1   
POL(isNePal(x1)) = 1 + x1   
POL(isPal(x1)) = 2 + x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 1   
POL(tt) = 0   
POL(u) = 1   
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)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → 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__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 0   
POL(a____(x1, x2)) = 1 + x1 + x2   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(and(x1, x2)) = x1 + x2   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))


(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(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 2 + 2·x1 + x2   
POL(a) = 0   
POL(a____(x1, x2)) = 2 + 2·x1 + x2   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = 2·x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(and(x1, x2)) = x1 + x2   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = 2·x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 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(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 2·x1 + 2·x2   
POL(a) = 1   
POL(a____(x1, x2)) = 2·x1 + 2·x2   
POL(a__and(x1, x2)) = 2·x1 + 2·x2   
POL(a__isList(x1)) = 2·x1   
POL(a__isNeList(x1)) = 2·x1   
POL(a__isNePal(x1)) = 2 + x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = 2·x1   
POL(and(x1, x2)) = 2·x1 + 2·x2   
POL(e) = 0   
POL(i) = 2   
POL(isList(x1)) = 2·x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = 2 + x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = 2·x1   
POL(nil) = 1   
POL(o) = 2   
POL(tt) = 1   
POL(u) = 1   
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(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(i) → i
mark(o) → o
mark(u) → u


(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(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isPal(X)) → a__isPal(X)
mark(e) → e
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

(9) 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__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = 1 + x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(and(x1, x2)) = x1 + x2   
POL(e) = 0   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = 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:

a__isNePal(X) → isNePal(X)


(10) 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(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isPal(X)) → a__isPal(X)
mark(e) → e
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isPal(X) → isPal(X)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 2·x1 + x2   
POL(a____(x1, x2)) = 2·x1 + x2   
POL(a__and(x1, x2)) = 2 + 2·x1 + x2   
POL(a__isList(x1)) = 2 + 2·x1   
POL(a__isNeList(x1)) = 2·x1   
POL(a__isPal(x1)) = 2 + 2·x1   
POL(a__isQid(x1)) = 2 + 2·x1   
POL(and(x1, x2)) = 2 + 2·x1 + x2   
POL(e) = 1   
POL(isList(x1)) = 2 + x1   
POL(isNeList(x1)) = 2·x1   
POL(isPal(x1)) = 2 + x1   
POL(isQid(x1)) = 2 + x1   
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:

mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isQid(X)) → a__isQid(X)
mark(isPal(X)) → a__isPal(X)
mark(e) → e


(12) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNeList(X)) → a__isNeList(X)
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isPal(X) → isPal(X)

Q is empty.

(13) 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__isList(x1)) = 1 + x1   
POL(a__isNeList(x1)) = x1   
POL(a__isPal(x1)) = 1 + x1   
POL(a__isQid(x1)) = 1 + x1   
POL(and(x1, x2)) = x1 + x2   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = 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:

a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isQid(X) → isQid(X)
a__isPal(X) → isPal(X)


(14) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNeList(X)) → a__isNeList(X)
a____(X1, X2) → __(X1, X2)
a__isNeList(X) → isNeList(X)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(a____(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(a__isNeList(x1)) = 2 + 2·x1   
POL(isNeList(x1)) = 1 + x1   
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)
a__isNeList(X) → isNeList(X)


(16) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNeList(X)) → a__isNeList(X)

Q is empty.

(17) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x1, x2)) = 1 + x1 + x2   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isNeList(x1)) = x1   
POL(isNeList(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(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isNeList(X)) → a__isNeList(X)


(18) Obligation:

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

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) RisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE