Termination w.r.t. Q proof of /tmp/tmpTDJiUg/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 PisEmptyProof (⇔)

    ↳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) → 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, n____(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)) → __(activate(X1), activate(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, n____(P, I))) → U71¹(isQid(activate(I)), activate(P))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(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(activate(X1), activate(X2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(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, n____(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)) → __(activate(X1), activate(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 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, n____(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)) → __(activate(X1), activate(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) = x1
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11
tt = tt
U21(x1, x2) = x1
U22(x1) = U22
isList(x1) = isList
activate(x1) = activate(x1)
U31(x1) = U31
U41(x1, x2) = U41(x2)
U42(x1) = U42
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51
U52(x1) = U52
U61(x1) = U61(x1)
U71(x1, x2) = U71
U72(x1) = U72
isPal(x1) = isPal
U81(x1) = x1
n__nil = n__nil
n____(x1, x2) = n____(x1, x2)
isQid(x1) = isQid
isNePal(x1) = isNePal
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:

[activate₁, U31, isNeList₁, isPal] > __₂ > n_{_{_{_₂}}} > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > isList > U11 > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > U41₁ > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > U51 > U52 > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > [U71, isNePal] > U61₁ > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > [U71, isNePal] > U72 > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > [n_{_e}, e] > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > [n_{_u}, u] > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > a > n_{_a} > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]
[activate₁, U31, isNeList₁, isPal] > o > n_{_o} > [nil, tt, U22, U42, n_{_nil}, isQid, n_{_i}, i]

Status:

__₂: [1,2]
nil: multiset
U11: multiset
tt: multiset
U22: multiset
isList: multiset
activate₁: [1]
U31: []
U41₁: multiset
U42: multiset
isNeList₁: [1]
U51: multiset
U52: []
U61₁: [1]
U71: multiset
U72: []
isPal: multiset
n_{_nil}: multiset
n_{_{_{_₂}}}: multiset
isQid: multiset
isNePal: 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, n____(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)) → __(activate(X1), activate(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, n____(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)) → __(activate(X1), activate(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____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

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, n____(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)) → __(activate(X1), activate(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.

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = x1
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U21(x1, x2) = U21
U22(x1) = U22
isList(x1) = isList(x1)
activate(x1) = x1
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1)
U52(x1) = U52
U61(x1) = U61
U71(x1, x2) = U71
U72(x1) = U72
isPal(x1) = isPal
U81(x1) = U81
n__nil = n__nil
isQid(x1) = isQid
isNePal(x1) = isNePal
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}, isList₁, U41, isNeList₁] > U21 > [tt, U22, U31, U52, isPal, U81, isQid]
[n_{_{_{_₂}}}, _{_₂}, isList₁, U41, isNeList₁] > U42 > [tt, U22, U31, U52, isPal, U81, isQid]
[n_{_{_{_₂}}}, _{_₂}, isList₁, U41, isNeList₁] > U51₁
[n_{_{_{_₂}}}, _{_₂}, isList₁, U41, isNeList₁] > [U71, U72] > [tt, U22, U31, U52, isPal, U81, isQid]
[nil, n_{_nil}] > [tt, U22, U31, U52, isPal, U81, isQid]
[U61, isNePal] > [U71, U72] > [tt, U22, U31, U52, isPal, U81, isQid]
[n_{_a}, a]
[n_{_e}, e]
[n_{_i}, i] > [tt, U22, U31, U52, isPal, U81, isQid]
[n_{_o}, o]
[n_{_u}, u] > [tt, U22, U31, U52, isPal, U81, isQid]

Status:

n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
nil: multiset
tt: multiset
U21: []
U22: []
isList₁: [1]
U31: []
U41: []
U42: multiset
isNeList₁: [1]
U51₁: [1]
U52: []
U61: []
U71: multiset
U72: multiset
isPal: []
U81: []
n_{_nil}: multiset
isQid: []
isNePal: []
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, n____(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)) → __(activate(X1), activate(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:
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, n____(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)) → __(activate(X1), activate(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) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) 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, n____(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, n____(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)) → __(activate(X1), activate(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.

