Termination w.r.t. Q proof of Transformed_CSR_04_PALINDROME_complete

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

0 QTRS

↳1 QTRSRRRProof (⇔, 420 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 42 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 0 ms)

↳6 QTRS

↳7 RisEmptyProof (⇔, 0 ms)

↳8 YES

(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) = a__U62(x1)
a__U71(x1, x2) = a__U71(x1, x2)
a__U72(x1) = a__U72(x1)
a__isNePal(x1) = a__isNePal(x1)
a__and(x1, x2) = a__and(x1, x2)
a__isPalListKind(x1) = x1
isPalListKind(x1) = 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) = U62(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isNePal(x1) = isNePal(x1)

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

[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_U11₂}, U11₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U53₁}, U53₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U21₃}, U21₃] > [a_{_U22₂}, U22₂] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_and₂}, and₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_U11₂}, U11₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U53₁}, U53₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_and₂}, and₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U41₃}, U41₃] > [a_{_U42₂}, U42₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_U11₂}, U11₂] > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U53₁}, U53₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
[a_{_{_{_₂}}}, _{_₂}] > [a_{_U51₃}, U51₃] > [a_{_isList₁}, a_{_U52₂}, isList₁, U52₂] > [a_{_and₂}, and₂]
[a_{_{_{_₂}}}, _{_₂}] > [isPal₁, a_{_isPal₁}] > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
[a_{_{_{_₂}}}, _{_₂}] > [isPal₁, a_{_isPal₁}] > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
nil > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
nil > tt > [a_{_U53₁}, U53₁]
nil > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
nil > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
a > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
a > tt > [a_{_U53₁}, U53₁]
a > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
a > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
e > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
e > tt > [a_{_U53₁}, U53₁]
e > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
e > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
i > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
i > tt > [a_{_U53₁}, U53₁]
i > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
i > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
o > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
o > tt > [a_{_U53₁}, U53₁]
o > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
o > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]
u > tt > [a_{_isNeList₁}, isNeList₁] > [a_{_U31₂}, U31₂]
u > tt > [a_{_U53₁}, U53₁]
u > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U61₂}, U61₂] > [a_{_U62₁}, U62₁]
u > tt > [a_{_U71₂}, a_{_isNePal₁}, U71₂, isNePal₁] > [a_{_U72₁}, U72₁]

Status:

a_{_{_{_₂}}}: [1,2]
__₂: [1,2]
nil: multiset
a_{_U11₂}: [2,1]
tt: multiset
a_{_isNeList₁}: [1]
a_{_U21₃}: multiset
a_{_U22₂}: multiset
a_{_isList₁}: [1]
a_{_U31₂}: [1,2]
a_{_U41₃}: multiset
a_{_U42₂}: multiset
a_{_U51₃}: multiset
a_{_U52₂}: [2,1]
a_{_U53₁}: [1]
a_{_U61₂}: multiset
a_{_U62₁}: multiset
a_{_U71₂}: [2,1]
a_{_U72₁}: multiset
a_{_isNePal₁}: [1]
a_{_and₂}: [2,1]
and₂: [2,1]
isPal₁: [1]
a_{_isPal₁}: [1]
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U11₂: [2,1]
isNeList₁: [1]
U21₃: multiset
U22₂: multiset
isList₁: [1]
U31₂: [1,2]
U41₃: multiset
U42₂: multiset
U51₃: multiset
U52₂: [2,1]
U53₁: [1]
U61₂: multiset
U62₁: multiset
U71₂: [2,1]
U72₁: multiset
isNePal₁: [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__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

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

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
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
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)

(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(i) → i

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) RisEmptyProof (EQUIVALENT transformation)

(8) YES