YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 75 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 4 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 16 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 2 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(and(tt, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   __(active(X1), X2) -> __(X1, X2)
   __(X1, active(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)
   isNePal(active(X)) -> isNePal(X)

The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(__(x_1, x_2)) = x_1 + x_2
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(isNePal(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(and(tt, X)) -> mark(X)




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)
   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   __(active(X1), X2) -> __(X1, X2)
   __(X1, active(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)
   isNePal(active(X)) -> isNePal(X)

The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(__(x_1, x_2)) = 2 + x_1 + x_2
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNePal(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(isNePal(__(I, __(P, I)))) -> mark(tt)




----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   __(active(X1), X2) -> __(X1, X2)
   __(X1, active(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)
   isNePal(active(X)) -> isNePal(X)

The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 2*x_1 + x_2
   POL(isNePal(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))




----------------------------------------

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   __(active(X1), X2) -> __(X1, X2)
   __(X1, active(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)
   isNePal(active(X)) -> isNePal(X)

The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(__(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(active(x_1)) = 2 + x_1
   POL(and(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(isNePal(x_1)) = 2 + 2*x_1
   POL(mark(x_1)) = 2*x_1
   POL(nil) = 2
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   __(active(X1), X2) -> __(X1, X2)
   __(X1, active(X2)) -> __(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNePal(active(X)) -> isNePal(X)




----------------------------------------

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)

The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
mark_1 > [___2, active_1]
mark_1 > nil
mark_1 > and_2
mark_1 > tt
mark_1 > isNePal_1


Status:
mark_1: multiset status
___2: multiset status
active_1: multiset status
nil: multiset status
and_2: [2,1]
tt: multiset status
isNePal_1: multiset status

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

   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
   mark(nil) -> active(nil)
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(isNePal(X)) -> active(isNePal(mark(X)))
   __(mark(X1), X2) -> __(X1, X2)
   __(X1, mark(X2)) -> __(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   isNePal(mark(X)) -> isNePal(X)




----------------------------------------

(10)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   active(__(__(x0, x1), x2))
   active(__(x0, nil))
   active(__(nil, x0))
   active(and(tt, x0))
   active(isNePal(__(x0, __(x1, x0))))
   mark(__(x0, x1))
   mark(nil)
   mark(and(x0, x1))
   mark(tt)
   mark(isNePal(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   __(active(x0), x1)
   __(x0, active(x1))
   and(mark(x0), x1)
   and(x0, mark(x1))
   and(active(x0), x1)
   and(x0, active(x1))
   isNePal(mark(x0))
   isNePal(active(x0))


----------------------------------------

(11) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(12)
YES