/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 1447 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 174 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 126 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt, V) -> a__U12(a__isPalListKind(V), V)
   a__U12(tt, V) -> a__U13(a__isNeList(V))
   a__U13(tt) -> tt
   a__U21(tt, V1, V2) -> a__U22(a__isPalListKind(V1), V1, V2)
   a__U22(tt, V1, V2) -> a__U23(a__isPalListKind(V2), V1, V2)
   a__U23(tt, V1, V2) -> a__U24(a__isPalListKind(V2), V1, V2)
   a__U24(tt, V1, V2) -> a__U25(a__isList(V1), V2)
   a__U25(tt, V2) -> a__U26(a__isList(V2))
   a__U26(tt) -> tt
   a__U31(tt, V) -> a__U32(a__isPalListKind(V), V)
   a__U32(tt, V) -> a__U33(a__isQid(V))
   a__U33(tt) -> tt
   a__U41(tt, V1, V2) -> a__U42(a__isPalListKind(V1), V1, V2)
   a__U42(tt, V1, V2) -> a__U43(a__isPalListKind(V2), V1, V2)
   a__U43(tt, V1, V2) -> a__U44(a__isPalListKind(V2), V1, V2)
   a__U44(tt, V1, V2) -> a__U45(a__isList(V1), V2)
   a__U45(tt, V2) -> a__U46(a__isNeList(V2))
   a__U46(tt) -> tt
   a__U51(tt, V1, V2) -> a__U52(a__isPalListKind(V1), V1, V2)
   a__U52(tt, V1, V2) -> a__U53(a__isPalListKind(V2), V1, V2)
   a__U53(tt, V1, V2) -> a__U54(a__isPalListKind(V2), V1, V2)
   a__U54(tt, V1, V2) -> a__U55(a__isNeList(V1), V2)
   a__U55(tt, V2) -> a__U56(a__isList(V2))
   a__U56(tt) -> tt
   a__U61(tt, V) -> a__U62(a__isPalListKind(V), V)
   a__U62(tt, V) -> a__U63(a__isQid(V))
   a__U63(tt) -> tt
   a__U71(tt, I, P) -> a__U72(a__isPalListKind(I), P)
   a__U72(tt, P) -> a__U73(a__isPal(P), P)
   a__U73(tt, P) -> a__U74(a__isPalListKind(P))
   a__U74(tt) -> tt
   a__U81(tt, V) -> a__U82(a__isPalListKind(V), V)
   a__U82(tt, V) -> a__U83(a__isNePal(V))
   a__U83(tt) -> tt
   a__U91(tt, V2) -> a__U92(a__isPalListKind(V2))
   a__U92(tt) -> tt
   a__isList(V) -> a__U11(a__isPalListKind(V), V)
   a__isList(nil) -> tt
   a__isList(__(V1, V2)) -> a__U21(a__isPalListKind(V1), V1, V2)
   a__isNeList(V) -> a__U31(a__isPalListKind(V), V)
   a__isNeList(__(V1, V2)) -> a__U41(a__isPalListKind(V1), V1, V2)
   a__isNeList(__(V1, V2)) -> a__U51(a__isPalListKind(V1), V1, V2)
   a__isNePal(V) -> a__U61(a__isPalListKind(V), V)
   a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), I, P)
   a__isPal(V) -> a__U81(a__isPalListKind(V), V)
   a__isPal(nil) -> tt
   a__isPalListKind(a) -> tt
   a__isPalListKind(e) -> tt
   a__isPalListKind(i) -> tt
   a__isPalListKind(nil) -> tt
   a__isPalListKind(o) -> tt
   a__isPalListKind(u) -> tt
   a__isPalListKind(__(V1, V2)) -> a__U91(a__isPalListKind(V1), V2)
   a__isQid(a) -> tt
   a__isQid(e) -> tt
   a__isQid(i) -> tt
   a__isQid(o) -> tt
   a__isQid(u) -> tt
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isPalListKind(X)) -> a__isPalListKind(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(isNeList(X)) -> a__isNeList(X)
   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
   mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3)
   mark(U23(X1, X2, X3)) -> a__U23(mark(X1), X2, X3)
   mark(U24(X1, X2, X3)) -> a__U24(mark(X1), X2, X3)
   mark(U25(X1, X2)) -> a__U25(mark(X1), X2)
   mark(isList(X)) -> a__isList(X)
   mark(U26(X)) -> a__U26(mark(X))
   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(isQid(X)) -> a__isQid(X)
   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
   mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3)
   mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3)
   mark(U45(X1, X2)) -> a__U45(mark(X1), X2)
   mark(U46(X)) -> a__U46(mark(X))
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3)
   mark(U53(X1, X2, X3)) -> a__U53(mark(X1), X2, X3)
   mark(U54(X1, X2, X3)) -> a__U54(mark(X1), X2, X3)
   mark(U55(X1, X2)) -> a__U55(mark(X1), X2)
   mark(U56(X)) -> a__U56(mark(X))
   mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
   mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
   mark(U63(X)) -> a__U63(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
   mark(U73(X1, X2)) -> a__U73(mark(X1), X2)
   mark(isPal(X)) -> a__isPal(X)
   mark(U74(X)) -> a__U74(mark(X))
   mark(U81(X1, X2)) -> a__U81(mark(X1), X2)
   mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
   mark(U83(X)) -> a__U83(mark(X))
   mark(isNePal(X)) -> a__isNePal(X)
   mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
   mark(U92(X)) -> a__U92(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   mark(a) -> a
   mark(e) -> e
   mark(i) -> i
   mark(o) -> o
   mark(u) -> u
   a____(X1, X2) -> __(X1, X2)
   a__U11(X1, X2) -> U11(X1, X2)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isPalListKind(X) -> isPalListKind(X)
   a__U13(X) -> U13(X)
   a__isNeList(X) -> isNeList(X)
   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
   a__U22(X1, X2, X3) -> U22(X1, X2, X3)
   a__U23(X1, X2, X3) -> U23(X1, X2, X3)
   a__U24(X1, X2, X3) -> U24(X1, X2, X3)
   a__U25(X1, X2) -> U25(X1, X2)
   a__isList(X) -> isList(X)
   a__U26(X) -> U26(X)
   a__U31(X1, X2) -> U31(X1, X2)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__isQid(X) -> isQid(X)
   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
   a__U43(X1, X2, X3) -> U43(X1, X2, X3)
   a__U44(X1, X2, X3) -> U44(X1, X2, X3)
   a__U45(X1, X2) -> U45(X1, X2)
   a__U46(X) -> U46(X)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__U52(X1, X2, X3) -> U52(X1, X2, X3)
   a__U53(X1, X2, X3) -> U53(X1, X2, X3)
   a__U54(X1, X2, X3) -> U54(X1, X2, X3)
   a__U55(X1, X2) -> U55(X1, X2)
   a__U56(X) -> U56(X)
   a__U61(X1, X2) -> U61(X1, X2)
   a__U62(X1, X2) -> U62(X1, X2)
   a__U63(X) -> U63(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__U72(X1, X2) -> U72(X1, X2)
   a__U73(X1, X2) -> U73(X1, X2)
   a__isPal(X) -> isPal(X)
   a__U74(X) -> U74(X)
   a__U81(X1, X2) -> U81(X1, X2)
   a__U82(X1, X2) -> U82(X1, X2)
   a__U83(X) -> U83(X)
   a__isNePal(X) -> isNePal(X)
   a__U91(X1, X2) -> U91(X1, X2)
   a__U92(X) -> U92(X)

