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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 83 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 18 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) CSRRRRProof [EQUIVALENT, 0 ms]
(12) CSR
(13) CSRRRRProof [EQUIVALENT, 0 ms]
(14) CSR
(15) CSRRRRProof [EQUIVALENT, 3 ms]
(16) CSR
(17) RisEmptyProof [EQUIVALENT, 0 ms]
(18) 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(U11(tt)) -> mark(tt)
   active(U21(tt, V2)) -> mark(U22(isList(V2)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNeList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt, P)) -> mark(U72(isPal(P)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(isList(V)) -> mark(U11(isNeList(V)))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2))
   active(isNeList(V)) -> mark(U31(isQid(V)))
   active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2))
   active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2))
   active(isNePal(V)) -> mark(U61(isQid(V)))
   active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P))
   active(isPal(V)) -> mark(U81(isNePal(V)))
   active(isPal(nil)) -> mark(tt)
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X)) -> U11(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X)) -> U31(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X)) -> U61(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X)) -> U81(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X)) -> mark(U11(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X)) -> mark(U81(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X)) -> U11(proper(X))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(isList(X)) -> isList(proper(X))
   proper(U31(X)) -> U31(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X)) -> U61(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X)) -> ok(U11(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   isList(ok(X)) -> ok(isList(X))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isPal(ok(X)) -> ok(isPal(X))
   U81(ok(X)) -> ok(U81(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(isList(x0))
   active(isNeList(x0))
   active(isNePal(x0))
   active(isPal(x0))
   active(isQid(a))
   active(isQid(e))
   active(isQid(i))
   active(isQid(o))
   active(isQid(u))
   active(__(x0, x1))
   active(U11(x0))
   active(U21(x0, x1))
   active(U22(x0))
   active(U31(x0))
   active(U41(x0, x1))
   active(U42(x0))
   active(U51(x0, x1))
   active(U52(x0))
   active(U61(x0))
   active(U71(x0, x1))
   active(U72(x0))
   active(U81(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   U11(mark(x0))
   U21(mark(x0), x1)
   U22(mark(x0))
   U31(mark(x0))
   U41(mark(x0), x1)
   U42(mark(x0))
   U51(mark(x0), x1)
   U52(mark(x0))
   U61(mark(x0))
   U71(mark(x0), x1)
   U72(mark(x0))
   U81(mark(x0))
   proper(__(x0, x1))
   proper(nil)
   proper(U11(x0))
   proper(tt)
   proper(U21(x0, x1))
   proper(U22(x0))
   proper(isList(x0))
   proper(U31(x0))
   proper(U41(x0, x1))
   proper(U42(x0))
   proper(isNeList(x0))
   proper(U51(x0, x1))
   proper(U52(x0))
   proper(U61(x0))
   proper(U71(x0, x1))
   proper(U72(x0))
   proper(isPal(x0))
   proper(U81(x0))
   proper(isQid(x0))
   proper(isNePal(x0))
   proper(a)
   proper(e)
   proper(i)
   proper(o)
   proper(u)
   __(ok(x0), ok(x1))
   U11(ok(x0))
   U21(ok(x0), ok(x1))
   U22(ok(x0))
   isList(ok(x0))
   U31(ok(x0))
   U41(ok(x0), ok(x1))
   U42(ok(x0))
   isNeList(ok(x0))
   U51(ok(x0), ok(x1))
   U52(ok(x0))
   U61(ok(x0))
   U71(ok(x0), ok(x1))
   U72(ok(x0))
   isPal(ok(x0))
   U81(ok(x0))
   isQid(ok(x0))
   isNePal(ok(x0))
   top(mark(x0))
   top(ok(x0))


