Termination w.r.t. Q proof of /tmp/tmpMwuzIt/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 RisEmptyProof (⇔)

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

Status:

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

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
o → n__o
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

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

Status:

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

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)
and(X1, X2) → n__and(X1, X2)
e → n__e
o → n__o

(4) Obligation:

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
u → n__u

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

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

nil > n_{_nil} > n_{_{_{_₂}}}
__₂ > n_{_{_{_₂}}}
a > n_{_a} > n_{_{_{_₂}}}
u > n_{_u} > n_{_{_{_₂}}}

Status:

a: multiset
__₂: multiset
n_{_u}: multiset
u: multiset
n_{_nil}: multiset
n_{_{_{_₂}}}: multiset
nil: multiset
n_{_a}: multiset

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)
a → n__a
u → n__u

(6) Obligation:

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

(7) 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) RisEmptyProof (EQUIVALENT transformation)

(8) TRUE