Q is empty.

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

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
a____/2(YES,YES)
__/2(YES,YES)
mark/1)YES(
nil/0)
a__U11/2(YES,YES)
tt/0)
a__U12/2(YES,YES)
a__isPalListKind/1)YES(
a__U13/1)YES(
a__isNeList/1(YES)
a__U21/3(YES,YES,YES)
a__U22/3(YES,YES,YES)
a__U23/3(YES,YES,YES)
a__U24/3(YES,YES,YES)
a__U25/2(YES,YES)
a__isList/1(YES)
a__U26/1)YES(
a__U31/2(YES,YES)
a__U32/2(YES,YES)
a__U33/1)YES(
a__isQid/1(YES)
a__U41/3(YES,YES,YES)
a__U42/3(YES,YES,YES)
a__U43/3(YES,YES,YES)
a__U44/3(YES,YES,YES)
a__U45/2(YES,YES)
a__U46/1(YES)
a__U51/3(YES,YES,YES)
a__U52/3(YES,YES,YES)
a__U53/3(YES,YES,YES)
a__U54/3(YES,YES,YES)
a__U55/2(YES,YES)
a__U56/1(YES)
a__U61/2(YES,YES)
a__U62/2(YES,YES)
a__U63/1)YES(
a__U71/3(YES,YES,YES)
a__U72/2(YES,YES)
a__U73/2(YES,YES)
a__isPal/1(YES)
a__U74/1)YES(
a__U81/2(YES,YES)
a__U82/2(YES,YES)
a__U83/1)YES(
a__isNePal/1(YES)
a__U91/2(YES,YES)
a__U92/1)YES(
a/0)
e/0)
i/0)
o/0)
u/0)
U11/2(YES,YES)
U12/2(YES,YES)
isPalListKind/1)YES(
U13/1)YES(
isNeList/1(YES)
U21/3(YES,YES,YES)
U22/3(YES,YES,YES)
U23/3(YES,YES,YES)
U24/3(YES,YES,YES)
U25/2(YES,YES)
isList/1(YES)
U26/1)YES(
U31/2(YES,YES)
U32/2(YES,YES)
U33/1)YES(
isQid/1(YES)
U41/3(YES,YES,YES)
U42/3(YES,YES,YES)
U43/3(YES,YES,YES)
U44/3(YES,YES,YES)
U45/2(YES,YES)
U46/1(YES)
U51/3(YES,YES,YES)
U52/3(YES,YES,YES)
U53/3(YES,YES,YES)
U54/3(YES,YES,YES)
U55/2(YES,YES)
U56/1(YES)
U61/2(YES,YES)
U62/2(YES,YES)
U63/1)YES(
U71/3(YES,YES,YES)
U72/2(YES,YES)
U73/2(YES,YES)
isPal/1(YES)
U74/1)YES(
U81/2(YES,YES)
U82/2(YES,YES)
U83/1)YES(
isNePal/1(YES)
U91/2(YES,YES)
U92/1)YES(

Quasi precedence:
[a_____2, ___2] > [a__U21_3, U21_3] > [a__U22_3, U22_3] > [a__U23_3, U23_3] > [a__U24_3, U24_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U21_3, U21_3] > [a__U22_3, U22_3] > [a__U23_3, U23_3] > [a__U24_3, U24_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
[a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_3, U42_3] > [a__U43_3, U43_3] > [a__U44_3, U44_3] > [a__U45_2, U45_2] > [a__U46_1, U46_1] > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_3, U42_3] > [a__U43_3, U43_3] > [a__U44_3, U44_3] > [a__U45_2, U45_2] > [a__U46_1, U46_1] > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
[a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_3, U42_3] > [a__U43_3, U43_3] > [a__U44_3, U44_3] > [a__U45_2, U45_2] > [a__U46_1, U46_1] > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
[a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_3, U42_3] > [a__U43_3, U43_3] > [a__U44_3, U44_3] > [a__U45_2, U45_2] > [a__U46_1, U46_1] > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_3, U52_3] > [a__U53_3, U53_3] > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_3, U52_3] > [a__U53_3, U53_3] > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
[a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_3, U52_3] > [a__U53_3, U53_3] > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
[a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_3, U52_3] > [a__U53_3, U53_3] > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U71_3, U71_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
[a_____2, ___2] > [a__U71_3, U71_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
[a_____2, ___2] > [a__U91_2, U91_2]
a > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
a > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
a > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
a > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
e > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
e > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
e > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
e > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
i > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
i > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
i > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
i > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
u > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U11_2, U11_2] > [a__U12_2, a__isNeList_1, U12_2, isNeList_1] > [a__U31_2, U31_2] > [a__U32_2, U32_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]
u > [nil, tt, a__U54_3, o, U54_3] > [a__U25_2, a__isList_1, a__U55_2, U25_2, isList_1, U55_2] > [a__U56_1, U56_1]
u > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__U73_2, U73_2]
u > [nil, tt, a__U54_3, o, U54_3] > [a__U72_2, U72_2] > [a__isPal_1, isPal_1] > [a__U81_2, U81_2] > [a__U82_2, U82_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] > [a__isQid_1, a__U62_2, isQid_1, U62_2]


