/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 92 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 10 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 4 ms]
(10) CSR
(11) CSRRRRProof [EQUIVALENT, 12 ms]
(12) CSR
(13) CSRRRRProof [EQUIVALENT, 0 ms]
(14) CSR
(15) CSRRRRProof [EQUIVALENT, 14 ms]
(16) CSR
(17) CSRRRRProof [EQUIVALENT, 0 ms]
(18) CSR
(19) CSRRRRProof [EQUIVALENT, 0 ms]
(20) CSR
(21) CSRRRRProof [EQUIVALENT, 4 ms]
(22) CSR
(23) CSRRRRProof [EQUIVALENT, 0 ms]
(24) CSR
(25) CSRRRRProof [EQUIVALENT, 0 ms]
(26) CSR
(27) CSRRRRProof [EQUIVALENT, 0 ms]
(28) CSR
(29) RisEmptyProof [EQUIVALENT, 0 ms]
(30) 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, V)) -> mark(U12(isPalListKind(V), V))
   active(U12(tt, V)) -> mark(U13(isNeList(V)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isPalListKind(V1), V1, V2))
   active(U22(tt, V1, V2)) -> mark(U23(isPalListKind(V2), V1, V2))
   active(U23(tt, V1, V2)) -> mark(U24(isPalListKind(V2), V1, V2))
   active(U24(tt, V1, V2)) -> mark(U25(isList(V1), V2))
   active(U25(tt, V2)) -> mark(U26(isList(V2)))
   active(U26(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isPalListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isQid(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isPalListKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isPalListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isPalListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isList(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNeList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isPalListKind(V1), V1, V2))
   active(U52(tt, V1, V2)) -> mark(U53(isPalListKind(V2), V1, V2))
   active(U53(tt, V1, V2)) -> mark(U54(isPalListKind(V2), V1, V2))
   active(U54(tt, V1, V2)) -> mark(U55(isNeList(V1), V2))
   active(U55(tt, V2)) -> mark(U56(isList(V2)))
   active(U56(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isPalListKind(V), V))
   active(U62(tt, V)) -> mark(U63(isQid(V)))
   active(U63(tt)) -> mark(tt)
   active(U71(tt, I, P)) -> mark(U72(isPalListKind(I), P))
   active(U72(tt, P)) -> mark(U73(isPal(P), P))
   active(U73(tt, P)) -> mark(U74(isPalListKind(P)))
   active(U74(tt)) -> mark(tt)
   active(U81(tt, V)) -> mark(U82(isPalListKind(V), V))
   active(U82(tt, V)) -> mark(U83(isNePal(V)))
   active(U83(tt)) -> mark(tt)
   active(U91(tt, V2)) -> mark(U92(isPalListKind(V2)))
   active(U92(tt)) -> mark(tt)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(isPalListKind(V1), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(isPalListKind(V1), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(isPalListKind(V1), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), I, P))
   active(isPal(V)) -> mark(U81(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(U91(isPalListKind(V1), V2))
   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(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(U23(X1, X2, X3)) -> U23(active(X1), X2, X3)
   active(U24(X1, X2, X3)) -> U24(active(X1), X2, X3)
   active(U25(X1, X2)) -> U25(active(X1), X2)
   active(U26(X)) -> U26(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
   active(U45(X1, X2)) -> U45(active(X1), X2)
   active(U46(X)) -> U46(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3)
   active(U53(X1, X2, X3)) -> U53(active(X1), X2, X3)
   active(U54(X1, X2, X3)) -> U54(active(X1), X2, X3)
   active(U55(X1, X2)) -> U55(active(X1), X2)
   active(U56(X)) -> U56(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(U63(X)) -> U63(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(U72(X1, X2)) -> U72(active(X1), X2)
   active(U73(X1, X2)) -> U73(active(X1), X2)
   active(U74(X)) -> U74(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X1, X2)) -> U82(active(X1), X2)
   active(U83(X)) -> U83(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   U23(mark(X1), X2, X3) -> mark(U23(X1, X2, X3))
   U24(mark(X1), X2, X3) -> mark(U24(X1, X2, X3))
   U25(mark(X1), X2) -> mark(U25(X1, X2))
   U26(mark(X)) -> mark(U26(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
   U45(mark(X1), X2) -> mark(U45(X1, X2))
   U46(mark(X)) -> mark(U46(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3))
   U53(mark(X1), X2, X3) -> mark(U53(X1, X2, X3))
   U54(mark(X1), X2, X3) -> mark(U54(X1, X2, X3))
   U55(mark(X1), X2) -> mark(U55(X1, X2))
   U56(mark(X)) -> mark(U56(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   U63(mark(X)) -> mark(U63(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   U72(mark(X1), X2) -> mark(U72(X1, X2))
   U73(mark(X1), X2) -> mark(U73(X1, X2))
   U74(mark(X)) -> mark(U74(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X1), X2) -> mark(U82(X1, X2))
   U83(mark(X)) -> mark(U83(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(U23(X1, X2, X3)) -> U23(proper(X1), proper(X2), proper(X3))
   proper(U24(X1, X2, X3)) -> U24(proper(X1), proper(X2), proper(X3))
   proper(U25(X1, X2)) -> U25(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U26(X)) -> U26(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
   proper(U46(X)) -> U46(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3))
   proper(U53(X1, X2, X3)) -> U53(proper(X1), proper(X2), proper(X3))
   proper(U54(X1, X2, X3)) -> U54(proper(X1), proper(X2), proper(X3))
   proper(U55(X1, X2)) -> U55(proper(X1), proper(X2))
   proper(U56(X)) -> U56(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(U63(X)) -> U63(proper(X))
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
   proper(U73(X1, X2)) -> U73(proper(X1), proper(X2))
   proper(isPal(X)) -> isPal(proper(X))
   proper(U74(X)) -> U74(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X1, X2)) -> U82(proper(X1), proper(X2))
   proper(U83(X)) -> U83(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(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(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   U13(ok(X)) -> ok(U13(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   U23(ok(X1), ok(X2), ok(X3)) -> ok(U23(X1, X2, X3))
   U24(ok(X1), ok(X2), ok(X3)) -> ok(U24(X1, X2, X3))
   U25(ok(X1), ok(X2)) -> ok(U25(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U26(ok(X)) -> ok(U26(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
   U46(ok(X)) -> ok(U46(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3))
   U53(ok(X1), ok(X2), ok(X3)) -> ok(U53(X1, X2, X3))
   U54(ok(X1), ok(X2), ok(X3)) -> ok(U54(X1, X2, X3))
   U55(ok(X1), ok(X2)) -> ok(U55(X1, X2))
   U56(ok(X)) -> ok(U56(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   U63(ok(X)) -> ok(U63(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
   U73(ok(X1), ok(X2)) -> ok(U73(X1, X2))
   isPal(ok(X)) -> ok(isPal(X))
   U74(ok(X)) -> ok(U74(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X1), ok(X2)) -> ok(U82(X1, X2))
   U83(ok(X)) -> ok(U83(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

(1) QTRSToCSRProof (EQUIVALENT)
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, V)) -> mark(U12(isPalListKind(V), V))
   active(U12(tt, V)) -> mark(U13(isNeList(V)))
   active(U13(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isPalListKind(V1), V1, V2))
   active(U22(tt, V1, V2)) -> mark(U23(isPalListKind(V2), V1, V2))
   active(U23(tt, V1, V2)) -> mark(U24(isPalListKind(V2), V1, V2))
   active(U24(tt, V1, V2)) -> mark(U25(isList(V1), V2))
   active(U25(tt, V2)) -> mark(U26(isList(V2)))
   active(U26(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isPalListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isQid(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isPalListKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isPalListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isPalListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isList(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNeList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isPalListKind(V1), V1, V2))
   active(U52(tt, V1, V2)) -> mark(U53(isPalListKind(V2), V1, V2))
   active(U53(tt, V1, V2)) -> mark(U54(isPalListKind(V2), V1, V2))
   active(U54(tt, V1, V2)) -> mark(U55(isNeList(V1), V2))
   active(U55(tt, V2)) -> mark(U56(isList(V2)))
   active(U56(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isPalListKind(V), V))
   active(U62(tt, V)) -> mark(U63(isQid(V)))
   active(U63(tt)) -> mark(tt)
   active(U71(tt, I, P)) -> mark(U72(isPalListKind(I), P))
   active(U72(tt, P)) -> mark(U73(isPal(P), P))
   active(U73(tt, P)) -> mark(U74(isPalListKind(P)))
   active(U74(tt)) -> mark(tt)
   active(U81(tt, V)) -> mark(U82(isPalListKind(V), V))
   active(U82(tt, V)) -> mark(U83(isNePal(V)))
   active(U83(tt)) -> mark(tt)
   active(U91(tt, V2)) -> mark(U92(isPalListKind(V2)))
   active(U92(tt)) -> mark(tt)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(isPalListKind(V1), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(isPalListKind(V1), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(isPalListKind(V1), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), I, P))
   active(isPal(V)) -> mark(U81(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(U91(isPalListKind(V1), V2))
   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(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(U23(X1, X2, X3)) -> U23(active(X1), X2, X3)
   active(U24(X1, X2, X3)) -> U24(active(X1), X2, X3)
   active(U25(X1, X2)) -> U25(active(X1), X2)
   active(U26(X)) -> U26(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
   active(U45(X1, X2)) -> U45(active(X1), X2)
   active(U46(X)) -> U46(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3)
   active(U53(X1, X2, X3)) -> U53(active(X1), X2, X3)
   active(U54(X1, X2, X3)) -> U54(active(X1), X2, X3)
   active(U55(X1, X2)) -> U55(active(X1), X2)
   active(U56(X)) -> U56(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(U63(X)) -> U63(active(X))
   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
   active(U72(X1, X2)) -> U72(active(X1), X2)
   active(U73(X1, X2)) -> U73(active(X1), X2)
   active(U74(X)) -> U74(active(X))
   active(U81(X1, X2)) -> U81(active(X1), X2)
   active(U82(X1, X2)) -> U82(active(X1), X2)
   active(U83(X)) -> U83(active(X))
   active(U91(X1, X2)) -> U91(active(X1), X2)
   active(U92(X)) -> U92(active(X))
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   U23(mark(X1), X2, X3) -> mark(U23(X1, X2, X3))
   U24(mark(X1), X2, X3) -> mark(U24(X1, X2, X3))
