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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 134 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 16 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) 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) -> 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)
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U21(x_1, x_2)) = 2*x_1 + x_2
   POL(U22(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U41(x_1, x_2)) = 1 + x_1 + x_2
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = 1 + x_1 + x_2
   POL(U52(x_1)) = 1 + x_1
   POL(U61(x_1)) = x_1
   POL(U71(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(U72(x_1)) = x_1
   POL(U81(x_1)) = 2*x_1
   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(a) = 0
   POL(activate(x_1)) = x_1
   POL(e) = 0
   POL(i) = 0
   POL(isList(x_1)) = x_1
   POL(isNeList(x_1)) = x_1
   POL(isNePal(x_1)) = x_1
   POL(isPal(x_1)) = 1 + 2*x_1
   POL(isQid(x_1)) = x_1
   POL(n____(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(n__a) = 0
   POL(n__e) = 0
   POL(n__i) = 0
   POL(n__nil) = 0
   POL(n__o) = 0
   POL(n__u) = 0
   POL(nil) = 0
   POL(o) = 0
   POL(tt) = 0
   POL(u) = 0
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
   U41(tt, V2) -> U42(isNeList(activate(V2)))
   U52(tt) -> tt
   U71(tt, P) -> U72(isPal(activate(P)))
   isList(n____(V1, V2)) -> U21(isList(activate(V1)), activate(V2))
   isNeList(n____(V1, V2)) -> U41(isList(activate(V1)), activate(V2))
   isNeList(n____(V1, V2)) -> U51(isNeList(activate(V1)), activate(V2))
   isNePal(n____(I, __(P, I))) -> U71(isQid(activate(I)), activate(P))
   isPal(V) -> U81(isNePal(activate(V)))
   isPal(n__nil) -> tt




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

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

   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(activate(V2)))
   U22(tt) -> tt
   U31(tt) -> tt
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(activate(V2)))
   U61(tt) -> tt
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(activate(V)))
   isList(n__nil) -> tt
   isNeList(V) -> U31(isQid(activate(V)))
   isNePal(V) -> U61(isQid(activate(V)))
   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.

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

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

   POL(U11(x_1)) = x_1
   POL(U21(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(U22(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U42(x_1)) = 1 + x_1
   POL(U51(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1)) = x_1
   POL(U72(x_1)) = 1 + 2*x_1
   POL(U81(x_1)) = 1 + x_1
   POL(__(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(a) = 0
   POL(activate(x_1)) = x_1
   POL(e) = 2
   POL(i) = 0
   POL(isList(x_1)) = 2 + 2*x_1
   POL(isNeList(x_1)) = 1 + 2*x_1
   POL(isNePal(x_1)) = 1 + 2*x_1
   POL(isQid(x_1)) = 2*x_1
   POL(n____(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(n__a) = 0
   POL(n__e) = 2
   POL(n__i) = 0
   POL(n__nil) = 0
   POL(n__o) = 2
   POL(n__u) = 0
   POL(nil) = 0
   POL(o) = 2
   POL(tt) = 0
   POL(u) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U42(tt) -> tt
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(activate(V)))
   isList(n__nil) -> tt
   isNeList(V) -> U31(isQid(activate(V)))
   isNePal(V) -> U61(isQid(activate(V)))
   isQid(n__e) -> tt
   isQid(n__o) -> tt




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

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

   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(activate(V2)))
   U22(tt) -> tt
   U31(tt) -> tt
   U51(tt, V2) -> U52(isList(activate(V2)))
   U61(tt) -> tt
   isQid(n__a) -> tt
   isQid(n__i) -> 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.

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

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

   POL(U11(x_1)) = 1 + 2*x_1
   POL(U21(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U22(x_1)) = 1 + x_1
   POL(U31(x_1)) = 1 + 2*x_1
   POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1)) = 1 + 2*x_1
   POL(__(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(a) = 1
   POL(activate(x_1)) = 2 + x_1
   POL(e) = 2
   POL(i) = 2
   POL(isList(x_1)) = 2*x_1
   POL(isQid(x_1)) = 2 + x_1
   POL(n____(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(n__a) = 0
   POL(n__e) = 2
   POL(n__i) = 1
   POL(n__nil) = 0
   POL(n__o) = 1
   POL(n__u) = 0
   POL(nil) = 1
   POL(o) = 2
   POL(tt) = 2
   POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(activate(V2)))
   U22(tt) -> tt
   U31(tt) -> tt
   U61(tt) -> tt
   isQid(n__i) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   a -> n__a
   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




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

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

   U51(tt, V2) -> U52(isList(activate(V2)))
   isQid(n__a) -> tt
   isQid(n__u) -> tt
   e -> n__e

Q is empty.

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

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

   POL(U51(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U52(x_1)) = 2 + x_1
   POL(activate(x_1)) = 2*x_1
   POL(e) = 1
   POL(isList(x_1)) = x_1
   POL(isQid(x_1)) = 1 + x_1
   POL(n__a) = 0
   POL(n__e) = 1
   POL(n__u) = 0
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U51(tt, V2) -> U52(isList(activate(V2)))




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

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

   isQid(n__a) -> tt
   isQid(n__u) -> tt
   e -> n__e

Q is empty.

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

(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:n__a > e > n__e > n__u > isQid_1 > tt

and weight map:

   n__a=1
   tt=2
   n__u=1
   e=1
   n__e=1
   isQid_1=1

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

   isQid(n__a) -> tt
   isQid(n__u) -> tt
   e -> n__e




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

(10)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(12)
YES