Termination w.r.t. Q proof of /tmp/tmpFhhq_1/PALINDROME_nokinds

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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → 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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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:
Lexicographic path order with status [LPO].
Quasi-Precedence:

[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > n_{_nil}
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > n_{_i}
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > n_{_o}
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > a > n_{_a}
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > e > n_{_e}
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > [nil, activate₁, i, o] > u > [tt, n_{_u}]
[_{_₂}, n_{_{_{_₂}}}] > [and₂, isList₁, n_{_isList₁}] > [isNeList₁, n_{_isNeList₁}] > isQid₁ > [tt, n_{_u}]
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > n_{_nil}
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > n_{_i}
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > n_{_o}
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > a > n_{_a}
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > e > n_{_e}
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > [nil, activate₁, i, o] > u > [tt, n_{_u}]
[_{_₂}, n_{_{_{_₂}}}] > [n_{_isPal₁}, isPal₁] > isNePal₁ > isQid₁ > [tt, n_{_u}]

Status:

i: []
n_{_u}: []
__₂: [1,2]
n_{_i}: []
activate₁: [1]
n_{_nil}: []
and₂: [2,1]
n_{_a}: []
tt: []
n_{_isList₁}: [1]
nil: []
a: []
isList₁: [1]
n_{_isPal₁}: [1]
e: []
n_{_e}: []
isNePal₁: [1]
o: []
n_{_{_{_₂}}}: [1,2]
isQid₁: [1]
n_{_isNeList₁}: [1]
isPal₁: [1]
n_{_o}: []
u: []
isNeList₁: [1]

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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → 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
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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

(2) Obligation:

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

__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

__₂ > [n_{_{_{_₂}}}, n_{_isList₁}, n_{_isNeList₁}, n_{_isPal₁}]
isList₁ > [n_{_{_{_₂}}}, n_{_isList₁}, n_{_isNeList₁}, n_{_isPal₁}]
isNeList₁ > [n_{_{_{_₂}}}, n_{_isList₁}, n_{_isNeList₁}, n_{_isPal₁}]
isPal₁ > [n_{_{_{_₂}}}, n_{_isList₁}, n_{_isNeList₁}, n_{_isPal₁}]

Status:

isPal₁: [1]
isList₁: [1]
__₂: [2,1]
n_{_isPal₁}: [1]
isNeList₁: [1]
n_{_isList₁}: [1]
n_{_{_{_₂}}}: [2,1]
n_{_isNeList₁}: [1]

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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) RisEmptyProof (EQUIVALENT transformation)

(6) TRUE