Status:
a_____2: [1,2]
___2: [1,2]
nil: multiset status
a__U11_2: multiset status
tt: multiset status
a__U12_2: multiset status
a__isNeList_1: multiset status
a__U21_3: multiset status
a__U22_3: multiset status
a__U23_3: multiset status
a__U24_3: multiset status
a__U25_2: multiset status
a__isList_1: multiset status
a__U31_2: multiset status
a__U32_2: multiset status
a__isQid_1: multiset status
a__U41_3: multiset status
a__U42_3: multiset status
a__U43_3: multiset status
a__U44_3: multiset status
a__U45_2: multiset status
a__U46_1: multiset status
a__U51_3: multiset status
a__U52_3: multiset status
a__U53_3: multiset status
a__U54_3: multiset status
a__U55_2: multiset status
a__U56_1: multiset status
a__U61_2: multiset status
a__U62_2: multiset status
a__U71_3: multiset status
a__U72_2: multiset status
a__U73_2: multiset status
a__isPal_1: multiset status
a__U81_2: multiset status
a__U82_2: multiset status
a__isNePal_1: multiset status
a__U91_2: multiset status
a: multiset status
e: multiset status
i: multiset status
o: multiset status
u: multiset status
U11_2: multiset status
U12_2: multiset status
isNeList_1: multiset status
U21_3: multiset status
U22_3: multiset status
U23_3: multiset status
U24_3: multiset status
U25_2: multiset status
isList_1: multiset status
U31_2: multiset status
U32_2: multiset status
isQid_1: multiset status
U41_3: multiset status
U42_3: multiset status
U43_3: multiset status
U44_3: multiset status
U45_2: multiset status
U46_1: multiset status
U51_3: multiset status
U52_3: multiset status
U53_3: multiset status
U54_3: multiset status
U55_2: multiset status
U56_1: multiset status
U61_2: multiset status
U62_2: multiset status
U71_3: multiset status
U72_2: multiset status
U73_2: multiset status
isPal_1: multiset status
U81_2: multiset status
U82_2: multiset status
isNePal_1: multiset status
U91_2: multiset status

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

   a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z)))
   a____(X, nil) -> mark(X)
   a____(nil, X) -> mark(X)
   a__U11(tt, V) -> a__U12(a__isPalListKind(V), V)
   a__U12(tt, V) -> a__U13(a__isNeList(V))
   a__U21(tt, V1, V2) -> a__U22(a__isPalListKind(V1), V1, V2)
   a__U22(tt, V1, V2) -> a__U23(a__isPalListKind(V2), V1, V2)
   a__U23(tt, V1, V2) -> a__U24(a__isPalListKind(V2), V1, V2)
   a__U24(tt, V1, V2) -> a__U25(a__isList(V1), V2)
   a__U25(tt, V2) -> a__U26(a__isList(V2))
   a__U31(tt, V) -> a__U32(a__isPalListKind(V), V)
   a__U32(tt, V) -> a__U33(a__isQid(V))
   a__U41(tt, V1, V2) -> a__U42(a__isPalListKind(V1), V1, V2)
   a__U42(tt, V1, V2) -> a__U43(a__isPalListKind(V2), V1, V2)
   a__U43(tt, V1, V2) -> a__U44(a__isPalListKind(V2), V1, V2)
   a__U44(tt, V1, V2) -> a__U45(a__isList(V1), V2)
   a__U45(tt, V2) -> a__U46(a__isNeList(V2))
   a__U46(tt) -> tt
   a__U51(tt, V1, V2) -> a__U52(a__isPalListKind(V1), V1, V2)
   a__U52(tt, V1, V2) -> a__U53(a__isPalListKind(V2), V1, V2)
   a__U53(tt, V1, V2) -> a__U54(a__isPalListKind(V2), V1, V2)
   a__U54(tt, V1, V2) -> a__U55(a__isNeList(V1), V2)
   a__U55(tt, V2) -> a__U56(a__isList(V2))
   a__U56(tt) -> tt
   a__U61(tt, V) -> a__U62(a__isPalListKind(V), V)
   a__U62(tt, V) -> a__U63(a__isQid(V))
   a__U71(tt, I, P) -> a__U72(a__isPalListKind(I), P)
   a__U72(tt, P) -> a__U73(a__isPal(P), P)
   a__U73(tt, P) -> a__U74(a__isPalListKind(P))
   a__U81(tt, V) -> a__U82(a__isPalListKind(V), V)
   a__U82(tt, V) -> a__U83(a__isNePal(V))
   a__U91(tt, V2) -> a__U92(a__isPalListKind(V2))
   a__isList(V) -> a__U11(a__isPalListKind(V), V)
   a__isList(nil) -> tt
   a__isList(__(V1, V2)) -> a__U21(a__isPalListKind(V1), V1, V2)
   a__isNeList(V) -> a__U31(a__isPalListKind(V), V)
   a__isNeList(__(V1, V2)) -> a__U41(a__isPalListKind(V1), V1, V2)
   a__isNeList(__(V1, V2)) -> a__U51(a__isPalListKind(V1), V1, V2)
   a__isNePal(V) -> a__U61(a__isPalListKind(V), V)
   a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), I, P)
   a__isPal(V) -> a__U81(a__isPalListKind(V), V)
   a__isPal(nil) -> tt
   a__isPalListKind(a) -> tt
   a__isPalListKind(e) -> tt
   a__isPalListKind(i) -> tt
   a__isPalListKind(u) -> tt
   a__isPalListKind(__(V1, V2)) -> a__U91(a__isPalListKind(V1), V2)
   a__isQid(a) -> tt
   a__isQid(e) -> tt
   a__isQid(i) -> tt
   a__isQid(o) -> tt
   a__isQid(u) -> tt