   U25(mark(X1), X2) -> mark(U25(X1, X2))
   U26(mark(X)) -> mark(U26(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
   U45(mark(X1), X2) -> mark(U45(X1, X2))
   U46(mark(X)) -> mark(U46(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3))
   U53(mark(X1), X2, X3) -> mark(U53(X1, X2, X3))
   U54(mark(X1), X2, X3) -> mark(U54(X1, X2, X3))
   U55(mark(X1), X2) -> mark(U55(X1, X2))
   U56(mark(X)) -> mark(U56(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   U63(mark(X)) -> mark(U63(X))
   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
   U72(mark(X1), X2) -> mark(U72(X1, X2))
   U73(mark(X1), X2) -> mark(U73(X1, X2))
   U74(mark(X)) -> mark(U74(X))
   U81(mark(X1), X2) -> mark(U81(X1, X2))
   U82(mark(X1), X2) -> mark(U82(X1, X2))
   U83(mark(X)) -> mark(U83(X))
   U91(mark(X1), X2) -> mark(U91(X1, X2))
   U92(mark(X)) -> mark(U92(X))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(U23(X1, X2, X3)) -> U23(proper(X1), proper(X2), proper(X3))
   proper(U24(X1, X2, X3)) -> U24(proper(X1), proper(X2), proper(X3))
   proper(U25(X1, X2)) -> U25(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U26(X)) -> U26(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
   proper(U46(X)) -> U46(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3))
   proper(U53(X1, X2, X3)) -> U53(proper(X1), proper(X2), proper(X3))
   proper(U54(X1, X2, X3)) -> U54(proper(X1), proper(X2), proper(X3))
   proper(U55(X1, X2)) -> U55(proper(X1), proper(X2))
   proper(U56(X)) -> U56(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(U63(X)) -> U63(proper(X))
   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
   proper(U73(X1, X2)) -> U73(proper(X1), proper(X2))
   proper(isPal(X)) -> isPal(proper(X))
   proper(U74(X)) -> U74(proper(X))
   proper(U81(X1, X2)) -> U81(proper(X1), proper(X2))
   proper(U82(X1, X2)) -> U82(proper(X1), proper(X2))
   proper(U83(X)) -> U83(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(U91(X1, X2)) -> U91(proper(X1), proper(X2))
   proper(U92(X)) -> U92(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(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   U13(ok(X)) -> ok(U13(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   U23(ok(X1), ok(X2), ok(X3)) -> ok(U23(X1, X2, X3))
   U24(ok(X1), ok(X2), ok(X3)) -> ok(U24(X1, X2, X3))
   U25(ok(X1), ok(X2)) -> ok(U25(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U26(ok(X)) -> ok(U26(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
   U46(ok(X)) -> ok(U46(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3))
   U53(ok(X1), ok(X2), ok(X3)) -> ok(U53(X1, X2, X3))
   U54(ok(X1), ok(X2), ok(X3)) -> ok(U54(X1, X2, X3))
   U55(ok(X1), ok(X2)) -> ok(U55(X1, X2))
   U56(ok(X)) -> ok(U56(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   U63(ok(X)) -> ok(U63(X))
   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
   U73(ok(X1), ok(X2)) -> ok(U73(X1, X2))
   isPal(ok(X)) -> ok(isPal(X))
   U74(ok(X)) -> ok(U74(X))
   U81(ok(X1), ok(X2)) -> ok(U81(X1, X2))
   U82(ok(X1), ok(X2)) -> ok(U82(X1, X2))
   U83(ok(X)) -> ok(U83(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   U91(ok(X1), ok(X2)) -> ok(U91(X1, X2))
   U92(ok(X)) -> ok(U92(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
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
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U71: {1}
U72: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
----------------------------------------

