Termination w.r.t. Q proof of /tmp/tmpDCk4Fn/PALINDROME_nokinds-noand

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 DependencyGraphProof (⇔)

    ↳19 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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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)
U21¹(tt, V2) → U22¹(isList(activate(V2)))
U21¹(tt, V2) → ISLIST(activate(V2))
U21¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, V2) → U42¹(isNeList(activate(V2)))
U41¹(tt, V2) → ISNELIST(activate(V2))
U41¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V2) → U52¹(isList(activate(V2)))
U51¹(tt, V2) → ISLIST(activate(V2))
U51¹(tt, V2) → ACTIVATE(V2)
U71¹(tt, P) → U72¹(isPal(activate(P)))
U71¹(tt, P) → ISPAL(activate(P))
U71¹(tt, P) → ACTIVATE(P)
ISLIST(V) → U11¹(isNeList(activate(V)))
ISLIST(V) → ISNELIST(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U21¹(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → U31¹(isQid(activate(V)))
ISNELIST(V) → ISQID(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → U41¹(isList(activate(V1)), activate(V2))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → U51¹(isNeList(activate(V1)), activate(V2))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → U61¹(isQid(activate(V)))
ISNEPAL(V) → ISQID(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → U71¹(isQid(activate(I)), activate(P))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → U81¹(isNePal(activate(V)))
ISPAL(V) → ISNEPAL(activate(V))
ISPAL(V) → ACTIVATE(V)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __¹(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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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 3 SCCs with 32 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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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, x2)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U21(x1, x2) = x2
U22(x1) = U22
isList(x1) = isList(x1)
activate(x1) = activate(x1)
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal
U81(x1) = U81
n__nil = n__nil
n____(x1, x2) = n____(x1, x2)
isQid(x1) = isQid
isNePal(x1) = isNePal(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

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

[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > [isList₁, activate₁, o] > nil > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > [isList₁, activate₁, o] > n_{_o} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > [isList₁, activate₁, o] > e > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > [isList₁, activate₁, o] > i > n_{_i} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > [isList₁, activate₁, o] > u > n_{_u} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U41₂ > U42₁ > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > [isList₁, activate₁, o] > nil > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > [isList₁, activate₁, o] > n_{_o} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > [isList₁, activate₁, o] > e > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > [isList₁, activate₁, o] > i > n_{_i} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > [isList₁, activate₁, o] > u > n_{_u} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[_{_^1₂}, _{_₂}, n_{_{_{_₂}}}] > U51₂ > U52₁ > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[U61₁, isNePal₁] > [isList₁, activate₁, o] > nil > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[U61₁, isNePal₁] > [isList₁, activate₁, o] > n_{_o} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[U61₁, isNePal₁] > [isList₁, activate₁, o] > e > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[U61₁, isNePal₁] > [isList₁, activate₁, o] > i > n_{_i} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[U61₁, isNePal₁] > [isList₁, activate₁, o] > u > n_{_u} > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]
[n_{_a}, a] > [tt, U22, isPal, U81, n_{_nil}, isQid, n_{_e}]

Status:

__^1₂: [1,2]
__₂: [1,2]
nil: multiset
tt: multiset
U22: []
isList₁: multiset
activate₁: multiset
U41₂: multiset
U42₁: multiset
U51₂: multiset
U52₁: multiset
U61₁: multiset
isPal: []
U81: []
n_{_nil}: multiset
n_{_{_{_₂}}}: [1,2]
isQid: []
isNePal₁: multiset
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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:

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

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2) = U71¹(x1, x2)
tt = tt
ISPAL(x1) = ISPAL(x1)
activate(x1) = x1
ISNEPAL(x1) = ISNEPAL(x1)
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
isQid(x1) = isQid(x1)
nil = nil
U11(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1) = x1
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = x1
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72
isPal(x1) = isPal(x1)
U81(x1) = U81
n__nil = n__nil
isNePal(x1) = isNePal(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