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

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

   a__U13(tt) -> tt
   a__U26(tt) -> tt
   a__U33(tt) -> tt
   a__U63(tt) -> tt
   a__U74(tt) -> tt
   a__U83(tt) -> tt
   a__U92(tt) -> tt
   a__isPalListKind(nil) -> tt
   a__isPalListKind(o) -> tt
   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isPalListKind(X)) -> a__isPalListKind(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(isNeList(X)) -> a__isNeList(X)
   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
   mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3)
   mark(U23(X1, X2, X3)) -> a__U23(mark(X1), X2, X3)
   mark(U24(X1, X2, X3)) -> a__U24(mark(X1), X2, X3)
   mark(U25(X1, X2)) -> a__U25(mark(X1), X2)
   mark(isList(X)) -> a__isList(X)
   mark(U26(X)) -> a__U26(mark(X))
   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(isQid(X)) -> a__isQid(X)
   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
   mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3)
   mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3)
   mark(U45(X1, X2)) -> a__U45(mark(X1), X2)
   mark(U46(X)) -> a__U46(mark(X))
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3)
   mark(U53(X1, X2, X3)) -> a__U53(mark(X1), X2, X3)
   mark(U54(X1, X2, X3)) -> a__U54(mark(X1), X2, X3)
   mark(U55(X1, X2)) -> a__U55(mark(X1), X2)
   mark(U56(X)) -> a__U56(mark(X))
   mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
   mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
   mark(U63(X)) -> a__U63(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
   mark(U73(X1, X2)) -> a__U73(mark(X1), X2)
   mark(isPal(X)) -> a__isPal(X)
   mark(U74(X)) -> a__U74(mark(X))
   mark(U81(X1, X2)) -> a__U81(mark(X1), X2)
   mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
   mark(U83(X)) -> a__U83(mark(X))
   mark(isNePal(X)) -> a__isNePal(X)
   mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
   mark(U92(X)) -> a__U92(mark(X))
   mark(nil) -> nil
   mark(tt) -> tt
   mark(a) -> a
   mark(e) -> e
   mark(i) -> i
   mark(o) -> o
   mark(u) -> u
   a____(X1, X2) -> __(X1, X2)
   a__U11(X1, X2) -> U11(X1, X2)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isPalListKind(X) -> isPalListKind(X)
   a__U13(X) -> U13(X)
   a__isNeList(X) -> isNeList(X)
   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
   a__U22(X1, X2, X3) -> U22(X1, X2, X3)
   a__U23(X1, X2, X3) -> U23(X1, X2, X3)
   a__U24(X1, X2, X3) -> U24(X1, X2, X3)
   a__U25(X1, X2) -> U25(X1, X2)
   a__isList(X) -> isList(X)
   a__U26(X) -> U26(X)
   a__U31(X1, X2) -> U31(X1, X2)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__isQid(X) -> isQid(X)
   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
   a__U43(X1, X2, X3) -> U43(X1, X2, X3)
   a__U44(X1, X2, X3) -> U44(X1, X2, X3)
   a__U45(X1, X2) -> U45(X1, X2)
   a__U46(X) -> U46(X)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__U52(X1, X2, X3) -> U52(X1, X2, X3)
   a__U53(X1, X2, X3) -> U53(X1, X2, X3)
   a__U54(X1, X2, X3) -> U54(X1, X2, X3)
   a__U55(X1, X2) -> U55(X1, X2)
   a__U56(X) -> U56(X)
   a__U61(X1, X2) -> U61(X1, X2)
   a__U62(X1, X2) -> U62(X1, X2)
   a__U63(X) -> U63(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__U72(X1, X2) -> U72(X1, X2)
   a__U73(X1, X2) -> U73(X1, X2)
   a__isPal(X) -> isPal(X)
   a__U74(X) -> U74(X)
   a__U81(X1, X2) -> U81(X1, X2)
   a__U82(X1, X2) -> U82(X1, X2)
   a__U83(X) -> U83(X)
   a__isNePal(X) -> isNePal(X)
   a__U91(X1, X2) -> U91(X1, X2)
   a__U92(X) -> U92(X)

Q is empty.