(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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U22(tt, V1, V2) -> U23(isPalListKind(V2), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U25(tt, V2) -> U26(isList(V2))
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isPalListKind(V1), V1, V2)
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U43(tt, V1, V2) -> U44(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U55(tt, V2) -> U56(isList(V2))
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U71(tt, I, P) -> U72(isPalListKind(I), P)
   U72(tt, P) -> U73(isPal(P), P)
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(nil) -> tt
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNeList(__(V1, V2)) -> U41(isPalListKind(V1), V1, V2)
   isNeList(__(V1, V2)) -> U51(isPalListKind(V1), V1, V2)
   isNePal(V) -> U61(isPalListKind(V), V)
   isNePal(__(I, __(P, I))) -> U71(isQid(I), I, P)
   isPal(V) -> U81(isPalListKind(V), V)
   isPal(nil) -> tt
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)
   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
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U71: {1}
U72: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U22(tt, V1, V2) -> U23(isPalListKind(V2), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U25(tt, V2) -> U26(isList(V2))
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U41(tt, V1, V2) -> U42(isPalListKind(V1), V1, V2)
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U43(tt, V1, V2) -> U44(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U55(tt, V2) -> U56(isList(V2))
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U71(tt, I, P) -> U72(isPalListKind(I), P)
   U72(tt, P) -> U73(isPal(P), P)
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(nil) -> tt
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNeList(__(V1, V2)) -> U41(isPalListKind(V1), V1, V2)
   isNeList(__(V1, V2)) -> U51(isPalListKind(V1), V1, V2)
   isNePal(V) -> U61(isPalListKind(V), V)
   isNePal(__(I, __(P, I))) -> U71(isQid(I), I, P)
   isPal(V) -> U81(isPalListKind(V), V)
   isPal(nil) -> tt
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)
   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
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U71: {1}
U72: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U22(x_1, x_2, x_3)) = 1 + x_1 + x_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 + x_2 + x_3
   POL(U25(x_1, x_2)) = 1 + x_1 + x_2
   POL(U26(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U43(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U44(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U45(x_1, x_2)) = x_1 + x_2
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U52(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U53(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U54(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U55(x_1, x_2)) = x_1 + x_2
   POL(U56(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U63(x_1)) = x_1
   POL(U71(x_1, x_2, x_3)) = 1 + x_1 + x_3
   POL(U72(x_1, x_2)) = x_1 + x_2
   POL(U73(x_1, x_2)) = x_1
   POL(U74(x_1)) = x_1
   POL(U81(x_1, x_2)) = x_1 + x_2
   POL(U82(x_1, x_2)) = x_1 + x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = x_1
   POL(U92(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)) = x_1
   POL(isPal(x_1)) = x_1
   POL(isPalListKind(x_1)) = 0
   POL(isQid(x_1)) = x_1
   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:

   __(X, nil) -> X
   __(nil, X) -> X
   U25(tt, V2) -> U26(isList(V2))
   U41(tt, V1, V2) -> U42(isPalListKind(V1), V1, V2)
   U71(tt, I, P) -> U72(isPalListKind(I), P)
   isList(nil) -> tt
   isNeList(__(V1, V2)) -> U51(isPalListKind(V1), V1, V2)
   isNePal(__(I, __(P, I))) -> U71(isQid(I), I, P)
   isPal(nil) -> tt
   isQid(a) -> tt
   isQid(e) -> tt
   isQid(i) -> tt
   isQid(o) -> tt
   isQid(u) -> tt




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U22(tt, V1, V2) -> U23(isPalListKind(V2), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U43(tt, V1, V2) -> U44(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U55(tt, V2) -> U56(isList(V2))
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U72(tt, P) -> U73(isPal(P), P)
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNeList(__(V1, V2)) -> U41(isPalListKind(V1), V1, V2)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U72: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U22(tt, V1, V2) -> U23(isPalListKind(V2), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U43(tt, V1, V2) -> U44(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U55(tt, V2) -> U56(isList(V2))
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U72(tt, P) -> U73(isPal(P), P)
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNeList(__(V1, V2)) -> U41(isPalListKind(V1), V1, V2)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U72: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U22(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U23(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U25(x_1, x_2)) = x_1 + x_2
   POL(U26(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U43(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U44(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U45(x_1, x_2)) = x_1 + x_2
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U52(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U53(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U54(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U55(x_1, x_2)) = 1 + x_1 + x_2
   POL(U56(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U63(x_1)) = x_1
   POL(U72(x_1, x_2)) = 1 + x_1 + x_2
   POL(U73(x_1, x_2)) = x_1
   POL(U74(x_1)) = x_1
   POL(U81(x_1, x_2)) = x_1 + x_2
   POL(U82(x_1, x_2)) = x_1 + x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = x_1
   POL(U92(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)) = 1 + x_1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 1
   POL(isQid(x_1)) = 1 + 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:

   U22(tt, V1, V2) -> U23(isPalListKind(V2), V1, V2)
   U43(tt, V1, V2) -> U44(isPalListKind(V2), V1, V2)
   U55(tt, V2) -> U56(isList(V2))
   U72(tt, P) -> U73(isPal(P), P)
   isNeList(__(V1, V2)) -> U41(isPalListKind(V1), V1, V2)