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

(1) QTRSToCSRProof (SOUND)
The following Q TRS is given: 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(U11(tt)) -> mark(tt)
   active(U21(tt, V2)) -> mark(U22(isList(V2)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNeList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt)) -> mark(tt)
   active(U71(tt, P)) -> mark(U72(isPal(P)))
   active(U72(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(isList(V)) -> mark(U11(isNeList(V)))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2))
   active(isNeList(V)) -> mark(U31(isQid(V)))
   active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2))
   active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2))
   active(isNePal(V)) -> mark(U61(isQid(V)))
   active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P))
   active(isPal(V)) -> mark(U81(isNePal(V)))
   active(isPal(nil)) -> mark(tt)
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X)) -> U11(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X)) -> U31(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X)) -> U61(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X)) -> U81(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X)) -> mark(U11(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X)) -> mark(U81(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X)) -> U11(proper(X))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(isList(X)) -> isList(proper(X))
   proper(U31(X)) -> U31(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X)) -> U61(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X)) -> ok(U11(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   isList(ok(X)) -> ok(isList(X))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isPal(ok(X)) -> ok(isPal(X))
   U81(ok(X)) -> ok(U81(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(isList(x0))
   active(isNeList(x0))
   active(isNePal(x0))
   active(isPal(x0))
   active(isQid(a))
   active(isQid(e))
   active(isQid(i))
   active(isQid(o))
   active(isQid(u))
   active(__(x0, x1))
   active(U11(x0))
   active(U21(x0, x1))
   active(U22(x0))
   active(U31(x0))
   active(U41(x0, x1))
   active(U42(x0))
   active(U51(x0, x1))
   active(U52(x0))
   active(U61(x0))
   active(U71(x0, x1))
   active(U72(x0))
   active(U81(x0))
   __(mark(x0), x1)
   __(x0, mark(x1))
   U11(mark(x0))
   U21(mark(x0), x1)
   U22(mark(x0))
   U31(mark(x0))
   U41(mark(x0), x1)
   U42(mark(x0))
   U51(mark(x0), x1)
   U52(mark(x0))
   U61(mark(x0))
   U71(mark(x0), x1)
   U72(mark(x0))
   U81(mark(x0))
   proper(__(x0, x1))
   proper(nil)
   proper(U11(x0))
   proper(tt)
   proper(U21(x0, x1))
   proper(U22(x0))
   proper(isList(x0))
   proper(U31(x0))
   proper(U41(x0, x1))
   proper(U42(x0))
   proper(isNeList(x0))
   proper(U51(x0, x1))
   proper(U52(x0))
   proper(U61(x0))
   proper(U71(x0, x1))
   proper(U72(x0))
   proper(isPal(x0))
   proper(U81(x0))
   proper(isQid(x0))
   proper(isNePal(x0))
   proper(a)
   proper(e)
   proper(i)
   proper(o)
   proper(u)
   __(ok(x0), ok(x1))
   U11(ok(x0))
   U21(ok(x0), ok(x1))
   U22(ok(x0))
   isList(ok(x0))
   U31(ok(x0))
   U41(ok(x0), ok(x1))
   U42(ok(x0))
   isNeList(ok(x0))
   U51(ok(x0), ok(x1))
   U52(ok(x0))
   U61(ok(x0))
   U71(ok(x0), ok(x1))
   U72(ok(x0))
   isPal(ok(x0))
   U81(ok(x0))
   isQid(ok(x0))
   isNePal(ok(x0))
   top(mark(x0))
   top(ok(x0))

Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U21: {1}
U22: {1}
isList: empty set
U31: {1}
U41: {1}
U42: {1}
isNeList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
----------------------------------------

(2)
Obligation:
Context-sensitive 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(V2))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(V2))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isList(nil) -> tt
   isList(__(V1, V2)) -> U21(isList(V1), V2)
   isNeList(V) -> U31(isQid(V))
   isNeList(__(V1, V2)) -> U41(isList(V1), V2)
   isNeList(__(V1, V2)) -> U51(isNeList(V1), V2)
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isPal(nil) -> tt
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U21: {1}
U22: {1}
isList: empty set
U31: {1}
U41: {1}
U42: {1}
isNeList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive 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(V2))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(V2))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isList(nil) -> tt
   isList(__(V1, V2)) -> U21(isList(V1), V2)
   isNeList(V) -> U31(isQid(V))
   isNeList(__(V1, V2)) -> U41(isList(V1), V2)
   isNeList(__(V1, V2)) -> U51(isNeList(V1), V2)
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isPal(nil) -> tt
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U21: {1}
U22: {1}
isList: empty set
U31: {1}
U41: {1}
U42: {1}
isNeList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U21(x_1, x_2)) = x_1 + x_2
   POL(U22(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1 + x_2
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1 + x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1)) = x_1
   POL(U71(x_1, x_2)) = x_1
   POL(U72(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(__(x_1, x_2)) = 1 + x_1 + x_2
   POL(a) = 1
   POL(e) = 1
   POL(i) = 1
   POL(isList(x_1)) = x_1
   POL(isNeList(x_1)) = x_1
   POL(isNePal(x_1)) = 0
   POL(isPal(x_1)) = 0
   POL(isQid(x_1)) = 0
   POL(nil) = 0
   POL(o) = 1
   POL(tt) = 0
   POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   __(X, nil) -> X
   __(nil, X) -> X
   isList(__(V1, V2)) -> U21(isList(V1), V2)
   isNeList(__(V1, V2)) -> U41(isList(V1), V2)
   isNeList(__(V1, V2)) -> U51(isNeList(V1), V2)