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

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

   POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U12(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U13(x_1)) = 1 + 2*x_1
   POL(U21(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U22(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U23(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U24(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U25(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U26(x_1)) = 1 + 2*x_1
   POL(U31(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U32(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U33(x_1)) = 1 + 2*x_1
   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U43(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U44(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U45(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U46(x_1)) = 1 + x_1
   POL(U51(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U52(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U53(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U54(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U55(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U56(x_1)) = 1 + x_1
   POL(U61(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U62(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U63(x_1)) = 1 + 2*x_1
   POL(U71(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(U72(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U73(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U74(x_1)) = 1 + 2*x_1
   POL(U81(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U82(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U83(x_1)) = 1 + 2*x_1
   POL(U91(x_1, x_2)) = 1 + x_1 + x_2
   POL(U92(x_1)) = 1 + 2*x_1
   POL(__(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a) = 1
   POL(a__U11(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U12(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U13(x_1)) = 2 + 2*x_1
   POL(a__U21(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U22(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U23(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U24(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U25(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U26(x_1)) = 2 + 2*x_1
   POL(a__U31(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U32(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U33(x_1)) = 2 + 2*x_1
   POL(a__U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U43(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U44(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U45(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U46(x_1)) = 2 + x_1
   POL(a__U51(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U52(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U53(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U54(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U55(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U56(x_1)) = 2 + x_1
   POL(a__U61(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U62(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U63(x_1)) = 2 + 2*x_1
   POL(a__U71(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(a__U72(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U73(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U74(x_1)) = 2 + 2*x_1
   POL(a__U81(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U82(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U83(x_1)) = 2 + 2*x_1
   POL(a__U91(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__U92(x_1)) = 2 + 2*x_1
   POL(a____(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(a__isList(x_1)) = 2 + 2*x_1
   POL(a__isNeList(x_1)) = 2 + 2*x_1
   POL(a__isNePal(x_1)) = 2 + 2*x_1
   POL(a__isPal(x_1)) = 2 + 2*x_1
   POL(a__isPalListKind(x_1)) = 2 + 2*x_1
   POL(a__isQid(x_1)) = 2 + 2*x_1
   POL(e) = 1
   POL(i) = 1
   POL(isList(x_1)) = 1 + 2*x_1
   POL(isNeList(x_1)) = 1 + 2*x_1
   POL(isNePal(x_1)) = 1 + 2*x_1
   POL(isPal(x_1)) = 1 + 2*x_1
   POL(isPalListKind(x_1)) = 1 + 2*x_1
   POL(isQid(x_1)) = 1 + 2*x_1
   POL(mark(x_1)) = 2*x_1
   POL(nil) = 1
   POL(o) = 1
   POL(tt) = 1
   POL(u) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__U13(tt) -> tt
   a__U26(tt) -> tt
   a__U33(tt) -> tt
   a__U63(tt) -> tt
   a__U74(tt) -> tt
   a__U83(tt) -> tt
   a__U92(tt) -> tt
   a__isPalListKind(nil) -> tt
   a__isPalListKind(o) -> tt
   mark(nil) -> nil
   mark(tt) -> tt
   mark(a) -> a
   mark(e) -> e
   mark(i) -> i
   mark(o) -> o
   mark(u) -> u
   a____(X1, X2) -> __(X1, X2)
   a__U11(X1, X2) -> U11(X1, X2)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isPalListKind(X) -> isPalListKind(X)
   a__U13(X) -> U13(X)
   a__isNeList(X) -> isNeList(X)
   a__U21(X1, X2, X3) -> U21(X1, X2, X3)
   a__U22(X1, X2, X3) -> U22(X1, X2, X3)
   a__U23(X1, X2, X3) -> U23(X1, X2, X3)
   a__U24(X1, X2, X3) -> U24(X1, X2, X3)
   a__U25(X1, X2) -> U25(X1, X2)
   a__isList(X) -> isList(X)
   a__U26(X) -> U26(X)
   a__U31(X1, X2) -> U31(X1, X2)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__isQid(X) -> isQid(X)
   a__U41(X1, X2, X3) -> U41(X1, X2, X3)
   a__U42(X1, X2, X3) -> U42(X1, X2, X3)
   a__U43(X1, X2, X3) -> U43(X1, X2, X3)
   a__U44(X1, X2, X3) -> U44(X1, X2, X3)
   a__U45(X1, X2) -> U45(X1, X2)
   a__U46(X) -> U46(X)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__U52(X1, X2, X3) -> U52(X1, X2, X3)
   a__U53(X1, X2, X3) -> U53(X1, X2, X3)
   a__U54(X1, X2, X3) -> U54(X1, X2, X3)
   a__U55(X1, X2) -> U55(X1, X2)
   a__U56(X) -> U56(X)
   a__U61(X1, X2) -> U61(X1, X2)
   a__U62(X1, X2) -> U62(X1, X2)
   a__U63(X) -> U63(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__U72(X1, X2) -> U72(X1, X2)
   a__U73(X1, X2) -> U73(X1, X2)
   a__isPal(X) -> isPal(X)
   a__U74(X) -> U74(X)
   a__U81(X1, X2) -> U81(X1, X2)
   a__U82(X1, X2) -> U82(X1, X2)
   a__U83(X) -> U83(X)
   a__isNePal(X) -> isNePal(X)
   a__U91(X1, X2) -> U91(X1, X2)
   a__U92(X) -> U92(X)




