Termination w.r.t. Q proof of Transformed_CSR_04_PALINDROME_complete

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

0 QTRS

↳1 QTRSRRRProof (⇔, 190 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 0 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 0 ms)

↳6 QTRS

↳7 RisEmptyProof (⇔, 0 ms)

↳8 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(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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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) = x1
isNeList(x1) = isNeList(x1)
activate(x1) = x1
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2) = U22(x1, x2)
isList(x1) = isList(x1)
U23(x1) = x1
U31(x1, x2) = U31(x1, x2)
U32(x1) = x1
isQid(x1) = isQid(x1)
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2) = U42(x1, x2)
U43(x1) = x1
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2) = U52(x1, x2)
U53(x1) = x1
U61(x1, x2) = U61(x1, x2)
U62(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isNePal(x1) = isNePal(x1)
and(x1, x2) = and(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
n__nil = n__nil
n____(x1, x2) = n____(x1, x2)
n__isPalListKind(x1) = n__isPalListKind(x1)
n__and(x1, x2) = n__and(x1, x2)
n__isPal(x1) = n__isPal(x1)
isPal(x1) = isPal(x1)
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}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > U21₃ > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > U41₃ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > U41₃ > U42₂ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > U51₃ > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > [U71₂, isNePal₁, and₂, n_{_and₂}] > U61₂ > [U31₂, isQid₁]
[_{_₂}, n_{_{_{_₂}}}, n_{_isPal₁}, isPal₁] > [U71₂, isNePal₁, and₂, n_{_and₂}] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[nil, n_{_nil}] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[nil, n_{_nil}] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_a}, a] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_a}, a] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_e}, e] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_e}, e] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_i}, i] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_i}, i] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_o}, o] > [tt, n_{_u}, u] > U22₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]
[n_{_o}, o] > [tt, n_{_u}, u] > U52₂ > isList₁ > [U11₂, isNeList₁] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [U31₂, isQid₁]

Status:

__₂: [1,2]
nil: multiset
U11₂: [2,1]
tt: multiset
isNeList₁: [1]
U21₃: [3,1,2]
U22₂: multiset
isList₁: multiset
U31₂: [2,1]
isQid₁: [1]
U41₃: [3,2,1]
U42₂: [2,1]
U51₃: [2,3,1]
U52₂: [2,1]
U61₂: multiset
U71₂: multiset
isNePal₁: multiset
and₂: [1,2]
isPalListKind₁: [1]
n_{_nil}: multiset
n_{_{_{_₂}}}: [1,2]
n_{_{isPalListKind₁}}: [1]
n_{_and₂}: [1,2]
n_{_isPal₁}: [1]
isPal₁: [1]
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(isNeList(activate(V)))
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U31(tt, V) → U32(isQid(activate(V)))
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U61(tt, V) → U62(isQid(activate(V)))
U71(tt, V) → U72(isNePal(activate(V)))
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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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

(2) Obligation:

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

U12(tt) → tt
U23(tt) → tt
U32(tt) → tt
U43(tt) → tt
U53(tt) → tt
U62(tt) → tt
U72(tt) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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:
Polynomial interpretation [POLO]:

POL(U12(x₁)) = 1 + x₁
POL(U23(x₁)) = 2·x₁
POL(U32(x₁)) = 2 + 2·x₁
POL(U43(x₁)) = 2 + 2·x₁
POL(U53(x₁)) = 2 + 2·x₁
POL(U62(x₁)) = 2 + 2·x₁
POL(U72(x₁)) = 2·x₁
POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a) = 1
POL(activate(x₁)) = 1 + 2·x₁
POL(and(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(e) = 2
POL(i) = 2
POL(isPal(x₁)) = 2 + 2·x₁
POL(isPalListKind(x₁)) = 2 + 2·x₁
POL(n____(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(n__a) = 0
POL(n__and(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(n__e) = 1
POL(n__i) = 2
POL(n__isPal(x₁)) = 2 + x₁
POL(n__isPalListKind(x₁)) = 2 + x₁
POL(n__nil) = 2
POL(n__o) = 1
POL(n__u) = 2
POL(nil) = 2
POL(o) = 2
POL(tt) = 1
POL(u) = 2

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

U12(tt) → tt
U23(tt) → tt
U32(tt) → tt
U43(tt) → tt
U53(tt) → tt
U62(tt) → tt
U72(tt) → tt
a → n__a
e → n__e
o → n__o
activate(n__nil) → nil
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(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)
isPal(X) → n__isPal(X)
i → n__i
u → n__u
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__a) → a

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

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

activate₁ > a > nil > n_{_u} > u > n_{_i} > n_{_isPal₁} > isPal₁ > n_{_{isPalListKind₁}} > n_{_a} > and₂ > i > n_{_and₂} > isPalListKind₁ > __₂ > n_{_nil} > n_{_{_{_₂}}}

and weight map:

nil=2
n__nil=1
i=2
n__i=1
u=2
n__u=1
n__a=1
a=1
isPalListKind_1=2
n__isPalListKind_1=1
isPal_1=2
n__isPal_1=1
activate_1=0
___2=0
n_____2=0
and_2=1
n__and_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:

nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
i → n__i
u → n__u
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__a) → a

(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) YES