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

(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: Combined order from the following AFS and order.
ISNEPAL(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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: Combined order from the following AFS and order.
ISPAL(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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: Combined order from the following AFS and order.
ISQID(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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: Combined order from the following AFS and order.
ISLIST(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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: Combined order from the following AFS and order.
ISNELIST(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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: Combined order from the following AFS and order.
ISPALLISTKIND(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U92^1₁

Status:

mark₁: [1]
U92^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

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

U92¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U91^1₂

Status:

U91^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U83^1₁

Status:

U83^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(51) Obligation:

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

U83¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U82^1₂

Status:

ok₁: [1]
U82^1₂: [1,2]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U81^1₂

Status:

U81^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U74^1₁

Status:

mark₁: [1]
U74^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

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

U74¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U73^1₂

Status:

U73^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U72^1₂

Status:

ok₁: [1]
U72^1₂: [1,2]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U71^1₂

Status:

U71^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U71^1₂

Status:

U71^1₂: [2,1]
mark₁: [1]

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U63^1₁

Status:

U63^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

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

U63¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U62^1₂

Status:

ok₁: [1]
U62^1₂: [1,2]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U61^1₂

Status:

U61^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U56^1₁

Status:

U56^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

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

U56¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U55^1₂

Status:

U55^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U54^1₂

Status:

ok₁: [1]
U54^1₂: [1,2]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U54^1₂

Status:

mark₁: [1]
U54^1₂: [2,1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U53^1₂

Status:

U53^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U53^1₂

Status:

U53^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U52^1₂

Status:

U52^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U52^1₂

Status:

U52^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U51^1₂

Status:

U51^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U51^1₂

Status:

U51^1₂: [2,1]
mark₁: [1]

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U46^1₁

Status:

U46^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

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

U46¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U45^1₂

Status:

U45^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U44^1₂

Status:

ok₁: [1]
U44^1₂: [1,2]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U44^1₂

Status:

mark₁: [1]
U44^1₂: [2,1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U43^1₂

Status:

U43^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U43^1₂

Status:

U43^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U42^1₂

Status:

U42^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U42^1₂

Status:

U42^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U41^1₂

Status:

U41^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U41^1₂

Status:

U41^1₂: [2,1]
mark₁: [1]

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U33^1₁

Status:

U33^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(205) Obligation:

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

U33¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U32^1₂

Status:

U32^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U31^1₂

Status:

U31^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U26^1₁

Status:

mark₁: [1]
U26^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

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

U26¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U25^1₂

Status:

ok₁: [1]
U25^1₂: [1,2]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U24^1₂

Status:

U24^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U24^1₂

Status:

U24^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U23^1₂

Status:

U23^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U23^1₂

Status:

U23^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U22^1₂

Status:

U22^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U22^1₂

Status:

U22^1₂: [2,1]
mark₁: [1]

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(x1, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U21^1₂

Status:

U21^1₂: [1,2]
ok₁: [1]

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, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U21^1₂

Status:

U21^1₂: [2,1]
mark₁: [1]

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(mark(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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U13^1₁

Status:

mark₁: [1]
U13^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

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

U13¹(ok(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(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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U12^1₂

Status:

U12^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

ok₁ > U11^1₂

Status:

U11^1₂: [1,2]
ok₁: [1]

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

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(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)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
__^11: [1]

The following usable rules [FROCOS05] were oriented: none

(289) Obligation:

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

__¹(X1, mark(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.

(290) 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)

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

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
__^12: [1,2]

The following usable rules [FROCOS05] were oriented: none

(291) 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.

(292) 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)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(293) 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.

(294) PisEmptyProof (EQUIVALENT transformation)

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

(295) TRUE

(296) 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.

(297) 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(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(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(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(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)

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)  =  U26(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)  =  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)  =  x1
U81(x1, x2)  =  U81(x1, x2)
U82(x1, x2)  =  U82(x1, x2)
U83(x1)  =  U83(x1)
isNePal(x1)  =  isNePal(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

U21₃ > PROPER₁
U31₂ > PROPER₁
U55₂ > PROPER₁
isNePal₁ > PROPER₁

Status:

U91₂: [1,2]
U62₂: [1,2]
_₂: [1,2]
U41₃: [1,2,3]
U25₂: [1,2]
U32₂: [1,2]
U24₃: [1,2,3]
U12₂: [1,2]
U43₃: [1,2,3]
U26₁: [1]
U22₃: [1,2,3]
U42₃: [1,2,3]
U45₂: [1,2]
U21₃: [1,2,3]
U51₃: [1,2,3]
U54₃: [1,2,3]
U83₁: [1]
PROPER₁: [1]
U55₂: [1,2]
U73₂: [1,2]
U31₂: [1,2]
U72₂: [1,2]
isNePal₁: [1]
U11₂: [1,2]
U53₃: [1,2,3]
U52₃: [1,2,3]
U23₃: [1,2,3]
U82₂: [1,2]
U44₃: [1,2,3]
U61₂: [1,2]
U81₂: [1,2]
U46₁: [1]
U71₃: [1,2,3]

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(U33(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(U56(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
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.

(299) 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)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  x1
U33(x1)  =  x1
isQid(x1)  =  isQid(x1)
U56(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U74(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROPER₁: [1]
isQid₁: [1]

The following usable rules [FROCOS05] were oriented: none

(300) 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(U33(X)) → PROPER(X)
PROPER(U56(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
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.

(301) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U56(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
U33(x1)  =  x1
U56(x1)  =  U56(x1)
U63(x1)  =  x1
isPal(x1)  =  x1
U74(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROPER₁: [1]
U56₁: [1]

The following usable rules [FROCOS05] were oriented: none

(302) 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(U33(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
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.

(303) 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)  =  x1
isPalListKind(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  x1
isList(x1)  =  isList(x1)
U33(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U74(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

isList₁: [1]

The following usable rules [FROCOS05] were oriented: none

(304) 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(U33(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
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.

(305) 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)  =  x1
isPalListKind(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  isNeList(x1)
U33(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U74(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

isNeList₁: [1]

The following usable rules [FROCOS05] were oriented: none

(306) Obligation:

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

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U74(X)) → PROPER(X)
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.

(307) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U74(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)  =  x1
U13(x1)  =  x1
U33(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  x1
U74(x1)  =  U74(x1)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U74₁: [1]

The following usable rules [FROCOS05] were oriented: none

(308) Obligation:

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

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(U33(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
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.

(309) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
U33(x1)  =  U33(x1)
U63(x1)  =  x1
isPal(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROPER₁: [1]
U33₁: [1]

The following usable rules [FROCOS05] were oriented: none

(310) Obligation:

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

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
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.

(311) 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)
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)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
U63(x1)  =  x1
isPal(x1)  =  isPal(x1)
U92(x1)  =  U92(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

PROPER₁: [1]
isPal₁: [1]
U92₁: [1]

The following usable rules [FROCOS05] were oriented: none

(312) Obligation:

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

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(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.

(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: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isPalListKind(x1)  =  x1
U13(x1)  =  x1
U63(x1)  =  U63(x1)

Lexicographic path order with status [LPO].
Precedence:

U63₁ > PROPER₁

Status:

PROPER₁: [1]
U63₁: [1]

The following usable rules [FROCOS05] were oriented: none

(314) Obligation:

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

PROPER(isPalListKind(X)) → PROPER(X)
PROPER(U13(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.

(315) 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)
isPalListKind(x1)  =  x1
U13(x1)  =  U13(x1)

Lexicographic path order with status [LPO].
Precedence:

U13₁ > PROPER₁

Status:

PROPER₁: [1]
U13₁: [1]

The following usable rules [FROCOS05] were oriented: none

(316) Obligation:

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

PROPER(isPalListKind(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.

(317) 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)

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)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

isPalListKind₁: [1]

The following usable rules [FROCOS05] were oriented: none

(318) 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.

(319) PisEmptyProof (EQUIVALENT transformation)

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

(320) TRUE

(321) 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.

(322) 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)

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)  =  x1
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)  =  x1
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)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

_₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(323) 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(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.

(324) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U73(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)  =  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)  =  x1
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)  =  x1
U72(x1, x2)  =  x1
U73(x1, x2)  =  U73(x1, x2)
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U73₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(325) 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(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(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.

(326) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U51(X1, X2, X3)) → 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)  =  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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
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)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U51₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(327) 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(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(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(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.

(328) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U53(X1, X2, X3)) → 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)  =  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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U52(x1, x2, x3)  =  x1
U53(x1, x2, x3)  =  U53(x1, x2, x3)
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)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U53₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(329) 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(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(U52(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(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.

(330) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U71(X1, X2, X3)) → 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)  =  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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U52(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, x2, x3)
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

U71₃ > ACTIVE₁

Status:

ACTIVE₁: [1]
U71₃: [1,2,3]

The following usable rules [FROCOS05] were oriented: none

(331) 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(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(U52(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(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.

(332) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U52(X1, X2, X3)) → 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)  =  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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U54(x1, x2, x3)  =  x1
U55(x1, x2)  =  x1
U56(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U52₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(333) 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(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(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(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.

(334) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U55(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)  =  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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U54(x1, x2, x3)  =  x1
U55(x1, x2)  =  U55(x1, x2)
U56(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U55₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(335) 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(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(U54(X1, X2, X3)) → 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(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.

(336) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U56(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)
U11(x1, x2)  =  x1
U12(x1, x2)  =  U12(x1, x2)
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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U54(x1, x2, x3)  =  x1
U56(x1)  =  U56(x1)
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U12₂: [1,2]
ACTIVE₁: [1]
U56₁: [1]

The following usable rules [FROCOS05] were oriented: none

(337) Obligation:

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

ACTIVE(U11(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(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(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.

(338) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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
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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U54(x1, x2, x3)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U82(x1, x2)  =  U82(x1, x2)
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U82₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(339) Obligation:

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

ACTIVE(U11(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(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(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.

(340) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U23(X1, X2, X3)) → 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
U13(x1)  =  x1
U21(x1, x2, x3)  =  x1
U22(x1, x2, x3)  =  x1
U23(x1, x2, x3)  =  U23(x1, x2, x3)
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)  =  x1
U43(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  x1
U46(x1)  =  x1
U54(x1, x2, x3)  =  x1
U61(x1, x2)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  x1
U81(x1, x2)  =  x1
U83(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U23₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(341) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(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(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(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.

(342) 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(U24(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U25(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(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U54(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U74(X)) → ACTIVE(X)
ACTIVE(U81(X1, X2)) → ACTIVE(X1)
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)  =  x1
U11(x1, x2)  =  x1
U13(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U25(x1, x2)  =  U25(x1, x2)
U26(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x3)
U42(x1, x2, x3)  =  x1
U43(x1, x2, x3)  =  U43(x1, x2)
U44(x1, x2, x3)  =  x1
U45(x1, x2)  =  U45(x1, x2)
U46(x1)  =  x1
U54(x1, x2, x3)  =  U54(x1, x3)
U61(x1, x2)  =  x1
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  x1
U72(x1, x2)  =  x1
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
U83(x1)  =  x1
U91(x1, x2)  =  U91(x1)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U62₂: [1,2]
U32₂: [1,2]
U25₂: [1,2]
U24₃: [1,2,3]
U54₂: [1,2]
U81₂: [1,2]
U22₃: [1,2,3]
U41₂: [1,2]
U91₁: [1]
U45₂: [1,2]
U43₂: [1,2]
U21₃: [1,2,3]
U74₁: [1]

The following usable rules [FROCOS05] were oriented: none

(343) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
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.

(344) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U31(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
U13(x1)  =  x1
U26(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U33(x1)  =  x1
U42(x1, x2, x3)  =  x1
U44(x1, x2, x3)  =  x1
U46(x1)  =  x1
U61(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U83(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U31₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(345) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
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.

(346) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U42(X1, X2, X3)) → 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
U13(x1)  =  x1
U26(x1)  =  x1
U33(x1)  =  x1
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U44(x1, x2, x3)  =  x1
U46(x1)  =  x1
U61(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U83(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U42₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(347) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
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.

(348) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U44(X1, X2, X3)) → 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
U13(x1)  =  x1
U26(x1)  =  x1
U33(x1)  =  x1
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U46(x1)  =  x1
U61(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U83(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U44₃: [1,2,3]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(349) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U13(X)) → ACTIVE(X)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U83(X)) → ACTIVE(X)
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.

(350) 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)
ACTIVE(U83(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U11(x1, x2)  =  x1
U13(x1)  =  U13(x1)
U26(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  x1
U61(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  x1
U83(x1)  =  U83(x1)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U13₁: [1]
U83₁: [1]

The following usable rules [FROCOS05] were oriented: none

(351) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U26(X)) → ACTIVE(X)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U72(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.

(352) 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)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U72(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)  =  x1
U11(x1, x2)  =  x1
U26(x1)  =  U26(x1)
U33(x1)  =  x1
U46(x1)  =  U46(x1)
U61(x1, x2)  =  x1
U63(x1)  =  x1
U72(x1, x2)  =  U72(x1, x2)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U26₁: [1]
U46₁: [1]
U72₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(353) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
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.

(354) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U61(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)  =  x1
U11(x1, x2)  =  x1
U33(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U63(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U61₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(355) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
ACTIVE(U63(X)) → ACTIVE(X)
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.

(356) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U63(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U11(x1, x2)  =  x1
U33(x1)  =  x1
U63(x1)  =  U63(x1)
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U63₁: [1]

The following usable rules [FROCOS05] were oriented: none

(357) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U33(X)) → ACTIVE(X)
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.

(358) 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)

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)
U33(x1)  =  x1
U92(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

U11₂ > ACTIVE₁

Status:

U11₂: [1,2]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(359) Obligation:

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

ACTIVE(U33(X)) → ACTIVE(X)
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.

(360) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)
U33(x1)  =  x1
U92(x1)  =  U92(x1)

Lexicographic path order with status [LPO].
Precedence:

U92₁ > ACTIVE₁

Status:

U92₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(361) Obligation:

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

ACTIVE(U33(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.

(362) 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)  =  x1
U33(x1)  =  U33(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U33₁: [1]

The following usable rules [FROCOS05] were oriented: none

(363) 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.

(364) PisEmptyProof (EQUIVALENT transformation)

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

(365) TRUE

(366) 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.

(367) 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
U92(x1)  =  U92(x1)
U91(x1, x2)  =  U91(x1, x2)
U83(x1)  =  U83(x1)
U82(x1, x2)  =  U82(x1, x2)
tt  =  tt
U11(x1, x2)  =  U11(x1, x2)
nil  =  nil
__(x1, x2)  =  __(x1, x2)
isNeList(x1)  =  isNeList(x1)
U13(x1)  =  U13(x1)
isPalListKind(x1)  =  isPalListKind(x1)
U12(x1, x2)  =  U12(x1, x2)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U23(x1, x2, x3)  =  U23(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U26(x1)  =  U26(x1)
U31(x1, x2)  =  U31(x1, x2)
U25(x1, x2)  =  U25(x1, x2)
isList(x1)  =  isList(x1)
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  U33(x1)
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U45(x1, x2)  =  U45(x1, x2)
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U43(x1, x2, x3)  =  U43(x1, x2, x3)
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)
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)  =  U63(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
U73(x1, x2)  =  U73(x1, x2)
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
e  =  e
a  =  a
i  =  i
isNePal(x1)  =  isNePal(x1)
isPal(x1)  =  isPal(x1)
o  =  o
u  =  u

Lexicographic path order with status [LPO].
Precedence:

_₂ > U91₂ > U92₁ > mark₁
_₂ > U91₂ > U92₁ > tt
_₂ > U91₂ > isPalListKind₁ > mark₁
_₂ > U91₂ > isPalListKind₁ > tt
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > U26₁ > mark₁
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > U26₁ > tt
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > mark₁
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > tt
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > mark₁
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > tt
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > mark₁
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > tt
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > U13₁ > mark₁
_₂ > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > U13₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > U13₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > U13₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > isPalListKind₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > isPalListKind₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > isQid > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > isQid > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > tt
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > U46₁ > mark₁
_₂ > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > U46₁ > tt
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > mark₁
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > isPalListKind₁ > tt
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > mark₁
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > isQid > tt
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > mark₁
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > tt
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > U13₁ > mark₁
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > U13₁ > tt
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > U56₁ > mark₁
_₂ > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > U56₁ > tt
_₂ > U71₃ > U72₂ > U73₂ > isPalListKind₁ > mark₁
_₂ > U71₃ > U72₂ > U73₂ > isPalListKind₁ > tt
_₂ > U71₃ > U72₂ > U73₂ > U74₁ > mark₁
_₂ > U71₃ > U72₂ > U73₂ > U74₁ > tt
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > U83₁ > mark₁
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > U83₁ > tt
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > isPalListKind₁ > mark₁
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > isPalListKind₁ > tt
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > isQid > mark₁
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > isQid > tt
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > U63₁ > mark₁
_₂ > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > U63₁ > tt
a > tt
i > mark₁
i > tt
u > tt

Status:

i: []
U62₂: [2,1]
U41₃: [1,2,3]
U32₂: [1,2]
U25₂: [1,2]
U24₃: [1,2,3]
mark₁: [1]
U12₂: [1,2]
U56₁: [1]
tt: []
U22₃: [1,2,3]
isPalListKind₁: [1]
U21₃: [1,2,3]
U54₃: [3,2,1]
nil: []
U55₂: [1,2]
U31₂: [1,2]
e: []
isNePal₁: [1]
o: []
U53₃: [2,3,1]
U23₃: [1,3,2]
U81₂: [1,2]
U44₃: [3,1,2]
U33₁: [1]
U74₁: [1]
U71₃: [2,1,3]
U91₂: [2,1]
_₂: [1,2]
U43₃: [1,2,3]
U26₁: [1]
U42₃: [1,3,2]
U92₁: [1]
U45₂: [2,1]
TOP₁: [1]
isQid: []
U51₃: [1,2,3]
U83₁: [1]
U73₂: [2,1]
a: []
isList₁: [1]
U72₂: [1,2]
U11₂: [1,2]
U63₁: [1]
U52₃: [1,3,2]
isPal₁: [1]
U82₂: [2,1]
U61₂: [1,2]
u: []
U46₁: [1]
isNeList₁: [1]
U13₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(368) 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.

(369) 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)  =  TOP(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U92(x1)  =  x1
mark(x1)  =  mark
U91(x1, x2)  =  U91(x1, x2)
U83(x1)  =  x1
U82(x1, x2)  =  U82(x1, x2)
U12(x1, x2)  =  U12(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U21(x1, x2, x3)  =  U21(x2, x3)
U13(x1)  =  x1
U23(x1, x2, x3)  =  U23(x1, x2, x3)
U22(x1, x2, x3)  =  x1
U25(x1, x2)  =  U25(x1)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U31(x1, x2)  =  U31(x2)
U26(x1)  =  x1
U33(x1)  =  U33(x1)
U32(x1, x2)  =  x1
U42(x1, x2, x3)  =  U42(x3)
U41(x1, x2, x3)  =  U41(x3)
U44(x1, x2, x3)  =  U44(x3)
U43(x1, x2, x3)  =  x2
U45(x1, x2)  =  U45(x2)
U46(x1)  =  U46(x1)
U51(x1, x2, x3)  =  U51(x1, x3)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U53(x1, x2, x3)  =  U53(x2)
U54(x1, x2, x3)  =  U54(x1, x2, x3)
U55(x1, x2)  =  x1
U56(x1)  =  U56(x1)
U61(x1, x2)  =  x1
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  U63(x1)
U71(x1, x2, x3)  =  U71(x1, x3)
U72(x1, x2)  =  x1
U73(x1, x2)  =  x2
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1)
__(x1, x2)  =  __(x1, x2)
isNeList(x1)  =  isNeList(x1)
isPalListKind(x1)  =  x1
isList(x1)  =  x1
tt  =  tt
nil  =  nil
e  =  e
a  =  a
i  =  i
isNePal(x1)  =  x1
isQid(x1)  =  isQid
isPal(x1)  =  x1
o  =  o
u  =  u

Lexicographic path order with status [LPO].
Precedence:

TOP₁ > mark
U91₂ > ok₁ > mark
U11₂ > U12₂ > ok₁ > mark
U21₂ > ok₁ > mark
U24₃ > U25₁ > ok₁ > mark
U41₁ > U42₁ > ok₁ > mark
U51₂ > U52₃ > ok₁ > mark
U71₂ > ok₁ > mark
U74₁ > ok₁ > mark
U81₁ > U82₂ > ok₁ > mark
_₂ > ok₁ > mark
tt > U82₂ > ok₁ > mark
tt > U12₂ > ok₁ > mark
tt > U23₃ > ok₁ > mark
tt > U25₁ > ok₁ > mark
tt > U33₁ > ok₁ > mark
tt > U42₁ > ok₁ > mark
tt > U44₁ > ok₁ > mark
tt > U45₁ > isNeList₁ > U31₁ > ok₁ > mark
tt > U46₁ > ok₁ > mark
tt > U52₃ > ok₁ > mark
tt > U53₁ > ok₁ > mark
tt > U54₃ > isNeList₁ > U31₁ > ok₁ > mark
tt > U56₁ > ok₁ > mark
tt > U62₂ > U63₁ > ok₁ > mark
nil > mark
e > mark
a > mark
i > mark
isQid > ok₁ > mark
o > mark
u > mark

Status:

U91₂: [2,1]
i: []
U62₂: [2,1]
_₂: [2,1]
U41₁: [1]
U81₁: [1]
U24₃: [3,1,2]
U45₁: [1]
U12₂: [1,2]
U21₂: [1,2]
U51₂: [2,1]
U56₁: [1]
U71₂: [2,1]
tt: []
TOP₁: [1]
U44₁: [1]
isQid: []
U54₃: [2,3,1]
nil: []
a: []
e: []
ok₁: [1]
U11₂: [2,1]
U63₁: [1]
o: []
U52₃: [3,1,2]
U23₃: [2,3,1]
U82₂: [1,2]
U31₁: [1]
U53₁: [1]
u: []
U42₁: [1]
U46₁: [1]
isNeList₁: [1]
U25₁: [1]
mark: []
U33₁: [1]
U74₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(370) 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.

(371) PisEmptyProof (EQUIVALENT transformation)

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

(372) TRUE