Termination w.r.t. Q proof of /tmp/tmpATjaX8/PALINDROME_complete_noand

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 RisEmptyProof (⇔)

    ↳18 TRUE

  ↳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__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(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, x2) = a__U12(x1, x2)
a__isPalListKind(x1) = a__isPalListKind(x1)
a__U13(x1) = x1
a__isNeList(x1) = a__isNeList(x1)
a__U21(x1, x2, x3) = a__U21(x1, x2, x3)
a__U22(x1, x2, x3) = a__U22(x1, x2, x3)
a__U23(x1, x2, x3) = a__U23(x1, x2, x3)
a__U24(x1, x2, x3) = a__U24(x1, x2, x3)
a__U25(x1, x2) = a__U25(x1, x2)
a__isList(x1) = a__isList(x1)
a__U26(x1) = a__U26(x1)
a__U31(x1, x2) = a__U31(x1, x2)
a__U32(x1, x2) = a__U32(x1, x2)
a__U33(x1) = x1
a__isQid(x1) = x1
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
a__U42(x1, x2, x3) = a__U42(x1, x2, x3)
a__U43(x1, x2, x3) = a__U43(x1, x2, x3)
a__U44(x1, x2, x3) = a__U44(x1, x2, x3)
a__U45(x1, x2) = a__U45(x1, x2)
a__U46(x1) = x1
a__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__U52(x1, x2, x3) = a__U52(x1, x2, x3)
a__U53(x1, x2, x3) = a__U53(x1, x2, x3)
a__U54(x1, x2, x3) = a__U54(x1, x2, x3)
a__U55(x1, x2) = a__U55(x1, x2)
a__U56(x1) = x1
a__U61(x1, x2) = a__U61(x1, x2)
a__U62(x1, x2) = a__U62(x1, x2)
a__U63(x1) = a__U63(x1)
a__U71(x1, x2, x3) = a__U71(x1, x2, x3)
a__U72(x1, x2) = a__U72(x1, x2)
a__U73(x1, x2) = a__U73(x1, x2)
a__isPal(x1) = a__isPal(x1)
a__U74(x1) = x1
a__U81(x1, x2) = a__U81(x1, x2)
a__U82(x1, x2) = a__U82(x1, x2)
a__U83(x1) = x1
a__isNePal(x1) = a__isNePal(x1)
a__U91(x1, x2) = a__U91(x1, x2)
a__U92(x1) = x1
a = a
e = e
i = i
o = o
u = u
U11(x1, x2) = U11(x1, x2)
U12(x1, x2) = U12(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
U13(x1) = x1
isNeList(x1) = isNeList(x1)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2, x3) = U22(x1, x2, x3)
U23(x1, x2, x3) = U23(x1, x2, x3)
U24(x1, x2, x3) = U24(x1, x2, x3)
U25(x1, x2) = U25(x1, x2)
isList(x1) = isList(x1)
U26(x1) = U26(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1, x2) = U32(x1, x2)
U33(x1) = x1
isQid(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x1, x2, x3)
U43(x1, x2, x3) = U43(x1, x2, x3)
U44(x1, x2, x3) = U44(x1, x2, x3)
U45(x1, x2) = U45(x1, x2)
U46(x1) = x1
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
U53(x1, x2, x3) = U53(x1, x2, x3)
U54(x1, x2, x3) = U54(x1, x2, x3)
U55(x1, x2) = U55(x1, x2)
U56(x1) = x1
U61(x1, x2) = U61(x1, x2)
U62(x1, x2) = U62(x1, x2)
U63(x1) = U63(x1)
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2) = U72(x1, x2)
U73(x1, x2) = U73(x1, x2)
isPal(x1) = isPal(x1)
U74(x1) = x1
U81(x1, x2) = U81(x1, x2)
U82(x1, x2) = U82(x1, x2)
U83(x1) = x1
isNePal(x1) = isNePal(x1)
U91(x1, x2) = U91(x1, x2)
U92(x1) = x1

