Termination w.r.t. Q proof of Transformed_CSR_04_PALINDROME_complete-noand

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

0 QTRS

↳1 QTRSRRRProof (⇔, 241 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 0 ms)

↳4 QTRS

↳5 RisEmptyProof (⇔, 0 ms)

↳6 YES

(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:
Combined order from the following AFS and order.
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11(x1, x2)
tt = tt
U12(x1, x2) = U12(x1, x2)
isPalListKind(x1) = x1
activate(x1) = x1
U13(x1) = x1
isNeList(x1) = isNeList(x1)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2, x3) = U22(x1, x2, x3)
U23(x1, x2, x3) = U23(x1, x2, x3)
U24(x1, x2, x3) = U24(x1, x2, x3)
U25(x1, x2) = U25(x1, x2)
isList(x1) = isList(x1)
U26(x1) = U26(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1, x2) = U32(x1, x2)
U33(x1) = x1
isQid(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x1, x2, x3)
U43(x1, x2, x3) = U43(x1, x2, x3)
U44(x1, x2, x3) = U44(x1, x2, x3)
U45(x1, x2) = U45(x1, x2)
U46(x1) = U46(x1)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
U53(x1, x2, x3) = U53(x1, x2, x3)
U54(x1, x2, x3) = U54(x1, x2, x3)
U55(x1, x2) = U55(x1, x2)
U56(x1) = U56(x1)
U61(x1, x2) = U61(x1, x2)
U62(x1, x2) = U62(x1, x2)
U63(x1) = x1
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2) = U72(x1, x2)
U73(x1, x2) = U73(x1, x2)
isPal(x1) = isPal(x1)
U74(x1) = x1
U81(x1, x2) = U81(x1, x2)
U82(x1, x2) = U82(x1, x2)
U83(x1) = x1
isNePal(x1) = isNePal(x1)
U91(x1, x2) = U91(x1, x2)
U92(x1) = U92(x1)
n__nil = n__nil
n____(x1, x2) = n____(x1, x2)
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
n__u = n__u
a = a
e = e
i = i
o = o
u = u

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

[_{_₂}, U71₃, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > [isList₁, U55₂] > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > [isList₁, U55₂] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > [isList₁, U55₂] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > [isList₁, U55₂] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > [isList₁, U55₂] > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > [isList₁, U55₂] > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U51₃ > U52₃ > U53₃ > U54₃ > [isList₁, U55₂] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U72₂ > U73₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U72₂ > isPal₁ > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U72₂ > isPal₁ > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U72₂ > isPal₁ > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U91₂ > U92₁ > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U91₂ > U92₁ > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[_{_₂}, U71₃, n_{_{_{_₂}}}] > U91₂ > U92₁ > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[nil, n_{_nil}] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[nil, n_{_nil}] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[nil, n_{_nil}] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[n_{_e}, e] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[n_{_e}, e] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[n_{_e}, e] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[n_{_i}, i] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[n_{_i}, i] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[n_{_i}, i] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[n_{_o}, o] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[n_{_o}, o] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[n_{_o}, o] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]
[n_{_u}, u] > [tt, n_{_a}, a] > U32₂ > [U26₁, U56₁]
[n_{_u}, u] > [tt, n_{_a}, a] > U46₁ > [U26₁, U56₁]
[n_{_u}, u] > [tt, n_{_a}, a] > isNePal₁ > U61₂ > U62₂ > [U26₁, U56₁]

Status:

__₂: [1,2]
nil: multiset
U11₂: [2,1]
tt: multiset
U12₂: [2,1]
isNeList₁: multiset
U21₃: [3,1,2]
U22₃: [3,1,2]
U23₃: [3,2,1]
U24₃: [1,2,3]
U25₂: multiset
isList₁: multiset
U26₁: [1]
U31₂: [2,1]
U32₂: [2,1]
U41₃: multiset
U42₃: [2,1,3]
U43₃: [2,1,3]
U44₃: multiset
U45₂: multiset
U46₁: multiset
U51₃: [3,1,2]
U52₃: [3,2,1]
U53₃: [2,3,1]
U54₃: multiset
U55₂: multiset
U56₁: [1]
U61₂: [1,2]
U62₂: [1,2]
U71₃: [1,2,3]
U72₂: [1,2]
U73₂: [1,2]
isPal₁: [1]
U81₂: [1,2]
U82₂: multiset
isNePal₁: [1]
U91₂: multiset
U92₁: multiset
n_{_nil}: multiset
n_{_{_{_₂}}}: [1,2]
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: 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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
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)))
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)))
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)))
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
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__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__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt

(2) Obligation:

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

U13(tt) → tt
U33(tt) → tt
U63(tt) → tt
U74(tt) → tt
U83(tt) → tt
isPalListKind(n__a) → tt
isQid(n__a) → 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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

U83₁ > activate₁ > n_{_u} > u > n_{_o} > U13₁ > o > n_{_i} > i > n_{_e} > e > a > n_{_a} > n_{_{_{_₂}}} > __₂ > n_{_nil} > U33₁ > nil > isQid₁ > isPalListKind₁ > U74₁ > U63₁ > tt

and weight map:

tt=2
n__a=1
nil=2
n__nil=1
a=2
e=2
n__e=1
i=2
n__i=1
o=2
n__o=1
u=2
n__u=1
U13_1=1
U33_1=1
U63_1=1
U74_1=1
U83_1=0
isPalListKind_1=1
isQid_1=1
activate_1=1
___2=1
n_____2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U13(tt) → tt
U33(tt) → tt
U63(tt) → tt
U74(tt) → tt
U83(tt) → tt
isPalListKind(n__a) → tt
isQid(n__a) → 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

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) RisEmptyProof (EQUIVALENT transformation)

(6) YES