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U33(tt) -> tt
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U46(tt) -> tt
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U62(tt, V) -> U63(isQid(V))
   U63(tt) -> tt
   U73(tt, P) -> U74(isPalListKind(P))
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U23: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
U73: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U22(x_1, x_2, x_3)) = x_1 + x_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)) = x_1 + x_2 + x_3
   POL(U25(x_1, x_2)) = x_1 + x_2
   POL(U26(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = 1 + x_1
   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U43(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U44(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U45(x_1, x_2)) = 1 + x_1 + x_2
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U52(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U53(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U54(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U55(x_1, x_2)) = x_1 + x_2
   POL(U56(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U63(x_1)) = x_1
   POL(U73(x_1, x_2)) = 1 + x_1 + x_2
   POL(U74(x_1)) = x_1
   POL(U81(x_1, x_2)) = x_1 + x_2
   POL(U82(x_1, x_2)) = x_1 + x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = x_1
   POL(U92(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)) = 1 + x_1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 1
   POL(isQid(x_1)) = 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:

   U23(tt, V1, V2) -> U24(isPalListKind(V2), V1, V2)
   U33(tt) -> tt
   U42(tt, V1, V2) -> U43(isPalListKind(V2), V1, V2)
   U45(tt, V2) -> U46(isNeList(V2))
   U51(tt, V1, V2) -> U52(isPalListKind(V1), V1, V2)
   U62(tt, V) -> U63(isQid(V))
   U73(tt, P) -> U74(isPalListKind(P))
   isList(__(V1, V2)) -> U21(isPalListKind(V1), V1, V2)




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U46(tt) -> tt
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U63(tt) -> tt
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U44: {1}
U45: {1}
U46: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(9) 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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U46(tt) -> tt
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U56(tt) -> tt
   U61(tt, V) -> U62(isPalListKind(V), V)
   U63(tt) -> tt
   U74(tt) -> tt
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U83(tt) -> tt
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPal(V) -> U81(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U21: {1}
U22: {1}
U24: {1}
U25: {1}
isList: empty set
U26: {1}
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U44: {1}
U45: {1}
U46: {1}
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U56: {1}
U61: {1}
U62: {1}
U63: {1}
isPal: empty set
U74: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = x_1
   POL(U12(x_1, x_2)) = 2*x_1
   POL(U13(x_1)) = 2*x_1
   POL(U21(x_1, x_2, x_3)) = 2 + 2*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(U24(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(U25(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U26(x_1)) = 1 + x_1
   POL(U31(x_1, x_2)) = 2*x_1
   POL(U32(x_1, x_2)) = 2*x_1
   POL(U33(x_1)) = 2*x_1
   POL(U44(x_1, x_2, x_3)) = 2 + 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(U52(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(U53(x_1, x_2, x_3)) = x_1 + 2*x_2
   POL(U54(x_1, x_2, x_3)) = x_1 + 2*x_2
   POL(U55(x_1, x_2)) = x_1
   POL(U56(x_1)) = 1 + x_1
   POL(U61(x_1, x_2)) = 2*x_1
   POL(U62(x_1, x_2)) = x_1
   POL(U63(x_1)) = 1 + x_1
   POL(U74(x_1)) = 1 + 2*x_1
   POL(U81(x_1, x_2)) = 1 + x_1
   POL(U82(x_1, x_2)) = 1 + x_1
   POL(U83(x_1)) = 1 + x_1
   POL(U91(x_1, x_2)) = 2*x_1
   POL(U92(x_1)) = x_1
   POL(__(x_1, x_2)) = 2*x_1 + x_2
   POL(a) = 2
   POL(e) = 2
   POL(i) = 2
   POL(isList(x_1)) = x_1
   POL(isNeList(x_1)) = 0
   POL(isNePal(x_1)) = 0
   POL(isPal(x_1)) = 2 + 2*x_1
   POL(isPalListKind(x_1)) = 0
   POL(isQid(x_1)) = 0
   POL(nil) = 2
   POL(o) = 2
   POL(tt) = 0
   POL(u) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U21(tt, V1, V2) -> U22(isPalListKind(V1), V1, V2)
   U24(tt, V1, V2) -> U25(isList(V1), V2)
   U26(tt) -> tt
   U44(tt, V1, V2) -> U45(isList(V1), V2)
   U46(tt) -> tt
   U56(tt) -> tt
   U63(tt) -> tt
   U74(tt) -> tt
   U83(tt) -> tt
   isPal(V) -> U81(isPalListKind(V), V)