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

[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₃}, U22₃] > [a_{_U23₃}, U23₃] > [a_{_U24₃}, U24₃] > [a_{_U25₂}, U25₂] > [a_{_isList₁}, isList₁] > [a_{_U11₂}, U11₂] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₃}, U22₃] > [a_{_U23₃}, U23₃] > [a_{_U24₃}, U24₃] > [a_{_U25₂}, U25₂] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₃}, U22₃] > [a_{_U23₃}, U23₃] > [a_{_U24₃}, U24₃] > [a_{_U25₂}, U25₂] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₃}, U22₃] > [a_{_U23₃}, U23₃] > [a_{_U24₃}, U24₃] > [a_{_U25₂}, U25₂] > [a_{_U26₁}, U26₁] > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₃}, U22₃] > [a_{_U23₃}, U23₃] > [a_{_U24₃}, U24₃] > [a_{_U25₂}, U25₂] > [a_{_U26₁}, U26₁] > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_U42₃}, U42₃] > [a_{_U43₃}, U43₃] > [a_{_U44₃}, U44₃] > [a_{_isList₁}, isList₁] > [a_{_U11₂}, U11₂] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_U42₃}, U42₃] > [a_{_U43₃}, U43₃] > [a_{_U44₃}, U44₃] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_U42₃}, U42₃] > [a_{_U43₃}, U43₃] > [a_{_U44₃}, U44₃] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_U42₃}, U42₃] > [a_{_U43₃}, U43₃] > [a_{_U44₃}, U44₃] > [a_{_U45₂}, U45₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_U52₃}, U52₃] > [a_{_U53₃}, U53₃] > [a_{_U54₃}, U54₃] > [a_{_U55₂}, U55₂] > [a_{_isList₁}, isList₁] > [a_{_U11₂}, U11₂] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_U52₃}, U52₃] > [a_{_U53₃}, U53₃] > [a_{_U54₃}, U54₃] > [a_{_U55₂}, U55₂] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_U52₃}, U52₃] > [a_{_U53₃}, U53₃] > [a_{_U54₃}, U54₃] > [a_{_U55₂}, U55₂] > [a_{_isList₁}, isList₁] > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U71₃}, U71₃] > [a_{_U72₂}, a_{_isPal₁}, U72₂, isPal₁] > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U71₃}, U71₃] > [a_{_U72₂}, a_{_isPal₁}, U72₂, isPal₁] > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U71₃}, U71₃] > [a_{_U72₂}, a_{_isPal₁}, U72₂, isPal₁] > [a_{_U73₂}, U73₂] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U71₃}, U71₃] > [a_{_U72₂}, a_{_isPal₁}, U72₂, isPal₁] > [a_{_U81₂}, U81₂] > [a_{_U82₂}, U82₂] > [a_{_isNePal₁}, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U71₃}, U71₃] > [a_{_U72₂}, a_{_isPal₁}, U72₂, isPal₁] > [a_{_U81₂}, U81₂] > [a_{_U82₂}, U82₂] > [a_{_isNePal₁}, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U91₂}, U91₂] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
nil > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
nil > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
a > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
a > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
i > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
i > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
o > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
o > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]
u > [tt, e] > [a_{_U12₂}, U12₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_{isPalListKind₁}}, a_{_U31₂}, isPalListKind₁, U31₂] > [a_{_U32₂}, U32₂]
u > [tt, e] > [a_{_U62₂}, U62₂] > [a_{_U63₁}, U63₁]

Status:

