Termination w.r.t. Q proof of /tmp/tmpebwNHH/PALINDROME_nokinds-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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
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 [LPO].
Precedence:

[_{_₂}, n_{_{_{_₂}}}] > U21₂ > U22₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > U11₁ > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > [activate₁, a] > nil > n_{_nil} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > [activate₁, a] > n_{_a} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > [activate₁, a] > [n_{_u}, u] > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > [activate₁, a] > e > n_{_e} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > U31₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U21₂ > isList₁ > isNeList₁ > isQid₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > U42₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > [activate₁, a] > nil > n_{_nil} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > [activate₁, a] > n_{_a} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > [activate₁, a] > [n_{_u}, u] > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > [activate₁, a] > e > n_{_e} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > U31₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U41₂ > isNeList₁ > isQid₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > U11₁ > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > [activate₁, a] > nil > n_{_nil} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > [activate₁, a] > n_{_a} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > [activate₁, a] > [n_{_u}, u] > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > [activate₁, a] > e > n_{_e} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > U31₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U51₂ > isList₁ > isNeList₁ > isQid₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > U72₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > U81₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > [activate₁, a] > nil > n_{_nil} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > [activate₁, a] > n_{_a} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > [activate₁, a] > [n_{_u}, u] > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > [activate₁, a] > e > n_{_e} > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > U61₁ > tt > U52₁
[_{_₂}, n_{_{_{_₂}}}] > U71₂ > isPal₁ > isNePal₁ > isQid₁ > tt > U52₁
[n_{_i}, i] > tt > U52₁
[n_{_o}, o] > tt > U52₁

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) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
a → n__a
e → n__e
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:

__(X1, X2) → n____(X1, X2)
i → n__i
o → n__o
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:

__₂ > n_{_{_{_₂}}}
i > n_{_i} > n_{_{_{_₂}}}
o > n_{_o} > n_{_{_{_₂}}}
u > n_{_u} > n_{_{_{_₂}}}

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

__(X1, X2) → n____(X1, X2)
i → n__i
o → n__o
u → n__u

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)