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
isList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(11) 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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
isList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = 1 + x_1
   POL(U52(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U53(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U54(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U55(x_1, x_2)) = x_1 + x_2
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U81(x_1, x_2)) = x_1 + x_2
   POL(U82(x_1, x_2)) = x_1 + x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = x_1
   POL(U92(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)) = 1 + x_1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 1
   POL(isQid(x_1)) = 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:

   U53(tt, V1, V2) -> U54(isPalListKind(V2), V1, V2)




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
isList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(13) 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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isList(V) -> U11(isPalListKind(V), V)
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
isList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U52: {1}
U53: {1}
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = 2*x_1
   POL(U12(x_1, x_2)) = 2*x_1
   POL(U13(x_1)) = 2*x_1
   POL(U31(x_1, x_2)) = 2*x_1
   POL(U32(x_1, x_2)) = 2*x_1
   POL(U33(x_1)) = x_1
   POL(U52(x_1, x_2, x_3)) = 2 + 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)) = 2*x_1 + 2*x_2 + 2*x_3
   POL(U55(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U61(x_1, x_2)) = 2*x_1
   POL(U62(x_1, x_2)) = x_1
   POL(U81(x_1, x_2)) = x_1 + 2*x_2
   POL(U82(x_1, x_2)) = x_1
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = 2*x_1
   POL(U92(x_1)) = x_1
   POL(__(x_1, x_2)) = 2*x_1 + x_2
   POL(a) = 2
   POL(e) = 2
   POL(i) = 2
   POL(isList(x_1)) = 2 + 2*x_1
   POL(isNeList(x_1)) = 0
   POL(isNePal(x_1)) = 0
   POL(isPalListKind(x_1)) = 0
   POL(isQid(x_1)) = 0
   POL(nil) = 2
   POL(o) = 2
   POL(tt) = 0
   POL(u) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U52(tt, V1, V2) -> U53(isPalListKind(V2), V1, V2)
   isList(V) -> U11(isPalListKind(V), V)




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

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

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(15) 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, V) -> U12(isPalListKind(V), V)
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U81: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
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_2)) = 1 + x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U13(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = 1 + x_1
   POL(U54(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U55(x_1, x_2)) = x_1 + x_2
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U81(x_1, x_2)) = 1 + x_1 + x_2
   POL(U82(x_1, x_2)) = x_1 + x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = x_1
   POL(U92(x_1)) = x_1
   POL(__(x_1, x_2)) = 1 + x_1 + x_2
   POL(a) = 1
   POL(e) = 1
   POL(i) = 1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 1
   POL(isQid(x_1)) = 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:

   U11(tt, V) -> U12(isPalListKind(V), V)
   U81(tt, V) -> U82(isPalListKind(V), V)




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

(16)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set

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

(17) 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))
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U31(tt, V) -> U32(isPalListKind(V), V)
   U32(tt, V) -> U33(isQid(V))
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(u) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
tt: empty set
U12: {1}
isPalListKind: empty set
U13: {1}
isNeList: empty set
U31: {1}
U32: {1}
U33: {1}
isQid: empty set
U54: {1}
U55: {1}
U61: {1}
U62: {1}
U82: {1}
U83: {1}
isNePal: empty set
U91: {1}
U92: {1}
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U12(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(U13(x_1)) = 1 + x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = 1 + x_1
   POL(U54(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U55(x_1, x_2)) = x_1 + 2*x_2
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1
   POL(U82(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U83(x_1)) = x_1
   POL(U91(x_1, x_2)) = 1 + x_1 + x_2
   POL(U92(x_1)) = x_1
   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(a) = 2
   POL(e) = 2
   POL(i) = 2
   POL(isNeList(x_1)) = 2 + 2*x_1
   POL(isNePal(x_1)) = 2 + 2*x_1
   POL(isPalListKind(x_1)) = 2 + x_1
   POL(isQid(x_1)) = 0
   POL(nil) = 2
   POL(o) = 2
   POL(tt) = 2
   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))
   U12(tt, V) -> U13(isNeList(V))
   U13(tt) -> tt
   U32(tt, V) -> U33(isQid(V))
   U54(tt, V1, V2) -> U55(isNeList(V1), V2)
   U82(tt, V) -> U83(isNePal(V))
   U91(tt, V2) -> U92(isPalListKind(V2))
   isPalListKind(a) -> tt
   isPalListKind(e) -> tt
   isPalListKind(i) -> tt
   isPalListKind(nil) -> tt
   isPalListKind(o) -> tt
   isPalListKind(__(V1, V2)) -> U91(isPalListKind(V1), V2)




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

(18)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U31(tt, V) -> U32(isPalListKind(V), V)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(u) -> tt

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U32: {1}
U61: {1}
U62: {1}
isNePal: empty set
U92: {1}
u: empty set

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

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

   U31(tt, V) -> U32(isPalListKind(V), V)
   U61(tt, V) -> U62(isPalListKind(V), V)
   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(u) -> tt

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U32: {1}
U61: {1}
U62: {1}
isNePal: empty set
U92: {1}
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U61(x_1, x_2)) = 1 + x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(U92(x_1)) = x_1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 0
   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:

   U31(tt, V) -> U32(isPalListKind(V), V)
   U61(tt, V) -> U62(isPalListKind(V), V)




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

(20)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(u) -> tt

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set
U92: {1}
u: empty set

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

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

   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)
   isPalListKind(u) -> tt

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set
U92: {1}
u: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U31(x_1, x_2)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U92(x_1)) = x_1
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 1 + x_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:

   isPalListKind(u) -> tt




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

(22)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set
U92: {1}

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

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

   U92(tt) -> tt
   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)

The replacement map contains the following entries:

tt: empty set
isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set
U92: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U31(x_1, x_2)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U92(x_1)) = 2 + x_1
   POL(isNeList(x_1)) = 2 + 2*x_1
   POL(isNePal(x_1)) = 2 + 2*x_1
   POL(isPalListKind(x_1)) = 2 + 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:

   U92(tt) -> tt




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

(24)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)

The replacement map contains the following entries:

isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set

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

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

   isNeList(V) -> U31(isPalListKind(V), V)
   isNePal(V) -> U61(isPalListKind(V), V)

The replacement map contains the following entries:

isPalListKind: empty set
isNeList: empty set
U31: {1}
U61: {1}
isNePal: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(isNeList(x_1)) = 1 + x_1
   POL(isNePal(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNePal(V) -> U61(isPalListKind(V), V)




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

(26)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   isNeList(V) -> U31(isPalListKind(V), V)

The replacement map contains the following entries:

isPalListKind: empty set
isNeList: empty set
U31: {1}

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

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

   isNeList(V) -> U31(isPalListKind(V), V)

The replacement map contains the following entries:

isPalListKind: empty set
isNeList: empty set
U31: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(isNeList(x_1)) = 1 + x_1
   POL(isPalListKind(x_1)) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNeList(V) -> U31(isPalListKind(V), V)




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

(28)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

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

(30)
YES