Termination w.r.t. Q proof of /tmp/tmpNZ81Gg/PALINDROME_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 DependencyGraphProof (⇔)

    ↳14 TRUE

  ↳15 QDP

    ↳16 QDPOrderProof (⇔)

    ↳17 QDP

    ↳18 PisEmptyProof (⇔)

    ↳19 TRUE

  ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 DependencyGraphProof (⇔)

    ↳24 TRUE

(0) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
__¹(__(X, Y), Z) → __¹(Y, Z)
U11¹(tt, V) → U12¹(isNeList(activate(V)))
U11¹(tt, V) → ISNELIST(activate(V))
U11¹(tt, V) → ACTIVATE(V)
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V2) → U23¹(isList(activate(V2)))
U22¹(tt, V2) → ISLIST(activate(V2))
U22¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, V) → U32¹(isQid(activate(V)))
U31¹(tt, V) → ISQID(activate(V))
U31¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V2) → U43¹(isNeList(activate(V2)))
U42¹(tt, V2) → ISNELIST(activate(V2))
U42¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → U53¹(isList(activate(V2)))
U52¹(tt, V2) → ISLIST(activate(V2))
U52¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V) → U62¹(isQid(activate(V)))
U61¹(tt, V) → ISQID(activate(V))
U61¹(tt, V) → ACTIVATE(V)
U71¹(tt, V) → U72¹(isNePal(activate(V)))
U71¹(tt, V) → ISNEPAL(activate(V))
U71¹(tt, V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISLIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISLIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → U31¹(isPalListKind(activate(V)), activate(V))
ISNELIST(V) → ISPALLISTKIND(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNELIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISNELIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __¹(X1, X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ACTIVATE(n__a) → A
ACTIVATE(n__e) → E
ACTIVATE(n__i) → I
ACTIVATE(n__o) → O
ACTIVATE(n__u) → U

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 52 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = __¹(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = x1
tt = tt
U12(x1) = U12
isNeList(x1) = isNeList(x1)
activate(x1) = activate(x1)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2) = U22(x2)
isList(x1) = isList(x1)
U23(x1) = U23
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32
isQid(x1) = isQid
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2) = U42
U43(x1) = U43
U51(x1, x2, x3) = U51(x3)
U52(x1, x2) = U52
U53(x1) = U53
U61(x1, x2) = x1
U62(x1) = U62
U71(x1, x2) = x1
U72(x1) = U72
isNePal(x1) = isNePal(x1)
and(x1, x2) = and(x2)
isPalListKind(x1) = x1
n__nil = n__nil
n____(x1, x2) = n____(x1, x2)
n__isPalListKind(x1) = x1
n__and(x1, x2) = x2
isPal(x1) = isPal(x1)
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
n__u = n__u
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > __^1₁
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > [activate₁, and₁, isPal₁] > [n_{_a}, a] > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > [activate₁, and₁, isPal₁] > o > n_{_o} > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > [activate₁, and₁, isPal₁] > u > n_{_u} > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > [U22₁, U23] > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > isQid > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > U41₃ > U42 > [tt, U12, U43, U62, U72]
[_{_₂}, U21₃, isList₁, n_{_{_{_₂}}}] > [U51₁, U52, U53] > [tt, U12, U43, U62, U72]
[nil, n_{_nil}] > [tt, U12, U43, U62, U72]
isNeList₁ > [activate₁, and₁, isPal₁] > [n_{_a}, a] > [tt, U12, U43, U62, U72]
isNeList₁ > [activate₁, and₁, isPal₁] > o > n_{_o} > [tt, U12, U43, U62, U72]
isNeList₁ > [activate₁, and₁, isPal₁] > u > n_{_u} > [tt, U12, U43, U62, U72]
isNeList₁ > [U31₂, U32] > [tt, U12, U43, U62, U72]
isNeList₁ > U41₃ > U42 > [tt, U12, U43, U62, U72]
isNeList₁ > [U51₁, U52, U53] > [tt, U12, U43, U62, U72]
isNePal₁ > [activate₁, and₁, isPal₁] > [n_{_a}, a] > [tt, U12, U43, U62, U72]
isNePal₁ > [activate₁, and₁, isPal₁] > o > n_{_o} > [tt, U12, U43, U62, U72]
isNePal₁ > [activate₁, and₁, isPal₁] > u > n_{_u} > [tt, U12, U43, U62, U72]
isNePal₁ > isQid > [tt, U12, U43, U62, U72]
[n_{_e}, e] > [tt, U12, U43, U62, U72]
[n_{_i}, i] > [tt, U12, U43, U62, U72]

Status:

i: []
__₂: [1,2]
n_{_u}: []
U41₃: [2,3,1]
U42: []
n_{_i}: []
activate₁: [1]
n_{_nil}: []
n_{_a}: []
tt: []
U72: []
U23: []
isQid: []
U43: []
U21₃: [2,1,3]
nil: []
a: []
isList₁: [1]
U31₂: [2,1]
U62: []
U32: []
e: []
U52: []
U53: []
U12: []
n_{_e}: []
__^1₁: [1]
isNePal₁: [1]
n_{_{_{_₂}}}: [1,2]
o: []
isPal₁: [1]
U22₁: [1]
n_{_o}: []
u: []
isNeList₁: [1]
and₁: [1]
U51₁: [1]

