Termination w.r.t. Q proof of /tmp/tmp26wD2G/PALINDROME_complete-noand

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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(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)) → U91(isPalListKind(activate(V1)), 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)
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__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 > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > U13₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > U26₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > U13₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > U46₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > U13₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > U56₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U71₃ > U72₂ > U73₂ > U74₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > U83₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > U63₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[_{_₂}, n_{_{_{_₂}}}] > U91₂ > U92₁ > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[nil, n_{_nil}] > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[n_{_a}, a] > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[n_{_e}, e] > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[n_{_i}, i] > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]
[n_{_u}, u] > [tt, isPalListKind₁, activate₁, isQid₁, n_{_o}, o]

Status:

__₂: [1,2]
nil: []
U11₂: [2,1]
tt: []
U12₂: [1,2]
isPalListKind₁: [1]
activate₁: [1]
U13₁: [1]
isNeList₁: [1]
U21₃: [1,2,3]
U22₃: [1,3,2]
U23₃: [1,3,2]
U24₃: [1,2,3]
U25₂: [2,1]
isList₁: [1]
U26₁: [1]
U31₂: [2,1]
U32₂: [1,2]
U33₁: [1]
isQid₁: [1]
U41₃: [1,2,3]
U42₃: [1,3,2]
U43₃: [2,3,1]
U44₃: [1,2,3]
U45₂: [2,1]
U46₁: [1]
U51₃: [3,1,2]
U52₃: [1,2,3]
U53₃: [3,2,1]
U54₃: [1,2,3]
U55₂: [2,1]
U56₁: [1]
U61₂: [2,1]
U62₂: [1,2]
U63₁: [1]
U71₃: [1,3,2]
U72₂: [2,1]
U73₂: [1,2]
isPal₁: [1]
U74₁: [1]
U81₂: [2,1]
U82₂: [1,2]
U83₁: [1]
isNePal₁: [1]
U91₂: [2,1]
U92₁: [1]
n_{_nil}: []
n_{_{_{_₂}}}: [1,2]
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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(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)) → U91(isPalListKind(activate(V1)), 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__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)
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}}}
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]
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:

nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
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