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

isPal₁ > [U71^1₂, tt, ISPAL₁, ISNEPAL₁, nil, U72, U81, n_{_nil}, n_{_a}, n_{_e}, n_{_o}, n_{_u}, a, e, o, u] > [isQid₁, U21₂, isList₁, U41₁, isNeList₁, U51₂] > [n_{_{_{_₂}}}, _{_₂}]
isPal₁ > isNePal₁ > [isQid₁, U21₂, isList₁, U41₁, isNeList₁, U51₂] > [n_{_{_{_₂}}}, _{_₂}]
isPal₁ > isNePal₁ > U61₁ > [n_{_{_{_₂}}}, _{_₂}]
isPal₁ > isNePal₁ > U71₂ > [n_{_{_{_₂}}}, _{_₂}]
[n_{_i}, i] > [U71^1₂, tt, ISPAL₁, ISNEPAL₁, nil, U72, U81, n_{_nil}, n_{_a}, n_{_e}, n_{_o}, n_{_u}, a, e, o, u] > [isQid₁, U21₂, isList₁, U41₁, isNeList₁, U51₂] > [n_{_{_{_₂}}}, _{_₂}]

Status:

U71^1₂: [2,1]
tt: multiset
ISPAL₁: [1]
ISNEPAL₁: [1]
n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
isQid₁: [1]
nil: multiset
U21₂: [2,1]
isList₁: [1]
U41₁: [1]
isNeList₁: [1]
U51₂: [2,1]
U61₁: multiset
U71₂: multiset
U72: []
isPal₁: multiset
U81: []
n_{_nil}: multiset
isNePal₁: [1]
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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:

ISPAL(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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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:

U21¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(isList(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → U21¹(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → U51¹(isNeList(activate(V1)), activate(V2))
U51¹(tt, V2) → ISLIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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.

U21¹(tt, V2) → ISLIST(activate(V2))
ISNELIST(n____(V1, V2)) → U41¹(isList(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → U21¹(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → U51¹(isNeList(activate(V1)), activate(V2))
U51¹(tt, V2) → ISLIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2) = U21¹(x1, x2)
tt = tt
ISLIST(x1) = ISLIST(x1)
activate(x1) = activate(x1)
ISNELIST(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2) = U41¹(x1, x2)
isList(x1) = isList
U51¹(x1, x2) = U51¹(x1, x2)
isNeList(x1) = x1
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11
U21(x1, x2) = U21
U22(x1) = x1
U31(x1) = U31
U41(x1, x2) = x2
U42(x1) = U42
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72
isPal(x1) = x1
U81(x1) = U81
n__nil = n__nil
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(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

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

[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > nil > n_{_nil} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > n_{_u} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > a > n_{_a} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > e > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > i > n_{_i} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > o > n_{_o} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > nil > n_{_nil} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > n_{_u} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > a > n_{_a} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > e > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > i > n_{_i} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > o > n_{_o} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > nil > n_{_nil} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > n_{_u} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > a > n_{_a} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > e > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > i > n_{_i} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > [isList, U11, U21] > [ISLIST₁, activate₁, U71₂, U72, u] > o > n_{_o} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > nil > n_{_nil} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > n_{_u} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > a > n_{_a} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > e > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > i > n_{_i} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [ISLIST₁, activate₁, U71₂, U72, u] > o > n_{_o} > [tt, U31, U42, U81, n_{_e}]
[n_{_{_{_₂}}}, _{_₂}] > U51₂ > U52 > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > nil > n_{_nil} > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > n_{_u} > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > a > n_{_a} > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > e > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > i > n_{_i} > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > [ISLIST₁, activate₁, U71₂, U72, u] > o > n_{_o} > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > U61₁ > [tt, U31, U42, U81, n_{_e}]
isNePal₁ > isQid₁ > [tt, U31, U42, U81, n_{_e}]

Status:

U21^1₂: multiset
tt: multiset
ISLIST₁: multiset
activate₁: multiset
n_{_{_{_₂}}}: [1,2]
U41^1₂: multiset
isList: []
U51^1₂: [1,2]
__₂: [1,2]
nil: multiset
U11: []
U21: []
U31: []
U42: []
U51₂: multiset
U52: multiset
U61₁: multiset
U71₂: multiset
U72: multiset
U81: []
n_{_nil}: multiset
isQid₁: multiset
isNePal₁: [1]
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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:
The TRS P consists of the following rules:

ISLIST(V) → ISNELIST(activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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)
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__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) DependencyGraphProof (EQUIVALENT transformation)

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

(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) DependencyGraphProof (EQUIVALENT transformation)

(19) TRUE