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = x1
n__isPalListKind(x1) = n__isPalListKind(x1)
ISPALLISTKIND(x1) = ISPALLISTKIND(x1)
n____(x1, x2) = n____(x1, x2)
AND(x1, x2) = AND(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
activate(x1) = activate(x1)
tt = tt
n__and(x1, x2) = n__and(x1, x2)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11
U12(x1) = U12
isNeList(x1) = isNeList(x1)
U21(x1, x2, x3) = U21
U22(x1, x2) = U22
isList(x1) = isList
U23(x1) = U23
U31(x1, x2) = U31(x1, x2)
U32(x1) = x1
isQid(x1) = isQid
U41(x1, x2, x3) = U41(x1, x3)
U42(x1, x2) = x1
U43(x1) = U43
U51(x1, x2, x3) = U51
U52(x1, x2) = U52
U53(x1) = x1
U61(x1, x2) = U61(x1, x2)
U62(x1) = U62
U71(x1, x2) = x1
U72(x1) = U72
isNePal(x1) = isNePal(x1)
and(x1, x2) = and(x1, x2)
n__nil = n__nil
isPal(x1) = isPal(x1)
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
n__u = n__u
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > n_{_and₂} > AND₂
[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > nil > n_{_nil}
[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > e > n_{_e}
[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > i > n_{_i}
[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > u > n_{_u}
[U11, U12, isNeList₁, U21, U22, isList, U23, U41₂, U51, U52] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [tt, U43, U62, U72]
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > n_{_and₂} > AND₂
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > nil > n_{_nil}
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > e > n_{_e}
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > i > n_{_i}
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [activate₁, and₂] > u > n_{_u}
[isNePal₁, isPal₁] > [n_{_{isPalListKind₁}}, ISPALLISTKIND₁, n_{_{_{_₂}}}, isPalListKind₁, _{_₂}] > [U31₂, isQid] > [tt, U43, U62, U72]
[isNePal₁, isPal₁] > U61₂
[n_{_a}, a] > [tt, U43, U62, U72]
[n_{_o}, o] > [tt, U43, U62, U72]

Status:

i: []
__₂: [1,2]
U22: []
n_{_u}: []
n_{_i}: []
U11: []
activate₁: [1]
and₂: [2,1]
n_{_nil}: []
n_{_a}: []
isList: []
tt: []
isPalListKind₁: [1]
U72: []
AND₂: [1,2]
U23: []
isQid: []
U43: []
nil: []
U21: []
U51: []
a: []
U31₂: [1,2]
U62: []
e: []
U52: []
U12: []
n_{_e}: []
isNePal₁: [1]
n_{_{_{_₂}}}: [1,2]
o: []
n_{_and₂}: [1,2]
isPal₁: [1]
n_{_o}: []
U61₂: [1,2]
ISPALLISTKIND₁: [1]
U41₂: [2,1]
u: []
isNeList₁: [1]
n_{_{isPalListKind₁}}: [1]

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1) = x1
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
ISPAL(x1) = ISPAL(x1)
activate(x1) = activate(x1)
U71¹(x1, x2) = U71¹(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
tt = tt
nil = nil
U11(x1, x2) = U11(x2)
U12(x1) = U12
isNeList(x1) = isNeList(x1)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2) = x1
isList(x1) = isList(x1)
U23(x1) = U23
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32
isQid(x1) = isQid
U41(x1, x2, x3) = U41(x1, x2)
U42(x1, x2) = U42(x1)
U43(x1) = U43
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2) = U52(x2)
U53(x1) = x1
U61(x1, x2) = U61(x1, x2)
U62(x1) = x1
U71(x1, x2) = U71
U72(x1) = U72
isNePal(x1) = isNePal(x1)
and(x1, x2) = and(x1, x2)
n__nil = n__nil
n__isPalListKind(x1) = n__isPalListKind(x1)
n__and(x1, x2) = n__and(x1, x2)
isPal(x1) = isPal(x1)
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
n__u = n__u
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_nil} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_and₂}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_e} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > a > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > o > n_{_o} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > u > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > ISPAL₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > U11₁ > U12 > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_nil} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_and₂}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_e} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > a > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > o > n_{_o} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > u > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U21₃ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > U11₁ > U12 > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U31₂ > U32 > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U42₁ > U43 > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_nil} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_and₂}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > n_{_e} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > a > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > o > n_{_o} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > [activate₁, U71^1₂, nil, and₂, e] > u > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U52₁ > [isPalListKind₁, isList₁, n_{_{isPalListKind₁}}] > U11₁ > U12 > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > U61₂
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > n_{_nil} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > n_{_and₂}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > n_{_e} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > a > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > o > n_{_o} > [tt, U23, isQid, n_{_a}, n_{_u}]
[n_{_{_{_₂}}}, _{_₂}, isNeList₁, U41₂, U51₃, isNePal₁, isPal₁] > [U71, U72] > [activate₁, U71^1₂, nil, and₂, e] > u > [tt, U23, isQid, n_{_a}, n_{_u}]