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(V2))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(V2))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isList(nil) -> tt
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isPal(nil) -> tt
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U21: {1}
U22: {1}
isList: empty set
U31: {1}
U41: {1}
U42: {1}
isNeList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(V2))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(V2))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isList(nil) -> tt
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isPal(nil) -> tt
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U21: {1}
U22: {1}
isList: empty set
U31: {1}
U41: {1}
U42: {1}
isNeList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U21(x_1, x_2)) = 1 + 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)) = x_1
   POL(U61(x_1)) = 1 + x_1
   POL(U71(x_1, x_2)) = 1 + x_1
   POL(U72(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(__(x_1, x_2)) = 1 + x_1 + x_2
   POL(a) = 1
   POL(e) = 1
   POL(i) = 1
   POL(isList(x_1)) = x_1
   POL(isNeList(x_1)) = x_1
   POL(isNePal(x_1)) = 1
   POL(isPal(x_1)) = 1
   POL(isQid(x_1)) = 0
   POL(nil) = 1
   POL(o) = 1
   POL(tt) = 0
   POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U21(tt, V2) -> U22(isList(V2))
   U41(tt, V2) -> U42(isNeList(V2))
   U51(tt, V2) -> U52(isList(V2))
   U61(tt) -> tt
   isList(nil) -> tt
   isPal(nil) -> tt




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> tt
   U22(tt) -> tt
   U31(tt) -> tt
   U42(tt) -> tt
   U52(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U22: {1}
isList: empty set
U31: {1}
U42: {1}
isNeList: empty set
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> tt
   U22(tt) -> tt
   U31(tt) -> tt
   U42(tt) -> tt
   U52(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isPal(V) -> U81(isNePal(V))
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U22: {1}
isList: empty set
U31: {1}
U42: {1}
isNeList: empty set
U52: {1}
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U22(x_1)) = 2 + 2*x_1
   POL(U31(x_1)) = x_1
   POL(U42(x_1)) = 2*x_1
   POL(U52(x_1)) = 2*x_1
   POL(U61(x_1)) = x_1
   POL(U71(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U72(x_1)) = 1 + x_1
   POL(U81(x_1)) = x_1
   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(a) = 0
   POL(e) = 2
   POL(i) = 0
   POL(isList(x_1)) = 2 + 2*x_1
   POL(isNeList(x_1)) = 2 + 2*x_1
   POL(isNePal(x_1)) = 2 + 2*x_1
   POL(isPal(x_1)) = 2 + 2*x_1
   POL(isQid(x_1)) = 2 + 2*x_1
   POL(o) = 2
   POL(tt) = 1
   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))
   U22(tt) -> tt
   U42(tt) -> tt
   U52(tt) -> tt
   U72(tt) -> tt
   isNePal(__(I, __(P, I))) -> U71(isQid(I), P)
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt




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

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   U31(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isPal(V) -> U81(isNePal(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
isList: empty set
U31: {1}
isNeList: empty set
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set

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

(9) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   U31(tt) -> tt
   U71(tt, P) -> U72(isPal(P))
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isNeList(V) -> U31(isQid(V))
   isNePal(V) -> U61(isQid(V))
   isPal(V) -> U81(isNePal(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
isList: empty set
U31: {1}
isNeList: empty set
U61: {1}
U71: {1}
U72: {1}
isPal: empty set
U81: {1}
isQid: empty set
isNePal: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U31(x_1)) = x_1
   POL(U61(x_1)) = 2*x_1
   POL(U71(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U72(x_1)) = 1 + 2*x_1
   POL(U81(x_1)) = 2*x_1
   POL(isList(x_1)) = 2 + 2*x_1
   POL(isNeList(x_1)) = 1 + 2*x_1
   POL(isNePal(x_1)) = 0
   POL(isPal(x_1)) = 1 + x_1
   POL(isQid(x_1)) = 0
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U71(tt, P) -> U72(isPal(P))
   U81(tt) -> tt
   isList(V) -> U11(isNeList(V))
   isNeList(V) -> U31(isQid(V))
   isPal(V) -> U81(isNePal(V))




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

(10)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   U31(tt) -> tt
   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U31: {1}
U61: {1}
isQid: empty set
isNePal: empty set

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

(11) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   U31(tt) -> tt
   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U31: {1}
U61: {1}
isQid: empty set
isNePal: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U31(x_1)) = 1 + x_1
   POL(U61(x_1)) = x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isQid(x_1)) = 1 + x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U31(tt) -> tt




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

(12)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U61: {1}
isQid: empty set
isNePal: empty set

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

(13) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> tt
   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U61: {1}
isQid: empty set
isNePal: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = 2 + x_1
   POL(U61(x_1)) = x_1
   POL(isNePal(x_1)) = 1 + 2*x_1
   POL(isQid(x_1)) = 1 + 2*x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt) -> tt




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

(14)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U61: {1}
isQid: empty set
isNePal: empty set

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

(15) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   isNePal(V) -> U61(isQid(V))

The replacement map contains the following entries:

U61: {1}
isQid: empty set
isNePal: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U61(x_1)) = 1 + 2*x_1
   POL(isNePal(x_1)) = 2 + 2*x_1
   POL(isQid(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNePal(V) -> U61(isQid(V))




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

(16)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

(17) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(18)
YES