Termination w.r.t. Q proof of /tmp/tmpbu0JoA/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 QTRSRRRProof (⇔)

↳6 QTRS

↳7 QTRSRRRProof (⇔)

↳8 QTRS

↳9 RisEmptyProof (⇔)

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

Status:

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

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
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:

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

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

[nil, n_{_nil}]
[_{_₂}, n_{_{_{_₂}}}]
[isPalListKind₁, n_{_{isPalListKind₁}}]
[and₂, n_{_and₂}]
[a, n_{_a}]
[e, n_{_e}]
[i, n_{_i}]
[o, n_{_o}]
u > n_{_u}

Status:

nil: []
n_{_nil}: []
__₂: [1,2]
n_{_{_{_₂}}}: [1,2]
isPalListKind₁: [1]
n_{_{isPalListKind₁}}: [1]
and₂: [1,2]
n_{_and₂}: [1,2]
a: []
n_{_a}: []
e: []
n_{_e}: []
i: []
n_{_i}: []
o: []
n_{_o}: []
u: []
n_{_u}: []

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

u → n__u

(4) Obligation:

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

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

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

nil > n_{_nil}
__₂ > n_{_{_{_₂}}}
[isPalListKind₁, n_{_{isPalListKind₁}}]
and₂ > n_{_and₂}
[a, n_{_a}]
e > n_{_e}
i > n_{_i}
o > n_{_o}

Status:

nil: []
n_{_nil}: []
__₂: [1,2]
n_{_{_{_₂}}}: [1,2]
isPalListKind₁: [1]
n_{_{isPalListKind₁}}: [1]
and₂: [1,2]
n_{_and₂}: [1,2]
a: []
n_{_a}: []
e: []
n_{_e}: []
i: []
n_{_i}: []
o: []
n_{_o}: []

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
and(X1, X2) → n__and(X1, X2)
e → n__e
i → n__i
o → n__o

(6) Obligation:

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

isPalListKind(X) → n__isPalListKind(X)
a → n__a

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

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

isPalListKind₁ > n_{_{isPalListKind₁}}
a > n_{_a}

Status:

isPalListKind₁: [1]
n_{_{isPalListKind₁}}: [1]
a: []
n_{_a}: []

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

isPalListKind(X) → n__isPalListKind(X)
a → n__a

(8) Obligation:

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

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) RisEmptyProof (EQUIVALENT transformation)

(10) TRUE