i: multiset
U62₂: multiset
a_{_U11₂}: multiset
U41₃: multiset
a_{_isNePal₁}: [1]
U25₂: multiset
U32₂: multiset
U24₃: multiset
U12₂: multiset
a_{_U72₂}: multiset
a_{_U22₃}: multiset
a_{_U62₂}: multiset
tt: multiset
isPalListKind₁: [1]
U22₃: multiset
a_{_isList₁}: multiset
U21₃: multiset
nil: multiset
U54₃: multiset
a_{_U53₃}: [2,1,3]
a_{_{_{_₂}}}: [1,2]
U55₂: [2,1]
U31₂: [2,1]
a_{_isNeList₁}: [1]
e: multiset
a_{_U21₃}: multiset
isNePal₁: [1]
o: multiset
a_{_U23₃}: multiset
a_{_U52₃}: multiset
U53₃: [2,1,3]
U23₃: multiset
a_{_U31₂}: [2,1]
a_{_U63₁}: [1]
U44₃: multiset
U81₂: multiset
a_{_U45₂}: [2,1]
a_{_U73₂}: multiset
a_{_isPal₁}: multiset
a_{_U71₃}: multiset
U71₃: multiset
U91₂: multiset
__₂: [1,2]
a_{_{isPalListKind₁}}: [1]
a_{_U82₂}: [2,1]
U43₃: multiset
U26₁: multiset
U42₃: multiset
a_{_U55₂}: [2,1]
U45₂: [2,1]
U51₃: multiset
a_{_U43₃}: multiset
U73₂: multiset
a: multiset
a_{_U26₁}: multiset
a_{_U44₃}: multiset
a_{_U61₂}: multiset
isList₁: multiset
U72₂: multiset
U11₂: multiset
U63₁: [1]
a_{_U51₃}: multiset
a_{_U24₃}: multiset
U52₃: multiset
a_{_U41₃}: multiset
isPal₁: multiset
a_{_U42₃}: multiset
U82₂: [2,1]
U61₂: multiset
u: multiset
a_{_U54₃}: multiset
isNeList₁: [1]
a_{_U81₂}: multiset
a_{_U25₂}: multiset
a_{_U91₂}: multiset
a_{_U12₂}: multiset
a_{_U32₂}: multiset

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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), V2)
a__isQid(a) → 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__U13(tt) → tt
a__U33(tt) → tt
a__U46(tt) → tt
a__U56(tt) → tt
a__U74(tt) → tt
a__U83(tt) → tt
a__U92(tt) → tt
a__isQid(e) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U25(x₁, x₂)) = x₁ + x₂
POL(U26(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁, x₂)) = x₁ + x₂
POL(U33(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U44(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U45(x₁, x₂)) = x₁ + x₂
POL(U46(x₁)) = 1 + x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U52(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U53(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U54(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U55(x₁, x₂)) = x₁ + x₂
POL(U56(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁, x₂)) = x₁ + x₂
POL(U63(x₁)) = x₁
POL(U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U72(x₁, x₂)) = x₁ + x₂
POL(U73(x₁, x₂)) = x₁ + x₂
POL(U74(x₁)) = 1 + x₁
POL(U81(x₁, x₂)) = x₁ + x₂
POL(U82(x₁, x₂)) = x₁ + x₂
POL(U83(x₁)) = 1 + x₁
POL(U91(x₁, x₂)) = x₁ + x₂
POL(U92(x₁)) = 1 + x₁
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U13(x₁)) = 1 + x₁
POL(a__U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U25(x₁, x₂)) = x₁ + x₂
POL(a__U26(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁ + x₂
POL(a__U32(x₁, x₂)) = x₁ + x₂
POL(a__U33(x₁)) = 1 + x₁
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U44(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U45(x₁, x₂)) = x₁ + x₂
POL(a__U46(x₁)) = 1 + x₁
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U52(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U53(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U54(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U55(x₁, x₂)) = x₁ + x₂
POL(a__U56(x₁)) = 1 + x₁
POL(a__U61(x₁, x₂)) = x₁ + x₂
POL(a__U62(x₁, x₂)) = x₁ + x₂
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U72(x₁, x₂)) = x₁ + x₂
POL(a__U73(x₁, x₂)) = x₁ + x₂
POL(a__U74(x₁)) = 1 + x₁
POL(a__U81(x₁, x₂)) = x₁ + x₂
POL(a__U82(x₁, x₂)) = x₁ + x₂
POL(a__U83(x₁)) = 1 + x₁
POL(a__U91(x₁, x₂)) = x₁ + x₂
POL(a__U92(x₁)) = 1 + x₁
POL(a____(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(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) = 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__U13(tt) → tt
a__U33(tt) → tt
a__U46(tt) → tt
a__U56(tt) → tt
a__U74(tt) → tt
a__U83(tt) → tt
a__U92(tt) → tt
a__isQid(e) → 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = 2 + x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + 2·x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U22(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U23(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U25(x₁, x₂)) = 1 + x₁ + x₂
POL(U26(x₁)) = x₁
POL(U31(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(U32(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U44(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(U45(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(U46(x₁)) = 2 + 2·x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U52(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + x₃
POL(U53(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U54(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U55(x₁, x₂)) = x₁ + x₂
POL(U56(x₁)) = 1 + 2·x₁
POL(U61(x₁, x₂)) = 2 + x₁ + x₂
POL(U62(x₁, x₂)) = 2 + x₁ + x₂
POL(U63(x₁)) = 1 + x₁
POL(U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U72(x₁, x₂)) = 1 + x₁ + x₂
POL(U73(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U74(x₁)) = x₁
POL(U81(x₁, x₂)) = 1 + x₁ + x₂
POL(U82(x₁, x₂)) = 2 + x₁ + x₂
POL(U83(x₁)) = x₁
POL(U91(x₁, x₂)) = 1 + x₁ + x₂
POL(U92(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U12(x₁, x₂)) = x₁ + 2·x₂
POL(a__U13(x₁)) = 2 + x₁
POL(a__U21(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U22(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(a__U23(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U24(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(a__U25(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U26(x₁)) = x₁
POL(a__U31(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U32(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(a__U42(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(a__U43(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U44(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(a__U45(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a__U46(x₁)) = 2 + 2·x₁
POL(a__U51(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U52(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + x₃
POL(a__U53(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(a__U54(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U55(x₁, x₂)) = x₁ + 2·x₂
POL(a__U56(x₁)) = 2 + 2·x₁
POL(a__U61(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U62(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U63(x₁)) = 2 + x₁
POL(a__U71(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(a__U72(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U73(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U74(x₁)) = x₁
POL(a__U81(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U82(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U83(x₁)) = x₁
POL(a__U91(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U92(x₁)) = x₁
POL(a____(x₁, x₂)) = 2 + x₁ + x₂
POL(a__isList(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = 1 + 2·x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isPalListKind(x₁)) = 2·x₁
POL(a__isQid(x₁)) = 2 + 2·x₁
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = 1 + 2·x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 2·x₁
POL(isQid(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 1
POL(o) = 1
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:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNeList(X)) → a__isNeList(X)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(nil) → nil
mark(o) → o
a____(X1, X2) → __(X1, X2)
a__U13(X) → U13(X)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__isList(X) → isList(X)
a__isQid(X) → isQid(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U56(X) → U56(X)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__U81(X1, X2) → U81(X1, X2)
a__U91(X1, X2) → U91(X1, X2)