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

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

   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isPalListKind(X)) -> a__isPalListKind(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(isNeList(X)) -> a__isNeList(X)
   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
   mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3)
   mark(U23(X1, X2, X3)) -> a__U23(mark(X1), X2, X3)
   mark(U24(X1, X2, X3)) -> a__U24(mark(X1), X2, X3)
   mark(U25(X1, X2)) -> a__U25(mark(X1), X2)
   mark(isList(X)) -> a__isList(X)
   mark(U26(X)) -> a__U26(mark(X))
   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(isQid(X)) -> a__isQid(X)
   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
   mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3)
   mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3)
   mark(U45(X1, X2)) -> a__U45(mark(X1), X2)
   mark(U46(X)) -> a__U46(mark(X))
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3)
   mark(U53(X1, X2, X3)) -> a__U53(mark(X1), X2, X3)
   mark(U54(X1, X2, X3)) -> a__U54(mark(X1), X2, X3)
   mark(U55(X1, X2)) -> a__U55(mark(X1), X2)
   mark(U56(X)) -> a__U56(mark(X))
   mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
   mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
   mark(U63(X)) -> a__U63(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
   mark(U73(X1, X2)) -> a__U73(mark(X1), X2)
   mark(isPal(X)) -> a__isPal(X)
   mark(U74(X)) -> a__U74(mark(X))
   mark(U81(X1, X2)) -> a__U81(mark(X1), X2)
   mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
   mark(U83(X)) -> a__U83(mark(X))
   mark(isNePal(X)) -> a__isNePal(X)
   mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
   mark(U92(X)) -> a__U92(mark(X))

Q is empty.

