Termination w.r.t. Q proof of /tmp/tmpgRxFqP/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:
Lexicographic path order with status [LPO].
Quasi-Precedence:

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

Status:

n_{_and₂}: [1,2]
__₂: [1,2]
and₂: [1,2]
n_{_{_{_₂}}}: [1,2]

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)
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

(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)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

__₂ > n_{_{_{_₂}}}
and₂ > n_{_and₂}

Status:

trivial

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)

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)