(6) Obligation:

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(u) → u
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U92(X) → U92(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U23(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U25(x₁, x₂)) = x₁ + x₂
POL(U26(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁, x₂)) = x₁ + x₂
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U44(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U45(x₁, x₂)) = x₁ + x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U52(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U53(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U54(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U55(x₁, x₂)) = x₁ + x₂
POL(U56(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U62(x₁, x₂)) = x₁ + x₂
POL(U63(x₁)) = 1 + x₁
POL(U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U72(x₁, x₂)) = 1 + x₁ + x₂
POL(U73(x₁, x₂)) = 1 + x₁ + x₂
POL(U74(x₁)) = x₁
POL(U81(x₁, x₂)) = 1 + x₁ + x₂
POL(U82(x₁, x₂)) = x₁ + x₂
POL(U83(x₁)) = x₁
POL(U91(x₁, x₂)) = 1 + x₁ + x₂
POL(U92(x₁)) = x₁
POL(__(x₁, x₂)) = 3 + x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U25(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U26(x₁)) = x₁
POL(a__U31(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U32(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U44(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U45(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U46(x₁)) = 1 + x₁
POL(a__U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U52(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U53(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U54(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U55(x₁, x₂)) = x₁ + x₂
POL(a__U56(x₁)) = x₁
POL(a__U61(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U62(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U72(x₁, x₂)) = x₁ + x₂
POL(a__U73(x₁, x₂)) = x₁ + x₂
POL(a__U74(x₁)) = x₁
POL(a__U81(x₁, x₂)) = x₁ + x₂
POL(a__U82(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U83(x₁)) = x₁
POL(a__U91(x₁, x₂)) = x₁ + x₂
POL(a__U92(x₁)) = x₁
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = 1 + x₁
POL(a__isNePal(x₁)) = 1 + x₁
POL(a__isPal(x₁)) = 1 + x₁
POL(a__isPalListKind(x₁)) = 1 + x₁
POL(a__isQid(x₁)) = x₁
POL(e) = 0
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)) = 2 + x₁
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:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(isList(X)) → a__isList(X)
mark(isQid(X)) → a__isQid(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U56(X)) → a__U56(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(u) → u
a__U11(X1, X2) → U11(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isNeList(X) → isNeList(X)
a__U25(X1, X2) → U25(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__isPal(X) → isPal(X)
a__U82(X1, X2) → U82(X1, X2)
a__isNePal(X) → isNePal(X)