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

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

   POL(U11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U12(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(U13(x_1)) = 2 + 2*x_1
   POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3
   POL(U22(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3
   POL(U23(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U24(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3
   POL(U25(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U26(x_1)) = 2*x_1
   POL(U31(x_1, x_2)) = 2*x_1 + x_2
   POL(U32(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(U33(x_1)) = 2 + 2*x_1
   POL(U41(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3
   POL(U42(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U43(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U44(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3
   POL(U45(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U46(x_1)) = 2 + 2*x_1
   POL(U51(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U52(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U53(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U54(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3
   POL(U55(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(U56(x_1)) = 2 + 2*x_1
   POL(U61(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(U62(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(U63(x_1)) = 2 + 2*x_1
   POL(U71(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3
   POL(U72(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U73(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U74(x_1)) = 2 + 2*x_1
   POL(U81(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(U82(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U83(x_1)) = 2 + 2*x_1
   POL(U91(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U92(x_1)) = 2 + 2*x_1
   POL(__(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(a__U11(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U12(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(a__U13(x_1)) = 1 + x_1
   POL(a__U21(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(a__U22(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(a__U23(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U24(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
   POL(a__U25(x_1, x_2)) = x_1 + 2*x_2
   POL(a__U26(x_1)) = x_1
   POL(a__U31(x_1, x_2)) = x_1 + 2*x_2
   POL(a__U32(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(a__U33(x_1)) = 2*x_1
   POL(a__U41(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U42(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U43(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U44(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U45(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U46(x_1)) = 1 + x_1
   POL(a__U51(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U52(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(a__U53(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U54(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(a__U55(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U56(x_1)) = 1 + x_1
   POL(a__U61(x_1, x_2)) = x_1 + 2*x_2
   POL(a__U62(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U63(x_1)) = 1 + x_1
   POL(a__U71(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
   POL(a__U72(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U73(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U74(x_1)) = 1 + x_1
   POL(a__U81(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U82(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U83(x_1)) = 1 + x_1
   POL(a__U91(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(a__U92(x_1)) = 2*x_1
   POL(a____(x_1, x_2)) = x_1 + x_2
   POL(a__isList(x_1)) = 2*x_1
   POL(a__isNeList(x_1)) = 2 + 2*x_1
   POL(a__isNePal(x_1)) = 1 + 2*x_1
   POL(a__isPal(x_1)) = 1 + 2*x_1
   POL(a__isPalListKind(x_1)) = 2*x_1
   POL(a__isQid(x_1)) = 2*x_1
   POL(isList(x_1)) = x_1
   POL(isNeList(x_1)) = 2 + x_1
   POL(isNePal(x_1)) = 2*x_1
   POL(isPal(x_1)) = x_1
   POL(isPalListKind(x_1)) = 2*x_1
   POL(isQid(x_1)) = 2*x_1
   POL(mark(x_1)) = 1 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   mark(__(X1, X2)) -> a____(mark(X1), mark(X2))
   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isPalListKind(X)) -> a__isPalListKind(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(isNeList(X)) -> a__isNeList(X)
   mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3)
   mark(U22(X1, X2, X3)) -> a__U22(mark(X1), X2, X3)
   mark(U23(X1, X2, X3)) -> a__U23(mark(X1), X2, X3)
   mark(U24(X1, X2, X3)) -> a__U24(mark(X1), X2, X3)
   mark(U25(X1, X2)) -> a__U25(mark(X1), X2)
   mark(isList(X)) -> a__isList(X)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(isQid(X)) -> a__isQid(X)
   mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3)
   mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3)
   mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3)
   mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3)
   mark(U45(X1, X2)) -> a__U45(mark(X1), X2)
   mark(U46(X)) -> a__U46(mark(X))
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(U52(X1, X2, X3)) -> a__U52(mark(X1), X2, X3)
   mark(U53(X1, X2, X3)) -> a__U53(mark(X1), X2, X3)
   mark(U54(X1, X2, X3)) -> a__U54(mark(X1), X2, X3)
   mark(U55(X1, X2)) -> a__U55(mark(X1), X2)
   mark(U56(X)) -> a__U56(mark(X))
   mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
   mark(U62(X1, X2)) -> a__U62(mark(X1), X2)
   mark(U63(X)) -> a__U63(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
   mark(U73(X1, X2)) -> a__U73(mark(X1), X2)
   mark(U74(X)) -> a__U74(mark(X))
   mark(U81(X1, X2)) -> a__U81(mark(X1), X2)
   mark(U82(X1, X2)) -> a__U82(mark(X1), X2)
   mark(U83(X)) -> a__U83(mark(X))
   mark(U91(X1, X2)) -> a__U91(mark(X1), X2)
   mark(U92(X)) -> a__U92(mark(X))




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

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

   mark(U26(X)) -> a__U26(mark(X))
   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
   mark(isPal(X)) -> a__isPal(X)
   mark(isNePal(X)) -> a__isNePal(X)

Q is empty.

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

(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:a__U26_1 > mark_1 > U31_2 > a__isNePal_1 > a__isPal_1 > a__U31_2 > isPal_1 > isNePal_1 > U26_1

and weight map:

   mark_1=3
   U26_1=2
   a__U26_1=1
   isPal_1=1
   a__isPal_1=4
   isNePal_1=1
   a__isNePal_1=3
   U31_2=1
   a__U31_2=0

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

   mark(U26(X)) -> a__U26(mark(X))
   mark(U31(X1, X2)) -> a__U31(mark(X1), X2)
   mark(isPal(X)) -> a__isPal(X)
   mark(isNePal(X)) -> a__isNePal(X)




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

(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

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

(10)
YES