Status:

i: []
__₂: [1,2]
n_{_u}: []
U71^1₂: [2,1]
n_{_i}: []
activate₁: [1]
and₂: [2,1]
n_{_nil}: []
n_{_a}: []
tt: []
ISPAL₁: [1]
isPalListKind₁: [1]
U72: []
U52₁: [1]
U11₁: [1]
U23: []
isQid: []
U43: []
U21₃: [2,1,3]
U51₃: [2,3,1]
nil: []
a: []
isList₁: [1]
U31₂: [2,1]
U32: []
e: []
U12: []
n_{_e}: []
isNePal₁: [1]
n_{_{_{_₂}}}: [1,2]
U71: []
o: []
n_{_and₂}: [1,2]
isPal₁: [1]
n_{_o}: []
U61₂: [2,1]
U41₂: [1,2]
u: []
isNeList₁: [1]
U42₁: [1]
n_{_{isPalListKind₁}}: [1]

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(17) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(19) TRUE

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U52¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U22¹(tt, V2) → ISLIST(activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U41¹(tt, V1, V2) → ISLIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U52¹(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U41¹(tt, V1, V2) → ISLIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = x2
tt = tt
ISNELIST(x1) = x1
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x2, x3)
and(x1, x2) = x2
isPalListKind(x1) = isPalListKind
n__isPalListKind(x1) = n__isPalListKind
U42¹(x1, x2) = U42¹(x2)
isList(x1) = isList
U51¹(x1, x2, x3) = U51¹(x1, x2, x3)
U52¹(x1, x2) = U52¹(x1, x2)
isNeList(x1) = x1
ISLIST(x1) = x1
U21¹(x1, x2, x3) = U21¹(x1, x2, x3)
U22¹(x1, x2) = x2
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = x1
U12(x1) = U12
U21(x1, x2, x3) = U21
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2) = x2
U32(x1) = x1
isQid(x1) = x1
U41(x1, x2, x3) = U41
U42(x1, x2) = U42
U43(x1) = U43
U51(x1, x2, x3) = U51(x2, x3)
U52(x1, x2) = U52(x1, x2)
U53(x1) = U53(x1)
U61(x1, x2) = x2
U62(x1) = x1
U71(x1, x2) = x1
U72(x1) = U72
isNePal(x1) = x1
n__nil = n__nil
n__and(x1, x2) = x2
isPal(x1) = isPal(x1)
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
n__u = n__u
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[nil, n_{_nil}]
isPal₁ > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U41^1₂, U42^1₁]
isPal₁ > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51^1₃, U52^1₂]
isPal₁ > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > U21^1₃
isPal₁ > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51₂, U52₂, U53₁]
[n_{_e}, e] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U41^1₂, U42^1₁]
[n_{_e}, e] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51^1₃, U52^1₂]
[n_{_e}, e] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > U21^1₃
[n_{_e}, e] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51₂, U52₂, U53₁]
[n_{_i}, i] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U41^1₂, U42^1₁]
[n_{_i}, i] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51^1₃, U52^1₂]
[n_{_i}, i] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > U21^1₃
[n_{_i}, i] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51₂, U52₂, U53₁]
[n_{_o}, o] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U41^1₂, U42^1₁]
[n_{_o}, o] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51^1₃, U52^1₂]
[n_{_o}, o] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > U21^1₃
[n_{_o}, o] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51₂, U52₂, U53₁]
[n_{_u}, u] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U41^1₂, U42^1₁]
[n_{_u}, u] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51^1₃, U52^1₂]
[n_{_u}, u] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > U21^1₃
[n_{_u}, u] > [tt, n_{_{_{_₂}}}, isPalListKind, n_{_{isPalListKind}}, isList, _{_₂}, U12, U21, U41, U42, U43, U72, n_{_a}, a] > [U51₂, U52₂, U53₁]

Status:

i: []
U52^1₂: [2,1]
U52₂: [1,2]
__₂: [1,2]
n_{_u}: []
isPalListKind: []
U42: []
n_{_i}: []
n_{_nil}: []
U51₂: [1,2]
n_{_a}: []
isList: []
tt: []
U72: []
U41: []
U21^1₃: [2,1,3]
U43: []
nil: []
U21: []
a: []
e: []
U12: []
n_{_e}: []
n_{_{_{_₂}}}: [1,2]
o: []
U51^1₃: [3,2,1]
isPal₁: [1]
n_{_o}: []
U41^1₂: [2,1]
U53₁: [1]
n_{_{isPalListKind}}: []
u: []
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, V) → ISNELIST(activate(V))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U22¹(tt, V2) → ISLIST(activate(V2))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) TRUE

(15) Obligation:

(16) QDPOrderProof (EQUIVALENT transformation)

(17) Obligation:

(18) PisEmptyProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) TRUE