(8) Obligation:

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

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U26(X)) → a__U26(mark(X))
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U74(X)) → a__U74(mark(X))
mark(U83(X)) → a__U83(mark(X))
mark(U92(X)) → a__U92(mark(X))
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U26(X) → U26(X)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U74(X) → U74(X)
a__U83(X) → U83(X)
a__U92(X) → U92(X)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U12(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U24(x₁, x₂, x₃)) = 1 + 2·x₁ + x₂ + 2·x₃
POL(U26(x₁)) = 1 + x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U54(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U55(x₁, x₂)) = 2 + x₁ + x₂
POL(U74(x₁)) = 1 + x₁
POL(U83(x₁)) = 1 + x₁
POL(U92(x₁)) = 1 + x₁
POL(a__U12(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a__U21(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U22(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U24(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(a__U26(x₁)) = 2 + x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U42(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U43(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U54(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(a__U55(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U74(x₁)) = 2 + x₁
POL(a__U83(x₁)) = 2 + x₁
POL(a__U92(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:

mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U92(X)) → a__U92(mark(X))
a__U12(X1, X2) → U12(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U26(X) → U26(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U74(X) → U74(X)
a__U83(X) → U83(X)

(10) Obligation:

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

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U26(X)) → a__U26(mark(X))
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U74(X)) → a__U74(mark(X))
mark(U83(X)) → a__U83(mark(X))
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U33(X) → U33(X)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U92(X) → U92(X)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U12(x₁, x₂)) = 1 + x₁ + x₂
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U24(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U26(x₁)) = 1 + x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U54(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U55(x₁, x₂)) = x₁ + x₂
POL(U74(x₁)) = 1 + x₁
POL(U83(x₁)) = 1 + x₁
POL(U92(x₁)) = x₁
POL(a__U12(x₁, x₂)) = x₁ + x₂
POL(a__U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U22(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U24(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U26(x₁)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U54(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U55(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U74(x₁)) = x₁
POL(a__U83(x₁)) = x₁
POL(a__U92(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(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U26(X)) → a__U26(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U74(X)) → a__U74(mark(X))
mark(U83(X)) → a__U83(mark(X))
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U92(X) → U92(X)

(12) Obligation:

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

mark(U33(X)) → a__U33(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
a__U33(X) → U33(X)
a__U43(X1, X2, X3) → U43(X1, X2, X3)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U33(x₁)) = 2 + 2·x₁
POL(U43(x₁, x₂, x₃)) = 1 + 2·x₁ + x₂ + x₃
POL(a__U33(x₁)) = 2 + 2·x₁
POL(a__U43(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
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(U33(X)) → a__U33(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)

(14) Obligation:

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

a__U33(X) → U33(X)
a__U43(X1, X2, X3) → U43(X1, X2, X3)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U33(x₁)) = x₁
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U33(x₁)) = 1 + x₁
POL(a__U43(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃

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

a__U33(X) → U33(X)
a__U43(X1, X2, X3) → U43(X1, X2, X3)

(16) Obligation:

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

(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

(21) RisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) QTRSRRRProof (EQUIVALENT transformation)

(10) Obligation:

(11) QTRSRRRProof (EQUIVALENT transformation)

(12) Obligation:

(13) QTRSRRRProof (EQUIVALENT transformation)

(14) Obligation:

(15) QTRSRRRProof (EQUIVALENT transformation)

(16) Obligation:

(17) RisEmptyProof (EQUIVALENT transformation)

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

(20) TRUE

(21) RisEmptyProof (EQUIVALENT transformation)

(22) TRUE