U71¹(tt, P) → ISPAL(activate(P))
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, n____(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(x2)
tt = tt
ISPAL(x1) = ISPAL(x1)
activate(x1) = x1
ISNEPAL(x1) = ISNEPAL(x1)
n____(x1, x2) = n____(x1, x2)
isQid(x1) = isQid(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11
U21(x1, x2) = U21(x2)
U22(x1) = U22
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x1
U52(x1) = U52
U61(x1) = U61
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
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:

[U71^1₁, n_{_{_{_₂}}}, _{_₂}] > ISPAL₁ > ISNEPAL₁ > isQid₁
[U71^1₁, n_{_{_{_₂}}}, _{_₂}] > U41₁ > U42₁ > isQid₁
[U71^1₁, n_{_{_{_₂}}}, _{_₂}] > U41₁ > isNeList₁ > isQid₁
[nil, n_{_nil}] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > ISPAL₁ > ISNEPAL₁ > isQid₁
[nil, n_{_nil}] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > U42₁ > isQid₁
[nil, n_{_nil}] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > isNeList₁ > isQid₁
[U11, isList₁] > U21₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > ISPAL₁ > ISNEPAL₁ > isQid₁
[U11, isList₁] > U21₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > U42₁ > isQid₁
[U11, isList₁] > U21₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > isNeList₁ > isQid₁
[U71₂, isNePal₁] > isPal₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > ISPAL₁ > ISNEPAL₁ > isQid₁
[U71₂, isNePal₁] > isPal₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > U42₁ > isQid₁
[U71₂, isNePal₁] > isPal₁ > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > isNeList₁ > isQid₁
[n_{_u}, u] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > ISPAL₁ > ISNEPAL₁ > isQid₁
[n_{_u}, u] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > U42₁ > isQid₁
[n_{_u}, u] > [tt, U22, U52, U61, U81, n_{_a}, n_{_e}, n_{_i}, n_{_o}, a, e, i, o] > isNeList₁ > isQid₁

Status:

U71^1₁: [1]
tt: multiset
ISPAL₁: [1]
ISNEPAL₁: multiset
n_{_{_{_₂}}}: [1,2]
isQid₁: [1]
__₂: [1,2]
nil: multiset
U11: multiset
U21₁: multiset
U22: []
isList₁: multiset
U41₁: multiset
U42₁: multiset
isNeList₁: multiset
U52: []
U61: []
U71₂: [1,2]
isPal₁: [1]
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, n____(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)) → __(activate(X1), activate(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) → 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, n____(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)) → __(activate(X1), activate(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:

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, n____(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)) → __(activate(X1), activate(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.

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))
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) = x2
tt = tt
ISLIST(x1) = x1
activate(x1) = x1
ISNELIST(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2) = U41¹(x1, x2)
isList(x1) = isList
U51¹(x1, x2) = x2
isNeList(x1) = isNeList
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11
U21(x1, x2) = U21
U22(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41
U42(x1) = U42
U51(x1, x2) = x1
U52(x1) = x1
U61(x1) = U61
U71(x1, x2) = U71
U72(x1) = U72
isPal(x1) = isPal
U81(x1) = U81
n__nil = n__nil
isQid(x1) = isQid
isNePal(x1) = 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}}}, U41^1₂, _{_₂}, U71] > U72 > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[isNeList, U41, isQid] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[nil, n_{_nil}] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
isPal > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[n_{_a}, a] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[n_{_i}, i] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[n_{_o}, o] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]
[n_{_u}, u] > [tt, isList, U11, U21, U42, U61, U81, n_{_e}, e]

Status:

tt: multiset
n_{_{_{_₂}}}: [1,2]
U41^1₂: [1,2]
isList: multiset
isNeList: multiset
__₂: [1,2]
nil: multiset
U11: multiset
U21: multiset
U41: multiset
U42: multiset
U61: multiset
U71: []
U72: []
isPal: multiset
U81: multiset
n_{_nil}: multiset
isQid: 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, n____(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)) → __(activate(X1), activate(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:

U21¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(isList(activate(V1)), activate(V2))
U51¹(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) → 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, n____(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)) → __(activate(X1), activate(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 4 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) PisEmptyProof (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