Termination w.r.t. Q proof of /tmp/tmpzhwb3V/PALINDROME_complete

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 RisEmptyProof (⇔)

↳6 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) QTRSRRRProof (EQUIVALENT transformation)

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

[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₂ > U23₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₂ > U43₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₂ > isList₁ > U11₂ > isNeList₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₂ > U53₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > U72₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > U61₂ > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > U61₂ > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > U61₂ > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > U61₂ > U62₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > nil > n_{_nil} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > isQid₁ > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[_{_₂}, n_{_{_{_₂}}}] > isPal₁ > U71₂ > isNePal₁ > [and₂, n_{_and₂}] > [activate₁, U31₂, e] > i > n_{_i} > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[n_{_a}, a] > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[n_{_o}, o] > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]
[n_{_u}, u] > [tt, isPalListKind₁, n_{_e}] > n_{_{isPalListKind₁}} > [U12₁, U32₁]

Status:

i: multiset
U52₂: multiset
__₂: [1,2]
n_{_u}: multiset
U41₃: [2,3,1]
n_{_i}: multiset
activate₁: multiset
and₂: multiset
n_{_nil}: multiset
U71₂: multiset
U23₁: [1]
n_{_a}: multiset
tt: multiset
isPalListKind₁: multiset
U72₁: [1]
U12₁: [1]
U21₃: multiset
U51₃: [1,3,2]
nil: multiset
a: multiset
isList₁: [1]
U31₂: multiset
U22₂: multiset
U42₂: [2,1]
e: multiset
n_{_e}: multiset
isNePal₁: multiset
U11₂: [2,1]
n_{_{_{_₂}}}: [1,2]
o: multiset
isQid₁: [1]
U43₁: [1]
U62₁: multiset
n_{_and₂}: multiset
isPal₁: multiset
n_{_o}: multiset
U53₁: [1]
U61₂: [2,1]
u: multiset
isNeList₁: multiset
U32₁: multiset
n_{_{isPalListKind₁}}: multiset

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

__(__(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
isPalListKind(X) → n__isPalListKind(X)
e → n__e
i → n__i
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

(2) Obligation:

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

__(X1, X2) → n____(X1, X2)
and(X1, X2) → n__and(X1, X2)
a → n__a
o → n__o
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

__₂ > n_{_{_{_₂}}}
and₂ > n_{_and₂} > n_{_{_{_₂}}}
a > n_{_a} > n_{_{_{_₂}}}
o > n_{_o} > n_{_{_{_₂}}}
u > n_{_u} > n_{_{_{_₂}}}

Status:

n_{_and₂}: multiset
a: multiset
n_{_o}: multiset
__₂: multiset
n_{_u}: multiset
u: multiset
and₂: multiset
n_{_{_{_₂}}}: multiset
o: multiset
n_{_a}: multiset

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

__(X1, X2) → n____(X1, X2)
and(X1, X2) → n__and(X1, X2)
a → n__a
o → n__o
u → n__u

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) RisEmptyProof (EQUIVALENT transformation)

(6) TRUE