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

(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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(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__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  a__U11(x1, x2)
tt  =  tt
a__U12(x1)  =  x1
a__isNeList(x1)  =  a__isNeList(x1)
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2)  =  a__U22(x1, x2)
a__isList(x1)  =  a__isList(x1)
a__U23(x1)  =  x1
a__U31(x1, x2)  =  a__U31(x1, x2)
a__U32(x1)  =  x1
a__isQid(x1)  =  x1
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__U42(x1, x2)  =  a__U42(x1, x2)
a__U43(x1)  =  x1
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52(x1, x2)
a__U53(x1)  =  a__U53(x1)
a__U61(x1, x2)  =  a__U61(x1, x2)
a__U62(x1)  =  x1
a__U71(x1, x2)  =  a__U71(x1, x2)
a__U72(x1)  =  x1
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  a__and(x1, x2)
a__isPalListKind(x1)  =  a__isPalListKind(x1)
isPalListKind(x1)  =  isPalListKind(x1)
and(x1, x2)  =  and(x1, x2)
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  U11(x1, x2)
U12(x1)  =  x1
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22(x1, x2)
isList(x1)  =  isList(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  x1
isQid(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  U53(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1)  =  x1
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂, isPal₁, a_isPal1] > [a_U213, U21₃] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U213, U21₃] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U213, U21₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U213, U21₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U213, U21₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U531, U53₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U413, U41₃] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U413, U41₃] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U413, U41₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U413, U41₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U413, U41₃] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U531, U53₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U513, U51₃] > [a_U522, U52₂] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U513, U51₃] > [a_U522, U52₂] > [a_U222, a_isList1, U22₂, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U513, U51₃] > [a_U522, U52₂] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U513, U51₃] > [a_U522, U52₂] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U513, U51₃] > [a_U522, U52₂] > [a_U222, a_isList1, U22₂, isList₁] > [tt, a, e, o, u] > [a_U531, U53₁]
[a₂, ₂, isPal₁, a_isPal1] > [a_U712, a_isNePal1, a_and2, and₂, U71₂, isNePal₁] > [a_U612, U61₂]
[a₂, ₂, isPal₁, a_isPal1] > [a_U712, a_isNePal1, a_and2, and₂, U71₂, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
nil > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
nil > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
nil > [tt, a, e, o, u] > [a_U531, U53₁]
i > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_U312, U31₂]
i > [tt, a, e, o, u] > [a_U422, U42₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
i > [tt, a, e, o, u] > [a_U531, U53₁]

Status:

i: multiset
U52₂: multiset
a_U531: multiset
_₂: [1,2]
a_U422: multiset
a_U112: multiset
a_{isPalListKind1}: multiset
U41₃: multiset
a_isNePal1: [1]
and₂: multiset
U71₂: [2,1]
a_U222: multiset
tt: multiset
isPalListKind₁: multiset
a_isList1: multiset
U21₃: multiset
U51₃: multiset
nil: multiset
a₂: [1,2]
a: multiset
a_U612: multiset
isList₁: multiset
U31₂: multiset
a_and2: multiset
U22₂: multiset
U42₂: multiset
a_isNeList1: multiset
e: multiset
a_U213: multiset
isNePal₁: [1]
U11₂: multiset
o: multiset
a_U513: multiset
a_U712: [2,1]
a_U413: multiset
isPal₁: [1]
a_U312: multiset
a_U522: multiset
U53₁: multiset
U61₂: multiset
u: multiset
isNeList₁: multiset
a_isPal1: [1]

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)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt, V) → a__U12(a__isNeList(V))
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U31(tt, V) → a__U32(a__isQid(V))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U71(tt, V) → a__U72(a__isNePal(V))
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
a__isQid(i) → tt

(2) Obligation:

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

a__U12(tt) → tt
a__U23(tt) → tt
a__U32(tt) → tt
a__U43(tt) → tt
a__U62(tt) → tt
a__U72(tt) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(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__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = x₁ + x₂   
POL(U12(x₁)) = 1 + x₁   
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U22(x₁, x₂)) = x₁ + x₂   
POL(U23(x₁)) = 1 + x₁   
POL(U31(x₁, x₂)) = x₁ + x₂   
POL(U32(x₁)) = 1 + x₁   
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U42(x₁, x₂)) = x₁ + x₂   
POL(U43(x₁)) = 1 + x₁   
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U52(x₁, x₂)) = x₁ + x₂   
POL(U53(x₁)) = x₁   
POL(U61(x₁, x₂)) = x₁ + x₂   
POL(U62(x₁)) = 1 + x₁   
POL(U71(x₁, x₂)) = x₁ + x₂   
POL(U72(x₁)) = 1 + x₁   
POL(__(x₁, x₂)) = x₁ + x₂   
POL(a) = 1   
POL(a__U11(x₁, x₂)) = x₁ + x₂   
POL(a__U12(x₁)) = 1 + x₁   
POL(a__U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U22(x₁, x₂)) = x₁ + x₂   
POL(a__U23(x₁)) = 1 + x₁   
POL(a__U31(x₁, x₂)) = x₁ + x₂   
POL(a__U32(x₁)) = 1 + x₁   
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U42(x₁, x₂)) = x₁ + x₂   
POL(a__U43(x₁)) = 1 + x₁   
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(a__U52(x₁, x₂)) = x₁ + x₂   
POL(a__U53(x₁)) = x₁   
POL(a__U61(x₁, x₂)) = x₁ + x₂   
POL(a__U62(x₁)) = 1 + x₁   
POL(a__U71(x₁, x₂)) = x₁ + x₂   
POL(a__U72(x₁)) = 1 + x₁   
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)) = x₁   
POL(a__isPal(x₁)) = x₁   
POL(a__isPalListKind(x₁)) = x₁   
POL(a__isQid(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(e) = 1   
POL(i) = 0   
POL(isList(x₁)) = x₁   
POL(isNeList(x₁)) = x₁   
POL(isNePal(x₁)) = x₁   
POL(isPal(x₁)) = x₁   
POL(isPalListKind(x₁)) = 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__U12(tt) → tt
a__U23(tt) → tt
a__U32(tt) → tt
a__U43(tt) → tt
a__U62(tt) → tt
a__U72(tt) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(o) → tt
a__isQid(u) → tt

(4) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(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__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = 2 + x₁ + x₂   
POL(U12(x₁)) = 1 + x₁   
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + x₃   
POL(U22(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(U23(x₁)) = 2 + 2·x₁   
POL(U31(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(U32(x₁)) = x₁   
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U42(x₁, x₂)) = 1 + x₁ + x₂   
POL(U43(x₁)) = x₁   
POL(U51(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃   
POL(U52(x₁, x₂)) = 1 + x₁ + x₂   
POL(U53(x₁)) = x₁   
POL(U61(x₁, x₂)) = 1 + x₁ + x₂   
POL(U62(x₁)) = 2·x₁   
POL(U71(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(U72(x₁)) = x₁   
POL(__(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(a) = 2   
POL(a__U11(x₁, x₂)) = 2 + x₁ + x₂   
POL(a__U12(x₁)) = 1 + x₁   
POL(a__U21(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + x₃   
POL(a__U22(x₁, x₂)) = 2 + 2·x₁ + 2·x₂   
POL(a__U23(x₁)) = 2 + 2·x₁   
POL(a__U31(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(a__U32(x₁)) = x₁   
POL(a__U41(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃   
POL(a__U42(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U43(x₁)) = x₁   
POL(a__U51(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃   
POL(a__U52(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(a__U53(x₁)) = x₁   
POL(a__U61(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U62(x₁)) = 2·x₁   
POL(a__U71(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(a__U72(x₁)) = x₁   
POL(a____(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(a__and(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(a__isList(x₁)) = 2 + 2·x₁   
POL(a__isNeList(x₁)) = 2 + x₁   
POL(a__isNePal(x₁)) = 2 + 2·x₁   
POL(a__isPal(x₁)) = 2 + 2·x₁   
POL(a__isPalListKind(x₁)) = 1 + 2·x₁   
POL(a__isQid(x₁)) = 2·x₁   
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(e) = 1   
POL(i) = 2   
POL(isList(x₁)) = 2 + x₁   
POL(isNeList(x₁)) = 1 + x₁   
POL(isNePal(x₁)) = 2 + x₁   
POL(isPal(x₁)) = 1 + 2·x₁   
POL(isPalListKind(x₁)) = 1 + 2·x₁   
POL(isQid(x₁)) = x₁   
POL(mark(x₁)) = 2·x₁   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 2   
POL(u) = 2   

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__isNeList(X) → isNeList(X)
a__U22(X1, X2) → U22(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__isPal(X) → isPal(X)

