(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) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(__(__(X, Y), Z)) → __1(X, __(Y, Z))
ACTIVE(__(__(X, Y), Z)) → __1(Y, Z)
ACTIVE(U11(tt, V)) → U121(isPalListKind(V), V)
ACTIVE(U11(tt, V)) → ISPALLISTKIND(V)
ACTIVE(U12(tt, V)) → U131(isNeList(V))
ACTIVE(U12(tt, V)) → ISNELIST(V)
ACTIVE(U21(tt, V1, V2)) → U221(isPalListKind(V1), V1, V2)
ACTIVE(U21(tt, V1, V2)) → ISPALLISTKIND(V1)
ACTIVE(U22(tt, V1, V2)) → U231(isPalListKind(V2), V1, V2)
ACTIVE(U22(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U23(tt, V1, V2)) → U241(isPalListKind(V2), V1, V2)
ACTIVE(U23(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U24(tt, V1, V2)) → U251(isList(V1), V2)
ACTIVE(U24(tt, V1, V2)) → ISLIST(V1)
ACTIVE(U25(tt, V2)) → U261(isList(V2))
ACTIVE(U25(tt, V2)) → ISLIST(V2)
ACTIVE(U31(tt, V)) → U321(isPalListKind(V), V)
ACTIVE(U31(tt, V)) → ISPALLISTKIND(V)
ACTIVE(U32(tt, V)) → U331(isQid(V))
ACTIVE(U32(tt, V)) → ISQID(V)
ACTIVE(U41(tt, V1, V2)) → U421(isPalListKind(V1), V1, V2)
ACTIVE(U41(tt, V1, V2)) → ISPALLISTKIND(V1)
ACTIVE(U42(tt, V1, V2)) → U431(isPalListKind(V2), V1, V2)
ACTIVE(U42(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U43(tt, V1, V2)) → U441(isPalListKind(V2), V1, V2)
ACTIVE(U43(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U44(tt, V1, V2)) → U451(isList(V1), V2)
ACTIVE(U44(tt, V1, V2)) → ISLIST(V1)
ACTIVE(U45(tt, V2)) → U461(isNeList(V2))
ACTIVE(U45(tt, V2)) → ISNELIST(V2)
ACTIVE(U51(tt, V1, V2)) → U521(isPalListKind(V1), V1, V2)
ACTIVE(U51(tt, V1, V2)) → ISPALLISTKIND(V1)
ACTIVE(U52(tt, V1, V2)) → U531(isPalListKind(V2), V1, V2)
ACTIVE(U52(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U53(tt, V1, V2)) → U541(isPalListKind(V2), V1, V2)
ACTIVE(U53(tt, V1, V2)) → ISPALLISTKIND(V2)
ACTIVE(U54(tt, V1, V2)) → U551(isNeList(V1), V2)
ACTIVE(U54(tt, V1, V2)) → ISNELIST(V1)
ACTIVE(U55(tt, V2)) → U561(isList(V2))
ACTIVE(U55(tt, V2)) → ISLIST(V2)
ACTIVE(U61(tt, V)) → U621(isPalListKind(V), V)
ACTIVE(U61(tt, V)) → ISPALLISTKIND(V)
ACTIVE(U62(tt, V)) → U631(isQid(V))
ACTIVE(U62(tt, V)) → ISQID(V)
ACTIVE(U71(tt, I, P)) → U721(isPalListKind(I), P)
ACTIVE(U71(tt, I, P)) → ISPALLISTKIND(I)
ACTIVE(U72(tt, P)) → U731(isPal(P), P)
ACTIVE(U72(tt, P)) → ISPAL(P)
ACTIVE(U73(tt, P)) → U741(isPalListKind(P))
ACTIVE(U73(tt, P)) → ISPALLISTKIND(P)
ACTIVE(U81(tt, V)) → U821(isPalListKind(V), V)
ACTIVE(U81(tt, V)) → ISPALLISTKIND(V)
ACTIVE(U82(tt, V)) → U831(isNePal(V))
ACTIVE(U82(tt, V)) → ISNEPAL(V)
ACTIVE(U91(tt, V2)) → U921(isPalListKind(V2))
ACTIVE(U91(tt, V2)) → ISPALLISTKIND(V2)
ACTIVE(isList(V)) → U111(isPalListKind(V), V)
ACTIVE(isList(V)) → ISPALLISTKIND(V)
ACTIVE(isList(__(V1, V2))) → U211(isPalListKind(V1), V1, V2)
ACTIVE(isList(__(V1, V2))) → ISPALLISTKIND(V1)
ACTIVE(isNeList(V)) → U311(isPalListKind(V), V)
ACTIVE(isNeList(V)) → ISPALLISTKIND(V)
ACTIVE(isNeList(__(V1, V2))) → U411(isPalListKind(V1), V1, V2)
ACTIVE(isNeList(__(V1, V2))) → ISPALLISTKIND(V1)
ACTIVE(isNeList(__(V1, V2))) → U511(isPalListKind(V1), V1, V2)
ACTIVE(isNePal(V)) → U611(isPalListKind(V), V)
ACTIVE(isNePal(V)) → ISPALLISTKIND(V)
ACTIVE(isNePal(__(I, __(P, I)))) → U711(isQid(I), I, P)
ACTIVE(isNePal(__(I, __(P, I)))) → ISQID(I)
ACTIVE(isPal(V)) → U811(isPalListKind(V), V)
ACTIVE(isPal(V)) → ISPALLISTKIND(V)
ACTIVE(isPalListKind(__(V1, V2))) → U911(isPalListKind(V1), V2)
ACTIVE(isPalListKind(__(V1, V2))) → ISPALLISTKIND(V1)
ACTIVE(__(X1, X2)) → __1(active(X1), X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(__(X1, X2)) → __1(X1, active(X2))
ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → U121(active(X1), X2)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → U131(active(X))
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3)) → U211(active(X1), X2, X3)
ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3)) → U221(active(X1), X2, X3)
ACTIVE(U22(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3)) → U231(active(X1), X2, X3)
ACTIVE(U23(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U24(X1, X2, X3)) → U241(active(X1), X2, X3)
ACTIVE(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U25(X1, X2)) → U251(active(X1), X2)
ACTIVE(U25(X1, X2)) → ACTIVE(X1)
ACTIVE(U26(X)) → U261(active(X))
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → U321(active(X1), X2)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → U331(active(X))
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → U421(active(X1), X2, X3)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U43(X1, X2, X3)) → U431(active(X1), X2, X3)
ACTIVE(U43(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → U441(active(X1), X2, X3)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U45(X1, X2)) → U451(active(X1), X2)
ACTIVE(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U46(X)) → U461(active(X))
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → U521(active(X1), X2, X3)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U53(X1, X2, X3)) → U531(active(X1), X2, X3)
ACTIVE(U53(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U54(X1, X2, X3)) → U541(active(X1), X2, X3)
ACTIVE(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U55(X1, X2)) → U551(active(X1), X2)
ACTIVE(U55(X1, X2)) → ACTIVE(X1)
ACTIVE(U56(X)) → U561(active(X))
ACTIVE(U56(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → U611(active(X1), X2)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → U621(active(X1), X2)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → U631(active(X))
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → U711(active(X1), X2, X3)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → U721(active(X1), X2)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U73(X1, X2)) → U731(active(X1), X2)
ACTIVE(U73(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → U741(active(X))
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → U811(active(X1), X2)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → U821(active(X1), X2)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → U831(active(X))
ACTIVE(U83(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → U911(active(X1), X2)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(U92(X)) → U921(active(X))
ACTIVE(U92(X)) → ACTIVE(X)
__1(mark(X1), X2) → __1(X1, X2)
__1(X1, mark(X2)) → __1(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U121(mark(X1), X2) → U121(X1, X2)
U131(mark(X)) → U131(X)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U231(mark(X1), X2, X3) → U231(X1, X2, X3)
U241(mark(X1), X2, X3) → U241(X1, X2, X3)
U251(mark(X1), X2) → U251(X1, X2)
U261(mark(X)) → U261(X)
U311(mark(X1), X2) → U311(X1, X2)
U321(mark(X1), X2) → U321(X1, X2)
U331(mark(X)) → U331(X)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2, X3) → U421(X1, X2, X3)
U431(mark(X1), X2, X3) → U431(X1, X2, X3)
U441(mark(X1), X2, X3) → U441(X1, X2, X3)
U451(mark(X1), X2) → U451(X1, X2)
U461(mark(X)) → U461(X)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U521(mark(X1), X2, X3) → U521(X1, X2, X3)
U531(mark(X1), X2, X3) → U531(X1, X2, X3)
U541(mark(X1), X2, X3) → U541(X1, X2, X3)
U551(mark(X1), X2) → U551(X1, X2)
U561(mark(X)) → U561(X)
U611(mark(X1), X2) → U611(X1, X2)
U621(mark(X1), X2) → U621(X1, X2)
U631(mark(X)) → U631(X)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U721(mark(X1), X2) → U721(X1, X2)
U731(mark(X1), X2) → U731(X1, X2)
U741(mark(X)) → U741(X)
U811(mark(X1), X2) → U811(X1, X2)
U821(mark(X1), X2) → U821(X1, X2)
U831(mark(X)) → U831(X)
U911(mark(X1), X2) → U911(X1, X2)
U921(mark(X)) → U921(X)
PROPER(__(X1, X2)) → __1(proper(X1), proper(X2))
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(__(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → U121(proper(X1), proper(X2))
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isPalListKind(X)) → ISPALLISTKIND(proper(X))
PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → U131(proper(X))
PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → ISNELIST(proper(X))
PROPER(isNeList(X)) → PROPER(X)
PROPER(U21(X1, X2, X3)) → U211(proper(X1), proper(X2), proper(X3))
PROPER(U21(X1, X2, X3)) → PROPER(X1)
PROPER(U21(X1, X2, X3)) → PROPER(X2)
PROPER(U21(X1, X2, X3)) → PROPER(X3)
PROPER(U22(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
PROPER(U22(X1, X2, X3)) → PROPER(X1)
PROPER(U22(X1, X2, X3)) → PROPER(X2)
PROPER(U22(X1, X2, X3)) → PROPER(X3)
PROPER(U23(X1, X2, X3)) → U231(proper(X1), proper(X2), proper(X3))
PROPER(U23(X1, X2, X3)) → PROPER(X1)
PROPER(U23(X1, X2, X3)) → PROPER(X2)
PROPER(U23(X1, X2, X3)) → PROPER(X3)
PROPER(U24(X1, X2, X3)) → U241(proper(X1), proper(X2), proper(X3))
PROPER(U24(X1, X2, X3)) → PROPER(X1)
PROPER(U24(X1, X2, X3)) → PROPER(X2)
PROPER(U24(X1, X2, X3)) → PROPER(X3)
PROPER(U25(X1, X2)) → U251(proper(X1), proper(X2))
PROPER(U25(X1, X2)) → PROPER(X1)
PROPER(U25(X1, X2)) → PROPER(X2)
PROPER(isList(X)) → ISLIST(proper(X))
PROPER(isList(X)) → PROPER(X)
PROPER(U26(X)) → U261(proper(X))
PROPER(U26(X)) → PROPER(X)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X1, X2)) → U321(proper(X1), proper(X2))
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U33(X)) → U331(proper(X))
PROPER(U33(X)) → PROPER(X)
PROPER(isQid(X)) → ISQID(proper(X))
PROPER(isQid(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2, X3)) → U421(proper(X1), proper(X2), proper(X3))
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X1, X2, X3)) → U431(proper(X1), proper(X2), proper(X3))
PROPER(U43(X1, X2, X3)) → PROPER(X1)
PROPER(U43(X1, X2, X3)) → PROPER(X2)
PROPER(U43(X1, X2, X3)) → PROPER(X3)
PROPER(U44(X1, X2, X3)) → U441(proper(X1), proper(X2), proper(X3))
PROPER(U44(X1, X2, X3)) → PROPER(X1)
PROPER(U44(X1, X2, X3)) → PROPER(X2)
PROPER(U44(X1, X2, X3)) → PROPER(X3)
PROPER(U45(X1, X2)) → U451(proper(X1), proper(X2))
PROPER(U45(X1, X2)) → PROPER(X1)
PROPER(U45(X1, X2)) → PROPER(X2)
PROPER(U46(X)) → U461(proper(X))
PROPER(U46(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → U521(proper(X1), proper(X2), proper(X3))
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(U53(X1, X2, X3)) → U531(proper(X1), proper(X2), proper(X3))
PROPER(U53(X1, X2, X3)) → PROPER(X1)
PROPER(U53(X1, X2, X3)) → PROPER(X2)
PROPER(U53(X1, X2, X3)) → PROPER(X3)
PROPER(U54(X1, X2, X3)) → U541(proper(X1), proper(X2), proper(X3))
PROPER(U54(X1, X2, X3)) → PROPER(X1)
PROPER(U54(X1, X2, X3)) → PROPER(X2)
PROPER(U54(X1, X2, X3)) → PROPER(X3)
PROPER(U55(X1, X2)) → U551(proper(X1), proper(X2))
PROPER(U55(X1, X2)) → PROPER(X1)
PROPER(U55(X1, X2)) → PROPER(X2)
PROPER(U56(X)) → U561(proper(X))
PROPER(U56(X)) → PROPER(X)
PROPER(U61(X1, X2)) → U611(proper(X1), proper(X2))
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → U621(proper(X1), proper(X2))
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → U631(proper(X))
PROPER(U63(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → U711(proper(X1), proper(X2), proper(X3))
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2)) → U721(proper(X1), proper(X2))
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(U73(X1, X2)) → U731(proper(X1), proper(X2))
PROPER(U73(X1, X2)) → PROPER(X1)
PROPER(U73(X1, X2)) → PROPER(X2)
PROPER(isPal(X)) → ISPAL(proper(X))
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → U741(proper(X))
PROPER(U74(X)) → PROPER(X)
PROPER(U81(X1, X2)) → U811(proper(X1), proper(X2))
PROPER(U81(X1, X2)) → PROPER(X1)
PROPER(U81(X1, X2)) → PROPER(X2)
PROPER(U82(X1, X2)) → U821(proper(X1), proper(X2))
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U83(X)) → U831(proper(X))
PROPER(U83(X)) → PROPER(X)
PROPER(isNePal(X)) → ISNEPAL(proper(X))
PROPER(isNePal(X)) → PROPER(X)
PROPER(U91(X1, X2)) → U911(proper(X1), proper(X2))
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(U92(X)) → U921(proper(X))
PROPER(U92(X)) → PROPER(X)
__1(ok(X1), ok(X2)) → __1(X1, X2)
U111(ok(X1), ok(X2)) → U111(X1, X2)
U121(ok(X1), ok(X2)) → U121(X1, X2)
ISPALLISTKIND(ok(X)) → ISPALLISTKIND(X)
U131(ok(X)) → U131(X)
ISNELIST(ok(X)) → ISNELIST(X)
U211(ok(X1), ok(X2), ok(X3)) → U211(X1, X2, X3)
U221(ok(X1), ok(X2), ok(X3)) → U221(X1, X2, X3)
U231(ok(X1), ok(X2), ok(X3)) → U231(X1, X2, X3)
U241(ok(X1), ok(X2), ok(X3)) → U241(X1, X2, X3)
U251(ok(X1), ok(X2)) → U251(X1, X2)
ISLIST(ok(X)) → ISLIST(X)
U261(ok(X)) → U261(X)
U311(ok(X1), ok(X2)) → U311(X1, X2)
U321(ok(X1), ok(X2)) → U321(X1, X2)
U331(ok(X)) → U331(X)
ISQID(ok(X)) → ISQID(X)
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U421(ok(X1), ok(X2), ok(X3)) → U421(X1, X2, X3)
U431(ok(X1), ok(X2), ok(X3)) → U431(X1, X2, X3)
U441(ok(X1), ok(X2), ok(X3)) → U441(X1, X2, X3)
U451(ok(X1), ok(X2)) → U451(X1, X2)
U461(ok(X)) → U461(X)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)
U531(ok(X1), ok(X2), ok(X3)) → U531(X1, X2, X3)
U541(ok(X1), ok(X2), ok(X3)) → U541(X1, X2, X3)
U551(ok(X1), ok(X2)) → U551(X1, X2)
U561(ok(X)) → U561(X)
U611(ok(X1), ok(X2)) → U611(X1, X2)
U621(ok(X1), ok(X2)) → U621(X1, X2)
U631(ok(X)) → U631(X)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
U721(ok(X1), ok(X2)) → U721(X1, X2)
U731(ok(X1), ok(X2)) → U731(X1, X2)
ISPAL(ok(X)) → ISPAL(X)
U741(ok(X)) → U741(X)
U811(ok(X1), ok(X2)) → U811(X1, X2)
U821(ok(X1), ok(X2)) → U821(X1, X2)
U831(ok(X)) → U831(X)
ISNEPAL(ok(X)) → ISNEPAL(X)
U911(ok(X1), ok(X2)) → U911(X1, X2)
U921(ok(X)) → U921(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 46 SCCs with 156 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNEPAL(ok(X)) → ISNEPAL(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNEPAL(ok(X)) → ISNEPAL(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNEPAL1

Status:
ok1: multiset
ISNEPAL1: multiset


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISPAL(ok(X)) → ISPAL(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISPAL(ok(X)) → ISPAL(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISPAL1

Status:
ISPAL1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISQID(ok(X)) → ISQID(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISQID(ok(X)) → ISQID(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISQID1

Status:
ISQID1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(18) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(19) TRUE

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISLIST(ok(X)) → ISLIST(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISLIST(ok(X)) → ISLIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISLIST1

Status:
ok1: multiset
ISLIST1: multiset


The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) TRUE

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNELIST(ok(X)) → ISNELIST(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNELIST(ok(X)) → ISNELIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNELIST1

Status:
ISNELIST1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISPALLISTKIND(ok(X)) → ISPALLISTKIND(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISPALLISTKIND(ok(X)) → ISPALLISTKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISPALLISTKIND1

Status:
ISPALLISTKIND1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(33) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(34) TRUE

(35) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U921(ok(X)) → U921(X)
U921(mark(X)) → U921(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(ok(X)) → U921(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U921(x1)  =  U921(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U92^11, ok1]

Status:
ok1: multiset
U92^11: multiset


The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U921(mark(X)) → U921(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(mark(X)) → U921(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U92^11

Status:
mark1: multiset
U92^11: multiset


The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(40) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(41) TRUE

(42) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U911(ok(X1), ok(X2)) → U911(X1, X2)
U911(mark(X1), X2) → U911(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(ok(X1), ok(X2)) → U911(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U911(x1, x2)  =  U911(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U91^11
mark > U91^11

Status:
U91^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U911(mark(X1), X2) → U911(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(mark(X1), X2) → U911(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U911(x1, x2)  =  U911(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U91^11, mark1]

Status:
U91^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(47) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(48) TRUE

(49) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U831(ok(X)) → U831(X)
U831(mark(X)) → U831(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U831(ok(X)) → U831(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U831(x1)  =  U831(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U83^11, ok1]

Status:
U83^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(51) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U831(mark(X)) → U831(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U831(mark(X)) → U831(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U83^11

Status:
U83^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(54) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(55) TRUE

(56) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U821(ok(X1), ok(X2)) → U821(X1, X2)
U821(mark(X1), X2) → U821(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U821(ok(X1), ok(X2)) → U821(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U821(x1, x2)  =  U821(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U82^11
mark > U82^11

Status:
mark: multiset
ok1: multiset
U82^11: multiset


The following usable rules [FROCOS05] were oriented: none

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U821(mark(X1), X2) → U821(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U821(mark(X1), X2) → U821(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U821(x1, x2)  =  U821(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U82^11, mark1]

Status:
mark1: [1]
U82^11: multiset


The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(61) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(62) TRUE

(63) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U811(ok(X1), ok(X2)) → U811(X1, X2)
U811(mark(X1), X2) → U811(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U811(ok(X1), ok(X2)) → U811(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U811(x1, x2)  =  U811(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U81^11
mark > U81^11

Status:
U81^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(65) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U811(mark(X1), X2) → U811(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U811(mark(X1), X2) → U811(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U811(x1, x2)  =  U811(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U81^11, mark1]

Status:
U81^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(68) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(69) TRUE

(70) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U741(ok(X)) → U741(X)
U741(mark(X)) → U741(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U741(ok(X)) → U741(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U741(x1)  =  U741(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U74^11, ok1]

Status:
ok1: multiset
U74^11: multiset


The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U741(mark(X)) → U741(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U741(mark(X)) → U741(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U74^11

Status:
mark1: multiset
U74^11: multiset


The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(75) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(76) TRUE

(77) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U731(ok(X1), ok(X2)) → U731(X1, X2)
U731(mark(X1), X2) → U731(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U731(ok(X1), ok(X2)) → U731(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U731(x1, x2)  =  U731(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U73^11
mark > U73^11

Status:
U73^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U731(mark(X1), X2) → U731(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U731(mark(X1), X2) → U731(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U731(x1, x2)  =  U731(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U73^11, mark1]

Status:
U73^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(82) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(83) TRUE

(84) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U721(ok(X1), ok(X2)) → U721(X1, X2)
U721(mark(X1), X2) → U721(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(ok(X1), ok(X2)) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  U721(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U72^11
mark > U72^11

Status:
mark: multiset
ok1: multiset
U72^11: multiset


The following usable rules [FROCOS05] were oriented: none

(86) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U721(mark(X1), X2) → U721(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(mark(X1), X2) → U721(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U721(x1, x2)  =  U721(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U72^11, mark1]

Status:
mark1: [1]
U72^11: multiset


The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(89) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(90) TRUE

(91) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U71^12, ok1]

Status:
U71^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U711(mark(X1), X2, X3) → U711(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(mark(X1), X2, X3) → U711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U71^11, mark1]

Status:
U71^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(96) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(97) TRUE

(98) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U631(ok(X)) → U631(X)
U631(mark(X)) → U631(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(ok(X)) → U631(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U631(x1)  =  U631(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U63^11, ok1]

Status:
U63^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U631(mark(X)) → U631(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(mark(X)) → U631(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U63^11

Status:
U63^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(103) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(104) TRUE

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U621(ok(X1), ok(X2)) → U621(X1, X2)
U621(mark(X1), X2) → U621(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(ok(X1), ok(X2)) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  U621(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U62^11
mark > U62^11

Status:
mark: multiset
ok1: multiset
U62^11: multiset


The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U621(mark(X1), X2) → U621(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(mark(X1), X2) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  U621(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U62^11, mark1]

Status:
mark1: [1]
U62^11: multiset


The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(110) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(111) TRUE

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U611(ok(X1), ok(X2)) → U611(X1, X2)
U611(mark(X1), X2) → U611(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(ok(X1), ok(X2)) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2)  =  U611(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U61^11
mark > U61^11

Status:
U61^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U611(mark(X1), X2) → U611(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2)  =  U611(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U61^11, mark1]

Status:
U61^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(117) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(118) TRUE

(119) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U561(ok(X)) → U561(X)
U561(mark(X)) → U561(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U561(ok(X)) → U561(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U561(x1)  =  U561(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U56^11, ok1]

Status:
U56^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U561(mark(X)) → U561(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U561(mark(X)) → U561(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U56^11

Status:
U56^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(124) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(125) TRUE

(126) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U551(ok(X1), ok(X2)) → U551(X1, X2)
U551(mark(X1), X2) → U551(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U551(ok(X1), ok(X2)) → U551(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U551(x1, x2)  =  U551(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U55^11
mark > U55^11

Status:
mark: multiset
U55^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U551(mark(X1), X2) → U551(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U551(mark(X1), X2) → U551(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U551(x1, x2)  =  U551(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U55^11, mark1]

Status:
mark1: [1]
U55^11: multiset


The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(131) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(132) TRUE

(133) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U541(ok(X1), ok(X2), ok(X3)) → U541(X1, X2, X3)
U541(mark(X1), X2, X3) → U541(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(134) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U541(ok(X1), ok(X2), ok(X3)) → U541(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U541(x1, x2, x3)  =  U541(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U54^12, ok1]

Status:
ok1: multiset
U54^12: multiset


The following usable rules [FROCOS05] were oriented: none

(135) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U541(mark(X1), X2, X3) → U541(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U541(mark(X1), X2, X3) → U541(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U541(x1, x2, x3)  =  U541(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U54^11, mark1]

Status:
mark1: multiset
U54^11: multiset


The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(138) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(139) TRUE

(140) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U531(ok(X1), ok(X2), ok(X3)) → U531(X1, X2, X3)
U531(mark(X1), X2, X3) → U531(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U531(ok(X1), ok(X2), ok(X3)) → U531(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U531(x1, x2, x3)  =  U531(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U53^12, ok1]

Status:
U53^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(142) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U531(mark(X1), X2, X3) → U531(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U531(mark(X1), X2, X3) → U531(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U531(x1, x2, x3)  =  U531(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U53^11, mark1]

Status:
mark1: multiset
U53^11: multiset


The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(145) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(146) TRUE

(147) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)
U521(mark(X1), X2, X3) → U521(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(148) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2, x3)  =  U521(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U52^12, ok1]

Status:
U52^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(149) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U521(mark(X1), X2, X3) → U521(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(150) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(mark(X1), X2, X3) → U521(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2, x3)  =  U521(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U52^11, mark1]

Status:
U52^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(151) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(152) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(153) TRUE

(154) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^12, ok1]

Status:
U51^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(156) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U511(mark(X1), X2, X3) → U511(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2, X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^11, mark1]

Status:
mark1: multiset
U51^11: multiset


The following usable rules [FROCOS05] were oriented: none

(158) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(159) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(160) TRUE

(161) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U461(ok(X)) → U461(X)
U461(mark(X)) → U461(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(162) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U461(ok(X)) → U461(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U461(x1)  =  U461(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U46^11, ok1]

Status:
U46^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U461(mark(X)) → U461(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U461(mark(X)) → U461(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U46^11

Status:
U46^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(166) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(167) TRUE

(168) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U451(ok(X1), ok(X2)) → U451(X1, X2)
U451(mark(X1), X2) → U451(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(169) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U451(ok(X1), ok(X2)) → U451(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U451(x1, x2)  =  U451(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U45^11
mark > U45^11

Status:
U45^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(170) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U451(mark(X1), X2) → U451(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U451(mark(X1), X2) → U451(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U451(x1, x2)  =  U451(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U45^11, mark1]

Status:
U45^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(173) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(174) TRUE

(175) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U441(ok(X1), ok(X2), ok(X3)) → U441(X1, X2, X3)
U441(mark(X1), X2, X3) → U441(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(176) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U441(ok(X1), ok(X2), ok(X3)) → U441(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U441(x1, x2, x3)  =  U441(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U44^12, ok1]

Status:
ok1: multiset
U44^12: multiset


The following usable rules [FROCOS05] were oriented: none

(177) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U441(mark(X1), X2, X3) → U441(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U441(mark(X1), X2, X3) → U441(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U441(x1, x2, x3)  =  U441(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U44^11, mark1]

Status:
mark1: multiset
U44^11: multiset


The following usable rules [FROCOS05] were oriented: none

(179) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(180) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(181) TRUE

(182) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U431(ok(X1), ok(X2), ok(X3)) → U431(X1, X2, X3)
U431(mark(X1), X2, X3) → U431(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(183) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U431(ok(X1), ok(X2), ok(X3)) → U431(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U431(x1, x2, x3)  =  U431(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U43^12, ok1]

Status:
U43^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(184) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U431(mark(X1), X2, X3) → U431(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(185) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U431(mark(X1), X2, X3) → U431(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U431(x1, x2, x3)  =  U431(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U43^11, mark1]

Status:
mark1: multiset
U43^11: multiset


The following usable rules [FROCOS05] were oriented: none

(186) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(187) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(188) TRUE

(189) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U421(ok(X1), ok(X2), ok(X3)) → U421(X1, X2, X3)
U421(mark(X1), X2, X3) → U421(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U421(ok(X1), ok(X2), ok(X3)) → U421(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U421(x1, x2, x3)  =  U421(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U42^12, ok1]

Status:
U42^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(191) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U421(mark(X1), X2, X3) → U421(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U421(mark(X1), X2, X3) → U421(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U421(x1, x2, x3)  =  U421(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U42^11, mark1]

Status:
mark1: multiset
U42^11: multiset


The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(194) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(195) TRUE

(196) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(197) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^12, ok1]

Status:
U41^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(198) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(mark(X1), X2, X3) → U411(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(mark(X1), X2, X3) → U411(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^11, mark1]

Status:
U41^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(201) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(202) TRUE

(203) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U331(ok(X)) → U331(X)
U331(mark(X)) → U331(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(204) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U331(ok(X)) → U331(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U331(x1)  =  U331(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U33^11, ok1]

Status:
U33^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(205) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U331(mark(X)) → U331(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(206) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U331(mark(X)) → U331(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U33^11

Status:
U33^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(207) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(208) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(209) TRUE

(210) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(ok(X1), ok(X2)) → U321(X1, X2)
U321(mark(X1), X2) → U321(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(211) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(ok(X1), ok(X2)) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U32^11
mark > U32^11

Status:
mark: multiset
U32^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(212) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U321(mark(X1), X2) → U321(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(213) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(mark(X1), X2) → U321(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U32^11, mark1]

Status:
mark1: [1]
U32^11: multiset


The following usable rules [FROCOS05] were oriented: none

(214) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(215) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(216) TRUE

(217) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(ok(X1), ok(X2)) → U311(X1, X2)
U311(mark(X1), X2) → U311(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(218) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(ok(X1), ok(X2)) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  U311(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U31^11
mark > U31^11

Status:
U31^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(219) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(mark(X1), X2) → U311(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(220) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(mark(X1), X2) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  U311(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U31^11, mark1]

Status:
U31^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(221) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(222) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(223) TRUE

(224) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U261(ok(X)) → U261(X)
U261(mark(X)) → U261(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(225) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U261(ok(X)) → U261(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U261(x1)  =  U261(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U26^11, ok1]

Status:
ok1: multiset
U26^11: multiset


The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U261(mark(X)) → U261(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(227) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U261(mark(X)) → U261(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U26^11

Status:
mark1: multiset
U26^11: multiset


The following usable rules [FROCOS05] were oriented: none

(228) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(229) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(230) TRUE

(231) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U251(ok(X1), ok(X2)) → U251(X1, X2)
U251(mark(X1), X2) → U251(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(232) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U251(ok(X1), ok(X2)) → U251(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U251(x1, x2)  =  U251(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U25^11
mark > U25^11

Status:
U25^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(233) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U251(mark(X1), X2) → U251(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U251(mark(X1), X2) → U251(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U251(x1, x2)  =  U251(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U25^11, mark1]

Status:
U25^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(236) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(237) TRUE

(238) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U241(ok(X1), ok(X2), ok(X3)) → U241(X1, X2, X3)
U241(mark(X1), X2, X3) → U241(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(239) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U241(ok(X1), ok(X2), ok(X3)) → U241(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U241(x1, x2, x3)  =  U241(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U24^12, ok1]

Status:
U24^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(240) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U241(mark(X1), X2, X3) → U241(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(241) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U241(mark(X1), X2, X3) → U241(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U241(x1, x2, x3)  =  U241(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U24^11, mark1]

Status:
U24^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(243) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(244) TRUE

(245) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U231(ok(X1), ok(X2), ok(X3)) → U231(X1, X2, X3)
U231(mark(X1), X2, X3) → U231(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(246) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U231(ok(X1), ok(X2), ok(X3)) → U231(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U231(x1, x2, x3)  =  U231(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U23^12, ok1]

Status:
U23^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(247) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U231(mark(X1), X2, X3) → U231(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(248) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U231(mark(X1), X2, X3) → U231(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U231(x1, x2, x3)  =  U231(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U23^11, mark1]

Status:
mark1: multiset
U23^11: multiset


The following usable rules [FROCOS05] were oriented: none

(249) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(250) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(251) TRUE

(252) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U221(ok(X1), ok(X2), ok(X3)) → U221(X1, X2, X3)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(253) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(ok(X1), ok(X2), ok(X3)) → U221(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1, x2, x3)  =  U221(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U22^12, ok1]

Status:
U22^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(254) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U221(mark(X1), X2, X3) → U221(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(255) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(mark(X1), X2, X3) → U221(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1, x2, x3)  =  U221(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U22^11, mark1]

Status:
U22^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(256) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(257) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(258) TRUE

(259) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(ok(X1), ok(X2), ok(X3)) → U211(X1, X2, X3)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(260) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(ok(X1), ok(X2), ok(X3)) → U211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2, x3)  =  U211(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U21^12, ok1]

Status:
U21^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(261) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U211(mark(X1), X2, X3) → U211(X1, X2, X3)

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.
We have to consider all minimal (P,Q,R)-chains.

(262) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X1), X2, X3) → U211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2, x3)  =  U211(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U21^11, mark1]

Status:
mark1: multiset
U21^11: multiset


The following usable rules [FROCOS05] were oriented: none

(263) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(264) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(265) TRUE

(266) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U131(ok(X)) → U131(X)
U131(mark(X)) → U131(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(ok(X)) → U131(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131(x1)  =  U131(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U13^11, ok1]

Status:
ok1: multiset
U13^11: multiset


The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U131(mark(X)) → U131(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(mark(X)) → U131(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U13^11

Status:
mark1: multiset
U13^11: multiset


The following usable rules [FROCOS05] were oriented: none

(270) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(271) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(272) TRUE

(273) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(ok(X1), ok(X2)) → U121(X1, X2)
U121(mark(X1), X2) → U121(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(274) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(ok(X1), ok(X2)) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U12^11
mark > U12^11

Status:
U12^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(275) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(mark(X1), X2) → U121(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(mark(X1), X2) → U121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U12^11, mark1]

Status:
U12^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(277) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(278) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(279) TRUE

(280) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(ok(X1), ok(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(281) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(ok(X1), ok(X2)) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U11^11
mark > U11^11

Status:
U11^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(282) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(mark(X1), X2) → U111(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(283) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(X1), X2) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U11^11, mark1]

Status:
U11^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(284) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(285) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(286) TRUE

(287) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__1(X1, mark(X2)) → __1(X1, X2)
__1(mark(X1), X2) → __1(X1, X2)
__1(ok(X1), ok(X2)) → __1(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(288) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


__1(X1, mark(X2)) → __1(X1, X2)
__1(mark(X1), X2) → __1(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__1(x1, x2)  =  __1(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
mark1: multiset
_^12: [2,1]


The following usable rules [FROCOS05] were oriented: none

(289) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__1(ok(X1), ok(X2)) → __1(X1, X2)

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.
We have to consider all minimal (P,Q,R)-chains.

(290) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


__1(ok(X1), ok(X2)) → __1(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__1(x1, x2)  =  __1(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[^11, ok1]

Status:
ok1: [1]
_^11: multiset


The following usable rules [FROCOS05] were oriented: none

(291) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(292) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(293) TRUE

(294) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(U21(X1, X2, X3)) → PROPER(X1)
PROPER(U21(X1, X2, X3)) → PROPER(X2)
PROPER(U21(X1, X2, X3)) → PROPER(X3)
PROPER(U22(X1, X2, X3)) → PROPER(X1)
PROPER(U22(X1, X2, X3)) → PROPER(X2)
PROPER(U22(X1, X2, X3)) → PROPER(X3)
PROPER(U23(X1, X2, X3)) → PROPER(X1)
PROPER(U23(X1, X2, X3)) → PROPER(X2)
PROPER(U23(X1, X2, X3)) → PROPER(X3)
PROPER(U24(X1, X2, X3)) → PROPER(X1)
PROPER(U24(X1, X2, X3)) → PROPER(X2)
PROPER(U24(X1, X2, X3)) → PROPER(X3)
PROPER(U25(X1, X2)) → PROPER(X1)
PROPER(U25(X1, X2)) → PROPER(X2)
PROPER(isList(X)) → PROPER(X)
PROPER(U26(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U33(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X1, X2, X3)) → PROPER(X1)
PROPER(U43(X1, X2, X3)) → PROPER(X2)
PROPER(U43(X1, X2, X3)) → PROPER(X3)
PROPER(U44(X1, X2, X3)) → PROPER(X1)
PROPER(U44(X1, X2, X3)) → PROPER(X2)
PROPER(U44(X1, X2, X3)) → PROPER(X3)
PROPER(U45(X1, X2)) → PROPER(X1)
PROPER(U45(X1, X2)) → PROPER(X2)
PROPER(U46(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(U53(X1, X2, X3)) → PROPER(X1)
PROPER(U53(X1, X2, X3)) → PROPER(X2)
PROPER(U53(X1, X2, X3)) → PROPER(X3)
PROPER(U54(X1, X2, X3)) → PROPER(X1)
PROPER(U54(X1, X2, X3)) → PROPER(X2)
PROPER(U54(X1, X2, X3)) → PROPER(X3)
PROPER(U55(X1, X2)) → PROPER(X1)
PROPER(U55(X1, X2)) → PROPER(X2)
PROPER(U56(X)) → PROPER(X)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(U73(X1, X2)) → PROPER(X1)
PROPER(U73(X1, X2)) → PROPER(X2)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
PROPER(U81(X1, X2)) → PROPER(X1)
PROPER(U81(X1, X2)) → PROPER(X2)
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U83(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(U92(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(295) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(U21(X1, X2, X3)) → PROPER(X1)
PROPER(U21(X1, X2, X3)) → PROPER(X2)
PROPER(U21(X1, X2, X3)) → PROPER(X3)
PROPER(U22(X1, X2, X3)) → PROPER(X1)
PROPER(U22(X1, X2, X3)) → PROPER(X2)
PROPER(U22(X1, X2, X3)) → PROPER(X3)
PROPER(U23(X1, X2, X3)) → PROPER(X1)
PROPER(U23(X1, X2, X3)) → PROPER(X2)
PROPER(U23(X1, X2, X3)) → PROPER(X3)
PROPER(U24(X1, X2, X3)) → PROPER(X1)
PROPER(U24(X1, X2, X3)) → PROPER(X2)
PROPER(U24(X1, X2, X3)) → PROPER(X3)
PROPER(U25(X1, X2)) → PROPER(X1)
PROPER(U25(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X1, X2, X3)) → PROPER(X1)
PROPER(U43(X1, X2, X3)) → PROPER(X2)
PROPER(U43(X1, X2, X3)) → PROPER(X3)
PROPER(U44(X1, X2, X3)) → PROPER(X1)
PROPER(U44(X1, X2, X3)) → PROPER(X2)
PROPER(U44(X1, X2, X3)) → PROPER(X3)
PROPER(U45(X1, X2)) → PROPER(X1)
PROPER(U45(X1, X2)) → PROPER(X2)
PROPER(U46(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(U53(X1, X2, X3)) → PROPER(X1)
PROPER(U53(X1, X2, X3)) → PROPER(X2)
PROPER(U53(X1, X2, X3)) → PROPER(X3)
PROPER(U54(X1, X2, X3)) → PROPER(X1)
PROPER(U54(X1, X2, X3)) → PROPER(X2)
PROPER(U54(X1, X2, X3)) → PROPER(X3)
PROPER(U55(X1, X2)) → PROPER(X1)
PROPER(U55(X1, X2)) → PROPER(X2)
PROPER(U56(X)) → PROPER(X)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(U73(X1, X2)) → PROPER(X1)
PROPER(U73(X1, X2)) → PROPER(X2)
PROPER(U74(X)) → PROPER(X)
PROPER(U81(X1, X2)) → PROPER(X1)
PROPER(U81(X1, X2)) → PROPER(X2)
PROPER(U82(X1, X2)) → PROPER(X1)
PROPER(U82(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(U92(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
__(x1, x2)  =  __(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U12(x1, x2)  =  U12(x1, x2)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U23(x1, x2, x3)  =  U23(x1, x2, x3)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U25(x1, x2)  =  U25(x1, x2)
isList(x1)  =  x1
U26(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  x1
isQid(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U43(x1, x2, x3)  =  U43(x1, x2, x3)
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U45(x1, x2)  =  U45(x1, x2)
U46(x1)  =  U46(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U53(x1, x2, x3)  =  U53(x1, x2, x3)
U54(x1, x2, x3)  =  U54(x1, x2, x3)
U55(x1, x2)  =  U55(x1, x2)
U56(x1)  =  U56(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
U73(x1, x2)  =  U73(x1, x2)
isPal(x1)  =  x1
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
U82(x1, x2)  =  U82(x1, x2)
U83(x1)  =  x1
isNePal(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
U461 > [PROPER1, U122, U223, U243, U252, U452, U543, U921]
U561 > [PROPER1, U122, U223, U243, U252, U452, U543, U921]
U741 > [PROPER1, U122, U223, U243, U252, U452, U543, U921]

Status:
U912: [1,2]
U622: multiset
_2: multiset
U413: multiset
U252: multiset
U322: [1,2]
U243: multiset
U122: multiset
U433: [3,2,1]
U561: [1]
U223: multiset
U423: multiset
U921: [1]
U452: multiset
U213: [3,2,1]
U513: multiset
U543: multiset
PROPER1: [1]
U552: multiset
U732: multiset
U312: multiset
U722: multiset
U112: multiset
U533: multiset
U523: multiset
U233: multiset
U822: multiset
U443: multiset
U612: multiset
U812: multiset
U461: multiset
U741: multiset
U713: multiset


The following usable rules [FROCOS05] were oriented: none

(296) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U26(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U83(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(297) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNePal(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  x1
U26(x1)  =  x1
U33(x1)  =  x1
isQid(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U83(x1)  =  x1
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isNePal1]

Status:
PROPER1: multiset
isNePal1: multiset


The following usable rules [FROCOS05] were oriented: none

(298) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U26(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U83(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(299) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U83(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isPalListKind(x1)  =  isPalListKind(x1)
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  x1
U26(x1)  =  x1
U33(x1)  =  x1
isQid(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U83(x1)  =  U83(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
isPalListKind1: multiset
U831: multiset


The following usable rules [FROCOS05] were oriented: none

(300) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U26(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(301) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U26(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  x1
U26(x1)  =  U26(x1)
U33(x1)  =  U33(x1)
isQid(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U331 > [PROPER1, U261]

Status:
U261: multiset
PROPER1: multiset
U331: multiset


The following usable rules [FROCOS05] were oriented: none

(302) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(303) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isPal(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  x1
isQid(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  isPal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
isPal1: multiset
PROPER1: multiset


The following usable rules [FROCOS05] were oriented: none

(304) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(U13(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(305) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U13(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U13(x1)  =  U13(x1)
isNeList(x1)  =  x1
isList(x1)  =  x1
isQid(x1)  =  x1
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, U131]

Status:
PROPER1: multiset
U131: multiset


The following usable rules [FROCOS05] were oriented: none

(306) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(307) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isQid(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNeList(x1)  =  x1
isList(x1)  =  x1
isQid(x1)  =  isQid(x1)
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isQid1]

Status:
PROPER1: multiset
isQid1: multiset


The following usable rules [FROCOS05] were oriented: none

(308) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(isNeList(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(309) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNeList(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNeList(x1)  =  isNeList(x1)
isList(x1)  =  x1
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isNeList1]

Status:
PROPER1: multiset
isNeList1: multiset


The following usable rules [FROCOS05] were oriented: none

(310) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(isList(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(311) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isList(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isList(x1)  =  isList(x1)
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isList1]

Status:
PROPER1: multiset
isList1: multiset


The following usable rules [FROCOS05] were oriented: none

(312) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(U63(X)) → PROPER(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(313) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U63(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
U631 > PROPER1

Status:
PROPER1: multiset
U631: multiset


The following usable rules [FROCOS05] were oriented: none

(314) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(315) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(316) TRUE

(317) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U25(X1, X2)) → ACTIVE(X1)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U43(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U53(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U55(X1, X2)) → ACTIVE(X1)
ACTIVE(U56(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U73(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(U92(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(318) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U92(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
__(x1, x2)  =  __(x1, x2)
U11(x1, x2)  =  x1
U12(x1, x2)  =  x1
U13(x1)  =  x1
U21(x1, x2, x3)  =  x1
U22(x1, x2, x3)  =  x1
U23(x1, x2, x3)  =  x1
U24(x1, x2, x3)  =  x1
U25(x1, x2)  =  x1
U26(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1, x2)  =  x1
U33(x1)  =  x1
U41(x1, x2, x3)  =  x1
U42(x1, x2, x3)  =  U42(x1, x2)
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U51(x1, x2, x3)  =  x1
U52(x1, x2, x3)  =  x1
U53(x1, x2, x3)  =  x1
U54(x1, x2, x3)  =  x1
U55(x1, x2)  =  x1
U56(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1, x3)
U72(x1, x2)  =  x1
U73(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
U712 > ACTIVE1

Status:
_2: multiset
U422: multiset
U921: multiset
ACTIVE1: [1]
U712: [1,2]


The following usable rules [FROCOS05] were oriented: none

(319) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U25(X1, X2)) → ACTIVE(X1)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U43(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U53(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U55(X1, X2)) → ACTIVE(X1)
ACTIVE(U56(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U73(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)

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.
We have to consider all minimal (P,Q,R)-chains.

(320) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U23(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U25(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U43(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U53(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U55(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U73(X1, X2)) → ACTIVE(X1)
ACTIVE(U82(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2)  =  x1
U12(x1, x2)  =  x1
U13(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U23(x1, x2, x3)  =  U23(x1, x2, x3)
U24(x1, x2, x3)  =  x1
U25(x1, x2)  =  U25(x1, x2)
U26(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1, x2)  =  U32(x1)
U33(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U43(x1, x2, x3)  =  U43(x1, x2, x3)
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U45(x1, x2)  =  U45(x1)
U46(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2, x3)  =  U52(x1, x3)
U53(x1, x2, x3)  =  U53(x1, x2, x3)
U54(x1, x2, x3)  =  U54(x1, x3)
U55(x1, x2)  =  U55(x1, x2)
U56(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  U62(x1)
U63(x1)  =  x1
U72(x1, x2)  =  U72(x1, x2)
U73(x1, x2)  =  U73(x1)
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  U82(x1, x2)
U83(x1)  =  x1
U91(x1, x2)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U213 > [ACTIVE1, U413, U443, U451, U533]
U433 > [ACTIVE1, U413, U443, U451, U533]
U621 > [ACTIVE1, U413, U443, U451, U533]

Status:
U522: multiset
U552: multiset
U312: multiset
U413: multiset
U252: multiset
U451: multiset
U722: multiset
U433: multiset
U533: multiset
ACTIVE1: [1]
U542: multiset
U621: multiset
U233: [3,2,1]
U822: multiset
U223: multiset
U443: multiset
U731: multiset
U321: multiset
U213: [3,1,2]
U513: multiset


The following usable rules [FROCOS05] were oriented: none

(321) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U56(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)

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.
We have to consider all minimal (P,Q,R)-chains.

(322) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2)  =  U11(x1, x2)
U12(x1, x2)  =  U12(x1, x2)
U13(x1)  =  x1
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U26(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  x1
U56(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U63(x1)  =  U63(x1)
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
U83(x1)  =  U83(x1)
U91(x1, x2)  =  U91(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
U112 > [ACTIVE1, U243, U631]
U122 > [ACTIVE1, U243, U631]
U612 > [ACTIVE1, U243, U631]
U812 > [ACTIVE1, U243, U631]

Status:
U912: multiset
U612: multiset
U812: multiset
U243: multiset
U122: multiset
U112: [2,1]
U631: multiset
ACTIVE1: multiset
U741: multiset
U831: multiset


The following usable rules [FROCOS05] were oriented: none

(323) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U56(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(324) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U13(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  U13(x1)
U26(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  x1
U56(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U131]

Status:
U131: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(325) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U56(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(326) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U46(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U26(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  U46(x1)
U56(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U461]

Status:
U461: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(327) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U56(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(328) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U26(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U26(x1)  =  U26(x1)
U33(x1)  =  x1
U56(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U261]

Status:
U261: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(329) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U56(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(330) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U33(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U33(x1)  =  U33(x1)
U56(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U331]

Status:
U331: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(331) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U56(X)) → ACTIVE(X)

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.
We have to consider all minimal (P,Q,R)-chains.

(332) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U56(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
U561 > ACTIVE1

Status:
U561: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(333) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(334) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(335) TRUE

(336) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

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.
We have to consider all minimal (P,Q,R)-chains.

(337) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TOP(mark(X)) → TOP(proper(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1)  =  TOP(x1)
ok(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)
proper(x1)  =  x1
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
U73(x1, x2)  =  U73(x1, x2)
isPal(x1)  =  isPal(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
U62(x1, x2)  =  U62(x1)
U63(x1)  =  U63(x1)
U56(x1)  =  U56(x1)
U61(x1, x2)  =  U61(x1, x2)
U54(x1, x2, x3)  =  U54(x1, x2, x3)
U55(x1, x2)  =  U55(x1, x2)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U53(x1, x2, x3)  =  U53(x1, x2, x3)
U46(x1)  =  U46(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
isNeList(x1)  =  isNeList(x1)
U13(x1)  =  U13(x1)
isPalListKind(x1)  =  isPalListKind(x1)
U12(x1, x2)  =  U12(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
__(x1, x2)  =  __(x1, x2)
u  =  u
o  =  o
i  =  i
e  =  e
a  =  a
U92(x1)  =  U92(x1)
U91(x1, x2)  =  U91(x1, x2)
isNePal(x1)  =  isNePal(x1)
U83(x1)  =  U83(x1)
U82(x1, x2)  =  U82(x1, x2)
nil  =  nil
tt  =  tt
U43(x1, x2, x3)  =  U43(x1, x2, x3)
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U45(x1, x2)  =  U45(x1, x2)
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U33(x1)  =  U33(x1)
U32(x1, x2)  =  U32(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
isQid(x1)  =  isQid
isList(x1)  =  isList(x1)
U25(x1, x2)  =  U25(x1, x2)
U31(x1, x2)  =  U31(x1, x2)
U26(x1)  =  U26(x1)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U23(x1, x2, x3)  =  U23(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > U831 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U812 > [isNePal1, U822] > U831 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U713 > [isPal1, U722] > U732 > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U513 > U523 > U533 > U543 > U552 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U912 > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U912 > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U912 > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U912 > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U912 > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U912 > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U413 > U423 > U433 > U443 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > U631 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U621 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > [isNeList1, U452] > [U612, isPalListKind1, U312] > U321 > isQid > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
_2 > U213 > U223 > U233 > U243 > [isList1, U252] > U112 > U122 > U131 > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
o > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
o > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
i > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
i > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
e > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
e > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]
nil > [mark1, U741, U561, U461, u, U921, U331, U261] > TOP1
nil > [mark1, U741, U561, U461, u, U921, U331, U261] > [a, tt]

Status:
i: multiset
U413: multiset
U252: multiset
U243: [1,3,2]
mark1: [1]
U122: [1,2]
U561: [1]
tt: multiset
U223: [2,3,1]
isPalListKind1: [1]
U213: [2,1,3]
nil: multiset
U543: [2,1,3]
U552: [1,2]
U312: [2,1]
e: multiset
isNePal1: [1]
o: multiset
U533: [2,3,1]
U621: [1]
U233: [1,3,2]
U443: [3,1,2]
U812: [1,2]
U321: [1]
U331: [1]
U741: [1]
U713: [3,2,1]
U912: [2,1]
_2: [1,2]
U433: [2,1,3]
U261: [1]
U423: [2,1,3]
U921: [1]
U452: multiset
TOP1: multiset
isQid: []
U513: [2,3,1]
U831: [1]
U732: [2,1]
a: multiset
isList1: multiset
U722: [2,1]
U112: [2,1]
U631: multiset
U523: [3,2,1]
isPal1: [1]
U822: [2,1]
U612: [2,1]
u: multiset
isNeList1: multiset
U461: [1]
U131: multiset


The following usable rules [FROCOS05] were oriented:

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

(338) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))

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.
We have to consider all minimal (P,Q,R)-chains.

(339) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TOP(ok(X)) → TOP(active(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  x1
isNeList(x1)  =  isNeList
U13(x1)  =  U13(x1)
isPalListKind(x1)  =  isPalListKind
U12(x1, x2)  =  U12(x1, x2)
U11(x1, x2)  =  x2
__(x1, x2)  =  __(x2)
U82(x1, x2)  =  x1
mark(x1)  =  mark
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
U72(x1, x2)  =  x2
U73(x1, x2)  =  x2
U74(x1)  =  x1
U81(x1, x2)  =  x1
isNePal(x1)  =  isNePal
U32(x1, x2)  =  x1
U33(x1)  =  U33(x1)
isQid(x1)  =  isQid(x1)
U41(x1, x2, x3)  =  x3
U42(x1, x2, x3)  =  x3
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  U44(x2, x3)
U45(x1, x2)  =  x1
U21(x1, x2, x3)  =  U21(x1)
U22(x1, x2, x3)  =  x1
U23(x1, x2, x3)  =  x1
U24(x1, x2, x3)  =  x2
U25(x1, x2)  =  x2
isList(x1)  =  x1
U26(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U63(x1)  =  x1
U62(x1, x2)  =  U62(x1, x2)
U71(x1, x2, x3)  =  x1
isPal(x1)  =  isPal
U51(x1, x2, x3)  =  x3
U46(x1)  =  x1
U53(x1, x2, x3)  =  U53(x2)
U52(x1, x2, x3)  =  U52(x2, x3)
U55(x1, x2)  =  U55(x1)
U54(x1, x2, x3)  =  x2
U61(x1, x2)  =  x2
U56(x1)  =  x1
nil  =  nil
tt  =  tt
a  =  a
o  =  o
u  =  u
i  =  i
e  =  e

Recursive path order with status [RPO].
Quasi-Precedence:
U122 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
_1 > U211 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U921 > tt > isNePal > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U921 > tt > U442 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U921 > tt > U622 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U921 > tt > U551 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U331 > tt > isNePal > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U331 > tt > U442 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U331 > tt > U622 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U331 > tt > U551 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
isQid1 > tt > isNePal > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
isQid1 > tt > U442 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
isQid1 > tt > U622 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
isQid1 > tt > U551 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U522 > isPalListKind > tt > isNePal > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U522 > isPalListKind > tt > U442 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U522 > isPalListKind > tt > U622 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
U522 > isPalListKind > tt > U551 > [ok1, U131, U311, U531] > [isNeList, mark, isPal, a]
nil > [isNeList, mark, isPal, a]
o > [isNeList, mark, isPal, a]
u > [isNeList, mark, isPal, a]
i > [isNeList, mark, isPal, a]
e > [isNeList, mark, isPal, a]

Status:
i: multiset
isPal: []
U522: [2,1]
U622: [2,1]
isPalListKind: multiset
isNeList: []
U122: [1,2]
U551: [1]
_1: [1]
tt: multiset
U921: multiset
isNePal: multiset
nil: multiset
a: multiset
e: multiset
ok1: [1]
U442: [2,1]
o: multiset
isQid1: multiset
U531: [1]
U311: [1]
u: multiset
U131: [1]
mark: []
U331: [1]
U211: [1]


The following usable rules [FROCOS05] were oriented:

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

(340) Obligation:

Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.

(341) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(342) TRUE