(6) 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)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U43(X)) → a__U43(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(o) → o
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = x₁ + x₂   
POL(U12(x₁)) = x₁   
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U22(x₁, x₂)) = 1 + x₁ + x₂   
POL(U23(x₁)) = x₁   
POL(U31(x₁, x₂)) = x₁ + x₂   
POL(U32(x₁)) = x₁   
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U42(x₁, x₂)) = x₁ + x₂   
POL(U43(x₁)) = x₁   
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U52(x₁, x₂)) = x₁ + x₂   
POL(U53(x₁)) = x₁   
POL(U61(x₁, x₂)) = x₁ + x₂   
POL(U62(x₁)) = x₁   
POL(U71(x₁, x₂)) = 1 + x₁ + x₂   
POL(U72(x₁)) = x₁   
POL(__(x₁, x₂)) = 3 + x₁ + x₂   
POL(a__U11(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U12(x₁)) = 1 + x₁   
POL(a__U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__U22(x₁, x₂)) = x₁ + x₂   
POL(a__U23(x₁)) = 1 + x₁   
POL(a__U31(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U32(x₁)) = x₁   
POL(a__U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__U42(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U43(x₁)) = x₁   
POL(a__U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(a__U52(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U53(x₁)) = x₁   
POL(a__U61(x₁, x₂)) = 1 + x₁ + x₂   
POL(a__U62(x₁)) = x₁   
POL(a__U71(x₁, x₂)) = x₁ + x₂   
POL(a__U72(x₁)) = x₁   
POL(a____(x₁, x₂)) = x₁ + x₂   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(a__isList(x₁)) = 1 + x₁   
POL(a__isNeList(x₁)) = x₁   
POL(a__isNePal(x₁)) = 1 + x₁   
POL(a__isPal(x₁)) = x₁   
POL(a__isPalListKind(x₁)) = 1 + x₁   
POL(a__isQid(x₁)) = 1 + x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(isList(x₁)) = x₁   
POL(isNeList(x₁)) = x₁   
POL(isNePal(x₁)) = x₁   
POL(isPal(x₁)) = x₁   
POL(isPalListKind(x₁)) = x₁   
POL(isQid(x₁)) = x₁   
POL(mark(x₁)) = 2 + x₁   
POL(nil) = 0   
POL(o) = 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(isNeList(X)) → a__isNeList(X)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isQid(X)) → a__isQid(X)
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(o) → o
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2) → U61(X1, X2)
a__isNePal(X) → isNePal(X)
a__isPalListKind(X) → isPalListKind(X)

(8) Obligation:

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

mark(U32(X)) → a__U32(mark(X))
mark(U43(X)) → a__U43(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U72(X)) → a__U72(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U32(X) → U32(X)
a__U43(X) → U43(X)
a__U53(X) → U53(X)
a__U62(X) → U62(X)
a__U72(X) → U72(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U32(x₁)) = 1 + x₁   
POL(U43(x₁)) = 1 + x₁   
POL(U53(x₁)) = 1 + x₁   
POL(U62(x₁)) = 2 + x₁   
POL(U72(x₁)) = 1 + x₁   
POL(a__U32(x₁)) = 2 + x₁   
POL(a__U43(x₁)) = 1 + x₁   
POL(a__U53(x₁)) = 2 + x₁   
POL(a__U62(x₁)) = 2 + x₁   
POL(a__U72(x₁)) = 2 + x₁   
POL(a__and(x₁, x₂)) = 2·x₁ + x₂   
POL(and(x₁, x₂)) = 2·x₁ + 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:

mark(U43(X)) → a__U43(mark(X))
mark(U62(X)) → a__U62(mark(X))
a__U32(X) → U32(X)
a__U53(X) → U53(X)
a__U72(X) → U72(X)

(10) Obligation:

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

mark(U32(X)) → a__U32(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U72(X)) → a__U72(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U43(X) → U43(X)
a__U62(X) → U62(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U32(x₁)) = 1 + x₁   
POL(U43(x₁)) = x₁   
POL(U53(x₁)) = 1 + x₁   
POL(U62(x₁)) = x₁   
POL(U72(x₁)) = 1 + x₁   
POL(a__U32(x₁)) = x₁   
POL(a__U43(x₁)) = 1 + x₁   
POL(a__U53(x₁)) = x₁   
POL(a__U62(x₁)) = 1 + x₁   
POL(a__U72(x₁)) = x₁   
POL(a__and(x₁, x₂)) = x₁ + x₂   
POL(and(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(U32(X)) → a__U32(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U72(X)) → a__U72(mark(X))
a__U43(X) → U43(X)
a__U62(X) → U62(X)

(12) Obligation:

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

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(X1, X2) → and(X1, X2)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > a_and2 > and₂

Status:

a_and2: multiset
mark₁: [1]
and₂: multiset

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)
a__and(X1, X2) → and(X1, X2)

(14) Obligation:

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

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE